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LECTURES &
  ADVANCED QUANTITATIVE TECHNIQUES   NOTES




          Lectures & Notes

  ADVANCED QUANTITATIVE TECHNIQUES
      (COURSE FOR PHD STUDENTS)




                 By
         Dr. Anwar F. Chishti
             Professor
Faculty of Management & Social
            Sciences

                                              1
LECTURES &
              ADVANCED QUANTITATIVE TECHNIQUES                                      NOTES




           ADVANCED QUANTITATIVE TECHNIQUES

                                     Course Plan
                                 Fall Semester 2012
Course Instructor                   Professor Dr. Anwar F. Chishti
       Contacts:              Phone Phone: 0346-9096046
                              Email anwar@jinnah.edu.pk; chishti_anwar@yahoo.com
Class venue                         Computer Laboratory

                                     Course contents
Topic 1:      Simple/Two-Variable Regression Analysis:
                    • An introduction of estimated model and its interpretation,
                    • Regression Coefficients and Related Diagnostic Statistics:
                        Computational Formulas
                    • Evaluating the results of regression analysis
                    • Standard assumptions, BLUE properties of the estimator.
                    • Take-home assignment - 1
Topic 2:      Simple Regression to Multiple Regression Analysis
                    • Shortcomings of simple/two-variables regression analysis
                    • An example of multiple regression analysis
                    • Use of Likert-scale type questionnaire, raw-data entry, reliability test
                        and generation of variables
                    • Estimation of multiple regression model
                    • Evaluation of the estimated model in terms of F-statistic, R2 and t-
                        statistic/p-value
                    • Take-home assignment - 2
Topic 3:      Multiple Regression: Model specification
                    • 3.1(a) Conceiving research ideas and converting it into research
                        projects: a procedure
                    • 3.1(b) Incorporating theory as the base of your research: econometrics
                        theory & economics/management theory
                    • Take-home assignment – 3(a)
                    • 3.2 (a) Specification of an econometric model: mathematical
                        specification




                                                                                             2
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            ADVANCED QUANTITATIVE TECHNIQUES                                        NOTES



                   •   3.2(b) Some practical examples of mathematical specification:
                       production-function specification, cost-function specification, revenue-
                       function specification
                   • Take-home assignment – 3(b)
                   • 3.3(a) Conceptual/econometric modeling: (a) Examples in Finance; (b)
                       Examples in Marketing; (c) Examples in HRM
                   • 3.3(b) Incorporating theory as the base of your research: econometrics
                       theory & economics/management theory
                   • Take-home assignment: adopting, adapting and developing a new
                       questionnaire
Topic 4:    Analyzing mean values
                   • Analyzing mean value, using one-sample t-test
                   • Comparing mean-differences of two or more groups
                   • Comparing two groups
                                  * Independent samples t test
                                  * Paired-sample t test
                   • Comparing more-than-two groups
                                  * One-Way ANOVA
                                  * Repeated ANOVA
                   • Take-home assignment – 4
Topic 5:    Uses of estimated econometric models
                   • Some examples
                   • Take-home assignment – 5
Topic 6:    Relaxing of Standard Assumptions: Normality Assumption and its testing
                   • Normality assumption
                   • Its testing
                   • Take-home assignment – 6
Topic 7:    Problem of Multicollinearity: What Happens if Regressors are Correlated?
                   • Consequences, tests for detection and solutions/remedies
                   • Take-home assignment - 7
Topic 8:    Problem of Heteroscadasticity: What Happens if the Error Variance is
                    nonconstant?
                   • Consequences, tests for detection and solutions/remedies
                   • Take-home assignment - 8
Topic 9:    Problem of Autocorrelation: What Happens if the Error terms are
            correlated?
                   • Consequences, tests for detection and solutions/remedies
                   • Take-home assignment - 9
Topic 10:   Mediation and moderation analysis - I
                   • Estimating and testing mediation


                                                                                             3
LECTURES &
              ADVANCED QUANTITATIVE TECHNIQUES                                     NOTES



                    • Take-home assignment – 10
Topic 11:    Mediation and moderation analysis - II
                    • Estimating and testing moderation
                    • Take-home assignment – 9
Topic 12:    Time-series analysis - I
                    • Unit root analysis
                    • Take-home assignment – 10
Topic 13:    Time-series analysis - II
                    • Unit root, co-integration and error correction modeling (ECM)
                    • Take-home assignment – 11
Topic 14     Panel data analysis, Simultaneous equation models/Structural equation
             models
                    • Panl data analysis
                    • SEM, ILS, 2SLS and 3SLS
                    • Take-home assignment – 12
Topic 15     Qualitative response regression models (when dependent variables are
             binary/dummy) and Optimization
                    • LPM, Logit model and Probit Model
                    • Take-home assignment – 13(a)
                    • * Optimization: minimization and maximization
                    • Take-home assignment – 13(b)
Topic 16     Welfare analysis: maximization of producer and consumer surpluses and
             minimization of social costs

Required Text & Recommended Reading
      The prescribed textbooks for this course are:

      Gujarati, Damodar N. Basic Econometrics, 4th Edition. McGraw-Hill. 2007

      Stock, J. H. and Watson, M.W. Introduction to Econometrics, 3/E. Pearson Education,
      2011

Reference Books/Materials

      Studenmund, A.H. Using Econometrics: A Practical Guide, 6/E, Prentice Hall

      Asteriou, D. and Hall, S.G. Applied Econometrics – A Modern Approach. Palgrave
      Macmillan, 2007.



                                                                                            4
LECTURES &
              ADVANCED QUANTITATIVE TECHNIQUES                                      NOTES



      Andren, Thomas. (2007). Econometrics. Bookboon.com

      Salvatore, D and Reagle, D. Statistics and Econometrics, 2nd Ed. Schaum’s Outlines.

      Instructor’s class-notes (hard copy at photo-copier shop)

Assessment Criteria

 Details                    Due Date                                    Weighting
                            10 best weekly assignments (out of total
 Individual Assignments     13 - 15, each having 2 marks) will be              20 %
                            counted toward total 20% marks.
                            A group of 2 students will select a topic,
 Group research on selected carry out research, complete a research
                                                                               20 %
 research topics            study, and make presentation in during
                            the last classes of the semester
 Mid-term Examination       As per University’s announcement                   20 %
 Final Examination          As per University’s announcement                   40 %
                                                                  Total marks: 100




                                                                                             5
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                ADVANCED QUANTITATIVE TECHNIQUES                                          NOTES



                                  Topic 1
                   Simple/Two-Variable Regression Analysis
1.1    Simple regression analysis: an example
Assuming a survey of 10 families yields the following data on their consumption expenditure (Y)
and income (X).
                      Y (Thousands)         X (Thousands)
                      70                    80
                      65                    100
                      90                    120
                      95                    140
                     110                    160
                     115                    180
                     120                    200
                     140                    220
                     155                    240
                     150                    260
The theory suggests that families’ consumption (Y) depends on their income (X); hence,
econometric model may be specified, as follows.
       Y = f(X)                      (General form)                                     (1a)
Or     Y = β0 + β1X + e              (Linear form)                                      (1b)
The above stated regression analysis model contains two variables (one independent variable X
and one dependent variable Y); this model is therefore called Two-variables or Simple
regression analysis model.
Is this type of Simple or Two-variable model justified? We will discuss this question later on;
let’s first estimate this model, using the Statistical Package for Social Sciences’ software SPSS.
The estimated model & interpretation
               Y       =   24.4530 + 0.5091 X                                               (2a)
                           (6.4140)  (0.0357)                  (Standard Error)             (2b)
                           (3.8124) (14.2445)                  (t-statistic)                (2c)
                            (0.005)  (0.000)                   (p-value/sig. level)         (2d)

               R= 0.981     R2 = 0.9621                R2adjusted = 0.957
               F = 203.082 (p-value = 0.000)           DW = 2.6809             N = 10       (2e)


1.2    Regression analysis: computational formulas
The econometric model specified in (1) is estimated in the form of estimated model (2a) along
with all its diagnostic statistics 2(b – e), using the formulas provided, as follows.


                                                                                                     6
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                 ADVANCED QUANTITATIVE TECHNIQUES                                                       NOTES



The coefficients ßs
             ∧               ∧
             β0 = Y − β1 X                                                                                (3)
             ∧
             β1 =
                    ∑ ( Xi − X ) (Yi −Y )
                                                                                                          (4)
                      ∑ ( Xi − X )
                                                                2



             ∧
             β1 =
                    ∑x y      i            i
                                                                                                          (5)
                    ∑x
                                      2
                                  i

Variances (σ 2) and Standard Errors (S.E):
                                                                                                    2
                                                                                   ∧
                                                                                      
              ∧
                             ∑e
                                               2                        ∑Y
                                                                         
                                                                                 −Yi 
                                                                                      
                                                                                        i
                                                                                                          (6)
            σ =2
                                                            =
                             ( N − 2)                                        ( N − 2)
                    ∧
            Var ( β 0 )          =             σ
                                                   ∧
                                                    2
                                                            =
                                                                        ∑ X .σ      i
                                                                                     2          2

                                                                                                           (7)
                                                                         N ∑x
                                                   β0                                       2
                                                                                            i

                      ∧                            ∧                        ∧
             S .E ( β0 ) = σ β0                             =           σ    β0
                                                                                2                         (8)
                        ∧                          ∧
                                                                        σ2
             Var ( β1 ) = σ β1 =
                            2
                                                                                                          (9)
                                                                    ∑x          2
                                                                                i

                    ∧                              ∧                        ∧
             S .E ( β1 )          = σ β1                    =           σ    β1
                                                                                2                         (10)
   T-ratios:
                             ∧
                            β0
             Tβ0 =           ∧                                                                            (11)
                            σβ    0

                             ∧
                            β1
              Tβ1 =          ∧
                            σβ        11


                                                            (12)
   The Coefficient of Determination ( R2 ):
                                                               ∧
                                                                           
                  ESS                            ∑Y
                                                  
                                                  
                                                                    i   −Y 
                                                                           
             R2 =                              =
                  TSS                             (
                                                 ∑Y                 i   −Y      )
                                                            (13)
                                                            RSS
                                               = 1−
                                                            TSS

                                               =1 −
                                                             ∑ e                        2
                                                                                        i


                                                            ∑Y −Y )
                                                             (                                  2
                                                                         i

   F – Statistics:

            F =
                 ESS df
                                                        =
                                                                   ( R ) ( K −1)2


                 RSS df                                         (1 − R ) ( N − K )  2


                                                            (14)


                                                                                                                 7
LECTURES &
                  ADVANCED QUANTITATIVE TECHNIQUES                                                                       NOTES



      Durban-Watson (D.W) Statistics:
                                                           2

                            ∑(e                − et −1 )
                             N

                                       t
                            t =2
               d =                     N

                                   ∑e
                                   t =1
                                                   2
                                                   t


                                                (15)

1.3     Estimation of the model using computational formulas
We now use formula provided in (3) to (15), make computations like Table 3.3 (Gujarati,
2007) and resolve the model, as follows.
              Yi = ßo + ß1 Xi + ℮i …….. Linear model                                                                       (16)
Regression Coefficients ( ß i ):
            ˆ
           β1 =
                ∑ xi . yi = 16800 = 0.5091
                                                                                                                           (17)
                 ∑ xi2       33000
                  ∧          ∧
              β0 = Y − β1 X = 111 − 0.5091 (170 )
                                                                                                                           (18)
                                            = 24.453
Variances (σ 2) and Standard Errors (S.E):
                            ∑e
                                   2
                  ∧
                                                            337.25
              σ =  2
                                                       =           = 42.15625                                     (19)
                            ( N − 2)                        10 − 2
                       ∧
              Var ( β0 )      =        σβ
                                               ∧
                                               2
                                                       =
                                                               ∑X .σ   i
                                                                        2       2

                                                                                    =
                                                                                        ( 322,000 ) ( 42.15625)
                                                0
                                                               N ∑x         2
                                                                            i               ( 10 ) ( 33,000 )


                                                                                    = 41.13428
                                                (20)
                       ∧                   ∧                   ∧
              S .E ( β0 )     = σ β0                   =   σ    β0
                                                                   2
                                                                       =     41.13428 = 6.4140
                                                (21)
                       ∧               ∧
                                                           σ2
                                                           ˆ                42.15625
              Var ( β1 ) = σ β1 =       2
                                                                       =                    = 0.001277
                                                           ∑x   2
                                                                i            33,000
                                                (22)
                       ∧                   ∧                   ∧
              S .E ( β1 )     = σ β1                   =   σ    β1
                                                                   2
                                                                       =     0.001277 = 0.03574
                                                (23)

      T-ratios:




                                                                                                                                  8
LECTURES &
                  ADVANCED QUANTITATIVE TECHNIQUES                                                             NOTES


                         ∧
                        β0                 42.453
               Tβ0 =     ∧
                                       =          = 3.8124
                       σβ                  6.414
                             0


                                                (24)
                         ∧
                        β1                      0.5091
              Tβ1 =      ∧
                                       =               = 14.2445
                       σβ                      0.03574
                             11


                                                (25)
      The Coefficient of Determination ( R2 ):

              R 2 = 1−
                               ∑e               2
                                                i
                                                              =1 −
                                                                      337.25
                                                                             = 0.9621
                              ∑(Y −Y )
                                                        2
                                                                       8890
                                           i

                                                (26)
      F – Statistics:
               F=
                         ( R ) ( K − 1)
                             2
                                                    =
                                                             0.9621 ( 2 − 1 )
                      (1 − R ) ( N − K )
                                  2
                                                            0.0379 (10 − 2 )
                                                                                                        (27)
                                                                0.9621
                                                        =                       =    203.082
                                                              0.0047375
The estimated model:
              Y        =              24.4530 + 0.5091X
                                      (6.414)  (0.0357)                          S.E.
                                      (3.812) (14.244)                           t-ratio
                                      (0.005)               (0.0000)            (p-valuel)

                                      R2 = 0.9621                    F = 203.082               N = 10            (28)

1.4     Regression analysis: the underlying theory
The above reported formulas reflect how various needed computations are carried out in
regression analysis. Specifically, formula (4) estimates the coefficient (β 1) of explanatory
variable X:
              ∧
              β1 =
                     ∑ ( Xi − X ) (Yi −Y )
                       ∑ ( Xi − X )
                                                    2




That is: ‘the deviations of individual observation on Xi from its mean, multiplied by deviations of
respective Yi from its mean (cross-deviation), divided by the squares of the variations of Xi’; so
it is the ratio between cross-deviations of X – Y variables and X variable. Theoretically, β1




                                                                                                                        9
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                ADVANCED QUANTITATIVE TECHNIQUES                                          NOTES



measures ‘total cross deviations/variations per unit of variation in X-variable’. The intercept β0
measures ‘mean value of Y minus total contribution of mean of X’.
          ∧         ∧
          β0 = Y − β1 X



1.5    Error term: its estimation and importance
When an econometric model, like 1(b), is specified:
               Y = β0 + β1X + e                                                              (29a)

It contains an error or residual term (e); but when model is estimated like 2(a):
              Y = 24.4530 + 0.5091X                                                          (29b)
The error term (e) seems to disappear; where does the error term go?
In fact the estimated model like 29(b) is valid only for the mean/average values of X and Y, and
equality in 29(b) does not hold when values other-than-mean values are used; we can compute
values of error terms or residuals, using the following formula.
               Yi – Ŷ = e                                                                   (30a)
               Yi – (24.4530 + 0.5091Xi) = e                                                 (30b)
Putting individual-observation values from the original data, that is:
                         Y                      X
                         70                     80
                         65                     100
                         90                     120
                         95                     140
                        110                     160
                        115                     180
                        120                     200
                        140                     220
                        155                     240
                        150                     260

       Yi –    (24.4530 + 0.5091Xi)     =   e
       70 – (24.4530 + 0.5091*80 = 4.8181                                                    (30c)
       65 – (24.4530 + 0.5091*100) = -10.3636                                                (30d)
       90 – (24.4530 + 0.5091*120 = 4.4545                                                   (30e)
       95 – (24.4530 + 0.5091*140) = -0.7272                                                 (30f)
       110 – (24.4530 + 0.5091*160) = 4.0909                                                 (30g)


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                  ADVANCED QUANTITATIVE TECHNIQUES                                           NOTES



         115 – (24.4530 + 0.5091*180) = -1.0909                                                (30i)
         120 – (24.4530 + 0.5091*200) = -6.2727                                                (30j)
         140 – (24.4530 + 0.5091*220) = 3.5454                                                 (30k)
         155 – (24.4530 + 0.5091*240) = 8.3636                                                 (30l)
         150 – (24.4530 + 0.5091*260) = -6.8181                                                (30m)
As reflects from the above computations, error term reflects how much an individual Y deviates
from its estimated value. The values of error terms play important role in determining the size of
variance Ϭ2 (computational formula 6), which further affects a number of other computations.
A characteristic of error or residual term is that, once we add or take its mean value, it turns out
equal to zero, in both cases.


1.6      Evaluating the estimated model
After running regression, the results are reported usually reported in the following form.
                Y       =    24.4530 + 0.5091X                                                 (31a)
                             (6.4140)  (0.0357)                    (Standard error)            (31b)
                             (3.8124) (14.2445)                    (t-statistic)               (31c)
                              (0.005)  (0.000)                     (p-value/sig. level)        (31d)

                 R= 0.981     R2 = 0.9621                  R2adjusted = 0.957
                 F = 202.868 (p-value = 0.000)             DW = 2.6809              N = 10     (31e)

The econometric model is specified in the form of 1 (a or b), estimated in the form of 31 (a) and
evaluated, using the diagnostic statistic provided in 31(b – e). The estimated model’s evaluation
is carried out, using three distinct criteria, namely:
         (a) Economic/management theory criteria (expected signs carrying with the coefficients
             of X-variables)
         (b) Statistical theory criteria (t statistic or p-value, F statistic, and R2)
         (c) Econometrics theory criteria (Autocorrelation, Heteroscadasticity &
             Multicollinearity)
Economic theory criteria
      Questions:
      a) Are these results in accordance with the economic theory?
      b) Are they in accordance with our prior expectation?



                                                                                                       11
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                 ADVANCED QUANTITATIVE TECHNIQUES                                                 NOTES



    c) Do the coefficients carry correct sign?
     Answer: Yes, we expected a positive relationship between the income of a family and its
consumption expenditure. The coefficient of income variable, X, is positive.




Statistical theory criteria
    Question 1:
    a) Are the estimated regression coefficients significant?
    b) Are the estimated regression coefficients ßs individually statistically significant?
    d) Are the estimated regression coefficients ßs individually statistically different from zero?
     Answer: Here, we need to test the hypothesis:
         HO:    ß1 = 0
         H1 :   ß1 ≠ 0
                        ß− 0
                   t=
                        S .E = (.5091 – 0) / .0357 = .5091 / .0357 = 14.2605               (32)


Our t calculated = 14.2605 > t tabulated = 1.86 at .05 level of significance, with df (N – k) = 8; hence, we
reject the null hypothesis; the coefficient ß1 is statistically significant. Another way of checking
the significance level of ßi coefficients is to check its respective p-value (Sig. level). In case of
the coefficient of X-variable, the p-value = 0.00, suggesting that coefficient ß 1 is statistically
significant at p < 0.01. In this second case, we do not need to check the statistical significance
level, using the t-distribution table appended at the end of some econometrics book; we can
directly check p-value provided next to the t-value in the output of the solved problem.
        Question 2:
        a) Are the estimated regression coefficients collectively significant?
        b) Do the data support the hypothesis that
                  ß1 = ß2 = ß3 = 0
        Here, we need to test the hypothesis:
         HO: ß1 = ß2 = ß3 = 0
         H1:    ßi are not equal to 0
        Answer: Here, we use F-stattistic, namely:


                                                                                                         12
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                 ADVANCED QUANTITATIVE TECHNIQUES                                              NOTES




                F =
                      ESS df
                                 =
                                      ESS / K − 1
                                                      =
                                                                 ( R ) ( K −1)
                                                                   2


                      RSS df          RSS / N − K             (1 − R ) ( N − K )
                                                                       2                         (33)

                  = 202.868
Our F statistic (F = 202.868 > F 1, 8; .05 = 5.32) suggests that the overall model is statistically
significant. Like in case of t-statistics, the significance level of F-statistic can also be checked
from p-value given next to Fcalculated in the output of the solved problem.
        Question 3: Does the model give a good fit?
        Answer: Yes; our R2 = 0.9621 suggests that 96.21% variation in the dependent variable
(Y) has been explained by variations in explanatory variable (X).
Econometrics theory criteria
        1) No Autocorrelation Criteria          (We will discuss
        2) No Heteroscadasticity Criteria       (these criteria in detail
        3) No Multicollinearity Criteria        (later on in the course


1.7     Interpreting the results of regression analysis
        The estimated results suggests that if there is one unit change in explanatory variable X
(family’s income), there will be about half unit (.5091) change in dependent variable Y (family’s
consumption expenditure). If X and Y both are in rupees, then it means that there will be 51
paisas increase in consumption expenditure if the family’s income increases by one rupee.


1.8     Standard assumptions of Least-Square estimation techniques
The linear regression model is based on certain assumptions; if these assumptions are not
fulfilled, then we have certain problems to deal with. These assumptions are:
1.    Error term μ i is a random variable, and has a mean value of zero.
      ===> μ i may assume any (+), (-) or zero value in any one observation/
      period, and the value it assume depends on chance.
      The mean value of μ i for some particular period, however, is zero, i.e.,
                            ∑ (μ i / xi) = 0
2.    The variance of μ I is constant in each period, i.e.,
                         Var (μ i ) = б2



                                                                                                        13
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                   ADVANCED QUANTITATIVE TECHNIQUES                                          NOTES



       This is normally referred to as homoscedasticity assumption, and if this
        Assumption is violated, then we face the problem of heteroscedasticity.
3.      Based on assumption 1 and 2 , we can say that variable μ i has a normal
        distribution, i.e.,
                              μ i ~ N(0, б2)
4.      Error term for one observation is independent of the error term of other
        observation, i.e., μ i and μ j are not correlated, or
                              Cov (μ i and μ j ) = 0
        This is no-serial-autocorrelation assumption, and if this assumption is
        violated, then we have autocorrelation problem.
5.       μ i is independent if the explanatory variables (X), that is, the μ i and μ j are
         not correlated.
                              Cov (X μ ) = ∑{[Xi - ∑ (Xi)] [ μ i -∑ (μ i)]}       =    0
      6. The explanatory variable (Xi) are not linearly correlated to each other; they
         do not affect each other. If this assumption is violated, then we face the
         multicolinearity problem.
7.       There is no specification problem, that is,
         a)   Model is specified correctly, mathematically, from the economic
              theory point of view.
         b)    Functional form of the model ( i.e., linear or log-linear or any other
               form) is correct.
         c)   Data on dependent and independent variables have correctly collected,
              i.e., there is no measurement error.


1.9      BLUE properties of estimator:
       Given the aforementioned assumptions of the classical linear regression model, the Least -
Square estimator (β) possess some ideal properties.
              1. It is linear.
              2. It is unbiased, i.e., its average or expected value is equal to its true
                 value.
                                 ˆ
                              Ε( βi ) = βi



                                                                                                    14
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                 ADVANCED QUANTITATIVE TECHNIQUES                                        NOTES



                Biasness can be measured as:
                        Bias          ˆ
                                 = Ε( βi ) − βi

                        − −−          ˆ
                                    Ε( βi ) = βi   if   Bias = 0

            3. It is minimum- variance, i.e. it has minimum variance in the class of all such Linear
unbiased estimators.
           4. It is efficient. An unbiased estimator with the least variance is known as an
Efficient estimator. From properly (2) and (3), our OLS estimator is unbiased and minimum
variance, so it is an efficient estimator.
            5. It is BLUE, i.e., Best-linear-unbiased estimator.
There is a famous theorem known as “Gaus-Markov Theorem” which tells:
             “Given the assumptions of the classical linear regression model, the least-square
Estimators, in the class of unbiased linear estimators, have minimum variance, So they are
best-linear unbiased estimators, BLUE”.




                                       Assignment 1
                                   (Due in the next class)
You have already received Gujarati’s (2007) ‘Basic Econometric’; study its relevant section to
solve the following assignment.
.
    1. Study sections 1.4 & 1.5: How does regression differ from correlation?
    2. Read section 1.6: What are some other names used for dependent and independent
        variables?
    3. Study section 1.7: What are different types of data? Explain each type in one or two
        sentences.
    4. Study example 6.1 (page 168-169): Which of the two estimated model (6.1.12 & 6.1.13)
        is better and why? What do you learn from this example, in general.




                                                                                                 15
LECTURES &
                ADVANCED QUANTITATIVE TECHNIQUES                                                   NOTES



                                               Topic 2
             Simple Regression to Multiple Regression Analysis
2.1    Shortcomings of two-variable regression analysis
In spite of providing the base for general regression, the simple or two-variable regression has
certain limitations; it gives biased results (of Least-Square Estimators, βs) if specified model
excludes some relevant explanatory variables (namely X2, X3, …..).
Let’s revisit to our first topic’s example of “Families’ Consumption’, wherein model was
specified and run, as follows.
               Y         =       β0 + β1X + e
                         =       24.4530 + 0.5091 X
                                 (6.4140)   (0.0357)          (Standard Error)
                                 (3.8124) (14.2445)           (t-statistic)
                                 (0.005)   (0.000)            (p-value/sig. level)

               R= 0.981     R2 = 0.9621               R2adjusted = 0.957
               F = 203.082 (p-value = 0.000)          DW = 2.6809                   N = 10           (2.1)

If we recall, the results of this estimated model, while we evaluated in terms of economic theory
(sign of the coefficient carrying with X) and statistical theory criteria (t-statistic/p-value, F-
statistic and R2), were turned out to be reasonably acceptable. But, while we reconsider the
specification of the model, we will find that we had misspecified the model at the first place;
according to the theory, consumption (Y) depends on income (X1), as well as, wealth of the
families    (X2),   prices       of   consumption    items     (X3),       prices     of     the      related
products/substitutes/complements (X4), and so on. Hence, in spite of the fact that results
provided in (2.1) are apparently seem reasonable in light of the diagnostic statistic used, the
estimated model provides biased results as it does not include some very important and relevant
explanatory variables.
Solution then lies in the Multiple regression analysis, wherein all relevant explanatory variables
need to be included, like the following one.
               Y         =       β0 + β1X1 + β2X2 + β3X3 + …………. + βNXN + e                          (2.2)

Let’s take a practical example of using multiple regression analysis (see next sub-section 2.2).




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2.2        An example of multiple regression analysis
In case, research topic is:
   “Organizational justice and employees’ job satisfaction: a case of Pakistani organizations”
Knowing that ‘organizational justice’ has 4 well identified facets, namely:
           1. Distributive justice (JS)
           2. Procedural justice (PS)
           3. Interactive justice (IJ), and
           4. Informational justice (INJ)
Assuming that, if organizational justice prevails in Pakistani organizations, then employees
would be satisfied (job satisfaction, JS); hence, respective econometric model may be specified,
as follows.
           JS     =       f(DJ, PJ, IJ, INJ)                                                     (2.3)
We may estimate this model in linear and/or log-linear form, that is:
           JS     = α0 + α1DJ + α2PJ + α3IJ + α 4INJ + ei                 (Linear model)          (2.4)
           lnJB   = β0 + β1lnDJ + β2lnPJ + β3lnIJ + β4lnINJ + μi          (Log-linear model)      (2.5)
                                    (Note: ‘ln’ stands for natural log)
Steps (to be taken):
For estimation of linear model
      1. As per requirements of the model specified in (2.3), we need to develop a questionnaire,
           like the one placed at Annex – I; and then collect the required data.
      2. Enter the data collected on the employees’ responses in SPSS, using data editor
           (spreadsheet like that of EXCEL-spreadsheet). Check how data has been entered in file
           named: CLASS-EXERCISE-DATA_1.
      3. Estimate reliability test (Chronbach’s Alpha) of the raw-data on employees’ responses,
           separately for each of the constructs used (JS, DJ, PJ, IJ & INJ).
      4.   Try to understand what reliability, validity and generalizability concepts stand for (see
           Annex – II). Interpret the results of reliability test (See ANNEX – III)
      5. Generate data on variables of interest, namely: JS, DJ, PJ, IJ & INJ.
      6. Run regression model specified in (2.4), and report the results.
                  JS =    2.371 + 0.098DJ - 0.021PJ + 0.076IJ + 0.292INJ - 0.005AEE
                          (9.882) (2.199) (-0.509)     (1.905) (4.472)      (-1.636)
                          (0.000) (0.029) (0.611)     (0.058) (0.000)        (0.103)


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              R= 0.506     R2 = 0.2560               R2adjusted = 0.2410
              F = 17.71 (p-value = 0.000)            DW = 1.5930             N = 264          (2.6)

     (Figures in the first and second parentheses, respectively, are t-statistics and p-values)
Note: AEE stands for the combined figures of age, education and experience of the employees,
and have been included to capture the combined effects of these variables.


For estimation of log-linear model
   7. Convert newly generated data on JS, DJ, PJ, IJ & INJ and AEE into their logs
   8. Run model 2.5, and report the results
       lnJS = 0.943 + 0.156lnDJ - 0.015lnPJ + 0.080lnIJ + 0.308lnINJ - 0.084lnAEE
              (4.594) (2.829)    (-0.308)     (1.554)     (4.506)      (-1.645)
              (0.000) (0.005)     (0.758)     (0.122)     (0.000)       (0.101)

              R= 0.522     R2 = 0.2720               R2adjusted = 0.2580
              F = 19.309 (p-value = 0.000)           DW = 1.618              N = 264          (2.7)

Evaluation and interpretation of the estimated models
Linear model 2.6
       (a) Model is found statistically significant (F = 17.71, p < 0.01); though all the
           explanatory variables included in the model seem to have explained around 25
           percent variance in the dependent variable (R2 = 0.2560; R2adjusted = 0.2410).
       (b) Variable PJ appears to be highly statistically insignificant (p = 0.611), compared to
           variables INJ and DJ with highly statistically significant contribution (p < 0.01 & p <
           0.05 ) and variable IJ and AEE with moderately statistically significant contribution
           (p = 0.058 & p = 0.103).
       (c) Results suggest that variables INJ, DJ and IJ positively contribute towards
           determination of employees’ job satisfaction, AEE negatively contributes while PJ
           does not contribute. The negative relationship of AEE with JB suggests that
           employees of higher age, with relatively higher education and experience, are less
           satisfied from their jobs.




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Log-linear model 2.7
      (a) Since the two formulations of the data (nominal-data and log-data), used in linear and
          log-linear models, differ from each other, we cannot compare results of one model
          with that of the other. However, we expect relatively better results from a log-linear
          model; so we can discuss whether or not the results have been improved. Yes, results
          are relatively improved, especially in terms of F-statistic and t-statistic/p-values.
          Model is found statistically significant (F = 19.309, p < 0.01); the explanatory
          variables explain around 27 percent variance in the dependent variable (R 2 = 0.2720;
          R2adjusted = 0.2580).
      (b) Log-linear model reinforces the results regarding signs and significance values of the
          individual explanatory variables.
      (c) Results (of the both models) suggest that facets like informational justice, distributive
          justice and informational justice appear to be positively contributing towards
          employees job satisfaction, as compared to the procedural justice, which needs to be
          taken care of for an overall satisfaction of Pakistani organizational employees. In
          addition, the senior, more educated and more experienced employees also need
          attention as they appear to be mostly dissatisfied from their jobs.




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                Assignment 2 (Due in the Next Class)
1. Briefly explain (in bullet-points) what the major contribution is that of simple/two-
   variables regression model, and why we have to resort to multiple regression analysis.
2. Go through the steps suggested for estimation of a linear-regression model; what is the
   difference between a linear and log-linear model? (a) How do the steps of estimation of a
   log-linear model differ from that of linear model? (b) How do the interpretations of the
   two model differ?
3. What is reliability? How is reliability test run in SPSS? Why is the running of reliability
   test important?
4. What is the procedure of generating data on variables of interest? How is a Likert-scale
   questionnaire used for generation of data on variables of interest?
5. How are and for what purposes, F-statistic, R2 and t-statistic/p-values used for the
   evaluation and interpretation of estimated models?

6. Study material (entitled “Formulating and clarifying a research topic”) provided in Annex
   – IV:
   (a) In Part – I (of Annex – IV), the answers of the following two questions have been
       provided:
       1. What are three major attributes of a good research topic?
       2. How can we turn research ideas into research projects?
   (b) In Part – II, you have been provided two lengthy lists of research topics proposed by
       my MS ARM’s class students of section 2 & 3. You please select one topic of your
       choice (select topic in light of what you have learnt from materials provided in Part –
       I), develop 2 – 3 research questions and 4 – 5 research objectives, and submit me
       through email (anwar@jinnah.edu.pk & chishti_anwar@yahoo.com), latest by
       12.00 (Noon) Monday; please note: we will discuss your selected topic along with
       research questions and objectives in Monday’s evening class (along with the
       remaining/leftover part of previous Lecture – 2).

       Please also note: you may suggest a topic of your own (not already enlisted), along
       with research questions and objectives. Whether you select a topic from our list or
       suggest the one from your own side, two students of my ARM class will assist you to
       carry out research on that topic, as part of your AQT class requirements, for a 20%
       marks.




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                                         ANNEX – I (Questionnaire)
                                                Section I
Your Organization (Tick 1 or zero):       Government = 1                                     2. Private = 0
Your gender (Tick 1 or zero):             Male = 1                                           2. Female = 0
Your age (in years like 25 years, 29 years,)
Your education (actual total years of schooling, like 14 years; 18 years)
Your area of specialization:
Your job title in this organization:
Experience: Working years in this organization:
                                               Section II
Strongly disagree – 1          Disagree = 2        Not disagree/neither agreed = 3               Agreed = 4       Strongly
                                                      agreed = 5
JS: Job satisfaction (Agho et al. 1993; Aryee, Fields & Luk (1999))                          1       2        3       4        5
1    I am often bored with my job (R)
2    I am fairly well satisfied with my present job
3    I am satisfied with my job for the time being
4    Most of the day, I am enthusiastic about my job
5    I like my job better than the average worker does
6    I find real enjoyment in my work
                       Organizational Justice (Niehoff and Moorman (1993))
    Strongly disagreed = 1 Slightly disagree = 2 Disagree = 3 Neutral (Not disagree/neither
            agreed) = 4 Agreed = 5 Slightly more agreed = 6 Strongly agreed = 7
                      Distributive justice items (DJ)                            1   2       3       4        5       6        7
1   My work schedule is fair
2   I think that my level of pay is fair
3   I consider my workload to be quite fair
4   Overall, the rewards I receive here are quite fair
5   I feel that my job responsibilities are fair

                         Procedural justice items (PJ)                           1       2       3       4        5       6    7
1    Job decisions are made by my supervisor in an unbiased manner
2   My supervisor makes sure that all employee concerns are heard before
    job decisions are made
3   To make formal job decisions, supervisor collects accurate & complete
    information
4   My supervisor clarifies decisions and provides additional information
    when requested by employees
5   All job decisions are applied consistently across all affected employees
6   Employees are allowed to challenge or appeal job decisions made by
    the supervisor
                                                Interactive justice items (IJ)
1   When decisions are made about my job, the supervisor treats me with
    kindness and consideration
2    When decisions are made about my job, the supervisor treats me with
    respect & dignity
3   When decisions are made about my job, supervisor is sensitive to my
    own needs
4   When decisions are made about my job, the supervisor deals with me
    in truthful manner


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5   When decisions are made about my job, the supervisor shows concern
    for my rights as an employee
6   Concerning decisions about my job, the supervisor discusses the
    implications of the decisions with me
7   My supervisor offers adequate justification for decisions made about
    my job
8   When decisions are made about my job, the supervisor offers
    explanations that make sense to me
9   My supervisor explains very clearly any decision made about my job
    Strongly disagree – 1    Disagree = 2 Not disagree/neither agreed = 3 Agreed = 4   Strongly agreed = 5
                           Informational justice items (INJ)                       1        2     3        4    5
1   Your supervisor has been open in his/her communications with you
2    Your supervisor has explained the procedures thoroughly
3   Your supervisor explanations regarding the procedures are reasonable
4   Your supervisor has communicated details in a timely manner
5   Your supervisor has seemed to tailor (his/her) communications to individuals’
    specific needs.




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                                          ANNEX - II
            Credibility of research findings: important considerations
                    (Reliability? Validity? Generalizability?)
Reliability: Reliability can be assessed by posing three questions:
   1. Will the measure yield the same results on other occasions?
   2. Will similar observations be reached by other observers?
   3. Is the measure/instrument stable and consistent across time and space in yielding
       findings?
4-Threats to reliability
       (i) Subject/participant error
       (ii) Subject/participant bias
       (iii) Observer error and
       (iv) Observer’s bias


Validity: Whether the findings are really about what they appear to be about.
Validity depends upon:
       History (same history or not),
       Testing (if respondents know they are being tested),
       Mortality (participants’ dropping out),
       Maturation (tiring up), and
       Ambiguity (about causal direction).


Generalizability:
       The extent to which research results are generalizable.




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                                            ANNEX – III
                              Reliability test and interpretation
Reliability test results
Responses on the elements of all five constructs (JS, DJ, PJ, Ij & INJ) were entered on SPSS’s
data editor and reliability tests were conducted; the following Cronbach’s Alphas were
estimated.
                                 Table 4.4 Results of reliability test
                        Construct                          Cronbach’s Alpha
                        Job Satisfaction (JS)                 0.739
                        Distributive Justice (DJ)             0.828
                        Procedural Justice (PJ)               0.890
                        Interactional Justice (IJ)            0.920
                        Informational Justice (INJ)           0.834

Interpretation
According to Uma Sekaran (2003), the closer the reliability coefficient Cronbach’s Alpha gets to
1.0, the better is the reliability. In general, reliability less than 0.60 is considered to be poor, that
in the 0.70 range, acceptable, and that over 0.80 and 0.90 are good and very good. The reliability
tests of our constructs happened to be in the acceptable to good and very good ranges.




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                                                 ANNEX - IV
                    Formulating and clarifying a research topic1
Part – I:        Two major questions:
    3. What are three major attributes of a good research topic?
    4. How we can turn research ideas into research projects
                    Three major attributes of a good research topic are
    •   Is it feasible?
    •   Is it worthwhile?
    •   Is it relevant?
Capability: is it feasible?
  » Are you fascinated by the topic?
  » Do you have the necessary research skills?
  » Can you complete the project in the time available?
  » Will the research still be current when you finish?
  » Do you have sufficient financial and other resources?
  » Will you be able to gain access to data?

Appropriateness: is it worthwhile?
  » Will the examining institute's standards be met?
  » Does the topic contain issues with clear links to theory?
  » Are the research questions and objectives clearly stated?
  » Will the proposed research provide fresh insights into the topic?
  » Are the findings likely to be symmetrical?
  » Does the research topic match your career goals?

Relevancy: is it relevant?
   » Does the topic relate clearly to an idea you were given - possibly by your organisation?


                        Turning research ideas into research projects
    •   Conceive some research idea
    •   Think about research topic (having attributes stated above)
    •   Write research questions
    •   Develop research objectives


1
 This discussion is based on materials contained in chapter 2 of Saunders, M., Lewis, P. and Thornhill, A. (2011)
Research Methods for Business Students 5th Edition. Pearson Education


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Part – II:     Research topics proposed by MS-ARM students
ARM (section – 2)

Performance appraisal as a tool to motivate employees: a comparison of public-private sector
organization
Performance appraisal in ……………….. (name of organization)
Marketing communication and brand loyalty
Implementation of Integrated Management System (IMS) in Pakistan Civil Aviation Authority
Information technology and financial services
Capital structure and firms profitability
Interest rates, imports, exports and GDP
Intra-Group Conflict and Group Performance
HR practices across public and private organizations
HR practices across SMEs and large companies
HR practices across manufacturing and services sector companies
Corporate governance practices in banking sector of Pakistan
Corporate governance practices in textile industry
Corporate governance practices in pharmaceutical industry
Effects of working capital management on profitability
Working capital with relationship to size of firm
Working capital and capital structure
Optimizing working capital
Dividend policy and stock prices
Sales, debt-to-equity ratio and cash flows
Relationship between KSE’s, LSE’s and ISE’s stock prices
Gold prices and stock exchange indices
Interest rates, bank deposits and private investments
Security Market Line (SML) & Capital Market Line (CML) at KSE
Relationship between stock market returns and rate of inflation


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Relationship between CPI and Bond price
Pakistan’s exchange rates with relation to major global currency regimes: an analysis


ARM (section – 3)
Trade deficit, budget deficit and national income
Performance appraisal and its outcomes
Impact of compensation on employee’s job satisfaction
Human resource management & outsourcing
Advertising and brand image
Performance management in public sector organizations
Impact of training on employees’ motivation and retention
Impact of performance appraisal
Financial returns, returns on shares, equity returns and share prices
Factors contributing towards employee turnover intention
Antecedents of employees’ retention
Employees’ retention policies and employees’ turnover
Impact of training and development on employees’ motivation and turnover intention
Outsourcing human resource function in Pakistani organizations
Exploring the impact of human resources management on employees’ performance
Service orientation, job satisfaction and intention to quit
Brand equity and customer loyalty: a case of …….. (name of orhanization)
PTCL privatization: effects on employees’ morale
PTCL privatization: effects on employees’ efficiency
PTCL privatization: effects in terms of profitability
Electronic and traditional banking: how do customers’ perceive?
FPI and FDI in Pakistan: a comparative analysis
Stock market indices: KSE, LSE and ISE compared
Work family conflict and employee job satisfaction: moderating role of supervisor’s support




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                                                 Topic 3
                        Multiple regression: model specification
3.1(a) Conceiving research ideas and converting it into research projects: a
       Procedure
          Procedure:       Research ideas à research topic à research questions à research
                           Objectives à research hypotheses
Your Take-home Assignment 2’s question 6 has set the example how research ideas and topics
are converted in to research projects, adopting the procedure detailed above. Students have also
provided details of their chosen topics; let’s discuss those topics and clarify them further,
judging them in light of the relevant theories (section 3.1b).


3.1(b) Incorporating theory as the base of your research
Econometrics theory
Please study section 7.2 and 7.3 of Andren (2007)2 and try to understand what difference it
creates when we omit a relevant explanatory variable or include an irrelevant one in an
econometrics model.


Economics/management theory
Let us evaluate whether the research projects you have proposed are based on the relevant
economic/management theory, and if not, then how you can incorporate the relevant theory into
your projects.

                   Discussion on your proposed research projects
            (You need to take notes on suggestions for improvements, and submit the
         improved version of your research project as part of your next assignment 3 (a).


                          (See Annexure – I for topics for discussion
                                            Assignment 3 (a)
1. You must have taken the notes on suggestions made during our class discussion on your
   respective research projects; you please refine your topics and research questions and
   objectives, in light of the discussions as well as what the following research articles suggest
2
    Andren, Thomas. (2007). Econometrics. Bookboon.com, pp.74-77


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regarding basing your research on relevant theory (soft copies of papers are provided on
AQT-Class Yahoo Group).

   Article/Note: ‘Formulating a Research Question’

   Rogelberg, Adelman & Askay (2009). Crafting a Successful Manuscript: Lessons from
   131 Reviews. J Bus Psychol (2009) 24:117–121 (Study only 8-points given under
   heading ‘Conceptual and/or theoretical rationale’.)

   Thomas, Cuervo-Cazurra & Brannen (2009). From the Editors: Explaining theoretical
   relationships in international business research: Focusing on the arrows, NOT the boxes.
   Journal of International Business Studies (2011) 42, 1073–1078 (Read only ‘Abstract’
   and ‘Introduction’ sections, and try to understand Figure 1 (Typical conceptual
   diagram).

    Andren, Thomas. (2007). Econometrics. Bookboon.com (Read only sections 72 & 73,
                                          pp.74-77)




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         Topic 3 Multiple regression: model specification….continues
In sub-section 3.1(a), we carried out an exercise on how a conceived research idea can be
converted in to a research projects (Research ideas à research topic à research questions à
research objectives). In sub-section 3.1(b), we tried to learn how much important the
econometrics (omission and inclusion of relevant and irrelevant explanatory variables) and
economics/management theories are for specification of an econometrics model. In this new
subsection 3.2, we will try to learn what role different mathematical formulations can play in
econometrics modeling


       3.2 Specifying an Econometric Model: Mathematical Specification
This section further consists of two subsections, namely:
       3.2(a) Specification of an econometric model: mathematical formulation in general
       3.2(b) Some practical examples of mathematical formulations/specifications: production
               function, cost-function and revenue function
3.2(a) Specification of an econometric model: mathematical formulation in general
Our discussion in earlier sections on simple regression and multiple regression analysis clarifies
two major points, namely:
   1. The simple and multiple regression analysis assumes that variable Y depends on variable
       X, but for this phenomenon of dependence or causation, the researcher takes insights
       from the basic theory (economics/management).
   2. Previous discussion further emphasizes that it is the researcher’s responsibility to specify
       an econometric model such that it contains all major relevant explanatory variables as
       independent variables; otherwise, empirical results obtained in terms of estimated
       coefficients would be biased.
While specifying a model, the researcher has to take the above points in to consideration.
Additionally, the researcher has to decide which mathematical formulation of the model he/she
should use so that the true relationship between dependent and independent variables is captured
to the maximum extent. This is how an econometric model is/should be specified.




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Let’s proceed further, taking some practical examples of mathematical formulations of the
model. In case, we have the following type of relationship between Y – X variables:

        Y                                     Y                                  Y




                               X                                      X                                 X
                Case 1 (a)                        Case 1 (b)                         Case 1 (c)

Case 1a is a general linear relationship, and can be measured, as follows.
        Y = β0 + β1X1 + e                                                                    (3.1)

In 3.1, we expect β1to carry positive sign.


The case 1(b) represents an exponential case, and can be measured, as follows:
                           2
        Y = β0 + β1X1 + β2X 1 + e                                                            (3.2)

Specially, the parameters β1and β2 will carry positive signs.


In case of a cubic-type of relationship like 1(c), the following mathematical formulation will have to be
adopted.
                           2       3
        Y = β0 + β1X1 + β2X 1 + β3X 1 + e                                                    (3.3)

The coefficients β1and β2 will carry positive but β3 negative sign.


In other words, it means that if we have to measure the stated type of relationships between
our Y – X variables, we need to use the relevant type of mathematical formulations while
specifying our econometrics model.


In certain other cases/on certain occasions, we have to adopt some other mathematical
formulations like the following ones:
        Y = β0 + β1X1 + β2X1X2 + β3X2 + e                                                    (3.4)
                            2                      2
        Y = β0 + β1X1 + β2X 1 + β3X1X2 + β4X2 + β5X 2 + e                                    (3.5)



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Equation 3.4 measures linear relationship, but includes an interaction term (X1X2). β2 can take

any sign (+, - or 0); a positive sign would show positive effect of the interaction of X 1 and X2 on Y, a

negative sign would mean negative effect of interaction of these two variables and zero effect
would mean zero effect on dependent variable Y. Let’s visit some practical examples where we
can use some of the above stated mathematical formulations (next section).


3.2(b) Some practical examples: production, cost and revenue functions

Production function
In case, we have data on production of product Y, wherein two major inputs used are X 1 and X2:

                                 Y                  X1                 X2
                                2500                 1                150
                                2525                 2                152
                                2555                 3                155
                                2592                 4                159
                                2635                 5                161
                                2677                 6                169
                                2718                 7                174
                                2745                 8                178
                                2766                 9                181
                                2781                10                182

Let’s check relationship between Y – X1, and Y – X2 (separately), using mathematical formulation given
in (3.3), using data provided in above table.
     Do this as Take-home Assignment 3b (Question 1); show the estimated
                          relationship through hand-drawn graph


Let’s check relationship between Y and X1 & X2, using mathematical formulation given in (3.4), using
data provided in the above table.
    Do this as Take-home Assignment 3b (Question 2); interpret the results,
                            including that of the interaction term




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Cost Function
Cost function can be developed when you have data like the following one:

                                               Y          TC
                                                    1      193
                                                    2      226
                                                    3      240
                                                    4      244
                                                    5      257
                                                    6      260
                                                    7      274
                                                    8      297
                                                    9      350
                                                   10      420

Mathematical formulation of a typical cost-function is:
                           2     3
        TC = β0 + β1Y - β1Y + β1Y + e                                                      (3.6)

Did you notice the signs of a typical cost-function are opposite to that of a typical production-function
(given in 3.3).


  Estimate cost-function 3.6 as Take-home Assignment 3b (Question 3); show
                  the estimated relationship through hand-drawn graph


                     Assignment 3b: Question 4
     Download 8 – 10 published research articles on the area of
   research/topic you have chosen for your class research project,
  study the conceptual models tried in these research articles, and
 develop your own model, including the mathematical one as part of
 your Take-home Assignment 3(b), due in next class; be ready for a
                       class presentation also.




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        Topic 3 Multiple regression: model specification….continues




                     3.3 Conceptual/econometric modeling

                             3.3 (a) Examples in Finance

                            3.3 (b) Examples in Marketing

                               3.3 (c) Examples in HRM


3.3 (a) Examples in Finance: summary

Example 1: Interest rates and GDP: a case of Pakistan
Example 2: Capturing effects of interest rates on Pakistani economy
Example 3: Exchange rates and Pakistan’s trade: an analysis
Example 4: Exchange rates and Pakistan’s economy: an analysis
Example 5: Research on Working Capital (WC)
      Proposal 1: “Relationship between Profitability and Working Capital
      Management”, using econometric technique

      Proposal 2: “Liquidity-profitability trade-off”, using Goal programming (of
Operations Research)




                                                                                       34
LECTURES &
                   ADVANCED QUANTITATIVE TECHNIQUES                                                      NOTES




3.3 (a) Examples in Finance

Example 1: Interest rates and GDP: a case of Pakistan3
Though we are interested in analyzing the effect of interest rates on Pakistan’s national income,
but we know that interest rates do not affect GDP directly, rather these affect saving (bank
deposits) and private investments, and as a consequence GDP is affected; so we conceptualize
the path of the effect, as follows:
         Interest rates (↑↓) à bank deposits (↑↓) & private investments (↓↑)
         à GDP (↓↑)
The above path of the effect (of interest rates) can be captured, through econometrics model,
postulated, as follows.
         Private investment = ƒ(Interest rates)                                                             (3.7a)
         GDP = ƒ(Private investments_predicted in equation 7a)                                              (3.7b)
Theory tells us that private investment (PI) is influenced not only by the interest rate (R) but is
also affected by openness of the economy (OE) and, especially the costs and taxes (C&T).
Hence, equation 3.7a would change to:
         PI = ƒ(R, OE, C&T)                                                                                 (3.8a)
                                                                  ̂
The private investment predicted on the basis of equation 3.8a (PI) is not the only determinant of
GDP, government expenditure (GE) or budget spending is another determining variable; while in
Pakistani context, Foreign Direct Investment (FDI) and Pakistan’s productive population, that is,
the active labor force (LF) are two other factors should be considered as determinants of
Pakistan’s national income (GDP). Hence, model 3.7b would change, as follows.
                   ̂
         GDP = ƒ(PI, GE, FDI, LF)                                                                           (3.8b)
The model postulated in 3.8 (a – b) still needs improvement; government expenditure (GE) and
FDI are not autonomous in nature, the former depends on government revenues (GR) and
government borrowing from foreign (FB) and domestic (DB) sources, and the latter depends




3
  Students are urged to think over the difference between topic of this Example 1 and that of Example 2, and then try
to understand how conceptual/econometric modeling can be differently developed to take care of the differences
which the two topics necessitate.


                                                                                                                  35
LECTURES &
                   ADVANCED QUANTITATIVE TECHNIQUES                                                      NOTES



    upon economy’s openness (OE) and cost of production and taxes (C&P). To incorporate these
effects, the model would therefore adopt the following form.
          PI = ƒ(R, OE, C&P)                                                                                (3.9a)
          GE = ƒ(GR, FB, DB)                                                                                (3.9b)
          FDI = ƒ(OE, C&P)                                                                                  (3.9c)
                    ̂   ̂    ̂
          GDP = ƒ(PI, GE, FDI, LF)                                                                          (3.9d)
Model 3.9 (a – d) represents what we need to do for a piece of research conducted under title
“Interest rates and GDP: a case of Pakistan”. In case we extend the scope of our research to what
is needed under title “Capturing effects of interest rates on Pakistani economy”, we will then
have to adopt the model specified in the following Example 2.


Example 2: Capturing effects of interest rates on Pakistani economy
Notice the difference between the two topics (Example 1 and 2); the first topic requires
analyzing the effect of exchange rates on GDP, while the second topic asks for looking in to the
same thing from a little broader perspective, that is, from the point of view of whole economy.
Since the model specified for the first topic covers largely the methodology needed for the
second topic, we can use the same first example model 3.9 (a – d), with an additional equation
for analyzing the effect of interest rates on bank deposits, which can be assumed to be
determined by money supply in the country (M), in addition to the interest rates (R).
          Bank deposit = ƒ(R, M)                                                                            (3.9e)
Hence, model 3.9 (a – e) will be used for the piece of research identified in example 2.


Example 3: Exchange rates and Pakistan’s trade: an analysis4
According to the theory, the appreciation or depreciation of exchange rates (ER) affects the
country’s trade; appreciation of a country’s currency makes exports expensive and imports
cheap, and depreciation makes exports cheap and imports expensive. This stated phenomenon is
true for the two trade partners, but is also affected by certain other situations prevailing in the
two trading countries. The foreign country’s exchange rates with respect to her other major trade

4
  Students are urged to think over the difference between topic of this Example 3 and that of Example 4, and then try
to understand how conceptual/econometric modeling can be differently developed to take in to account the
differences which the two topics necessitate.



                                                                                                                  36
LECTURES &
                    ADVANCED QUANTITATIVE TECHNIQUES                                                   NOTES



partners, availability and prices of the substitutes in foreign country and world over, consumers’
income, trade openness and political situations are some other important factors affecting export
and import trade.
    Tracing and finding out the effects of the determinants of export and import trade might be easy
when trade of certain known commodities between two specific countries is analyzed; but the
case becomes cumbersome, and needs extra care when analysis of trade is required at aggregate
level, for instance the topic of this piece of research - Exchange rates and Pakistan’s trade: an
analysis.
We can think primarily about some very simple questions like what the exchange rates are
(definition), how these are determined (or are autonomous in nature), they affect what and how,
and specifically what relationship they have with trade – its two components, imports and
exports. And since we are analyzing the exchange rates of Pakistan and her trade, we should
think over the answers of such questions in the context of Pakistan’s economy.
Exchange rates (ER) are not autonomous in nature, these are determined by the forces of demand
for and supply of major medium of currency (US dollar in Pakistan) used in imports and exports
trade. Value of imports seems to be the major factor to determine demand for US dollar in
Pakistan, and while value of exports, workers’ remittances (WR), foreign direct investment
(FDI) and foreign borrowings (FB) appear to be the major determinants of supply of dollar.
Hence, these demand and supply factors determine exchange rates in Pakistan, which in turn
affect volumes of import and export.
           ER = ƒ(IM, EX, WR, FDI, FB)                                                                   (3.10)
                    ̂
           IM = ƒ(ER)                                                                                    (3.11)
                    ̂
           EX = ƒ(ER)                                                                                    (3.12)
But ER̂ is not the only determinant of import (IM). Imports in Pakistan have historically been
largely composed of capital goods (28% in 1980-81 and 24% in 2010-11) and industrial raw
materials (58% in 1980-81 and 60% in 2010-11)5; the value of the share of Pakistan GDP’s
manufacturing sector (GDPM) may therefore be included in equation 3.11 as proxy to represent
the demand for imports, in addition to the population or its growth rate (POP) as proxy for the
size of the market. Hence, equation 3.11 adopts new form, namely:
                    ̂
           IM = ƒ(ER, GDPM, POP)                                                                         (3.13)

5
    Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Statistical Appendix Table 8.5B


                                                                                                              37
LECTURES &
                    ADVANCED QUANTITATIVE TECHNIQUES                                     NOTES



In case of exports, primary commodities and semi-manufactured and manufactured products
have been the major components, with share of 44% in 1980-81 and 18% in 2010-11, 11% in
1980-81 and 13% in 2010-11 and 45% in 1980-81 and 69% in 2010-11, respectively 6. The values
of the primary (GDPP) and secondary/manufacturing sectors’ contributions to GDP (GDPM)
may therefore be included in equation 3.12 as proxies to represent major supplying sectors of
exports. The demand for Pakistani exports has come from both developed (60.8% in 1990-91 and
44.5% in 2010-11) and developing (39.2% in 190-91 and 55.5% in 2010-11) countries 7, the
world’s GDP can be taken as proxy to represent demand from the whole world (GDPW). Hence,
equation 3.12 adopts the new form, namely:
                    ̂
           EX = ƒ(ER, GDPP, GDPM, GDPW)                                                    (3.14)
Summarizing the model,
           ER = ƒ(IM, EX, WR, FDI, FB)                                                     (3.15a)
                    ̂
           IM = ƒ(ER, GDPM, POP)                                                           (3.15b)
                    ̂
           EX = ƒ(ER, GDPP, GDPM, GDPW)                                                    (3.15c)
We can add even some other relevant variables and improve the model (model 3.15), and
reviewing the relevant literature on respective topics and sub-topics, with special reference to
Pakistan, would help us in this regards.
Please note that model 15 (a – c) will restrict research to the analysis of the effects of exchange
rates on Pakistan’s trade; in case, if someone is interested to analyze the exchange rates’ effects
on Pakistan economy (or GDP), then model specified in following Example 4 should be used.


Example 4: Exchange rates and Pakistan’s economy: an analysis
Model specified in 3.15 (a – c) will work as the base to analyze the effect of exchange rates on
import and export trade, and incorporation of an additional equation (3.15d), which transfers the
                      ̂                ̂
effects of imports (IM) and exports (EX) to GDP will help complete a model for the analysis
necessary for new topic.
                      ̂   ̂
           GDP = ƒ (IM, EX, POP)                                                           (3.15d)
The effect of the size of population (POP) has been included as a proxy for the effect of domestic
consumption on country’s GDP.

6
    Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Table 8.5A
7
    Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Table 8.7


                                                                                                38
LECTURES &
                    ADVANCED QUANTITATIVE TECHNIQUES                                               NOTES



Example 5: Research on Working Capital (WC)
Working capital: in general
Working capital is defined as8:
           Working Capital (WC) = current assets (CA) - current liabilities (CL)
           (3.16a) Where Current assets are cash and other assets that can be converted to cash
within a year, and Current liabilities are obligations that the company plans to pay off within the year.
Working capital indicates the assets the company has at its disposal for current expenses. The
process of managing the WC efficiently is called Working capital Management. An excess of
working capital many mean that the company is not managing its assets efficiently. It's not using its
assets to get a bigger return or better profit. An aggressive company may keep its working capital
smaller. But a very low working capital may mean the company may not be suited well enough to
payoff its short term obligations.
This decision of how to manage the working capital of the company depends on the Working
capital policy of the company. An important factor that determines the policy is the industry in which
the company operates. For Example, an IT service company may not have a lot of shot-debt in
terms of inventory but it still needs to pay wages, insurances and other expenses like rent. The
company needs to have a policy that makes sure it sets targets were it gets paid as the project
progresses so it can keep paying its staff in time. The company has to manage its account
receivables according to this policy. Some industries operate in a high profit margin that they can
afford to have a longer term on the account receivables because the higher cash balance part of the
current assets. The Collection Ratio helps project this aspect of a company; The collection ratio is
defined as:
           Collection Ratio = Accounts Receivable / (Revenue/ 365)                                    3.16b)
Collection ratio tells us the average number of days it takes a company to collect unpaid invoices. A
ratio which is very near to 30 days is very good since it means that the company is getting paid on a
monthly basis.
Sales is another attribute that strongly impacts working capital. It is the ability of a company to sell its
products fast enough to get the money back to put back into operations or supplies for producing
more materials. Moving inventory fast is always a good plan for a company. It also helps in reducing
costs associated with holding and moving inventory. A good ratio that helps put the attribute in
perspective is inventory turnover ratio, which is defined as:
           Inventory turnover ratio = sales / inventory
Or         Inventory turnover ratio = Cost of goods sold / inventory                                  (3.16c)

8
    The following material is based on http://www.business.com/finance/working-capital/; downloaded on October
    12, 2012.


                                                                                                            39
LECTURES &
                   ADVANCED QUANTITATIVE TECHNIQUES                                                     NOTES


This ratio shows the efficiency the company has in selling its products. The higher the ratio the better
the company is able to move the products. Again this could be dictated by the industry, for example,
a daily products company is usually forced to sell its products fast enough or lose it. The ratio also
provides a good insight into how a company is doing within an industry. The direct ratio of
companies can be compared to see how well the company is able to sell the products in comparison
to its competitors.
Financing is another attribute of Working Capital management. Debt - Asset ratio provides a good
insight into how much of the company's assets are being financed though debt. The debt – asset
ratio is defuned as:
        Debt-asset ratio = Total liabilities / Total assets                                                (3.16d)
Working capital management becomes a very important aspect for a company since it is the first line
of defense against market downturn cycles and recession. A company with cash is usually in a good
position to make better use of the opportunities the markets provide. Its can spend the money on
R&D for coming up with better products. Increase in current assets, especially, increase in account
receivables due to growth is sales have to be managed efficiently. Ability to control working capital
plays a significant role in the survival of the company.


Research on Working Capital
Let us see how the above information on working capital (WC) and working capital management
(WCM) has been used by different researchers to carry out research on the topic under study.
Lazaridis and Tryfonidis’s (2006)9 and Gill, Biger and Mathur (2010)10 analyzed the relationship
between profitability and working capital management, using about the same model, and
measuring and generating the dependent and independent variables in the following way:
        No. of Days A/R = (Accounts Receivables/Sales) x 365
        No. of Days A/P = (Accounts Payables/Cost of Goods Sold) x 365
        No. of Days Inventory = (Inventory/Cost of Goods Sold) x 365
        Cash Conversion Cycle = (No. of Days A/R + No. of Days Inventory) – No. of Days A/P
        Firm Size = Natural Logarithm of Sales
        Financial Debt Ratio = (Short-Term Loans + Long-Term Loans)/Total Assets
        Fixed Financial Asset Ratio = Fixed Financial Assets/Total assets
        Profit = (Sales - Cost of Goods Sold) / (Total Assets - Financial Assets)



9
  Lazaridis I, and Tryfonidis D, (2006). Relationship between working capital management and profitability of listed
companies in the Athens stock exchange. Journal of Financial Management and Analysis, 19: 26-25.
10
   Gill, A., Biger, N. and Mathur, N. (2010). The Relationship Between Working Capital Management And
Profitability: Evidence From The United States. Business and Economics Journal, Volume 2010: BEJ-10


                                                                                                                 40
LECTURES &
                  ADVANCED QUANTITATIVE TECHNIQUES                                                  NOTES



Raheman A. and Nasr, M. (2007)11 used similar methodology but measured the required
variables in somewhat different way, namely:
        NOPit = β0 + β1(ACPit) + β2 (ITIDit) + β3 (APPit) + β4(CCCit) + β5(CRit) + β6(DRit)
        + β7(LOSit) + β8(FATAit) + ε                                                                    (3.17)
Where:
        NOP : Net Operating Profitability
        ACP : Average Collection Period
        ITID : Inventory Turnover in Days’
        APP : Average Payment Period
        CCC : Cash Conversion Cycle
        CR : Current Ratio
        DR : Debt Ratio
        LOS : Natural logarithm of Sales
        FATA: Financial Assets to Total Assets
        ε : The error term.
Researchers have estimated/generated variables, using the following definitions.
Net Operating Profitability (NOP) which is a measure of Profitability of the firm is used as
dependant variable. It is defined as Operating Income plus depreciation, and divided by
total assets minus financial assets.
Average Collection Period (ACP) used as proxy for the Collection Policy is an
independent variable. It is calculated by dividing account receivable by sales and multiplying
the result by 365 (number of days in a year).
Inventory turnover in days (ITID) used as proxy for the Inventory Policy is also an
independent variable. It is calculated by dividing inventory by cost of goods sold and
multiplying with 365 days.
Average Payment Period (APP) used as proxy for the Payment Policy is also an
independent variable. It is calculated by dividing accounts payable by purchases and
multiplying the result by 365.




11
  Raheman A. and Nasr, M. (2007). Working capital management and profitability – case of Pakistani firms.
International Review of Business Research Papers, 3: 279-300.


                                                                                                            41
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Aqt instructor-notes-final

  • 1. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Lectures & Notes ADVANCED QUANTITATIVE TECHNIQUES (COURSE FOR PHD STUDENTS) By Dr. Anwar F. Chishti Professor Faculty of Management & Social Sciences 1
  • 2. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES ADVANCED QUANTITATIVE TECHNIQUES Course Plan Fall Semester 2012 Course Instructor Professor Dr. Anwar F. Chishti Contacts: Phone Phone: 0346-9096046 Email anwar@jinnah.edu.pk; chishti_anwar@yahoo.com Class venue Computer Laboratory Course contents Topic 1: Simple/Two-Variable Regression Analysis: • An introduction of estimated model and its interpretation, • Regression Coefficients and Related Diagnostic Statistics: Computational Formulas • Evaluating the results of regression analysis • Standard assumptions, BLUE properties of the estimator. • Take-home assignment - 1 Topic 2: Simple Regression to Multiple Regression Analysis • Shortcomings of simple/two-variables regression analysis • An example of multiple regression analysis • Use of Likert-scale type questionnaire, raw-data entry, reliability test and generation of variables • Estimation of multiple regression model • Evaluation of the estimated model in terms of F-statistic, R2 and t- statistic/p-value • Take-home assignment - 2 Topic 3: Multiple Regression: Model specification • 3.1(a) Conceiving research ideas and converting it into research projects: a procedure • 3.1(b) Incorporating theory as the base of your research: econometrics theory & economics/management theory • Take-home assignment – 3(a) • 3.2 (a) Specification of an econometric model: mathematical specification 2
  • 3. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES • 3.2(b) Some practical examples of mathematical specification: production-function specification, cost-function specification, revenue- function specification • Take-home assignment – 3(b) • 3.3(a) Conceptual/econometric modeling: (a) Examples in Finance; (b) Examples in Marketing; (c) Examples in HRM • 3.3(b) Incorporating theory as the base of your research: econometrics theory & economics/management theory • Take-home assignment: adopting, adapting and developing a new questionnaire Topic 4: Analyzing mean values • Analyzing mean value, using one-sample t-test • Comparing mean-differences of two or more groups • Comparing two groups * Independent samples t test * Paired-sample t test • Comparing more-than-two groups * One-Way ANOVA * Repeated ANOVA • Take-home assignment – 4 Topic 5: Uses of estimated econometric models • Some examples • Take-home assignment – 5 Topic 6: Relaxing of Standard Assumptions: Normality Assumption and its testing • Normality assumption • Its testing • Take-home assignment – 6 Topic 7: Problem of Multicollinearity: What Happens if Regressors are Correlated? • Consequences, tests for detection and solutions/remedies • Take-home assignment - 7 Topic 8: Problem of Heteroscadasticity: What Happens if the Error Variance is nonconstant? • Consequences, tests for detection and solutions/remedies • Take-home assignment - 8 Topic 9: Problem of Autocorrelation: What Happens if the Error terms are correlated? • Consequences, tests for detection and solutions/remedies • Take-home assignment - 9 Topic 10: Mediation and moderation analysis - I • Estimating and testing mediation 3
  • 4. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES • Take-home assignment – 10 Topic 11: Mediation and moderation analysis - II • Estimating and testing moderation • Take-home assignment – 9 Topic 12: Time-series analysis - I • Unit root analysis • Take-home assignment – 10 Topic 13: Time-series analysis - II • Unit root, co-integration and error correction modeling (ECM) • Take-home assignment – 11 Topic 14 Panel data analysis, Simultaneous equation models/Structural equation models • Panl data analysis • SEM, ILS, 2SLS and 3SLS • Take-home assignment – 12 Topic 15 Qualitative response regression models (when dependent variables are binary/dummy) and Optimization • LPM, Logit model and Probit Model • Take-home assignment – 13(a) • * Optimization: minimization and maximization • Take-home assignment – 13(b) Topic 16 Welfare analysis: maximization of producer and consumer surpluses and minimization of social costs Required Text & Recommended Reading The prescribed textbooks for this course are: Gujarati, Damodar N. Basic Econometrics, 4th Edition. McGraw-Hill. 2007 Stock, J. H. and Watson, M.W. Introduction to Econometrics, 3/E. Pearson Education, 2011 Reference Books/Materials Studenmund, A.H. Using Econometrics: A Practical Guide, 6/E, Prentice Hall Asteriou, D. and Hall, S.G. Applied Econometrics – A Modern Approach. Palgrave Macmillan, 2007. 4
  • 5. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Andren, Thomas. (2007). Econometrics. Bookboon.com Salvatore, D and Reagle, D. Statistics and Econometrics, 2nd Ed. Schaum’s Outlines. Instructor’s class-notes (hard copy at photo-copier shop) Assessment Criteria Details Due Date Weighting 10 best weekly assignments (out of total Individual Assignments 13 - 15, each having 2 marks) will be 20 % counted toward total 20% marks. A group of 2 students will select a topic, Group research on selected carry out research, complete a research 20 % research topics study, and make presentation in during the last classes of the semester Mid-term Examination As per University’s announcement 20 % Final Examination As per University’s announcement 40 % Total marks: 100 5
  • 6. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Topic 1 Simple/Two-Variable Regression Analysis 1.1 Simple regression analysis: an example Assuming a survey of 10 families yields the following data on their consumption expenditure (Y) and income (X). Y (Thousands) X (Thousands) 70 80 65 100 90 120 95 140 110 160 115 180 120 200 140 220 155 240 150 260 The theory suggests that families’ consumption (Y) depends on their income (X); hence, econometric model may be specified, as follows. Y = f(X) (General form) (1a) Or Y = β0 + β1X + e (Linear form) (1b) The above stated regression analysis model contains two variables (one independent variable X and one dependent variable Y); this model is therefore called Two-variables or Simple regression analysis model. Is this type of Simple or Two-variable model justified? We will discuss this question later on; let’s first estimate this model, using the Statistical Package for Social Sciences’ software SPSS. The estimated model & interpretation Y = 24.4530 + 0.5091 X (2a) (6.4140) (0.0357) (Standard Error) (2b) (3.8124) (14.2445) (t-statistic) (2c) (0.005) (0.000) (p-value/sig. level) (2d) R= 0.981 R2 = 0.9621 R2adjusted = 0.957 F = 203.082 (p-value = 0.000) DW = 2.6809 N = 10 (2e) 1.2 Regression analysis: computational formulas The econometric model specified in (1) is estimated in the form of estimated model (2a) along with all its diagnostic statistics 2(b – e), using the formulas provided, as follows. 6
  • 7. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES The coefficients ßs ∧ ∧ β0 = Y − β1 X (3) ∧ β1 = ∑ ( Xi − X ) (Yi −Y ) (4) ∑ ( Xi − X ) 2 ∧ β1 = ∑x y i i (5) ∑x 2 i Variances (σ 2) and Standard Errors (S.E): 2  ∧  ∧ ∑e 2 ∑Y  −Yi   i (6) σ =2 = ( N − 2) ( N − 2) ∧ Var ( β 0 ) = σ ∧ 2 = ∑ X .σ i 2 2 (7) N ∑x β0 2 i ∧ ∧ ∧ S .E ( β0 ) = σ β0 = σ β0 2 (8) ∧ ∧ σ2 Var ( β1 ) = σ β1 = 2 (9) ∑x 2 i ∧ ∧ ∧ S .E ( β1 ) = σ β1 = σ β1 2 (10) T-ratios: ∧ β0 Tβ0 = ∧ (11) σβ 0 ∧ β1 Tβ1 = ∧ σβ 11 (12) The Coefficient of Determination ( R2 ):  ∧  ESS ∑Y   i −Y   R2 = = TSS ( ∑Y i −Y ) (13) RSS = 1− TSS =1 − ∑ e 2 i ∑Y −Y ) ( 2 i F – Statistics: F = ESS df = ( R ) ( K −1)2 RSS df (1 − R ) ( N − K ) 2 (14) 7
  • 8. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Durban-Watson (D.W) Statistics: 2 ∑(e − et −1 ) N t t =2 d = N ∑e t =1 2 t (15) 1.3 Estimation of the model using computational formulas We now use formula provided in (3) to (15), make computations like Table 3.3 (Gujarati, 2007) and resolve the model, as follows. Yi = ßo + ß1 Xi + ℮i …….. Linear model (16) Regression Coefficients ( ß i ): ˆ β1 = ∑ xi . yi = 16800 = 0.5091 (17) ∑ xi2 33000 ∧ ∧ β0 = Y − β1 X = 111 − 0.5091 (170 ) (18) = 24.453 Variances (σ 2) and Standard Errors (S.E): ∑e 2 ∧ 337.25 σ = 2 = = 42.15625 (19) ( N − 2) 10 − 2 ∧ Var ( β0 ) = σβ ∧ 2 = ∑X .σ i 2 2 = ( 322,000 ) ( 42.15625) 0 N ∑x 2 i ( 10 ) ( 33,000 ) = 41.13428 (20) ∧ ∧ ∧ S .E ( β0 ) = σ β0 = σ β0 2 = 41.13428 = 6.4140 (21) ∧ ∧ σ2 ˆ 42.15625 Var ( β1 ) = σ β1 = 2 = = 0.001277 ∑x 2 i 33,000 (22) ∧ ∧ ∧ S .E ( β1 ) = σ β1 = σ β1 2 = 0.001277 = 0.03574 (23) T-ratios: 8
  • 9. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES ∧ β0 42.453 Tβ0 = ∧ = = 3.8124 σβ 6.414 0 (24) ∧ β1 0.5091 Tβ1 = ∧ = = 14.2445 σβ 0.03574 11 (25) The Coefficient of Determination ( R2 ): R 2 = 1− ∑e 2 i =1 − 337.25 = 0.9621 ∑(Y −Y ) 2 8890 i (26) F – Statistics: F= ( R ) ( K − 1) 2 = 0.9621 ( 2 − 1 ) (1 − R ) ( N − K ) 2 0.0379 (10 − 2 ) (27) 0.9621 = = 203.082 0.0047375 The estimated model: Y = 24.4530 + 0.5091X (6.414) (0.0357)  S.E. (3.812) (14.244)  t-ratio (0.005) (0.0000) (p-valuel) R2 = 0.9621 F = 203.082 N = 10 (28) 1.4 Regression analysis: the underlying theory The above reported formulas reflect how various needed computations are carried out in regression analysis. Specifically, formula (4) estimates the coefficient (β 1) of explanatory variable X: ∧ β1 = ∑ ( Xi − X ) (Yi −Y ) ∑ ( Xi − X ) 2 That is: ‘the deviations of individual observation on Xi from its mean, multiplied by deviations of respective Yi from its mean (cross-deviation), divided by the squares of the variations of Xi’; so it is the ratio between cross-deviations of X – Y variables and X variable. Theoretically, β1 9
  • 10. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES measures ‘total cross deviations/variations per unit of variation in X-variable’. The intercept β0 measures ‘mean value of Y minus total contribution of mean of X’. ∧ ∧ β0 = Y − β1 X 1.5 Error term: its estimation and importance When an econometric model, like 1(b), is specified: Y = β0 + β1X + e (29a) It contains an error or residual term (e); but when model is estimated like 2(a): Y = 24.4530 + 0.5091X (29b) The error term (e) seems to disappear; where does the error term go? In fact the estimated model like 29(b) is valid only for the mean/average values of X and Y, and equality in 29(b) does not hold when values other-than-mean values are used; we can compute values of error terms or residuals, using the following formula. Yi – Ŷ = e (30a) Yi – (24.4530 + 0.5091Xi) = e (30b) Putting individual-observation values from the original data, that is: Y X 70 80 65 100 90 120 95 140 110 160 115 180 120 200 140 220 155 240 150 260 Yi – (24.4530 + 0.5091Xi) = e 70 – (24.4530 + 0.5091*80 = 4.8181 (30c) 65 – (24.4530 + 0.5091*100) = -10.3636 (30d) 90 – (24.4530 + 0.5091*120 = 4.4545 (30e) 95 – (24.4530 + 0.5091*140) = -0.7272 (30f) 110 – (24.4530 + 0.5091*160) = 4.0909 (30g) 10
  • 11. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES 115 – (24.4530 + 0.5091*180) = -1.0909 (30i) 120 – (24.4530 + 0.5091*200) = -6.2727 (30j) 140 – (24.4530 + 0.5091*220) = 3.5454 (30k) 155 – (24.4530 + 0.5091*240) = 8.3636 (30l) 150 – (24.4530 + 0.5091*260) = -6.8181 (30m) As reflects from the above computations, error term reflects how much an individual Y deviates from its estimated value. The values of error terms play important role in determining the size of variance Ϭ2 (computational formula 6), which further affects a number of other computations. A characteristic of error or residual term is that, once we add or take its mean value, it turns out equal to zero, in both cases. 1.6 Evaluating the estimated model After running regression, the results are reported usually reported in the following form. Y = 24.4530 + 0.5091X (31a) (6.4140) (0.0357) (Standard error) (31b) (3.8124) (14.2445) (t-statistic) (31c) (0.005) (0.000) (p-value/sig. level) (31d) R= 0.981 R2 = 0.9621 R2adjusted = 0.957 F = 202.868 (p-value = 0.000) DW = 2.6809 N = 10 (31e) The econometric model is specified in the form of 1 (a or b), estimated in the form of 31 (a) and evaluated, using the diagnostic statistic provided in 31(b – e). The estimated model’s evaluation is carried out, using three distinct criteria, namely: (a) Economic/management theory criteria (expected signs carrying with the coefficients of X-variables) (b) Statistical theory criteria (t statistic or p-value, F statistic, and R2) (c) Econometrics theory criteria (Autocorrelation, Heteroscadasticity & Multicollinearity) Economic theory criteria Questions: a) Are these results in accordance with the economic theory? b) Are they in accordance with our prior expectation? 11
  • 12. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES c) Do the coefficients carry correct sign? Answer: Yes, we expected a positive relationship between the income of a family and its consumption expenditure. The coefficient of income variable, X, is positive. Statistical theory criteria Question 1: a) Are the estimated regression coefficients significant? b) Are the estimated regression coefficients ßs individually statistically significant? d) Are the estimated regression coefficients ßs individually statistically different from zero? Answer: Here, we need to test the hypothesis: HO: ß1 = 0 H1 : ß1 ≠ 0 ß− 0 t= S .E = (.5091 – 0) / .0357 = .5091 / .0357 = 14.2605 (32) Our t calculated = 14.2605 > t tabulated = 1.86 at .05 level of significance, with df (N – k) = 8; hence, we reject the null hypothesis; the coefficient ß1 is statistically significant. Another way of checking the significance level of ßi coefficients is to check its respective p-value (Sig. level). In case of the coefficient of X-variable, the p-value = 0.00, suggesting that coefficient ß 1 is statistically significant at p < 0.01. In this second case, we do not need to check the statistical significance level, using the t-distribution table appended at the end of some econometrics book; we can directly check p-value provided next to the t-value in the output of the solved problem. Question 2: a) Are the estimated regression coefficients collectively significant? b) Do the data support the hypothesis that ß1 = ß2 = ß3 = 0 Here, we need to test the hypothesis: HO: ß1 = ß2 = ß3 = 0 H1: ßi are not equal to 0 Answer: Here, we use F-stattistic, namely: 12
  • 13. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES F = ESS df = ESS / K − 1 = ( R ) ( K −1) 2 RSS df RSS / N − K (1 − R ) ( N − K ) 2 (33) = 202.868 Our F statistic (F = 202.868 > F 1, 8; .05 = 5.32) suggests that the overall model is statistically significant. Like in case of t-statistics, the significance level of F-statistic can also be checked from p-value given next to Fcalculated in the output of the solved problem. Question 3: Does the model give a good fit? Answer: Yes; our R2 = 0.9621 suggests that 96.21% variation in the dependent variable (Y) has been explained by variations in explanatory variable (X). Econometrics theory criteria 1) No Autocorrelation Criteria (We will discuss 2) No Heteroscadasticity Criteria (these criteria in detail 3) No Multicollinearity Criteria (later on in the course 1.7 Interpreting the results of regression analysis The estimated results suggests that if there is one unit change in explanatory variable X (family’s income), there will be about half unit (.5091) change in dependent variable Y (family’s consumption expenditure). If X and Y both are in rupees, then it means that there will be 51 paisas increase in consumption expenditure if the family’s income increases by one rupee. 1.8 Standard assumptions of Least-Square estimation techniques The linear regression model is based on certain assumptions; if these assumptions are not fulfilled, then we have certain problems to deal with. These assumptions are: 1. Error term μ i is a random variable, and has a mean value of zero. ===> μ i may assume any (+), (-) or zero value in any one observation/ period, and the value it assume depends on chance. The mean value of μ i for some particular period, however, is zero, i.e., ∑ (μ i / xi) = 0 2. The variance of μ I is constant in each period, i.e., Var (μ i ) = б2 13
  • 14. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES This is normally referred to as homoscedasticity assumption, and if this Assumption is violated, then we face the problem of heteroscedasticity. 3. Based on assumption 1 and 2 , we can say that variable μ i has a normal distribution, i.e., μ i ~ N(0, б2) 4. Error term for one observation is independent of the error term of other observation, i.e., μ i and μ j are not correlated, or Cov (μ i and μ j ) = 0 This is no-serial-autocorrelation assumption, and if this assumption is violated, then we have autocorrelation problem. 5. μ i is independent if the explanatory variables (X), that is, the μ i and μ j are not correlated. Cov (X μ ) = ∑{[Xi - ∑ (Xi)] [ μ i -∑ (μ i)]} = 0 6. The explanatory variable (Xi) are not linearly correlated to each other; they do not affect each other. If this assumption is violated, then we face the multicolinearity problem. 7. There is no specification problem, that is, a) Model is specified correctly, mathematically, from the economic theory point of view. b) Functional form of the model ( i.e., linear or log-linear or any other form) is correct. c) Data on dependent and independent variables have correctly collected, i.e., there is no measurement error. 1.9 BLUE properties of estimator: Given the aforementioned assumptions of the classical linear regression model, the Least - Square estimator (β) possess some ideal properties. 1. It is linear. 2. It is unbiased, i.e., its average or expected value is equal to its true value. ˆ Ε( βi ) = βi 14
  • 15. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Biasness can be measured as: Bias ˆ = Ε( βi ) − βi − −−  ˆ Ε( βi ) = βi if Bias = 0 3. It is minimum- variance, i.e. it has minimum variance in the class of all such Linear unbiased estimators. 4. It is efficient. An unbiased estimator with the least variance is known as an Efficient estimator. From properly (2) and (3), our OLS estimator is unbiased and minimum variance, so it is an efficient estimator. 5. It is BLUE, i.e., Best-linear-unbiased estimator. There is a famous theorem known as “Gaus-Markov Theorem” which tells: “Given the assumptions of the classical linear regression model, the least-square Estimators, in the class of unbiased linear estimators, have minimum variance, So they are best-linear unbiased estimators, BLUE”. Assignment 1 (Due in the next class) You have already received Gujarati’s (2007) ‘Basic Econometric’; study its relevant section to solve the following assignment. . 1. Study sections 1.4 & 1.5: How does regression differ from correlation? 2. Read section 1.6: What are some other names used for dependent and independent variables? 3. Study section 1.7: What are different types of data? Explain each type in one or two sentences. 4. Study example 6.1 (page 168-169): Which of the two estimated model (6.1.12 & 6.1.13) is better and why? What do you learn from this example, in general. 15
  • 16. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Topic 2 Simple Regression to Multiple Regression Analysis 2.1 Shortcomings of two-variable regression analysis In spite of providing the base for general regression, the simple or two-variable regression has certain limitations; it gives biased results (of Least-Square Estimators, βs) if specified model excludes some relevant explanatory variables (namely X2, X3, …..). Let’s revisit to our first topic’s example of “Families’ Consumption’, wherein model was specified and run, as follows. Y = β0 + β1X + e = 24.4530 + 0.5091 X (6.4140) (0.0357) (Standard Error) (3.8124) (14.2445) (t-statistic) (0.005) (0.000) (p-value/sig. level) R= 0.981 R2 = 0.9621 R2adjusted = 0.957 F = 203.082 (p-value = 0.000) DW = 2.6809 N = 10 (2.1) If we recall, the results of this estimated model, while we evaluated in terms of economic theory (sign of the coefficient carrying with X) and statistical theory criteria (t-statistic/p-value, F- statistic and R2), were turned out to be reasonably acceptable. But, while we reconsider the specification of the model, we will find that we had misspecified the model at the first place; according to the theory, consumption (Y) depends on income (X1), as well as, wealth of the families (X2), prices of consumption items (X3), prices of the related products/substitutes/complements (X4), and so on. Hence, in spite of the fact that results provided in (2.1) are apparently seem reasonable in light of the diagnostic statistic used, the estimated model provides biased results as it does not include some very important and relevant explanatory variables. Solution then lies in the Multiple regression analysis, wherein all relevant explanatory variables need to be included, like the following one. Y = β0 + β1X1 + β2X2 + β3X3 + …………. + βNXN + e (2.2) Let’s take a practical example of using multiple regression analysis (see next sub-section 2.2). 16
  • 17. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES 2.2 An example of multiple regression analysis In case, research topic is: “Organizational justice and employees’ job satisfaction: a case of Pakistani organizations” Knowing that ‘organizational justice’ has 4 well identified facets, namely: 1. Distributive justice (JS) 2. Procedural justice (PS) 3. Interactive justice (IJ), and 4. Informational justice (INJ) Assuming that, if organizational justice prevails in Pakistani organizations, then employees would be satisfied (job satisfaction, JS); hence, respective econometric model may be specified, as follows. JS = f(DJ, PJ, IJ, INJ) (2.3) We may estimate this model in linear and/or log-linear form, that is: JS = α0 + α1DJ + α2PJ + α3IJ + α 4INJ + ei (Linear model) (2.4) lnJB = β0 + β1lnDJ + β2lnPJ + β3lnIJ + β4lnINJ + μi (Log-linear model) (2.5) (Note: ‘ln’ stands for natural log) Steps (to be taken): For estimation of linear model 1. As per requirements of the model specified in (2.3), we need to develop a questionnaire, like the one placed at Annex – I; and then collect the required data. 2. Enter the data collected on the employees’ responses in SPSS, using data editor (spreadsheet like that of EXCEL-spreadsheet). Check how data has been entered in file named: CLASS-EXERCISE-DATA_1. 3. Estimate reliability test (Chronbach’s Alpha) of the raw-data on employees’ responses, separately for each of the constructs used (JS, DJ, PJ, IJ & INJ). 4. Try to understand what reliability, validity and generalizability concepts stand for (see Annex – II). Interpret the results of reliability test (See ANNEX – III) 5. Generate data on variables of interest, namely: JS, DJ, PJ, IJ & INJ. 6. Run regression model specified in (2.4), and report the results. JS = 2.371 + 0.098DJ - 0.021PJ + 0.076IJ + 0.292INJ - 0.005AEE (9.882) (2.199) (-0.509) (1.905) (4.472) (-1.636) (0.000) (0.029) (0.611) (0.058) (0.000) (0.103) 17
  • 18. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES R= 0.506 R2 = 0.2560 R2adjusted = 0.2410 F = 17.71 (p-value = 0.000) DW = 1.5930 N = 264 (2.6) (Figures in the first and second parentheses, respectively, are t-statistics and p-values) Note: AEE stands for the combined figures of age, education and experience of the employees, and have been included to capture the combined effects of these variables. For estimation of log-linear model 7. Convert newly generated data on JS, DJ, PJ, IJ & INJ and AEE into their logs 8. Run model 2.5, and report the results lnJS = 0.943 + 0.156lnDJ - 0.015lnPJ + 0.080lnIJ + 0.308lnINJ - 0.084lnAEE (4.594) (2.829) (-0.308) (1.554) (4.506) (-1.645) (0.000) (0.005) (0.758) (0.122) (0.000) (0.101) R= 0.522 R2 = 0.2720 R2adjusted = 0.2580 F = 19.309 (p-value = 0.000) DW = 1.618 N = 264 (2.7) Evaluation and interpretation of the estimated models Linear model 2.6 (a) Model is found statistically significant (F = 17.71, p < 0.01); though all the explanatory variables included in the model seem to have explained around 25 percent variance in the dependent variable (R2 = 0.2560; R2adjusted = 0.2410). (b) Variable PJ appears to be highly statistically insignificant (p = 0.611), compared to variables INJ and DJ with highly statistically significant contribution (p < 0.01 & p < 0.05 ) and variable IJ and AEE with moderately statistically significant contribution (p = 0.058 & p = 0.103). (c) Results suggest that variables INJ, DJ and IJ positively contribute towards determination of employees’ job satisfaction, AEE negatively contributes while PJ does not contribute. The negative relationship of AEE with JB suggests that employees of higher age, with relatively higher education and experience, are less satisfied from their jobs. 18
  • 19. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Log-linear model 2.7 (a) Since the two formulations of the data (nominal-data and log-data), used in linear and log-linear models, differ from each other, we cannot compare results of one model with that of the other. However, we expect relatively better results from a log-linear model; so we can discuss whether or not the results have been improved. Yes, results are relatively improved, especially in terms of F-statistic and t-statistic/p-values. Model is found statistically significant (F = 19.309, p < 0.01); the explanatory variables explain around 27 percent variance in the dependent variable (R 2 = 0.2720; R2adjusted = 0.2580). (b) Log-linear model reinforces the results regarding signs and significance values of the individual explanatory variables. (c) Results (of the both models) suggest that facets like informational justice, distributive justice and informational justice appear to be positively contributing towards employees job satisfaction, as compared to the procedural justice, which needs to be taken care of for an overall satisfaction of Pakistani organizational employees. In addition, the senior, more educated and more experienced employees also need attention as they appear to be mostly dissatisfied from their jobs. 19
  • 20. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Assignment 2 (Due in the Next Class) 1. Briefly explain (in bullet-points) what the major contribution is that of simple/two- variables regression model, and why we have to resort to multiple regression analysis. 2. Go through the steps suggested for estimation of a linear-regression model; what is the difference between a linear and log-linear model? (a) How do the steps of estimation of a log-linear model differ from that of linear model? (b) How do the interpretations of the two model differ? 3. What is reliability? How is reliability test run in SPSS? Why is the running of reliability test important? 4. What is the procedure of generating data on variables of interest? How is a Likert-scale questionnaire used for generation of data on variables of interest? 5. How are and for what purposes, F-statistic, R2 and t-statistic/p-values used for the evaluation and interpretation of estimated models? 6. Study material (entitled “Formulating and clarifying a research topic”) provided in Annex – IV: (a) In Part – I (of Annex – IV), the answers of the following two questions have been provided: 1. What are three major attributes of a good research topic? 2. How can we turn research ideas into research projects? (b) In Part – II, you have been provided two lengthy lists of research topics proposed by my MS ARM’s class students of section 2 & 3. You please select one topic of your choice (select topic in light of what you have learnt from materials provided in Part – I), develop 2 – 3 research questions and 4 – 5 research objectives, and submit me through email (anwar@jinnah.edu.pk & chishti_anwar@yahoo.com), latest by 12.00 (Noon) Monday; please note: we will discuss your selected topic along with research questions and objectives in Monday’s evening class (along with the remaining/leftover part of previous Lecture – 2). Please also note: you may suggest a topic of your own (not already enlisted), along with research questions and objectives. Whether you select a topic from our list or suggest the one from your own side, two students of my ARM class will assist you to carry out research on that topic, as part of your AQT class requirements, for a 20% marks. 20
  • 21. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES ANNEX – I (Questionnaire) Section I Your Organization (Tick 1 or zero): Government = 1 2. Private = 0 Your gender (Tick 1 or zero): Male = 1 2. Female = 0 Your age (in years like 25 years, 29 years,) Your education (actual total years of schooling, like 14 years; 18 years) Your area of specialization: Your job title in this organization: Experience: Working years in this organization: Section II Strongly disagree – 1 Disagree = 2 Not disagree/neither agreed = 3 Agreed = 4 Strongly agreed = 5 JS: Job satisfaction (Agho et al. 1993; Aryee, Fields & Luk (1999)) 1 2 3 4 5 1 I am often bored with my job (R) 2 I am fairly well satisfied with my present job 3 I am satisfied with my job for the time being 4 Most of the day, I am enthusiastic about my job 5 I like my job better than the average worker does 6 I find real enjoyment in my work Organizational Justice (Niehoff and Moorman (1993)) Strongly disagreed = 1 Slightly disagree = 2 Disagree = 3 Neutral (Not disagree/neither agreed) = 4 Agreed = 5 Slightly more agreed = 6 Strongly agreed = 7 Distributive justice items (DJ) 1 2 3 4 5 6 7 1 My work schedule is fair 2 I think that my level of pay is fair 3 I consider my workload to be quite fair 4 Overall, the rewards I receive here are quite fair 5 I feel that my job responsibilities are fair Procedural justice items (PJ) 1 2 3 4 5 6 7 1 Job decisions are made by my supervisor in an unbiased manner 2 My supervisor makes sure that all employee concerns are heard before job decisions are made 3 To make formal job decisions, supervisor collects accurate & complete information 4 My supervisor clarifies decisions and provides additional information when requested by employees 5 All job decisions are applied consistently across all affected employees 6 Employees are allowed to challenge or appeal job decisions made by the supervisor Interactive justice items (IJ) 1 When decisions are made about my job, the supervisor treats me with kindness and consideration 2 When decisions are made about my job, the supervisor treats me with respect & dignity 3 When decisions are made about my job, supervisor is sensitive to my own needs 4 When decisions are made about my job, the supervisor deals with me in truthful manner 21
  • 22. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES 5 When decisions are made about my job, the supervisor shows concern for my rights as an employee 6 Concerning decisions about my job, the supervisor discusses the implications of the decisions with me 7 My supervisor offers adequate justification for decisions made about my job 8 When decisions are made about my job, the supervisor offers explanations that make sense to me 9 My supervisor explains very clearly any decision made about my job Strongly disagree – 1 Disagree = 2 Not disagree/neither agreed = 3 Agreed = 4 Strongly agreed = 5 Informational justice items (INJ) 1 2 3 4 5 1 Your supervisor has been open in his/her communications with you 2 Your supervisor has explained the procedures thoroughly 3 Your supervisor explanations regarding the procedures are reasonable 4 Your supervisor has communicated details in a timely manner 5 Your supervisor has seemed to tailor (his/her) communications to individuals’ specific needs. 22
  • 23. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES ANNEX - II Credibility of research findings: important considerations (Reliability? Validity? Generalizability?) Reliability: Reliability can be assessed by posing three questions: 1. Will the measure yield the same results on other occasions? 2. Will similar observations be reached by other observers? 3. Is the measure/instrument stable and consistent across time and space in yielding findings? 4-Threats to reliability (i) Subject/participant error (ii) Subject/participant bias (iii) Observer error and (iv) Observer’s bias Validity: Whether the findings are really about what they appear to be about. Validity depends upon: History (same history or not), Testing (if respondents know they are being tested), Mortality (participants’ dropping out), Maturation (tiring up), and Ambiguity (about causal direction). Generalizability: The extent to which research results are generalizable. 23
  • 24. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES ANNEX – III Reliability test and interpretation Reliability test results Responses on the elements of all five constructs (JS, DJ, PJ, Ij & INJ) were entered on SPSS’s data editor and reliability tests were conducted; the following Cronbach’s Alphas were estimated. Table 4.4 Results of reliability test Construct Cronbach’s Alpha Job Satisfaction (JS) 0.739 Distributive Justice (DJ) 0.828 Procedural Justice (PJ) 0.890 Interactional Justice (IJ) 0.920 Informational Justice (INJ) 0.834 Interpretation According to Uma Sekaran (2003), the closer the reliability coefficient Cronbach’s Alpha gets to 1.0, the better is the reliability. In general, reliability less than 0.60 is considered to be poor, that in the 0.70 range, acceptable, and that over 0.80 and 0.90 are good and very good. The reliability tests of our constructs happened to be in the acceptable to good and very good ranges. 24
  • 25. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES ANNEX - IV Formulating and clarifying a research topic1 Part – I: Two major questions: 3. What are three major attributes of a good research topic? 4. How we can turn research ideas into research projects Three major attributes of a good research topic are • Is it feasible? • Is it worthwhile? • Is it relevant? Capability: is it feasible? » Are you fascinated by the topic? » Do you have the necessary research skills? » Can you complete the project in the time available? » Will the research still be current when you finish? » Do you have sufficient financial and other resources? » Will you be able to gain access to data? Appropriateness: is it worthwhile? » Will the examining institute's standards be met? » Does the topic contain issues with clear links to theory? » Are the research questions and objectives clearly stated? » Will the proposed research provide fresh insights into the topic? » Are the findings likely to be symmetrical? » Does the research topic match your career goals? Relevancy: is it relevant? » Does the topic relate clearly to an idea you were given - possibly by your organisation? Turning research ideas into research projects • Conceive some research idea • Think about research topic (having attributes stated above) • Write research questions • Develop research objectives 1 This discussion is based on materials contained in chapter 2 of Saunders, M., Lewis, P. and Thornhill, A. (2011) Research Methods for Business Students 5th Edition. Pearson Education 25
  • 26. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Part – II: Research topics proposed by MS-ARM students ARM (section – 2) Performance appraisal as a tool to motivate employees: a comparison of public-private sector organization Performance appraisal in ……………….. (name of organization) Marketing communication and brand loyalty Implementation of Integrated Management System (IMS) in Pakistan Civil Aviation Authority Information technology and financial services Capital structure and firms profitability Interest rates, imports, exports and GDP Intra-Group Conflict and Group Performance HR practices across public and private organizations HR practices across SMEs and large companies HR practices across manufacturing and services sector companies Corporate governance practices in banking sector of Pakistan Corporate governance practices in textile industry Corporate governance practices in pharmaceutical industry Effects of working capital management on profitability Working capital with relationship to size of firm Working capital and capital structure Optimizing working capital Dividend policy and stock prices Sales, debt-to-equity ratio and cash flows Relationship between KSE’s, LSE’s and ISE’s stock prices Gold prices and stock exchange indices Interest rates, bank deposits and private investments Security Market Line (SML) & Capital Market Line (CML) at KSE Relationship between stock market returns and rate of inflation 26
  • 27. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Relationship between CPI and Bond price Pakistan’s exchange rates with relation to major global currency regimes: an analysis ARM (section – 3) Trade deficit, budget deficit and national income Performance appraisal and its outcomes Impact of compensation on employee’s job satisfaction Human resource management & outsourcing Advertising and brand image Performance management in public sector organizations Impact of training on employees’ motivation and retention Impact of performance appraisal Financial returns, returns on shares, equity returns and share prices Factors contributing towards employee turnover intention Antecedents of employees’ retention Employees’ retention policies and employees’ turnover Impact of training and development on employees’ motivation and turnover intention Outsourcing human resource function in Pakistani organizations Exploring the impact of human resources management on employees’ performance Service orientation, job satisfaction and intention to quit Brand equity and customer loyalty: a case of …….. (name of orhanization) PTCL privatization: effects on employees’ morale PTCL privatization: effects on employees’ efficiency PTCL privatization: effects in terms of profitability Electronic and traditional banking: how do customers’ perceive? FPI and FDI in Pakistan: a comparative analysis Stock market indices: KSE, LSE and ISE compared Work family conflict and employee job satisfaction: moderating role of supervisor’s support 27
  • 28. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Topic 3 Multiple regression: model specification 3.1(a) Conceiving research ideas and converting it into research projects: a Procedure Procedure: Research ideas à research topic à research questions à research Objectives à research hypotheses Your Take-home Assignment 2’s question 6 has set the example how research ideas and topics are converted in to research projects, adopting the procedure detailed above. Students have also provided details of their chosen topics; let’s discuss those topics and clarify them further, judging them in light of the relevant theories (section 3.1b). 3.1(b) Incorporating theory as the base of your research Econometrics theory Please study section 7.2 and 7.3 of Andren (2007)2 and try to understand what difference it creates when we omit a relevant explanatory variable or include an irrelevant one in an econometrics model. Economics/management theory Let us evaluate whether the research projects you have proposed are based on the relevant economic/management theory, and if not, then how you can incorporate the relevant theory into your projects. Discussion on your proposed research projects (You need to take notes on suggestions for improvements, and submit the improved version of your research project as part of your next assignment 3 (a). (See Annexure – I for topics for discussion Assignment 3 (a) 1. You must have taken the notes on suggestions made during our class discussion on your respective research projects; you please refine your topics and research questions and objectives, in light of the discussions as well as what the following research articles suggest 2 Andren, Thomas. (2007). Econometrics. Bookboon.com, pp.74-77 28
  • 29. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES regarding basing your research on relevant theory (soft copies of papers are provided on AQT-Class Yahoo Group). Article/Note: ‘Formulating a Research Question’ Rogelberg, Adelman & Askay (2009). Crafting a Successful Manuscript: Lessons from 131 Reviews. J Bus Psychol (2009) 24:117–121 (Study only 8-points given under heading ‘Conceptual and/or theoretical rationale’.) Thomas, Cuervo-Cazurra & Brannen (2009). From the Editors: Explaining theoretical relationships in international business research: Focusing on the arrows, NOT the boxes. Journal of International Business Studies (2011) 42, 1073–1078 (Read only ‘Abstract’ and ‘Introduction’ sections, and try to understand Figure 1 (Typical conceptual diagram). Andren, Thomas. (2007). Econometrics. Bookboon.com (Read only sections 72 & 73, pp.74-77) 29
  • 30. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Topic 3 Multiple regression: model specification….continues In sub-section 3.1(a), we carried out an exercise on how a conceived research idea can be converted in to a research projects (Research ideas à research topic à research questions à research objectives). In sub-section 3.1(b), we tried to learn how much important the econometrics (omission and inclusion of relevant and irrelevant explanatory variables) and economics/management theories are for specification of an econometrics model. In this new subsection 3.2, we will try to learn what role different mathematical formulations can play in econometrics modeling 3.2 Specifying an Econometric Model: Mathematical Specification This section further consists of two subsections, namely: 3.2(a) Specification of an econometric model: mathematical formulation in general 3.2(b) Some practical examples of mathematical formulations/specifications: production function, cost-function and revenue function 3.2(a) Specification of an econometric model: mathematical formulation in general Our discussion in earlier sections on simple regression and multiple regression analysis clarifies two major points, namely: 1. The simple and multiple regression analysis assumes that variable Y depends on variable X, but for this phenomenon of dependence or causation, the researcher takes insights from the basic theory (economics/management). 2. Previous discussion further emphasizes that it is the researcher’s responsibility to specify an econometric model such that it contains all major relevant explanatory variables as independent variables; otherwise, empirical results obtained in terms of estimated coefficients would be biased. While specifying a model, the researcher has to take the above points in to consideration. Additionally, the researcher has to decide which mathematical formulation of the model he/she should use so that the true relationship between dependent and independent variables is captured to the maximum extent. This is how an econometric model is/should be specified. 30
  • 31. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Let’s proceed further, taking some practical examples of mathematical formulations of the model. In case, we have the following type of relationship between Y – X variables: Y Y Y X X X Case 1 (a) Case 1 (b) Case 1 (c) Case 1a is a general linear relationship, and can be measured, as follows. Y = β0 + β1X1 + e (3.1) In 3.1, we expect β1to carry positive sign. The case 1(b) represents an exponential case, and can be measured, as follows: 2 Y = β0 + β1X1 + β2X 1 + e (3.2) Specially, the parameters β1and β2 will carry positive signs. In case of a cubic-type of relationship like 1(c), the following mathematical formulation will have to be adopted. 2 3 Y = β0 + β1X1 + β2X 1 + β3X 1 + e (3.3) The coefficients β1and β2 will carry positive but β3 negative sign. In other words, it means that if we have to measure the stated type of relationships between our Y – X variables, we need to use the relevant type of mathematical formulations while specifying our econometrics model. In certain other cases/on certain occasions, we have to adopt some other mathematical formulations like the following ones: Y = β0 + β1X1 + β2X1X2 + β3X2 + e (3.4) 2 2 Y = β0 + β1X1 + β2X 1 + β3X1X2 + β4X2 + β5X 2 + e (3.5) 31
  • 32. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Equation 3.4 measures linear relationship, but includes an interaction term (X1X2). β2 can take any sign (+, - or 0); a positive sign would show positive effect of the interaction of X 1 and X2 on Y, a negative sign would mean negative effect of interaction of these two variables and zero effect would mean zero effect on dependent variable Y. Let’s visit some practical examples where we can use some of the above stated mathematical formulations (next section). 3.2(b) Some practical examples: production, cost and revenue functions Production function In case, we have data on production of product Y, wherein two major inputs used are X 1 and X2: Y X1 X2 2500 1 150 2525 2 152 2555 3 155 2592 4 159 2635 5 161 2677 6 169 2718 7 174 2745 8 178 2766 9 181 2781 10 182 Let’s check relationship between Y – X1, and Y – X2 (separately), using mathematical formulation given in (3.3), using data provided in above table. Do this as Take-home Assignment 3b (Question 1); show the estimated relationship through hand-drawn graph Let’s check relationship between Y and X1 & X2, using mathematical formulation given in (3.4), using data provided in the above table. Do this as Take-home Assignment 3b (Question 2); interpret the results, including that of the interaction term 32
  • 33. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Cost Function Cost function can be developed when you have data like the following one: Y TC 1 193 2 226 3 240 4 244 5 257 6 260 7 274 8 297 9 350 10 420 Mathematical formulation of a typical cost-function is: 2 3 TC = β0 + β1Y - β1Y + β1Y + e (3.6) Did you notice the signs of a typical cost-function are opposite to that of a typical production-function (given in 3.3). Estimate cost-function 3.6 as Take-home Assignment 3b (Question 3); show the estimated relationship through hand-drawn graph Assignment 3b: Question 4 Download 8 – 10 published research articles on the area of research/topic you have chosen for your class research project, study the conceptual models tried in these research articles, and develop your own model, including the mathematical one as part of your Take-home Assignment 3(b), due in next class; be ready for a class presentation also. 33
  • 34. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Topic 3 Multiple regression: model specification….continues 3.3 Conceptual/econometric modeling 3.3 (a) Examples in Finance 3.3 (b) Examples in Marketing 3.3 (c) Examples in HRM 3.3 (a) Examples in Finance: summary Example 1: Interest rates and GDP: a case of Pakistan Example 2: Capturing effects of interest rates on Pakistani economy Example 3: Exchange rates and Pakistan’s trade: an analysis Example 4: Exchange rates and Pakistan’s economy: an analysis Example 5: Research on Working Capital (WC) Proposal 1: “Relationship between Profitability and Working Capital Management”, using econometric technique Proposal 2: “Liquidity-profitability trade-off”, using Goal programming (of Operations Research) 34
  • 35. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES 3.3 (a) Examples in Finance Example 1: Interest rates and GDP: a case of Pakistan3 Though we are interested in analyzing the effect of interest rates on Pakistan’s national income, but we know that interest rates do not affect GDP directly, rather these affect saving (bank deposits) and private investments, and as a consequence GDP is affected; so we conceptualize the path of the effect, as follows: Interest rates (↑↓) à bank deposits (↑↓) & private investments (↓↑) à GDP (↓↑) The above path of the effect (of interest rates) can be captured, through econometrics model, postulated, as follows. Private investment = ƒ(Interest rates) (3.7a) GDP = ƒ(Private investments_predicted in equation 7a) (3.7b) Theory tells us that private investment (PI) is influenced not only by the interest rate (R) but is also affected by openness of the economy (OE) and, especially the costs and taxes (C&T). Hence, equation 3.7a would change to: PI = ƒ(R, OE, C&T) (3.8a) ̂ The private investment predicted on the basis of equation 3.8a (PI) is not the only determinant of GDP, government expenditure (GE) or budget spending is another determining variable; while in Pakistani context, Foreign Direct Investment (FDI) and Pakistan’s productive population, that is, the active labor force (LF) are two other factors should be considered as determinants of Pakistan’s national income (GDP). Hence, model 3.7b would change, as follows. ̂ GDP = ƒ(PI, GE, FDI, LF) (3.8b) The model postulated in 3.8 (a – b) still needs improvement; government expenditure (GE) and FDI are not autonomous in nature, the former depends on government revenues (GR) and government borrowing from foreign (FB) and domestic (DB) sources, and the latter depends 3 Students are urged to think over the difference between topic of this Example 1 and that of Example 2, and then try to understand how conceptual/econometric modeling can be differently developed to take care of the differences which the two topics necessitate. 35
  • 36. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES upon economy’s openness (OE) and cost of production and taxes (C&P). To incorporate these effects, the model would therefore adopt the following form. PI = ƒ(R, OE, C&P) (3.9a) GE = ƒ(GR, FB, DB) (3.9b) FDI = ƒ(OE, C&P) (3.9c) ̂ ̂ ̂ GDP = ƒ(PI, GE, FDI, LF) (3.9d) Model 3.9 (a – d) represents what we need to do for a piece of research conducted under title “Interest rates and GDP: a case of Pakistan”. In case we extend the scope of our research to what is needed under title “Capturing effects of interest rates on Pakistani economy”, we will then have to adopt the model specified in the following Example 2. Example 2: Capturing effects of interest rates on Pakistani economy Notice the difference between the two topics (Example 1 and 2); the first topic requires analyzing the effect of exchange rates on GDP, while the second topic asks for looking in to the same thing from a little broader perspective, that is, from the point of view of whole economy. Since the model specified for the first topic covers largely the methodology needed for the second topic, we can use the same first example model 3.9 (a – d), with an additional equation for analyzing the effect of interest rates on bank deposits, which can be assumed to be determined by money supply in the country (M), in addition to the interest rates (R). Bank deposit = ƒ(R, M) (3.9e) Hence, model 3.9 (a – e) will be used for the piece of research identified in example 2. Example 3: Exchange rates and Pakistan’s trade: an analysis4 According to the theory, the appreciation or depreciation of exchange rates (ER) affects the country’s trade; appreciation of a country’s currency makes exports expensive and imports cheap, and depreciation makes exports cheap and imports expensive. This stated phenomenon is true for the two trade partners, but is also affected by certain other situations prevailing in the two trading countries. The foreign country’s exchange rates with respect to her other major trade 4 Students are urged to think over the difference between topic of this Example 3 and that of Example 4, and then try to understand how conceptual/econometric modeling can be differently developed to take in to account the differences which the two topics necessitate. 36
  • 37. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES partners, availability and prices of the substitutes in foreign country and world over, consumers’ income, trade openness and political situations are some other important factors affecting export and import trade. Tracing and finding out the effects of the determinants of export and import trade might be easy when trade of certain known commodities between two specific countries is analyzed; but the case becomes cumbersome, and needs extra care when analysis of trade is required at aggregate level, for instance the topic of this piece of research - Exchange rates and Pakistan’s trade: an analysis. We can think primarily about some very simple questions like what the exchange rates are (definition), how these are determined (or are autonomous in nature), they affect what and how, and specifically what relationship they have with trade – its two components, imports and exports. And since we are analyzing the exchange rates of Pakistan and her trade, we should think over the answers of such questions in the context of Pakistan’s economy. Exchange rates (ER) are not autonomous in nature, these are determined by the forces of demand for and supply of major medium of currency (US dollar in Pakistan) used in imports and exports trade. Value of imports seems to be the major factor to determine demand for US dollar in Pakistan, and while value of exports, workers’ remittances (WR), foreign direct investment (FDI) and foreign borrowings (FB) appear to be the major determinants of supply of dollar. Hence, these demand and supply factors determine exchange rates in Pakistan, which in turn affect volumes of import and export. ER = ƒ(IM, EX, WR, FDI, FB) (3.10) ̂ IM = ƒ(ER) (3.11) ̂ EX = ƒ(ER) (3.12) But ER̂ is not the only determinant of import (IM). Imports in Pakistan have historically been largely composed of capital goods (28% in 1980-81 and 24% in 2010-11) and industrial raw materials (58% in 1980-81 and 60% in 2010-11)5; the value of the share of Pakistan GDP’s manufacturing sector (GDPM) may therefore be included in equation 3.11 as proxy to represent the demand for imports, in addition to the population or its growth rate (POP) as proxy for the size of the market. Hence, equation 3.11 adopts new form, namely: ̂ IM = ƒ(ER, GDPM, POP) (3.13) 5 Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Statistical Appendix Table 8.5B 37
  • 38. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES In case of exports, primary commodities and semi-manufactured and manufactured products have been the major components, with share of 44% in 1980-81 and 18% in 2010-11, 11% in 1980-81 and 13% in 2010-11 and 45% in 1980-81 and 69% in 2010-11, respectively 6. The values of the primary (GDPP) and secondary/manufacturing sectors’ contributions to GDP (GDPM) may therefore be included in equation 3.12 as proxies to represent major supplying sectors of exports. The demand for Pakistani exports has come from both developed (60.8% in 1990-91 and 44.5% in 2010-11) and developing (39.2% in 190-91 and 55.5% in 2010-11) countries 7, the world’s GDP can be taken as proxy to represent demand from the whole world (GDPW). Hence, equation 3.12 adopts the new form, namely: ̂ EX = ƒ(ER, GDPP, GDPM, GDPW) (3.14) Summarizing the model, ER = ƒ(IM, EX, WR, FDI, FB) (3.15a) ̂ IM = ƒ(ER, GDPM, POP) (3.15b) ̂ EX = ƒ(ER, GDPP, GDPM, GDPW) (3.15c) We can add even some other relevant variables and improve the model (model 3.15), and reviewing the relevant literature on respective topics and sub-topics, with special reference to Pakistan, would help us in this regards. Please note that model 15 (a – c) will restrict research to the analysis of the effects of exchange rates on Pakistan’s trade; in case, if someone is interested to analyze the exchange rates’ effects on Pakistan economy (or GDP), then model specified in following Example 4 should be used. Example 4: Exchange rates and Pakistan’s economy: an analysis Model specified in 3.15 (a – c) will work as the base to analyze the effect of exchange rates on import and export trade, and incorporation of an additional equation (3.15d), which transfers the ̂ ̂ effects of imports (IM) and exports (EX) to GDP will help complete a model for the analysis necessary for new topic. ̂ ̂ GDP = ƒ (IM, EX, POP) (3.15d) The effect of the size of population (POP) has been included as a proxy for the effect of domestic consumption on country’s GDP. 6 Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Table 8.5A 7 Government of Pakistan (2012). Pakistan Economic Survey 2011-12. Table 8.7 38
  • 39. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Example 5: Research on Working Capital (WC) Working capital: in general Working capital is defined as8: Working Capital (WC) = current assets (CA) - current liabilities (CL) (3.16a) Where Current assets are cash and other assets that can be converted to cash within a year, and Current liabilities are obligations that the company plans to pay off within the year. Working capital indicates the assets the company has at its disposal for current expenses. The process of managing the WC efficiently is called Working capital Management. An excess of working capital many mean that the company is not managing its assets efficiently. It's not using its assets to get a bigger return or better profit. An aggressive company may keep its working capital smaller. But a very low working capital may mean the company may not be suited well enough to payoff its short term obligations. This decision of how to manage the working capital of the company depends on the Working capital policy of the company. An important factor that determines the policy is the industry in which the company operates. For Example, an IT service company may not have a lot of shot-debt in terms of inventory but it still needs to pay wages, insurances and other expenses like rent. The company needs to have a policy that makes sure it sets targets were it gets paid as the project progresses so it can keep paying its staff in time. The company has to manage its account receivables according to this policy. Some industries operate in a high profit margin that they can afford to have a longer term on the account receivables because the higher cash balance part of the current assets. The Collection Ratio helps project this aspect of a company; The collection ratio is defined as: Collection Ratio = Accounts Receivable / (Revenue/ 365) 3.16b) Collection ratio tells us the average number of days it takes a company to collect unpaid invoices. A ratio which is very near to 30 days is very good since it means that the company is getting paid on a monthly basis. Sales is another attribute that strongly impacts working capital. It is the ability of a company to sell its products fast enough to get the money back to put back into operations or supplies for producing more materials. Moving inventory fast is always a good plan for a company. It also helps in reducing costs associated with holding and moving inventory. A good ratio that helps put the attribute in perspective is inventory turnover ratio, which is defined as: Inventory turnover ratio = sales / inventory Or Inventory turnover ratio = Cost of goods sold / inventory (3.16c) 8 The following material is based on http://www.business.com/finance/working-capital/; downloaded on October 12, 2012. 39
  • 40. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES This ratio shows the efficiency the company has in selling its products. The higher the ratio the better the company is able to move the products. Again this could be dictated by the industry, for example, a daily products company is usually forced to sell its products fast enough or lose it. The ratio also provides a good insight into how a company is doing within an industry. The direct ratio of companies can be compared to see how well the company is able to sell the products in comparison to its competitors. Financing is another attribute of Working Capital management. Debt - Asset ratio provides a good insight into how much of the company's assets are being financed though debt. The debt – asset ratio is defuned as: Debt-asset ratio = Total liabilities / Total assets (3.16d) Working capital management becomes a very important aspect for a company since it is the first line of defense against market downturn cycles and recession. A company with cash is usually in a good position to make better use of the opportunities the markets provide. Its can spend the money on R&D for coming up with better products. Increase in current assets, especially, increase in account receivables due to growth is sales have to be managed efficiently. Ability to control working capital plays a significant role in the survival of the company. Research on Working Capital Let us see how the above information on working capital (WC) and working capital management (WCM) has been used by different researchers to carry out research on the topic under study. Lazaridis and Tryfonidis’s (2006)9 and Gill, Biger and Mathur (2010)10 analyzed the relationship between profitability and working capital management, using about the same model, and measuring and generating the dependent and independent variables in the following way: No. of Days A/R = (Accounts Receivables/Sales) x 365 No. of Days A/P = (Accounts Payables/Cost of Goods Sold) x 365 No. of Days Inventory = (Inventory/Cost of Goods Sold) x 365 Cash Conversion Cycle = (No. of Days A/R + No. of Days Inventory) – No. of Days A/P Firm Size = Natural Logarithm of Sales Financial Debt Ratio = (Short-Term Loans + Long-Term Loans)/Total Assets Fixed Financial Asset Ratio = Fixed Financial Assets/Total assets Profit = (Sales - Cost of Goods Sold) / (Total Assets - Financial Assets) 9 Lazaridis I, and Tryfonidis D, (2006). Relationship between working capital management and profitability of listed companies in the Athens stock exchange. Journal of Financial Management and Analysis, 19: 26-25. 10 Gill, A., Biger, N. and Mathur, N. (2010). The Relationship Between Working Capital Management And Profitability: Evidence From The United States. Business and Economics Journal, Volume 2010: BEJ-10 40
  • 41. LECTURES & ADVANCED QUANTITATIVE TECHNIQUES NOTES Raheman A. and Nasr, M. (2007)11 used similar methodology but measured the required variables in somewhat different way, namely: NOPit = β0 + β1(ACPit) + β2 (ITIDit) + β3 (APPit) + β4(CCCit) + β5(CRit) + β6(DRit) + β7(LOSit) + β8(FATAit) + ε (3.17) Where: NOP : Net Operating Profitability ACP : Average Collection Period ITID : Inventory Turnover in Days’ APP : Average Payment Period CCC : Cash Conversion Cycle CR : Current Ratio DR : Debt Ratio LOS : Natural logarithm of Sales FATA: Financial Assets to Total Assets ε : The error term. Researchers have estimated/generated variables, using the following definitions. Net Operating Profitability (NOP) which is a measure of Profitability of the firm is used as dependant variable. It is defined as Operating Income plus depreciation, and divided by total assets minus financial assets. Average Collection Period (ACP) used as proxy for the Collection Policy is an independent variable. It is calculated by dividing account receivable by sales and multiplying the result by 365 (number of days in a year). Inventory turnover in days (ITID) used as proxy for the Inventory Policy is also an independent variable. It is calculated by dividing inventory by cost of goods sold and multiplying with 365 days. Average Payment Period (APP) used as proxy for the Payment Policy is also an independent variable. It is calculated by dividing accounts payable by purchases and multiplying the result by 365. 11 Raheman A. and Nasr, M. (2007). Working capital management and profitability – case of Pakistani firms. International Review of Business Research Papers, 3: 279-300. 41