1. Econometric ModelingEconometric Modeling
Research MethodsResearch Methods
Professor Lawrence W. LanProfessor Lawrence W. Lan
Email:Email: lawrencelan@mdu.edu.twlawrencelan@mdu.edu.tw
http://140.116.6.5/mdu/http://140.116.6.5/mdu/
Institute of ManagementInstitute of Management

2. OutlineOutline
• Overview
• Single-equation Regression Models
• Simultaneous-equation Regression
Models
• Time-Series Models

3. OverviewOverview
• Objectives
• Model building
• Types of models
• Criteria of a good model
• Data
• Desirable properties of estimators
• Methods of estimation
• Software packages and books

4. ObjectivesObjectives
• Empirical verification of the theories in business,
economics, management and related disciplines is
becoming increasingly quantitative.
• Econometrics, or economic measurement, is a social
science in which the tools of economic theory,
mathematical statistics are applied to the analysis of
economic phenomena.
• Focus on models that can be expressed in equation form
and relating variables quantitatively.
• Data are used to estimate the parameters of the
equations, and the theoretical relationships are tested
statistically.
• Used for policy analysis and forecasting.

5. Model BuildingModel Building
• Model building is a science and art, which
serves for policy analysis and forecasting.
– science: consists of a set of quantitative tools
used to construct and test mathematical
representations of the real world problems.
– art: consists of intuitive judgments that occur
during the modeling process. No clear-cut
rules for making these judgments.

6. Types of Models (1/4)Types of Models (1/4)
• Time-series models
– Examine the past behavior of a time series in
order to infer something about its future
behavior, without knowing about the causal
relationships that affect the variable we are
trying to forecast.
– Deterministic models (e.g. linear
extrapolation) vs. stochastic models (e.g.
ARIMA, SARIMA).

7. Types of Models (2/4)Types of Models (2/4)
• Single-equation models
– With causal relationships (based on
underlying theory) in which the variable (Y)
under study is explained by a single function
(linear or nonlinear) of a number of variables
(Xs)
– Y: explained or dependent variable
– Xs: explanatory or independent variables

8. Types of Models (3/4)Types of Models (3/4)
• Simultaneous-equation models (or multi-
equation simulation models)
– With causal relationships (based on
underlying theory) in which the dependent
variables (Ys) under study are related to each
other as well as to a set of equations (linear or
nonlinear) with a number of explanatory
variables (Xs)

9. Types of Models (4/4)Types of Models (4/4)
• Combination of time-series and regression
models
– Single-input vs. multiple-input transfer
function models
– Linear vs. rational transfer functions
– Simultaneous-equation transfer functions
– Transfer functions with interventions or
outliers

10. Criteria of a Good ModelCriteria of a Good Model
• Parsimony
• Identifiability
• Goodness of fit
• Theoretical consistency
• Predictive power

11. DataData
• Sample data: the set of observations from the
measurement of variables, which may come
from any number of sources and in a variety of
forms.
• Time-series data: describe the movement of any
variable over time.
• Cross-section data: describe the activities of any
individual or group at a given point in time.
• Pooled data: a combination of time-series and
cross-section data, also known as panel data,
longitudinal or micropanel data.

12. Desirable Properties of EstimatorsDesirable Properties of Estimators
• Unbiased: the mean or expected value of an
estimator is equal to the true value.
• Efficient (best): the variance of an estimator is
smaller than any other ones.
• Minimum mean square error (MSE): to trade off
bias and variance. MSE is equal to the square of
the bias and the variance of the estimator.
• Consistent: the probability limit of an estimator
gets close to the true value. It is a large-sample
or asymptotic property.

13. Methods of EstimationMethods of Estimation
• Ordinary least squares (OLS)
• Maximum likelihood (ML)
• Weighted least squares (WLS)
• Generalized least squares (GLS)
• Instrumental variable (IV)
• Two-stage least squares (2SLS)
• Indirect least squares (ILS)
• Three-stage least squares (3SLS)

14. Software Packages and BooksSoftware Packages and Books
• LIMDEP: single-equation and
simultaneous-equation regression models
• SCA: time series models
• Textbooks
– (1) Damodar Gujarati, Essentials of Econometrics,
2nd ed. McGraw-Hill, 1999.
– (2) Robert S. Pindyck and Daniel L. Rubinfeld,
Econometric Models and Economic Forecasts, 4th
ed. McGraw-Hill, 1997.

15. Single-equation Regression ModelsSingle-equation Regression Models
• Assumptions
• Best Linear Unbiased Estimation (BLUE)
• Hypothesis testing
• Violations for assumptions 1 ~ 5
• Forecasting

16. AssumptionsAssumptions
• A1: (i) The relationship between Y and X is truly
existent and correctly specified. (ii) Xs are
nonstochastic variables whose values are fixed.
(iii) Xs are not linearly correlated.
• A2: The error term has zero expected value for
all observations.
• A3: The error term has constant variance for all
observations
• A4: The error terms are statistically independent.
• A5: The error term is normally distributed.

17. Best Linear Unbiased EstimationBest Linear Unbiased Estimation
• Gauss-Markov (GM) Theorem: Given
assumptions 1, 2, 3, and 4, the estimation of the
regression parameters by least squares (OLS)
method are the best (most efficient) linear
unbiased estimators. (BLUE)
• GM theorem applies only to linear estimators
where the estimators can be written as a
weighted average of the individual observations
on Y.

18. Hypothesis TestingHypothesis Testing
• Normal, Chi-square, t, and F distributions
• Goodness of fit
• Testing the regression coefficients (single
equation)
• Testing the regression equation (joint
equations)
• Testing for structural stability or
transferability of regression models

19. A1(i) Violation -- Specification ErrorA1(i) Violation -- Specification Error
• Omitting irrelevant variables  biased and
inconsistent estimators
• Inclusion of irrelevant variables 
unbiased but inefficient estimators
• Incorrect functional form (nonlinearities,
structural changes)  biased and
inconsistent estimators

20. A1(ii) Violation – Xs Correlated with ErrorA1(ii) Violation – Xs Correlated with Error
• OLS leads to biased and inconsistent estimators
• Criteria of good instrumental (proxy) variables
• Instrumental-variables estimation  consistent,
but no guarantee for unbiased or unique
estimators
• Two-stage least squares (2SLS) estimation 
optimal instrumental variable, unique consistent
estimators

21. A1(iii) Violation -- MulticollinearityA1(iii) Violation -- Multicollinearity
• Perfect collinearity between any of Xs 
no solution will exist
• Near or imperfect multicollinearity  large
standard error of OLS estimators or wider
confidence intervals; high R2
but few
significant t values; wrong signs for
regression coefficients; difficulty in
explaining or assessing the individual
contribution of Xs to Y.

22. Detection of MulticollinearityDetection of Multicollinearity
• Testing the significance of R-i
2
from the various
auxiliary regressions. F=[R-i
2
/(k-1)]/[(1-R-i
2
)/(n-k)],
where n=number of observations, k=number of
explanatory variables including the intercept.
Check if F-value is significantly different from zero. If
yes (F-value > F-table), X-i and Xi are significantly
collinear with each other.
• Variance inflation factor (VIF = 1/(1-R-i
2
): VIF=1
representing no collinearity; if VIF>10 then high degree
of multicollinearity

23. A2 Violation – Measurement Error in YA2 Violation – Measurement Error in Y
• OLS will result in biased intercept;
however, the estimated slope parameters
are still unbiased and consistent.
• Correction for the dependent variable

24. A3 Violation -- HeteroscedasticityA3 Violation -- Heteroscedasticity
• It happens mostly for cross-sectional data;
sometimes for time-series data.
• OLS will lead to inefficient estimation, but still
unbiased.
• Can be corrected by weighted least squares
(WLS) method
• Detection: Goldfeld-Quandt test, Breusch-Pagan
test, White test, Park-Glejser test, Bartlett test,
Peak test, Spearman’s rank correlation test, etc.

25. A4 Violation -- AutocorrelationA4 Violation -- Autocorrelation
• It happens mostly for time-series data;
sometimes for cross-sectional data.
• OLS will lead to inefficient estimation, but still
unbiased.
• Can be corrected by generalized least squares
(GLS) method
• Detection: Durbin-Watson test, runs test. (For
lagged dependent variable, DW2 even when
serial correlation, do not use DW test, use h test
or t test instead)

26. A5 Violation – Non-normalityA5 Violation – Non-normality
• Chi-square, t, F tests are not valid;
however, these tests are still valid for large
sample.
• Detection: Shapiro-Wilk test, Anderson-
Darling test, Jarque-Bera (JB) test.
JB=(n/6)[S2
+ (K-3)2
/4] where n=sample
size, K=kurtosis, S=skewness. (For
normal, K=3, S=0) JB~ Chi-square
distribution with 2 d.f.

27. ForecastingForecasting
• Ex post vs. ex ante forecast
• Unconditional forecasting
• Conditional forecasting
• Evaluation of ex post forecast errors
– means: root-mean-square error, root-mean-square
percent error, mean error, mean percent error, mean
absolute error, mean absolute percent error, Theil’s
inequality coefficient
– variances: Akaike information criterion (AIC), Schwarz
information criterion (SIC)

28. Simultaneous-equationsSimultaneous-equations
Regression ModelsRegression Models
• Simultaneous-equation models
• Seemly unrelated equation models
• Identification problem

29. Simultaneous-equations ModelsSimultaneous-equations Models
• Endogenous variables exist on both sides of the
equations
• Structural model vs. reduced form model
• OLS will lead to biased and inconsistent
estimation; indirect least squares (ILS) method
can be used to obtain consistent estimation
• Three-stage least squares (3SLS) method will
result in consistent estimation
• 3SLS often performs better than 2SLS in terms
of estimation efficiency

30. Seemly Unrelated Equation ModelsSeemly Unrelated Equation Models
• Endogenous variables appear only on the
left hand side of equations
• OLS usually results in unbiased but
inefficient estimation
• Generalized least squares (GLS) method
is used to improve the efficiency  Zellner
method

31. Identification ProblemIdentification Problem
• Unidentified vs. identified (over identified
and exactly identified)
• Order condition
• Rank condition

32. Time-series ModelsTime-series Models
• Time-series data
• Univariate time series models
• Box-Jenkins modeling approach
• Transfer function models

33. Time-series DataTime-series Data
• Yt: A sequence of data observed at equally
spaced time interval
• Stationary vs. non-stationary time series
• Homogeneous vs. non-homogeneous time
series
• Seasonal vs. non-seasonal time series

34. Univariate Time Series ModelsUnivariate Time Series Models
• Types of models: white noise model,
autoregressive (AR) models, moving-average
(MA) models, autoregressive-moving average
(ARMA) models, integrated autoregressive-
moving average (ARIMA) models, seasonal
ARIMA models
• Model identification: MA(q)  sample
autocorrelation function (ACF) cuts off; AR(p) 
sample partial autocorrelation function (PACF)
cuts off; ARMA(p,q)  both ACF and PACF die
out

35. Box-Jenkins Modeling ApproachBox-Jenkins Modeling Approach
• Tentative model identification (p, q)  extended
sample autocorrelation function (EACF)
• Estimation (maximum likelihood estimation 
conditional or exact)
• Diagnostic checking (t, R2
, Q tests, sample ACF
of residuals, residual plots, outlier analysis)
• Application (using minimum mean squared error
forecasts)

36. Transfer Function ModelsTransfer Function Models
• Single input (X) vs. multiple input (Xs) models
• Linear transfer function (LTF) vs. rational
transfer function (RTF) models
• Model identification (variables to be used; b, s, r
for each input variable using corner table
method; ARMA model for the noise)
• Model estimation: maximum likelihood
estimation (conditional or exact)
• Diagnostic checking: cross correlation function
(CCF)
• Forecasting: simultaneous forecasting

37. Simultaneous Transfer FunctionSimultaneous Transfer Function
(STF) Models(STF) Models
• Purposes (to facilitate forecasting and
structural analysis of a system, and to
improve forecast accuracy)
• Yt and Xt can be both endogenous
variables in the system
• Use LTF method for model identification,
FIML for estimation, CCM (cross
correlation matrices) for diagnostic
checking, simultaneous forecasting

38. Transfer Function Models withTransfer Function Models with
Interventions or OutliersInterventions or Outliers
• Additive Outlier (AO)
• Level Shift (LS)
• Temporary Change (TC)
• Innovational Outlier (IO)
• Intervention models