Time series analysis of output and factors of production, Japan and US 1900-2000.

1. IIASA-TITECH Technical Meeting
18-19 Sept, Laxenburg
Benjamin Warr and Robert Ayres
Center for the Management of Environmental Resources (CMER)
INSEAD
Boulevard de Constance
Fontainebleau
77300
http://benjamin.warr.insead.edu
Time series analysis of output and factors of
production, Japan and US 1900-2000.

2. Coal fractions of fossil fuel exergy apparent consumption,
Japan 1900-2000
Electricity
100%
Heat (Steam coals for space heating and coking coal for steel production)
90%
Non-fuel (includes industrial transformation processes)
80% Other prime movers (steam locomotives)
70%
60%
percent
50%
40%
30%
20%
10%
0%
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
year

3. Petroleum products fractions of fossil fuel exergy apparent
consumption, Japan 1900-2000
Electricity (Heavy Oil)
100%
Heat (Residential and Commercial uses of Heavy Oil and LPG)
90%
Light (Kerosene)
80%
Non-fuel (Machinery Oil, Lubricants, Asphalt)
70%
Other prime movers (Gasoline, Light Oil, Heavy Oil, LPG, Jet Oil, Kerosene)
60%
percent
50%
40%
30%
20%
10%
0%
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
year

4. Technical efficiency of primary work services from exergy
sources, Japan 1900-2000
coal
45%
petroleum
40%
technical efficiency (%)
natural gas
35% nuclear, hydroelectric, thermal
fuelwood, charcoal
30%
25%
20%
15%
10%
5%
0%
19
19
19
19
19
19
19
19
19
19
00
10
20
30
40
50
60
70
80
90
year

5. Exergy to work conversion efficiencies, Japan 1900-2000
50%
High Temperature Industrial Heat
45%
Medium Temperature Industrial Heat
40% Low Temperature Space Heat
35% Electric Power Generation and Distribution
30% Other Mechanical Work
efficiency
25%
20%
15%
10%
5%
0%
1900 1920 1940 1960 1980 2000
year

6. Comparison of the technical efficiency of primary work (exergy)
services from exergy sources,
Japan and US 1900-2000
25%
technical efficiency (%)
20% Japan - f(Ub)
15%
US - f( Ub)
10%
5%
0%
19
19
19
19
19
19
19
19
19
19
00
10
20
30
40
50
60
70
80
90
year

7. LINEX fits for GDP, Japan and US
1900-2000.
8000
empirical GDP, Japan
7000
predicted GDP, Japan
GDP (thousand billion 1992$)
6000
empirical GDP, US
5000 predicted GDP, US
4000
3000
2000
1000
0
1900 1920 1940 1960 1980
year

8. Estimates of GDP, UK 1960-2000
3
Y
LINEX
2.5 Time Dependent CD
Time Average CD
2
output (1960=1)
1.5
1
0.5
0
1963 1968 1973 1978 1983 1988 1993

9. Estimates of GDP, France 1960-2000
4
Y
3.5 LINEX
Time Dependent CD
Time Average CD
3
2.5
output (1960=1)
2
1.5
1
0.5
0
1963 1968 1973 1978 1983 1988 1993

10. Elasticities of factors of production*,
US 1960-2000
GDP=Capital*alpha*Labour*beta*Work*gamma
1.0
alpha
0.9
beta
0.8 gamma
0.7
0.6
elasticity
0.5
0.4
0.3
0.2
0.1
0.0
1900 1910 1920 1930 1940
year
* derived from optimisation of the LINEX function.

11. Elasticities of factors of production*,
Japan 1960-2000
GDP=Capital*alpha*Labour*beta*Work*gamma
1.0 alpha
0.9 beta
gamma
0.8
0.7
0.6
elasticity
0.5
0.4
0.3
0.2
0.1
0.0
1960 1970 1980 1990 2000
year
* derived from optimisation of the LINEX function.

12. Some problems using econometric
time series in OLS
• Multicollinearity
• Stationarity
• Unit roots – explosive behaviour.

13. Multicollinearity
• Variables highly correlated. Usual procedure
take logs and increments or ratios.
lny lnk lnl lnu
lny 1.00 0.97 0.98 0.99
lnk 0.97 1.00 0.96 0.96
lnl 0.98 0.96 1.00 0.96
lnu 0.99 0.96 0.96 1.00
dlny dlnk dlnl dlnu
dlny 1.00 -0.0012 0.78 0.78
dlnk -0.0012 1.00 0.19 0.11
dlnl 0.78 0.19 1.00 0.80
dlnu 0.78 0.11 0.80 1.00

14. Stationarity
• Stationarity describes the situation where the data generating
stochastic process is invariant over time. If the distribution of a
variable depends on time, the sequence is non-stationary and is
said to be controlled by a trend. Being dependent upon time, the
mean, variance and autocovariance do not converge to finite values
as the number of samples increases.
• The formal definition of a stationary time series is defined by,
E ( yt ) = µ Equation 10
•
• [
E ( yt − µ ) = γ 0
2
] Equation 11
• E [( yt − µ )( yt −k − µ )] = γ k Equation 12
• for all t=1,2,…,n
• and for all k=,…,-2,-1,0,1,2,…
• Formal tests for 10 require an estimate of 11 which in turn depends
on the validity of 10. In practice this is troublesome.

15. Unit Roots
• A unit root test is a statistical test for the
proposition that in a autoregressive times
Y(t+1)=ay(t)+other terms
that a = 1.
• For values smaller than 1, the time series is
mean reverting and shocks are transitory.
• For values larger than 1 the shock is
permanent causing a change in the mean value
of value of Yt
• A process having a unit root is non-stationary

16. log(y) = α log(k)+β log(l)+γ log(u)
Japan
Estimate Std. Error t value Pr(>|t|)
lnk 0.31493 0.02146 14.677 <2e-16 ***
lnl 0.28453 0.16495 1.725 0.0877 .
lnu 0.45467 0.03473 13.091 <2e-16 ***
• Multiple R-Squared: 0.9992, Adjusted R-squared: 0.9991
USA
Estimate Std. Error t value Pr(>|t|)
lnk 0.52414 0.07439 7.045 2.59e-10 ***
lnl 0.07243 0.15769 0.459 0.647
lnu 0.77385 0.07556 10.241 < 2e-16 ***
• Multiple R-Squared: 0.9962, Adjusted R-squared: 0.9961

17. Diagnostic plots: model 1
US Japan
Residuals vs Fitted Normal Q-Q plot Residuals vs Fitted Normal Q-Q plot
0.3
0.1
Standardized residuals
Standardized residuals
2
1
0.1
0
1
Residuals
Residuals
-0.1
0
-0.1
-2
-1
50
52 50
52
-0.3
-2
-0.3
-4
34
33
22 51
33
22 34 51
0.0 1.0 2.0 3.0 -2 -1 0 1 2 0 1 2 3 4 -2 -1 0 1 2
Fitted values Theoretical Quantiles Fitted values Theoretical Quantiles
Scale-Location plot Cook's distance plot Scale-Location plot Cook's distance plot
0.12
Standardized residuals
Standardized residuals
0.04
22 51
0.0 0.5 1.0 1.5 2.0
1.5
34
33 45 51
34
33 52
Cook's distance
Cook's distance
50 50
0.08
1.0
46
0.02
0.04
0.5
0.00
0.00
0.0
0.0 1.0 2.0 3.0 0 20 40 60 80 100 0 1 2 3 4 0 20 40 60 80 100
Fitted values Obs. number Fitted values Obs. number

18. Other models tested
1. log(y) = α log(k)+β log(l)+γ log(u)
good fit, US R2= 0.99, JP R2= 0.99 possible spurious
regression
2. ∆log(y) = α ∆ log(k)+β ∆ log(l)+γ ∆ log(u)
poor fit, US R2= 0.70, JP R2= 0.69
3. log(y) = α log(k)+β log(l)+ α log(u)
+β (l+u)/k+γ (l/u)
good fit US R2= 0.997, JP R2= 0.99 and k, l, l/u not significant
4. log(y) = α log(u)+β (l+u)/k
good fit US R2= 0.997, JP R2= 0.999

19. Diagnostic plots: model 4
US Japan
Residuals vs Fitted Normal Q-Q plot Residuals vs Fitted Normal Q-Q plot
0.2
1 2 3
3
Standardized residuals
Standardized residuals
2 2 46 46
0.2
2
Residuals
Residuals
1
0.0
-1 0
0.0
0
-2 -1
-0.2
52
-0.2
52
-3
22
21 51
21 22 51
0.0 1.0 2.0 3.0 -2 -1 0 1 2 0 1 2 3 4 -2 -1 0 1 2
Fitted values Theoretical Quantiles Fitted values Theoretical Quantiles
Scale-Location plot Cook's distance plot Scale-Location plot Cook's distance plot
Standardized residuals
Standardized residuals
2 51
1.5
21
22 2 46
0.20
0.04
1.5
Cook's distance
Cook's distance
46 52
1
1.0
51
1.0
3
0.10
72
0.02
0.5
0.5
0.00
0.00
0.0
0.0
0.0 1.0 2.0 3.0 0 20 40 60 80 100 0 1 2 3 4 0 20 40 60 80 100
Fitted values Obs. number Fitted values Obs. number

20. Regression Procedure.
• Application of OLS to non-stationary, multicollinear time series
leads to spurious regression, parameter bias and uncertainty
problems if applying ordinary least squares (OLS).
• Differencing renders the time series stationary, but also
reduces the goodness of fit. OLS regression shows that only
labour and work are significant.
• When LINEX ratios are introduced work remains significant,
but now the ratio labour and work to capital is also significant.
Labour alone is no longer significant.
• Only work is significant for the differenced version of this
model.
• The residuals from the estimates suggest the presence of a
structural break. We tested this using ZA tests.
• We then redo the OLS regression over the two periods and
compare the parameter values.

21. Cointegration
• Conventionally nonstationary variables should
be differenced to make them stationary before
including them in multivariate models.
• Engle and Granger (1987 « Cointegration and
Error correction »Econometrica, 55, 251-76),
showed that it is possible for a linear
combination of integrated variables to be
stationary. They are cointegrated.
• Cointegrated variables show common stochastic
trends.

22. JOHANSEN PROCEDURE: Under the null hypotheses the series
has X unit roots. The null hypothesis is rejected when the value of
the test statistic is smaller than the critical value.
• US
test 10% 5% 1%
r <= 3 | 2.70 2.82 3.96 6.94
r <= 2 | 12.38 13.34 15.20 19.31
r <= 1 | 42.08 26.79 29.51 35.40
r = 0 | 80.10 43.96 47.18 53.79
• Evidence of cointegration rank 1 for US.

23. Time series plot of y1 Time series plot of y2
3.0
2.0
1.5
1.0
0.0
0.0
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
Cointegration relation of 1. variable Cointegration relation of 2. variable
0.0
-0.1
-2.0 -1.0
-0.4
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
Time series plot of y3 Time series plot of y4
2.0
0.0 0.4 0.8
1.0
0.0
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
Cointegration relation of 3. variable Cointegration relation of 4. variable
0.4
0.0
0.0
-0.3
-0.4
0 20 40 60 80 100 0 20 40 60 80 100
Time Time

24. Residuals of 1. VAR regression Residuals of 2. VAR regression
0.10
0.2 0.3
0.00
0.1
0.0
-0.10
0 20 40 60 80 100 0 20 40 60 80 100
Autocorrelations of Residuals Partial Autocorrelations of Residuals Autocorrelations of Residuals Partial Autocorrelations of Residuals
0.2
0.2
1.0
1.0
0.1
Partial ACF
Partial ACF
0.6
0.6
0.0
ACF
ACF
-0.2 -0.1 0.0
0.2
0.2
-0.2
-0.2
-0.2
0 5 10 15 5 10 15 0 5 10 15 5 10 15
Lag Lag Lag Lag
Residuals of 3. VAR regression Residuals of 4. VAR regression
0.10
0.00
0.00
-0.10
-0.10
0 20 40 60 80 100 0 20 40 60 80 100
Autocorrelations of Residuals Partial Autocorrelations of Residuals Autocorrelations of Residuals Partial Autocorrelations of Residuals
0.2
1.0
1.0
0.0 0.1 0.2
Partial ACF
Partial ACF
0.6
0.6
0.0
ACF
ACF
0.2
0.2
-0.2
-0.2
-0.2
0 5 10 15 5 10 15 0 5 10 15 -0.2 5 10 15
Lag Lag Lag Lag

25. JOHANSEN PROCEDURE: Under the null hypotheses the series
has X unit roots. The null hypothesis is rejected when the value of
the test statistic is smaller than the critical value.
• Japan
test 10% 5% 1%
r <= 3 | 0.27 2.82 3.96 6.94
r <= 2 | 8.50 13.34 15.20 19.31
r <= 1 | 31.89 26.79 29.51 35.40
r = 0 | 65.41 43.96 47.18 53.79
• Evidence of cointegration rank 1 for
Japan.

26. Time series plot of y1 Time series plot of y2
4
3
4
2
2
1
0
0
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
Cointegration relation of 1. variable Cointegration relation of 2. variable
0.1
1.0
-0.1
0.0
-0.3
-1.0
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
Time series plot of y3 Time series plot of y4
0.6
0 1 2 3 4
0.3
0.0
0 20 40 60 80 100 0 20 40 60 80 100
Time Time
Cointegration relation of 3. variable Cointegration relation of 4. variable
0.5
-0.5
-0.5
-1.5
0 20 40 60 80 100 0 20 40 60 80 100
Time Time

27. Conclusions
• A long run equilibrium exists between factor
inputs and GDP.
• However significant deviations from the
equilibrium exist as evidenced by the
cointegration relations.
• The LINEX function, by using ratios captures the
deviations from equilibrium.
• Using LINEX we avoid re-calibration.
• We are able to use the same parameters even
after unforseen and dramatic perturbations.