Warr 7th Iiasa Titech Technical Meeting
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Transcript of Warr 7th Iiasa Titech Technical Meeting
IIASA-TITECH Technical Meeting18-19 Sept, Laxenburg
Benjamin Warr and Robert AyresCenter for the Management of Environmental Resources (CMER)
INSEADBoulevard de Constance
Fontainebleau77300
http://benjamin.warr.insead.edu
Time series analysis of output and factors of production, Japan and US 1900-2000.
Coal fractions of fossil fuel exergy apparent consumption, Japan 1900-2000
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000year
per
cen
t
Electricity
Heat (Steam coals for space heating and coking coal for steel production)
Non-fuel (includes industrial transformation processes)
Other prime movers (steam locomotives)
Petroleum products fractions of fossil fuel exergy apparent consumption, Japan 1900-2000
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000year
per
cen
t
Electricity (Heavy Oil)
Heat (Residential and Commercial uses of Heavy Oil and LPG)
Light (Kerosene)
Non-fuel (Machinery Oil, Lubricants, Asphalt)
Other prime movers (Gasoline, Light Oil, Heavy Oil, LPG, Jet Oil, Kerosene)
Technical efficiency of primary work services from exergy sources, Japan 1900-2000
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
year
tech
nic
al e
ffic
ien
cy (
%)
coalpetroleumnatural gas
nuclear, hydroelectric, thermalfuelwood, charcoal
Exergy to work conversion efficiencies, Japan 1900-2000
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
1900 1920 1940 1960 1980 2000
year
effi
cien
cy
High Temperature Industrial Heat
Medium Temperature Industrial Heat
Low Temperature Space Heat
Electric Power Generation and Distribution
Other Mechanical Work
Comparison of the technical efficiency of primary work (exergy) services from exergy sources,
Japan and US 1900-2000
0%
5%
10%
15%
20%
25%
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
year
tech
nic
al e
ffic
ien
cy (
%)
Japan - f(Ub)
US - f( Ub)
LINEX fits for GDP, Japan and US 1900-2000.
0
1000
2000
3000
4000
5000
6000
7000
8000
1900 1920 1940 1960 1980
year
GD
P (
tho
usa
nd
bill
ion
199
2$)
empirical GDP, Japan
predicted GDP, Japan
empirical GDP, US
predicted GDP, US
Estimates of GDP, UK 1960-2000
0
0.5
1
1.5
2
2.5
3
1963 1968 1973 1978 1983 1988 1993
ou
tpu
t (1
960=
1)YLINEXTime Dependent CDTime Average CD
Estimates of GDP, France 1960-2000
0
0.5
1
1.5
2
2.5
3
3.5
4
1963 1968 1973 1978 1983 1988 1993
ou
tpu
t (1
960=
1)YLINEXTime Dependent CDTime Average CD
Elasticities of factors of production*, US 1960-2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1900 1910 1920 1930 1940
year
elas
tici
tyalpha
beta
gamma
GDP=Capital*alpha*Labour*beta*Work*gamma
* derived from optimisation of the LINEX function.
Elasticities of factors of production*, Japan 1960-2000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1960 1970 1980 1990 2000
year
elas
tici
ty
alpha
beta
gamma
GDP=Capital*alpha*Labour*beta*Work*gamma
* derived from optimisation of the LINEX function.
Some problems using econometrictime series in OLS
• Multicollinearity• Stationarity• Unit roots – explosive behaviour.
Multicollinearity
• Variables highly correlated. Usual proceduretake logs and increments or ratios.
lny lnk lnl lnulny 1.00 0.97 0.98 0.99lnk 0.97 1.00 0.96 0.96lnl 0.98 0.96 1.00 0.96lnu 0.99 0.96 0.96 1.00
dlny dlnk dlnl dlnudlny 1.00 -0.0012 0.78 0.78dlnk -0.0012 1.00 0.19 0.11dlnl 0.78 0.19 1.00 0.80dlnu 0.78 0.11 0.80 1.00
• 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, themean, 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,
Equation 10•• Equation 11
• 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.
( ) µ=tyE
( )[ ] 02 γµ =−tyE
( )( )[ ] kktt yyE γµµ =−− −
Stationarity
Unit Roots
• A unit root test is a statistical test for theproposition that in a autoregressive timesY(t+1)=ay(t)+other termsthat a = 1.
• For values smaller than 1, the time series ismean reverting and shocks are transitory.
• For values larger than 1 the shock ispermanent causing a change in the mean value of value of Yt
• A process having a unit root is non-stationary
log(y) = α log(k)+β log(l)+γ log(u)
JapanEstimate 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
USAEstimate 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
Diagnostic plots: model 1US Japan
0.0 1.0 2.0 3.0
-0.3
-0.1
0.1
0.3
Fitted values
Res
idua
ls
Residuals vs Fitted
223433
-2 -1 0 1 2
-2-1
01
2
Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
Normal Q-Q plot
22 3433
0.0 1.0 2.0 3.0
0.0
0.5
1.0
1.5
Fitted values
Sta
ndar
dize
d re
sidu
als
Scale-Location plot223433
0 20 40 60 80 100
0.00
0.04
0.08
0.12
Obs. number
Coo
k's
dist
ance
Cook's distance plot45
3433
0 1 2 3 4
-0.3
-0.1
0.1
Fitted values
Res
idua
ls
Residuals vs Fitted
51
5250
-2 -1 0 1 2
-4-2
01
Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
Normal Q-Q plot
51
5250
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
Fitted values
Sta
ndar
dize
d re
sidu
als
Scale-Location plot51
5250
0 20 40 60 80 100
0.00
0.02
0.04
Obs. number
Coo
k's
dist
ance
Cook's distance plot51
50
46
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)/kgood fit US R2= 0.997, JP R2= 0.999
Diagnostic plots: model 4US Japan
0 1 2 3 4
-0.2
0.0
0.2
Fitted values
Res
idua
ls
Residuals vs Fitted
51
46
52
-2 -1 0 1 2
-3-1
01
23
Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
Normal Q-Q plot
51
46
52
0 1 2 3 4
0.0
0.5
1.0
1.5
Fitted values
Sta
ndar
dize
d re
sidu
als
Scale-Location plot51
46 52
0 20 40 60 80 100
0.00
0.02
0.04
Obs. number
Coo
k's
dist
ance
Cook's distance plot46
51
72
0.0 1.0 2.0 3.0
-0.2
0.0
0.2
Fitted values
Res
idua
ls
Residuals vs Fitted2
2122
-2 -1 0 1 2
-2-1
01
23
Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
Normal Q-Q plot2
21 22
0.0 1.0 2.0 3.0
0.0
0.5
1.0
1.5
Fitted values
Sta
ndar
dize
d re
sidu
als
Scale-Location plot2 2122
0 20 40 60 80 100
0.00
0.10
0.20
Obs. number
Coo
k's
dist
ance
Cook's distance plot2
1
3
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.
Cointegration
• Conventionally nonstationary variables shouldbe differenced to make them stationary beforeincluding them in multivariate models.
• Engle and Granger (1987 « Cointegration andError correction »Econometrica, 55, 251-76), showed that it is possible for a linearcombination of integrated variables to bestationary. They are cointegrated.
• Cointegrated variables show common stochastictrends.
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 19.31r <= 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.
Time series plot of y1
Time
0 20 40 60 80 100
0.0
1.5
3.0
Cointegration relation of 1. variable
Time
0 20 40 60 80 100
-0.4
-0.1
Time series plot of y2
Time
0 20 40 60 80 100
0.0
1.0
2.0
Cointegration relation of 2. variable
Time
0 20 40 60 80 100
-2.0
-1.0
0.0
Time series plot of y3
Time
0 20 40 60 80 100
0.0
0.4
0.8
Cointegration relation of 3. variable
Time
0 20 40 60 80 100
-0.4
0.0
0.4
Time series plot of y4
Time
0 20 40 60 80 100
0.0
1.0
2.0
Cointegration relation of 4. variable
Time
0 20 40 60 80 100
-0.3
0.0
Residuals of 1. VAR regression
0 20 40 60 80 100
-0.1
00.
000.
10
0 5 10 15
-0.2
0.2
0.6
1.0
Lag
AC
F
Autocorrelations of Residuals
5 10 15
-0.2
0.0
0.2
Lag
Par
tial A
CF
Partial Autocorrelations of Residuals
Residuals of 2. VAR regression
0 20 40 60 80 100
0.0
0.1
0.2
0.3
0 5 10 15
-0.2
0.2
0.6
1.0
Lag
AC
F
Autocorrelations of Residuals
5 10 15
-0.2
-0.1
0.0
0.1
0.2
Lag
Par
tial A
CF
Partial Autocorrelations of Residuals
Residuals of 3. VAR regression
0 20 40 60 80 100
-0.1
00.
000.
10
0 5 10 15
-0.2
0.2
0.6
1.0
Lag
AC
F
Autocorrelations of Residuals
5 10 15
-0.2
0.0
0.2
Lag
Par
tial A
CF
Partial Autocorrelations of Residuals
Residuals of 4. VAR regression
0 20 40 60 80 100-0
.10
0.00
0 5 10 15
-0.2
0.2
0.6
1.0
Lag
AC
F
Autocorrelations of Residuals
5 10 15-0
.20.
00.
10.
2
Lag
Par
tial A
CF
Partial Autocorrelations of Residuals
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.
• Japantest 10% 5% 1%
r <= 3 | 0.27 2.82 3.96 6.94r <= 2 | 8.50 13.34 15.20 19.31r <= 1 | 31.89 26.79 29.51 35.40r = 0 | 65.41 43.96 47.18 53.79• Evidence of cointegration rank 1 for
Japan.
Time series plot of y1
Time
0 20 40 60 80 100
01
23
4
Cointegration relation of 1. variable
Time
0 20 40 60 80 100
-1.0
0.0
1.0
Time series plot of y2
Time
0 20 40 60 80 100
02
4
Cointegration relation of 2. variable
Time
0 20 40 60 80 100
-0.3
-0.1
0.1
Time series plot of y3
Time
0 20 40 60 80 100
0.0
0.3
0.6
Cointegration relation of 3. variable
Time
0 20 40 60 80 100
-1.5
-0.5
Time series plot of y4
Time
0 20 40 60 80 100
01
23
4Cointegration relation of 4. variable
Time
0 20 40 60 80 100
-0.5
0.5
Conclusions
• A long run equilibrium exists between factorinputs and GDP.
• However significant deviations from theequilibrium exist as evidenced by thecointegration relations.
• The LINEX function, by using ratios captures thedeviations from equilibrium.
• Using LINEX we avoid re-calibration.• We are able to use the same parameters even
after unforseen and dramatic perturbations.