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%

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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

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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

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Fitted values

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ndar

dize

d re

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als

Scale-Location plot51

5250

0 20 40 60 80 100

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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

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0.2

Fitted values

Res

idua

ls

Residuals vs Fitted

51

46

52

-2 -1 0 1 2

-3-1

01

23

Theoretical Quantiles

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ndar

dize

d re

sidu

als

Normal Q-Q plot

51

46

52

0 1 2 3 4

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1.5

Fitted values

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sidu

als

Scale-Location plot51

46 52

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Obs. number

Coo

k's

dist

ance

Cook's distance plot46

51

72

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idua

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Residuals vs Fitted2

2122

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01

23

Theoretical Quantiles

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d re

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als

Normal Q-Q plot2

21 22

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Fitted values

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Scale-Location plot2 2122

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Obs. number

Coo

k's

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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

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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

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0.4

0.8

Cointegration relation of 3. variable

Time

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-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

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-0.3

0.0

Residuals of 1. VAR regression

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00.

000.

10

0 5 10 15

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0.2

0.6

1.0

Lag

AC

F

Autocorrelations of Residuals

5 10 15

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0.0

0.2

Lag

Par

tial A

CF

Partial Autocorrelations of Residuals

Residuals of 2. VAR regression

0 20 40 60 80 100

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Lag

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F

Autocorrelations of Residuals

5 10 15

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-0.1

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Lag

Par

tial A

CF

Partial Autocorrelations of Residuals

Residuals of 3. VAR regression

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00.

000.

10

0 5 10 15

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Lag

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F

Autocorrelations of Residuals

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Lag

Par

tial A

CF

Partial Autocorrelations of Residuals

Residuals of 4. VAR regression

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.10

0.00

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Lag

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Autocorrelations of Residuals

5 10 15-0

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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

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-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.