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

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- 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.

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