1. FUNCTIONAL FORMS OF REGRESSION MODELS Application 5

2. LOG-LIN OR GROWTH MODELS  The rate of growth of real GDP: can be transformed into a linear model by taking natural logs of both sides:  Letting B 1 = ln RGDP 1960 and B 2 = ln (l+r), this can be rewritten as: ln RGDP t = B 1 +B 2 t  B 2 is considered a semi-elasticity or an instantaneous growth rate.  The compound growth rate (r) is equal to (e B2 – 1).

3. The Cobb-Douglas production function  The Cobb-Douglas Production Function: can be transformed into a linear model by taking natural logs of both sides:  The slope coefficients can be interpreted as elasticities.  If (B 2 + B 3 ) = 1, we have constant returns to scale.  If (B 2 + B 3 ) > 1, we have increasing returns to scale.  If (B 2 + B 3 ) < 1, we have decreasing returns to scale.

4. The Cobb-Douglas production function for the USA, Empirical results Dependent Variable: LNOUTPUT Method: Least Squares Date: 11/12/15 Time: 12:53 Sample: 1 51 Included observations: 51 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C3.8876050.30331812.816930.0000 LNLABOR0.4683330.1243633.7658510.0005 LNCAPITAL0.5212780.1083964.8090110.0000 R-squared0.964175 Mean dependent var16.94139 Adjusted R-squared0.962683 S.D. dependent var1.380869 S.E. of regression0.266752 Akaike info criterion0.252027 Sum squared resid3.415516 Schwarz criterion0.365664 Log likelihood-3.426687 Hannan-Quinn criter.0.295451 F-statistic645.9312 Durbin-Watson stat1.946388 Prob(F-statistic)0.000000 Wald F-statistic1039.863 Prob(Wald F-statistic)0.000000

5. The linear production function for the USA, Empirical results Dependent Variable: OUTPUT Method: Least Squares Date: 11/12/15 Time: 12:56 Sample: 1 51 Included observations: 51 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C233621.61034963.0.2257290.8224 LABOR47.9873610.758634.4603590.0000 CAPITAL9.9518911.3991117.1130110.0000 R-squared0.981065 Mean dependent var43217548 Adjusted R-squared0.980276 S.D. dependent var44863661 S.E. of regression6300694. Akaike info criterion34.20724 Sum squared resid1.91E+15 Schwarz criterion34.32088 Log likelihood-869.2846 Hannan-Quinn criter.34.25066 F-statistic1243.514 Durbin-Watson stat1.684519 Prob(F-statistic)0.000000 Wald F-statistic1905.011 Prob(Wald F-statistic)0.000000

6. The Cobb-Douglas production function with linear restriction for the USA, Empirical results Dependent Variable: LNOUTLAB Method: Least Squares Date: 11/12/15 Time: 13:01 Sample: 1 51 Included observations: 51 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C3.7562420.18664420.125170.0000 LNCAPLAB0.5237560.1091034.8005780.0000 R-squared0.378823 Mean dependent var4.749135 Adjusted R-squared0.366146 S.D. dependent var0.332104 S.E. of regression0.264405 Akaike info criterion0.215754 Sum squared resid3.425582 Schwarz criterion0.291512 Log likelihood-3.501734 Hannan-Quinn criter.0.244704 F-statistic29.88248 Durbin-Watson stat1.936841 Prob(F-statistic)0.000002 Wald F-statistic23.04555 Prob(Wald F-statistic)0.000015

7. LOG-LIN OR GROWTH MODELS  The rate of growth of real GDP: can be transformed into a linear model by taking natural logs of both sides:  Letting B 1 = ln RGDP 1960 and B 2 = ln (l+r), this can be rewritten as: ln RGDP t = B 1 +B 2 t  B 2 is considered a semi-elasticity or an instantaneous growth rate.  The compound growth rate (r) is equal to (e B2 – 1).

8. Rate of growth of real GDP, USA, 1960-2007 Dependent Variable: LNRGDP Method: Least Squares Date: 11/12/15 Time: 13:06 Sample: 1 48 Included observations: 48 VariableCoefficientStd. Errort-StatisticProb. C7.8756620.009759807.00680.0000 TIME0.0314900.00034790.816530.0000 R-squared0.994454 Mean dependent var8.647157 Adjusted R-squared0.994333 S.D. dependent var0.442081 S.E. of regression0.033280 Akaike info criterion-3.926968 Sum squared resid0.050947 Schwarz criterion-3.849002 Log likelihood96.24724 Hannan-Quinn criter.-3.897505 F-statistic8247.643 Durbin-Watson stat0.347739 Prob(F-statistic)0.000000

9. Trend in real US GDP, 1960-2007 Dependent Variable: RGDP Method: Least Squares Date: 11/12/15 Time: 13:08 Sample: 1 48 Included observations: 48 VariableCoefficientStd. Errort-StatisticProb. C1664.218131.999012.607810.0000 TIME186.99394.68988639.871740.0000 R-squared0.971878 Mean dependent var6245.569 Adjusted R-squared0.971267 S.D. dependent var2655.520 S.E. of regression450.1314 Akaike info criterion15.09773 Sum squared resid9320440. Schwarz criterion15.17570 Log likelihood-360.3455 Hannan-Quinn criter.15.12719 F-statistic1589.756 Durbin-Watson stat0.069409 Prob(F-statistic)0.000000

10. LIN-LOG MODELS  Lin-log models follow this general form:  Note that B 2 is the absolute change in Y responding to a percentage (or relative) change in X  If X increases by 100%, predicted Y increases by B 2 units  Used in Engel expenditure functions: “The total expenditure that is devoted to food tends to increase in arithmetic progression as total expenditure increases in geometric proportion.”

11. Lin-log model of expenditure on food Dependent Variable: SFDHO (SHARE OF FOOD EXPENDITURE) Method: Least Squares Date: 11/12/15 Time: 13:12 Sample: 1 869 Included observations: 869 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C0.9303870.04486220.738630.0000 LOG(EXPEND)-0.0777370.004298-18.086460.0000**TOTAL EXPENDITURE R-squared0.350876 Mean dependent var0.144736 Adjusted R-squared0.350127 S.D. dependent var0.085283 S.E. of regression0.068750 Akaike info criterion-2.514368 Sum squared resid4.097984 Schwarz criterion-2.503396 Log likelihood1094.493 Hannan-Quinn criter.-2.510170 F-statistic468.6456 Durbin-Watson stat1.968386 Prob(F-statistic)0.000000 Wald F-statistic327.1202 Prob(Wald F-statistic)0.000000

12. SFDHO and log of expenditure

13. RECIPROCAL MODELS  Lin-log models follow this general form:  Note that:  As X increases indefinitely, the term approaches zero and Y approaches the limiting or asymptotic value B 1.  The slope is:  Therefore, if B 2 is positive, the slope is negative throughout, and if B 2 is negative, the slope is positive throughout.

14. Reciprocal model of food expenditure Dependent Variable: SFDHO Method: Least Squares Date: 11/12/15 Time: 13:43 Sample: 1 869 Included observations: 869 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C0.0772630.00513715.039400.0000 1/EXPEND1331.338115.930611.483920.0000 R-squared0.333236 Mean dependent var0.144736 Adjusted R-squared0.332467 S.D. dependent var0.085283 S.E. of regression0.069678 Akaike info criterion-2.487556 Sum squared resid4.209346 Schwarz criterion-2.476584 Log likelihood1082.843 Hannan-Quinn criter.-2.483357 F-statistic433.3100 Durbin-Watson stat1.997990 Prob(F-statistic)0.000000 Wald F-statistic131.8804 Prob(Wald F-statistic)0.000000

15. Share of food expenditure in total expenditure

16. POLYNOMIAL REGRESSION MODELS  The following regression predicting GDP is an example of a quadratic function, or more generally, a second-degree polynomial in the variable time: The slope is nonlinear and equal to:

17. Polynomial model of US GDP, 1960-2007 Dependent Variable: RGDP Method: Least Squares Date: 11/12/15 Time: 13:48 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C2651.38178.4738933.786790.0000 TIME68.534366.9247889.8969610.0000 TIME22.4175420.12563019.243290.0000 R-squared0.996787 Mean dependent var6245.569 Adjusted R-squared0.996644 S.D. dependent var2655.520 S.E. of regression153.8419 Akaike info criterion12.97019 Sum squared resid1065030. Schwarz criterion13.08714 Log likelihood-308.2845 Hannan-Quinn criter.13.01438 F-statistic6979.430 Durbin-Watson stat0.462850 Prob(F-statistic)0.000000 Wald F-statistic10003.42 Prob(Wald F-statistic)0.000000

18. Polynomial model of log US GDP, 1960-2007 Dependent Variable: LNRGDP Method: Least Squares Date: 11/12/15 Time: 13:50 Sample: 1 48 Included observations: 48 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C7.8334800.017778440.61900.0000 TIME0.0365510.00145025.202490.0000 TIME2-0.0001032.55E-05-4.0519710.0002 R-squared0.996095 Mean dependent var8.647157 Adjusted R-squared0.995921 S.D. dependent var0.442081 S.E. of regression0.028234 Akaike info criterion-4.236106 Sum squared resid0.035873 Schwarz criterion-4.119156 Log likelihood104.6665 Hannan-Quinn criter.-4.191911 F-statistic5738.826 Durbin-Watson stat0.471705 Prob(F-statistic)0.000000 Wald F-statistic7597.878 Prob(Wald F-statistic)0.000000

19. SUMMARY OF FUNCTIONAL FORMS