FUNCTIONAL FORMS OF REGRESSION MODELS Application 5

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).

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.

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. C LNLABOR LNCAPITAL R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Wald F-statistic Prob(Wald F-statistic)

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. C LABOR CAPITAL R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid1.91E+15 Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Wald F-statistic Prob(Wald F-statistic)

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. C LNCAPLAB R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Wald F-statistic Prob(Wald F-statistic)

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).

Rate of growth of real GDP, USA, Dependent Variable: LNRGDP Method: Least Squares Date: 11/12/15 Time: 13:06 Sample: 1 48 Included observations: 48 VariableCoefficientStd. Errort-StatisticProb. C TIME R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic)

Trend in real US GDP, Dependent Variable: RGDP Method: Least Squares Date: 11/12/15 Time: 13:08 Sample: 1 48 Included observations: 48 VariableCoefficientStd. Errort-StatisticProb. C TIME R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic)

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.”

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: Included observations: 869 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C LOG(EXPEND) **TOTAL EXPENDITURE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Wald F-statistic Prob(Wald F-statistic)

SFDHO and log of expenditure

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.

Reciprocal model of food expenditure Dependent Variable: SFDHO Method: Least Squares Date: 11/12/15 Time: 13:43 Sample: Included observations: 869 White heteroskedasticity-consistent standard errors & covariance VariableCoefficientStd. Errort-StatisticProb. C /EXPEND R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Wald F-statistic Prob(Wald F-statistic)

Share of food expenditure in total expenditure

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:

Polynomial model of US GDP, 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. C TIME TIME R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Wald F-statistic Prob(Wald F-statistic)

Polynomial model of log US GDP, 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. C TIME TIME E R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Wald F-statistic Prob(Wald F-statistic)

SUMMARY OF FUNCTIONAL FORMS