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 B1 = ln RGDP1960 and B2 = ln (l+r), this can be rewritten as: ln RGDPt = B1 +B2 t B2 is considered a semi-elasticity or an instantaneous growth

rate. The compound growth rate (r) is equal to (eB2 – 1).

1960 (1 )ttRGDP RGDP r

1960ln ln ln(1 )tRGDP RGDP t r

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 (B2 + B3) = 1, we have constant returns to scale.

If (B2 + B3) > 1, we have increasing returns to scale.

If (B2 + B3) < 1, we have decreasing returns to scale.

321

BBi i iQ B L K

1 2 3ln ln ln lni i iQ B B L B K

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

• Variable Coefficient Std. Error t-Statistic Prob. • C 3.887605 0.303318 12.81693 0.0000• LNLABOR 0.468333 0.124363 3.765851 0.0005• LNCAPITAL 0.521278 0.108396 4.809011 0.0000• R-squared 0.964175 Mean dependent var 16.94139• Adjusted R-squared 0.962683 S.D. dependent var 1.380869• S.E. of regression 0.266752 Akaike info criterion 0.252027• Sum squared resid 3.415516 Schwarz criterion 0.365664• Log likelihood -3.426687 Hannan-Quinn criter. 0.295451• F-statistic 645.9312 Durbin-Watson stat 1.946388• Prob(F-statistic) 0.000000 Wald F-statistic 1039.863• Prob(Wald F-statistic) 0.000000

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• Variable Coefficient Std. Error t-Statistic Prob. •

• C 233621.6 1034963. 0.225729 0.8224• LABOR 47.98736 10.75863 4.460359 0.0000• CAPITAL 9.951891 1.399111 7.113011 0.0000• R-squared 0.981065 Mean dependent var 43217548• Adjusted R-squared 0.980276 S.D. dependent var 44863661• S.E. of regression 6300694. Akaike info criterion 34.20724• Sum squared resid 1.91E+15 Schwarz criterion 34.32088• Log likelihood -869.2846 Hannan-Quinn criter. 34.25066• F-statistic 1243.514 Durbin-Watson stat 1.684519• Prob(F-statistic) 0.000000 Wald F-statistic 1905.011• Prob(Wald F-statistic) 0.000000

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• Variable Coefficient Std. Error t-Statistic Prob. • C 3.756242 0.186644 20.12517 0.0000• LNCAPLAB 0.523756 0.109103 4.800578 0.0000•

•

• R-squared 0.378823 Mean dependent var 4.749135• Adjusted R-squared 0.366146 S.D. dependent var 0.332104• S.E. of regression 0.264405 Akaike info criterion 0.215754• Sum squared resid 3.425582 Schwarz criterion 0.291512• Log likelihood -3.501734 Hannan-Quinn criter. 0.244704• F-statistic 29.88248 Durbin-Watson stat 1.936841• Prob(F-statistic) 0.000002 Wald F-statistic 23.04555• Prob(Wald F-statistic) 0.000015•

•

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 B1 = ln RGDP1960 and B2 = ln (l+r), this can be rewritten as: ln RGDPt = B1 +B2 t B2 is considered a semi-elasticity or an instantaneous growth

rate. The compound growth rate (r) is equal to (eB2 – 1).

1960 (1 )ttRGDP RGDP r

1960ln ln ln(1 )tRGDP RGDP t r

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•

•

• Variable Coefficient Std. Error t-Statistic Prob. • C 7.875662 0.009759 807.0068 0.0000• TIME 0.031490 0.000347 90.81653 0.0000

• R-squared 0.994454 Mean dependent var 8.647157• Adjusted R-squared 0.994333 S.D. dependent var 0.442081• S.E. of regression 0.033280 Akaike info criterion-3.926968• Sum squared resid 0.050947 Schwarz criterion -3.849002• Log likelihood 96.24724 Hannan-Quinn criter. -3.897505• F-statistic 8247.643 Durbin-Watson stat 0.347739• Prob(F-statistic) 0.000000

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• Variable Coefficient Std. Error t-Statistic Prob. • C 1664.218 131.9990 12.60781 0.0000• TIME 186.9939 4.689886 39.87174 0.0000•

•

• R-squared 0.971878 Mean dependent var 6245.569• Adjusted R-squared 0.971267 S.D. dependent var 2655.520• S.E. of regression 450.1314 Akaike info criterion 15.09773• Sum squared resid 9320440. Schwarz criterion 15.17570• Log likelihood -360.3455 Hannan-Quinn criter. 15.12719• F-statistic 1589.756 Durbin-Watson stat 0.069409• Prob(F-statistic) 0.000000•

LIN-LOG MODELS

Lin-log models follow this general form:

Note that B2 is the absolute change in Y responding to a percentage (or relative) change in X If X increases by 100%, predicted Y increases by B2 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.”

1 2 lni i iY B B X u

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• Variable Coefficient Std. Error t-Statistic Prob. • C 0.930387 0.044862 20.73863 0.0000• LOG(EXPEND) -0.077737 0.004298 -18.08646 0.0000 **TOTAL EXPENDITURE• R-squared 0.350876 Mean dependent var 0.144736• Adjusted R-squared 0.350127 S.D. dependent var 0.085283• S.E. of regression 0.068750 Akaike info criterion -2.514368• Sum squared resid 4.097984 Schwarz criterion -2.503396• Log likelihood 1094.493 Hannan-Quinn criter. -2.510170• F-statistic 468.6456 Durbin-Watson stat 1.968386• Prob(F-statistic) 0.000000 Wald F-statistic 327.1202• Prob(Wald F-statistic) 0.000000

SFDHO and log of expenditure

.0

.1

.2

.3

.4

.5

.6

.7

.8

8 9 10 11 12

LOG(EXPEND)

sfdh

o

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 B1.

The slope is:

Therefore, if B2 is positive, the slope is negative throughout, and if B2 is negative, the slope is positive throughout.

1 2

1( )i i

i

Y B B uX

2 2

1( )

dYB

dX X

2

1( )

i

BX

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•

• Variable Coefficient Std. Error t-Statistic Prob. •

• C 0.077263 0.005137 15.03940 0.0000• 1/EXPEND 1331.338 115.9306 11.48392 0.0000• R-squared 0.333236 Mean dependent var 0.144736• Adjusted R-squared 0.332467 S.D. dependent var 0.085283• S.E. of regression 0.069678 Akaike info criterion -2.487556• Sum squared resid 4.209346 Schwarz criterion -2.476584• Log likelihood 1082.843 Hannan-Quinn criter. -2.483357• F-statistic 433.3100 Durbin-Watson stat 1.997990• Prob(F-statistic) 0.000000 Wald F-statistic 131.8804• Prob(Wald F-statistic) 0.000000

Share of food expenditure in total expenditure

.0

.1

.2

.3

.4

.5

.6

.7

.8

0 20,000 60,000 100,000 140,000

expend

sfdh

o

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:

21 2 3t tRGDP A A time A time u

2 32dRGDP

A A timetime

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• Variable Coefficient Std. Error t-Statistic Prob. •

• C 2651.381 78.47389 33.78679 0.0000• TIME 68.53436 6.924788 9.896961 0.0000• TIME2 2.417542 0.125630 19.24329 0.0000• R-squared 0.996787 Mean dependent var 6245.569• Adjusted R-squared 0.996644 S.D. dependent var 2655.520• S.E. of regression 153.8419 Akaike info criterion 12.97019• Sum squared resid 1065030. Schwarz criterion 13.08714• Log likelihood -308.2845 Hannan-Quinn criter. 13.01438• F-statistic 6979.430 Durbin-Watson stat 0.462850• Prob(F-statistic) 0.000000 Wald F-statistic 10003.42• Prob(Wald F-statistic) 0.000000

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

• Variable Coefficient Std. Error t-Statistic Prob.

• C 7.833480 0.017778 440.6190 0.0000• TIME 0.036551 0.001450 25.20249 0.0000• TIME2 -0.000103 2.55E-05 -4.051971 0.0002• R-squared 0.996095 Mean dependent var 8.647157• Adjusted R-squared 0.995921 S.D. dependent var 0.442081• S.E. of regression 0.028234 Akaike info criterion -4.236106• Sum squared resid 0.035873 Schwarz criterion -4.119156• Log likelihood 104.6665 Hannan-Quinn criter. -4.191911• F-statistic 5738.826 Durbin-Watson stat 0.471705• Prob(F-statistic) 0.000000 Wald F-statistic 7597.878• Prob(Wald F-statistic) 0.000000

SUMMARY OF FUNCTIONAL FORMS

MODEL FORM SLOPE ELASTICITY

(dY

dX) .

dY X

dX Y

Linear Y =B1 + B2 X 2B )(2 Y

XB

Log-linear lnY =B1 + ln X 2 ( )Y

BX

2B

Log-lin lnY =B1 + B2 X 2 ( )B Y )(2 XB

Lin-log 1 2 lnY B B X 2

1( )B

X )

1(2 Y

B

Reciprocal 1 2

1( )Y B B

X 2 2

1( )B

X )

1(2 XY

B