Sub TopicsBackground:Definition:Equation:Diagnostic Tests:Estimation of parameters in regression:

Cobb-douglas production function

Developed by Paul Douglas and C. W. Cobb in the 1930’s.

Background:

Cobb Douglas is a Mathematical Formula that relates Labor Capital and Output.Cobb- Douglas equation: Q=AKL

Definition:

Q K L100 100 100101 100 105112 107 110122 114 118124 122 123122 131 116143 138 125152 149 133151 163 138126 176 121155 185 140159 198 144153 208 145177 216 152184 226 154169 236 149189 244 154225 266 182227 298 196223 335 200280 366 193231 387 193179 407 147240 417 161

Cobb-douglas data file

Following Formula is Used for Log Transformation=(log(A)) so on….

Log transformation for regression

Log Q Log K Log L2 2 2

2.004321 2

2.021189

2.049218

2.029384

2.041393

2.08636

2.056905

2.071882

2.093422

2.08636

2.089905

2.08636

2.117271

2.064458

2.155336

2.139879

2.09691

2.181844

2.173186

2.123852

2.178977

2.212188

2.139879

2.100371

2.245513

2.082785

2.190332

2.267172

2.146128

2.201397

2.296665

2.158362

2.184691

2.318063

2.161368

2.247973

2.334454

2.181844

2.264818

2.354108

2.187521

2.227887

2.372912

2.173186

2.276462

2.38739

2.187521

2.352183

2.424882

2.260071

2.356026

2.474216

2.292256

2.348305

2.525045

2.30103

2.447158

2.563481

2.285557

2.363612

2.587711

2.285557

2.252853

2.609594

2.167317

2.380211

2.620136

2.206826

Results of normality test:

As p(Q,K,L)0.05 so series is Normally Distributed.

Diagnostic tests:

Q K L Mean  2.209588  2.299855  2.155283 Median  2.195864  2.307364  2.159865 Maximum  2.447158  2.620136  2.301030 Minimum  2.000000  2.000000  2.000000 Std. Dev.  0.124198  0.198841  0.087327 Skewness  0.058511  0.078104  0.107486 Kurtosis  2.104485  1.870039  2.151359

        Jarque-Bera  0.815641  1.301213  0.766404 Probability  0.665098  0.521729  0.681675

        Sum  53.03012  55.19651  51.72680 Sum Sq. Dev.  0.354779  0.909371  0.175396

        Observations  24  24  24

Stationarity of “Q”: As t-statt-crit 4.363.63 at 5% significance level. OR p0.050.010.05So Q is a stationary series.

Stationarity:Null Hypothesis: Q has a unit root  Exogenous: Constant, Linear Trend  Lag Length: 1 (Automatic based on SIC, MAXLAG=5)

                        t-Statistic   Prob.*                  

Augmented Dickey-Fuller test statistic -4.369442  0.0116Test critical values: 1% level   -4.440739  

  5% level   -3.632896  

 10% level   -3.254671  

                  

Null Hypothesis: K has a unit root  

Exogenous: Constant, Linear Trend  

Lag Length: 1 (Automatic based on SIC, MAXLAG=4)

         

         

      t-Statistic   Prob.*

         

         Augmented Dickey-Fuller test statistic

-4.007282  0.0250

Test critical values: 1% level  

-4.467895  

  5% level  -

3.644963  

 10% level  

-3.261452  

         

Stationary of “k” :

As t-statt-crit 4.0073.6 at 5% significance level. OR

p0.050.0250.05 So, K is a stationary series.

9

As t-statt-crit 4.19>3.69 at 5% significance level. OR p0.050.020.05 So, Q is a stationary series.

Stationary of “L” :Null Hypothesis: L has a unit root  

Exogenous: Constant, Linear Trend  

Lag Length: 4 (Automatic based on SIC, MAXLAG=4)

         

         

      t-Statistic   Prob.*

         

         Augmented Dickey-Fuller test statistic

-4.196949  0.0200

Test critical values: 1% level  

-4.571559  

  5% level  -

3.690814  

 10% level  

-3.286909  

Results of correlation test:Results of Correlation Test show that there Exists high Multicolinearity betweenQ,K and L.

Multicolinearity:

Q K L

Q 1.00000

0 0.91955

8 0.95942

8

K 0.91955

8 1.00000

0 0.87844

0

L 0.95942

8 0.87844

0 1.00000

0

Here, α=0.206925 & Adding α & β : i.e α + β0.206925+0.952008 =1.158>1→ Industry exhibits increasing returns to scale

 

Regression Analysis of C0bb-Douglas P.F:In E-views:

  Variable Coefficient Std. Error t-Statistic Prob.  

           

           

  C -0.318155 0.197991 -1.606916 0.1230

  LOGK 0.206925 0.065080 3.179549 0.0045

  LOGL 0.952008 0.148186 6.424416 0.0000

           

         

R-squared 0.953241    Mean dependent var 2.209588

Adjusted R-squared 0.948788    S.D. dependent var 0.124198

S.E. of regression 0.028106    Akaike info criterion -4.189186

Sum squared resid 0.016589    Schwarz criterion -4.041929

Log likelihood 53.27023    Hannan-Quinn criter. -4.150119

F-statistic 214.0556    Durbin-Watson stat 2.247216

Prob(F-statistic) 0.000000      

         

         

R-squared is a statistical measure. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression.

 It is the percentage of the response variable variation that is explained by a linear model.

→ R-squared = Explained variation / Total variation R-squared is always between 0 and 100%: 0% indicates that the model explains none of the

variability of the response data around its mean. 100% indicates that the model explains all the

variability of the response data around its mean.

Interpretation of r-square:

95%variation in Dependent variable are Explained by independent variable.

R-square=0.953241           

         

R-squared 0.953241    Mean dependent var 2.209588

Adjusted R-squared 0.948788    S.D. dependent var 0.124198

S.E. of regression 0.028106    Akaike info criterion-

4.189186

Sum squared resid 0.016589    Schwarz criterion-

4.041929

Log likelihood 53.27023    Hannan-Quinn criter.-

4.150119

F-statistic 214.0556    Durbin-Watson stat 2.247216

Prob(F-statistic) 0.000000      

         

         

Model is significant because : Tkcal>Tcrit

3.179549>1.70 and TLcal>Tcrit 6.424416>1.70

Checking the signifacance of the model:

fcal>fcrit

F>F(k-1,n-k)

214.0556>4.35

Hence the model is good

Checking the goodness of the model:

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.976341

R Square0.953

241Adjusted R Square

0.948788

Standard Error

0.028106

Observations 24

ANOVA

  df SS MS FSignificance F

Regression 2 0.33819

0.169095

214.0556

1.08E-14

Residual 210.01658

90.00079

Total 230.35477

9      

 Coefficients

Standard Error t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

-0.318

150.19799

1

-1.60692

0.123006 -0.7299

0.09359

-0.7299

0.09359

Log K0.206

925 0.065083.17

95490.004

5130.0715

840.342

2650.0715

840.3422

65

Log L0.952

0080.14818

66.42

44162.29E

-060.6438

381.260

1770.6438

381.2601

77

REGRESSION ANALYSIS OF COBB-DOUGLAS IN EXCEL:

Results are same like in e-views.

Q=-0.31+0.20K+o.95L

The End…. ☺