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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…. ☺
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