Academy of Economic StudiesFaculty of Business Administration

Estimating the aggregate production function in UK

Prof: Adriana Agapie

Bucharest2015

ContentsPros and cons for the estimation of an aggregated production function3Testing the stationarity of the variables concerned3Regression using differences of logarithms17We see now if the production function has constant increasing or decreasing returns to scale.17

Pros and cons for the estimation of an aggregated production functionCautam pe net motive pro si contra

Testing the stationarity of the variables concerned

In order to verify the stationarity of the series, we must compute the unit root test and see if the Augmented Dickey-Fuller test value is lower than the critical 1% level, for labour, capital and rgnp.

Null Hypothesis: RGNP has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic0.5997770.9993

Test critical values:1% level-4.148465

5% level-3.500495

10% level-3.179617

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(RGNP)

Method: Least Squares

Date: 05/24/15 Time: 12:53

Sample (adjusted): 2 52

Included observations: 51 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

RGNP(-1)0.0180350.0300690.5997770.5515

C37.1986050.211810.7408340.4624

@TREND(1)2.2933696.1021860.3758270.7087

R-squared0.355224Mean dependent var212.1864

Adjusted R-squared0.328358S.D. dependent var147.8580

S.E. of regression121.1752Akaike info criterion12.48937

Sum squared resid704804.8Schwarz criterion12.60301

Log likelihood-315.4791F-statistic13.22221

Durbin-Watson stat1.544039Prob(F-statistic)0.000027

T-Statistic Dickey-Fuller value 0.59 is higher than the 1% level therefore this series is not stationary.1st difference is stationary t Stat is indeed lower than 1% level therefore RGNP is stationary.

Null Hypothesis: D(RGNP) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-5.4062710.0003

Test critical values:1% level-4.152511

5% level-3.502373

10% level-3.180699

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(RGNP,2)

Method: Least Squares

Date: 05/24/15 Time: 12:58

Sample (adjusted): 3 52

Included observations: 50 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(RGNP(-1))-0.7617320.140898-5.4062710.0000

C36.8686435.878331.0276020.3094

@TREND(1)4.7784471.4128843.3820530.0015

R-squared0.384464Mean dependent var5.106063

Adjusted R-squared0.358271S.D. dependent var147.7293

S.E. of regression118.3429Akaike info criterion12.44317

Sum squared resid658237.4Schwarz criterion12.55790

Log likelihood-308.0794F-statistic14.67812

Durbin-Watson stat1.874641Prob(F-statistic)0.000011

As for capital upon performing the 1st difference unit root test t-Stat is still higher than 1% is not stationary.Null Hypothesis: D(CAPITAL) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-4.0684450.0126

Test critical values:1% level-4.156734

5% level-3.504330

10% level-3.181826

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(CAPITAL,2)

Method: Least Squares

Date: 05/24/15 Time: 13:07

Sample (adjusted): 4 52

Included observations: 49 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(CAPITAL(-1))-0.3524740.086636-4.0684450.0002

D(CAPITAL(-1),2)0.5350920.1271574.2081370.0001

C93.0234028.966603.2114030.0024

@TREND(1)4.4707831.2251763.6490950.0007

R-squared0.363456Mean dependent var13.80499

Adjusted R-squared0.321019S.D. dependent var75.12702

S.E. of regression61.90488Akaike info criterion11.16718

Sum squared resid172449.7Schwarz criterion11.32162

Log likelihood-269.5960F-statistic8.564734

Durbin-Watson stat1.760140Prob(F-statistic)0.000131

Upon performing the 2nd difference we can see that the series is stationary. T-Stat is lower than 1%.

Null Hypothesis: D(CAPITAL) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-4.0684450.0126

Test critical values:1% level-4.156734

5% level-3.504330

10% level-3.181826

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(CAPITAL,2)

Method: Least Squares

Date: 05/24/15 Time: 13:07

Sample (adjusted): 4 52

Included observations: 49 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(CAPITAL(-1))-0.3524740.086636-4.0684450.0002

D(CAPITAL(-1),2)0.5350920.1271574.2081370.0001

C93.0234028.966603.2114030.0024

@TREND(1)4.4707831.2251763.6490950.0007

R-squared0.363456Mean dependent var13.80499

Adjusted R-squared0.321019S.D. dependent var75.12702

S.E. of regression61.90488Akaike info criterion11.16718

Sum squared resid172449.7Schwarz criterion11.32162

Log likelihood-269.5960F-statistic8.564734

Durbin-Watson stat1.760140Prob(F-statistic)0.000131

Labour After computing the first unit root test we can see that the series is not stationary as t-Stat is higher than 1% level.

Null Hypothesis: LABOR has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-3.8176390.0237

Test critical values:1% level-4.152511

5% level-3.502373

10% level-3.180699

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LABOR)

Method: Least Squares

Date: 05/24/15 Time: 13:08

Sample (adjusted): 3 52

Included observations: 50 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

LABOR(-1)-0.2491910.065274-3.8176390.0004

D(LABOR(-1))0.4131250.1225003.3724450.0015

C14.957003.7364744.0029730.0002

@TREND(1)0.4249170.1091783.8919780.0003

R-squared0.344121Mean dependent var1.537123

Adjusted R-squared0.301346S.D. dependent var1.273701

S.E. of regression1.064630Akaike info criterion3.039749

Sum squared resid52.13805Schwarz criterion3.192711

Log likelihood-71.99373F-statistic8.044958

Durbin-Watson stat1.948348Prob(F-statistic)0.000205

Upon performing the second unit root test, we can see that t-Stat is lower than 1% therefore is stationary.

Null Hypothesis: D(LABOUR) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-5.0820330.0007

Test critical values:1% level-4.156734

5% level-3.504330

10% level-3.181826

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LABOUR,2)

Method: Least Squares

Date: 05/24/15 Time: 13:10

Sample (adjusted): 4 52

Included observations: 49 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(LABOUR(-1))-0.8396040.165210-5.0820330.0000

D(LABOUR(-1),2)0.2693040.1438261.8724230.0677

C0.9633920.4166522.3122250.0254

@TREND(1)0.0123100.0121751.0111170.3174

R-squared0.379038Mean dependent var0.019436

Adjusted R-squared0.337641S.D. dependent var1.461682

S.E. of regression1.189596Akaike info criterion3.263212

Sum squared resid63.68126Schwarz criterion3.417647

Log likelihood-75.94870F-statistic9.156077

Durbin-Watson stat2.028437Prob(F-statistic)0.000076

At log_rgnp we can see that is not stationary.

Null Hypothesis: LOG_RGNP has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-2.3271510.4121

Test critical values:1% level-4.152511

5% level-3.502373

10% level-3.180699

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LOG_RGNP)

Method: Least Squares

Date: 05/24/15 Time: 13:12

Sample (adjusted): 3 52

Included observations: 50 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

LOG_RGNP(-1)-0.1870350.080371-2.3271510.0244

D(LOG_RGNP(-1))0.2003010.1397971.4328020.1587

C1.4850800.6248742.3766080.0217

@TREND(1)0.0061150.0026672.2929390.0265

R-squared0.122164Mean dependent var0.032969

Adjusted R-squared0.064913S.D. dependent var0.020478

S.E. of regression0.019802Akaike info criterion-4.929452

Sum squared resid0.018037Schwarz criterion-4.776490

Log likelihood127.2363F-statistic2.133853

Durbin-Watson stat1.873090Prob(F-statistic)0.108850

The next test shows that the series is stationary.

Null Hypothesis: D(LOG_RGNP) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-6.2534660.0000

Test critical values:1% level-4.152511

5% level-3.502373

10% level-3.180699

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LOG_RGNP,2)

Method: Least Squares

Date: 05/24/15 Time: 13:13

Sample (adjusted): 3 52

Included observations: 50 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(LOG_RGNP(-1))-0.8835020.141282-6.2534660.0000

C0.0310180.0081883.7882980.0004

@TREND(1)-7.45E-050.000205-0.3643250.7172

R-squared0.455256Mean dependent var-0.000728

Adjusted R-squared0.432075S.D. dependent var0.027483

S.E. of regression0.020711Akaike info criterion-4.858151

Sum squared resid0.020161Schwarz criterion-4.743429

Log likelihood124.4538F-statistic19.63950

Durbin-Watson stat1.884766Prob(F-statistic)0.000001

Log_labours first unit root test shows that the series is not stationary as t-Stat is higher than 1% level as shown in the table below

Null Hypothesis: LOG_LABOUR has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-2.3512050.3997

Test critical values:1% level-4.152511

5% level-3.502373

10% level-3.180699

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LOG_LABOUR)

Method: Least Squares

Date: 05/24/15 Time: 13:14

Sample (adjusted): 3 52

Included observations: 50 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

LOG_LABOUR(-1)-0.1756160.074692-2.3512050.0231

D(LOG_LABOUR(-1))0.4032140.1421722.8360990.0068

C0.7379340.3087712.3899100.0210

@TREND(1)0.0029230.0012672.3076350.0256

R-squared0.185441Mean dependent var0.015320

Adjusted R-squared0.132317S.D. dependent var0.012963

S.E. of regression0.012075Akaike info criterion-5.918781

Sum squared resid0.006707Schwarz criterion-5.765819

Log likelihood151.9695F-statistic3.490748

Durbin-Watson stat1.894808Prob(F-statistic)0.022964

But after the 1st difference we can see that it is stationary.

Null Hypothesis: D(LOG_LABOUR) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-5.3389680.0003

Test critical values:1% level-4.156734

5% level-3.504330

10% level-3.181826

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LOG_LABOUR,2)

Method: Least Squares

Date: 05/24/15 Time: 13:14

Sample (adjusted): 4 52

Included observations: 49 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(LOG_LABOUR(-1))-0.9125500.170922-5.3389680.0000

D(LOG_LABOUR(-1),2)0.2718930.1427061.9052640.0631

C0.0154680.0049003.1567870.0028

@TREND(1)-5.68E-050.000126-0.4491360.6555

R-squared0.408022Mean dependent var-0.000103

Adjusted R-squared0.368557S.D. dependent var0.015634

S.E. of regression0.012424Akaike info criterion-5.860332

Sum squared resid0.006946Schwarz criterion-5.705897

Log likelihood147.5781F-statistic10.33878

Durbin-Watson stat1.998336Prob(F-statistic)0.000027

Log_capital unit root test level shows that the series is not stationary.

Null Hypothesis: LOG_CAPITAL has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 2 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-1.3936000.8506

Test critical values:1% level-4.156734

5% level-3.504330

10% level-3.181826

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LOG_CAPITAL)

Method: Least Squares

Date: 05/24/15 Time: 13:15

Sample (adjusted): 4 52

Included observations: 49 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

LOG_CAPITAL(-1)-0.0172500.012378-1.3936000.1704

D(LOG_CAPITAL(-1))0.9837800.1351007.2818490.0000

D(LOG_CAPITAL(-2))-0.3951880.135135-2.9243960.0054

C0.1710970.1112891.5374140.1314

@TREND(1)0.0004080.0003841.0615370.2942

R-squared0.766794Mean dependent var0.031118

Adjusted R-squared0.745594S.D. dependent var0.006744

S.E. of regression0.003402Akaike info criterion-8.432702

Sum squared resid0.000509Schwarz criterion-8.239659

Log likelihood211.6012F-statistic36.16863

Durbin-Watson stat1.874476Prob(F-statistic)0.000000

The 1st difference shown in the table below shows that the series is not stationary.

Null Hypothesis: D(LOG_CAPITAL) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-3.9632980.0165

Test critical values:1% level-4.156734

5% level-3.504330

10% level-3.181826

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LOG_CAPITAL,2)

Method: Least Squares

Date: 05/24/15 Time: 13:16

Sample (adjusted): 4 52

Included observations: 49 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(LOG_CAPITAL(-1))-0.4110390.103711-3.9632980.0003

D(LOG_CAPITAL(-1),2)0.4162420.1356863.0676750.0036

C0.0161160.0042643.7798080.0005

@TREND(1)-0.0001244.72E-05-2.6193190.0120

R-squared0.291777Mean dependent var-0.000231

Adjusted R-squared0.244563S.D. dependent var0.003954

S.E. of regression0.003437Akaike info criterion-8.430325

Sum squared resid0.000532Schwarz criterion-8.275891

Log likelihood210.5430F-statistic6.179783

Durbin-Watson stat1.863267Prob(F-statistic)0.001312

Upon computing the 2nd difference we can see that the series is stationary.

Null Hypothesis: D(LOG_CAPITAL,2) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-StatisticProb.*

Augmented Dickey-Fuller test statistic-6.2689110.0000

Test critical values:1% level-4.161144

5% level-3.506374

10% level-3.183002

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LOG_CAPITAL,3)

Method: Least Squares

Date: 05/24/15 Time: 13:17

Sample (adjusted): 5 52

Included observations: 48 after adjustments

VariableCoefficientStd. Errort-StatisticProb.

D(LOG_CAPITAL(-1),2)-1.0972580.175032-6.2689110.0000

D(LOG_CAPITAL(-1),3)0.3824240.1391292.7486940.0086

C-0.0001830.001195-0.1535460.8787

@TREND(1)-2.06E-063.88E-05-0.0530890.9579

R-squared0.486542Mean dependent var7.64E-05

Adjusted R-squared0.451534S.D. dependent var0.005023

S.E. of regression0.003720Akaike info criterion-8.270737

Sum squared resid0.000609Schwarz criterion-8.114804

Log likelihood202.4977F-statistic13.89784

Durbin-Watson stat2.104704Prob(F-statistic)0.000002

We have generated a regression with the logarithms and we can notice r-squared having almost a perfect value (0.99 almost 1) and durbin Watson has a low value resulting in a spurious regression.

Dependent Variable: LOG_RGNP

Method: Least Squares

Date: 05/24/15 Time: 13:27

Sample: 1 52

Included observations: 52

LOG_RGNP=C(1)+C(2)*LOG_CAPITAL+C(3)*LOG_LABOUR

CoefficientStd. Errort-StatisticProb.

C(1)-1.5066650.140143-10.750880.0000

C(2)0.8678480.0905699.5821270.0000

C(3)0.3636430.1684402.1588910.0358

R-squared0.996586Mean dependent var8.664846

Adjusted R-squared0.996447S.D. dependent var0.502731

S.E. of regression0.029967Akaike info criterion-4.121487

Sum squared resid0.044003Schwarz criterion-4.008915

Log likelihood110.1587Durbin-Watson stat0.285118

The spurious regression resulted because of the fact that log_rgnp and log_labour variables are stationary in 1st difference, but log_capital is not stationary in 1st difference.

Regression using differences of logarithmsUpon performing the second regression we receive the following:

Dependent Variable: DIF_LOG_RGNP

Method: Least Squares

Date: 05/24/15 Time: 13:44

Sample (adjusted): 2 52

Included observations: 51 after adjustments

DIF_LOG_RGNP=C(1)+C(2)*DIF_LOG_LABOUR+C(3)

*DIF_LOG_CAPITAL

CoefficientStd. Errort-StatisticProb.

C(1)0.0032130.0092050.3490310.7286

C(2)1.1449800.1667676.8657470.0000

C(3)0.4025470.3184641.2640270.2123

R-squared0.611766Mean dependent var0.033678

Adjusted R-squared0.595590S.D. dependent var0.020894

S.E. of regression0.013287Akaike info criterion-5.746996

Sum squared resid0.008474Schwarz criterion-5.633360

Log likelihood149.5484Durbin-Watson stat1.856282

The regression is not spurious because R-squared is not perfect anymore and moreover Durbin Watson has a decent value.We see now if the production function has constant increasing or decreasing returns to scale.

Wald Test:

Equation: REGRESSION_2

Test StatisticValuedfProbability

F-statistic3.897829(1, 48)0.0541

Chi-square3.89782910.0483

Null Hypothesis Summary:

Normalized Restriction (= 0)ValueStd. Err.

-1 + C(2) + C(3)0.5475260.277328

Restrictions are linear in coefficients.

Increasing returns to scale after performing the Wald test f statistic compared to 1:>1 increasing