SUR.RPF

SUR.RPF is an example of (linear) systems estimation. It computes joint GLS estimates for a two company subset for Grunfeld’s investment equations.

The Grunfeld data is a common “textbook” data set for small panels. The full data set has annual data from 1935 to 1954 on 11 companies with investment (the dependent variable), capital stock and the firm’s market value (explanatory variables). The two countries included in the example are GE and Westinghouse.

Because the instruction SUR operates off EQUATIONs, the following are used to define them:

equation geeq ige

# constant fge cge

equation westeq iwest

# constant fwest cwest

Since this is also doing OLS estimates, it’s also possible to estimate by LINREG and define the define the equation at the same time:

linreg(define=geeq) ige

# constant fge cge

If the EQUATIONs have already been defined, the LINREGs can be done with:

linreg(equation=geeq) ige

linreg(equation=westeq) iwest

There are two ways to get the description of the overall model into the SUR instruction: with supplementary cards or with a MODEL. With just two equations, the former is probably easier, but we’ll show both. The unrestricted SUR can be done with

sur(vcv) 2

# geeq

# westeq

or with

group grunfeld geeq westeq

sur(model=grunfeld)

This uses RESTRICT to test for equality between the “F” (market value) coefficients. In the stacked coefficient vector, the FGE coefficient is position 2 and the FWEST is position 5:

restrict(title="Test of equality of F coefficients") 1

# 2 5

# 1 -1 0

and this estimates the system subject to equality constraints across all the equations by using the EQUATE option:

sur(model=grunfeld,equate=||2,3||)

which forces the second (Fxx) and third (Cxx) coefficients to be equal across equations. Note that this rearranges the coefficient vector, so the coefficients which are forced equal are listed first. In this case, there are actually only four free coefficients: Fxx, Cxx, CONSTANT in GE and CONSTANT in WEST. The output will display the information in the usual equation by equation form, but you will note that the coefficient numbers are out-of-sequence.

Full Program

cal 1935
open data grunfeld.wks
data(org=obs,format=wks) 1935:1 1954:1 ige fge cge iwest fwest cwest
*
* Define equations
*
equation geeq ige
# constant fge cge
equation westeq iwest
# constant fwest cwest
*
* Estimate by OLS
*
linreg(equation=geeq) ige
linreg(equation=westeq) iwest
*
* Now by SUR
*
sur(vcv) 2
# geeq
# westeq
*
* Same thing defining a MODEL
*
group grunfeld geeq westeq
sur(model=grunfeld)
*
* Test that the "F" coefficients are identical
*
restrict(title="Test of equality of F coefficients") 1
# 2 5
# 1 -1 0
*
* Forcing the 2nd and 3rd coefficients to be equal across equations.
*
sur(model=grunfeld,equate=||2,3||)

Output

Linear Regression - Estimation by Least Squares

Dependent Variable IGE

Annual Data From 1935:01 To 1954:01

Usable Observations 20

Degrees of Freedom 17

Centered R^2 0.7053067

R-Bar^2 0.6706369

Uncentered R^2 0.9479894

Mean of Dependent Variable 102.29000000

Std Error of Dependent Variable 48.58449937

Standard Error of Estimate 27.88272475

Sum of Squared Residuals 13216.587770

Regression F(2,17) 20.3435

Significance Level of F 0.0000309

Log Likelihood -93.3137

Durbin-Watson Statistic 1.0721

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -9.95630645 31.37424914 -0.31734 0.75484994

2. FGE 0.02655119 0.01556610 1.70571 0.10626510

3. CGE 0.15169387 0.02570408 5.90155 0.00001742

Linear Regression - Estimation by Least Squares

Dependent Variable IWEST

Annual Data From 1935:01 To 1954:01

Usable Observations 20

Degrees of Freedom 17

Centered R^2 0.7444461

R-Bar^2 0.7143810

Uncentered R^2 0.9594526

Mean of Dependent Variable 42.891500000

Std Error of Dependent Variable 19.110188596

Standard Error of Estimate 10.213122845

Sum of Squared Residuals 1773.2339304

Regression F(2,17) 24.7611

Significance Level of F 0.0000092

Log Likelihood -73.2271

Durbin-Watson Statistic 1.4130

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -0.509390184 8.015288941 -0.06355 0.95006800

2. FWEST 0.052894126 0.015706501 3.36766 0.00365476

3. CWEST 0.092406492 0.056098974 1.64720 0.11787433

Linear Systems - Estimation by Seemingly Unrelated Regressions

Iterations Taken 2

Annual Data From 1935:01 To 1954:01

Usable Observations 20

Log Likelihood -158.3196

Dependent Variable IGE

Mean of Dependent Variable 102.29000000

Std Error of Dependent Variable 48.58449937

Standard Error of Estimate 26.25678563

Sum of Squared Residuals 13788.375833

Durbin-Watson Statistic 0.9856

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -27.71931712 27.03282800 -1.02539 0.30517701

2. FGE 0.03831021 0.01329011 2.88261 0.00394396

3. CGE 0.13903627 0.02303559 6.03572 0.00000000

Dependent Variable IWEST

Mean of Dependent Variable 42.891500000

Std Error of Dependent Variable 19.110188596

Standard Error of Estimate 9.490260477

Sum of Squared Residuals 1801.3008785

Durbin-Watson Statistic 1.3647

Variable Coeff Std Error T-Stat Signif

************************************************************************************

4. Constant -1.251988228 6.956346688 -0.17998 0.85716997

5. FWEST 0.057629796 0.013411012 4.29720 0.00001730

6. CWEST 0.063978067 0.048900998 1.30832 0.19076540

Covariance\Correlation Matrix of Coefficients

Constant FGE CGE Constant FWEST CWEST

Constant 730.7737897 -0.91648771 -0.23484453 0.67515453 -0.62391596 0.29692502

FGE -0.3292660 0.0001766 -0.11121555 -0.56990847 0.67323119 -0.49976868

CGE -0.1462417 -0.0000340 0.0005306 -0.24725669 -0.05461557 0.52805012

Constant 126.9626218 -0.0526884 -0.0396213 48.3907592 -0.85776714 0.33400076

FWEST -0.2261930 0.0001200 -0.0000169 -0.0800225 0.0001799 -0.72367292

CWEST 0.3925148 -0.0003248 0.0005948 0.1136178 -0.0004746 0.0023913

Covariance\Correlation Matrix of Residuals

IGE IWEST

IGE 689.41879166 0.76504294

IWEST 190.63625609 90.06504392

Linear Systems - Estimation by Seemingly Unrelated Regressions

Iterations Taken 2

Annual Data From 1935:01 To 1954:01

Usable Observations 20

Log Likelihood -158.3196

Dependent Variable IGE

Mean of Dependent Variable 102.29000000

Std Error of Dependent Variable 48.58449937

Standard Error of Estimate 26.25678563

Sum of Squared Residuals 13788.375833

Durbin-Watson Statistic 0.9856

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -27.71931712 27.03282800 -1.02539 0.30517701

2. FGE 0.03831021 0.01329011 2.88261 0.00394396

3. CGE 0.13903627 0.02303559 6.03572 0.00000000

Dependent Variable IWEST

Mean of Dependent Variable 42.891500000

Std Error of Dependent Variable 19.110188596

Standard Error of Estimate 9.490260477

Sum of Squared Residuals 1801.3008785

Durbin-Watson Statistic 1.3647

Variable Coeff Std Error T-Stat Signif

************************************************************************************

4. Constant -1.251988228 6.956346688 -0.17998 0.85716997

5. FWEST 0.057629796 0.013411012 4.29720 0.00001730

6. CWEST 0.063978067 0.048900998 1.30832 0.19076540

Covariance\Correlation Matrix of Residuals

IGE IWEST

IGE 689.41879166 0.76504294

IWEST 190.63625609 90.06504392

Test of equality of F coefficients

Chi-Squared(1)= 3.203911 with Significance Level 0.07346239

Linear Systems - Estimation by Seemingly Unrelated Regressions

Iterations Taken 2

Annual Data From 1935:01 To 1954:01

Usable Observations 20

Log Likelihood -160.6472

Dependent Variable IGE

Mean of Dependent Variable 102.29000000

Std Error of Dependent Variable 48.58449937

Standard Error of Estimate 26.03373641

Sum of Squared Residuals 13555.108631

Durbin-Watson Statistic 1.0062

Variable Coeff Std Error T-Stat Signif

************************************************************************************

3. Constant -22.47292135 18.95280698 -1.18573 0.23572874

1. FGE 0.03521313 0.00808344 4.35621 0.00001323

2. CGE 0.14095059 0.02296517 6.13758 0.00000000

Dependent Variable IWEST

Mean of Dependent Variable 42.891500000

Std Error of Dependent Variable 19.110188596

Standard Error of Estimate 9.761225716

Sum of Squared Residuals 1905.6305498

Durbin-Watson Statistic 1.4008

Variable Coeff Std Error T-Stat Signif

************************************************************************************

4. Constant 7.1956492033 6.1594381566 1.16823 0.24271343

1. FWEST 0.0352131317 0.0080834405 4.35621 0.00001323

2. CWEST 0.1409505908 0.0229651663 6.13758 0.00000000

Covariance\Correlation Matrix of Residuals

IGE IWEST

IGE 677.75543154 0.70480098

IWEST 179.10485379 95.28152749