The following example was used by Zellner in his classic 1962 paper on seemingly unrelated regressions. Different stock prices often move in the same direction at a given point in time. The SUR technique might provide more efficient estimates than OLS in this situation.
The following statements read the data. (The prefix GE stands for General Electric and WH stands for Westinghouse.)
*---------Zellner's Seemingly Unrelated Technique------------* | A. Zellner, "An Efficient Method of Estimating Seemingly | | Unrelated Regressions and Tests for Aggregation Bias," | | JASA 57(1962) pp.348-364 | | | | J.C.G. Boot, "Investment Demand: an Empirical Contribution | | to the Aggregation Problem," IER 1(1960) pp.3-30. | | | | Y. Grunfeld, "The Determinants of Corporate Investment," | | Unpublished thesis, Chicago, 1958 | *------------------------------------------------------------*; data grunfeld; input year ge_i ge_f ge_c wh_i wh_f wh_c; label ge_i = 'Gross Investment, GE' ge_c = 'Capital Stock Lagged, GE' ge_f = 'Value of Outstanding Shares Lagged, GE' wh_i = 'Gross Investment, WH' wh_c = 'Capital Stock Lagged, WH' wh_f = 'Value of Outstanding Shares Lagged, WH'; datalines; 1935 33.1 1170.6 97.8 12.93 191.5 1.8 ... more lines ...
The following statements compute the SUR estimates for the Grunfeld model.
proc syslin data=grunfeld sur; ge: model ge_i = ge_f ge_c; westing: model wh_i = wh_f wh_c; run;
The PROC SYSLIN output is shown in Output 29.2.1 through Output 29.2.5.
Output 29.2.1: PROC SYSLIN Output for SUR
Model | GE |
---|---|
Dependent Variable | ge_i |
Label | Gross Investment, GE |
Analysis of Variance | |||||
---|---|---|---|---|---|
Source | DF | Sum of Squares | Mean Square | F Value | Pr > F |
Model | 2 | 31632.03 | 15816.02 | 20.34 | <.0001 |
Error | 17 | 13216.59 | 777.4463 | ||
Corrected Total | 19 | 44848.62 |
Root MSE | 27.88272 | R-Square | 0.70531 |
---|---|---|---|
Dependent Mean | 102.29000 | Adj R-Sq | 0.67064 |
Coeff Var | 27.25850 |
Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Variable | DF | Parameter Estimate |
Standard Error | t Value | Pr > |t| | Variable Label |
Intercept | 1 | -9.95631 | 31.37425 | -0.32 | 0.7548 | Intercept |
ge_f | 1 | 0.026551 | 0.015566 | 1.71 | 0.1063 | Value of Outstanding Shares Lagged, GE |
ge_c | 1 | 0.151694 | 0.025704 | 5.90 | <.0001 | Capital Stock Lagged, GE |
Output 29.2.2: PROC SYSLIN Output for SUR
Model | WESTING |
---|---|
Dependent Variable | wh_i |
Label | Gross Investment, WH |
Analysis of Variance | |||||
---|---|---|---|---|---|
Source | DF | Sum of Squares | Mean Square | F Value | Pr > F |
Model | 2 | 5165.553 | 2582.776 | 24.76 | <.0001 |
Error | 17 | 1773.234 | 104.3079 | ||
Corrected Total | 19 | 6938.787 |
Root MSE | 10.21312 | R-Square | 0.74445 |
---|---|---|---|
Dependent Mean | 42.89150 | Adj R-Sq | 0.71438 |
Coeff Var | 23.81153 |
Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Variable | DF | Parameter Estimate |
Standard Error | t Value | Pr > |t| | Variable Label |
Intercept | 1 | -0.50939 | 8.015289 | -0.06 | 0.9501 | Intercept |
wh_f | 1 | 0.052894 | 0.015707 | 3.37 | 0.0037 | Value of Outstanding Shares Lagged, WH |
wh_c | 1 | 0.092406 | 0.056099 | 1.65 | 0.1179 | Capital Stock Lagged, WH |
Output 29.2.3: PROC SYSLIN Output for SUR
Cross Model Covariance | ||
---|---|---|
GE | WESTING | |
GE | 777.446 | 207.587 |
WESTING | 207.587 | 104.308 |
Cross Model Correlation | ||
---|---|---|
GE | WESTING | |
GE | 1.00000 | 0.72896 |
WESTING | 0.72896 | 1.00000 |
Cross Model Inverse Correlation | ||
---|---|---|
GE | WESTING | |
GE | 2.13397 | -1.55559 |
WESTING | -1.55559 | 2.13397 |
Cross Model Inverse Covariance | ||
---|---|---|
GE | WESTING | |
GE | 0.002745 | -.005463 |
WESTING | -.005463 | 0.020458 |
Output 29.2.4: PROC SYSLIN Output for SUR
System Weighted MSE | 0.9719 |
---|---|
Degrees of freedom | 34 |
System Weighted R-Square | 0.6284 |
Model | GE |
---|---|
Dependent Variable | ge_i |
Label | Gross Investment, GE |
Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Variable | DF | Parameter Estimate |
Standard Error | t Value | Pr > |t| | Variable Label |
Intercept | 1 | -27.7193 | 29.32122 | -0.95 | 0.3577 | Intercept |
ge_f | 1 | 0.038310 | 0.014415 | 2.66 | 0.0166 | Value of Outstanding Shares Lagged, GE |
ge_c | 1 | 0.139036 | 0.024986 | 5.56 | <.0001 | Capital Stock Lagged, GE |
Output 29.2.5: PROC SYSLIN Output for SUR
Model | WESTING |
---|---|
Dependent Variable | wh_i |
Label | Gross Investment, WH |
Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Variable | DF | Parameter Estimate |
Standard Error | t Value | Pr > |t| | Variable Label |
Intercept | 1 | -1.25199 | 7.545217 | -0.17 | 0.8702 | Intercept |
wh_f | 1 | 0.057630 | 0.014546 | 3.96 | 0.0010 | Value of Outstanding Shares Lagged, WH |
wh_c | 1 | 0.063978 | 0.053041 | 1.21 | 0.2443 | Capital Stock Lagged, WH |