Two-Stage Least Squares Estimation

The SYSLIN Procedure

Two-Stage Least Squares Estimation

In the supply and demand model, P is an endogenous variable, and consequently the OLS estimates are biased. The following example estimates this model using two-stage least squares:


   proc syslin data=in 2sls;
      endogenous  p;
      instruments y u s;
      demand: model q = p y s;
      supply: model q = p u;
   run;

The 2SLS option on the PROC SYSLIN statement specifies the two-stage least-squares method. The ENDOGENOUS statement specifies that P is an endogenous regressor for which first-stage predicted values are substituted. You only need to declare an endogenous variable in the ENDOGENOUS statement if it is used as a regressor; thus although Q is endogenous in this model, it is not necessary to list it in the ENDOGENOUS statement.

Usually, all predetermined variables that appear in the system are used as instruments. The INSTRUMENTS statement specifies that the exogenous variables Y, U, and S are used as instruments for the first-stage regression to predict P.

The 2SLS results are shown in Figure 19.3 and Figure 19.4. The first-stage regressions are not shown. To see the first-stage regression results, use the FIRST option on the MODEL statement.

The SYSLIN Procedure

Two-Stage Least Squares Estimation

Model	DEMAND
Dependent Variable	q
Label	Quantity

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	3	9.670882	3.223627	115.58	<.0001
Error	56	1.561944	0.027892
Corrected Total	59	10.03723

Root MSE	0.16701	R-Square	0.86095
Dependent Mean	1.30095	Adj R-Sq	0.85350
Coeff Var	12.83740

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|	Variable Label
Intercept	1	1.901040	1.171224	1.62	0.1102	Intercept
p	1	-1.11518	0.607391	-1.84	0.0717	Price
y	1	0.419544	0.117954	3.56	0.0008	Income
s	1	0.331475	0.088472	3.75	0.0004	Price of Substitutes

Figure 19.3: 2SLS Results for Demand Equation

The SYSLIN Procedure

Two-Stage Least Squares Estimation

Model	SUPPLY
Dependent Variable	q
Label	Quantity

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	2	9.646098	4.823049	253.96	<.0001
Error	57	1.082503	0.018991
Corrected Total	59	10.03723

Root MSE	0.13781	R-Square	0.89910
Dependent Mean	1.30095	Adj R-Sq	0.89556
Coeff Var	10.59291

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|	Variable Label
Intercept	1	-0.51878	0.490999	-1.06	0.2952	Intercept
p	1	1.333080	0.059271	22.49	<.0001	Price
u	1	-1.14623	0.243491	-4.71	<.0001	Unit Cost

Figure 19.4: 2SLS Results for Supply Equation

The 2SLS output is similar in form to the OLS output. However, the 2SLS results are based on predicted values for the endogenous regressors from the first stage instrumental regressions. This makes the analysis of variance table and the R² statistics difficult to interpret. See the sections "ANOVA Table for Instrumental Variables Methods" and "The R² Statistics" later in this chapter for details.

Note that, unlike the OLS results, the 2SLS estimate for the P coefficient in the demand equation (-1.115) is negative.

Chapter Contents
Previous
Next
Top