Two-Stage Least Squares Fit


Description Usage Options Output Method See Also

Description

The Two-Stage Least Squares Fit can be used in place of the Ordinary Least Squares in cases where the error term resulting from the regression is correlated with one of the explanatory variables.  Two-Stage Least Squares uses instrumental variables to generate proxy variables for the regression of interest.  The regression is performed in two stages.

Stage 1 determines the proxy variables by performing an ordinary least squares regression of the variable for which a proxy is being determined using the instrumental variables as the explanatory variables in the regression.  

Stage 2 preforms an ordinary least squares regression using the original regression specification, but replacing all non-instrumental variables with their respective proxies.  

Two-Stage Least Squares will eliminate biases occurring in coefficients due to the violation of the Least Squares Fit assumption of an uncorrelated error term.  However, a bias may still be present in the coefficients for small sample sizes.

Usage

Two-Stage Least Squares Window

To perform a two-stage least squares regression, the appropriate data must be chosen.  To select a variable as either dependent or independent, click the check box associated with each in the least squares window.  Only one variable may be chosen as the dependent variable.

Compute Regression Menu
After selecting the desired options from the "Options" menu, the regression can be performed by selecting "Compute Regression..." from the "Perform Fit" menu.  Once performed, a text window will open, showing the results of the regression.

After performing the regression, the independent and dependent check boxes in the least squares window will no longer be changeable.  Also, the "Options" menu associated with this regression will be deactivated.  The results window can be closed at any time; to reopen the results, simply select the "Compute Regression..." menu item again.

Options

Variances

The least squares fit allows for two types of variances to be computed for output. The Ordinary Least Squares variances are determined from the product of the variance of the residuals and the solution matrix, (X'X)-1.  The White's Robust Variance Estimates compute the variances based on a more conservative equation.

Include Constant

By unchecking this box, the constant term is omitted from the regression procedure.

Ignore Matrix Condition

Selecting this option disables the safety check for matrix conditioning prior to attempting to solve the least-squares problem.  More information can be found on the Condition page.  The solution matrix tends to appear poorly condition when colinearity exists in series.  The consequence of proceeding when conditioning is poor can range from highly-biased results to complete software failure.  This option, however, tends to be common when dealing with Two-Stage Least Squares Fits due to the occurrence of correlated explanatory variables often appearing during the Stage 1 regression(s).

Output

The two-stage least squares fit will generate a table of coefficient values as well as the errors associated with each.  The results of the fit, including the R2 value are also output.  Independent variables that are not marked as instruments will be labeled with a "_HAT" suffix, denoting that a proxy variable was used in stage 2 of the regression process.  An example from a least squares fit is shown below:

Results of the Two-Stage Least Squares Regression
Regression Variable: CO
---------------------------------------------------------------
Sum Squared of the Residuals:  3.3633E04
Standard Error of the Fit   :    34.6582
R-Squared Value             :    0.99723
Adjusted R-Squared Value    :    0.99703
---------------------------------------------------------------
Constant: -145.29034    Std Err:   29.87929    t-Score:   -4.86258
Coef 0 (CO_lagged):     0.0308    Std Err:    0.02789    t-Score:    1.10442
Coef 1 (YD_HAT):    0.91049    Std Err:    0.03041    t-Score:   29.93612

Further information can be generated after the regression.  By selecting "Generate Column Data" from the "Perform Fit" menu, the estimated dependent variable data will be output as a new variable in the Data window.  The resulting covariance matrix can be viewed by selecting "View Covariance Matrix..." from the "Supporting Data Menu."  The residuals from the regression can also be output to the Data window as a new variable by selecting "Output Residuals to Column" from the "Supporting Data Menu."

Method

The Two-Stage Least Squares Fit makes use of the Least Squares Fit procedure for each stage.  In stage 1, any endogenous, non-instrument variables are regressed against all instruments.  The resulting regressions from stage 1 are used to generate complete proxy variables, which are used in the stage 2 regression.  All residuals are recomputed with respect to the true variables (not the proxy variables); these residuals are used in the calculation of the covariance matrix and all derived values.

See Also

Least Squares Fit
Iteration Controls

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