## Forecast Friday Topic: Two-Stage Least Squares

(Thirty-first in a series)

In the last two Forecast Friday posts, we set up the discussion of how to solve systems of simultaneous equations when forecasting by first mentioning the identification problem and then breaking equation systems down from structural into reduced forms. Today we finalize that discussion with a talk about performing two-stage least squares (2SLS) regression. Quite often, when we forecast, we build one regression model using one or more variables as our explanatory variables and another as our dependent variable. However, many of our explanatory variables may in fact be dependent variables in another regression model that is highly related to the model we develop, particularly if one or more of our explanatory variables is contained in those other regressions. Because of this, our results may suffer from simultaneous equation bias. To minimize this bias, 2SLS regression is necessary.

Two-Stage Least Squares

As mentioned in the last two posts, I will not be going into too much mathematical detail in our discussions of simultaneous equations and 2SLS. This is a brief theoretical discussion to help you recognize situations that warrant a 2SLS approach. In our last post, we talked about how to generate the reduced form of the equations in our system. Hence the first stage of 2SLS is:

Perform OLS on the Reduced Form Equations

Recall our discussion on endogenous and exogenous variables. For each endogenous variable in your system, you must have one reduced form equation, and each reduced form equation must have all its exogenous variables on the right side of the equation. While tedious, this process isn’t always as tedious as it could be. If in your system, you have seven endogenous variables, you do not need to run OLS on all seven of the resulting reduced form equations, only on those whose endogenous variables appear on the right side of the structural equations you want to estimate. Hence, if you’re trying to forecast consumption (an endogenous variable), of which disposable income (another endogenous variable) is an independent variable, then you would need only perform OLS on the reduced form equation for disposable income, since it is the only endogenous variable that appears on the right side of the consumption equation. You do not need to do OLS on both variables’ reduced form.

Performing OLS on the reduced form equations gives you the fitted values to use in the second-stage regressions. Only the R2 and fitted values of each equation are the important pieces of information provided by the first stage. T-ratios are of no value, since the likelihood of significant multicollinearity will be strong. But the R2 statistic is important. A low R2 suggests little or no correlation between the fitted values and the endogenous variables. The fitted values, after all, are intended to replace the endogenous variables, so you want a high correlation, via a high R2. Furthermore, a low R2 can lead to biased standard errors of the parameter estimates in the second stage, resulting in coefficients that are inefficient. Hence, a correction factor for the standard errors is necessary.

Once you have performed OLS on the reduced form equations, the next stage of 2SLS is:

Perform OLS on the Structural Equations

If you performed OLS on the reduced form equation for disposable income, you would then substitute the fitted values for each regression into value of disposable income for the structural consumption equation. At this stage, everything works like regular OLS. However, you must be on heightened alert for autocorrelation, since you are using time series data, and because you are using several time series equations, which can increase the likelihood of autocorrelation.

Next Forecast Friday Topic: Leading Economic Indicators

In next week’s Forecast Friday post, we will discuss some of the more interesting aspects of economic forecasting, leading economic indicators. We often hear about leading economic indicators in the news, and we will be discussing the theory behind them, and the role that expectations play. We will see how leading economic indicators impact forecasting as well. Don’t miss it!

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