Forecast Friday Topic: Structural and Reduced Forms

(Thirtieth in a series)

Last week, we discussed the identification problem – a common occurrence in forecasting when consistent estimators of the parameters of the equation or model in which we are interested don’t exist. We also discussed how identifying variables unique to one equation but not to the other is the first step to alleviating the identification problem.

Today, we will briefly discuss the next step in solving the identification problem: structural and reduced forms. Because the math can get a little complicated, we won’t be focusing on it here. Like last week, this week’s – and next week’s – post will be theoretical in nature.

Structural and reduced forms get their origin from matrix algebra and involve systems of equations. Indeed, the equations contained within a system are called structural equations because, together, they are developed to explain the hypothesized structure of a given market. Structural equations are based on economic theory and are used to derive the reduced form equations for two-stage least squares regression.

To derive the reduced form equations, one endogenous variable must be placed on the left side of the equation, while all exogenous variables must be placed on the right. You must have one reduced form equation for each endogenous variable present in the system. So, if your system of equations has five endogenous variables, then you must have five reduced form equations.

The process for reducing the form of the structural equations follows that of solving for a system of linear equations:

  1. Set one equation equal to another;
  2. Subtract the endogenous parameter term (estimate times variable) and error term from each side of equation;
  3. Factor both sides;
  4. Divide to solve for the endogenous variable. This gives you the first reduced form equation.
  5. Find the next reduced form equation by substituting the right side of the first reduced form equation into one of the original structural equations.

Essentially, it’s best to think of endogenous variables as dependent variables and of exogenous variables as independent variables; this way, you get the result of the reduced form having precisely the same format as multiple regression models. Given assumptions about future values of exogenous variables, the reduced form can facilitate computation of conditional forecasts of future values of the endogenous variables.

Forecast Friday Resumes Two Weeks From Today

Forecast Friday will not be published next Thursday, in observance of Thanksgiving.  We here at Analysights are very thankful for readers like you who check in every week, and look forward to your continued visits to Insight Central.  Our Forecast Friday post will resume two weeks from today, December 2, in which we will conclude our discussion of simulataneous equations with a post on Two-Stage Least Squares regression analysis.  We here at Analysights wishes you and your family a Happy Thanksgiving.

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