## Posts Tagged ‘two stage least squares regression’

### Forecast Friday Topic: Structural and Reduced Forms

November 18, 2010

(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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### Forecast Friday Topic: The Identification Problem

November 11, 2010

(Twenty-ninth in a series)

When we work with regression analysis, it is assumed that outside factors determine each of the independent variables in the model; these factors are said to be exogenous to the system. This is especially of interest to economists, who have long used econometric models to forecast demand and supply for various goods. The price the market will bear for a good or service, for example, is not determined by a single equation, but by the interaction of the equations for both supply and demand. If price was what we were trying to forecast, then a single equation would do us little good. In fact, since price is part of a multi-equation system, performing regression analysis for just demand without supply or vice-versa will result in biased parameter estimates.

This post begins our three-part “series within a series” on “Simultaneous Equations and Two-Stage Least Squares Regression”. Although this topic sounds intimidating, I will not be covering it in much technical detail. My purpose in discussing it is to make you aware of these concepts, so that you can determine when to look beyond a simple regression analysis.

Hence, we start with the most basic concept of simultaneous equations: the Identification problem. Let’s assume that you are the supply chain manager for a beer company. You need to forecast the price of barley, so your company can budget how much money it needs to spend in order to have enough barley to produce its beer; determine whether the price is on an upward trend, so that it could purchase derivatives to hedge its risk; and determine the final price for its beer.

You have statistics for the price and traded quantity of barley for the last several years. You also remember three concepts from your college economics class:

1. The price and quantity supplied of a good have a direct relationship – producers supply more as the price goes up and less as the price goes down;
2. The price and quantity demanded of a good have an inverse relationship – consumers purchase less as the price goes up and vice-versa; and
3. The market price is determined by the interaction of the supply and demand equations.

Since price and quantity are positively sloped for supply and negatively sloped for demand, with only the two variables of quantity and price, you cannot determine – that is identify – the supply and demand equations using regression analysis; the information is insufficient. However, if you can identify variables that are in one equation and not the other, you will be able to identify the individual relations.

In agriculture, the supply of a crop is greatly affected by weather. If you can obtain information on the amount of rainfall in barley producing regions during the years for which you have data, you might be able to identify the different equations. Moreover, production costs impact supply. So if you can obtain information on the costs of planting and harvesting the barley, that too would help. On the demand side, barley’s quantity can be influenced by changes in tastes. If beer demand goes up, so too will the demand for barley; if farm animal raising increases, farmers may need to purchase more barley for animal fodder; and various health fads may emerge, increasing the demands for barley breads and soups. If you can obtain these kinds of information, you are on your way to identifying the supply and demand curves.

Exogenous and Endogenous Variables

Since rainfall affects the supply of barley, but the barley market does not influence the amount of rainfall, rainfall is said to be an exogenous variable, because its value is determined by factors outside of the equation system. Since the demand for beer helps derive the demand for barley, but not the other way around, beer demand is an exogenous variable.

Because price and quantity of barley are part of a demand and supply system, they are determined by the interaction of the two equations – that is by the equation system – so they are said to be endogenous variables.

Identifying an Equation

If you are trying to identify an equation that is part of a multi-equation system, you must have a minimum of one less variable than you do equations excluded from that equation. Hence, if you have a two-equation system, you must have at least one variable excluded from the model you’re trying to identify, that is included in the other equation; if your system has three equations, you need to have at least two variables excluded from the model you want to identify, and so on.

When you have just enough exogenous variables in one equation that is not in the other equation(s), then your equation is just identified. You can use several econometric techniques to estimate just identified systems, however they are quite rare in practice. When you have no exogenous variables that are unique to one equation in the system, your equations are under identified and cannot be estimated with any econometric techniques. Most often, equations are over identified, because there are more exogenous variables excluded from one equation than required by the number of equations in the system. When over identification is the case, then two-stage least squares (the topic of the third post of this miniseries) is required in order to tell which of the variables is causing your supply (or demand) curve to shift along the fixed demand (or supply) curve.

Next Forecast Friday Topic: Structural and Reduced Forms

Next week’s Forecast Friday topic builds on today’s topic with a discussion of structural and reduced forms of equations. These are the first steps in Two-Stage Least Squares Regression analysis, and are part of the effort to solve the identification problem.

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