One of the most common problems complicating the research of an economist is created by the inclusion of endogenous variables as an explanatory variable. The variable on the left-hand-side of a regression is an endogenous variable; its level is determined by the levels of the explanatory variables—that is, the variables on the right-hand-side of the equation. In OLS we assume that the explanatory variables are independent of the error term. However, if the level of one of these explanatory variables is determined by the levels of the other variables in the model, that explanatory variable actually is an endogenous variable. In a nutshell the problem with having endogenous explanatory variables is that these endogenous variables cause the error term in the model to be correlated with the explanatory variables thus causing the OLS estimator to be biased. This problem is also known as simultaneous equation bias and it is a problem that is subtly different from sample selection bias. See "What is the difference between 'endogeneity' and 'sample selection bias"'?" for an excellent discussion of the difference between these two econometric problems.

In this module we explore both the statistical and algebraic issues raised by the inclusion of endogenous explanatory variables in a model. This introduction is too sketchy to give you a thorough understanding of the many problems raised by simultaneous equation bias. Hopefully, by the time you finish the module along with the problem set, you will have an least an intuitive understanding of the problem and will be able to recognize it when you come across the problem in your own research. If you think the model you are estimating may have simultaneous equation bias, you should seek the advice of an econometrician.

Imagine we know with certainty that the following model fully describes the true state of the supply and demand for wheat. First, the demand for wheat in any year, q_t, is a function of the price of wheat, p t w , p t w , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaWG0baabaGaam4DaaaakiaacYcaaaa@39C5@ the income of the average individual, I_t, and the price of corn, p t c . p t c . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaWG0baabaGaam4yaaaakiaac6caaaa@39B3@ Second, in any year the price of wheat is a function of the amount of wheat brought to market, q_t, and a weather index, W_t, that is positively related to the amount of wheat that is harvested. Third, the error terms in the supply and demand functions are due purely to measurement errors—that is, there are no omitted variables in the model. Thus, we have the following two equation model:

and

Supply:

p t w = β 0 + β 1 q t + β 2 W t + η t . p t w = β 0 + β 1 q t + β 2 W t + η t . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaWG0baabaGaam4Daaaakiabg2da9iabek7aInaaBaaaleaacaaIWaaabeaakiabgUcaRiabek7aInaaBaaaleaacaaIXaaabeaakiaadghadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaWGxbWaaSbaaSqaaiaadshaaeqaaOGaey4kaSIaeq4TdG2aaSbaaSqaaiaadshaaeqaaOGaaiOlaaaa@4C33@

We assume that the error terms each are normally distributed with a mean of zero and a constant variance. Moreover, we assume that the two error terms are independent of each other—that is, we are assuming that:

Finally, we assume that income, the price of corn, and the weather index are non-stochastic variables—i.e., these variables are independent of the two error terms. Clearly, the price of wheat and the quantity of wheat are stochastic variables.1

What we have here is an ideal model in the sense that we know and can measure all of the variables in the model. The model as written has two endogenous variables—q_t and p t w p t w MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaWG0baabaGaam4Daaaaaaa@390B@ —and three exogenous variables— I t , I t , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMeadaWgaaWcbaGaamiDaaqabaGccaGGSaaaaa@3896@ p t c , p t c , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaqhaaWcbaGaamiDaaqaaiaadogaaaGccaGGSaaaaa@39A6@ and W t . W t . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEfadaWgaaWcbaGaamiDaaqabaGccaGGUaaaaa@38A6@ Equations (1) and (2) are known as structural equations. What makes this model useful for our purposes is that there is an endogenous explanatory variable in each of the two structural equations.

What we ultimately want to know is if we can use ordinary least squares (OLS) to obtain unbiased estimates of the parameters in Equations (1) and (2). One of the assumptions of OLS is that each of the explanatory variables are independent of the error term, ε t ; ε t ; MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew7aLnaaBaaaleaacaWG0baabeaakiaacUdaaaa@397E@ if this assumption is violated, OLS will produce biased estimates of the slope parameters. Thus, what we need to do is see if the error term in each equation is independent of the endogenous variable on the right-hand-side of that equation. That is, we want to see if E( ε t p t w )=0 E( ε t p t w )=0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiabew7aLnaaBaaaleaacaWG0baabeaakiaadchadaqhaaWcbaGaamiDaaqaaiaadEhaaaaakiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@3FF3@ and E( η t q t )=0. E( η t q t )=0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiabeE7aOnaaBaaaleaacaWG0baabeaakiaadghadaWgaaWcbaGaamiDaaqabaaakiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaaaa@3FAE@

It is convenient in answering our question to use the two structural equations to find what are known as the reduced form equations—that is, one equation for each endogenous variable in which the endogenous variable is written as a function solely of exogenous variables and error terms. We can find the reduce form equations by solving the structural equations simultaneously for the endogenous variables. Substituting (2) into (1), we get:

q t = α 0 + α 1 ( β 0 + β 1 q t + β 2 W t + η t )+ α 2 I t + α 3 p t c + ε t q t = α 0 + α 1 ( β 0 + β 1 q t + β 2 W t + η t )+ α 2 I t + α 3 p t c + ε t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBaaaleaacaWG0baabeaakiabg2da9iabeg7aHnaaBaaaleaacaaIWaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIXaaabeaakmaabmaabaGaeqOSdi2aaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaamyCamaaBaaaleaacaWG0baabeaakiabgUcaRiabek7aInaaBaaaleaacaaIYaaabeaakiaadEfadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaH3oaAdaWgaaWcbaGaamiDaaqabaaakiaawIcacaGLPaaacqGHRaWkcqaHXoqydaWgaaWcbaGaaGOmaaqabaGccaWGjbWaaSbaaSqaaiaadshaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaaiodaaeqaaOGaamiCamaaDaaaleaacaWG0baabaGaam4yaaaakiabgUcaRiabew7aLnaaBaaaleaacaWG0baabeaaaaa@61AE@

q t = α 0 + α 1 β 0 + α 1 β 1 q t + α 1 β 2 W t + α 1 η t + α 2 I t + α 3 p t c + ε t q t = α 0 + α 1 β 0 + α 1 β 1 q t + α 1 β 2 W t + α 1 η t + α 2 I t + α 3 p t c + ε t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBaaaleaacaWG0baabeaakiabg2da9iabeg7aHnaaBaaaleaacaaIWaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIWaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaakiaadghadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaHXoqydaWgaaWcbaGaaGymaaqabaGccqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaWGxbWaaSbaaSqaaiaadshaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeq4TdG2aaSbaaSqaaiaadshaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaamysamaaBaaaleaacaWG0baabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIZaaabeaakiaadchadaqhaaWcbaGaamiDaaqaaiaadogaaaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamiDaaqabaaaaa@67D5@

q t − α 1 β 1 q t =( α 0 + α 1 β 0 )+ α 1 β 2 W t + α 2 I t + α 3 p t c +( ε t + α 1 η t ) q t − α 1 β 1 q t =( α 0 + α 1 β 0 )+ α 1 β 2 W t + α 2 I t + α 3 p t c +( ε t + α 1 η t ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6AFC@

or

Substituting (1) into (2) yields:

p t w = β 0 + β 1 ( α 0 + α 1 p t w + α 2 I t + α 3 p t c + ε t )+ β 2 W t + η t p t w = β 0 + β 1 ( α 0 + α 1 p t w + α 2 I t + α 3 p t c + ε t )+ β 2 W t + η t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@63A6@

p t w = β 0 + β 1 α 0 + α 1 β 1 p t w + α 2 β 1 I t + α 3 β 1 p t c + β 1 ε t + β 2 W t + η t p t w = β 0 + β 1 α 0 + α 1 β 1 p t w + α 2 β 1 I t + α 3 β 1 p t c + β 1 ε t + β 2 W t + η t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6C65@

or

Equations (4) and (5) are the reduced form equations for this model. We can use them to calculate E( ε t p t w )=0 E( ε t p t w )=0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeqyTdu2aaSbaaSqaaiaadshaaeqaaOGaamiCamaaDaaaleaacaWG0baabaGaam4DaaaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3FFE@ and E( η t q t )=0. E( η t q t )=0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeq4TdG2aaSbaaSqaaiaadshaaeqaaOGaamyCamaaBaaaleaacaWG0baabeaaaOGaayjkaiaawMcaaiabg2da9iaaicdacaGGUaaaaa@3FB9@ In particular,

E( ε t p t w )=E[ ε t ( β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t + β 1 ε t + η t 1− α 1 β 1 ) ] E( ε t p t w )=E[ ε t ( β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t + β 1 ε t + η t 1− α 1 β 1 ) ] MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@91D1@

E( ε t p t w )=E[ ε t ( β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t )+ ε t ( β 1 ε t + η t 1− α 1 β 1 ) ] E( ε t p t w )=E[ ε t ( β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t )+ ε t ( β 1 ε t + η t 1− α 1 β 1 ) ] MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeqyTdu2aaSbaaSqaaiaadshaaeqaaOGaamiCamaaDaaaleaacaWG0baabaGaam4DaaaaaOGaayjkaiaawMcaaiabg2da9iaadweadaWadaqaaiabew7aLnaaBaaaleaacaWG0baabeaakmaabmaabaWaaSaaaeaacqaHYoGydaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGymaaqabaGccqaHXoqydaWgaaWcbaGaaGimaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiabgUcaRmaalaaabaGaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaGccaWGjbWaaSbaaSqaaiaadshaaeqaaOGaey4kaSYaaSaaaeaacqaHXoqydaWgaaWcbaGaaG4maaqabaGccqaHYoGydaWgaaWcbaGaaGymaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiaadchadaqhaaWcbaGaamiDaaqaaiaadogaaaGccqGHRaWkdaWcaaqaaiabek7aInaaBaaaleaacaaIYaaabeaaaOqaaiaaigdacqGHsislcqaHXoqydaWgaaWcbaGaaGymaaqabaGccqaHYoGydaWgaaWcbaGaaGymaaqabaaaaOGaam4vamaaBaaaleaacaWG0baabeaaaOGaayjkaiaawMcaaiabgUcaRiabew7aLnaaBaaaleaacaWG0baabeaakmaabmaabaWaaSaaaeaacqaHYoGydaWgaaWcbaGaaGymaaqabaGccqaH1oqzdaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaH3oaAdaWgaaWcbaGaamiDaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@9630@

or

Factoring out the non-stochastic terms from the expected value operators gives:

E( ε t p t w )=( β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t )E[ ε t ]+ β 1 E( ε t 2 ) 1− α 1 β 1 + E( η t ε t ) 1− α 1 β 1 . E( ε t p t w )=( β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t )E[ ε t ]+ β 1 E( ε t 2 ) 1− α 1 β 1 + E( η t ε t ) 1− α 1 β 1 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A196@

Moreover, by assumption E( ε t )=0, E( ε t )=0, MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeqyTdu2aaSbaaSqaaiaadshaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaiaacYcaaaa@3D8D@ E( η t ε t )=0, E( η t ε t )=0, MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeq4TdG2aaSbaaSqaaiaadshaaeqaaOGaeqyTdu2aaSbaaSqaaiaadshaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaiaacYcaaaa@4068@ and E( ε t 2 )= σ ε 2 . E( ε t 2 )= σ ε 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeqyTdu2aa0baaSqaaiaadshaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaeyypa0Jaeq4Wdm3aa0baaSqaaiabew7aLbqaaiaaikdaaaGccaGGUaaaaa@41EF@ Thus, we get:

A similar analysis yields:

Equations (6) and (7) are what create the endogeneity problem (or simultaneous equation bias)—using OLS to estimate the parameters of equations that have an endogenous variable as an explanatory variable yields biased estimates of the unknown parameters. Figure 1 illustrates the endogeneity problem. In this figure we have demand and supply equations that have both risen due to changes in exogenous variables. What the researcher observes are two (red) points: (1) the intersection of the old demand and supply curves and (2) the intersection of the new demand and supply curves.

Figure 1: The simultaneous equation problem.

The thick red line shows the regression that would result from using OLS to estimate either of the two structural equations. As illustrated, an OLS estimate of the slope estimate will be biased. We need to use some other estimation technique than OLS.

As noted earlier, the basic problem created by the endogeneity problem is that the endogenous explanatory variable is correlated with the error term. The most logical approach would be to replace this variable with one that is not correlated with the error term but highly correlated with the endogenous variable. Consider the value of the price predicted by the reduced form equation (5):

where γ ⌢ i γ ⌢ i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbambadaWgaaWcbaGaamyAaaqabaaaaa@38CF@ is the OLS estimate of γ 0 = β 0 + β 1 α 0 1− α 1 β 1 , γ 0 = β 0 + β 1 α 0 1− α 1 β 1 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaOGaeyypa0ZaaSaaaeaacqaHYoGydaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGymaaqabaGccqaHXoqydaWgaaWcbaGaaGimaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiaacYcaaaa@49AF@ γ 1 = α 2 β 1 1− α 1 β 1 , γ 1 = α 2 β 1 1− α 1 β 1 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacqaHXoqydaWgaaWcbaGaaGOmaaqabaGccqaHYoGydaWgaaWcbaGaaGymaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiaacYcaaaa@463F@ γ 2 = α 3 β 1 1− α 1 β 1 , γ 2 = α 3 β 1 1− α 1 β 1 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacqaHXoqydaWgaaWcbaGaaG4maaqabaGccqaHYoGydaWgaaWcbaGaaGymaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiaacYcaaaa@4641@ and γ 3 = β 2 1− α 1 β 1 . γ 3 = β 2 1− α 1 β 1 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaSaaaeaacqaHYoGydaWgaaWcbaGaaGOmaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiaac6caaaa@43B2@

Clearly, p ⌢ t w p ⌢ t w MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaataWaa0baaSqaaiaadshaaeaacaWG3baaaaaa@3925@ is correlated with p t w . p t w . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaWG0baabaGaam4Daaaakiaac6caaaa@39C7@ It also is true that the covariance between p ⌢ t w p ⌢ t w MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaataWaa0baaSqaaiaadshaaeaacaWG3baaaaaa@3925@ and ε t ε t MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaSbaaSqaaiaadshaaeqaaaaa@38C0@ goes to zero as the sample size increasing. Thus, we can use (8) to construct a variable that will produce a consistent estimator of α 1 . α 1 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaaiOlaaaa@3936@ It is this conclusion that underlies the strategy of both two-stage least squares (TSLQ) and instrumental variable (IV) estimators.

There is an additional issue that arises with estimating systems of equations—identification. Essentially, identification is an algebraic problem. Consider the reduced form equations given earlier in (4) and (5):

q t = α 0 + α 1 β 0 1− α 1 β 1 + α 1 β 2 1− α 1 β 1 W t + α 2 1− α 1 β 1 I t + α 3 1− α 1 β 1 p t c + ε t + α 1 η t 1− α 1 β 1 q t = α 0 + α 1 β 0 1− α 1 β 1 + α 1 β 2 1− α 1 β 1 W t + α 2 1− α 1 β 1 I t + α 3 1− α 1 β 1 p t c + ε t + α 1 η t 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@81EE@

and

p t w = β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t + β 1 ε t + η t 1− α 1 β 1 . p t w = β 0 + β 1 α 0 1− α 1 β 1 + α 2 β 1 1− α 1 β 1 I t + α 3 β 1 1− α 1 β 1 p t c + β 2 1− α 1 β 1 W t + β 1 ε t + η t 1− α 1 β 1 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8582@

OLS estimation of both of these equations yields unbiased estimates of the parameters in the reduced form equations. Identification asks if we can retrieve the parameters of the structural equations from the reduced form equations. Say, for instance, that we re-write the reduced form equations as:

and

Table 1 shows each of the parameters in (11) and (12) in terms of the parameters of the two reduced form equations. We can recover the parameters of the structural equations by algebraic manipulation of the relationships in Table 1. (This method of estimation—that is, estimating the reduced form equations of a model using OLS and then solving algebraically for the parameters of the structural equations is referred to in the literature as indirect least squares.) For instance,

δ 21 δ 12 = ( α 2 β 1 1− α 1 β 1 ) ( α 2 1− α 1 β 1 ) = β 1 δ 21 δ 12 = ( α 2 β 1 1− α 1 β 1 ) ( α 2 1− α 1 β 1 ) = β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@59C5@

and

δ 11 δ 23 = ( α 1 β 2 1− α 1 β 1 ) ( β 2 1− α 1 β 1 ) = α 1 . δ 11 δ 23 = ( α 1 β 2 1− α 1 β 1 ) ( β 2 1− α 1 β 1 ) = α 1 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5A82@

One can continue in a likewise manner to find formulae for other of the structural parameters. However, an interesting problem does arrive in that it is also true that β 1 = δ 22 δ 13 . β 1 = δ 22 δ 13 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacqaH0oazdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeqiTdq2aaSbaaSqaaiaaigdacaaIZaaabeaaaaGccaGGUaaaaa@40F3@ Since there is no a priori reason to believe that δ 22 δ 13 = δ 21 δ 12 , δ 22 δ 13 = δ 21 δ 12 , MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqaH0oazdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeqiTdq2aaSbaaSqaaiaaigdacaaIZaaabeaaaaGccqGH9aqpdaWcaaqaaiabes7aKnaaBaaaleaacaaIYaGaaGymaaqabaaakeaacqaH0oazdaWgaaWcbaGaaGymaiaaikdaaeqaaaaakiaacYcaaaa@4513@ we have two estimates of β 1 . β 1 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaa@387C@ This result illustrates the point that there are three possibilities when calculating the structural parameters from the reduced form equations—first, there may be more than one formula for a structural parameter; second, there may be only one formula for a structural parameter; or third, there may be no formula for a structural parameter. We say in the first case that the equation is over-identified; is exactly identified in the second case; and is under-identified in the third case. It turns out that in the case of an over-identified equation we can to use TSLS to estimate the structural parameters. However, in the case of an exactly identified equation, the TSLS estimators are equal to the indirect-least-squares estimators that can be calculated using estimates of the reduced form equations. Finally, an under-identified equation cannot be estimated by any technique.

Clearly, we need to know how to identify if an equation is either over-identified, exactly identified, or under-identified. A necessary rule is that the number of exogenous variables in a system of equation that are not included in a particular regression must be greater than or equal to the number of endogenous variables on the right-hand-side of the equation for the equation to be either exactly or over identified. Consider the following three-equation model, where the endogenous variables are y 1 , y 1 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaaaaa@37D9@ y 2 y 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@37DA@ , and y 3 y 3 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIZaaabeaaaaa@37DB@ and the exogenous variables are represented by x 1 x 1 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaaaa@37D8@ with i=1,…,5: i=1,…,5: MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2da9iaaigdacaGGSaGaeSOjGSKaaiilaiaaiwdacaGG6aaaaa@3CA2@

The error terms in these three equations are omitted because they are irrelevant to determining if an equation is identified—remember, identification is an algebraic problem, not a statistical issue. There are 3 endogenous variables in the system and 3 equations in the system. Also, there are 5 exogenous variables in the system of equations. Equation (13) is exactly identified; Equation (14) is over-identified; and Equation (15) is under-identified. What this means is (1) Equation (13) can be estimated directly from the reduced form equation (using indirect-least-squares) or using TSLS; (2) Equation (14) must be estimated using TSLS; and Equation (15) cannot be estimated. Table 2 summarizes how to determine if an equation is or is not identified. Basically, if the number in column 2 equals the number in column 3, the equation is exactly identified. If the number in column 2 is less than the number in column 3, the equation is over-identified. Finally, if the number in column 2 is greater than the number in column 3, the equation is under-identified.3

One other thing to notice is the similarity of TSLS to IV estimation. The exogenous variables play the role of instruments in TSLS estimation. By implication, the instruments in an IV estimation must not include any of the exogenous variables in the equation.4 Similarly, one of the

ways to isolate potential instruments in a regression is to think of what system of equation the equation is and then ask what exogenous variables in that system are not included in the equation. These excluded exogenous variables are potential instruments.

The command for estimating an equation in Stata using two-stages least squares (TSLS) is a bit tricky. Assume that you want to estimate equations (13) and (14) in the model discussed above.5 For simplicity assume that each variable assumes the name for it in Table 2. Thus, in our Stata commands Y1 refers to variable Thus, in our Stata commands Y1 refers to variable y 1 y 1 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaaaaa@37D9@ and so on. The command to estimate either a TSLS or an IV regression is the same.6 The command, ivreg, consists of three major parts—(1) the name of the dependent variable is followed by (2) a list of the names of the exogenous variables that are being used as explanatory variables and then followed in parentheses by (3) the information needed to estimate the first stage (the list of the endogenous variables that are explanatory variables along with the names of the exogenous variables in the system that are excluded from the equation or, in the case of IV, a list of the instruments).7

Example 1

An example from Stata. The Stata manual offers the following example analysis. Assume that you want to use state level data from the 1980 census to estimate the following system of equations:

hsngval= α 0 + α 1 fainc+ α 2 reg2+ α 3 reg3+ α 4 reg4+ε hsngval= α 0 + α 1 fainc+ α 2 reg2+ α 3 reg3+ α 4 reg4+ε MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaadohacaWGUbGaam4zaiaadAhacaWGHbGaamiBaiabg2da9iabeg7aHnaaBaaaleaacaaIWaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIXaaabeaakiaadAgacaWGHbGaamyAaiaad6gacaWGJbGaey4kaSIaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaamOCaiaadwgacaWGNbGaaGOmaiabgUcaRiabeg7aHnaaBaaaleaacaaIZaaabeaakiaadkhacaWGLbGaam4zaiaaiodacqGHRaWkcqaHXoqydaWgaaWcbaGaaGinaaqabaGccaWGYbGaamyzaiaadEgacaaI0aGaey4kaSIaeqyTdugaaa@5FAD@

(16)

and

rent= β 0 + β 1 hsngval+ β 2 pcturban+ν, rent= β 0 + β 1 hsngval+ β 2 pcturban+ν, MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaadwgacaWGUbGaamiDaiabg2da9iabek7aInaaBaaaleaacaaIWaaabeaakiabgUcaRiabek7aInaaBaaaleaacaaIXaaabeaakiaadIgacaWGZbGaamOBaiaadEgacaWG2bGaamyyaiaadYgacqGHRaWkcqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaWGWbGaam4yaiaadshacaWG1bGaamOCaiaadkgacaWGHbGaamOBaiabgUcaRiabe27aUjaacYcaaaa@55A7@

(17)

where hsngval is the median dollar value of owner-occupied housing; rent is the median monthly gross rent; fainc is family income; pcturban is the percent of the state population living in an urban area; and reg2, reg3, and reg4 are dummy variables that designate the region of the country where the state is located. In this example we focus on estimating (17).

We begin by loading the data set and describing the data.

. use http://www.stata-press.com/data/r8/hsng2

(1980 Census housing data)

.describe

Table 4: Description of the Stata data set used in the example.

Contains data from http://www.stata-press.com/data/r8/hsng2.dta

obs: 50 1980 Census housing data vars: 16 3 Sep 2002 12:25

size: 3,600 (99.7% of memory free)

variable name storage type display format value label variable label state str14 % 14s   State division int % 8.0g division Census division region int % 8.0g region Region pop long % 10.0g   Population in 1980 popgrow float % 6.1f   Pop. growth 1970-80 popden int % 6.1f   Pop/sq. mile pcturban float % 8.1f   Percent urban faminc long % 8.2f   Median family inc., 1979 hsng long % 10.0g   Hsng units 1980 hsnggrow float % 8.1f   % housing growth hsngval long % 9.2f   Median hsng value rent long % 6.2f   Median gross rent reg1 float % 9.0g     reg2 float % 9.0g     reg3 float % 9.0g     reg4 float % 9.0g

Sorted by: state

Now we estimate equation (17) using TSLS as shown in Figure 2.

Figure 2:

Two-stages least square estimate of the example.

The manual continues the example to include some testing of the model including the Hausman test. Students using TSLS and IV should read the discussion in the Stata manual thoroughly.

In this exercise we address this issue by using data used originally by Richard Schmalensee (1972) in his Ph.D. dissertation. You will use these data to estimate a simple two-equation model of the cigarette advertising industry.

We use annual data for the period 1955 to 1967 to estimate the impact of cigarette advertising on aggregate demand for cigarettes and the impact of cigarette consumption on cigarette advertising. We begin with a model of the demand for cigarettes. We assume that the demand for cigarettes is given by:

where

q_t = cigarettes consumed per person over age 15,

pc_t = retail price of cigarettes,

y_t = real disposable personal income per capita (1958 dollars),

A_t = real advertising expenditures per individual over age 15 (1960 dollars), and

D64 = a dummy variable equal to 1 for the years 1964 through 1967 and zero otherwise.

We include the dummy variable for years after 1964 to pick up the negative impact on cigarette sales of the 1964 report of the US Surgeon General’s Advisory Committee (1964) announcing that the government believed that there was enough evidence available to conclude that cigarette smoking causes cancer. We expect the signs of the parameters with the price of cigarettes and the dummy variable to be negative. We expect that the sign of the parameters with income and advertising to be positive.

Next we turn to a model of the supply of advertising. We assume:

where:

pa_t = advertising price index, and

m_t = gross profits as a percentage of gross sales.

The last variable needs a bit of explaining. The amount of advertising in the industry should be a function of degree of competition in the industry. If the market were perfectly competitive, there would be no reason for any firm to advertise. If the firm were a monopoly, there also would be no reason to advertise. However, if the market is an oligopoly, then a firm would advertise in an effort to gain market share by differentiating its product from the product of its competitors.

The traditional measure of the degree of monopoly power that a firm has is the ratio of its marginal profits to its marginal cost:

where p is output price, mc is marginal cost, and m is the measure of monopoly power. Since we cannot observe the firms’ marginal costs, we approximate m by the ratio of gross profits to gross sales. We expect the impact of the degree of monopoly to have a non-linear impact on advertising expenditures.

The data used to estimate our two equations are listed in Table 5 and are available in the MS Excel file Cigarette sales and advertising data.xls. These data are with the exception of disposable personal income from Schmalensee (1972: 273-290). The disposable personal income data are from the Department of Commerce (1975: Table F26, page 225).

Specification of the Model. Equations (18) and (19) are, as written, very general and need further specification before they can be estimated. We will assume that the two equations take a log-log form. In particular, we assume that we want to estimate:

and

Answer the following six questions:

a) Which variables in the model are exogenous and which are endogenous?

b) Check and see if equations (18) and (19) are underidentified, exactly identified, or overidentified.

c) Estimate equations (21) and (22) using ordinary least squares.

d) Estimate equations (21) and (22) using two-stage least squares. Present the results in a table that for comparison reasons includes the results from the OLS estimation. Be sure to include the R² and the Durbin-Watson statistic.

e) Which side of the advertising-sales controversy do your results appear to support?

f) How well-specified does your model appear to be? Why?

Q_t = total commercial loans (billions of dollars)

R_t = average prime rate charged by banks

RS_t = 3-month Treasury bill rate (represents an alternative rate of return for banks)

RD_t = Aaa corporate bond rate (represents the price of alternative financing to firms)

X_t = industrial production index (represents firms’ expectation about future economic activity)

y_t = total bank deposits (billions of dollars) (represents a scale variable).

The demand and supply equations to be estimated, respectively, are as follows:

and

Questions

a) What are the endogenous and exogenous variables in this model?

b) Solve for the two “reduced form” equations of this model. Estimate these two equations using the data in Table 6.

c) Check the “order” condition for identification of each equation of the model.

d) Estimate equations (23) and (24) using ordinary least squares using the data in Table 6.

e) Estimate equations (23) and (24) using two-stage least squares. Report the results of the estimations for part 4 and 5 in a single table. Be sure to include the t-ratios, R²’s, and Durbin-Watson statistics for each of the equations estimated.

f) Perform the Hausman Specification Test on both equations.9

g) When presenting this model, Maddala notes “[T]he model postulated here is not necessarily the right model for the problem of analyzing the commercial loan market.” Is there anything in the results reported above that suggests that the model may be mis-specified?