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# Endogenous explanatory variables

Module by: Christopher Curran. E-mail the author

Summary: This module is a short introduction to the problems that arise when there are explanatory variables in a model that are endogenous. It is intended for the use of advanced undergraduate economics majors who have completed at least one semester of econometrics.

## Endogenous Explanatory Variables

### Introduction

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.

### The Statistical Problem

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, qt, 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, It, 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, qt, and a weather index, Wt, 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:

#### Demand:

q t = α 0 + α 1 p t w + α 2 I t + α 3 p t c + ε t q t = α 0 + α 1 p t w + α 2 I t + α 3 p t c + ε t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBaaaleaacaWG0baabeaakiabg2da9iabeg7aHnaaBaaaleaacaaIWaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIXaaabeaakiaadchadaqhaaWcbaGaamiDaaqaaiaadEhaaaGccqGHRaWkcqaHXoqydaWgaaWcbaGaaGOmaaqabaGccaWGjbWaaSbaaSqaaiaadshaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaaiodaaeqaaOGaamiCamaaDaaaleaacaWG0baabaGaam4yaaaakiabgUcaRiabew7aLnaaBaaaleaacaWG0baabeaaaaa@51DF@
(1)

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:

ε t ~N( 0, σ ε 2 ),  η t ~N( 0, σ η 2 ), and  E( ε t η t )=0. ε t ~N( 0, σ ε 2 ),  η t ~N( 0, σ η 2 ), and  E( ε t η t )=0. MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaH1oqzdaWgaaWcbaGaamiDaaqabaGccaGG+bGaamOtamaabmaabaGaaGimaiaacYcacqaHdpWCdaqhaaWcbaGaeqyTdugabaGaaGOmaaaaaOGaayjkaiaawMcaaiaacYcacaqGGaaabaGaeq4TdG2aaSbaaSqaaiaadshaaeqaaOGaaiOFaiaad6eadaqadaqaaiaaicdacaGGSaGaeq4Wdm3aa0baaSqaaiabeE7aObqaaiaaikdaaaaakiaawIcacaGLPaaacaGGSaGaaeiiaiaabggacaqGUbGaaeizaiaabccaaeaacaWGfbWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamiDaaqabaGccqaH3oaAdaWgaaWcbaGaamiDaaqabaaakiaawIcacaGLPaaacqGH9aqpcaaIWaGaaiOlaaaaaa@5E76@
(2)

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—qt 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=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@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=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@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

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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@82AA@
(3)

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

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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@863E@
(4)

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=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@9630@

or

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 ) ]+E( β 1 ε t 2 + η t ε 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 ) ]+E( β 1 ε t 2 + η t ε t 1 α 1 β 1 ). MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@9869@
(5)

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:

E( ε t p t w )= β 1 σ ε 2 1 α 1 β 1 0. E( ε t p t w )= β 1 σ ε 2 1 α 1 β 1 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeqyTdu2aaSbaaSqaaiaadshaaeqaaOGaamiCamaaDaaaleaacaWG0baabaGaam4DaaaaaOGaayjkaiaawMcaaiabg2da9maalaaabaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaeq4Wdm3aa0baaSqaaiabew7aLbqaaiaaikdaaaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiabgcMi5kaaicdacaGGUaaaaa@5040@
(6)

A similar analysis yields:

E( η t q t )= α 1 σ η 2 1 α 1 β 1 0. E( η t q t )= α 1 σ η 2 1 α 1 β 1 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabmaabaGaeq4TdG2aaSbaaSqaaiaadshaaeqaaOGaamyCamaaBaaaleaacaWG0baabeaaaOGaayjkaiaawMcaaiabg2da9maalaaabaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeq4Wdm3aa0baaSqaaiabeE7aObqaaiaaikdaaaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaakiabgcMi5kaaicdacaGGUaaaaa@4F4C@
(7)

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.

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.

### Estimation

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):

(8)

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.

#### Two-stages least squares

The easiest way to understand two-stage least squares is to think of the estimation process as being in the following two steps (although the computer programs calculate the estimators in one step):

Stage 1: obtain a OLS predictions for any endogenous variable on the right-hand side of the equation to be estimated using as the explanatory variables all of the exogenous variables in the system.

Stage 2: estimate the parameters of the equation using OLS and replacing the endogenous variable on the right-hand side of the equation by the its predictions as obtained in step 1.

For obvious reasons he TSLS method works best when the full model is specified or when you know and can measure all of the exogenous variables in the system.

#### Instrumental variables (IV)

While the use of instrumental variable (IV) estimators is appropriate in a large number of situations, the two situations where they are most commonly used are (1) in the presence of endogenous explanatory variables and (2) in cases when errors arise in the measurement of an explanatory variable (or the errors-in-variables problem). Since I have already described the endogeneity problem, I now turn to a brief discussion of errors-in-variables.

Consider the following simple model:

y i = β 1 x i + ε i and x i = x i + μ i . y i = β 1 x i + ε i and x i = x i + μ i . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbaabeaakiabg2da9iabek7aInaaBaaaleaacaaIXaaabeaakiaadIhadaqhaaWcbaGaamyAaaqaaiabgEHiQaaakiabgUcaRiabew7aLnaaBaaaleaacaWGPbaabeaakiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaamiEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadIhadaqhaaWcbaGaamyAaaqaaiabgEHiQaaakiabgUcaRiabeY7aTnaaBaaaleaacaWGPbaabeaakiaac6caaaa@5113@
(9)

In this model the researcher observes x i x i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaaaa@380B@ but not the desired x i x i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDaaaleaacaWGPbaabaGaey4fIOcaaaaa@38FB@ because of some random measurement error. Using OLS to estimate (9) using the observable x i x i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaaaa@380B@ instead of the correct x i x i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDaaaleaacaWGPbaabaGaey4fIOcaaaaa@38FB@ is equivalent to estimating:

y i = β 1 x i +( ε i β 1 μ i ). y i = β 1 x i +( ε i β 1 μ i ). MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbaabeaakiabg2da9iabek7aInaaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHRaWkdaqadaqaaiabew7aLnaaBaaaleaacaWGPbaabeaakiabgkHiTiabek7aInaaBaaaleaacaaIXaaabeaakiabeY7aTnaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiaac6caaaa@4A0F@
(10)

The important thing to note in estimating (10) using OLS is that the explanatory variable, x i x i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaaaa@380B@ , is correlated with the error term, ( ε i β 1 μ i ). ( ε i β 1 μ i ). MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamyAaaqabaGccqGHsislcqaHYoGydaWgaaWcbaGaaGymaaqabaGccqaH8oqBdaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacaGGUaaaaa@4152@ As was the case with the endogeneity problem, the OLS estimate of β 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaa@387C@ is biased. Murray (2006) summarizes the situation as follows:

In both examples, ordinary least squares estimation is biased because an explanatory variable in the regression is correlated with the error term in the regression. Such a correlation can result from an endogenous explanator, a mismeasured explanator, an omitted explanator, or a lagged dependent variable among the explanators. I call all such explanators “troublesome.” Instrumental variable estimation can consistently estimate coefficients when ordinary least squares cannot—that is, the instrumental variable estimate of the coefficient will almost certainly be very close to the coefficient’s true value if the sample is sufficiently large—despite troublesome explanators. [Murray (2006a): 112]

Consider a regression that includes a “troublesome explanator,” like x i x i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDaaaleaacaWGPbaabaGaey4fIOcaaaaa@38FB@ in (9). Assume that there exists a variable z i z i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBaaaleaacaWGPbaabeaaaaa@380D@ (or set of variables) that (1) is correlated with the “troublesome explanator,” (2) is uncorrelated with the error term—like ε i ε i MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaSbaaSqaaiaadMgaaeqaaaaa@38B5@ in (9), and (3) is not one of the explanatory variables in the equation to be estimated. Greene (1990: 300) offers the following example of such a variable. Self-reported income tends to be a very “noisy” variable because sometimes people forget to report minor sources of income and sometimes they deliberately misreport their income. If the regression you are estimating uses income as explanatory variable of consumption, OLS will yield biased estimates. On the other hand, the number of checks written in a month by the household head might serve as an instrumental variable. Clearly, the number of checks written might well be positively correlated with income and there is no reason to assume that it is correlated with the error term in the consumption equation.2

It is usually fairly easy to identify instances when IV estimation methods are appropriate. This is especially true when one of the explanatory variables is possibly an endogenous variable. The real problem arises in finding an instrumental variable or a set of instrumental variables. However, assuming you have one or more instrumental variables, the IV method follows the same steps as described above for TSLS. In the first stage you estimate a regression of the “troublesome variable” as a function of the instruments and the exogenous variables in the equation—i.e., you estimate the reduced form equation. In the second stage you use OLS to estimate the original equation with the value of the “troublesome variable” predicted by the first stage regression substituted for the actual values of the “troublesome variable.”

In a sense TSLS is a IV estimation. The exogenous variables not in a particular regression play the role of the instruments. Thus, in the IV estimation of (1), the weather index is the instrument. In the estimation of (2) the price of corn and the income level are the IVs. Thus, in a fully specified model, the exogenous variables excluded from the regression play the role of instrumental variables. In other situations the choice of an appropriate instrument can be very difficult. The selection process demands creativity both in finding the instrument and in defending the choice.

The use either of IV or TSLS comes at a cost. First, the OLS estimators are more precise (i.e., have a smaller standard error) than the TSLS or IV estimators. Second, selecting invalid or weak instruments can create results that are not meaningful. So how does one know if they have chosen a good set of instruments? There is no easy answer to this question. Murray (2006a: 116-117) discusses some possible tests of the validity of an instrumental variable. In the end, however, the “success” of your instrument may depend more on how convincing your justifications are than any statistical test. Some economists, like Steven Levitt, make a living coming up with and justifying the use of some very creative instrumental variables. Murray (2006a) offers a detailed discussion of IV and should be read by any student planning to make use either of TSLS or IV regression estimators.

### The identification problem

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:

q t = δ 10 + δ 11 W t + δ 12 I t + δ 13 p t c + γ 1 q t = δ 10 + δ 11 W t + δ 12 I t + δ 13 p t c + γ 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBaaaleaacaWG0baabeaakiabg2da9iabes7aKnaaBaaaleaacaaIXaGaaGimaaqabaGccqGHRaWkcqaH0oazdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaam4vamaaBaaaleaacaWG0baabeaakiabgUcaRiabes7aKnaaBaaaleaacaaIXaGaaGOmaaqabaGccaWGjbWaaSbaaSqaaiaadshaaeqaaOGaey4kaSIaeqiTdq2aaSbaaSqaaiaaigdacaaIZaaabeaakiaadchadaqhaaWcbaGaamiDaaqaaiaadogaaaGccqGHRaWkcqaHZoWzdaWgaaWcbaGaaGymaaqabaGccaqGGaGaaeyyaiaab6gacaqGKbaaaa@56F8@
(11)

and

p t w = δ 20 + δ 21 I t + δ 22 p t c + δ 23 W t + δ 2 . p t w = δ 20 + δ 21 I t + δ 22 p t c + δ 23 W t + δ 2 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaWG0baabaGaam4Daaaakiabg2da9iabes7aKnaaBaaaleaacaaIYaGaaGimaaqabaGccqGHRaWkcqaH0oazdaWgaaWcbaGaaGOmaiaaigdaaeqaaOGaamysamaaBaaaleaacaWG0baabeaakiabgUcaRiabes7aKnaaBaaaleaacaaIYaGaaGOmaaqabaGccaWGWbWaa0baaSqaaiaadshaaeaacaWGJbaaaOGaey4kaSIaeqiTdq2aaSbaaSqaaiaaikdacaaIZaaabeaakiaadEfadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaH0oazdaWgaaWcbaGaaGOmaaqabaGccaGGUaaaaa@554A@
(12)

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@

 Explanatory variable Equation (11) Equation (12) Intercept δ 10 = α 0 + α 1 β 0 1− α 1 β 1 δ 10 = α 0 + α 1 β 0 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaigdacaaIWaaabeaakiabg2da9maalaaabaGaeqySde2aaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaicdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@49AB@ δ 20 = β 0 + β 1 α 0 1− α 1 β 1 δ 20 = β 0 + β 1 α 0 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaikdacaaIWaaabeaakiabg2da9maalaaabaGaeqOSdi2aaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaeqySde2aaSbaaSqaaiaaicdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@49AE@ I t I t MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBaaaleaacaWG0baabeaaaaa@37E7@ δ 11 = α 1 β 2 1− α 1 β 1 δ 11 = α 1 β 2 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaigdacaaIXaaabeaakiabg2da9maalaaabaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaikdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@463D@ δ 21 = α 2 β 1 1− α 1 β 1 δ 21 = α 2 β 1 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaikdacaaIXaaabeaakiabg2da9maalaaabaGaeqySde2aaSbaaSqaaiaaikdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@463E@ p t c p t c MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaDaaaleaacaWG0baabaGaam4yaaaaaaa@38F7@ δ 12 = α 2 1− α 1 β 1 δ 12 = α 2 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9maalaaabaGaeqySde2aaSbaaSqaaiaaikdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@43AC@ δ 22 = α 3 β 1 1− α 1 β 1 δ 22 = α 3 β 1 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9maalaaabaGaeqySde2aaSbaaSqaaiaaiodaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@4640@ W t W t MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBaaaleaacaWG0baabeaaaaa@37F5@ δ 13 = α 3 1− α 1 β 1 δ 13 = α 3 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9maalaaabaGaeqySde2aaSbaaSqaaiaaiodaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@43AE@ δ 23 = β 2 1− α 1 β 1 δ 23 = β 2 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaikdacaaIZaaabeaakiabg2da9maalaaabaGaeqOSdi2aaSbaaSqaaiaaikdaaeqaaaGcbaGaaGymaiabgkHiTiabeg7aHnaaBaaaleaacaaIXaaabeaakiabek7aInaaBaaaleaacaaIXaaabeaaaaaaaa@43B0@ Error term γ 1 = ε t + α 1 η t 1− α 1 β 1 γ 1 = ε t + α 1 η t 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacqaH1oqzdaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaHXoqydaWgaaWcbaGaaGymaaqabaGccqaH3oaAdaWgaaWcbaGaamiDaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaaaaa@4984@ δ 2 = β 1 ε t + η t 1− α 1 β 1 δ 2 = β 1 ε t + η t 1− α 1 β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacqaHYoGydaWgaaWcbaGaaGymaaqabaGccqaH1oqzdaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaH3oaAdaWgaaWcbaGaamiDaaqabaaakeaacaaIXaGaeyOeI0IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaaaaaa@4985@

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@

y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 11 x 1 + α 12 x 2 + α 15 x 5 , y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 11 x 1 + α 12 x 2 + α 15 x 5 , MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5AC7@
(13)
y 2 = β 20 + β 21 y 1 + α 23 x 3 , and y 2 = β 20 + β 21 y 1 + α 23 x 3 , and MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaakiabg2da9iabek7aInaaBaaaleaacaaIYaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGOmaiaaigdaaeqaaOGaamyEamaaBaaaleaacaaIXaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIYaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@486B@
(14)
y 3 = β 30 + β 31 y 1 + α 31 x 1 + α 32 x 2 + α 33 x 3 + α 34 x 4 + α 35 x 5 . y 3 = β 30 + β 31 y 1 + α 31 x 1 + α 32 x 2 + α 33 x 3 + α 34 x 4 + α 35 x 5 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@60F3@
(15)

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

 Equation Number of endogenous variables on right-hand-side Number of exogenous variables excluded from the equation Identification y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 11 x 1 + α 12 x 2 + α 15 x 5 y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 11 x 1 + α 12 x 2 + α 15 x 5 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5AC7@ 2 2 Exactly y 2 = β 20 + β 21 y 1 + α 23 x 3 y 2 = β 20 + β 21 y 1 + α 23 x 3 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaakiabg2da9iabek7aInaaBaaaleaacaaIYaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGOmaiaaigdaaeqaaOGaamyEamaaBaaaleaacaaIXaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIYaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@486B@ 1 4 Over y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 12 x 2 + α 13 x 3 + α 15 x 5 y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 12 x 2 + α 13 x 3 + α 15 x 5 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5ACB@ 1 0 Under

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.

### TSLS and IV in Stata

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

 Equation to be estimated Stata command y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 12 x 2 + α 13 x 3 + α 15 x 5 y 1 = β 10 + β 12 y 2 + β 13 y 3 + α 12 x 2 + α 13 x 3 + α 15 x 5 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5ACB@ .ivreg y1 x2 x3 x5 (y2 y3 = x1 x4) y 2 = β 20 + β 21 y 1 + α 23 x 3 y 2 = β 20 + β 21 y 1 + α 23 x 3 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaakiabg2da9iabek7aInaaBaaaleaacaaIYaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGOmaiaaigdaaeqaaOGaamyEamaaBaaaleaacaaIXaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIYaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@486B@ .ivreg y2 x3 (y1 = x1 x2 x4 x5)

#### 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:

(16)

and

(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).

. 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.

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.

### Exercises

#### Exercise 1

Cigarette advertising and sales. A great deal of controversy exists over the issue of whether advertising expenditures affect sales. This controversy is particularly sharp when it affects policy decisions. An example of this phenomenon is the controversy over the impact of cigarette advertising on advertising sales. While many public policy experts advocate bans on cigarette advertising, a majority of economists caution against bans on cigarette advertising. The economists point out that there is little theoretical reasons to believe that cigarette advertising affects total demand for cigarettes. Instead, economists argue that cigarette advertising only affects brand choice and not the number of cigarettes that people smoke. Moreover, these economists point out that there is also little empirical evidence that supports the argument that cigarette advertising affects the demand for cigarettes. Given the negative impact advertising bans have on freedom of speech, most economists conclude that the negative effects of cigarette advertising bans outweigh the benefits of the bans.

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:

(18)

where

qt = cigarettes consumed per person over age 15,

pct = retail price of cigarettes,

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

At = 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:

(19)

where:

pat = advertising price index, and

mt = 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:

(20)

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:

ln( q t )= α 0 + α 1 ln( p c t )+ α 2 ln( y t )+ α 3 ln( A t )+ α 4 D 64 t ln( q t )= α 0 + α 1 ln( p c t )+ α 2 ln( y t )+ α 3 ln( A t )+ α 4 D 64 t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@61C9@
(21)

and

ln( A t )= β 0 + β 1 ln( q t )+ β 2 ln( p a t )+ β 3 m t + β 4 m t 2 . ln( A t )= β 0 + β 1 ln( q t )+ β 2 ln( p a t )+ β 3 m t + β 4 m t 2 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5E7C@
(22)
 Year Cigarettes Sold per Person Over Age 15 Retail Price of Cigarettes Real Advertising per Person Over Age 15 Advertising Price Index Degree of Monopoly Disposable Personal Income in 1958 dollars 1955 3163.090 93.9693 0.96100 95.4775 18.595 1659 1956 3230.517 94.7049 1.09969 94.3800 19.207 1673 1957 3313.033 94.2535 1.22180 96.2125 20.165 1683 1958 3479.063 94.7712 1.40471 97.8300 21.736 1666 1959 3584.930 98.1779 1.45816 98.2800 22.042 1735 1960 3676.912 100.0000 1.37863 100.0000 22.04 1749 1961 3743.354 99.8677 1.31871 102.0400 22.465 1756 1962 3733.504 99.6761 1.35467 102.9725 22.226 1814 1963 3775.886 101.3630 1.51345 103.9525 22.848 1867 1964 3648.211 102.3110 1.73665 103.4775 23.168 1948 1965 3710.075 105.7510 1.59761 103.7225 23.598 2047 1966 3689.386 108.0450 1.71062 104.2200 25.085 2127 1967 3652.016 109.2490 1.71444 104.6125 26.310 2164

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?

#### Exercise 2

Exercise 2. Demand and supply of commercial loans. We are interested in estimating the demand for commercial loans by business firms and the supply of commercial loans by banks. We have available in Table 6 monthly data from the U. S. commercial loan market for the period from January, 1979 through December, 1984 and available in the MS Excel file Exercise 2.xls.8 Define:

Qt = total commercial loans (billions of dollars)

Rt = average prime rate charged by banks

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

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

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

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

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

Q t = β 0 + β 1 R t + β 2 R D t + β 3 X t + μ t Q t = β 0 + β 1 R t + β 2 R D t + β 3 X t + μ t MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBaaaleaacaWG0baabeaakiabg2da9iabek7aInaaBaaaleaacaaIWaaabeaakiabgUcaRiabek7aInaaBaaaleaacaaIXaaabeaakiaadkfadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaWGsbGaamiramaaBaaaleaacaWG0baabeaakiabgUcaRiabek7aInaaBaaaleaacaaIZaaabeaakiaadIfadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaH8oqBdaWgaaWcbaGaamiDaaqabaaaaa@508C@
(23)

and

Q t = α 0 + α 1 R t + α 2 R S t + α 3 y t + ε t . Q t = α 0 + α 1 R t + α 2 R S t + α 3 y t + ε t . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBaaaleaacaWG0baabeaakiabg2da9iabeg7aHnaaBaaaleaacaaIWaaabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIXaaabeaakiaadkfadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaHXoqydaWgaaWcbaGaaGOmaaqabaGccaWGsbGaam4uamaaBaaaleaacaWG0baabeaakiabgUcaRiabeg7aHnaaBaaaleaacaaIZaaabeaakiaadMhadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamiDaaqabaaaaa@50A5@
(24)

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?

 N Date Q R RD X RS y 1 January-79 251.8 11.75 9.25 150.8 9.35 994.3 2 February-79 255.6 11.75 9.26 151.5 9.32 1002.5 3 March-79 259.8 11.75 9.37 152.0 9.48 994.0 4 April-79 264.7 11.75 9.38 153.0 9.46 997.4 5 May-79 268.8 11.75 9.50 150.8 9.61 1013.2 6 June-79 274.6 11.65 9.29 152.4 9.06 1015.6 7 July-79 276.9 11.54 9.20 152.6 9.24 1012.3 8 August-79 280.5 11.91 9.23 152.8 9.52 1020.9 9 September-79 288.1 12.90 9.44 151.6 10.26 1043.6 10 October-79 288.3 14.39 10.13 152.4 11.70 1062.6 11 November-79 287.9 15.55 10.76 152.4 11.79 1058.5 12 December-79 295.0 15.30 11.31 152.1 12.64 1076.3 13 January-80 295.1 15.25 11.86 152.2 13.50 1063.1 14 February-80 298.5 15.63 12.36 152.7 14.35 1070.0 15 March-80 301.7 18.31 12.96 152.6 15.20 1073.5 16 April-80 302.0 19.77 12.04 152.1 13.20 1101.1 17 May-80 298.1 16.57 10.99 148.3 8.58 1097.1 18 June-80 297.8 12.63 10.58 144.0 7.07 1088.7 19 July-80 301.2 11.48 11.07 141.5 8.06 1099.9 20 August-80 304.7 11.12 11.64 140.4 9.13 1111.1 21 September-80 308.1 12.23 12.02 141.8 10.27 1122.2 22 October-80 315.6 13.79 12.31 144.1 11.62 1161.4 23 November-80 323.1 16.06 11.94 146.9 13.73 1200.6 24 December-80 330.6 20.35 13.21 149.4 15.49 1239.9 25 January-81 330.9 20.16 12.81 151.0 15.02 1223.5 26 February-81 331.3 19.43 13.35 151.7 14.79 1207.1 27 March-81 331.6 18.04 13.33 151.5 13.36 1190.6 28 April-81 336.2 17.15 13.88 152.1 13.69 1206.0 29 May-81 340.9 19.61 14.32 151.9 16.30 1221.4 30 June-81 345.5 20.03 13.75 152.7 14.73 1236.7 31 July-81 350.3 20.39 14.38 152.9 14.95 1221.5 32 August-81 354.2 20.50 14.89 153.9 15.51 1250.3 33 September-81 366.3 20.08 15.49 153.6 14.70 1293.7 34 October-81 361.7 18.45 15.40 151.6 13.54 1224.6 35 November-81 365.5 16.84 14.22 149.1 10.86 1254.1 36 December-81 361.4 15.75 14.23 146.3 10.85 1288.7 37 January-82 359.8 15.75 15.18 143.4 12.28 1251.5 38 February-82 364.6 16.56 15.27 140.7 13.48 1258.3 39 March-82 372.4 16.50 14.58 142.7 12.68 1295.0 40 April-82 374.7 16.50 14.46 141.5 12.70 1272.1 41 May-82 379.3 16.50 14.26 140.2 12.09 1286.1 42 June-82 386.7 16.50 14.81 139.2 12.47 1325.8 43 July-82 384.4 16.26 14.61 138.7 11.35 1307.3 44 August-82 384.5 14.39 13.71 138.8 8.68 1321.7 45 September-82 395.0 13.50 12.94 138.4 7.92 1335.5 46 October-82 393.7 12.52 12.12 137.3 7.71 1345.2 47 November-82 398.9 11.85 11.68 135.7 8.07 1358.1 48 December-82 395.3 11.50 11.83 134.9 7.94 1409.7 49 January-83 392.4 11.16 11.79 135.2 7.86 1385.4 50 February-83 392.3 10.98 12.01 137.4 8.11 1412.6 51 March-83 395.9 10.50 11.73 138.1 8.35 1419.5 52 April-83 393.5 10.50 11.51 140.0 8.21 1411.0 53 May-83 391.7 10.50 11.46 142.6 8.19 1413.1 54 June-83 395.3 10.50 11.74 144.4 8.79 1443.8 55 July-83 397.7 10.50 12.15 146.4 9.08 1438.1 56 August-83 400.6 10.89 12.51 149.7 9.34 1461.4 57 September-83 402.7 11.00 12.37 151.8 9.00 1448.9 58 October-83 405.3 11.00 12.25 153.8 8.64 1459.0 59 November-83 412.0 11.00 12.41 155.0 8.76 1499.4 60 December-83 420.1 11.00 12.57 155.3 9.00 1508.9 61 January-84 424.4 11.00 12.20 156.2 8.90 1504.1 62 February-84 428.8 11.00 12.08 158.5 9.09 1499.3 63 March-84 433.1 11.21 12.57 160.0 9.52 1494.5 64 April-84 439.7 11.93 12.81 160.8 9.69 1501.5 65 May-84 447.3 12.39 13.28 162.1 9.83 1541.3 66 June-84 452.9 12.60 13.55 162.8 9.87 1532.9 67 July-84 454.4 13.00 13.44 164.4 10.12 1535.5 68 August-84 455.2 13.00 12.87 165.9 10.47 1539.0 69 September-84 459.9 12.97 12.66 166.0 10.37 1549.9 70 October-84 467.7 12.58 12.63 165.0 9.74 1578.9 71 November-84 468.7 11.77 12.29 164.4 8.61 1578.2 72 December-84 476.8 11.06 12.13 164.8 8.06 1631.2

### References

Angrist, Joshua D. and Alan B. Krueger (2001). Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments. Journal of Economic Perspectives 15(4): 69–85.

Berndt, Ernst R. (1991). The Practice of Econometrics (Reading, MA: Addison-Wesley Publishing Company).

Greene, William H. (1990). Econometric Analysis (New York: Macmillan Publishing Company).

Murray, Michael P. (2006a). Avoiding Invalid Instruments and Coping with Weak Instruments. Journal of Economic Perspectives 20(4): 111-132.

Murray, Michael P. (2006b). Econometrics: A Modern Introduction. (Boston: Addison-Wesley): Chapter 13.

Schmalensee, Richard (1972). The Economics of Advertising (Amsterdam: North-Holland Publishing Company).

StataCorp (2003). Stata Statistical Software: Release 8 (College Station, TX: Stata Corporation): Volume 2: Reference G-M, pages 186-194.

Stock, James H, and Mark W. Watson (2003). Introduction to Econometrics (Boston, MA: Addison-Wesley): Chapter 10.

US Department of Commerce (1975). Historical Statistics of the United States: Colonial Times to 1970 (Washington: Government Printing Office).

US Surgeon General’s Advisory Committee (1964). Smoking and Health (Washington: Government Printing Office).

## Footnotes

1. A stochastic variable is a random variable—i.e., a variable whose value is determined as a result of a process involving an uncertain outcome.
2. Greene suggested this example in 1990 when most people paid their bills with checks. Currently it would not be such a good example because of the development of electronic payment of bills.
3. In these notes I discuss only what is known in the literature as the order condition for identification. The order condition is necessary for identification. Another condition—the rank condition—is a sufficient condition. See Greene (1990: Chapter 19, especially pp. 600-609) for a fuller discussion of simultaneous-equation models and the identification problem.
4. Using one of the exogenous variables in an equation as an instrument will create perfect multicollinearity in the first stage regression.
5. We exclude Equation (15) from this discussion because it is under-identified and, thus, cannot be estimated.
6. The advantage of the ivreg command is that it allows you to estimate a single equation of a system of equations without fully specifying the equations in the rest of the model. Use the command reg3 if you want to specify the whole model or use Three-Stage Least Squares.
7. The description of the command “ivreg depvar [varlist1] (varlist2=varlist_iv)” in the Stata help file is “ivreg fits a linear regression model using instrumental variables (or two-stage least squares) of depvar on varlist1 and varlist2 using varlist_iv (along with varlist1) as instruments for varlist2. In the language of two-stage least squares, varlist1 and varlist_iv are the exogenous variables and varlist2 the endogenous variables.”
8. The model and data for this problem first appeared in Maddala, G. S. (1988) Introductory Econometrics (New York: Macmillan Publishing Company): 331-317.
9. See Berndt, Ernst R. (1991) The Practice of Econometrics (Reading, MA: Addison-Wesley Publishing Company): 375-380.

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