6Additional Single-Equation Topics
6.1 Estimation with Generated Regressors and Instruments
6.1.1 OLS with Generated Regressors
We often need to draw on results for OLS estimation when one or more of the regressors have been estimated from a ﬁrst-stage procedure. To illustrate the issues, consider the model
y ¼ b0 þ b1x1 þ þ bKxK þ gq þ u ð6:1Þ
We observe x1;...;xK, but q is unobserved. However, suppose that q is related to observable data through the function q ¼ fðw;dÞ, where f is a known function and w is a vector of observed variables, but the vector of parameters d is unknown (which is why q is not observed). Often, but not always, q will be a linear function of w and d. Suppose that we can consistently estimate d, and let d be the estimator. For each
observation i, qi ¼ fðwi;dÞ e¤ectively estimates qi. Pagan (1984) calls qi a generated regressor. It seems reasonable that, replacing qi with qi in running the OLS regression
yi on 1;xi1;xi2;...;xik;qi; i ¼ 1;...;N ð6:2Þ
should produce consistent estimates of all parameters, including g. The question is, What assumptions are su‰cient?
While we do not cover the asymptotic theory needed for a careful proof until Chapter 12 (which treats nonlinear estimation), we can provide some intuition here. Because plim d ¼ d, by the law of large numbers it is reasonable that
Nÿ1 Xqiui ! EðqiuiÞ; i¼1
Nÿ1 Xxijqi ! EðxijqiÞ i¼1
FromthisrelationitiseasilyshownthattheusualOLSassumptioninthepopulation— that u is uncorrelated with ðx1;x2;...;xK;qÞ—su‰ces for the two-step procedure to be consistent (along with the rank condition of Assumption OLS.2 applied to the expanded vector of explanatory variables). In other words, for consistency, replacing
qi with qi in an OLS regression causes no problems.
Things are not so simple when it comes to inference: the standard errors and test
statistics obtained from regression (6.2) are generally invalid because they ignore the sampling variation in d. Since d is also obtained using data—usually the same sample of data—uncertainty in the estimate should be accounted for in the second step. Nevertheless, there is at least one important case where the sampling variation of d can be ignored, at least asymptotically: if
116
E½‘ fðw;dÞ0u ¼ 0
Chapter 6
ð6:3Þ
g ¼ 0 ð6:4Þ then the pN-limiting distribution of the OLS estimators from regression (6.2) is the
same as the OLS estimators when q replaces q. Condition (6.3) is implied by the zero conditional mean condition
Eðujx;wÞ ¼ 0 ð6:5Þ
which usually holds in generated regressor contexts.
We often want to test the null hypothesis H0: g ¼ 0 before including q in the ﬁnal regression. Fortunately, the usual t statistic on q has a limiting standard normal dis-
tribution under H0, so it can be used to test H0. It simply requires the usual homo-skedasticity assumption, Eðu2 jx;qÞ ¼ s2. The heteroskedasticity-robust statistic
works if heteroskedasticity is present in u under H0.
Even if condition (6.3) holds, if g00, then an adjustment is needed for the asymptotic variances of all OLS estimators that are due to estimation of d. Thus, standard t statistics, F statistics, and LM statistics will not be asymptotically valid when g00. Using the methods of Chapter 3, it is not di‰cult to derive an ad-justment to the usual variance matrix estimate that accounts for the variability in
d (and also allows for heteroskedasticity). It is not true that replacing qi with qi simply introduces heteroskedasticity into the error term; this is not the correct way
to think about the generated regressors issue. Accounting for the fact that d depends on the same random sample used in the second-stage estimation is much di¤erent from having heteroskedasticity in the error. Of course, we might want to use a heteroskedasticity-robust standard error for testing H0: g ¼ 0 because heteroskedasticity in the population error u can always be a problem. However, just as with the usual OLS standard error, this is generally justiﬁed only under H0: g ¼ 0. A general formula for the asymptotic variance of 2SLS in the presence of gen-erated regressors is given in the appendix to this chapter; this covers OLS with gen-erated regressors as a special case. A general framework for handling these problems is given in Newey (1984) and Newey and McFadden (1994), but we must hold o¤
until Chapter 14 to give a careful treatment.
6.1.2 2SLS with Generated Instruments
In later chapters we will need results on 2SLS estimation when the instruments have been estimated in a preliminary stage. Write the population model as
Additional Single-Equation Topics 117
y ¼ xb þu ð6:6Þ
Eðz0uÞ ¼ 0 ð6:7Þ
where x is a 1 K vector of explanatory variables and z is a 1 L ðLbKÞ vector of intrumental variables. Assume that z ¼ gðw;lÞ, where gð;lÞ is a known function but l needs to be estimated. For each i, deﬁne the generated instruments zi 1gðwi;lÞ. What can we say about the 2SLS estimator when the zi are used as instruments?
By the same reasoning for OLS with generated regressors, consistency follows under weak conditions. Further, under conditions that are met in many applications, we can ignore the fact that the instruments were estimated in using 2SLS for infer-ence. Su‰cient are the assumptions that l is N-consistent for l and that
E½‘ gðw;lÞ0u ¼ 0 ð6:8Þ Under condition (6.8), which holds when EðujwÞ ¼ 0, the pN-asymptotic distribu-
tion of b is the same whether we use l or l in constructing the instruments. This fact greatly simpliﬁes calculation of asymptotic standard errors and test statistics. There-fore, if we have a choice, there are practical reasons for using 2SLS with generated instruments rather than OLS with generated regressors. We will see some examples in Part IV.
One consequence of this discussion is that, if we add the 2SLS homoskedasticity assumption (2SLS.3), the usual 2SLS standard errors and test statistics are asymp-totically valid. If Assumption 2SLS.3 is violated, we simply use the heteroskedasticity-robust standard errors and test statistics. Of course, the ﬁnite sample properties of the estimator using zi as instruments could be notably di¤erent from those using zi as instruments, especially for small sample sizes. Determining whether this is the case requires either more sophisticated asymptotic approximations or simulations on a case-by-case basis.
6.1.3 Generated Instruments and Regressors
We will encounter examples later where some instruments and some regressors are estimated in a ﬁrst stage. Generally, the asymptotic variance needs to be adjusted because of the generated regressors, although there are some special cases where the usual variance matrix estimators are valid. As a general example, consider the model
y ¼ xb þgfðw;dÞ þ u; Eðujz;wÞ ¼ 0
and we estimate d in a ﬁrst stage. If g ¼ 0, then the 2SLS estimator of ðb0;gÞ0 in the equation
118 Chapter 6
yi ¼ xib þ gfi þerrori
using instruments ðzi; fiÞ, has a limiting distribution that does not depend on the limiting distribution of Nðd ÿdÞ under conditions (6.3) and (6.8). Therefore, the
usual 2SLS t statistic for g, or its heteroskedsticity-robust version, can be used to test H0: g ¼ 0.
6.2 Some Speciﬁcation Tests
In Chapters 4 and 5 we covered what is usually called classical hypothesis testing for OLS and 2SLS. In this section we cover some tests of the assumptions underlying either OLS or 2SLS. These are easy to compute and should be routinely reported in applications.
6.2.1 Testing for Endogeneity
We start with the linear model and a single possibly endogenous variable. For nota-
tional clarity we now denote the dependent variable by y1 and the potentially endog-enous explanatory variable by y2. As in all 2SLS contexts, y2 can be continuous or binary, or it may have continuous and discrete characteristics; there are no restric-
tions. The population model is
y1 ¼ z1d1 þ a1y2 þu1 ð6:9Þ
where z1 is 1 L1 (including a constant), d1 is L1 1, and u1 is the unobserved dis-turbance. The set of all exogenous variables is denoted by the 1 L vector z, where z1 is a strict subset of z. The maintained exogeneity assumption is
Eðz0u1Þ ¼ 0 ð6:10Þ
It is important to keep in mind that condition (6.10) is assumed throughout this section. We also assume that equation (6.9) is identiﬁed when Eðy2u1Þ00, which requires that z have at least one element not in z1 (the order condition); the rank condition is that at least one element of z not in z1 is partially correlated with y2 (after netting out z1). Under these assumptions, we now wish to test the null hypothesis
that y2 is actually exogenous.
Hausman (1978) suggested comparing the OLS and 2SLS estimators of b 1 ðd1;a1Þ0 as a formal test of endogeneity: if y2 is uncorrelated with u1, the OLS and 2SLS estimators should di¤er only by sampling error. This reasoning leads to the
Hausman test for endogeneity.
Additional Single-Equation Topics 119
The original form of the statistic turns out to be cumbersome to compute because the matrix appearing in the quadratic form is singular, except when no exogenous variables are present in equation (6.9). As pointed out by Hausman (1978, 1983), there is a regression-based form of the test that turns out to be asymptotically equivalent to the original form of the Hausman test. In addition, it extends easily to other situations, including some nonlinear models that we cover in Chapters 15, 16, and 19.
To derive the regression-based test, write the linear projection of y2 on z in error form as
y2 ¼ zp2 þ v2 ð6:11Þ Eðz0v2Þ ¼ 0 ð6:12Þ
where p2 is L 1. Since u1 is uncorrelated with z, it follows from equations (6.11) and (6.12) that y2 is endogenous if and only if Eðu1v2Þ00. Thus we can test whether the structural error, u1, is correlated with the reduced form error, v2. Write the linear projection of u1 onto v2 in error form as
u1 ¼ r1v2 þe1 ð6:13Þ
where r1 ¼ Eðv2u1Þ=Eðv2Þ, Eðv2e1Þ ¼ 0, and Eðz0e1Þ ¼ 0 (since u1 and v2 are each orthogonal to z). Thus, y2 is exogenous if and only if r1 ¼ 0.
Plugging equation (6.13) into equation (6.9) gives the equation
y1 ¼ z1d1 þa1y2 þ r1v2 þe1 ð6:14Þ
The key is that e1 is uncorrelated with z1, y2, and v2 by construction. Therefore, a test of H0: r1 ¼ 0 can be done using a standard t test on the variable v2 in an OLS re-gression that includes z1 and y2. The problem is that v2 is not observed. Nevertheless, the reduced form parameters p2 are easily estimated by OLS. Let v2 denote the OLS residuals from the ﬁrst-stage reduced form regression of y2 on z—remember that z contains all exogenous variables. If we replace v2 with v2 we have the equation
y1 ¼ z1d1 þa1y2 þ r1^2 þerror ð6:15Þ
and d1, a1, and r1 can be consistently estimated by OLS. Now we can use the results on generated regressors in Section 6.1.1: the usual OLS t statistic for r1 is a valid test of H0: r1 ¼ 0, provided the homoskedasticity assumption Eðu2 jz;y2Þ ¼ s2 is sat-isﬁed under H0. (Remember, y2 is exogenous under H0.) A heteroskedasticity-robust t statistic can be used if heteroskedasticity is suspected under H0.
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