11More Topics in Linear Unobserved E¤ects Models
This chapter continues our treatment of linear, unobserved e¤ects panel data models. We ﬁrst cover estimation of models where the strict exogeneity Assumption FE.1 fails but sequential moment conditions hold. A simple approach to consistent esti-mation involves di¤erencing combined with instrumental variables methods. We also cover models with individual slopes, where unobservables can interact with explana-tory variables, and models where some of the explanatory variables are assumed to be orthogonal to the unobserved e¤ect while others are not.
The ﬁnal section in this chapter brieﬂy covers some non-panel-data settings where unobserved e¤ects models and panel data estimation methods can be used.
11.1 Unobserved E¤ects Models without the Strict Exogeneity Assumption
11.1.1 Models under Sequential Moment Restrictions
In Chapter 10 all the estimation methods we studied assumed that the explanatory variables were strictly exogenous (conditional on an unobserved e¤ect in the case of ﬁxed e¤ects and ﬁrst di¤erencing). As we saw in the examples in Section 10.2.3, strict
exogeneity rules out certain kinds of feedback from yit to future values of xit. Gen-erally, random e¤ects, ﬁxed e¤ects, and ﬁrst di¤erencing are inconsistent if an ex-
planatory variable in some time period is correlated with uit. While the size of the inconsistency might be small—something we will investigate further—in other cases it can be substantial. Therefore, we should have general ways of obtaining consistent estimators as N ! y with T ﬁxed when the explanatory variables are not strictly exogenous.
The model of interest can still be written as
yit ¼ xitb þ ci þ uit; t ¼ 1;2;...;T ð11:1Þ
but, in addition to allowing ci and xit to be arbitrarily correlated, we now allow uit to be correlated with future values of the explanatory variables, ðxi;tþ1;xi;tþ2;...;xiTÞ. We saw in Example 10.3 that uit and xi;tþ1 must be correlated because xi;tþ1 ¼ yit. Nevertheless, there are many models, including the AR(1) model, for which it is
reasonable to assume that uit is uncorrelated with current and past values of xit. Following Chamberlain (1992b), we introduce sequential moment restrictions:
Eðuit jxit;xi;tÿ1;...;xi1;ciÞ ¼ 0; t ¼ 1;2;...;T ð11:2Þ
When assumption (11.2) holds, we will say that the xit are sequentially exogenous conditional on the unobserved e¤ect.
Given model (11.1), assumption (11.2) is equivalent to
Eðyit jxit;xi;tÿ1;...;xi1;ciÞ ¼ Eðyit jxit;ciÞ ¼ xitb þ ci
which makes it clear what sequential exogeneity implies about the explanatory vari-ables: after xit and ci have been controlled for, no past values of xit a¤ect the expected
value of yit. This condition is more natural than the strict exogeneity assumption, which requires conditioning on future values of xit as well.
Example 11.1 (Dynamic Unobserved E¤ects Model): tional explanatory variables is
yit ¼ zitg þr1 yi;tÿ1 þ ci þ uit
An AR(1) model with addi-
and so xit 1ðzit; yi;tÿ1Þ. Therefore, ðxit;xi;tÿ1;...;xi1Þ ¼ ðzit; yi;tÿ1;zi;tÿ1;...;zi1; yi0Þ, and the sequential exogeneity assumption (11.3) requires
Eðyit jzit; yi;tÿ1;zi;tÿ1;...;zi1; yi0;ciÞ ¼ Eðyit jzit; yi;tÿ1;ciÞ
¼ zitg þ r1 yi;tÿ1 þ ci ð11:5Þ
An interesting hypothesis in this model is H0: r1 ¼ 0, which means that, after unob-served heterogeneity, ci, has been controlled for (along with current and past zit), yi;tÿ1 does not help to predict yit. When r1 00, we say that fyitg exhibits state de-pendence: the current state depends on last period’s state, even after controlling for ci and ðzit;...;zi1Þ.
In this example, assumption (11.5) is an example of dynamic completeness condi-tional on ci; we covered the unconditional version of dynamic completeness in Section
7.8.2. It means that one lag of yit is su‰cient to capture the dynamics in the con-ditional expectation; neither further lags of yit nor lags of zit are important once ðzit; yi;tÿ1;ciÞ have been controlled for. In general, if xit contains yi;tÿ1, then as-sumption (11.3) implies dynamic completeness conditional on ci.
Assumption (11.3) does not require that zi;tþ1 ...;ziT be uncorrelated with uit, so that feedback is allowed from yit to ðzi;tþ1;...;ziTÞ. If we think that zis is uncorre-lated with uit for all s, then additional orthogonality conditions can be used. Finally, we do not need to restrict the value of r1 in any way because we are doing ﬁxed-T asymptotics; the arguments from Section 7.8.3 are also valid here.
Example 11.2 (Static Model with Feedback):
yit ¼ zitg þdwit þ ci þ uit
Consider a static panel data model
where zit is strictly exogenous and wit is sequentially exogenous:
More Topics in Linear Unobserved E¤ects Models 301
Eðuit jzi;wit;wi;tÿ1;...;wi1;ciÞ ¼ 0 ð11:7Þ
However, wit is inﬂuenced by past yit, as in this case:
wit ¼ zitx þ r1 yi;tÿ1 þ cci þrit ð11:8Þ
For example, let yit be per capita condom sales in city i during year t, and let wit be the HIV infection rate for year t. Model (11.6) can be used to test whether condom
usage is inﬂuenced by the spread of HIV. The unobserved e¤ect ci contains city-speciﬁc unobserved factors that can a¤ect sexual conduct, as well as the incidence of HIV. Equation (11.8) is one way of capturing the fact that the spread of HIV is in-ﬂuenced by past condom usage. Generally, if Eðri;tþ1uitÞ ¼ 0, it is easy to show that
Eðwi;tþ1uitÞ ¼ r1EðyituitÞ ¼ r1EðuitÞ > 0 under equations (11.7) and (11.8), and so strict exogeneity fails unless r1 ¼ 0.
Lagging variables that are thought to violate strict exogeneity can mitigate but
does not usually solve the problem. Suppose we use wi;tÿ1 in place of wit in equation (11.6) because we think wit might be correlated with uit. For example, let yit be the percentage of ﬂights canceled by airline i during year t, and let wi;tÿ1 be airline proﬁts during the previous year. In this case xi;tþ1 ¼ ðzi;tþ1;witÞ, and so xi;tþ1 is correlated with uit; this fact results in failure of strict exogeneity. In the airline example this issue may be important: poor airline performance this year (as measured by canceled
ﬂights) can a¤ect proﬁts in subsequent years. Nevertheless, the sequential exogeneity condition (11.2) is reasonable.
Keane and Runkle (1992) argue that panel data models for testing rational expectations using individual-level data generally do not satisfy the strict exogeneity requirement. But they do satisfy sequential exogeneity: in fact, in the conditioning set in assumption (11.2), we can include all variables observed at time t ÿ1.
What happens if we apply the standard ﬁxed e¤ects estimator when the strict exo-geneity assumption fails? Generally,
" T #ÿ1" T # plimðbFEÞ ¼ b þ Tÿ1 Eðx0 €itÞ Tÿ1 Eðx0 uitÞ
where xit ¼ xit ÿ xi, as in Chapter 10 (i is a random draw from the cross section). Now, under sequential exogeneity, EðxituitÞ ¼ E½ðxit ÿ xiÞ0uit ¼ ÿEðxiuitÞ because Eðx0 uitÞ ¼ 0, and so Tÿ1 t¼1 Eðx0 uitÞ ¼ ÿTÿ1 t¼1 EðxiuitÞ ¼ ÿEðxiuiÞ. We can bound the size of the inconsistency as a function of T if we assume that the time series
process is appropriately stable and weakly dependent. Under such assumptions, Tÿ1 t¼1 Eðx0 €itÞ is bounded. Further, VarðxiÞ and VarðuiÞ are of order Tÿ1. By the
302 Chapter 11
Cauchy-Schwartz inequality (for example, Davidson, 1994, Chapter 9), jEðxijuiÞja ½VarðxijÞVarðuiÞ1=2 ¼ OðTÿ1Þ. Therefore, under bounded moments and weak de-
pendence assumptions, the inconsistency from using ﬁxed e¤ects when the strict exogeneity assumption fails is of order Tÿ1. With large T the bias may be minimal. See Hamilton (1994) and Wooldridge (1994) for general discussions of weak depen-dence for time series processes.
Hsiao (1986, Section 4.2) works out the inconsistency in the FE estimator for the AR(1) model. The key stability condition su‰cient for the bias to be of order Tÿ1 is
jr1j < 1. However, for r1 close to unity, the bias in the FE estimator can be sizable, even with fairly large T. Generally, if the process fxitg has very persistent elements— which is often the case in panel data sets—the FE estimator can have substantial
If our choice were between ﬁxed e¤ects and ﬁrst di¤erencing, we would tend to prefer ﬁxed e¤ects because, when T > 2, FE can have less bias as N ! y. To see this point, write
" T #ÿ1" T #
plimðbFDÞ ¼ b þ Tÿ1 EðDx0 DxitÞ Tÿ1 EðDx0 DuitÞ ð11:9Þ t¼1 t¼1
If fxitg is weakly dependent, so is fDxitg, and so the ﬁrst average in equation (11.9) is bounded as a function of T. (In fact, under stationarity, this average does not depend on T.) Under assumption (11.2), we have
EðDxitDuitÞ ¼ EðxituitÞ þEðxi;tÿ1ui;tÿ1Þ ÿ Eðxi;tÿ1uitÞ ÿ Eðxitui;tÿ1Þ ¼ ÿEðxitui;tÿ1Þ
which is generally di¤erent from zero. Under stationarity, Eðxitui;tÿ1Þ does not de-pend on t, and so the second average in equation (11.9) is constant. This result shows
not only that the FD estimator is inconsistent, but also that its inconsistency does not depend on T. As we showed previously, the time demeaning underlying FE results in its bias being on the order of Tÿ1. But we should caution that this analysis assumes
that the original series, fðxit; yitÞ: t ¼ 1;...;Tg, is weakly dependent. Without this assumption, the inconsistency in the FE estimator cannot be shown to be of order Tÿ1.
If we make certain assumptions, we do not have to settle for estimators that are inconsistent with ﬁxed T. A general approach to estimating equation (11.1) under assumption (11.2) is to use a transformation to remove ci, but then search for in-strumental variables. The FE transformation can be used provided that strictly ex-ogenous instruments are available (see Problem 11.9). For models under sequential exogeneity assumptions, ﬁrst di¤erencing is more attractive.
More Topics in Linear Unobserved E¤ects Models 303
First di¤erencing equation (11.1) gives
Dyit ¼ Dxitb þDuit; t ¼ 2;3;...;T ð11:10Þ
Now, under assumption (11.2),
EðxisuitÞ ¼ 0; s ¼ 1;2;...;t ð11:11Þ
Assumption (11.11) implies the orthogonality conditions
EðxisDuitÞ ¼ 0; s ¼ 1;2;...;t ÿ 1 ð11:12Þ
so at time t we can use xi;tÿ1 as potential instruments for Dxit, where
xit 1ðxi1;xi2;...;xitÞ ð11:13Þ
The fact that xi;tÿ1 is uncorrelated with Duit opens up a variety of estimation procedures. For example, a simple estimator uses Dxi;tÿ1 as the instruments for Dxit: EðDxi;tÿ1DuitÞ ¼ 0 under assumption (11.12), and the rank condition rank EðDxi;tÿ1DxitÞ ¼ K is usually reasonable. Then, the equation
Dyit ¼ Dxitb þDuit; t ¼ 3;...;T ð11:14Þ
can be estimated by pooled 2SLS using instruments Dxi;tÿ1. This choice of instru-ments loses an additional time period. If T ¼ 3, estimation of equation (11.14) becomes 2SLS on a cross section: ðxi2 ÿ xi1Þ is used as instruments for ðxi3 ÿ xi2Þ. When T > 3, equation (11.14) is a pooled 2SLS procedure. There is a set of assumptions—the sequential exogeneity analogues of Assumptions FD.1–FD.3— under which the usual 2SLS statistics obtained from the pooled 2SLS estimation are valid; see Problem 11.8 for details. With Dxi;tÿ1 as the instruments, equation (11.14) is just identiﬁed.
Rather than use changes in lagged xit as instruments, we can use lagged levels of xit. For example, choosing ðxi;tÿ1;xi;tÿ2Þ as instruments at time t is no less e‰cient than the procedure that uses Dxi;tÿ1, as the latter is a linear combination of the for-mer. It also gives K overidentifying restrictions that can be used to test assumption (11.2). (There will be fewer than K if xit contains time dummies.)
When T ¼ 2, b may be poorly identiﬁed. The equation is Dyi2 ¼ Dxi2b þ Dui2, and, under assumption (11.2), xi1 is uncorrelated with Dui2. This is a cross section equation that can be estimated by 2SLS using xi1 as instruments for Dxi2. The esti-mator in this case may have a large asymptotic variance because the correlations between xi1, the levels of the explanatory variables, and the di¤erences Dxi2 ¼ xi2 ÿ xi1 are often small. Of course, whether the correlation is su‰cient to yield small enough standard errors depends on the application.
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