Identification of a Nonparametric Panel Data Model with Unobserved Heterogeneity and Lagged Dependent Variables∗ Ne¸se Yıldız†
Abstract Panel data are often used to allow for unobserved individual heterogeneity in econometric models. Recently, in an impressive paper Evdokimov (2010) provided conditions for nonparametric identification and estimation of all structural elements using short panels of a model that allows for heterogeneous marginal effects. In this paper, we show how, under an additional testable assumption, the lagged dependent variables can be allowed to have certain types of explicit ceteris paribus effects on the current period dependent variable in basically the same model studied in Evdokimov (2010). JEL Numbers: C3, C14. KEYWORDS: Dynamic panel data; semiparametric methods; identification. ∗
I would like to thank Kirill Evdokimov for valuable questions and comments. Department of Economics, University of Rochester, 231 Harkness Hall, Rochester, NY 14627; Email:
[email protected]; Phone: 585-275-5782; Fax: 585-256-2309. †
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Introduction
Panel data are often used to allow for unobserved individual heterogeneity in econometric models. Recently, in an impressive paper Evdokimov (2010) provided conditions for nonparametric identification and estimation of all structural elements using short panels of a model that allows for heterogeneous marginal effects. The analysis in Evdokimov (2010) is done under a certain strict exogeneity assumption (Assumption ID(ii)), which rules out lagged dependent variables as regressors.1 In this paper, we show how, under an additional testable assumption, the lagged dependent variables can be allowed to have certain types of explicit ceteris paribus effects on the current period dependent variable in basically the same model studied in Evdokimov (2010). Thus, the model we suggest allows for persistence in the dependent variable as a result of true state dependence in addition to persistence due to unobserved individual heterogeneity. See, for example, Heckman (1981) for a discussion of this. Specifically, the model we study is the following: Yit = m(Xit , αi ) + γYit−1 + Uit ,
i = 1, 2, ..., n,
t = 1, 2, ..., T,
(1)
where Xit is a vector of explanatory variables, Yit is a scalar outcome variable, αi is a scalar random variable representing unobserved individual heterogeneity, and Uit is a scalar idiosyncratic disturbance term. In addition, we assume Yi0 is observed. As in Evdokimov (2010) αi could be a random effect (i.e. independent of Xit ) or a fixed effect (i.e. correlated with Xit ). Finally, since γYit−1 enters additively into the model, censored and truncated regression models as well as binary response models are ruled out.
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Identification Results:
2.1
Identification of the Baseline Model:
In this section we give conditions under which γ is identified. To do so, we need to introduce some additional notation. Let X denote the support of Xit . For random vectors Ait for i = 1, 2, ..., n and t = 1, 2, ..., T , let Ai := (ATi1 , ATi2 , ..., ATiT )T , and A˜i := (ATi1 , ATi2 , ..., ATit−1 , ATit+1 , ..., ATiT )T . Let a ˜i be similarly defined. Assumption 2.1. Suppose that (i) T ≥ 2 and {Xi , Ui , αi , Yi0 }ni=1 is a random sample; (ii) fUit |Xit ,αi ,Xi,−t ,Ui,−t (u|x, α, x˜, u˜) = fUit |Xit (u|x) for all (u, x, α, x˜, u˜) ∈ R×X ×R×X T −1 × RT −1 and t = 1, 2, ..., T ; (iii) E[Uit |Xit = x] = 0, for all x ∈ X and t = 1, 2, ..., T ; 1
This assumption also rules out serial correlation in Uit below. Section 6.2 in the Appendix of Evdokimov (2010) provides conditions for identification of the model when Uit follows either an AR(1) or an MA(1) process.
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(iv) the joint density of (Xit , Xit−1 ) satisfies fXit ,Xit−1 (x, x) > 0 for all x ∈ X , where for discrete components of Xit the density is taken with respect to the counting measure. Assumption 2.1(i) is almost identical to Assumption ID(i) of Evdokimov (2010). We require that Yi0 are i.i.d. across individuals, in addition. This is because we have lagged Y in our model described by equation 1. Assumptions 2.1(ii)-(iv) are almost the same as Assumptions ID (ii), (iii) and (vi) in Evdokimov (2010), respectively. The difference is that in his case, Xit denotes all the observable regressors. In this paper, Xit denotes all the observable regressors except lagged dependent variables (or their higher order moments). In some situations γ (or γ(Xit ) below) may be the main parameter of interest. For example, one might be interested in testing habit formation or learning by doing, while allowing for unobserved individual heterogeneity to enter the model somewhat flexibly. If the goal is identification of γ (or γ(Xit )) only then Assumptions 2.1(ii) and 2.1(iii), which together constitute a strict exogeneity assumption, can be replaced by the following exchangability assumption: Assumption 2.2. E[Uit |Xit = x, Xit−1 = x] = E[Uit−1 |Xit = x, Xit−1 = x]. for each x ∈ X . Now we are ready to state our result of identification of γ. Theorem 2.1. Suppose Assumption 2.1 holds. If in addition E[Yi1 |Xi2 = Xi1 = x] 6= E[Yi0 |Xi2 = Xi1 = x],
(2)
for some x, then γ is identified. Proof. E[Yit − Yit−1 |Xit = x, Xit−1 = x, αi ] = γ E[Yit−1 − Yit−2 |Xit = x, Xit−1 = x, αi ] + E[Uit |Xit = x, Xit−1 = x, αi ] − E[Uit−1 |Xit = x, Xit−1 = x, αi ]
= γ E[Yit−1 − Yit−2 |Xit = x, Xit−1 = x, αi ] + E[Uit |Xit = x] − E[Uit−1 |Xit−1 = x] = γ (E[Yit−1 − Yit−2 |Xit = x, Xit−1 = x, αi ]) . The first equality follows from Assumption 2.1(iv), the second from 2.1(ii), and the third from 2.1(iii). Integrating out αi we get E[Yit − Yit−1 |Xit = x, Xit−1 = x] = γ E[Yit−1 − Yit−2 |Xit = x, Xit−1 = x] .
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This theorem makes the extra assumption that, for some x, E[Yi1 |Xi2 = Xi1 = x] 6= E[Yi0 |Xi2 = Xi1 = x]. Suppose |γ| < 1, m(x, α) = xT β + α, and the initial period was not 0, but t = −∞. Then under strict exogeneity, or assumption 2.1(ii), we could write "∞ # X T k E[Yi1 − Yi0 |Xi1 = Xi2 = x] = β E γ (Xi,1−k − Xi,−k )|Xi1 = Xi2 = x . k=0
If, in addition, Xit are i.i.d. over time, then we have E[Yi1 − Yi0 |Xi1 = Xi2 = x] = β T (x − E(Xit )), and assumption 2 is satisfied.2 Once γ is identified, the remaining structural elements of the model, in particular m(Xit , αi ) and the distribution of αi can be identified under the conditions given in Evdokimov (2010) by applying the analysis there to Y˜it := Yit − γYit−1 . Remark 1: With T = 2, if E[Yi1 |Xi2 = Xi1 = x] 6= E[Yi0 |Xi2 = Xi1 = x] for a.e. x ∈ X , then γ can be estimated by minimizing n
2 1 1 X [Yi2 − Yi1 ] − c[Yi1 − Yi0 ] K n i=1 hn
Xi2 − Xi1 hn
(3)
with respect to c, where K denotes the kernel function and hn denotes the bandwidth as in Fan and Gijbels (1996). Remark 2: Higher order moments of Yit could be included in the model if stricter “rank 2 2 conditions” hold. For example, if E[Yit−1 −Yit−2 |Xit = x, Xit−1 = x] and E[Yit−1 −Yit−2 |Xit = x, Xit−1 = x] are linearly independent, then ρ1 and ρ2 can be identified by the same strategy given in the proof of the above theorem in the model 2 Yit = m(Xit , αi ) + ρ1 Yit−1 + ρ2 Yit−1 + Uit .
Remark 3: Similarly, further lags of the dependent variable could be included in the model. Consider, for example, the model Yit = m(Xit , αi ) + γ1 Yit−1 + γ2 Yit−2 + Uit . This model will be identified if T ≥ 3 and E[Yit−1 − Yit−2 |Xit = x, Xit−1 = x] and E[Yit−2 − Yit−3 |Xit = x, Xit−1 = x] are linearly independent using the same differencing approach 2
I would like to thank the referee for these arguments.
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above. Remark 4: In this paper, as in Evdokimov (2010), Assumption 2.1(iv) is critical to the identification result. This assumption rules out covariates that vary over time deterministically. Remarks 4 and 5 Evdokimov (2010) discuss how variables that violate this assumption could be incorporated in the model. We could use the same approach as in his Remark 5 (i.e. under the same additional assumptions) here as well if T ≥ 3 and there are additional covariates, Zit , that are different from Xit , are independent of αi conditional on Xi , that satisfy E(Uit |Xit , Xit−1 , Zit , Zit−1 , αi ) = 0 for each t, and whose joint density satisfies fZit ,Zit−1 (z, z) = 0 for each z, to identify the model Yit = m(Xit , αi ) + g(Xit , Zit ) + γYit−1 + Uit . To do so, normalize g(x, z0 ) = 0 for some known z0 and for each x ∈ X as in Evdokimov (2010). Then for s 6= t, we have ∆E[Yit |Xit = Xit−1 = x, Zit = z, Zit−1 = z0 ]−∆E[Yis |Xis = Xis−1 = x, Zis = z, Zis−1 = z0 ] = γ ∆E[(Yit−1 |Xit = x, Xit−1 = x, Zit = z, Zit−1 = z0 ] − ∆E[(Yis−1 |Xis = x, Xis−1 = x, Zis = z, Zis−1 = z0 ] , where ∆E[Yit |Xit = Xit−1 = x, Zit = z, Zit−1 = z0 ] = E[Yit |Xit = Xit−1 = x, Zit = z, Zit−1 = z0 ] − E[Yit |Xit = Xit−1 = x, Zit = z, Zit−1 = z0 ]. Thus, γ could be identified and estimated under these assumptions provided that T ≥ 4 and ∆E[(Yit−1 |Xit = x, Xit−1 = x, Zit = z, Zit−1 = z0 ] − ∆E[(Yis−1 |Xis = x, Xis−1 = x, Zis = z, Zis−1 = z0 ] 6= 0. Note that this last condition is equivalent to (γ − 1)(E[Yi1 |Xi3 = Xi2 = x, Zi3 = z, Zi2 = z0 ] − E[Yi0 |Xi2 = Xi1 = x, Zi2 = z, Zi1 = z0 ]) 6= E[m(x, αi )|Xi3 = Xi2 = x, Zi3 = z, Zi2 = z0 ]−E[m(x, αi )|Xi2 = Xi1 = x, Zi2 = z, Zi1 = z0 ] In addition, higher order moments of Yit−1 or further lags of the dependent variable can also be included in this model; their coefficients could be identified using basically the same arguments as in Remarks 3 and 4 above.
2.2
Identification of the Extended Model:
In this section we discuss identification of the extended model Yit = m(Xit , αi ) + γ(Xit )Yit−1 + Uit ,
i = 1, 2, ..., n,
t = 1, 2, ..., T,
(4)
under the same assumptions given in the previous subsection. We start the analysis by studying when we have P E[Yit−1 |Xit = x, Xit−1 = x] − E[Yit−2 |Xit = x, Xit−1 = x] = 6 0 > 0. 5
To see when E[Yit−1 |Xit = x, Xit−1 = x] − E[Yit−2 |Xit = x, Xit−1 = x] 6= 0, suppose t = 0, 1, 2 and note that E[Yi2 |Xi2 = x, Xi1 = x, αi = a] = m(x, a) + γ(x)m(x, a) + γ 2 (x)E[Yi0 |Xi2 = x, Xi1 = x, αi = a] + γ(x)E[Ui1 |Xi2 = x, Xi1 = x, αi = a] + E[Ui2 |Xi2 = x, Xi1 = x, αi = a], (5) E[Yi1 |Xi2 = x, Xi1 = x, αi = a] = m(x, a) + γ(x)E[Yi0 |Xi2 = x, Xi1 = x, αi = a] + E[Ui1 |Xi2 = x, Xi1 = x, αi = a]. (6) By assumption E[Ui2 |Xi2 = x, Xi1 = x, αi = a] = E[Ui2 |Xi2 = x] = 0, and E[Ui1 |Xi2 = x, Xi1 = x, αi = a] = E[Ui1 |Xi1 = x] = 0. As a result, the difference between 5 and 6 equals ∆E[Yi2 |Xi2 = x, Xi1 = x, αi = a] = γ(x)m(x, a) + γ(x)[γ(x) − 1]E[Yi0 |Xi2 = x, Xi1 = x, αi = a]. (7) Similarly, ∆E[Yi1 |Xi2 = x, Xi1 = x, αi = a] = m(x, a) + [γ(x) − 1]E[Yi0 |Xi2 = x, Xi1 = x, αi = a]. (8) Thus, ∆E[Yi1 |Xi2 = x, Xi1 = x] 6= 0 ⇐⇒ E[m(Xi1 , αi )|Xi2 = x, Xi1 = x] 6= [1 − γ(x)]E[Yi0 |Xi2 = x, Xi1 = x]. (9) Theorem 2.2. Suppose Assumption 2.1 and Condition 9 hold for each x ∈ X . Then γ(x) is identified for each x ∈ X . Proof. Integrating αi out in the equations 7 and 8 above, we have γ(x) =
∆E[Yi2 |Xi2 = x, Xi1 = x] . ∆E[Yi1 |Xi2 = x, Xi1 = x]
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If for all x ∈ X , E[Yit−1 |Xit = x, Xit−1 = x] − E[Yit−2 |Xit = x, Xit−1 = x] 6= 0 then γ(x) is identified, and we can define Y˜it := Yit − γ(Xit )Yit−1 and apply the analysis in Evdokimov (2010) to identify other structural parameters in the model. If E[Yit−1 |Xit = x, Xit−1 = x] − E[Yit−2 |Xit = x, Xit−1 = x] 6= 0 for some x but not all, then we may extend identification of γ at each x by parametric assumptions. Remark: The model Yit = m(Xit , αi ) + g(Xit , Zit ) + γYit−1 + Uit . can be identified under the same additional assumptions as in Remark 4 above. Similarly, models that have higher order moments of Yit−1 or further lags of the dependent variable, similar to those described in Remarks 2 and 3, can be identified under the same additional assumptions described in those remarks.
References [1] Evdokimov, K. 2010, “Identification and Estimation of a Nonparametric Panel Data Model with Unobserved Heterogeneity,” unpublished manuscript, Princeton University. [2] Fan, J. and I. Gijbels (1996): Local polynomial modelling and its applications. CRC Press. [3] Heckman, J J. 1981, ”Statistical models for discrete panel data”, in: C F Manski and D McFadden, eds , Structural Analysis of Discrete Panel Data with Econometric Applications (MIT Press).
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