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Heteroskedasticity and Spatiotemporal Dependence Robust Inference for Linear Panel Models with Fixed E ¤ ects
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kim, Min Seong Sun, Yixiao |
| Copyright Year | 2012 |
| Abstract | This paper studies robust inference for linear panel models with xed e¤ects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. We propose a bivariate kernel covariance estimator that is exible to nest existing estimators as special cases with certain choices of bandwidths. For distributional approximations, we embed the level of smoothing and the sample size in two di¤erent limiting sequences. In the rst case where the level of smoothing increases with the sample size, the proposed covariance estimator is consistent and the associated Wald statistic converges to a 2 distribution. We show that our covariance estimator improves upon existing estimators in terms of robustness and e¢ ciency. In the second case where the level of smoothing is xed, the covariance estimator has a random limit. We derive an asymptotically equivalent distribution of the Wald statistic and we show by asymptotic expansion that it depends on the bandwidth parameters, the kernel function, and the number of restrictions being tested. As the asymptotically equivalent distribution is nonstandard, we establish the validity of a convenient F -approximation to this distribution. For bandwidth selection, we employ and optimize a modi ed mean square error criterion. The exibility of our estimator and the proposed bandwidth selection procedure make our estimator adaptive to the dependence structure. This adaptiveness e¤ectively automates the selection of covariance estimators. Simulation results show that our proposed testing procedure works well in nite samples. Keywords: Adaptiveness, HAC estimator, F -approximation, Fixed-smoothing asymptotics, Fixed-e¤ects 2SLS, Increasing-smoothing asymptotics, Panel data, Optimal bandwidth, Robust inference, Spatiotemporal dependence JEL Classi cation Number : C13, C14, C23 Email: minseong.kim@ryerson.ca and yisun@ucsd.edu. We thank Brendan Beare, Alan Bester, Otilia Boldea, Pierre-Andre Chiappori, Tim Conley, Gordon Dahl, Feico Drost, Graham Elliott, Patrik Guggenberger, Jinyong Hahn, James Hamilton, Christian Hansen, Hiroaki Kaido, Ivana Komunjer, Esfandiar Maasoumi, Jan Magnus, Bertrand Melenberg, Leo Michelis, Choon-Geol Moon, Philip Neary, Benoit Perron, Elena Pesavento, Ingmar Prucha, Andres Santos, Halbert White and the participants of Panel Data Conference, Econometric Society World Congress, CESG and seminars at Tilburg, UCSD, Emory, UC Davis, Chicago Booth, Ryerson, Maryland, KIPF, Hanyang, and UWO. Sun gratefully acknowledges partial research support from NSF under Grant No. SES0752443. 1 Introduction This paper studies robust inference for linear panel models with xed e¤ects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. As economic data is potentially heterogeneous and correlated in unknown ways across individuals and time, robust inference in the panel setting is an important issue. See, for example, Betrand, Duo and Mullainathan (2004) and Petersen (2009). The main interest in this problem lies in (i) how to construct covariance estimators that take the correlation structure into account; (ii) how to approximate the sampling distribution of the associated test statistic; and (iii) how to select smoothing parameters in nite samples. Regarding covariance estimation, we propose a bivariate kernel estimator. In order to utilize the kernel in the spatial dimension, we need a priori knowledge about the dependence structure. It is often assumed that the covariance of two random variables at locations i and j is a decreasing function of an observable distance measure dij between them. The idea of using a distance measure to characterize spatial dependence is common in the spatial econometrics literature. See, for example, Conley (1999), Kelejian and Prucha (2007), Bester, Conley, Hansen and Vogelsang (2011, BCHV hereafter) and Kim and Sun (2011). There are several robust covariance estimators with correlated panel data. Arellano (1987) proposes the clustered covariance estimator (CCE) by extending the White standard error (White, 1980) to account for serial correlation. Wooldridge (2003) provides a concise review on the CCE. Driscoll and Kraay (1998, DK hereafter) suggest a di¤erent approach that uses a time series HAC estimator (e.g. Newey and West, 1987) applied to cross sectional averages of moment conditions. Gonçalves (2011) examines the properties of this estimator in linear panel models with xed e¤ects. Another approach considered in this paper is an extension of the spatial HAC estimator applied to time series averages of moment conditions, which we name the DK estimator. This is symmetric to the original DK estimator. Conley (1999) is among the rst to propose the spatial HAC estimator. Kelejian and Prucha (2007) argue that it can be extended to the panel setting with xed T . Our estimator includes these existing estimators as special cases, reducing to each of them with certain bandwidth choice. We refer to this as exibility. If the sequence of the bandwidth in the spatial dimension, dn; increases at a fast enough rate with the cross sectional sample size n, then our estimator with the rectangular kernel is asymptotically equivalent to the DK estimator. Similarly, if the sequence of the bandwidth in the time dimension, dT ; increases fast enough relative to the time series sample size T , then our estimator with the rectangular kernel is asymptotically equivalent to the DK estimator. On the other hand, if dn is assumed to approach zero, our estimator reduces to a generalized CCE de ned later in the paper. For distributional approximations, we consider two types of asymptotics: the increasingsmoothing asymptotics and the xed-smoothing asymptotics. The di¤erence lies in whether the level of smoothing increases or stays xed as the sample size increases. Let `i;n denote the number of individuals whose distance from individual i is less than or equal to dn and `n be the average of `i;n across i. We also de ne `t;T and `T in the same way along the time dimension. If dn; dT !1 as (n; T )!1 but slowly so that nT= (`n`T )!1, then the level of smoothing increases with the sample size. Under this increasing-smoothing asymptotics, our covariance estimator is consistent and the limiting distribution of the associated Wald statistic is a 2 distribution. The alternative estimators are also consistent under the increasing-smoothing asymptotics, but each estimator has an important limitation in practice. The performance of the CCE heavily depends on spatial correlation. While this estimator is quite e¢ cient in the presence of spatial |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://economics.ryerson.ca/workingpapers/wp029.pdf |
| Alternate Webpage(s) | http://econ.ucsb.edu/~doug/245a/Papers/Spatiotemporal%20Dependence.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
