Nonparametric multidimensional fixed effects panel data models

Abstract

Multidimensional panel datasets are routinely employed to identify marginal effects in empirical research. Fixed effects estimators are typically used to deal with potential correlation between unobserved effects and regressors. Nonparametric estimators for one-way fixed effects models exist, but are cumbersome to employ in practice as they typically require iteration, marginal integration or profile estimation. We develop a nonparametric estimator that works for essentially any dimension fixed effects model, has a closed form solution and can be estimated in a single step. A cross-validation bandwidth selection procedure is proposed and asymptotic properties (for either a fixed or large time dimension) are given. Finite sample properties are shown via simulations, as well as with an empirical application, which further extends our model to the partially linear setting.

Keywords:

• Fixed effects
• multidimensional
• nonparametric
• panel

JEL Classification:

• C14
• C23

Notes

1 Although our interest is in the gradient, researchers may also be interested the conditional mean. We discuss how to estimate the unknown function in Section 2.4.

2 In a previous version of the article, a fixed effects estimator using a within transformation to deal with the presence of the individual effects was proposed. However, a high-dimensional kernel weight was required to deal with the well-known non-negligible asymptotic bias of this type of differencing estimator. The resulting nonparametric estimator was subject to a large variance and slow rate of convergence. The theoretical and simulation results for our multidimensional within estimator are available upon request.

3 For example, a fourth-order model can be given as $Y_{\mathit{\text{ijlt}}}=X_{\mathit{\text{ijlt}}}^{\top }\beta +{\mu }_i+{\gamma }_j+{\eta }_l+{\lambda }_t+{ϵ}_{\mathit{\text{ijlt}}},$ where l represents a third cross-sectional dimension and η_l is its corresponding fixed effect.

4 See Mundra (2005), Lee et al. (2019), Qian and Wang (2012) or Rodriguez-Poo and Soberon (2015) for nonparametric estimation in a two-dimensional setting.

5 Given that the fixed effects are additively separable, it can be argued that this is a semiparametric model. We nonetheless follow the literature and refer to this as a nonparametric fixed effects model.

6 An interesting paper by Freyberger (2018) proposes a nonparametric panel data model with two-dimensional, unobserved (interactive) individual effects that enter non-additively with a fixed time dimension. In the case where the individual effects enter in this particular non-additive structure, this estimator would be preferable. That being said, our primary interest lies in higher-dimensional panel and while it may be feasible to extend this estimator to the higher-dimensional panel data models we have in mind, this is beyond the scope of this article.

7 Alternative transformations are proposed in the literature for this type of multi-dimensional problem. For a fully parametric model, Balazsi et al. (2017) suggest the following transformation

${\tilde{Y}}_{\mathit{\text{ijt}}}=Y_{\mathit{\text{ijt}}}-{\overline{Y}}_{i··}-{\overline{Y}}_{·j·}-{\overline{Y}}_t+2\overline{Y},$

where ${\overline{Y}}_{i··}={(N_{2}T)}^{-1}\sum _{\mathit{\text{jt}}}{Y}_{\mathit{\text{ijt}}}$ and ${\overline{Y}}_{·j·}={(N_{1}T)}^{-1}\sum _{it}{Y}_{\mathit{\text{ijt}}}.$ However, its extension to the nonparametric framework is not straightforward as we would have to use a kernel weight that controls the distance among all time periods and cross-sectional units to avoid the non-negligible asymptotic bias pointed out in Mundra (2005) for these types of differencing specifications.

8 See Fan and Gijbels (1995b) or Ruppert and Wand (1994) for further details.

9 Using this notation makes the theory more general for additional dimensions.

10 Note that in the above minimization problem, we only use non-overlapped cross-sections of j. More precisely, we use only the differencing between j and j – 1, but we do not include the difference between other distinct pairs of j₁ and j₂, where $|j_1-j_2|>1.$ Therefore, we have dropped more than half of the sample related with the individuals j, so an efficiency effect over the resulting estimator is expected. If we included all possible differences, the resulting samples are no longer independent in cross sections and it would be necessary to resort to U-statistics techniques to obtain the main asymptotic properties of the local linear estimator. Therefore, if the estimation of the level function was the primary objective of the article, the U-statistics solution would be recommended.

11 In practice, authors use a diagonal bandwidth matrix.

12 For the four-dimensional panel we proposed before, the transformation would be ${\tilde{Y}}_{\text{ijls}}=Y_{\mathit{\text{ijlt}}}-Y_{\text{ijls}}-{\overline{Y}}_t+{\overline{Y}}_s,$ for t < s, where ${\overline{Y}}_t$ and ${\overline{Y}}_s$ are the corresponding cross-sectional means.

13 A potentially interesting extension would be to consider factor models. However, this is beyond the scope of this paper.

14 The use of bounded kernel functions was purely for convenience in the asymptotic development. It is worth noting that we also ran each of the simulations with Gaussian kernels and our estimators performed well. In fact, we found non-trivial improvements as compared with Lee et al. (2019). When using an Epanechnikov kernel function, the performance of the two estimators was nearly indistinguishable. This bodes well for the pairwise estimator as it is obtained under less restrictive assumptions. These results are available upon request.

15 Eriksen and Ross (2015) attempt to further model heterogeneity by interacting the log of vouchers with a fifth-order polynomial of the log of the supply elasticity.

16 It is arguable that the MSA fixed effect is redundant given that rental units do not change MSAs over time. We also attempted to model a two-way fixed effects specification and found that the results are qualitatively similar. These are available upon request.

17 The supply elastic and inelastic regions are only available for populations in excess of 500,000. This only leaves us with 94 of the 135 MSA areas. See Eriksen and Ross (2015) for more details.

Additional information

Funding

Financial support comes from the Programa Estatal de Generación de Conocimiento y Fortalecimiento Científico y Tecnológico del Sistema de I + D + i y del Programa Estatl de I + D + i Orientada a los Retos de la Sociedad/Spanish Ministry of Science and Innovation (Ref. PID2019-105986GB-C22). This work is part of the Research Project APIE 1/2015-17: “New methods for the empirical analysis of financial markets” of the Santander Financial Institute (SANFI) of UCEIF Foundation resolved by the University of Cantabria and funded with sponsorship from Banco Santander.