[Paper Review] Identification and Estimation of Time-Varying Nonseparable Panel Data Models without Stayers

Takuya Ishihara|arXiv (Cornell University)|Dec 26, 2017

Spatial and Panel Data AnalysisEconomics, Econometrics and Finance10 references3 citations

TL;DR

This paper proposes a nonparametric identification and estimation method for time-varying nonseparable panel data models without requiring stayers or time-invariant structural functions. By assuming the structural function is strictly increasing in a scalar unobservable and that unobservable distributions are time-invariant, the authors establish identification under weak support conditions and develop a consistent, asymptotically normal estimator for parametric models, with Monte Carlo evidence of strong finite-sample performance and extensions to discrete outcomes via partial identification.

ABSTRACT

This paper explores the identification and estimation of nonseparable panel data models. We show that the structural function is nonparametrically identified when it is strictly increasing in a scalar unobservable variable, the conditional distributions of unobservable variables do not change over time, and the joint support of explanatory variables satisfies some weak assumptions. To identify the target parameters, existing studies assume that the structural function does not change over time, and that there are "stayers", namely individuals with the same regressor values in two time periods. Our approach, by contrast, allows the structural function to depend on the time period in an arbitrary manner and does not require the existence of stayers. In estimation part of the paper, we consider parametric models and develop an estimator that implements our identification results. We then show the consistency and asymptotic normality of our estimator. Monte Carlo studies indicate that our estimator performs well in finite samples. Finally, we extend our identification results to models with discrete outcomes, and show that the structural function is partially identified.

Motivation & Objective

• To address the limitations of existing nonseparable panel models that require time-invariant structural functions and the existence of 'stayers'—individuals with identical regressor values across time periods.
• To develop a framework for identifying time-varying structural functions in nonseparable panel data models under weaker assumptions.
• To provide a consistent and asymptotically normal estimator for parametric specifications of the time-varying model.
• To extend the identification results to models with discrete outcomes, establishing partial identification of the structural function.

Proposed method

• The identification strategy relies on the structural function being strictly increasing in a scalar unobservable variable, with time-invariant conditional distributions of the unobservables.
• The joint support of the explanatory variables is required to satisfy weak regularity conditions, such as connectedness or full-dimensional support across time periods.
• A parametric model is assumed for the structural function, and a two-step estimation procedure is developed using conditional moment restrictions.
• The estimator is constructed via a generalized method of moments (GMM) approach, leveraging the monotonicity and time-invariance assumptions to identify parameters.
• Asymptotic normality is established using empirical process theory and bootstrap-based inference, with a novel bootstrap equicontinuity argument to handle the time-varying nature of the estimating equations.
• For discrete outcomes, the paper derives sharp bounds on the structural function using the same monotonicity and time-invariance assumptions, leading to partial identification.

Experimental results

Research questions

• RQ1 Can the structural function in a nonseparable panel data model be nonparametrically identified when it varies over time and no stayers exist?
• RQ2 Under what conditions can time-varying structural functions be identified without assuming time-invariant structural functions or the presence of individuals with identical regressor values across time periods?
• RQ3 How can a consistent and asymptotically normal estimator be constructed for parametric time-varying nonseparable panel models under these relaxed assumptions?
• RQ4 What are the finite-sample properties of the proposed estimator, and how does it compare to existing methods in terms of bias and mean squared error?
• RQ5 To what extent can the identification framework be extended to models with discrete outcomes, and what are the sharp bounds on the structural function in such cases?

Key findings

• The structural function is nonparametrically identified under the assumptions that it is strictly increasing in a scalar unobservable, the conditional distribution of the unobservable is time-invariant, and the support of the explanatory variables satisfies weak regularity conditions.
• The proposed estimator is consistent and asymptotically normal, with Monte Carlo simulations showing good finite-sample performance, including low bias and mean squared error that decrease with sample size.
• For $N=1600$, the mean squared error for $\theta_1$ drops to 0.0127, and for $\theta_2$ to 0.0122, indicating strong finite-sample convergence.
• In comparison to Hoderlein and White (2012), the proposed method yields lower bias and mean squared error in simulation 1(ii), with MSE decreasing from 0.0138 to 0.0073 as sample size increases from 500 to 1000.
• For discrete outcomes, the structural function is partially identified, with sharp bounds derived using the monotonicity and time-invariance assumptions, as shown in Table 4 for various $x$ values.
• The method successfully accommodates models with time-varying macroeconomic shocks that directly interact with unobservables—such as Engel functions with time-varying preferences—where prior models relying on additive or multiplicative time effects fail.