Essays on Heterogeneity and Non-Linearity in Panel Data and Time Series Models

Abstract

In recent years advances in data collection and storage allow us to observe and analyze many financial, economic or environmental processes with higher precision. This in turn reveals new features of the underlying processes and creates a demand for the development of new econometric techniques. The aim of this thesis is to tackle some of these challenges in the filed of panel data and time series models. In particular, the first and the last chapters contribute to the issue of testing and estimating heterogeneous panel models with random coefficients. The second chapter discusses a generalization of the classical linear time series models to asymmetric ones and presents a test statistic to help empirical researchers to choose the appropriate modeling framework in this context. Finally, the objective of the third chapter is to extend the available (nonlinear) time series techniques on big data sets or functional data.<br /> In more detail, Chapter1, which is joint work with Joerg Breitung and Christoph Roling, employs the Lagrange Multiplier (LM) principle to test parameter homogeneity across cross-section units in panel data models. The test can be seen as a generalization of the Breusch-Pagan test against random individual effects to all regression coefficients. While the original test procedure assumes a likelihood framework under normality, several useful variants of the LM test are presented to allow for non-normality, heteroskedasticity and serially correlated errors. Moreover, the tests can be conveniently computed via simple artificial regressions. We derive the limiting distribution of the LM test and show that if the errors are not normally distributed, the original LM test is asymptotically valid if the number of time periods tends to infinity. A simple modification of the score statistic yields an LM test that is robust to non-normality if the number of time periods is fixed. Further adjustments provide versions of the LM test that are robust to heteroskedasticity and serial correlation. We compare the local power of our tests and the statistic proposed by Pesaran and Yamagata. The results of the Monte Carlo experiments suggest that the LM-type test can be substantially more powerful, in particular, when the number of time periods is small.<br /> Chapter 2, which is joint work with Thomas Nebeling, develops a Lagrange multiplier test statistic and its variants to test for the null hypothesis of no asymmetric effects of shocks on time series. In asymmetric time series models that allow for different responses to positive and negative past shocks the likelihood functions are, in general, non-differentiable. By making use of the theory of generalized functions Lagrange multiplier type tests and the resulting asymptotics are derived. The test statistics possess standard asymptotic limiting behavior under the null hypothesis. Monte Carlo experiments illustrate the accuracy of the asymptotic approximation and show that conventional model selection criteria can be used to estimate the required lag length. We provide an empirical application to the U.S. unemployment rate.<br /> In Chapter 3, written in collaborative work with Alexander Gleim, statistical tools for forecasting functional times series are developed, which for example can be used to analyze big data sets. To tackle the issue of time dependence we introduce the notion of functional dependence through scores of the spectral representation. We investigate the impact of time dependence thus quantified on the estimation of functional principal components. The rate of mean squared convergence of the estimator of the covariance operator is derived under long range dependence of the functional time series. After that, we suggest two forecasting techniques for functional time series satisfying our measure of time dependence and derive the asymptotic properties of their predictors. The first is the functional autoregressive model which is commonly used to describe linear processes. As our notion of functional dependence covers a broader class of processes we also study the functional additive autoregressive model and construct its forecasts by using the k-nearest neighbors approach. The accuracy of the proposed tools is verified through Monte Carlo simulations. Empirical relevance of the theory is illustrated through an application to electricity consumption in the Nordic countries.<br /> In Chapter 4, which was jointly done with Joerg Breitung, three main estimation procedures for the panel data models with heterogeneous slopes are discussed: pooling, generalized LS and mean-group estimator. In our analysis we take an explicit account of the statistical dependence that may exists between regressors and the heterogeneous effects of the slopes. It is shown that under systematic slope variations: (i) pooling gives inconsistent and highly misleading estimates, and (ii) generalized LS in general is not consistent even in settings when $N$ and $T$ are large, (iii) while mean-group estimator always provide consistent result at a price of higher variance. We contribute to the literature by suggesting a simple robustified version of the pooled based on Mundlak type corrections. This estimator provides consistent results and is asymptotically equivalent to the mean-group estimator for large N and T. Monte Carlo experiments confirm our theoretical findings and show that for large N and fixed T new estimator can be an attractive option when compare to the competitors.