Three Essays in Econometrics

Abstract

Economists examine extensive cross sectional and time series data. This information allows them to study economic decision making of households, firms and countries over time or to produce forecasts of financial time series to support portfolio allocation and risk management. When analyzing and interpreting these datasets, the linear regression model continues to be fundamental to sound empirical work. The three chapters in this thesis contribute to solving several econometric issues in linear regression analysis. In the first two chapters, a potentially large number of regression parameters arises in two distinct econometric frameworks. Estimating these parameters to produce accurate forecasts of an economic variable of interest is the objective in the first chapter. The inferential methods presented in the second chapter enable the researcher to decide whether estimating a high-dimensional parameter vector is appropriate or whether a more parsimonious regression model applies. In contrast, chapter 3 shifts attention to the predictability of economic time series under a general statistical loss function.<br /> In particular, chapter 1 examines forecasting regressions that employ many predictors, leading to the task of estimating a large number of parameters in a linear regression. Here, many regressors arise naturally due to a frequency mismatch between the series of interest and the series that is considered to have predictive power. These mixed frequency regression models arise often in macroeconomics and finance, if, for instance, quarterly g.d.p. or the monthly volatility of a return index is forecasted with daily observations of macroeconomic leading indicators or (intra-)daily observations of financial variables. A new estimation procedure for these mixed data sampling (MIDAS) regression models is proposed. The estimator is a modified ordinary least squares (OLS) estimator which assumes that the weights assigned to high-frequency regressors are a smooth function and complements the least squares criterion by a smoothness penalty, resulting in a penalized least squares (PLS) estimator. The estimation method does not rely on a particular parametric specification of the weighting function, but depends on a smoothing parameter. Several methods are presented to choose this parameter including a variant of the Akaike information criterion (AIC). A simulation study is conducted to evaluate the estimation accuracy as measured by the mean squared error of the modified OLS estimator and the parametric MIDAS approach, which requires estimation by non-linear least squares (NLS). The simulation results illustrate in which cases the PLS estimator produces more accurate estimates than the parametric NLS estimator, and in which cases the parametric approach performs better. The results show that the PLS estimator is flexible alternative method to estimate MIDAS regression models. These MIDAS approaches are then employed to forecast volatility of the German stock index (DAX). In addition to the mixed frequency models, the GARCH(1,1) model is used as a benchmark. Using current and lagged absolute returns as predictors, MIDAS-PLS provides more precise forecasts than MIDAS-NLS or the GARCH(1,1) model over biweekly or monthly forecasting horizons. In a companion paper, the PLS estimator is used to forecast the monthly German inflation rate, see Breitung et al. (2013).<br /> In chapter 2, which is joint work with Jörg Breitung and Nazarii Salish, the linear panel data model is studied, in which observations are available both in the cross section and in the time series dimension. When estimating a panel regression, it must be decided whether the economic relationship of interest is taken to be homogenous, such that the same parameter vector applies to all cross sectional units, or whether a heterogeneous empirical model is more appropriate, in which the regression parameters differ between cross sectional units. In a classical panel data setup, in which the cross section dimension is large relative to the time series dimension, modelling regression parameters to be indiviudal-specific introduces a large number of parameters to be estimated in the panel even if only a few explanatory variables are studied for each cross section unit individually. In this chapter, a statistical test is proposed to determine whether regression parameters are indiviudal-specific or common to all units in the cross section. Answering this question is important as a preparatory step in panel data analysis to select a parsimonious model if possible, and to choose the subsequent estimation procedure accordingly.<br /> To this end, the Lagrange Multiplier (LM) principle is employed 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 and heteroskedasticity. Moreover, the tests can be conveniently computed via simple artificial regressions. The limiting distribution of the LM test is derived and it is shown 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 normality if the time dimension is fixed. A further adjustment provides a heteroskedasticity-robust version of the LM test. The local power of these tests LM tests and the delta statistic proposed by Pesaran and Yamagata (2008) is then compared. The results of our 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 3, written in collaborative work with Matei Demetrescu, investigates the predictive regression model under a general statistical loss function, including mean squared error (MSE) loss as a special case. In this predictive regression, the coefficient of a predictor is tested for significance. While for stationary predictors this task does not pose difficulties, non-standard limiting distributions of standard inference methods arise once the regressors are endogenous, such that there is contemporaneous correlation between shocks of the regressor and the dependent variable, and persistent, so that the predictor is reverting very slowly to its long-run mean, if at all. With a more general loss function beyond squared error loss, endogeneity is loss-function specific. Thus, no endogeneity under OLS does not imply, and is not implied by, endogeneity under, say, an asymmetric quadratic loss function. Existent solutions for the endogeneity problem in predictive regressions with predictors of unknown persistence are valid for OLS-based inference only, and thus apply exclusively to MSE-optimal forecasting. An overidentified instrumental variable based test is proposed, using particular trending functions and transformations of the predictor as instruments. The test statistic is analogous to the Anderson- Rubin statistic but takes the relevant loss into account, and follows a chi-squared distribution asymptotically, irrespective of the degree of persistence of the predictor. The forward premium puzzle is then reexamined by providing evidence for deviations from MSE loss and by conducting robust inference of the rational expectations hypothesis. The analysis provides little evidence for failure of the rational expectations hypothesis, in contrast to early empirical tests in this literature.