Asymptotics for LS,GLS, and feasible GLS statistics in an AR(1) model with conditional heteroskedasticity

We consider a first-order autoregressive model with conditionally heteroskedastic innovations. The asymptotic distributions of least squares (LS), infeasible generalized least squares (GLS), and feasible GLS estimators and $t$ statistics are determined. The GLS procedures allow for misspecification of the form of the conditional heteroskedasticity and, hence, are referred to as quasi-GLS procedures. The asymptotic results are established for drifting sequences of the autoregressive parameter ${\rho }_n$ and the distribution of the time series of innovations. In particular, we consider the full range of cases in which ${\rho }_n$ satisfies $n(1?{\rho }_n)\to \infty$ and $n(1?{\rho }_n)\to h_1\in [0,\infty )$ as $n\to \infty$, where $n$ is the sample size. Results of this type are needed to establish the uniform asymptotic properties of the LS and quasi-GLS statistics.     

Mining big data using parsimonious factor,machine learning,variable selection and shrinkage methods

A number of recent studies in the economics literature have focused on the usefulness of factor models in the context of prediction using “big data” (see Bai and Ng, 2008; Dufour and Stevanovic, 2010; Forni, Hallin, Lippi, & Reichlin, 2000; Forni et al., 2005; Kim and Swanson, 2014a; Stock and Watson, 2002b, 2006, 2012, and the references cited therein). We add to this literature by analyzing whether “big data” are useful for modelling low frequency macroeconomic variables, such as unemployment, inflation and GDP. In particular, we analyze the predictive benefits associated with the use of principal component analysis (PCA), independent component analysis (ICA), and sparse principal component analysis (SPCA). We also evaluate machine learning, variable selection and shrinkage methods, including bagging, boosting, ridge regression, least angle regression, the elastic net, and the non-negative garotte. Our approach is to carry out a forecasting “horse-race” using prediction models that are constructed based on a variety of model specification approaches, factor estimation methods, and data windowing methods, in the context of predicting 11 macroeconomic variables that are relevant to monetary policy assessment. In many instances, we find that various of our benchmark models, including autoregressive (AR) models, AR models with exogenous variables, and (Bayesian) model averaging, do not dominate specifications based on factor-type dimension reduction combined with various machine learning, variable selection, and shrinkage methods (called “combination” models). We find that forecast combination methods are mean square forecast error (MSFE) “best” for only three variables out of 11 for a forecast horizon of $h=1$, and for four variables when $h=3$ or $12$. In addition, non-PCA type factor estimation methods yield MSFE-best predictions for nine variables out of 11 for $h=1$, although PCA dominates at longer horizons. Interestingly, we also find evidence of the usefulness of combination models for approximately half of our variables when $h>1$. Most importantly, we present strong new evidence of the usefulness of factor-based dimension reduction when utilizing “big data” for macroeconometric forecasting.     

Pythagorean generalization of testing the equality of two symmetric positive definite matrices

We provide a new test for equality of two symmetric positive-definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality or the sandwich asymptotic covariance matrix of the GMM estimator. The test relies on a new characterization of equality between two $k$ dimensional symmetric positive-definite matrices $A$ and $B$: the traces of $A{B}^{?1}$ and $B{A}^{?1}$ are equal to $k$ if and only if $A=B$. Using this simple criterion, we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions. A preferred test in the class with good omni-directional power is recommended for practical work. Monte Carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives. The test is applicable in many settings, including GMM estimation, SVAR models and high dimensional variance matrix settings.     

The ZD-GARCH model: A new way to study heteroscedasticity

This paper proposes a first-order zero-drift GARCH (ZD-GARCH(1, 1)) model to study conditional heteroscedasticity and heteroscedasticity together. Unlike the classical GARCH model, the ZD-GARCH(1, 1) model is always non-stationary regardless of the sign of the Lyapunov exponent ${\gamma }_0$, but interestingly it is stable with its sample path oscillating randomly between zero and infinity over time when ${\gamma }_0=0$. Furthermore, this paper studies the generalized quasi-maximum likelihood estimator (GQMLE) of the ZD-GARCH(1, 1) model, and establishes its strong consistency and asymptotic normality. Based on the GQMLE, an estimator for ${\gamma }_0$, a $t$-test for stability, a unit root test for the absence of the drift term, and a portmanteau test for model checking are all constructed. Simulation studies are carried out to assess the finite sample performance of the proposed estimators and tests. Applications demonstrate that a stable ZD-GARCH(1, 1) model is more appropriate than a non-stationary GARCH(1, 1) model in fitting the KV-A stock returns in Francq and Zakoïan (2012).