Bootstrap methods for testing and forecasting with estimated factors

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The number of potential predictors of a given variable of interest is often much larger than the number of time series observations. For this reason, factor-augmented regression models where some of the regressors include estimated factors have become increasingly popular in economics. The factors represent the common factors in a large panel factor model and are estimated in a first step using the method of principal components. In a second step, we regress the variable of interest on the estimated factors (and on additional observed regressors such as lags of the dependent variable). Although estimation of factor-augmented regression models is easy, inference is potentially complicated due to estimation of the factors. The general goal of this grant proposal is to contribute to the literature on inference for forecasting models with estimated factors. We can distinguish three main parts. Part A considers the problem of evaluating latent factors using the bootstrap. This is a problem of great interest in economics and finance since observed variables are often used as proxies for the latent common factors postulated by the theoretical models. We will propose bootstrap confidence intervals for the latent factors based on the method of principal components of Bai (2003). These intervals can be used to evaluate whether a given observed variable coincides with an estimated factor. As showed by Bai (2003), the asymptotic covariance matrix of the estimated factors at a given point in time depends on the cross sectional dependence of the idiosyncratic error term. Therefore, we will build on Goncalves and Perron (2013) and generalize their wild bootstrap method to accommodate cross sectional dependence of unknown form. At each point in time, our proposal is to obtain an N by 1 vector of bootstrap idiosyncratic error terms by multiplying the square root of an estimated covariance matrix of residuals by an i.i.d. draw of a random vector with mean zero and covariance matrix equal to the identity matrix. We will rely on the statistics literature on estimation of large covariance matrices to estimate this matrix, thus generalizing the existing estimator proposed by Bai and Ng (2006). Part B aims at developing bootstrap methods for inference on the regression parameters of multi-step ahead forecasting models. When the forecasting horizon is larger than one, the regression error term is typically serially correlated and heteroskedastic, rendering the wild bootstrap method of Goncalves and Perron (2013) invalid. Our proposal will be to rely on a two-step bootstrap algorithm as in Goncalves and Perron (2013), where the wild bootstrap method used to generate the bootstrap regression residuals is replaced by the dependent wild bootstrap of Shao (2010). We will establish the consistency of this method for factor-augmented regression models, where some of the regressors are estimated factors. Finally, part C of this research program considers the problem of evaluating predictions based on factor-augmented regression models. One goal is the construction of bootstrap prediction intervals in a multi-step environment, based on the dependent wild bootstrap proposed in part B. This method has the advantage of not requiring the Gaussianity assumption that justifies the asymptotic prediction intervals of Bai and Ng (2006). Another goal is to propose bootstrap methods for out-of-sample predictability tests that involve estimated factors. We will first provide conditions on the cross sectional dimension N and on the time series dimension T under which the existing asymptotic distributions apply. We will then relax these conditions and propose bootstrap methods that can capture the factors estimation uncertainty in an out-of-sample context.