Bootstrap methods for testing and forecasting with estimated factors

给定的感兴趣变量的潜在预测因子的数量往往比时间序列观测的数量多得多。因此，在经济学中，一些回归变量包含估计因素的因素扩展回归模型变得越来越流行。这些因子代表了大面板因子模型中的公共因子，并在第一步中使用主成分方法进行估计。在第二步中，我们将感兴趣的变量回归到估计因素(以及其他观察到的回归变量，如因变量的滞后)。虽然因子增广回归模型的估计很容易，但由于对因子的估计，推断可能会很复杂。这项拨款提案的总体目标是对带有估计因素的预测模型的推断的文献作出贡献。我们可以区分三个主要部分。 A部分考虑了使用自助法评估潜在因素的问题。这是经济学和金融学中非常感兴趣的问题，因为观测变量经常被用作理论模型假设的潜在公共因素的代理。我们将基于BAI(2003)的主成分方法提出潜在因素的Bootstrap可信区间。这些间隔可用于评估给定的观测变量是否与估计因子一致。正如Bai(2003)所表明的，估计因子在给定时间点的渐近协方差矩阵依赖于特性误差项的横截面相关性。因此，我们将在Goncalves和Perron(2013)的基础上推广他们的野生Bootstrap方法，以适应未知形式的截面依赖。在每个时间点，我们的建议是通过将估计的残差协方差矩阵的平方根乘以I.I.D.来获得自举特性误差项的N乘1向量。画一个均值为零、协方差矩阵等于单位矩阵的随机向量。我们将依靠关于大协方差矩阵估计的统计学文献来估计这个矩阵，从而推广了Bai和Ng(2006)提出的现有估计量。 B部分旨在开发Bootstrap方法来推断多步超前预测模型的回归参数。当预测周期大于1时，回归误差项通常是序列相关和异方差的，使得Goncalves和Perron(2013)的野生Bootstrap方法无效。我们的建议将依赖于Goncalves和Perron(2013)中的两步Bootstrap算法，其中用于生成Bootstrap回归残差的野生Bootstrap方法被Shao(2010)的依赖野生Bootstrap所取代。我们将为因子增广回归模型建立这种方法的一致性，其中一些回归变量是估计因子。 最后，本研究计划的C部分考虑了评估基于因素增强回归模型的预测的问题。一个目标是在多步环境下基于B部分中提出的依赖的野生自举构造自举预测区间。该方法的优点是不需要高斯假设来证明Bai和Ng(2006)的渐近预测区间的合理性。另一个目标是为涉及估计因素的样本外可预测性测试提出Bootstrap方法。我们将首先给出横截面维度N和时间序列维度T的条件，在这些条件下，现有的渐近分布是适用的。然后，我们将放宽这些条件，并提出Bootstrap方法，该方法可以在样本外的情况下捕捉因素估计的不确定性。

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.