Bootstrap methods for testing and forecasting with estimated factors

使用估计因素进行测试和预测的 Bootstrap 方法

• 批准号: RGPIN-2014-06482
• 负责人: Gonçalves, Sílvia
• 金额: $ 1.02万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=652135
• 关键词: Bootstrap; methods; testing; forecasting; estimated

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.

给定感兴趣的变量的潜在预测因子的数量通常比时间序列观测值的数量大得多。由于这个原因，因子增强回归模型，其中一些回归量包括估计的因素，已成为越来越流行的经济学。这些因子代表大面板因子模型中的共同因子，并在第一步使用主成分法进行估计。在第二步中，我们根据估计的因素（以及额外观察到的回归量，如因变量的滞后）对感兴趣的变量进行回归。虽然因子增强回归模型的估计很容易，但由于因子的估计，推理可能很复杂。这项拨款提案的总体目标是为具有估计因子的预测模型的推理文献做出贡献。我们可以区分出三个主要部分。* A部分考虑了使用bootstrap评估潜在因素的问题。这是经济学和金融学中一个非常有趣的问题，因为观察到的变量经常被用作理论模型假设的潜在共同因素的代理。我们将基于Bai（2003）的主成分法提出潜在因素的自举置信区间。这些区间可用于评估给定的观测变量是否与估计因子一致。如Bai（2003）所示，在给定时间点估计因子的渐近协方差矩阵取决于特质误差项的横截面依赖性。因此，我们将以Goncalves和Perron（2013）为基础，推广他们的野生自举方法，以适应未知形式的横截面依赖。在每个时间点，我们的建议是通过将残差估计的协方差矩阵的平方根乘以平均为零且协方差矩阵等于单位矩阵的随机向量的i.i.d绘制来获得一个N × 1的bootstrap特质误差项向量。我们将依靠关于估计大协方差矩阵的统计文献来估计这个矩阵，从而推广Bai和Ng（2006）提出的现有估计量。*Part B旨在开发对多步超前预测模型的回归参数进行推理的自举方法。当预测水平大于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bootstrap方法无效。我们的建议将依赖Goncalves和Perron（2013）中的两步自举算法，其中用于生成自举回归残差的野生自举方法被Shao（2010）的依赖野生自举方法所取代。我们将在因子增强回归模型中建立这种方法的一致性，其中一些回归因子是估计的因子。*最后，本研究计划的C部分考虑了基于因子增强回归模型评估预测的问题。其中一个目标是基于b部分中提出的依赖野生自举在多步环境中构建自举预测区间。该方法的优点是不需要证明Bai和Ng（2006）的渐近预测区间的高斯假设。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自举方法。我们将首先给出适用于现有渐近分布的横截面维数N和时间序列维数T的条件。然后，我们将放宽这些条件，并提出可以在样本外环境中捕获因素估计不确定性的自举方法。