Adaptive Testing for Cointegration With Nonstationary Volatility

This article develops a class of adaptive cointegration tests for multivariate time series with nonstationary volatility. Persistent changes in the innovation variance matrix of a vector autoregressive model lead to size distortions in conventional cointegration tests, which may be resolved using the wild bootstrap, as shown in recent work by Cavaliere, Rahbek, and Taylor. We show that it also leads to the possibility of constructing tests with higher power, by taking the time-varying volatilities and correlations into account in the formulation of the likelihood function and the resulting likelihood ratio test statistic. We find that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric volatility matrix estimation does not lead to a loss of asymptotic local power relative to the case where the volatilities are observed. The asymptotic null distribution of the test is nonstandard and depends on the volatility process; we show that various bootstrap implementations may be used to conduct asymptotically valid inference. Monte Carlo simulations show that the resulting test has good size properties, and higher power than existing tests. Empirical analyses of the U.S. term structure of interest rates and purchasing power parity illustrate the applicability of the tests.

本文针对具有非平稳波动率的多元时间序列，构建了一类自适应协整检验方法。向量自回归模型（vector autoregressive model）的新息方差矩阵发生持续性变动时，传统协整检验会出现显著性水平偏误；正如Cavaliere、Rahbek与Taylor在近期研究中所证明的，这类偏误可通过野Bootstrap（wild bootstrap）方法予以修正。本文证明，通过在似然函数（likelihood function）的构建以及最终似然比检验统计量（likelihood ratio test statistic）的推导中纳入时变波动率与时变相关性，可构建出具有更高检验功效的检验方法。研究发现，在适当条件下，可针对波动率过程实现自适应推断：相较于波动率可直接观测的情形，非参数波动率矩阵估计并不会导致渐近局部功效的损失。该检验的渐近原分布为非标准分布，且依赖于波动率过程；本文证明，可通过多种Bootstrap实现方式进行渐近有效的统计推断。蒙特卡洛（Monte Carlo）模拟结果表明，本文所提出的检验方法具有良好的显著性水平特性，且相较于现有检验方法拥有更高的检验功效。通过对美国利率期限结构（U.S. term structure of interest rates）与购买力平价（purchasing power parity, PPP）开展实证分析，验证了该检验方法的实际应用价值。