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Difference-based covariance matrix estimate in time series nonparametric regression with applications to specification tests

Lujia Bai, Weichi Wu

Abstract

Long-run covariance matrix estimation is the building block of time series inference. The corresponding difference-based estimator, which avoids detrending, has attracted considerable interest due to its robustness to both smooth and abrupt structural breaks and its competitive finite sample performance. However, existing methods mainly focus on estimators for the univariate process while their direct and multivariate extensions for most linear models are asymptotically biased. We propose a novel difference-based and debiased long-run covariance matrix estimator for functional linear models with time-varying regression coefficients, allowing time series non-stationarity, long-range dependence, state-heteroscedasticity and their mixtures. We apply the new estimator to (i) the structural stability test, overcoming the notorious non-monotonic power phenomena caused by piecewise smooth alternatives for regression coefficients, and (ii) the nonparametric residual-based tests for long memory, improving the performance via the residual-free formula of the proposed estimator. The effectiveness of the proposed method is justified theoretically and demonstrated by superior performance in simulation studies, while its usefulness is elaborated via real data analysis. Our method is implemented in the R package mlrv.

Difference-based covariance matrix estimate in time series nonparametric regression with applications to specification tests

Abstract

Long-run covariance matrix estimation is the building block of time series inference. The corresponding difference-based estimator, which avoids detrending, has attracted considerable interest due to its robustness to both smooth and abrupt structural breaks and its competitive finite sample performance. However, existing methods mainly focus on estimators for the univariate process while their direct and multivariate extensions for most linear models are asymptotically biased. We propose a novel difference-based and debiased long-run covariance matrix estimator for functional linear models with time-varying regression coefficients, allowing time series non-stationarity, long-range dependence, state-heteroscedasticity and their mixtures. We apply the new estimator to (i) the structural stability test, overcoming the notorious non-monotonic power phenomena caused by piecewise smooth alternatives for regression coefficients, and (ii) the nonparametric residual-based tests for long memory, improving the performance via the residual-free formula of the proposed estimator. The effectiveness of the proposed method is justified theoretically and demonstrated by superior performance in simulation studies, while its usefulness is elaborated via real data analysis. Our method is implemented in the R package mlrv.
Paper Structure (34 sections, 19 theorems, 269 equations, 11 figures, 4 tables)

This paper contains 34 sections, 19 theorems, 269 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The debiased difference-based estimator
  4. Consistency under smooth structural changes
  5. Applications
  6. Structural change detection with monotonic power
  7. Testing for long memory
  8. Simulation
  9. Setting
  10. Testing for structural changes
  11. Testing for long-range dependence
  12. Data analysis
  13. Conclusion
  14. Bias in the difference-based estimator
  15. Implementation details
  16. ...and 19 more sections

Key Result

Theorem 4.1

Under Assumptions A:nonpar, Ass-U, Ass-W, Ass-E, A:K and B:H_delta with constant $\kappa \geq 1$, suppose $m = O(n^{1/3})$, $m \to \infty$, $\tau_n \to 0$, $m / (n \tau_n^{3/2 + 2/\kappa}) \to 0$, $\tau_n^{3-1/\kappa}\surd{m} \to 0$, $m\tau_n \to \infty$, we have

Figures (11)

  • Figure 6.1: The empirical rejection rates of gradient-based structural change point tests as $\delta$ increases from $0$ to $1$ with sample size $n = 300$ and simulation times $2000$, under three scenarios CP1(blue), CP2(orange), CP4(red), using blocks of ordinary least squares residuals (small-dashes), and difference-based long-run covariance matrix estimator (solid). Left panel: nominal size 0.05; Right panel: nominal size 0.1.
  • Figure 6.2: Empirical rejection rates of KPSS(orange), K/S(blue), R/S(red), and V/S(green) tests under different $d$'s with sample size $1500$, using the plug-in method (small-dashes) and the difference-based method (solid). Left panel: nominal size 0.05; Right panel: nominal size 0.1.
  • Figure A.1: The sample path of $(x_{i+1}y_{i+1} - x_{i}y_{i})^2 - (x_{i+1}e_{i+1} - x_{i}e_{i})^2$ with stochastic trend (solid lines), and with only deterministic smooth trend (dotted lines). The left panel shows the gaps between A.1 and A.2 and the right panel shows that between B.1 and B.2, respectively.
  • Figure B.1: A boxplot of the selected smoothing parameter $m$ of V/S, R/S, KPSS, K/S tests and the test of structural stability.
  • Figure B.2: Empirical rejection rates of the structural stability test with triangular(solid), Epanechnikov (dotted), quartic(small dashes), triweight (dashes) and tricube (long dashes) kernels, respectively.
  • ...and 6 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Theorem 4.1
  • Theorem 5.1
  • Proposition 5.1
  • Theorem 5.2
  • Definition 5.1
  • Theorem 5.3
  • Definition C.1
  • Lemma C.1
  • proof
  • ...and 31 more