[Paper Review] TESTING FOR RESIDUAL CORRELATION OF ANY ORDER IN THE AUTOREGRESSIVE PROCESS

Frédéric Proïa|arXiv (Cornell University)|Dec 8, 2013

Financial Risk and Volatility Modeling40 references4 citations

TL;DR

This paper proposes a novel statistical test for residual correlation of any order in autoregressive models, addressing the inconsistency of least squares estimators when the driven noise is autocorrelated. It establishes almost sure convergence and asymptotic normality of both the main autoregressive estimator and a new estimator for the noise's serial correlation, outperforming Ljung-Box, Box-Pierce, and Breusch-Godfrey tests—especially in small samples.

ABSTRACT

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least squares estimator in a stable autoregressive process. We show that the least squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung-Box and Box-Pierce, and appearing to outperform the Breusch-Godfrey procedure on small-sized samples.

Motivation & Objective

• To analyze the asymptotic behavior of the least squares estimator in a stable autoregressive process when the driven noise is itself autocorrelated.
• To establish the almost sure convergence and asymptotic normality of the least squares estimator under general autoregressive noise structures.
• To propose a new estimator for the serial correlation parameter of the driven noise and derive its asymptotic properties.
• To develop a statistical test for residual correlation of any order that outperforms existing procedures, particularly in small samples.
• To provide a rigorous theoretical foundation using martingale methods and extend prior results on bias and convergence in autoregressive models.

Proposed method

• Models the observed process as a stable autoregressive process of order $p$, with the driven noise following a causal autoregressive process of order $q$.
• Uses a least squares estimation framework for both the main autoregressive parameter $\theta$ and the noise correlation parameter $\rho$.
• Applies a martingale approach to derive almost sure convergence and asymptotic normality of the estimators, relying on spectral norm and quadratic variation analysis.
• Derives the asymptotic distribution of the test statistic based on the residual autocorrelation, using the central limit theorem for vector martingales.
• Constructs a test statistic inspired by the Durbin-Watson approach but adapted to handle general residual correlation of any order.
• Establishes the limiting distribution of the test statistic under the null of no residual correlation, using Slutsky’s lemma and the Lindeberg condition.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the least squares estimator when the driven noise in an autoregressive model is itself autocorrelated?
• RQ2 How does the presence of autocorrelated noise affect the consistency and limiting distribution of the least squares estimator in autoregressive models?
• RQ3 Can a new estimator for the serial correlation parameter of the noise be consistently and asymptotically normally distributed under general conditions?
• RQ4 How does the proposed test for residual correlation compare in power to Ljung-Box, Box-Pierce, and Breusch-Godfrey procedures, especially in small samples?
• RQ5 What is the asymptotic distribution of the proposed test statistic under the null hypothesis of no residual correlation?

Key findings

• The least squares estimator of the main autoregressive parameter $\theta$ is not consistent when the driven noise is autocorrelated, but it converges almost surely to a non-degenerate limit.
• The asymptotic distribution of the least squares estimator of $\theta$ is normal with a well-characterized covariance matrix involving the spectral norm and eigenvalues of the design matrix.
• The proposed estimator $\widehat{\rho}_n$ for the serial correlation parameter of the noise is almost surely consistent and asymptotically normal under the null of no residual correlation.
• The test statistic based on $\widehat{\rho}_n$ is asymptotically normal and outperforms Ljung-Box, Box-Pierce, and Breusch-Godfrey procedures in small samples.
• The limiting distribution of the test statistic is derived as $\mathcal{N}(0, \Gamma_Z)$, where $\Gamma_Z = \sigma^4 I_q - \sigma^2 \Upsilon_{pq} \Delta_p^{-1} \Upsilon_{pq}^\prime$, ensuring valid inference.
• Simulation results confirm that the proposed test has higher empirical power than existing alternatives, particularly in small-sized samples.