Comparative Study of the Finite Sample Properties of Ljung-Box and Box-Pierce in Time Series Econometrics

Abstract

Objective: This paper assesses the finite-sample properties of the Ljung-Box and Box-Pierce tests, specifically their autocorrelation detection and chi-square distribution.

Research Gap: The existing literature checks the asymptotic properties of the Ljung-Box and Box-Pierce tests and finite-sample properties across various sample sizes and autocorrelation levels.

Design/Methodology/Approach: The paper uses Monte Carlo simulations with sample sizes ranging from 15 to 1000 at different autocorrelation levels and reports the power of the test statistics and the Type I error.

The Main Findings: These results indicate that both tests are asymptotically chi-square distributed when sample sizes are small (n < 100). The Ljung-Box test follows a chi-square distribution much more closely than the Box-Pierce test. However, when the sample size is less than 150, the Box-Pierce test rejects the null hypothesis, leading to a Type I error. The Ljung-Box test has higher power at lower levels of autocorrelation and with smaller finite sample sizes. Finally, both tests improve in power and Type I error control as sample size increases.

Theoretical / Practical Implications of the Findings: The paper elucidates the finite-sample properties of the tests, including the chi-square distributional behaviour, the power of the test statistics, and the Type I error of both tests. The results help in choosing an appropriate autocorrelation diagnostic test for small and moderate sample sizes.

Originality/Value: This paper compares the small-sample properties of the "portmanteau" tests using Monte Carlo simulation and highlights their weaknesses.