Introduction

In the realm of time series analysis, the evaluation of model adequacy and goodness of fit is of paramount importance. The Box-Pierce Q-statistic and the Ljung-Box Q statistic are statistical tests designed to assess the presence of autocorrelation in a time series. These tests are essential tools for determining whether the residuals or errors of a model exhibit significant autocorrelation beyond what would be expected by chance. Let’s delve into the intricacies of these tests and their significance in assessing the quality of time series models.

Box-Pierce Q-Statistic

The Box-Pierce Q-statistic is a measure used to test the null hypothesis that the first $h$ autocorrelations of a time series are all zero. In other words, it helps us determine whether the observed autocorrelation coefficients are significantly different from zero. Mathematically, the Box-Pierce Q-statistic is calculated as:

$$Q(h)=n\sum _{k=1}^h{\widehat{{\rho }_k}}^2$$

Where:

• $Q(h)$ is the Box-Pierce Q-statistic for lag $h$
• $n$ is the sample size
• $\widehat{{\rho }_k}$ is the sample autocorrelation at lag $k$

A higher Q-statistic indicates stronger evidence of autocorrelation in the time series.

Ljung-Box Q-Statistic

The Ljung-Box Q statistic is an extension of the Box-Pierce Q-statistic. It addresses potential issues related to the finiteness of the sample size. The Ljung-Box $Q$ statistic tests the null hypothesis that a group of $h$ autocorrelations are jointly zero. It is defined as:

$$Q^{\ast }(h)=n(n+2)\sum _{k=1}^h\frac{{\hat{\rho }}_k^2}{n-k}$$

Where:

• $Q^{\ast }(h)$ is the Ljung-Box Q-statistic for lag $h$

The Ljung-Box Q-statistic takes into account the degrees of freedom $n-k$, which adjusts for the decreasing number of available data points as lag increases.

Interpreting the Results

In both tests, the calculated Q-statistic is compared to the critical value from the chi-squared distribution with $h$ degrees of freedom. If the calculated Q-statistic exceeds the critical value, it suggests significant autocorrelation in the residuals, indicating that the model might not adequately capture all the patterns in the data. Conversely, a low Q-statistic suggests that the model’s residuals are not significantly autocorrelated.

Example: Consider a financial time series dataset of daily stock prices. After fitting a time series model, you calculate the Q-statistic for lag $h=10$ and find it to be 16.43 . The critical value for a significance level of 0.05 and 10 degrees of freedom is 18.307. Since $16.43<18.307$, you fail to reject the null hypothesis, indicating that the model’s residuals do not exhibit significant autocorrelation up to lag 10.