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6. Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.4. Univariate Time Series Models

Box-Ljung Test

Test for Lack of Fit The Box-Ljung test (1978) is a diagnostic tool used to test the lack of fit of a time series model

The test is applied to the residuals of a time series after fitting an ARMA(\(p,q\)) model to the data. The test examines \(m\) autocorrelations of the residuals. If the autocorrelations are very small, we conclude that the model does not exhibit significant lack of fit.

Section contains an example in which the Box-Ljung test is implemented using a residual time series.

Definition In general, the Box-Ljung test is defined as:
\(H_0\): The model does not exhibit lack of fit.
\(H_a\): The model exhibits lack of fit.
Test Statistic: Given a time series \(Y\) of length \(n\), the test statistic is defined as:

$$ Q = n(n+2)\sum_{k=1}^{m} \, \frac{\hat{r}^2_k}{n-k} $$

where \(\hat{r}_k\) is the estimated autocorrelation of the series at lag \(k\), and \(m\) is the number of lags being tested.

Significance Level: \(\alpha\)
Critical Region: The Box-Ljung test rejects the null hypothesis (indicating that the model has significant lack of fit) if
    $$ Q > \chi_{1-\alpha, \, h}^2 $$
where \(\chi_{1-\alpha, \, h}^2\) is the chi-square distribution table value with \(h\) degrees of freedom and significance level \(\alpha\).

Because the test is applied to residuals, the degrees of freedom must account for the estimated model parameters so that \(h = m-p-q\), where \(p\) and \(q\) indicate the number of parameters from the ARMA(\(p,q\)) model fit to the data.

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