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Run-Test 📂Statistical Test

Run-Test

Hypothesis Testing

Let us denote the ARMA model obtained from time series analysis as ARMA(p,q)ARMA(p,q) to be MM.

  • H0H_{0}: MM is fit.
  • H1H_{1}: MM is not fit.

Explanation

The Ljung-Box Test, also abbreviated as LBQ, is a test for determining the goodness-of-fit of an ARIMA model.

In 1970, Box and Pierce proposed the following test statistic QQ, through the sACF r^1,⋯ ,r^k\hat{r}_{1} , \cdots , \widehat{r}_{k} of residuals obtained from ARIMA models. Q=n(r^12+⋯+r^k2) Q = n \left( \hat{r}_{1}^{2} + \cdots + \widehat{r}_{k}^{2} \right) QQ follows a chi-squared distribution with degrees of freedom k−p−qk-p-q, allowing for goodness-of-fit testing, but could only be used when nn was sufficiently large. There were cases where it did not converge even when it was about n=100n=100, but in 1978 Ljung and Box proposed an improved test statistic as follows. Q∗=n(n+2)(r^12n−1+⋯+r^k2n−k)∼χ2(k−p−q) Q_{*} = n(n+2) \left( {{\hat{r}_{1}^{2}} \over {n-1}} + \cdots + {{\widehat{r}_{k}^{2}} \over {n-k}} \right) \sim \chi^{2} ( k - p - q )

See Also