# Check for White Noise Residuals for AR(1) Model

I was going through the SAS documentation for PROC ARIMA and got stuck at this point.

Autocorrelation Check of Residuals
To Lag  Chi-Square  DF  Pr>ChiSq    Autocorrelations
6       19.09       5   0.0019     0.327    -0.220  -0.128   0.068  -0.002  -0.096
12      22.90       11  0.0183     0.072     0.116  -0.042  -0.066   0.031  -0.091
18      31.63       17  0.0167    -0.233    -0.129  -0.024   0.056  -0.014  -0.008
24      32.83       23  0.0841     0.009    -0.057  -0.057  -0.001   0.049  -0.015


"The test statistics for the residuals series indicate whether the residuals are uncorrelated (white noise) or contain additional information that might be used by a more complex model. In this case, the test statistics reject the no-autocorrelation hypothesis at a high level of significance (p = 0.0019 for the first six lags.) This means that the residuals are not white noise, and so the AR(1) model is not a fully adequate model for this series. The ESTIMATE statement output also includes graphical analysis of the residuals. It is not shown here. The graphical analysis also reveals the inadequacy of the AR(1) model."

Isn't the null hypothesis the assumption that the residuals are white noise? If yes how are we rejecting the hypothesis since the p-values are significantly higher than the stated level of significance of 0.0019?

• You may have noticed that low $p$-values are associated with rejections of $H_0$, while high $p$-values are associated with non-rejections of $H_0$, whatever the test being considered. – Richard Hardy Nov 8 '15 at 20:08

Actually, the null hypothesis of a portmanteau test on residuals is "$\rho_1=\rho_2=...=\rho_L=0$" where $\rho_l$ is the correlation of the noise between $\epsilon_t$ and $\epsilon_{t-l}$. Thus, if $L=6$, then a small p-value (column "Pr>Chisq") indicates it is difficult to accept the hypothesis of an independently distributed noise (until 6 time lags). As a result, the model (here, an AR(1) process) does not seem to take into account all the dynamic of the time series.