A Complete Introduction To Time Series Analysis (with R):: Tests for Stationarity
In the last article, we saw how we could estimate autocovariance by using a slightly modified version of the typical covariance sample estimator. Further, we saw that in reasonably large samples, this converges in distribution to a normal random variable. In this article, we will use this fact to construct some useful hypothesis tests for stationarity, to check, for instance, whether our decomposition analysis of a series in trend + seasonal component, that is, the residuals after having estimated and removed these, are correct.
Confidence bounds for the ACF
Assuming the ACF follows an underlying normal distribution,
Lagwise Test
We can make direct use of the C.I. above to estimate whether a series is truly stationary: we know that a true stationary series should have 0 autocovariance and therefore 0 autocorrelation, so that we can employ the hypothesis
So in particular, if Ho is true, we should have that
That is, the estimated confidence interval should contain the value 0 for most lags.
Portmanteau Test
This test is also quite straightforward; consider the hypothesis
