Solved – ACF of a stationary time series

Two questions:

  1. Why for a stationary time series the ACF will drop to zero relatively quickly? Stationarity is a form of dependence and the ACF (and the PACF) measure the dependence between two r.v's, it's seems a contradiction to me (of course I'm wrong, but I don't understand why).

  2. We can use Ljung-box test to test stationarity. Stationarity is a form of dependence. If the series is stationary I'll refuse the null hypothesis, that is independence. Is it correct?

  1. It depends on what you mean by "drop to zero relatively quickly". For a stationary AR(1), $y_t = rho y_{t-1} + epsilon_t$, $|rho| < 1$, the ACF never goes to zero, but rather decays as $rho^k$. In contrast, for an MA(1), $y_t = alpha epsilon_{t-1} + epsilon_t$, the $E[y_t y_{t+k}] = 0$ for $k > 1$, so it drop to zero very quickly.

  2. Ljung-Box is not used for testing stationarity. Rather, it is used to test for the presence of serial correlation. For example, an AR(1) is stationary, but the Ljung-Box test will reject the null hypothesis due to the presence of serial correlation in the AR(1). If you apply the test to a non stationary time series you will also reject the null, but you can't conclude that the series is non-stationary just because the Ljung-Box test was significant.

Just a side note, an iid time series is still stationary (trivially), so just saying "if the series is stationary I'll reject the null hypothesis" is not correct.

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