探讨自回归(AR)与移动平均(MA)模型中相关系数与协方差的适配性

Let’s cut to the key distinction between Autoregressive (AR) and Moving Average (MA) models—it all hinges on how time series values correlate with each other across different time lags.

• Moving Average (MA) Models: Once the lag $n$ exceeds the order of the MA model, the correlation between $x(t)$ and $x(t-n)$ drops straight to zero. This isn’t a random quirk; it comes directly from the fact that the covariance between these two time points is zero (as demonstrated in the earlier example we covered). In simple terms, MA models only "remember" recent lags up to their specified order—anything beyond that has no impact on the current value whatsoever.

• Autoregressive (AR) Models: This is where things diverge sharply. In AR models, the correlation between $x(t)$ and $x(t-n)$ doesn’t just vanish after a certain threshold. Instead, it gradually decays as the lag $n$ grows larger. That means even older values in the time series still hold a (diminishing) influence on the current value, rather than cutting off entirely.

内容的提问来源于stack exchange，提问作者Trajan