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$y_{MA}$ = $ε_t$ + $ε_{t-1}$ <- stationary

$y_{AR}$ = $ε_t$ + $y_{AR_{t-1}}$

$y_{AR_{t-1}}$ = $ε_t$ + $ε_{t-1}$ + $y_{AR_{t-2}}$

$y_{AR_{t-2}}$ = $ε_t$ + $ε_{t-1}$ + $ε_{t-2}$ + $y_{AR_{t-3}}$ <- non-stationary?

etc.

The MA time series is stationary. This makes sense to me because you're summing up normally distributed mean 0 random variables, but then, an AR process, if the $y_{AR_{t-i}}$ terms are all also sums of normally distributed mean 0 random variables, why is it non-stationary? Again here, you're summing normally distributed mean 0 random variables. I'm obviously not understanding something here, where is the flaw in my logic?

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1 Answer 1

up vote 1 down vote accepted

For an AR process, the variance changes (and, in fact, increases) over time. Thus, for example,

$$\begin{align} Var(y_{AR_t}) &= Var(\epsilon_t) + Var(y_{AR_{t-1}}) \\ &> Var(y_{AR_{t-1}}). \end{align}$$

Since the definition of a stationary process is one whose probability distribution doesn't change under shifts in time, an AR process cannot be stationary.

You're right that the mean of $y_{AR_t}$ and $y_{AR_{t-1}}$ are both $0$, but stationary requires more than just the means being the same.

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I realized what I was missing this morning. In an MA process, each step consists of the sum of only two normally distributed mean 0 random variables, allowing the variance to stay constant. Thanks for the response, it helped me arrive my improved understanding :) –  Soo Mar 2 '12 at 1:35

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