# Conditional distributions of (higher-order) autoregressive processes

If we specify an $p$-th order autoregressive process in discrete time by its transition distribution $F_{t|t-1,\ldots,t-p}$, what can be said about lower order conditional distribution where we condition not on all relevant lags but only on some, e.g., $F_{t|t-1}$?

For illustration, consider an strictly stationary autoregressive process of order 2, $Y_t = a_1Y_{t-1}+a_2Y_{t-2}+V_t$, where $V_t\stackrel{iid}{\sim} F_{V}$. Clearly, the conditional distribution of $Y_t$ given the past is $F_{t|t-1,t-2}(y_t|y_{t-1},y_{t-2})=F_{V}(y_t-a_1y_{t-1}-a_2y_{t-2})$, i.e., this conditional distribution belongs to a location family. But what about $F_{t|t-1}(y_t|y_{t-1})$? Can we say anything about the properties of this conditional distribution? In particular, does $F_{t|t-1}(y_t|y_{t-1})$ also belong to a location family?

As far as I know, the only fact that is known is that if $V_t$ has a symmetric distribution (if the density of the error exist this means that $f_{V}(v) = f_{V}(-v)$), then any finite dimensional distribution is also symmetric, which implies that $f_{t|t-1}(y_t|y_{t-1}) = f_{t|t-1}(-y_{t}|-y_{t-1})$. I’ve tried several things to obtain more insights into the properties (especially by means of characteristic functions), but didn’t come up with a result. The basic problem is that we know $f_{t|t-1}(y_t|y_{t-1})$ only if we know $f_{t-1|t}(y_{t-1}|y_t)$ and vice versa, but both functions are unknown. (Note, $f_{t|t-1}(y_t|y_{t-1})= f_{t-1|t}(y_{t-1}|y_t)$ if and only if $V_t$ is normally distributed, as it is well known in the literature. Thus, the process is a Gaussian process and it is clear that each conditional distribution belongs to a location family.)

So has anybody an idea what the conditional distribution $F_{t|t-1}$ does look like or what can be said about its properties? Feel free to impose further restrictions on the process, for instance, that $V_t$ has a stable (symmetric) distribution, i.e., a Cauchy distribution.

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