# Deriving PDF in AutoRegressive Model

I hope that this is a right place and way to ask this question. I am trying to understand how to derive the probability density function of x(t) in an AR model of order K given (t-k) past observations.
I am primarily referring to this paper (Section 2.2) but it cites a Japanese book for this derivation. I could not find similar derivation in the other time series books(Shumway and stoffer, Pourahmadi). Can someone please provide me appropriate references or explain it.

Thanks for your time. iinception

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The link doesn't give us the full paper and so we can't look at Section 2.2. Can you give us the relevant part of the section that gives the derivation? –  Michael Chernick Sep 15 '12 at 1:32
There has to be additional assumptions other than the AR order. The marginal distribution for X(t) for a stationary AR model will be normalif the error term is normal. But other distributiona for the error term can lead to different marginals. –  Michael Chernick Sep 15 '12 at 1:36
@MichaelChernick thanks for the comment, I have updated the reference. Yes the error term is normal and I want to understand how they derived it. –  iinception Sep 15 '12 at 20:00

I don't like the notation used in the paper but they assume the error term is normal with mean 0 and constant variance. What they derive in section 2.2 is the conditional distribution of X(t) given the previous k observations for a kth order AR process. The result is really rather simple.

The model is as follows:

X(t)= a$_1$ X(t-1) + a$_2$ X(t-2)+...+ a$_k$ X(t-k) + e$_t$

where e$_(t)$ is normal mean 0 and variance σ$^2$. In this formulation the AR coefficients are given and the time series points X(t-j) for j=1,2,..., k are all given. So all the terms except e$_t$ are fixed values. So this means that you are just adding a constant to a single normal random variable.

So the conditional distribution of X(t) given X(t-1)=x(t-1), X(t-2)=x(t-2), ..., X(t-k)=x(t-k) is normal with mean a$_1$ x(t-1) + a$_2$ x(t-2)+...+ a$_k$ x(t-k) and variance σ$^2$.

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now i can understand it...thanks for the explanation @Michael Chernick ...:) –  iinception Sep 16 '12 at 17:24