Question

1. Find the ACF and PACF and plot the ACF ρk for k = 0, 1, 2, 3, 4, 5 for the following model where the wt is a Gaussian white noise process.

Zt = −0.5Zt−1 + wt

Answer 1

The AR(1) model given is as follows:

where is the  Gaussian white noise process.

The autocorrelation function (ACF) of a general AR(1) with the parameter is and the partial ACF(PACF) is given by

Thus the above model can be written as

thus,

Thus PCF will be given by

and the PACF will be

The ACF property defines a distinct pattern for the autocorrelations. For a positive value of , the ACF exponentially decreases to 0 as the lag k increases.  For negative, the ACF also exponentially decays to 0 as the lag increases, but the algebraic signs for the autocorrelations alternate between positive and negative.

I have used matlab plot:

>> k = 0:1:5;

>> y = 0.5.^k;

>> stem(k,y)