Consider an ARCH(1) model:(adsbygoogle = window.adsbygoogle || []).push({});

X_{t}= σ_{t}Z_{t}, where Z_{t}~ i.i.d. N(0,1)

σ_{t}^{2}= w_{0}+ w_{1}X_{t-1}^{2}

Find (i) E(X_{t})

and (ii) the autocovariance function γ_{X}(h) for h=0,1,2,3,..., assuming the process is second-order stationary.

Solution:

(i) E(X_{t}) = E[E(X_{t}|σ_{t}^{2})] =E[E(σ_{t}Z_{t}|σ_{t}^{2})]

=E[σ_{t}E(Z_{t}|σ_{t}^{2})] = E[σ_{t}E(Z_{t})] = E[σ_{t}* 0] = 0

(here I don't understand why E(Z_{t}|σ_{t}^{2})=E(Z_{t}).)

(ii) γ_{X}(0)=E(X_{t}^{2})

=E(σ_{t}^{2}Z_{t}^{2})=E(σ_{t}^{2})E(Z_{t}^{2})

=E(w_{0}+ w_{1}X_{t-1}^{2}) * 1

= w_{0}+ w_{1}γ_{X}(0)

Solve for γ_{X}(0) => γ_{X}(0) = w_{0}/(1-w_{1})

γ_{X}(h)=E(X_{t}X_{t+h})

=E[E(X_{t}X_{t+h}|Z_{t+h-1},...,Z_{t+1})] = E[X_{t}E(X_{t+h}|Z_{t+h-1},...,Z_{t+1})]

(here I don't understand why we can pull the X_{t}out of the expectation.)

=E[X_{t}E(σ_{t+h}Z_{t+h}|Z_{t+h-1},...,Z_{t+1})] = E[X_{t}σ_{t+h}E(Z_{t+h}|Z_{t+h-1},...,Z_{t+1})]

(here I don't understand why we can pull the σ_{t+h}out of the expectation.)

=E[X_{t}σ_{t+h}E(Z_{t+h})] = E[X_{t}σ_{t+h}* 0] = 0 for all h>0.

I'm cannot follow the reasoning of the three equalities labelled in red above. Can someone explain why they are true?

Any help would be much appreciated! :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Time Series: ARCH model properties

**Physics Forums | Science Articles, Homework Help, Discussion**