Time Series: ARCH model properties

[kingwinner]

Consider an ARCH(1) model:
X_t = σ_tZ_t, where Z_t~ i.i.d. N(0,1)
σ_t² = w₀ + w₁ X_t-1²
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_t²)] =E[E(σ_tZ_t|σ_t²)]
=E[σ_tE(Z_t|σ_t²)] = E[σ_tE(Z_t)] = E[σ_t * 0] = 0
(here I don't understand why E(Z_t|σ_t²)=E(Z_t).)

(ii) γ_X(0)=E(X_t²)
=E(σ_t² Z_t²)=E(σ_t²)E(Z_t²)
=E(w₀ + w₁ X_t-1²) * 1
= w₀ + w₁ γ_X(0)
Solve for γ_X(0) => γ_X(0) = w₀/(1-w₁)

γ_X(h)=E(X_tX_t+h)
=E[E(X_tX_t+h|Z_t+h-1,...,Z_t+1)] = E[X_tE(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_tE(σ_t+hZ_t+h|Z_t+h-1,...,Z_t+1)] = E[X_tσ_t+hE(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_tσ_t+hE(Z_t+h)] = E[X_tσ_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! :)

 

[Stephen Tashi]

(here I don't understand why E(Z_t|σ_t²)=E(Z_t).)

Z_t is independent of \sigma_t^2. I think this step is an example of the fact that if X and Y are independent random variables then E(X|Y) = E(X).

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

The conditional expectation with respect to the sequence of values Z_{h+t-1}...Z_{t+1} amounts to an integration with respect to the joint density of a sequence of independent random variables. These are events that happen after time t and X_t doesn't depend on them. I think this step is an example of the idea that \int g(x_3) f(x_1,x_2)) dx_1 dx_2 = g( x_3) \int f(x_1,x_2) dx_1 dx_2

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

I don't understand that step yet. I wonder if the phrase "assuming the process is second-order stationary" tells us anything important. The definition of the process looks very specific, so it isn't clear to me what that phase adds to it.

 

[Stephen Tashi]

=E[X_tE(σ_t+hZ_t+h|Z_t+h-1,...,Z_t+1)] = E[X_tσ_t+hE(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_tσ_t+hE(Z_t+h)] = E[X_tσ_t+h * 0] = 0 for all h>0.

I could understand the above if it said:

E(X_t E(\sigma_{t+h} X_{t+h}| Z_{t+h-1}...Z_{t+1} ) = E(X_t E(\sigma_{t+h}|Z_{t+h-1}...Z_{t+1}) E(Z_{t+h}|Z_{t+h-1}...Z_{t+1}))

since Z_{t+h} is independent of \sigma_{t+h}. The result will follow because the factor of zero still appears.