1. Business
  2. Finance
  3. consider the following arma1 process zt zt1...

Question: consider the following arma1 process zt zt1...

Question details

Consider the following ARMA(1) process: zt = γ + αzt1 + εt + θεt1, (1)

where εt is a zero-mean white noise process with variance σ2 , and assume |α|, |θ| < 1 and α+θ0, which together make sure zt is covariance stationary.

(a) Calculate the conditional and unconditional means of zt , that is, Et1 [zt ] and E [zt ].

(b) Set α = 0. Derive the autocovariance and autocorrelation function of this process for all lags as functions of the parameters θ and σ.

(c) Assume now aθ. Calculate the conditional and unconditional variances of zt , that is, Vart1 [zt ] and Var [zt ].

Hint: for the unconditional variance, you might want to start by deriving the unconditional covariance between the variable and the innovation term, i.e., Cov [zt , εt ].

(d)  Derive the autocovariance and autocorrelation for lags of 1 and 2 as functions of the parameters of the model. Hint: use the hint of part (c). 

Solution by an expert tutor
Blurred Solution