Further Econometrics Problem Set # 2 Semester 2 2021-22
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Further Econometrics
Semester 2 2021-22
Problem Set # 2
1. Let {wt} be a time series of of independently and identically distributed random variables with mean zero and variance σw(2) (that is, wt ~ i.i.d. (0, σw(2)) for t = 1, 2, . . . ,T). Define vt = wtwt− 1 (vt is a simple example of what is known as a bilinear time series process).
(a) Show that wt is a white noise process.
(b) Show that vt is a white noise process.
(c) Show that vt and vt− 1 are not independent and hence vt is not an i.i.d. process. Hint: Consider E[vt(2)vt(2)− 1] and E[vt(2)]E[vt(2)− 1] . If vt and vt− 1 are independent then E[vt(2)vt(2)− 1] = E[vt(2)]E[vt(2)− 1] ; if E[vt(2)vt(2)− 1] E[vt(2)]E[vt(2)− 1] then vt and vt− 1 are not independent.
2. Consider the AR(1) model:
vt = θvt− 1 + et ,
where et is i.i.d. with mean zero and variance σ2 , and |θ| < 1. In lectures, it is shown that E[vT+h| vT ,vT − 1 , . . . ,v1] = θhvT ,
and this forms the basis of our forecast of vT+h given the observed sample. In this question,
you explore the accuracy of this forecast by looking at the forecast mean square error (MSE)
mse(h) = E [vT+h − E[vT+h| vT ,vT − 1 , . . . ,v1]]2 .
(Notice that for the purposes of this calculation we treat θ as known whereas to be strictly correct we should take account of the parameter estimation error in calculation of the forecast MSE. This simplification makes the analysis easier without losing the basic the structure of how the forecast MSE behaves in large samples as h increases.)
(a) Show that mse(1) = σ 2 .
(b) Show that mse(2) = σ2 (1 + θ 2 ).
(c) Show that mse(h) = σ2 对 θ 2i
(d) What is limh→∞ mse(h)? Interpret this result.
3. Consider the AR(1) model:
vt = c + θvt− 1 + et ,
where et is white noise, c is a constant and |θ| < 1.
Show that:
(a) For any positive integer m,
m m
vt = 工 θic + θm+1vt−m − 1 +工 θiet−i .
i=0 i=0
(b) Using part (a), show that
vt = +工(∞)θiet−i .
(c) Using the representation for vt in part (b), show that E[vt] = c/(1 − θ) = µ, say.
(d) Use part (a) to show that
m
vt − µ = θm+1(vt−m − 1 − µ) +工 θiet−i .
i=0
and so that Cov[vt,vt−s] = θs Var[vt] for any positive integer s.
(e) Assume now that {et} are i.i.d. with mean zero. Suppose we have a sample of observa- tions {vt; t = 1, 2, . . . ,T} and wish to forecast vT+h where h is a positive integer. Derive E[vT+h| vT ,vT − 1 , . . . , 1]. What happens to E[vT+h| vT ,vT − 1 , . . . , 1] as h → ∞?
4. Consider the MA(2) process
vt = c + et + φ 1 et− 1 + φ2 et−2 ,
where et is white noise and φ2 0.
(a) Show that E[vt] = c.
(b) Show that Cov[vt,vt−2] 0 and Cov[vt,vt−k] = 0 for k > 2.
5. Consider the ARMA(1,1) model
vt = θvt− 1 + et + φet− 1 ,
where et is white noise and |θ| < 1 and θ = −φ .
(a) Use back substitution to show that for any positive integer m
vt − et = θm {vt−m− et−m}.
(b) Use part (a) to show that vt has the time series properties of white noise.
6. Consider the random walk process with drift:
vt = α + vt− 1 + et, t = 1, 2, . . . ,T
where v0 = 0 and et is white noise with Var[et] = σ 2 .
(a) Show that Var[vt] = tσ2 .
(b) Show that Cov[vt,vt−j] = (t − j)σ2 .
(c) Show that Corr[vt,vt−j] = ^(t − j)/t and hence that as t → ∞ Corr[vt,vt−j] → 1 for any fixed j .
(d) Contrast the behaviour of the autocorrelations of random walk with drift with those of a stationary AR(1) process.
2022-08-29