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Section A

This section is worth 40% of the exam. Answer either Q1 or Q2 but not both.

Q1 Let εt be an i.i.d. process with E(εt) = 0 and E(ε2t ) = 1. Let y t = ε t − 12 ε t − 1

(a) Show that yt is stationary. (10 marks) (b) Show that yt is invertible. (10 marks)

(c) Compute the variance along with the first and second autocovariances of yt. (10 marks) (d) What is the shape of the PACF of this process? (10 marks)

Q2 Letεt beani.i.d.processeswithE(εt)=0andE(ε2t)=1. Letx0 =y0 =0andlet xt = xt−1 + εt, yt = yt−1 + xt, for t = 1, 2, . . .

(a) Show that yt is I(2). (20 marks)
(b) Show that xt and yt are not cointegrated. (10 marks)

(c) Compute the h-period-ahead forecast of xt. (10 marks)

Section B

This section is worth 40% of the exam. Answer either Q3 or Q4 but not both.

Q3 Let ut be an i.i.d. process with E(ut) = 0 and E(u2t ) = 1. Let xt = 1 ut.

(4−L)(3−L) (a) Express xt in terms of ut, ut−1, . . . (10 marks)

(b) Compute the first four multiplier effects of ut on xt. (10 marks) (c) Compute the total multiplier effect of ut on xt. (10 marks)

(d) Compute the mean and the median lag of the multiplier effects of ut on xt. (10 marks) Q4 Let (ε ,ε ) be an i.i.d. process with E(ε ) = [0 ] and variance covariance matrix [1 0 ]. Let

1,t2,t t0 01 the initial values of xt be x1,0 = x2,0 = 0 and let

􏰀x1,t 􏰁 􏰀1 0􏰁􏰀x1,t−1 􏰁 􏰀ε1,t 􏰁 x=11x+ε.

2,t 2 2,t−1 2,t

1. (a)  Show that the process above is non-stationary. (10 marks)
2. (b)  Show that it is cointegrated and state its cointegration vector. (10 marks)
3. (c)  Howwouldyouestimatethecointegrationvectorgivenadataset(x1,1,x2,1),…,(x1,T,x2,T)? (10 marks)
4. (d)  How many common stochastic trends are there in this model? (10 marks) 1

Section C

This section is worth 20% of the exam. Answer Q5.

Q5 Let νt be an i.i.d. process with E(νt) = 0 and E(νt2) = 1. Let εt = 􏰂htνt

and let

ht = 12 + 14ε2t−1 + 18ε2t−2. Assume that ε2t is stationary.

1. (a)  Compute the 1-period-ahead and 2-period-ahead forecasts of volatility. (10 marks)
2. (b)  Compute the mean and variance of εt. (10 marks)