Further Questions for EC306

Sam Astill and Alex Karalis Isaac

These questions are designed to aid your study for EC 306. The answers can almost all be found

directly in the lecture notes. The questions therefore serve as a kind of tour through the notes,

to help you read them ‘actively’. Questions with a ∗ are harder, and can be considered optional.

They are there to test those students looking for very high marks, and applying to the top MSc

programmes. Further references include Verbeek A Guid to Modern Econometrics and Harris

and Solis Applied Time Series.

Answers: We will not be distributing typed answers. In part this is because most are found

in the lecture notes. Also, it is because too many students read the answers, feel ok, and then

realise shortly before the exam that they do not understand the answers/how to get them. If

you want us to check your work, or are uncertain about a particular question, either email or

visit our office hours.

1 What are the ACF and the SACF?

2 What are the PACF and the SPACF?

3 a yt follows a stable autoregression. Sketch the ACF and PACF. Write a general equation

for t. State any restrictions on the parameters.

b yt follows a stable moving average process. Sketch the ACF and PACF. Write a general

equation for t. State any restrictions on the parameters.

c yt follows a stable ARMA(p,q) process. Sketch the ACF and PACF. Write a general

equation for t. State any restrictions on the parameters.

4 State the assumptions required on εt, the errors in your above equations, in order to ensure that

the time series have the properties of ARMA models.

5 How would you decide formally if the error series εt in

yt = x′tβ + εt

displayed (G)ARCH effects?

6 Show that the shocks in a unit root process have permanent effects.

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7 How would you test for a unit root in

yt = φyt−1 + εt

8 Let yt = α+ φyt−1 + εt

a What is E[yt|yt−1]? Recall yt = yt, yt−1, yt−2, . . . .

b Assume |φ < 1|, what is E[yt]?

c Maintaining this assumption, show E[yt+k|yt]→ E[yt] as k →∞.

9 For the process in 7, with |φ| < 1,

a What is var(yt|yt−1)?

b What is var(yt)?

10 Let yt = α+ φyt−1 + γt+ εt. Put g = γ1−φ as the long run growth rate of the process.

a Re write the process for yt in terms of the Long Run parameter g.

b How would you test this process for a unit root?

c Show the deterministic trend gt drops out if φ = 1.

d∗ What if the trend were not restricted? ∆yt = α+ γt+ εt. Show by substituing back three

times, that you could write

yt = 3t+ 3γt− γ2∑i=0

i+2∑i=0

εt−i

Therefore show that

yt = αt+γt(t+ 1)

2+

t−1∑i=0

εt−i + y0

hint :∑n

i=0 = n(n+ 1)/2. Note: this derivation corrects an algebraic mistake on the board

in the final lecture.

e In practice, why is the specification in part d a bad idea for your unit root test?

11

yt = x′tb+ εt

ht = α0 + α1ε2t−1 + β1ht−1

εt = h1/2t · zt

a What assumptions on zt and εt ensure this is a GARCH(1,1) process?

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b Show ε2t = c0 + c1ε2t−1 + c2νt−1 + νt where E[νt|Ψt−1] = 0. What are the coefficients c0, c1,

c2?

12 a Write yt = εt + θ1εt−1 as an AR(∞) process, spcifying the φi in terms of θ1. What

assumption is needed on θ1 to do this?

b Write yt = φ1yt−1 + φ2yt−2 + εt as an MA(∞) process,specifying the θi in terms of the φj ,

j = 1, 2. What assumption on the φj is needed to be able to do this?

c What are E[yt+1|yt], E[yt+2|yt] and E[yt+k|yt] for the models in parts a and b?

d What is var(yt) for the process in a and for the process in b?

13 Make a cup of tea before the VAR stuff arrives

14 Let εt be an ARCH(1) process.

a What is the conditional variance of the process at date t, given information up to t− 1?

b Find σ2, the unconditional variance of the process.

c∗ Show εt has ‘excess kurtosis’.

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y1t = a0,1 + a011y1,t−1 + a012y2,t−1 + a111y1,t−2 + a112y2,t−2 + u1t

y2t = a0,1 + a021y1,t−1 + a022y2,t−1 + a121y1,t−2 + a122y2,t−2 + u2t

where E[ut] = 02.1 and E[utu′t] = Ω

a Write the VAR in matrix form.

b What are the four elements of Ω?

c Find E[yt+1|yt] and E[yt+2|yt]

d What is var(yt|yt−1)?

16 Let yt = c+ φ1yt−1 + φ2yt−2 + εt.

a Write the system as a VAR(1) in the two-variable vector yt = [yt, yt−1]′:

yt = c + Fyt−1 + ut

b∗ Find E[yt+2|yt] using this representation and compare your answer to 12 c.

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17 Let

mt = β0 + β1pt + β2yt + β3rt + e+ t

where mt is log money demand, yt is log income, pt is the log price level and rt is a short term

interest rate.

a From your knowledge, what is the likely order of integration of the series in this equation?

b The Book of Crazy Macroeconomic Theory suggests that this equation represents money

market equilibrium. What does this theory imply about the time series properties of the

errors et?

b How would you test this theory? What is the general name for this procedure?

18 Stack the four variables in the vector xt:

xt =

mt

pt

yt

rt

a Suggest a way to model the 4 variables that would overcome the endogeneity problems

likely to beset the money market equation.

b How will you choose the lag order in this estimation procedure?

c If you found a VAR(1) was appropriate (bit of hint that!), show the associated VECM is

∆xt = c + Πxt−1 + εt

What is Π in terms of the VAR parameters?

Say a VAR(3) was appropriate. Derive the associated VECM. Be careful to show your

working as well as the final result.

d What are the three possibilities for the rank of Π? How do they relate to cointegration?

e Imagine there are 2 cointegrating vectors, Π = αβ′. What is the rank of α and the rank of

β?

f In this case, how many restrictions are required to give an economically meaningful inter-

pretation to the cointegrationg vectors.

g∗ In terms of the cointegrating vector, how would you represent the theory that real money

demand moves in a one for one relationship with both income and interest rates, where the

effect of income is positive and interest rates is negative?

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19 Let yt = c + Φ1yt−1 + Φ2yt−2 + εt, with E[εtεt] = Ω.

a Write the VAR in terms of the lag polynomial Φ(L)yt = c + εt. Find E[yt].

b∗ Provided Φ(L) is stable, the VAR has a VMA(∞) representation.

yt = µ+ Ψ(L)εt

where analagously to the univariate case, Ψ(L) = (In + Ψ1L+ Ψ2L2 + Ψ3L

3 + . . . )

Find Ψj j = 1, 2, 3, . . . in terms of Φi, i = 1, 2. Hint, as in univariate case use Ψ(L) =

Φ(L)−1 and XX−1 = I

c What is∂yt+s∂εt

What do we call the series ψsij s = 0, 1, 2, . . . where ψsij is the row i column j elemetns of

Ψs?

d∗ How would you ‘orthogonalise the IRFs’?

e∗ Comment on your favourite computer programme’s default method of orthogonalising IRFs.

20 a Show that the reculsive least squares estimator of the mean xt = 1t

∑ti=1 xi has declining

gain.

b What weight does the constant gain estimator of the mean, xt = xt−1 + γ(xt− xt−1) assign

to data pont xt−i?

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