ETF5200: Time Series and Panel Data Econometrics
Information about Final Examination in Semester one 2022
Department of Econometrics and Business Statistics
Monash University
1. Exam instructions
• This is an individual assessment task.
• All responses must be included within this document and under individual
questions.
• Students are required to answer ALL questions.
• Your submission must occur within 2 hours and 40 minutes of the official
commencement of this assessment task (Australian Eastern Standard Time).
2. Examinable material
• The lecture material from Weeks 1–12 will all be examinable
• There are two questions in the time series section, and two more in the panel
data section
• Additional reading material will not be examinable, but some may help your
understanding of the lecture material
3. Possible types of exam questions
• Exam questions would be similar to those questions used in the assignments,
tutorials and examples
• There will be no questions involving ‘long’ derivations, but ‘short’ derivations
are possible
• This assessment has a hurdle requirement of 50% to pass the unit.
• Upon completion of this assessment task, please upload a separate PDF doc-
ument of your answers in either handwritten or typed form to Moodle using
the assignment submission link.
• There will be no questions involving ‘on–the–spot’ coding and/or computing
using a package
4. Wordings of exam questions
• If you are asked to derive an expression, you will need to provide sufficient
details for the derivation
• If you are asked to write down a formula, however, you will need only to write
down it (without necessarily providing the reasoning)
• If you are asked to derive and then write down a formula, you will need to
provide sufficient details for the derivation before you provide an expression
for the formula
Sample Questions from Past Exams
1. Question 1
• Let yt = µ+ ρ yt−1 + εt, for t = 1, 2, · · · , be an auto–regressive model of order
1 (AR(1)), where y0 is a constant, ρ is an unknown parameter, and {εt} is an
independent and identically distributed (i.i.d.) random error with E[ε1] = 0
and 0 < σ20 = E[ε
2
1] <∞.
– Derive the ordinary least squares (OLS) estimators of (µ, ρ).
– In the case of ρ = 1, find the covariance function cov(ys, yt).
– Is yt stationary ? Give your reasoning. If not, construct a stationary
version derived from yt.
2. Question 2
Define a general time series model of the form:
∆yt = θ0 + θ1 t+ γ yt−1 + εt, (1)
where ∆yt = yt − yt−1, {εt} is a stationary time series error with E[ε1] = 0 and
0 < σ20 = E[ε
2
1] <∞, and (θ0, θ1, γ) is a vector of unknown parameters.
Consider the null and alternative hypotheses: H0 : γ = 0 and H1 : γ < 0. Some of
the commonly used tests include the Dicky–Fuller (DF) test, the Augmented Dicky–
Fuller (ADF) test, the Phillips–Perron (PP) test, and the Kwiatkowski–Phillips–
Schmidt–Shin (KPSS) test.
• Discuss the similarities and differences among the DF, the ADF, the PP and
the KPSS tests.
• An econometrician considers model (1) and then applies the DF test to a
data set of a sample size of 50 observations. His/her calculation of the ADF
test yields an ADF test statistic = −4.57. Using the following table, can you
conclude that the null hypothesis of a unit root should be accepted at the 5%
level ? Give your reasoning.
Table: Critical values for the Dickey–Fuller t–distribution
Sample size case A: without trend case B: with trend
1% 5% 1% 5%
T = 25 -3.75 -3.00 -4.38 -3.60
T = 50 -3.58 -2.93 -4.15 -3.50
T = 100 -3.51 -2.89 -4.04 -3.45
T = 250 -3.46 -2.88 -3.99 -3.43
T = 500 -3.44 -2.87 -3.98 -3.42
T =∞ -3.43 -2.86 -3.96 -3.41
3. Question 3
• Consider the following two nonstationary models:
yt = θ0 + θ1 t+ ut and yt = µ0 + yt−1 + vt, (2)
for t = 1, 2, · · · , where y0 is a known constant, θ0, θ1 6= 0, and µ0 6= 0 are all
unknown parameters, and ut and vt are weakly stationary time series errors
with E[u1] = E[v1] = 0, and 0 < E[u
2
1] = σ
2
u <∞ and 0 < E[v21] = σ2v <∞.
– Find the mean and the covariance functions of yt for each of the models in
(2).
– Using the mean and the covariance functions for each case, explain why
the two models have different types of nonstationarity.
• Consider a smooth transition auto–regressive (STAR) model of the form:
yt = ρ yt−1 + δ yt−1G(st; γ, τ) + εt, (3)
where {εt} is a sequence of stationary time series errors with E[ε1] = 0 and
0 < σ20 = E[ε
2
1] < ∞, θ = (ρ, δ)′ is a vector of unknown parameters, and
G(st; γ, τ) is the transition function, in which st is the transition variable, γ is
the smooth transition parameter, and τ is the threshold parameter.
Suppose the data set {(yt, st) : t = 1, 2, · · · , T} becomes available to you.
– Consider the case where G(st; γ, τ) = I[st ≤ τ ], in which I[·], with I[x >
0] = 1 and I[x ≤ 0] = 0, is the conventional indicator function.
In this case, model (3) reduces to
yt = ρ yt−1 + δ yt−1 I[st ≤ τ ] + εt. (4)
∗ Outline the main steps about how to estimate (ρ, δ) and then τ .
∗ Outline the main steps about how to test H0 : δ = 0 versus H1 : δ 6= 0.
– A commonly used form of G(st; γ, τ) is the Logistic function: G(st; γ, τ) =
(1 + exp (−γ(st − τ)))−1 I[γ > 0]. The corresponding version of model (3)
is called the LSTAR model.
∗ For the LSTAR model, outline the main steps about how to test for
linearity versus nonlinearity.
∗ Explain how you might apply the LSTAR for an analysis of the ex-
change rate between the British pound and the US dollar.
4. Question 4
Consider a simple panel data model of the form:
Yit = β0 + β1Xit + β2X
2
it + αi + eit, (5)
for 1 ≤ i ≤ N and 1 ≤ t ≤ T , where {Xit} is an array of independent and identically
distributed (i.i.d.) observed variables, {eit} is an array of i.i.d. random errors with
E[eit] = 0 and E[e
2
it] = σ
2
1 < ∞, {αi} is the unobservable individual specific effect
with E[αi] = 0 and E[α
2
i ] = σ
2
2 < ∞, and {αi} and {eit} are independent of each
other, and βj for j = 0, 1, 2 are all unknown parameters.
• Discuss how to eliminate αi before estimating βj by β̂j for j = 0, 1, 2, respec-
tively.
• Using one of the estimation methods you have learned from the Lectures, write
down the expressions of β̂j for j = 0, 1, 2. Give your reasoning.
5. Question 5
Consider the following dynamic panel data model of the form:
Yit = µ+ θ t+ ρ Yi,t−1 + αi + εit (6)
for 1 ≤ i ≤ N, 2 ≤ t ≤ T , where µ, θ and ρ are unknown parameters, αi is the
fixed effects component, {εit : 1 ≤ i ≤ N ; 2 ≤ t ≤ T} is an array of i.i.d. random
errors with E[ε11] = 0 and 0 < E [ε
2
11] = σ
2
1 < ∞, and is independent of (αi, Yi,t−1)
with E[αi] = 0 and E[α
2
i ] = σ
2
2 <∞.
• Discuss a couple of different ways of estimating (µ, θ, ρ).
• Using one of the estimation methods you have learned from the Lectures, dis-
cuss how you would address possible endogeneity issues and then estimate the
unknown parameters consistently.
• Discuss how to test H0 : ρ = 0 and provide the main steps to show how to
implement the suggested test of yours.
• Outline four empirical scenarios where such binary models are applicable.
6. Question 6
Consider the following binary model:
Yit = I [X
′
itβi + αi − εit ≥ 0] =
1, X
′
itβi + αi ≥ εit,
0, X ′itβi + αi < εit,
(7)
where Xit, αi and εit are defined in a similar way to what has been discussed before.
• Discuss how to estimate (βi, αi) for the heterogeneity case.
• Discuss how to estimate β0 for the homogeneity case where β1 = β2 = · · · = β0.
• Outline three empirical scenarios where such binary models are applicable.