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 con
stant, 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 β j for j = 0, 1, 2, respec- tively. Using one of the estimation methods you have learned from the Lectures, write down the expressions of β j 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.