Økonomisk Kandidateksamen 2004 (II) Econometrics 2 June 14, 2004 This is a four hours closed-book exam (uden hjælpemidler). Answer all questions! The questions 1 to 4 have equal weight. Within each question, part (a) represents very basic material, part (b) requires a somewhat more detailed knowledge of the curriculum, and part (c) requires a deeper understanding, for example it may be technically demanding or it may require a good understanding for how to combine different theoretical results. A full and correct answer of all part (a) questions is sufficient for passing the exam. To be answered in Danish or English. Question 1 (a) We are given a linear panel data model with a fixed effect: y it = a i + βx it + u it. We observe each unit (i =1, 2,...,N)twice(sothatt =1, 2). Show that the first differenced estimator and the within estimator give identical estimates for β. Derive the estimator for β. (b) Explain what we mean by a difference-in-differences estimator and show how we can implement it using OLS. Give conditions under which the estimator is consistent. (c) Suppose that we have the following linear model with a trend which has a different effect on different people: y it = α i + βx it + δ i t + u it. We have a panel with more than three observations on each person (t =1, 2, 3,...,T). Discuss the problems of using a first differenced estimator to estimate β and suggest an alternative. 1
Question 2 (a) Consider the regression model Y t = δ + c 1 Y t 1 + c 2 Y t 2 + πy t 1 + t, t =1, 2,...,T, (2.1) where the initial values, Y 2,Y 1 and Y 0, are given and the error term is assumed to be independently and identically distributed with zero mean and constant variance. Showhowtheequationin(2.1)isrelatedtoanautoregressivemodelforY t. Now, let Y t be a monthly time series for the percentage yield of a 10 years US government bond covering the period t = 1955 : 1 2002 : 12. The results of an OLS estimation of equation (2.1) is reported in Table 2.1 below, while Table 2.2 is a copy of Table 8.1 in Verbeek (2000). Use the information in the tables to construct a test for the hypothesis that Y t has a unit root. Explain the test and the outcome. (b) Some macroeconomic theories predict that the real interest rate is constant in equilibrium. In a stochastic environment where shocks repeatedly hit the economy, a testable implication is that the real interest rate should be stationary. Let µ Pt Z t = 100 1 P t 12 be the year-on-year inflation rate where P t is the price level; and define the realized real interest rate as X t = Y t Z t. Given your conclusion regarding the order of integration of Y t, what does the theory suggest on the order of integration of the inflation rate, Z t? Using the consumer price index as a measure of P t, Table 2.3 reports the results of OLS applied to the regression model X t = e δ + ec 1 X t 1 + eπx t 1 + e t. Use the information in the tables to infer if the US data is in accordance with the theoretical prediction. (c) Instead of the real interest rate, X t = Y t Z t, with a unit coefficient to inflation, we could consider a different linear combination X t = Y t β 1 Z t. Explain how the parameter β 1 can be estimated and outline a procedure for testing if β =( 1 β 1 ) 0 is a cointegrating vector. 2
Table 2.1: Modelling Y t by OLS for 1955 : 4 2002 : 12 Coefficient Std.Error t value Constant 0.049972 0.03080 1.62 Y t 1 0.394837 0.04074 9.69 Y t 2 0.221742 0.04083 5.43 Y t 1 0.007030 0.00420 1.67 bσ 0.265579 RSS 40.13 R 2 0.153091 F (3, 569) 34.28 No. of observations 573 Table 2.2: 1% and 5% critical values for Dickey-Fuller tests. No constant Constant Constant No trend No trend Trend Sample size 1% 5% 1% 5% 1% 5% T =25 2.66 1.95 3.75 3.00 4.38 3.60 T =50 2.62 1.95 3.58 2.93 4.15 3.50 T =100 2.60 1.95 3.51 2.89 4.04 3.45 T =250 2.58 1.95 3.46 2.88 3.99 3.43 T =500 2.58 1.95 3.44 2.87 3.98 3.42 T = 2.58 1.95 3.43 2.86 3.96 3.41 Table 2.3: Modelling X t by OLS for 1956 : 3 2002 : 12 Coefficient Std.Error t value Constant 0.045980 0.02490 1.85 X t 1 0.252356 0.04094 6.16 X t 1 0.017406 0.00686 2.54 bσ 0.389264 RSS 84.70 R 2 0.070025 F (2, 559) 21.05 No. of observations 562 3
Question 3 (a) Imagine that a manufacturer of bulbs is interested in estimating the expected lifetime of a bulb. From an experiment he observes a sequence of n independent lifetimes, x 1,x 2,...,x n. From experience he knows that the lifetime of an individual bulb, x i, follows an exponential distribution with expectation θ, i.e. with a density function given by n f x (x i θ) =θ 1 exp x o i, i =1, 2,...,n, (3.1) θ where x i > 0andθ>0. Write the log-likelihood function for the set of observations, x 1,x 2,...,x n, and derive the maximum likelihood estimator, b θ ML. Assume that the manufacturer has observed n = 100 lifetimes, x 1,x 2,...,x 100, and that P 100 i=1 x i = 1000. Find the maximum likelihood estimate, b θ ML,forthis case. (b) Find the information, 2 log L i (θ) I(θ) = E, θ θ where log L i (θ) is the log of the likelihood contribution for observation i; and derive the asymptotic variance of b θ ML. Explain the Wald principle for hypotheses testing. Construct a Wald test for the hypothesis that the expected lifetime is 11.5 inthe numerical example in question (a). What do you conclude? (c) Consider the regression model y t = x 0 tβ + t, t =1, 2,...,T, (3.2) where the error term, t, conditional on x t is claimed to be distributed as t x t N 0,σ 2. (3.3) Discuss the following statement: The maximum likelihood estimator, β b ML,based on (3.2) and (3.3) is consistent even if the normality assumption in (3.3) is not correct. Question 4 (a) Consider the AR(2) model Y t = δ + θ 1 Y t 1 + θ 2 Y t 2 + t, (4.1) where the error term, t, is independently and identically distributed with zero mean and variance σ 2. Assume that the roots of the characteristic polynomial, A(L) =1 θ 1 L θ 2 L 2, are located outside the unit circle. 4
Derive the mean, µ = E [Y t ], the autocovariances γ k = E [(Y t µ)(y t k µ)] for k =1, 2, 3,... and the autocorrelations, ρ 1,ρ 2,ρ 3,... State the assumptions you use in the derivations. (b) Imagine that you have estimated a model like (4.1) and you suspect that there might be first order autocorrelation in the error term. Explain the Breusch-Godfrey LM test for first order autocorrelation. Discuss if the OLS estimators, b δ, b θ 1 and b θ 2, obtained from (4.1) are consistent in the presence of error autocorrelation. (c) Define the property of autoregressive conditional heteroscedasticity, ARCH. Show that an ARCH(2) model implies that the squared error, 2 t, follows an AR(2) model. Explain how the presence of ARCH(2) can be tested. 5