Eksamen på Økonomistudiet 2006-II Econometrics 2 June 9, 2006

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1 Eksamen på Økonomistudiet 2006-II Econometrics 2 June 9, 2006 This is a four hours closed-book exam (uden hjælpemidler). Please answer all questions. As a guiding principle 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) An interesting variable for economic analysis is the ratio between the price of existing houses and the price of new constructions, the so-called Tobin s Q. If the variable Q t is stable over time, it means that house prices have not developed systematically different from the cost of replacement buildings and there is no systematic price gain on housing investments. To analyze the stability of Tobin s Q, we consider a linear regression model for q t =log(q t ), q t = δ + πq t 1 + c 1 q t 1 + c 2 q t 2 + t, t =1, 2,...,T, (1.1) where t is an independently and identically distributed error term with mean zero and variance σ 2. Show how (1.1) is related to an autoregressive model for the levels of q t.explain what a unit root implies for the parameters of equation (1.1), and explain how the presence of a unit root can be tested with a Dickey-Fuller t test. Table 1.1 reports the results of an OLS estimation of (1.1) based on quarterly observations for q t for Denmark, t = 1971 : : 2. Use the Dickey-Fuller t test to assess the presence of a unit root using the critical values in Table 1.2. What do you conclude regarding the stability of Tobin s Q for Denmark over the considered period? (b) Now assume for simplicity that c 1 = c 2 =0in equation (1.1), so that q t follows an AR(1) model. First consider the case H A : 2 <π<0, and derive the solution for q t in terms of the initial value, q 0, the constant term, δ, and the error terms, 1

2 1, 2,..., t. Draw a sketch of a realization of the time series and briefly describe the properties. Next consider the AR(1) model for the case H 0 : π =0and derive the solution for q t. Draw a sketch of a realization of the time series under H 0 and briefly describe the properties. Finally consider the AR(1) model for the case H0 : π = δ =0and derive the solution for q t. Once again draw a sketch of a realization of the time series and briefly describe the properties. (c) Explain how the Dickey-Fuller t test is related to the three cases considered in question (b), and discuss why the treatment of the deterministic terms sometimes makes the test difficult to interpret. Explain how the joint hypothesis H0 can be tested. Table 1.1: Modelling q t by OLS for t = 1971 : : 2 Coefficient Std.Error t value Constant q t q t q t Log-likelihood bσ No. of observations 135 R Table 1.2: Critical values for the Dickey-Fuller test No constant Constant Constant No trend No trend Trend Sample size 1% 5% 1% 5% 1% 5% T = T = T = T = T = T =

3 Question 2 (a) Consider the time series regression model y t = x 0 tβ + t, t =1, 2,...,T, (2.1) where x t is a k 1 vector of explanatory variables not including lags of y t,andβ is a k 1 vector of corresponding parameters to be estimated. Assume that (y t,x 0 t) 0 is a jointly stationary and weakly dependent stochastic process. State the moment conditions that are sufficient to derive a consistent estimator of β, and derive the method of moments estimator, β b MM. Explain the difference between method of moments (MM) estimation and generalized method of moments (GMM) estimation. Discuss in particular why there is no role for a weight matrix in MM estimation, while it plays a role in GMM estimation. (b) Now you are told that there could be autocorrelation in the error term in the regression in (2.1). What are the implications of autocorrelation on an OLS estimation? How would your answer change if y t 1 was included as a regressor in (2.1)? Explain the Breusch-Godfrey Lagrange Multiplier (LM) test for no-autocorrelation. (c) After testing, it turns out that there are no indications of autocorrelation in t, but you find clear evidence of autoregressive conditional heteroskedasticity (ARCH). Explain the idea of ARCH, and augment the model in (2.1) to take the ARCH effects in the residuals into account. Question 3 (a) Let the binary variable y i (i =1, 2,...,n)denotethelabormarketsuccessofnewly educated economists, where y i =1indicates that student i is employed in a relevant position within three months after finishing university, while y i =0indicates that it takes more than three months. Assume that the probability of labor market success is fixed, θ =prob(y i =1), such that the density function for y i isgivenbythe binomial, f(y i θ) =θ yi (1 θ) 1 y i. (3.1) Assume furthermore that the random outcomes are independent. We want to estimate the probability θ. Write the log-likelihood function for the set of observations, y 1,y 2,...,y n,and derive the maximum likelihood estimator, b θ ML. Be careful to explain the steps in the likelihood analysis. In a given year n = 100 economics students finished university, and 50 of those where hired within three months. Find the maximum likelihood estimate of the probability of labor market success. 3

4 (b) Find the information, 2 log L i (θ) I(θ) = E, θ θ where log L i (θ) is the contribution to the likelihood function for a single observation. State the asymptotic distribution of the maximum likelihood estimator. The official goal of the university is that 60% of the students get a job within three months. Derive a Wald test for the hypothesis that θ =0.6 in the numerical example in question (a). What do you conclude? (c) Your lecturer suggests that the probability of labor market success is not fixed; it rather depends on the exam result in econometrics, denoted x i. How would you modify the binary model to test that hypothesis? Question 4 (a) Let r t denote the one week interest rate on the US money market, t =1, 2,...,T, and let b t denote the three month interest rate. Assume that the two interest rates behave as unit root non-stationary time series and consider the following linear regression model r t = µ + β b t + u t. (4.1) To analyze whether there is a relationship between the interest rates, a friend of yours suggests to estimate the linear regression in (4.1) using OLS and test the hypothesis H 0 : β =0, against the alternative H A : β 6= 0, by a standard t test. Explain why this approach is not a good idea. Define the concept of cointegration, and explain how the presence or absence of cointegration can be used to analyze the existence of an equilibrium relationship between the interest rates. How would you test the hypothesis of no-cointegration based on the static regression in (4.1)? (b) Now consider the autoregressive distributed lag (ADL) model, r t = δ + θ 1 r t 1 + φ 0 b t + φ 1 b t 1 + t, (4.2) where t is an independently and identically distributed error term. Derive the errorcorrection model (ECM) corresponding to (4.2) and state the long-run solution. Table 4.1 reports the results of an OLS estimation of the equation in (4.2) for monthly observations, t = 1982 : : 2. Derive an estimate of the long-run coefficient, β. Explain how you could test for no-cointegration based on the ADL model in Table 4.1. (c)usingequation(4.2)weassumethatitistheshortinterestrate,r t,thatcorrects deviations from equilibrium. How would you modify the analysis if it is primarily the long interest rate, b t, that error corrects? What if both interest rates error correct? 4

5 Table 4.1: Modelling r t by OLS for t = 1982 : : 2 Coefficient Std.Error t value Constant r t b t b t Log-likelihood bσ No. of observations 290 R