Økonomisk Kandidateksamen 2005(I) Econometrics 2 January 20, 2005

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1 Økonomisk Kandidateksamen 2005(I) Econometrics 2 January 20, 2005 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 In order to assess the overall welfare effects of a proposed change in the early retirement scheme ( efterlønsordningen ) we would like to obtain an estimate of the effect of retirement on a person s psychological well-being. For this we use a survey based on a random sample of n = 1000 individuals at the age of 58 in 1994 and construct an index of psychological well-being, y, for each person in the sample. The index combines a number of different factors such as being in good form, feeling depressed, feeling lonely, etc., into an single measure of psychological well-being. The index is constructed in such a way that an increase in y implies an improved well-being. The same individuals were reinterviewed in 1999 so we have a panel of 2000 observations in total. (a) The survey has no direct health information but it is fairly clear that bad health in general is harmful to a person s psychological well-being. Moreover, health problems are believed to increase the probability that a person retires. Consider two variables from the 1999-survey, the index of psychological well-being, y i,1999, and a dummy variable, r i,1999, which takes the value 1 if person i had retired and zero otherwise. Explain why an OLS regression of y i,1999 on a constant and r i,1999, is unlikely to estimate the true effect of retirement on well-being. Determine the expected direction of the bias. (b) Consider the following panel data model for the index of well-being, y it, for individual i at time t, with the retirement dummy, r it, and a time dummy for 1999 as 1

2 explanatory variables: y it = β 0 + δ 0 d99 t + β 1 r it + a i + u it, t = 1994, 1999, i =1, 2,...,n. Here a i is an unobserved effect which is specific to individual i and constant across time, whereas u it is an idiosyncratic error term that varies randomly across time and individuals. ExplainwhatwemeanbytheWithinestimator in this model. Show that it can be a consistent estimator of β 1 even if corr(a i,r it ) 6= 0for some t. (c) It is now suggested that one should just subtract the initial value of the well-being index from the value obtained in the second survey round, transform the retirement dummy and the time dummy in the same way, and apply OLS to the sample of transformed variables. However, there is a concern that the retirement decision might be influenced by unobserved time-varying variables which might affect consistency of the suggested estimation procedure. Based on register data we are able to add another piece of information to the data, the retirement status of the wife or husband of the person in the survey. Discuss how you would use this extra information to obtain a consistent estimate of the effect of retirement on well-being. Question 2 (a) Consider a first order autoregressive, AR(1), model given by Y t = δ + θy t 1 + t, t =1, 2,...,T, (2.1) where the initial value, Y 0, is given. Explain what is meant by generalized autoregressive conditional heteroscedasticity, GARCH, e.g. by referring to the equation σ 2 t = + α 2 t 1 + βσ 2 t 1, (2.2) where σ 2 t = E[ 2 t I t 1 ] denotes the variance of the error term conditional on the information set at time t 1. Now, let IBM t denote the month-on-month percentage change in the price of the IBM stock, recorded for the period t = 1926 : 1,...,1999 : 12. Table 2.1 reports the maximum likelihood estimates corresponding to the model in (2.1) and (2.2) under the assumption of conditional normality, t I t 1 N(0,σ 2 t ). Furthermore, Figure 2.1 graphs the estimated residuals, b t, together with ±1.96 bσ t. Comment on the estimated parameters of (2.1) and (2.2) and the graph. In particular explain whether the conditional heteroscedasticity is significant. 2

3 (b) Explain how the presence of ARCH effects in a regression like (2.1) can be tested using a LM test. Show that the GARCH(1,1) model implies that the squared residuals, 2 t,followan ARMA process. (c) State the stationarity condition for the ARMA process, and check if it is fulfilled in the estimated model for the IBM stock in Table 2.1. What does it mean if the ARMA process is close to having a unit root? Table 2.1: GARCH(1,1) estimation of IBM t,for1926 : : 12 Coefficient Std.Error t value δ θ α β log-likelihood No. of obs. 887 no. of parameters Residuals ±1.96 ^σ t Figure 2.1: Estimated residuals, b t, and the conditional standard deviation. 3

4 Question 3 (a) Assume that we have observed a univariate time series: y 0,y 1,...,y T. To model the time series a first order autoregressive model is suggested, i.e. the specification y t = θy t 1 + t, (3.1) where the error term is normal, t N(0,σ 2 ), i.e. given by the density function ¾ f ( t σ 2 1 )= ½ exp 2 t 2πσ 2 2σ 2. Formulate the likelihood function for the observations y 1,y 2,...,y T the initial value, y 0,i.e. conditional on L(θ, σ 2 )=f(y 1,y 2,..., y T y 0 ; θ, σ 2 ). Find the score vector, s(θ, σ 2 ), and derive the maximum likelihood (ML) estimators, b θml and bσ 2 ML. (b) Consider again the AR(1) model in (3.1) and assume that θ < 1. Derive the mean, µ = E[y t ]; the autocovariances, γ j = E[(y t µ)(y t j µ)], for j =0, 1, 2,...; and the autocorrelations, ρ j = γ j /γ 0. How would you suggest to use the obtained µ and γ 0 to formulate the likelihood function for the full set of observations, i.e. without conditioning on y 0. (c) Now consider the MA(1) model L Full (θ, σ 2 )=f(y 0,y 1,y 2,...,y T θ, σ 2 ), y t = t + α t 1, t =1, 2,...,T. (3.2) Assume that α < 1 and t N(0,σ 2 ). You might recall that ML estimation of the MA(1) model is more complicated than ML estimation of the AR(1) model, and a GMM type estimator could be a simple alternative. Remember that for the MA(1) model in (3.2) the autocovariances are given by γ 0 = 1+α 2 σ 2 (3.3) γ 1 = ασ 2. (3.4) Use the results in (3.3) and (3.4) to suggest two moment conditions, which could be used to construct simple method of moments (MM) estimators of the parameters α and σ 2. 4

5 Question 4 (a) Consider the following regression model x t = δ + c x t 1 + πx t 1 + t, t =2, 3,...,T, (4.1) where the initial values, x 0 and x 1, are given. Show the correspondence between the model in (4.1) and an autoregressive model for x t. Explain how the presence of a unit root can be tested against the stationary alternative. Let r t denote the effective US Federal funds rate (which is an overnight interest rate), and let b t denote a 1 year bond rate. Define the interest rate spread, measuring the slope of the short end of the yield curve, as x t = b t r t. Imagine that you are informed that both r t and b t areunitrootprocesses,and that we are interested in testing whether the slope of the yield curve behaves in a stationary manner. Table 4.1 contains the output of the regression in (4.1) for the interest rate spread 1988 : : 10, while Table 4.2 is similar to Table 8.1 in Verbeek (2004). Use the information in the tables to test the hypothesis that x t has a unit root. How is this related to the concept of cointegration. (b) Now consider the autoregressive distributed lag, ADL, model given by r t = δ + θ 1 r t 1 + θ 2 r t 2 + φ 0 b t + φ 1 b t 1 + φ 2 b t 2 + t. (4.2) Derive the corresponding error correction model (ECM), and explain how it is related to cointegration. (c) For a vector autoregressive (VAR) model of order 2, the error correction form is given by Y t = γβ 0 Y t 1 + δ + Γ 1 Y t 1 + t, where Y t is now a vector. For the two-dimensional case Y t =(r t,b t ) 0, we obtain the following estimation results: Ã! d rt µ (5.01) = r t d b t b t 1 (34.29) (0.18) Ã! ( 4.80) (4.26) (3.91) r t 1 + +, b t 1 ( 0.25) (0.40) (5.33) where the numbers in parentheses are t values. Interpret the estimated model. In particular, explain how the variables cointegrate and how the variables adjust to deviations from equilibrium. 5

6 Table 4.1: Modelling s t by OLS for 1988 : : 10 Coefficient Std.Error t value Constant s t s t bσ RSS R F (2, 199) No. of observations 202 Table 4.2: 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 = T = T = T = T = T =

Økonomisk Kandidateksamen 2004 (II) Econometrics 2 June 14, 2004

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,