Written Exam for the B.Sc. and M.Sc. in Economics 2007-I ECONOMETRICS 2. Final Exam January 5, 2007 (4-hour closed book exam)

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1 Written Exam for the B.Sc. and M.Sc. in Economics 2007-I ECONOMETRICS 2 Final Exam January 5, 2007 (4-hour closed book exam) PLEASE NOTE that the language used in your exam paper must correspond to the language of the title for which you registered during exam registration. I.e. if you registered for the English title of the course, you must write your exam paper in English. Likewise, if you registered for the Danish title of the course or if you registered for the English title which was followed by eksamen på dansk in brackets, you must write your exam paper in Danish. If you are in doubt about which title you registered for, please see the print of your exam registration from the students self-service system. PLEASE ANSWER ALL FOUR QUESTIONS. 1

2 Question 1 (a) According to simple theoretical growth models, the so-called great ratios, defined as consumption over income, c = C/Y, and investment over income, v = V/Y, should be stable; i.e. they should fluctuate around a constant mean. The empirical economic literature has for many years tried to assess the stability of the great ratios. Let x t (t =1, 2,...,T) denote a stochastic process and consider an autoregressive model with k =3lags, x t = δ + θ 1 x t 1 + θ 2 x t 2 + θ 3 x t 3 + t, (1.1) where t is independently and identically distributed. Define the concept of a unit root and explain how the presence of a unit root in (1.1) can be tested. Discuss what the conclusion regarding unit roots tells you about the stability of the great ratios. (b) Figure 1.1 shows quarterly time series for the ratios of private consumption to GDP (c t ) and gross fixed investments to GDP (v t ) recorded for the US economy for 1947 : : 3. Briefly describe the time series behavior and comment on the potential stationarity of the great ratios. Table 1.1 and Table 1.2 report augmented Dickey-Fuller tests for c t and v t based on equations like (1.1) for different values of the lag lengths k =1, 2,...,6. Usethe information in the tables to test the stability of the great ratios. Explain how you perform the test, i.e. which statistics you look at and which critical values you use. AretheconclusionsinlinewithyourimpressionfromFigure1.1? (c) One of your colleagues is an expert in unit root testing. She makes the following claim: The simple Dickey-Fuller t test for a unit root treats the constant term in (1.1) in an inappropriate way because the models do not have the same deterministic specification under the null and under the alternative. Explain what she could mean by that and suggest an alternative test with a more appropriate treatment of the constant term. 2

3 Figure 1.1: Time series graphs of the consumption to income ratio, c t = C t /Y t,andthegross investment to income ratio, v t = V t /Y t,fortheus,1947 : : Consumption to income, c t 0.18 Investment to income, v t Table 1.1: Augmented Dickey-Fuller tests for c t based on equation (1.1). Lag- t test for length Information criteria significance of k ADF test SC HQ AIC lag k, t θk = Note: T = 233 observations. Critical values for ADF test: 5% = 2.87 and 1% = Table 1.2: Augmented Dickey-Fuller tests for v t based on equation (1.1). Lag- t test for length Information criteria significance of k ADF test SC HQ AIC lag k, t θk = Note: T = 233 observations. Critical values for ADF test: 5% = 2.87 and 1% =

4 Question 2 (a) Let x t denote the number of patent applications received by the government patent office on a given day, t =1, 2,...,T. The variable x t is discrete and takes values 0, 1, 2,... Based on experience from the patent office we assume that the number of applications is random across days and that it follows a Poisson distribution with density function f(x t λ) = exp( λ)λxt. x t! Here the constant parameter, λ, is the expectation of x t, E[x t ] = λ > 0, and x t!=x t (x t 1) (x t 2)... 1 is the factorial of x t. Write the log-likelihood contribution for one observation, log L t (λ x t ),andthe log-likelihood function for the total set of observations, x 1,x 2,...,x T. Derive the maximum likelihood estimator of the expected number of applications, λ b ML. During the T = 200 opening days in the year 2005 the patent office received a total of 400 applications. Find the maximum likelihood estimate, λ b ML, for this year. (b) Now find the information 2 log L t (λ x t ) I(λ) = E, λ λ and state the asymptotic distribution of λ b ML. In 2005 the patent office was allocated resources to handle 1.5 applications per day. Construct a Wald test for the hypothesis that λ =1.5, and conclude whether the actual number of applications per day was significantly higher than planned. (c) One of the employees has the idea that the expected number of patent applications depends on the weekday. Let z 1t,...,z 4t denote dummy variables taking the value one on Tuesdays, Wednesdays, Thursdays, and Fridays, respectively, and zero otherwise. Suggest a modification of the Poisson regression model to take the dependence on weekdays into account, and state the modified log-likelihood function. Question 3 (a) Let y t (t =1, 2,...,T) denote a stochastic variable and consider the following model: y t = δ + θy t 1 + t (3.1) σ 2 t = + α 2 t 1 + βσ 2 t 1. (3.2) We assume that the initial value, y 0,isfixed and 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. Explain the idea of generalized autoregressive conditional heteroskedasticity (GARCH) by referring to the equations in (3.1) and (3.2). 4

5 Now, let CARLSBERG t denote the daily recorded stock price for a particular Danish firm and let y t = log(carlsberg t ) be the day-to-day change in the logarithm for the period 1/ to 13/ Column (A) in Table 3.1 reports the results of a likelihood analysis based on the assumption of a conditional normal distribution, i.e. t I t 1 N(0,σ 2 t ). Comment on all the estimated parameters of (3.1) and (3.2). (b) Now consider an augmented equation for the conditional mean given by y t = δ + θy t 1 + γσ 2 t + t. (3.3) Explain how the new term can be interpreted. Columns (B) in Table 3.1 presents the estimation results based on the equations in (3.3) and (3.2). Comment on the output and whether the fit of the model has improved. (c) The scaled residuals from the model in column (A), calculated as b t /bσ t, do not seem to follow a standard normal distribution. The Jarque-Bera test for a normal distribution is JB = , which is clearly rejected compared to the asymptotic χ 2 (2) distribution; and a graphical inspection suggests that the rejection is mainly due to excess kurtosis, i.e. too fat tails. Explain the potential implications for the likelihood analysis and suggest a solution to the problem. Table 3.1:GARCHmodelsfory t, 1/ to 13/ (A) (B) Model to be estimated: (3.1)+(3.2) (3.3)+(3.2) δ (1.56) θ ( 1.84) (1.66) ( 1.80) γ ( 12.50) (scaled by 10 6 ) (1.19) α (4.76) β (122.00) (0.92) (1.89) (57.70) log-likelihood No. of observations Note: t values in parentheses. 5

6 Question 4 (a) Let r t denote a short money market interest rate and let b t denote an interest rate with a longer maturity. Consider the following autoregressive distributed lag (ADL) model, 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, (4.1) where t is independently and identically distributed. Derive the error-correction representation of (4.1) and state the long-run solution. (b) Now you are informed that the two interest rates are unit root non-stationary. Define the concept of cointegration and explain how cointegration relates to the long-run solution and the error correction model in question (a). Suggest a way to test the hypothesis of no-cointegration based on the error correction model. (c) Now consider also the policy interest rate f t. You are told that f t and r t cointegrate with cointegration vector β =(1, 1) 0 so that changes in the policy interest rate are transmitted to the short market interest rate. You are also told that the conclusion from the above analysis is that r t and b t cointegrate with the same cointegration vector, β. Does that tell you anything at all on the potential cointegration between the policy rate f t and the long-run bond rate b t? Motivate your answer. 6

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