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[NBS8187] NEWCASTLE UNIVERSITY ————————————— SEMESTER 2 2014/2015 ————————————— TIME SERIES ECONOMETRICS Time allowed - 90 minutes Answer THREE out of the four questions. Each question is worth 1/3 of total marks. [Please turn over ] [NBS8187] Question 1 Consider the random variable Y with probability mass function Pr(Y = y) = exp(−λ)λy y! , λ > 0, y = 0, 1, 2, . . . You remember that E(Y ) = V(Y ) = λ. Let y1, y2, . . . , yn be a random sample from this distribution. a) What is the name of this distribution? For what type of applications can it be used? Give an example of a possible application. b) Derive the likelihood function and the log-likelihood function. c) Derive expressions for the score, the Hessian and the information. d) Show that the expectation of the score is zero. e) Derive the ML estimator for λ. f) Derive the Cramer-Rao lower bound of the ML estimator. g) Does the ML estimator reach the Cramer-Rao lower bound? h) Assume n = 20 and ∑20 i=1 yi = 20. Test whether λ = 2 using both the Wald and Score tests at a 5% significance level. 2 [NBS8187] Question 2 a) Define the terms autocorrelation function (acf) and partial autocorrelation function (pacf). b) What is the difference between an autocorrelation function (acf) and a sample auto- correlation function (sample acf)? c) You obtain the following sample autocorrelations and sample partical autocorrelations for a series of 225 observations: Lag 1 2 3 4 5 6 7 8 sample acf 0.61 0.61 0.47 0.45 0.39 0.37 0.32 0.32 sample pacf 0.61 0.38 0.00 0.15 0.05 0.04 0.04 0.05 i) Which of these sample autocorrelations and sample partial autocorrelations are significant at a 5% significance level? ii) Given your result from i), what would be a good time series process for this data? iii) Check whether the first two sample autocorrelation coefficients are jointly signifi- cantly different from zero using the Ljung-Box test at a 5 % significance level. d) You have estimated the following ARMA(1,2) model: yt = 1 2 yt−1 + εt + 1 8 εt−1 + 1 2 εt−2, εt iid∼ N(0, σ2). i) Is this model stationary? ii) Derive the 1-step-ahead and 2-step-ahead forecasts. iii) Derive the 1-step-ahead and 2-step-ahead forecast errors. iv) Are the forecast errors from iii) corellated? e) Consider the AR(1) process yt = βyt−1 + εt, εt iid∼ N(0, σ2) with −1 < β="">< 1.="" let="" wt="∆yt" =="" yt="" −="" yt−1.="" derive="" v(wt).="" [hint:="" v(a−b)="V(A)" +="" v(b)−="" 2="" cov(a,b)]="" 3="" [please="" turn="" over="" ]="" [nbs8187]="" question="" 3="" consider="" the="" garch(1,1)="" model:="" εt="νt" √="" ht,="" νt="" iid∼="" n(0,="" 1),="" ht="ω" +="" αε="" 2="" t−1="" +="" βht−1,="" with="" ω=""> 0, α ≥ 0 and β ≥ 0. a) Derive the conditional mean Et−1(εt) and the conditional variance Vt−1(ε 2 t ). b) Derive the (unconditional) mean E(εt) and the (unconditional) variance V(ε 2 t ). c) Define ηt = ε 2 t −ht and remember that ηt is an uncorrelated series with zero mean (you do not need to prove this). Which process does the {ε2t} sequence follow? d) Derive the 1-step-ahead and 2-step-ahead forecasts of the conditional variance. Next you fit a GARCH(1,1) model to daily logarithmic returns of the FTSE 100 index. Here is part of the output: Conditional Variance Dynamics ----------------------------------- GARCH Model : sGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm Optimal Parameters ------------------------------------ Estimate Std. Error t value Pr(>|t|) omega 0.000004 0.000001 7.5075 0 alpha1 0.065918 0.008479 7.7744 0 beta1 0.863779 0.018348 47.0778 0 LogLikelihood : 879.0964 Information Criteria ------------------------------------ Akaike -6.9257 Bayes -6.8838 Q-Statistics on Standardized Residuals ------------------------------------ statistic p-value Lag[1] 0.1236 0.7252 Lag[p+q+1][1] 0.1236 0.7252 Lag[p+q+5][5] 6.1507 0.2918 d.o.f=0 H0 : No serial correlation 4 [NBS8187] Q-Statistics on Standardized Squared Residuals ------------------------------------ statistic p-value Lag[1] 0.3071 0.5795 Lag[p+q+1][3] 1.5886 0.2075 Lag[p+q+5][7] 2.7330 0.7411 d.o.f=2 ARCH LM Tests ------------------------------------ Statistic DoF P-Value ARCH Lag[2] 0.9853 2 0.6110 ARCH Lag[5] 2.5346 5 0.7713 ARCH Lag[10] 9.6939 10 0.4677 Sign Bias Test ------------------------------------ t-value prob sig Sign Bias 1.8504 0.06545 * Negative Sign Bias 0.7197 0.47238 Positive Sign Bias 0.1367 0.89137 Joint Effect 5.0584 0.16757 e) Assess whether the GARCH(1,1) model adequately fits the data. Question 4 Discuss in detail univariate GARCH models. Make sure you address the following points: estimation, model selection, model evaluation and model extensions. 5 [Please turn over ] [NBS8187] Critical Values of the Chi-Square Distribution with ν Degrees of Freedom Significance Level ν 0.10 0.05 0.01 1 2.71 3.84 6.63 2 4.61 5.99 9.21 3 6.25 7.81 11.34 4 7.78 9.49 13.28 5 9.24 11.07 15.09 6 10.64 12.59 16.81 7 12.02 14.07 18.48 8 13.36 15.51 20.09 9 14.68 16.92 21.67 10 15.99 18.31 23.21 11 17.28 19.68 24.72 12 18.55 21.03 26.22 13 19.81 22.36 27.69 14 21.06 23.68 29.14 15 22.31 25.00 30.58 16 23.54 26.30 32.00 17 24.77 27.59 33.41 18 25.99 28.87 34.81 19 27.20 30.14 36.19 20 28.41 31.41 37.57 21 29.62 32.67 38.93 22 30.81 33.92 40.29 23 32.01 35.17 41.64 24 33.20 36.42 42.98 25 34.38 37.65 44.31 26 35.56 38.89 45.64 27 36.74 40.11 46.96 28 37.92 41.34 48.28 29 39.09 42.56 49.59 30 40.26 43.77 50.89 END OF PAPER [NBS8187] NEWCASTLE UNIVERSITY ————————————— SEMESTER 2 2016/2017 ————————————— TIME SERIES ECONOMETRICS Time allowed - 90 minutes Answer ONE question from Section A and ALL questions from Section B. Each question in Section A is worth 30% of total marks and each question in Section B is worth 35%. [Please turn over ] [NBS8187] Section A Answer ONE question in this section. Question 1 Consider the random variable Y with probability density function f(y) = y α2 exp ( −y α ) , y ≥ 0, α > 0. You know that E(Y ) = 2α and var(Y ) = 2α2. Let y1, y2, . . . , yn be a random sample from this distribution. (a) Define the term random sample. (b) Derive the likelihood function and the log-likelihood function. (c) Derive expressions for the score, the Hessian and the information. (d) Show that the expectation of the score is zero. (e) Derive the ML estimator for α. (f) Derive the Cramer-Rao lower bound of the ML estimator. (g) Does the ML estimator reach the Cramer-Rao lower bound in finite samples? (h) Assume n = 10 and ∑10 i=1 yi = 20. Test whether α = 2 using both the Wald and likelihood ratio (LR) tests. Choose a significance level of 5% (a table with critical values of the chi-square distribution can be found at the end of the exam). Question 2 Discuss the three classical tests in the maximum likelihood context. Clarify your points with a graphical illustration. 2 [NBS8187] Section B Answer ALL questions in this section. Question 3 Consider the GARCH(1,1) model: εt = νt √ ht, νt iid∼ N(0, 1), ht = ω + αε 2 t−1 + βht−1, where ω > 0, α ≥ 0 and β ≥ 0. Define ηt = ε2t − ht. (a) Write down the likelihood function for this model. (b) Show that E(ηt) = 0. (c) Show that E(ηtηt−k) = 0 for k > 0 and thus ηt is serially uncorrelated. (d) Show that ηt is also uncorrelated with past squared errors. (e) What process does the {ε2t} sequence follow? (f) Derive an expression for E(ε2t+2|εt, εt−1, . . . , ht, ht−1, . . .). (g) Assume you fit the model to a daily stock return series. What range of values are likely for the coefficients ω, α and β? (h) Discuss the leverage effect and describe a model that can account for it. 3 [Please turn over ] [NBS8187] Question 4 Consider the ARMA(2,1) model: yt = yt−1 − 1 4 yt−2 + εt + 1 2 εt−1 where {εt} is white noise with E(εt) = 0 and var(εt) = σ2. (a) Is this model stationary? (b) Is this model invertible? (c) Derive the 1-step-ahead and 2-step-ahead forecasts. (d) Derive the 1-step-ahead and 2-step-ahead forecast errors. Now consider the following ARMA(1,1) model: yt = 3 4 yt−1 + εt − 3 4 εt−1 where {εt} is white noise with E(εt) = 0 and var(εt) = 1. (e) You generate T = 500 observations from this model and plot the sam- ple autocorrelation function (ACF) and partial autocorrelation function (PACF). > y <- arima.sim(n="500," list(ar="0.75," ma="-0.75))"> acf(y) > pacf(y) 4 [NBS8187] 0 5 10 15 20 25 0. 0 0. 4 0. 8 Lag A C F 0 5 10 15 20 25 − 0. 05 0. 05 0. 10 P ar ti al A C F Define the terms ACF and PACF and comment on the plots. Do they look as expected? (f) Next you fit an ARMA(1,1) model to the simulated data and obtain the following results: Call: arima(x = y, order = c(1, 0, 1), include.mean = FALSE) Coefficients: ar1 ma1 -0.3984 0.3560 s.e. 0.4079 0.4137 sigma^2 estimated as 0.9997 log likelihood = -709.4 aic = 1424.81 What do you observe and how do you explain it? 5 [Please turn over ] [NBS8187] Critical Values of the Chi-Square Distribution with ν Degrees of Freedom Significance Level ν 0.10 0.05 0.01 1 2.71 3.84 6.63 2 4.61 5.99 9.21 3 6.25 7.81 11.34 4 7.78 9.49 13.28 5 9.24 11.07 15.09 6 10.64 12.59 16.81 7 12.02 14.07 18.48 8 13.36 15.51 20.09 9 14.68 16.92 21.67 10 15.99 18.31 23.21 11 17.28 19.68 24.72 12 18.55 21.03 26.22 13 19.81 22.36 27.69 14 21.06 23.68 29.14 15 22.31 25.00 30.58 16 23.54 26.30 32.00 17 24.77 27.59 33.41 18 25.99 28.87 34.81 19 27.20 30.14 36.19 20 28