MAT 3379 (Winter 2016) FINAL EXAM (PRACTICE)

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1 MAT 3379 (Winter 2016) FINAL EXAM (PRACTICE) 15 April 2016 (180 minutes) Professor: R. Kulik Student Number: Name: This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only non programmable calculators are permitted. There are FIFTEEN questions. Cellular phones, unauthorized electronic devices or course notes (unless an open-book exam) are not allowed during this exam. Phones and devices must be turned off and put away in your bag. Do not keep them in your possession, such as in your pockets. If caught with such a device or document, the following may occur: you will be asked to leave immediately the exam, academic fraud allegations will be filed which may result in you obtaining a 0 (zero) for the exam.by signing below, you acknowledge that you have ensured that you are complying with the above statement. Your signature: Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15 Total GOOD LUCK!!! Maximal no. of points Your Score 1

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3 3 Q1. Consider the sequence X t = Z t + (Z 2 t 1 1), t = 1, 2,..., where Z t, t = 1, 2,..., is a sequence of independent standard normal random variables. Show that E[X t ] = 0. Show that E[X t X t+h ] = 0 for h 0. Q2. Consider ARMA(2, 1) model given by Answer the following questions: (a) Is this process stationary? (b) Is this process causal? X t 0.75X t X t 2 = Z t Z t 1. Q3. Consider ARMA(1, 2) model X t ϕx t 1 = Z t + θz t 2, where ϕ < 1, θ 1, θ 2 R, and Z t are i.i.d. random variables with mean 0 and variance σ 2. For this model: (a) Derive the linear representation for X t, i.e. find the coefficients ψ j in X t = ψ j Z t j. j=0 (b) Using the linear representation, find γ X (1). Q4. Consider an AR(2) model X t = ϕ 1 X t 1 + ϕ 2 X t 2 + Z t, where Z t is an i.i.d sequence with mean 0 and variance σ 2 Z. For this model: (a) Find P n X n+1. Note: You need to check that your chosen predictor fulfills the Yule-Walker equations. (b) Apply the Yule-Walker procedure to show that P n X n+3 = ϕ 3 X n (three step prediction). Note: You need to check that your chosen predictor fulfills the Yule-Walker equations. (c) Compute the corresponding MSPE n (3). (d) Assume that a data set X is well-described by an AR(1) model with mean zero. Below you can find the following output for its : Autocorrelations of series MyTimeSeries, by lag Also, the estimated variance of the time series is It is also known that the last two observation are: X n = and X n 2 = Use this information to calculate P n X n+3 as well as MSPE n (3). Q5. Consider MA(1) model given by X t = Z t +θz t 1, where Z t are i.i.d. random variables with mean 0 and variance σ 2 Z. Find the best linear predictor of X 3 based on X 1, X 2. Q6. Consider a stationary AR(2) model X t = ϕ 1 X t 1 + ϕ 2 X t 2 + Z t, where Z t are i.i.d. normal random variables with mean zero and variance σ 2 Z. Assume that ϕ 1 and σ 2 Z are known. Derive the Maximum Likelihood Estimator for ϕ 2. Q7. (a) Time series X can be modelled by AR(1) sequence. Below you can find the output for the R command acf(x):

4 Also, the sample variance for the time series is 2.3 and the sample mean is 1.1. If the last observation is X 1000 = 3.33, predict the next observation. Calculate the Mean Squared Prediction Error. (b) Time series X can be modelled by AR(2) sequence with mean zero. The sample variance is 1. Below you can find the output for the R command acf(x): If the last two observations are X 999 = 1.88, X 1000 = 2.14, predict the next observation. Is the estimated model causal? Q8. The following graph shows and P of a time series: Partial (a) Argue that AR(2) is a good choice. (b) You have the following information given: n = 100, ˆγ X (0) = 0.05, X = 0.1, x99 = , x 100 = , Autocorrelations of series Data, by lag Find the Yule-Walker estimates of ϕ 1, ϕ 2 and σ 2 Z. Predict X 101. What is the estimated value of covariance at lag 7, i.e. ˆγ X (7)? Q9. We consider Yule-Walker estimation for AR(p) models. Recall that the Yule-Walker estimators ˆϕ p of ϕ p = (ϕ 1,..., ϕ p ) and ˆσ 2 of σ 2 are derived from ˆϕ p = ˆΓ 1 p ˆγ X,p, σ 2 = ˆγ X (0) ˆϕ T p ˆγ X,p,

5 5 where ˆΓ p and ˆγ X,p are obtained by replacing γ X with ˆγ X. Recall also that ˆϕ p is asymptotically normal with variance 1 n σ2 Γ 1 p. Consider AR(2) model X t = ϕ 1 X t 1 + ϕ 2 X t 2 + Z t, where Z t are i.i.d random variables with mean 0 and variance σ 2. (a) Derive confidence intervals for ˆϕ 1 and ˆϕ 2. (b) Assume that a data set X is well-described by an AR(2) model. Below you can find the following output for its : Autocorrelations of series X, by lag Also, the estimated variance of the time series is Use your results from part (a) to calculate the 95% confidence intervals for ˆϕ 1 and ˆϕ 2. Note: z = Q10. For a given time series X t we fitted AR(1) model. We estimated ϕ using the Yule-Walker estimator and computed residuals as R t = X t+1 ˆϕX t. The following graph shows and P of residuals. Is the fit appropriate? Series fit$resid[2:100] Series fit$resid[2:100] Partial

6 6 Q11. Consider the following ARCH(1) process X t = σ t Z t, σ 2 t = X 2 t 1, where Z t are i.i.d. random variables with mean 0 and variance σ 2 = 1. Its stationary representation is Xt 2 = 0.1 (0, 2) j Zt 2 Zt 1 2 Zt j. 2 j=0 (a) Calculate Var(X t ). (a) Assume furthermore that Z t are normal. Calculate E[X t 1 X 2 t ]. Q12. Consider the following ARCH(1) process X t = σ t Z t, σ 2 t = X 2 t 1, where Z t are i.i.d. standard normal random variables. (a) Calculate E[X 3 t ]. (b) Calculate Cov[X t 1, X 2 t+1]. Q13. To a data set we fitted an GARCH(1,1) model. The estimated parameters are as follows: a0 a1 b (a) (1 point) The following graph displays and P for the data set. Why an ARMA model should not be fitted here? Partial (b) (4 points) Is the model stationary? Why? (c) (5 points) Predict the next value of the squared volatility. Note that the last observation from the data sequence is , whereas the last fitted value of volatility is Q14. Consider the following graphs of for a time series X and for X 2. Explain why a GARCH model is a better choice than an ARMA model.

7 7 ^ Q15. Assume that Z = (Z 1, Z 2 ) is a random vector with the mean vector 0 and the covariance matrix [ ] σ11 σ Σ = 12. σ 21 σ 22 Let A be a deterministic matrix defined by A = Find the covariance matrix of the random vector AZ. [ ] a11 a 12. a 21 a 22