Booth School of Business, University of Chicago Business 41914, Spring Quarter 2013, Mr. Ruey S. Tsay. Midterm

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1 Booth School of Business, University of Chicago Business 41914, Spring Quarter 2013, Mr. Ruey S. Tsay Midterm Chicago Booth Honor Code: I pledge my honor that I have not violated the Honor Code during this examination. Signature: Name: UC. ID: Notes: Open notes and books. The exam has eight (8) pages and six (6) pages of output. Write your answer in the blank space provided for each question. Manage your time carefully and answer as many questions as you can. Unless stated otherwise, {a t } is a sequence of iid Gaussian random vectors with mean zero and positive-definite covariance matrix Σ a. This assumption applies to the univariate case too. For simplicity, ALL tests use the 5% significance level. In the exam, Γ i = Cov(z t, z t i ) denotes the lag-i autocovariance matrix of a k-dimensional stationary time series z t. Problem A: (45 points; 3 points per question) Answer briefly the following questions. 1. (For questions 1 to 8). Consider the following k-dimensional linear time series model φ(b)z t = φ 0 + θ(b)a t, where z t = (z 1t,., z kt ), φ(b) = I p i=1 φ i B i and θ(b) = I q i=1 θ i B i are two left-coprime matrix polynomials such that rank[φ p, θ q ] = k and p and q are non-negative integers. Give a sufficient condition for the invertibility of the process z t. 2. Assume that p = 2, z it are I(1) processes for 1 i k, and z t is co-integrated. Write down an Error-Correction form for z t. 1

2 3. Assume that φ 0 = 0, p = 2 and q = 1. Define y t = (z t, z t 1). Write down a VARMA(1,1) model for y t, including the covariance matrix of the innovations. 4. Assume that k = 2, p = 2 and q = 1. What is the implied univariate ARMA model for z 2t? 5. Assume that p = 1 and q = 2. Derive the moment equations for a stationary z t. 6. Assume that p = q = 1 and φ 0 = 0. Let z t = ψ(b)a t be the MA representation of z t. Derive the impulse response function ψ i for i = 1, 2, Assume, again, that p = 1, q = 2 and φ 0 = 0. Describe a method that can obtain consistent estimate of φ 1 via the ordinary least squares method. 8. Assume that k = 2. Describe a necessary and sufficient condition that z 1t serves as an input variable and z 2t as a dependent variable. That is, there exists a transfer function model with z 1t as the input variable. 9. Consider the MA(1) model z t = a t θa t 1. This model can be estimated by either the conditional or the exact likelihood methods. Describe briefly the difference between the two estimation methods. Briefly discuss the advantages and disadvantages of each method. 2

3 10. Give two situations under which VARMA modeling is preferred over the transfer function modeling even if we suspect the existence of the later model. 11. (For questions 11 and 12) Consider the 2-dimensional VARMA(1,1) model [ ] [ ] z t = z t 1 z t 2 [ ] [ ] a t + a t 1 + a t 2, Σ a = Is the model identifiable? Why? [ ]. 12. Let y t = z 2t z 1t. What is the univariate model for y t? 13. (For questions 13 to 15). Suppose that 2-dimensional series z t follows the model z t = [ Show that this model has a unit root. ] z t 1 + a t [ ] a t Show that both z 1t and z 2t are unit-root nonstationary. 15. Obtain a co-integrating vector for the system. 3

4 Problem B. (30 points) Consider a 2-dimensional time series z t = (z 1t, z 2t ) with 296 observations. Some analysis of the series is given in the attached R output. Answer the following questions: 1. (4 points) If VAR models are entertained. What VAR models are selected by the order specification methods discussed in the class? 2. (5 points) A VAR(6) model is estimated and refined. The refinement is to remove simultaneously all parameters with t-ratio less than 1.96 in modulus. Write down the refined model, including the residual covariance matrix. 3. (2 points) Provide a justification that one indeed can simultaneously remove all estimates with t-ratio less than 1.96 in modulus. 4. (3 points) Model checking shows that the refined VAR(6) model is adequate. Based on the model, is there any Granger causality between z 1t and z 2t? Why? 5. (7 points) Write down the transfer function model implied by the refined VAR(6) model. 4

5 6. (2 points) Based on the refined VAR(6) model, obtain an 95% interval forecast of z 2,t+6 at the forecast origin t = (2 points) The extended cross-correlation method is applied to z t. What is the model suggested by the method? 8. (3 points) A VARMA(3,1) model is fitted to the series. Write down the fitted model, including the residual covariance matrix. 9. (2 points) Does the fitted VARMA(3,1) model imply any Granger causality in z t? Why? 5

6 Problem C. (25 points total, 5 points each question) Simple derivation. 1. Consider the white noise series z t = a t. Let Γ 1 be the lag-1 sample cross covariance matrix. Derive the limiting covariance matrix of T vec( Γ 1 ). 2. Consider a k-dimensional VAR(p) model with p 1. Let y t = 1 z t be the sum of z it, where 1 is the k-dimensional vector of 1 s. Show that y t follows an ARMA model and obtain the maximum order of y t. 6

7 3. Consider a stationary and invertible VARMA(2,1) model φ(b)z t = θ(b)a t, Var(a t ) = Σ a > 0, where φ(b) = I k φ 1 B φ 2 B 2 and θ(b) = I k θ 1 B. Define the following vectors y t = z t z t 1 a t where 0 is a k-dimensional vector of zeros., b t = a t 0 a t, (a) Find the coefficient matrix Φ such that y t = Φy t 1 + b t. (b) Using the VAR(1) model of y t, derive an equation that can be used to obtain the auto-covariance matrices Γ l of z t, the given VARMA(2,1) model. 4. Consider the VAR(1) model z t = φ 1 z t 1 + a t. Let φ 1 be the ordinary least squares estimate of φ 1. What is the limiting distribution of T vec( φ 1 ) as the sample size T increases. 7

8 5. Consider the random walk series z t = z t 1 + a t, where a t is a sequence of iid random variables with mean zero, variance σ 2 and finite fourth moment. From the unit-root lecture, we define the partial sum X T (r) = 1 T σ S [T r], 0 r 1, where S t = t i=1 a i. By the Functional Center Limit Theorem and the standard Brownian motion by W (r), we have X T (r) W (r). Therefore, T 1/2 S [T r] σw (r). Consider the statistic T 3/2 T t=1 z t 1, which is approximately T 1/2 z. Show that T 3/2 T t=1 1 z t 1 σ W (r)dr. 0 8

9 R output ***** Problem B ****** > zt=read.table("xxxx.txt") > dim(zt) [1] > source("mts.r") > VARorder(zt) selected order: aic = 6 selected order: bic = 4 selected order: hq = 4 Summary table: p AIC BIC HQ M(p) p-value [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] > m1=var(zt,6) Constant term: Estimates: Std.Error: AR coefficient matrix AR( 1 )-matrix [1,] [2,] standard error [1,] [2,] AR( 6 )-matrix [1,] [2,] standard error 9

10 [1,] [2,] Residuals cov-mtx: [1,] [2,] det(sse) = AIC = BIC = HQ = > m2=refvar(m1,thres=1.96) Constant term: Estimates: Std.Error: AR coefficient matrix AR( 1 )-matrix [1,] [2,] standard error [1,] [2,] AR( 2 )-matrix [1,] [2,] standard error [1,] [2,] AR( 3 )-matrix [1,] [2,] standard error [1,] [2,] AR( 4 )-matrix [1,] [2,]

11 standard error [1,] [2,] AR( 5 )-matrix [1,] [2,] standard error [1,] [2,] AR( 6 )-matrix [1,] [2,] standard error [1,] [2,] Residuals cov-mtx: [1,] [2,] det(sse) = AIC = BIC = HQ = > MTSdiag(m2,adj=10) [1] "Covariance matrix:" V1 V2 V V CCM at lag: 0 [1,] [2,] Simplified matrix: CCM at lag: 1 CCM at lag: 2 11

12 CCM at lag: 3 CCM at lag: 4 CCM at lag: 5 CCM at lag: 6 CCM at lag: 7 CCM at lag: 8 CCM at lag: 9 CCM at lag: 10 > VARpred(m2,6) orig 296 Forecasts at origin: 296 V1 V2 [1,] [5,] [6,] Standard Errors of predictions: [1,] [5,] [6,] > Eccm(zt,maxp=8,maxq=5) p-values table of Extended Cross-correlation Matrices: Column: MA order Row : AR order

13 > m3=varma(zt,p=3,q=1) Number of parameters: 18 Coefficient(s): Estimate Std. Error t value Pr(> t ) V ** V V < 2e-16 *** V *** V e-09 *** V *** V * V ** V e-05 *** V < 2e-16 *** V e-06 *** V < 2e-16 *** V e-05 *** V e-07 *** *** e-05 *** e-07 *** --- Estimates in matrix form: Constant term: Estimates: AR coefficient matrix AR( 1 )-matrix [1,] [2,] AR( 2 )-matrix [1,] [2,] AR( 3 )-matrix 13

14 [1,] [2,] MA coefficient matrix MA( 1 )-matrix [1,] [2,] Residuals cov-matrix: [1,] [2,] aic= bic= > MTSdiag(m3) [1] "Covariance matrix:" V1 V2 V V Simplified matrix: CCM at lag: 1 CCM at lag: 2 CCM at lag: 3 CCM at lag: 4 CCM at lag: 5 CCM at lag: 6 CCM at lag: 7 CCM at lag: 8 14