Booth School of Business, University of Chicago Business 41914, Spring Quarter 2017, Mr. Ruey S. Tsay. Solutions to Midterm

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1 Booth School of Business, University of Chicago Business 41914, Spring Quarter 017, Mr Ruey S Tsay Solutions to Midterm Problem A: (51 points; 3 points per question) Answer briefly the following questions 1 Consider the transfer function models: (a) y t = 3B 3 B 4 1 3B+17B 04B 3 x t B a t, (b) y t = 06B+B x 1 07B+08B t + a t, where x t is an input variable independent of a t Are the models stable? Why? Compute the steady-state gain of any stable model A: System (a) is not stable, because the polynomial in the denominator contains a unit root System (b) is stable, because the solutions of 1 07x + 08x = 0 are greater than 1 in absolute The steady-state gain is ( 06 + )/( ) 17 Compute the first five (5) impulse response weights of the transfer function y t = B+05B x 1 13B+04B t + 1 a 1 07B t A: The weights are 0,, 31, 33, and 959 You may use the R script CornerFunR to compute the weights 3 (Questions 3 to 6) Consider the -dimensional model where φ 1 = z t = φ 1 z t 1 + φ z t + a t,, φ = , Let y t = (z t, z t 1) Write down a VAR model for y t A: The model is y t = y t Verify that z t is stationary at 0, Σ a =, Σ = A: Eigenvalues of the AR(1) coefficient matrix are 0719±0637i, 05, and 038 Their absolute values are all less than 1 so that z t is indeed stationary 5 Suppose z 100 = (1, 06) and z 99 = (05, 09) Compute the 1-step and -step ahead predictions at the forecast origin t = 100 A: z 100 (1) = (146, 065) and z 100 () = (138, 188) 1

2 6 Compute the covariance matrix of the -step ahead forecast error A: Cov = Σ a + φ 1 Σ 1 φ 1 = (Questions 7 and 8) Suppose {z 1,, z 100 } is a realization from a bivariate VAR(1) model, z t = φ 1 z t 1 + a t The sample covariance matrices are given below: Γ 0 =, 10 0 Γ = 07 1 Using the Yule-Walker equation to obtain the estimate of φ 1 A: φ 1 = Γ Γ = Obtain an estimate of Σ a A: Using Σ a = Γ 0 φ 1 Γ 1, we have Σ a = (Questions 9 to 16) 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 φ 1 B φ B and θ(b) = I θ 1 B are matrix polynomials with order and 1 Assume that φ(b) and θ(b) are left coprime and φ, θ 1 is of rank k What is the invertibility condition for z t? A: Eigenvalues of θ 1 are less than 1 in absolute values 10 Assume that z t is stationary What is E(z t )? A: E(z t ) = (I φ 1 φ ) 1 φ 0 11 Assume that k = 3, what is the implied univariate ARMA model for z it? A: The order is ARMA(6,5), ie (kp, (k 1)p + q) 1 Assume stationarity What are the Yule-Walker equations for z t? A: The equations are Γ = φ 1 Γ 1 + φ Γ 0 Γ 3 = φ 1 Γ 3 + φ Γ 1 13 Assume that z t is stationary Let z t µ = ψ(b)a t be the MA representation of z t, where µ = E(z t ) Derive the impulse response function ψ i for i = 1,, 3 A: By equating the coefficient matrices of φ(b)ψ(b) = θ(b), we have ψ 1 = φ 1 θ 1, ψ = φ 1 ψ 1 + φ, and ψ 3 = φ 1 ψ + φ ψ 1

3 14 Assume k = What is the necessary and sufficient condition that z 1t is an input variable and z t is a output variable? That is, there is a unidirectional relationship between z it A: Let k ij (B) be the (i, j)th element of the matrix polynomial k(b) The necessary and sufficient condition is φ (B)θ 1 (B) φ 1 (B)θ (B) = 0, and φ 11 (B)θ 1 (B) φ 1 (B)θ 11 (B) 0 15 Let w l = z l + z l 1 for l = 1,, It is easy to see that w l follows a VARMA(p, q) model What are the maximum values of p and q? A: p = ( 1) + 1 = 3 and q = ( 1) + 1 = 3 16 Let w t = z 1t +z t and k = It is also easy to see that w t follows a univariate ARMA(p, q) model What are the maximum values of p and q? A: p = 4 and q = 3 The determinant of φ(b) is a 4th-order polynomial and it applies to both component 17 Consider the k-dimensional MA() model z t = a t θ 1 a t 1 θ a t Show that the component z it also follows an MA() model A: z it = a it k j=1 θ (1) ij a j,t 1 k j=1 θ () ij a j,t, where θ (v) ij is the (i, j)th element of θ v From the independent of a t series, it is easy to see that Cov(z it, z i,t l ) = 0 for l > Problem B (34 points) Consider the US quarterly gross domestic income (gdi) and consumption of fixed capital from 1947I to 016IV for 80 observations The data are from FRED and in billions of US dollars, and log transformation was taken R output of some analysis is attached, where yt denotes the log series Answer the following questions 1 (4 points) State the VAR orders selected by the four order-selection statistics (ie, AIC, BIC, HQ and M(p)) A: The selected orders are 13, 3, 3, and 10, respectively, for AIC,BIC, HQ and M(p), where 5% type-i error is used for M(p) (4 points) Johansen s test is performed to test for co-integration between the two series Draw your conclusion using the 5% significance level A: There is co-integration because the null hypothesis of rank = 1 cannot be rejected 3 (4 points) An ECM is fitted Write down the simplified ECM model A: The fitted model is 008 t = w 0005 t where w t = y 1t 0911y t and cov(a t ) is t t + a t,

4 4 ( points) Is the fitted ECM model adequate? Why? A: No, the Ljung-Box statistics of the residuals show that there are serial correlations in the residuals 5 (4 points) A VAR(3) model is also entertained Write down the simplified VAR(3) model A: The fitted model is y t = + y t 1 y t y t 3 + a t, Σ a = ( points) Compared the VAR(3) and ECM models Which model do you prefer? Why? A: VAR(3) model for lower AIC 7 (4 points) Consider the first-differenced data, dyt Extended cross-correlation matrices are given What are the two most parsimonious models specified by the Eccm table? A: VAR(4) or VAR(5) 8 (4 points) A VAR(5) model is used for dyt series Write down the simplified model A: The fitted model is t = t t t t where the residual covariance matrix is ( points) Is the VAR(5) model adequate? Why? 10 5 t 5 + a t A: Yes, the model is adequate as the Ljung-Box statistics of the residuals confirm that there are no significant serial correlations in the residuals 10 (4 points) The VAR(5) model for dyt and the ECM model for yt seem to be inconsistent? Why? Which model do you preferred? Why? A: The VAR(5) model for t indicates there is no co-integration The VAR(5) model is preferred for lower AIC value Problem C (15 points) Briefly answer the following questions 4

5 1 (4 points) Consider the transfer function model (1 B)z 1t = ɛ 1t θɛ 1,t 1, z t = ωz 1t + ɛ t, where ɛ 1t and ɛ t are independent Guassian white noise series with mean zero and variances σ 1 and σ, respectively What is the univariate ARIMA model for z t? A: Applying (1 B) to z t, one can see that it is an ARIMA(0,1,1) model, ie the exponential smoothing model Let z t = (z 1t, z t ) Express the model of z t in the form (1 B)z t = (I ΘB)a t What is Θ? Is the resulting bivariate VARMA model invertible? Why? A: It is easy to see that (1 B)z t = 1 0 ω 1 ɛ t where the MA(1) coefficient matrix is θ θ = ωθ 1 1 ω 1 θ 1 0 ωθ = ɛ t 1 a t θa t 1, θ 1 0 ω(θ 1 1) 1 which clearly has an eigenvale beeing 1 Thus, the model is non-invertible (6 points) Consider the model z t = z t + a t, where {a t } is a Gaussian white noise series with mean zero and variance σ a Let {z 1,, z T } be a realization of the process z t, where, for simplicity, assume thay z 1 = z 0 = 0 Hint: You may use the results discussed in the lecture notes Obtain the limiting distribution of T T t=1 zt A: By definition, it is easy to see that { a1 + a z t = a l 1 if t = l 1, an odd number a + a a l if t = l, an even number Without loss of generality, assume T is even so that n = T/ Then T T t=1 n zt = T (zl 1 + zl) = 1 n 4 (n zl 1 + n, n zl) Applying the result of part (a) of Theorem of Lecture 5, we have T T t=1 zt ( d σa w 1(r)dr w (r)dr ), where w 1 (r) and w (r) are two independent standard Brownian motions 5

6 Obtain the limiting distribution of T 1 T t=1 z t a t A: You may apply the same techniques T ( T 1 n ) z t a t = T 1 n z l 3 a l 1 + z l a l t=1 = 1 ( n ) n n 1 z l 3 a l 1 + z l a l 1 σ a d w 1 (1) + w(1) ) = σ a 4 (χ ) 3 (5 points) Consider the bivariate time series VARMA(1,1) model z t = where a t is a white noise series z t 1 + a t a t 1, ( points) Show that each component z it is unit-root nonstationary A: There is a unit root in the system because φ(1) = Write the model in an Error-correction form A: Subtracting z t 1 from the equation, we have z t = = = z t 1 + a t a t ( 06, 08)z 075 t 1 + a t a t 1 6