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Questions and Exercises Questions and Exercises 1 CORSO DI MACROECONOMETRIA Facoltà di Economia prof. Mario Forni Facoltà di Economia Questions and Exercises Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part I: Questions Facoltà di Economia Questions and Exercises CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part I: Questions 6. What does it mean that xt and yt are orthogonal? 1. What is a stochastic process? 7. What is the content of the orthogonal decomposition theorem? 2. What is a stationary process? 3. What is a purely deterministic process? 4. Is the white noise process a member of the MA class? Is it a member of the AR class? {xt }∞ t=−∞ 5. Consider the stationary process Can we think of the variable xt as an element of a linear space? How can we define this space? What are the inner product and the norm? 8. Assume that xt ∈ H and S ⊂ H. We know that xt = Pt + t and we want to prove that Pt , which is an element of S, is the projection of xt onto S. What should we show? 9. The projection operator have two important properties. Which properties? 10. What is the Wold decomposition? 11. What is a regular process? Facoltà di Economia CORSO DI MACROECONOMETRIA Facoltà di Economia CORSO DI MACROECONOMETRIA Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part I: Exercises Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part II: Questions 1. Show that t + 2t−1 is stationary if t ∼ w .n.. 1. What is a two-sided filter in L? 2.* Consider the process xt = 0.5xt−1 + t , where t ⊥ xt−k , k > 0, E t = 0. Show that t is a white noise process. 2. What is the invertibility condition for a(L)? 3. Consider the process of Exercise 2. What is the projection of xt onto span(xt−1 )? What is the projection of xt onto span(xt−1 , xt−2 , . . .)? 4. What does it mean that xt = 0.5xt−1 + t , t ∼ w.n., is stationary? 4. Consider the process of Exercise 2. What is the best linear prediction of xt+1 , given the present and the past of xt ? What is the best linear prediction of xt+1 ? 5. Consider the process xt = 1 + 0.4xt−1 + ηt , where ηt = 0.1xt−1 + t , t ⊥ xt−1 . What is the projection of xt onto span(1, xt−1 )? Facoltà di Economia Questions and Exercises CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part II: Exercises 3. What is a stochastic difference equation? 5. Consider the MA(1) process xt = (1 + aL)t , t ∼ w.n. What does it mean that this representation is invertible? What is the invertibility condition? 6. What is the covariance generating function? What is the sample covariogram? 7. Describe a procedure for ARMA model selection and diagnostic checking. Facoltà di Economia Questions and Exercises CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part III: Questions 1. Find the stationary solution of xt = 1 + 0.5xt−1 + t , t w.n. What is the general solution? 1. What is the difference between a TS and a DS model? 2.* Consider the model in 1. Show that t ⊥ xt−k , k > 0. What is the best linear prediction for xt+3 , given the present and the past of xt ? 2. How can we interpret the number a(1) if ∆xt = a(L)t , a(0) = 1 is the Wold representation of ∆xt ? 3.* Factorize 1 − 1.5L + 0.5L2 . Is it invertible? Why? 3. What is an ARIMA(p,d,q) process? 4. Find the covariance generating function of the process xt = (1 + 2L)t , t w.n. 4. How can we test whether a process is DS or TS? 5. Consider the process (1 − 1/3L)xt = µ + (1 − L)t , t w.n. What is this process? Is it stationary? Is it regular? Is it invertible? Find the Wold representation. Facoltà di Economia CORSO DI MACROECONOMETRIA 5. What is the Beveridge and Nelson decomposition? What is a trend-cycle decomposition? Facoltà di Economia CORSO DI MACROECONOMETRIA Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part III: Exercises Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part IV: Questions 1. Consider the random walk with drift process ∆xt = a + t . What is the optimal prediction for xt+1 given span(xt , xt−1 , xt−2 , . . .)? What is the prediction for xt+k ? 2. Consider the TS process xt = b + at + t . t w.n. Compute x̂t+k . What is the difference with respect to the prediction in 1? 3.* Consider the process d(L)∆xt = b + c(L)t , with c(1) = 0. Is xt a DS or a TS process? Why? 4.* Write the Beveridge and Nelson decomposition for the process ∆xt = 4 + (1 + 0.5L − 0.5L2 )t . 5. Write the Beveridge and Nelson decomposition for ∆xt = 1 + (1 − L)t . Is the trend deterministic or stochastic? Facoltà di Economia Questions and Exercises 1. What does it mean that xt and yt are jointly stationary? 2. What is the stability condition for the Vector ARMA B(L)zt = C (L)t ? What is the invertibility condition? 3. What is the cross-covariogram γkxy , k = −∞, ∞? What is the relation between γ xy and γ yx ? 4. What is the law of iterated projections? 5. What does it mean that yt does not Granger-cause xt Write the Wold representation of zt = (yt xt )0 when yt does not Granger-cause xt . CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part IV: Exercises Facoltà di Economia Questions and Exercises CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part IV: Exercises 1.* Define the process ηt = t for t odd and ηt = ωt for t even, where (t ηt )0 is a vector white noise, var(t ) = var(ωt ) = 1, cov(t , ηt ) = 0. Is ηt a stationary process? are t and ηt jointly stationary? 2. Consider the w.n. t and set ηt = t−1 . Is ηt w.n.? Are t and ηt jointly w.n.? 5.* Show that ηt does not Granger-cause t in exercise 2, while t Granger-causes ηt . 3. Find the auto-covariances and cross-covariances of yt 1 L t = xt 0.5L 1 etat where σ2 = ση2 = 1 and ση = 0. 4. Find the AR(∞) representation for the process in 3. Facoltà di Economia CORSO DI MACROECONOMETRIA Facoltà di Economia CORSO DI MACROECONOMETRIA Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part V: Questions Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part V: Exercises 1. What does it mean that yt and xt are cointegrated? 2. What is the relation between the common-trend representation and cointegration? 3. Can we say something about cointegration by looking at the Wold representation of the first differences? 4. What is the relation between cointegration and the Error Correction Mechanism? 1. Consider the process ∆yt 1 0.5L t = ∆xt −L 1 − 1.5L ηt Are yt and xt cointegrated? why? 2. Find the common-trend representation for xt and yt starting from ∆yt 1 −L t = ∆xt −L 1 ηt 5. How can we test for cointegration? 3.* Find the ECM representation for the variables in 2. Facoltà di Economia Questions and Exercises CORSO DI MACROECONOMETRIA Facoltà di Economia Part I Part II Part III Part IV Part V Answers to the exercises Part V: Exercises Questions and Exercises CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part I 4.* Find the Wold representation for the first differences of the variables yt Tt = + t xt Tt ηt 1. Either compute the autocovariances and show that they do not depend on t, or simply argue that xt is a member of the Hilbert space spanned by t−k , all k. 2. Write t−k = xt−k + 0.5xt−k−1 and compute cov(t , t−k ) for k > 0. 3. 0.5xt−1 , since t ⊥ xt−1 ; 0.5xt−1 , since t ⊥ xt−k , k > 0. where ∆Tt = t . 4. x̂t+1 = 0.5xt ; x̂t+2 = 0.25xt (apply the chain rule of forecasting). 5. 1 + 0.5xt−1 . Facoltà di Economia CORSO DI MACROECONOMETRIA Facoltà di Economia CORSO DI MACROECONOMETRIA Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part II Part I Part II Part III Part IV Part V Answers to the exercises Questions and Exercises Part II 1. The stationary solution is xt = 2 + 1/(1 − 0.5L)t = 2 + (1 + 0.5L + 0.52 L2 + · · · )t . The term 0.5t c must be added to get the general solution. 2. We have xt−k = 2 + 1/(1 − 0.5L)t−k . Since t ⊥ t−k for k > 0, then t ⊥ xt−k , k > 0. We have x̂t+3 = 1 + 0.5x̂t+2 = 1 + 0.5 + 0.52 x̂t+1 = 1 + 0.5 + 0.52 + 0.53 xt = 1.75 + 0.125xt . q 3. The roots are z ∗ = 23 ± 94 − 2 = 32 ± 12 which is either 1 or 2. 5. It is ARMA(1,1). It is stationary, regular, not invertible. The Wold representation is xt = 3µ 1−L + t . 2 1 − L/3 Hence 1 − 1.5L + 0.5L2 = (1 − L)(1 − 0.5L), which is not invertible because of the factor (1 − L). 4. (1 + 2z)(1 + 2z −1 )σ2 = σ2 (5 + 2z −1 + 2z). Facoltà di Economia Questions and Exercises CORSO DI MACROECONOMETRIA Facoltà di Economia Part I Part II Part III Part IV Part V Answers to the exercises Part III CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Questions and Exercises Part IV 1. x̂t+1 = a + xt ; x̂t+k = ak + xt ; 2. x̂t+k = b + at + ak. The main difference is that here the level of xt does not matter. 3. It is a DS process, since we can write xt = a + c̃(L) b t+ t , d(1) d(L) 1. ηt is stationary since it is w.n. t and ηt are not jointly stationary, since cov(t , ηt ) = 1 for t odd and cov(t , ηt ) = 1 for t odd and cov(t , ηt ) = 0 for t even. 2. ηt is w.n.; the vector (ηt t )0 is not w.n., since cov(t , ηt−1 ) = σ2 6= 0. 3. 1 0.5z where c̃(L) = c(L)/(1 − L) and a is an arbitrary constant. 4. ∆xt = (4 + t ) + (1 − L)(0.5L)t . 5. ∆xt = 1 + (1 − L)t . The trend is the deterministic trend a + t. Facoltà di Economia CORSO DI MACROECONOMETRIA 4. z 1 1 z −1 0.5z −1 1 1 1−0.5L2 −L 2(1−0.5L2 ) −L 1−0.5L2 1 1−0.5L2 Facoltà di Economia 2 0.5z −1 + z −1 0.5z + z 1.25 = ! yt = t xt ηt CORSO DI MACROECONOMETRIA Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part IV Questions and Exercises Part I Part II Part III Part IV Part V Answers to the exercises Part V 1. Yes, they are. The determinant of the Wold impulse response function matrix is 1 − 1.5L + 0.5L2 which vanishes for L = 1. 5. The projection of t onto the space spanned by the past of and η is 0. The projection of ηt onto the same space is t−1 . 2. The Beveridge and Nelson decomposition is ∆yt 1 0.5 t 0 −0.5(1 − L) t = + ∆xt −1 −0.5 ηt 1 − L 1.5(1 − L) ηt Hence we have ∆yt ∆Tt 0 −0.5 t = + (1 − L) ∆xt −∆Tt 1 1.5 ηt where ∆Tt = t + 0.5ηt . Facoltà di Economia Questions and Exercises Facoltà di Economia CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part V Questions and Exercises CORSO DI MACROECONOMETRIA Part I Part II Part III Part IV Part V Answers to the exercises Part V 3. Premultiplying by the adjugate of C (L) we get 1 L yt = (1 + L) t L 1 xt ηt Let A(L) be the matrix on the RHS. By subtracting to both sides A(1)L(yt xt )0 we get ∆yt 1 =− s + (1 + L) t ∆xt 1 t−1 ηt where st = yt + xt . Facoltà di Economia CORSO DI MACROECONOMETRIA 4. We have ∆yt 2−L 0 t = t + (1 − L) t = ∆xt t ηt 1 1−L ηt The determinant of the matrix on the RHS, let it be A(L), does not have roots within the unit circle. Hence we have a fundamental representation. Since the Wold matrix in 0 is the identity matrix, the Wold matrix is C (L) = A(L)A(0)−1 and the residuals are A(0)(t ηt )0 , i.e. ∆yt 1 − 0.5L 0 2t = ∆xt 0.5L 1−L t + ηt Facoltà di Economia CORSO DI MACROECONOMETRIA