Winter 2004 / Economics, U.C. Davis
Problem Set 4 – Due march 11
Instructions
This problem set is divided into two parts: (1) Analytical Questions, and (2) Applied Questions. Part 1, Analytical Questions, should be attempted by each student individually. Part 2, Applied Questions, can be done in collaboration with another partner or alone. Please try to answer the questions rigorously by stating any implied assumptions and ensuring all the steps to your conclusion have been properly verified.
Part I – Analytical Questions
Problem 1: Consider a stationary autoregressive process A(L)Xt = t and its corresponding moving average representation, Xt = C(L)t , where .
(a) Find the moving average coefficients for an VAR(1) process.
(b) Show that the moving average coefficients for a VAR(2) can be found recursively by
Problem 2: Consider the following bivariate VAR,
with .
(a) Find a matrix H, which is lower triangular and ensures that if , then where D is a diagonal matrix.
(b) Given this matrix H calculate the structural representation of this VAR.
(c) Calculate the VMA representation for the reduced form of this VAR (notice that it is very simple in this case – don’t apply the usual formulas mechanically!)
(d) Calculate the VMA representation of the structural form of the VAR.
(e) Under what conditions will the reduced form and the structural form produce identical impulse response functions?
(f) Suppose you obtained the structural form as in part (a) but for a system that had the variable m ordered first. Under what conditions would these two structural identification schemes deliver the same impulse responses?
Problem 3: Consider the following bivariate VAR
with for t = and 0 other wise, for t = and 0 other wise, and for all t, and . Answer the following questions:
(a) Is this system covariance-stationary?
(b) Calculate for s = 0, 1, and 2. What is the limit as ?
(c) Calculate the fraction of the MSE of the two period-ahead forecast error for variable 1, , that is due to
Problem 4: Consider the process
(a) Derive where D denotes the density function and Hint: the system can be rewritten in matrix form as
(b) Assume that xt is stationary. Derive and show that is positive definite. What are the implications of this result?
Problem 5: Consider the Gaussian linear regression model,
with ut ~ i.i.d. N(0, 2) and ut independent of x for all t and . Define The log of the likelihood of (y1, …, yT) conditional on (x1,…,xT) is given by
(a) Show that the estimate is given by ’ where and and denote the maximum likelihood estimates.
(b) Show that the estimate is given by .
(c) Show that the where for
(d) Consider a set of m linear restrictions on of the form R = r for R a known matrix and r a known vector. Show that for , the Wald test statistic given by is identical to the Wald test form of the OLS test given by with the OLS estimate of the variance sT2 replaced with the MLE .
(e) Show that when the lower left and upper right blocks of are set to their plim of zero, then the quasi-maximum likelihood Wald test of R = r is identical to the heteroskedasticity-consistent form of the OLS test given by
Problem 6: Consider the following DGP for the cointegrated random variables z and y
where ~ N(0, I) with z0 = y0 = 0.
(a) Obtain the autoregressive representation of this DGP.
(b) Obtain the error-correction representation of this DGP.
(c) Deduce the long-run relation between z and y.
Problem 7: Consider the following DGP
with || < 1, and
where D denotes a generic distribution.
(a) Derive the degree of integratedness of the two series, xt and yt. Do your results depend on any restrictions on the values of , , and ? Discuss how.
(b) Under what coefficient restrictions are xt and yt cointegrated? What are the cointegrating vectors in such cases?
(c) Choose a particular set of coefficients that ensures xt and yt are cointegrated and derive the following representations:
- The moving-average.
- The autoregressive.
- The error-correction.
(d) Can all the cointegrated systems be represented as an error-correction model? What are the problem/s of analyzing a VAR in the differences when the system is cointegrated?
(e) Suppose that economic theory suggests that xt and yt should be cointegrated with cointegrating vector [1 + 0.5t]. Describe:
- How would you test whether this is indeed a cointegrating vector?
- What is the likely outcome of the test in short samples? Why?
- What is the likely outcome of the test asymptotically? Why?
Problem 8: Consider the bivariate VECM
where and Equation by equation, the system is given by
Answer the following questions:
(a) From the VECM representation above, derive the VECM representation
and the VAR(1) representation
(b) Based on the given values of the elements in and , determine , such that
(c) Using the Granger representation theorem determine that , where is the moving average polynomial corresponding to the VECM system above and I2 is the identity matrix of order 2. Hint: you may show this result by showing that is orthogonal to the cointegrating space.
(d) Using the Beveridge-Nelson decomposition and the result in (c), determine the common trend in the VECM system.
(e) Show that follows an AR(1) process and show that this AR(1) is stable provided that . What can you say about the system when 1 = 0?
Problem 9: Consider the following VAR
(a) Show that this VAR is not stationary.
(b) Find the cointegrating vector and derive the VECM representation.
(c) Transform the model so that it involves the error correction term (call it z) and a difference stationary variable (call it wt). w will be a linear combination of x and y but should not contain z. Hint: the weights in this linear combination will be related the coefficients of the error correction terms.
(d) Verify that y and x can be expressed as a linear combination of w and z. Give an interpretation as a decomposition of the vector (y x)’ into permanent and transitory components.
Part II – Empirical Question
There are two auxiliary files that you need for this exercise: “mfirfs.g” and “mfirfs.src.” These are gauss programs that allow you to estimate impulse responses from a traditional VAR, using lineal projection, and cubic projection methods as described in my paper “Model-Free Impulse Responses.” Your task for this assignment is to analyze a system of time series of your choosing and your interest. I recommend that you use no more than 6 variables and no less than 3. The simpler, probably the better.
You will need to report the following experiments:
(1) Display the impulse responses due to a shock in a variable of interest calculated from the GAUSS code. Display the usual impulse responses calculated with EViews to make sure the code is doing things correctly. Make sure that you comment on any differences of correspondences between the displays. Also, be sure of the following considerations: (a) how are you choosing the lag-length? Make sure the lag length is the same for both sets of impulse responses; (b) how are you choosing the size of the initial shock? If you choose the Cholesky decomposition, EViews chooses the standard error for that variable. I recommend calculating those responses first and then using the same size shock in the GAUSS code to get them to match; and (c) note that I am only asking for responses to a shock in one of the variables, not all of them.
(2) Lag-length misspecification: Check the robustness of the local projection method by using only two lags instead of the optimal lag-length. Display the responses calculated with GAUSS and EViews. Comment on the differences and similarities you find relative to the previous answer.
(3) The paper discusses that Variance decompositions can be easily calculated from the local projection method. Using linear projections, write the GAUSS code that will accomplish this and compare your answers to those obtained from the same procedure in EViews. Do your answers match?
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