Fall 2002 / Economics, U.C. Davis
Problem Set 5 – Solutions
Part I – Analytical Questions
Problem 1: Consider the following DGP for the cointegrated random variables z and y
where e ~ 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.
(a) Directly inverting the lagged matrices on the right-hand side, we get
(b) From the autoregressive representation
(c) From the ECM, the long run solution is y = 2z.
Problem 2: Consider the following DGP
with |r| < 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 a, b, and ? Discuss how.
If θ = 1, then xt and yt are I(1) since . In addition, given |r| < 1 one needs to impose the condition . The same linear combination of xt and yt cannot be simulatenously I(0) and I(1).
(b) Under what coefficient restrictions are xt and yt cointegrated? What are the cointegrating vectors in such cases?
. Cointegrating vector (1 a).
(c) Choose a particular set of coefficients that ensures xt and yt are cointegrated and derive the following representations:
I. The moving-average.
II. The autoregressive.
III. The error-correction.
I. MA
II. AR
III. ECM
Define the error correction term , then
(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?
From the Granger representation theorem we know the answer is yes. Analyzing the VAR in the differences omits the error correction term in the specification. Therefore we have the classical problems of omitted variable bias.
(e) Suppose that economic theory suggests that xt and yt should be cointegrated with cointegrating vector [1 a + 0.5t]. Describe:
I. How would you test whether this is indeed a cointegrating vector?
Run the OLS regression
and test the residuals with an ADF test.
II. What is the likely outcome of the test in short samples? Why?
In short-samples, the cointegrating vector (1 a+0.5t) will differ from the cointegrating vector (1 a). However, as the sample size gets larger, note that the bias 0.5t disappears very quickly.
III. What is the likely outcome of the test asymptotically? Why?
Asymptotically the bias disappears sufficiently quickly.
Problem 3: 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 a and b, 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.
Using the hint: Py(1)’ = 0. It is easy to show that
and therefore
(d) Using the Beveridge-Nelson decomposition and the result in (c), determine the common trend in the VECM system.
All you need to remember is that from the B-N decomposition, the trends are the linear combinations captured in y(1)yt, which in this case turns out to be b2y1t + y2t. Notice that this combination is orthogonal to the cointegrating vector.
(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 a1 = 0?
Let be the cointegrating vector. From the equations for y1 and y2 we have
Combining terms
which is an AR(1) whose stationarity requires that |a1 + 1| < 1 or the equivalent condition -2 < a1 < 0. When a1 = 0, zt is no longer stationary, so there is no cointegration for any value of b2. y1 and y2 are in this case two independent random walks.
Problem 4: Consider the following VAR
(a) Show that this VAR is not-stationary.
Stationarity requires that the values of z satisfying
lie outside the unit circle. For z = 1, notice
(b) Find the cointegrating vector and derive the VECM representation.
Notice that
so that
(c) Transform the model so that it involves the error correction term (call it z) and a difference stationary variable (call it Dwt). 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.
Given the ECM in part (b), notice
Next
(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.
From part (c)
taking the inverse
and therefore
wt is I(1) and zt is I(0), which is a version of the Beveridge-Nelson decomposition proposed by Gonzalo and Granger (1995).
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