Question

2. The dynamic behavior of two macroeconomic aggregates X and Y follows a VAR(1) model ?Xt? = ? 1.1 -0.2 ? ?Xt-1? + ??1t? ? Yt ? ? -0.2 1.4 ? ? Yt-1 ? ??2t? (a) Write the model in the error-correction form ?xt = ?xt-1 + ?t for x = (X, Y)', and particularly determine the matrix ?. This impact matrix should have a rank of 1. Can you confirm this? Answer: First the re-write: ??Xt? = ? 0.1 -0.2 ? ?Xt-1? + ??1t? ??Yt? ? -0.2 0.4 ? ? Yt-1 ? ??2t? The second row (column) of ? is a multiple of the first row (column), so the matrix indeed has a rank of 1. (b) Try and find a factorization of ? = ??'. Provide the cointegrating vector ?. [Hint: we assume that explosive roots and unit roots other than one are not present in this system.] Answer: A possible solution is ? 0.1 -0.2 ? = ? 0.1 ? [ 1 -2 ], ? -0.2 0.4 ? ? -0.2 ? with the implied cointegrating vector (1, -2)', which is the solution preferred by some software packages, restricting the coefficient of the first variable at 1. With any non-zero real number c, multiplying the vector ? by c and dividing ? by c yields other equally valid answers. (c) We find another macroeconomic variable Z that follows the AR(1) model Zt = 0.5Zt-1 + ?t. Provide the characteristic polynomial and its root (zero) to show that this variable is stable. Answer: This characteristic polynomial is 1 - 0.5z with the root ? = 2, which is greater one in modulus, so the variable Z is stable (asymptotically stationary). (d) Now suppose we are joining all three variables into a big system ?Xt? ? 1.1 -0.2 0 ? ?Xt-1? ??1t? |Yt| = | -0.2 1.4 0 | | Yt-1 | + |?2t| ?Zt? ? 0 0 0.5 ? ? Zt-1 ? ??3t? Try to re-write it in its error-correction form. What is the rank of the impact matrix ? now? Provide the cointegrating vectors for this system. Answer: First, the re-write ??Xt? ? 0.1 -0.2 0 ? ?Xt-1? ??1t? |?Yt| = | -0.2 0.4 0 | | Yt-1 | + |?2t|, ??Zt? ? 0 0 -0.5 ? ? Zt-1 ? ??3t? which also shows the matrix ?. In fact, the required (non-unique) decomposition is fairly trivial: ? 0.1 -0.2 0 ? ? 0.1 0 ? ? 1 -2 0 ? | -0.2 0.4 0 | = | -0.2 0 | ? 0 0 1 ?, ? 0 0 -0.5 ? ? 0 -0.5 ? with a unit vector as the second cointegration vector. This scheme shows that any stationary component can easily be integrated into the general error-correction model structure. For this reason, it is fairly safe to proceed with the system cointegration procedure, even if some components are borderline cases between I(0) and I(1). I(2) variables, however, are not admitted. Issue: Several participants failed to provide the cointegrating vectors (1, -2, 0)' and (0, 0, 1)' in this point.

          2. The dynamic behavior of two macroeconomic aggregates X and Y follows a VAR(1) model

?Xt? = ? 1.1 -0.2 ? ?Xt-1? + ??1t?
? Yt ?   ? -0.2  1.4 ? ? Yt-1 ?   ??2t?

(a) Write the model in the error-correction form ?xt = ?xt-1 + ?t for x = (X, Y)', and particularly determine the matrix ?. This impact matrix should have a rank of 1. Can you confirm this? Answer: First the re-write:

??Xt? = ? 0.1 -0.2 ? ?Xt-1? + ??1t?
??Yt?   ? -0.2  0.4 ? ? Yt-1 ?   ??2t?

The second row (column) of ? is a multiple of the first row (column), so the matrix indeed has a rank of 1.

(b) Try and find a factorization of ? = ??'. Provide the cointegrating vector ?. [Hint: we assume that explosive roots and unit roots other than one are not present in this system.] Answer: A possible solution is

? 0.1 -0.2 ? = ? 0.1 ? [ 1 -2 ],
? -0.2  0.4 ?   ? -0.2 ?

with the implied cointegrating vector (1, -2)', which is the solution preferred by some software packages, restricting the coefficient of the first variable at 1. With any non-zero real number c, multiplying the vector ? by c and dividing ? by c yields other equally valid answers.

(c) We find another macroeconomic variable Z that follows the AR(1) model Zt = 0.5Zt-1 + ?t. Provide the characteristic polynomial and its root (zero) to show that this variable is stable. Answer: This characteristic polynomial is 1 - 0.5z with the root ? = 2, which is greater one in modulus, so the variable Z is stable (asymptotically stationary).

(d) Now suppose we are joining all three variables into a big system

?Xt?   ? 1.1 -0.2  0  ? ?Xt-1?   ??1t?
|Yt| = | -0.2  1.4  0  | | Yt-1 | + |?2t|
?Zt?   ?  0    0   0.5 ? ? Zt-1 ?   ??3t?

Try to re-write it in its error-correction form. What is the rank of the impact matrix ? now? Provide the cointegrating vectors for this system. Answer: First, the re-write

??Xt?   ? 0.1 -0.2   0  ? ?Xt-1?   ??1t?
|?Yt| = | -0.2  0.4   0  | | Yt-1 | + |?2t|,
??Zt?   ?  0    0   -0.5 ? ? Zt-1 ?   ??3t?

which also shows the matrix ?. In fact, the required (non-unique) decomposition is fairly trivial:

? 0.1 -0.2   0  ?   ?  0.1    0  ? ? 1 -2  0 ?
| -0.2  0.4   0  | = | -0.2    0  | ? 0  0  1 ?,
?  0    0   -0.5 ?   ?   0   -0.5 ?

with a unit vector as the second cointegration vector. This scheme shows that any stationary component can easily be integrated into the general error-correction model structure. For this reason, it is fairly safe to proceed with the system cointegration procedure, even if some components are borderline cases between I(0) and I(1). I(2) variables, however, are not admitted. Issue: Several participants failed to provide the cointegrating vectors (1, -2, 0)' and (0, 0, 1)' in this point.

2. The dynamic behavior of two macroeconomic aggregates X and Y follows a VAR(1) model

?Xt? = ? 1.1 -0.2 ? ?Xt-1? + ??1t?
? Yt ? ? -0.2 1.4 ? ? Yt-1 ? ??2t?

(a) Write the model in the error-correction form ?xt = ?xt-1 + ?t for x = (X, Y)', and particularly determine the matrix ?. This impact matrix should have a rank of 1. Can you confirm this? Answer: First the re-write:

??Xt? = ? 0.1 -0.2 ? ?Xt-1? + ??1t?
??Yt? ? -0.2 0.4 ? ? Yt-1 ? ??2t?

The second row (column) of ? is a multiple of the first row (column), so the matrix indeed has a rank of 1.

(b) Try and find a factorization of ? = ??'. Provide the cointegrating vector ?. [Hint: we assume that explosive roots and unit roots other than one are not present in this system.] Answer: A possible solution is

? 0.1 -0.2 ? = ? 0.1 ? [ 1 -2 ],
? -0.2 0.4 ? ? -0.2 ?

with the implied cointegrating vector (1, -2)', which is the solution preferred by some software packages, restricting the coefficient of the first variable at 1. With any non-zero real number c, multiplying the vector ? by c and dividing ? by c yields other equally valid answers.

(c) We find another macroeconomic variable Z that follows the AR(1) model Zt = 0.5Zt-1 + ?t. Provide the characteristic polynomial and its root (zero) to show that this variable is stable. Answer: This characteristic polynomial is 1 - 0.5z with the root ? = 2, which is greater one in modulus, so the variable Z is stable (asymptotically stationary).

(d) Now suppose we are joining all three variables into a big system

?Xt? ? 1.1 -0.2 0 ? ?Xt-1? ??1t?
|Yt| = | -0.2 1.4 0 | | Yt-1 | + |?2t|
?Zt? ? 0 0 0.5 ? ? Zt-1 ? ??3t?

Try to re-write it in its error-correction form. What is the rank of the impact matrix ? now? Provide the cointegrating vectors for this system. Answer: First, the re-write

??Xt? ? 0.1 -0.2 0 ? ?Xt-1? ??1t?
|?Yt| = | -0.2 0.4 0 | | Yt-1 | + |?2t|,
??Zt? ? 0 0 -0.5 ? ? Zt-1 ? ??3t?

which also shows the matrix ?. In fact, the required (non-unique) decomposition is fairly trivial:

? 0.1 -0.2 0 ? ? 0.1 0 ? ? 1 -2 0 ?
| -0.2 0.4 0 | = | -0.2 0 | ? 0 0 1 ?,
? 0 0 -0.5 ? ? 0 -0.5 ?

with a unit vector as the second cointegration vector. This scheme shows that any stationary component can easily be integrated into the general error-correction model structure. For this reason, it is fairly safe to proceed with the system cointegration procedure, even if some components are borderline cases between I(0) and I(1). I(2) variables, however, are not admitted. Issue: Several participants failed to provide the cointegrating vectors (1, -2, 0)' and (0, 0, 1)' in this point.

Added by Lisa D.

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