variance-covariance matrix of the composite shocks
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- It may not he apparent at first glance, but you can obtain the *-elements in Equation (6) using a Cholesky decomposition of a variance-covariance matrix of the composite shocks. In order to do so, you need reorder the entries of II to account for a new variable ordering y’t. = (rf, pt, ay . We call this new variance-covariance matrix SY. Note that the same values occur in both matrices, they are just ordered differently (for example, Var(4) is the [1, *element in 12 and the [2, 2]-element in SY). Construct IT and print it to the command window.
- The identification scheme in Equation (6) entails for the reordered variables:
Et [E 141 Et ut f Utf ]
where
wr
- *
(7)
Obtain NV from a Cholesky decomposition of tif. Note that Wr holds the same elements as W. but they are in a different ordering. Hint: Use the MATLAB function chol to perform the decomposition and keep in mind that ciao]. returns an upper triagonal matrix.
- Reorder the elements of INTT to obtain W. Print W to the output window.
- Study the effects of one unit shocks in the idiosyncratic innovations on yt by platting impulse response functions. Proceed in the following way: To study the effects of a one unit shock in ith, set u1 = (1, 0, 0)1, and obtain the respective E 1 by means of Equation (5). Now, similar to what you already did in task 1.10, iterate the system forwards (consider s = 0, … , 10) by setting all future u = 0. Do the same for one unit shocks in ttf and ut and plot the resulting impulse response functions comprehensively. Interpret your results.
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