Inoue and Kilian, Journal of Econometrics 2013

[Solstice]

Dear Tom/all,

I was wondering if anybody had put together something for the Inoue and Kilian JoE (2013) paper on credible sets and modal model responses? Matlab and Fortan code can be found on Kilians website, but in particular, I was looking for an example pertaining to the Fortan code listed there which gives an example applied to the Uhlig JME identification strategy (I found the example of Fry and Pagan 2011 applied to Uhligs JME strategy most insightful by the way - many thanks for making that available).

Thanks for your time

S.

[TomDoan]

While the idea seems to be light years ahead of Fry and Pagan (referees should be required to read this paper before requiring papers to include FP), I'm not 100% sure of the correctness of the derivation of the posterior. (BTW, the Matlab code applies only to a sign-constrained model which generates a complete model; the more complicated Fortran code is used for partially identified models).

[TomDoan]

There were, and apparently still are, some technical issues with this. The problem in the original paper was that, while the unitary matrix U is being drawn "uniformly", that's uniform in an abstract space, not in the Cartesian coordinates used in computing the density function. (For instance, with a size 2 matrix, the distribution is uniform in the rotation angle on the unit circle, which is *not* the same thing as being uniform in (x,y)). They did a corrigendum that apparently fixed that, but unfortunately, the result in the paper cited seems to only work for U's that are "rotations" while the U's that are needed to do sign restrictions require both rotations and reflections.

[TomDoan]

The attached does an example of the Inoue and Kilian analysis for sign-restricted responses based upon an example from the Kilian and Luetkepohl textbook. As mentioned in the earlier posts, this is specific to a sign-restricted model which generates a full set of shocks. The original paper had a technical flaw, which was partially, but not fully, fixed. I believe this implements the original idea correctly. (There was an issue with orthogonal matrices which weren't pure rotations, which is 50% of them by random draw).

The idea behind Inoue and Kilian is to rank "models" (impact matrices) by their posterior densities and use Highest Posterior Density (HPD) intervals rather than the more traditional intervals which are generated by taking the middle x% of the draws done separately for each combination of horizon, variable, shock combination. This program graphs the top 1% of responses in the HPD Examples.PDF (with 5000 saved responses, that's 50 sets of lines), and also does a separate graph of the conventional 16-84% bands (Standard Display.PDF). Note that, in this case, the two are quite consistent---the HPD draws cover almost the same range as the 16-84 bands on pretty much all the graphs.

Note that, with the calculations corrected, there is no single "modal" response---all impact matrices corresponding to a given combination of VAR coefficients and covariance matrix produce the same density function (which probably means that the same idea can be applied to partial models without all the added complication from the IK paper).

Please note: This requires version 9.2 of RATS. When we first started working on this in 2017, we added functions (such as %SKEWNN and %LNN) for doing various manipulations of matrices to handle the matrix differentials.

figure13_7.rpf

Program File

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kiliandata.txt

Data file

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HPD Examples.PDF

Graph of HPD 1%

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Standard Display.PDF

Standard graph of 68% intervals, done entry by entry

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