vgxset command: Q parameter for resulting model object?

1
votes

Matlab's command for defining a vector time series model is vgxset, the formalism for which can be accessed by the command "doc vgxset". It says that the model parameter Q is "[a]n n-by-n symmetric innovations covariance matrix". No description of what it is for. I assumed that it was the covariance of the noise sources that show up in the equations for each times series in the archetypal representation of a vector time series, e.g., http://faculty.chicagobooth.edu/john.cochrane/research/papers/time_series_book.pdf.

I could be off about something (I often am), but this doesn't seem to match results from actually issuing the command to estimate a model's parameters. You can access the code that illustrates such estimation via the command "doc vgxvarx":

load Data_VARMA22
[EstSpec, EstStdErrors] = vgxvarx(vgxar(Spec), Y, [], Y0);

The object EstSpec contains the model, and the Q matrix is:

0.0518  0.0071
0.0071  0.0286

I would have expected that a covariance matrix as ones on the diagonal. Obviously, I misunderstand and/or mis-guessed at the purpose of Q. However, if you actually pull up the code for vgxset ("edit vgxset"), the comments explicitly describe Q as an "[i]nnovations covariance matrix".

I have 3 questions:

(1) What exactly is Q?

(2)Is there a Matlab reference document that I've failed to locate for fundamental parameters like this?

(3)If it isn't the covariance matrix for the noise sources, how does one actually supply actual noise source covariances to the model?

Please note that this question is specifically about Matlab's command for setting up the model, and as such, does not belong in the more concept-oriented Cross Validated Stack Exchange forum. I have posted this to:

(1) vgxset command: Q parameter for resulting model object?

(2) http://groups.google.com/forum/#!topic/comp.soft-sys.matlab/tg59h1wkRCw

1
matlabtime-series

1 Answers

1
votes

I will try to iterate to an answer, but being so many branches of discussion, i prefer to access directly onto this format. Whatever mean, this is a constructive process, as the purpose of this forum is...

Some previous "clarifications":

  1. The Output Covariance from EstSpec.Q after and before running the command vgxvarx are quite similar. Thus the command is outputting what he is shiningly expecting from itself.

  2. As an Output Covariance -or whatever other meaning for the Q parameter- is almost never to be a "mask" of the parameters to use, -i.e. an identity or a sparse zero-one matrix input parameter-. If you can assign it as a diagonal multiplied by some scalar univariate scalar is a different history. This is a covariance, plainly, just as in other MATLAB commands.

Hence:

(2) Is there a Matlab reference document that I've failed to locate for fundamental parameters like this?

No, Matlab ussualy don't give further explanations for "non popular" commands. Yes, this is, under some measure, "not popular", so i would not be impressed if the answer for this question is no.

Of course, the doctoral method is to check the provided references, on this case, those provided under doc vartovec. Which i dunno the hell where to find without order the books seeking the proper library or seeking the overall internet on five minutes...

Thus the obscure method is always better... check the code for the function by doing edit vgxvarx. Check the commented Section % Step 7 - Solve for parameters (Line 515, Matlab R2014b). There are calculations for a Q matrix through a function mvregress. At this point, both of us know, this is the core function.

This mvregress function (Line 62, Matlab R2014b) receives an input parameter called Covar0, which is described as a *D-by-D matrix to be used as the initial estimate for SIGMA*.

This antecedent leads to the answer for (1).

(1) What exactly is Q?

The MATLAB code has dozens of switch -both as options and auto-triggered- so i am actually not sure of which algorithm are you interested on, or based on your data, which ones are actually "triggered" :). Please read the previous answer, and place a Debug Point on the mvregress function:

Covar=Covar+CovAdj;  %Line 433, Matlab R2014b

and/or at:

Covar = (Covar + Resid'*Resid) / Count; % Line 439, Matlab R2014b

Having that, the exact meaning of Q, and as indicated by the mvregress help, would be an "Initial Matrix for the Estimate of the Output Covariance Matrix". The average is simply given by averaging the Counts...

But, for the provided data, making:

Spec.Q=[1 0.1;0.1 1];

and then running vgxvarx, the parameter Covar never got initialized!. Which for the presented unfortunate case, leads to a simply "Unused Parameter".

(3) If it isn't the covariance matrix for the noise sources, how does one actually supply actual noise source covariances to the model?

I've lost tons of manhours trying to gather the correct information from pre-built Matlab commands. Thus, my suggestion here, is to stick onto the concepts of system identification, and I would put my faith under one of the following alternatives:

  1. Keep believing, and dig a bit and debug inside the mvregress function, and check if some of the EstMethods -i.e. cwls ecm mvn under Line 195- leads to a proper filling of the Covar0 parameter,

  2. Stick to the vgxvarx command, but let the Q parameter go, and diagonalize | normalize the data properly, in order to let the algorithm identify the data as a Identically Distributed Gaussian Noise,

  3. Send vgxvarx to the hell, and use arx. I am not sure about the current stability of vgxvarx, but i am quite sure arx should be more "stable" on this regard...

Good Luck,

hypfco.

EDIT

A huge and comprehensive comment, i have nothing much to add.

  1. Indeed, it is quite probable the vgxvarx was run on the Matlab data sample. Hence the results lay explained,
  2. I tried to use the Q parameter on the vgxvarx, with no success by now. If any working code is found, it would be interesting to include it,

  3. The implementation of the noise transformation over the data should be really simple, of the form:

    Y1=(Y-Y0)*L

with L the left triangular cholesky for the inverse calculated covariance of Y, and Y0 the mean,

  1. I think the MA part is as critical as the AR part. Unless you have very good reasons, you usually cannot say you explained your data in a gaussian way.
  2. From your very last comment, I really suggest you to move onto a better, more established command for doing AR, MA, ARMA and such flavours. I am pretty sure they handle the MV case...
  3. Again, Matlab don't impress me on that behaviour...

Cheers...

hypfco