# RMT (Random Matrix Theory) issue with callibrating MP distribution -

I am seeing an issue when callibrating an MP distribution. Assume a log return series for the SP500 with the following dimensions

dim(xts.sp500.ret.stocksonly)

==> [1] 1133 478

sp500.cor <- cor.empirical(xts.sp500.ret.stocksonly)
sp500.eigens <- eigen(sp500.cor)\$values
sp500.eigen.density <-  density(sp500.eigens,n=5000)
plot(sp500.eigen.density,xlim=c(0,4),main="sp500 returns eigenvalue density")

I assume my 'Q' value is 1133/478 =

Problem: Even though the 'shape' and cutoffs seem OK -- the density value (y axis) seem vastly OFF. Peak of 1.5 for the real series - 5 or so for the theoretical, (note I am truncating the plot so the market eigenvalues are not shown, they are huge around 200).

Question: 1) Is this expected? 2) How does this affect callibration? Should I trust the results and simply look at the cutoffs? 3) Also when 'cleansing' the matrix I see most code (e.g. filter.RMT in tawny) simply replaces values below Lambda+ with the average, what about Lambda- though?

thanks much!

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1. The plot function is smoothing the plot. You should show the distribution of eigenvalues via a bar chart. Because a bar chart is discrete you can better discern the separation of the top-most eigenvaules. The top-most eigenvalue (representing the market factor) should be substantially greater than bulk of the eigenvalue distribution.

2. I assume by "calibration" you mean "eigenvalue cleansing". The approach would be to apply the RMT cleansing procedure on eigenvalues beneath the upper noise-band (lambda+)

3. The procedure of replacing eigenvalues below Lambda+ with the average will by definition also replace values below Lambda- as well. Replacing the eigenvalues by the average is one of many possible cleansing procedures.

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Clarification to the question: By callibration I mean how to fit the correct MP distribution to the eigenvalues density I get. Notice in the top plot the max "y" value I get is ~1.5, while for the MP distribution from theory the values are much higher. How do I know i've made a good fit of the MP parameters? thanks. – nxstock-trader Jan 21 '12 at 23:29
The empirical distribution of eigenvalues does not match the MP distribution because the former contains non-random factors. The theory shows what the eigenvalue distribution is for a random symmetric matrix for some choice of Q and Sigma. You can calculate Q and Sigma directly using the asymptotic definition of Q (which you have in your post), and sigma is simply the sigma of the elements of the de-meaned symmetric covariance matrix. So there is no need to "fit" the empirical distribution. – Ram Ahluwalia Jan 22 '12 at 1:47