Testing Standardized residuals

[mlaz]

Hi,
I am new to working with ARCH/GARCH and have recently purchased the course notes you are selling. After going over the univariate models, I have a couple of questions about testing the standardized residuals from a GARCH(1,1) and EGARCH(1,1).
First, in the notes from the ARCH/Garch course, it seems that that for the standard GARCH(1,1) the notes have the test as being:

Code: Select all

* GARCH(1,1) with Gaussian errors
*
garch(p=1,q=1,regressors,hseries=h) / y
# constant y{1}
set ustd = %resids/sqrt(h)
graph(footer="GARCH(1,1) Standardized Residuals")
# ustd
*
* Diagnostics
*
stats ustd
@regcorrs(number=10,dfc=1,nocrits,qstat,$
title="Standardized Residuals") ustd
disp "Q for Residual Serial Correlation" %qstat $
"significance level" %qsignif
set ustd2 = ustdˆ2
@regcorrs(number=10,dfc=2,nocrits,qstat,$
title="Standardized Squared Residuals") ustd2
disp "McLeod-Li for Residual ARCH=" %qstat $
"significance level" %qsignif

So the first test we have dfc=1 because of one lag, and in the test for the squared residuals McLeod-Li is 2 (the # of GARCH parameters (lagged variance/residual squared terms) for this case – in general I think this is just p+q)
I have four questions about the example/extensions:
1) How do we know that number=10 is the right value for the tests? Other examples use other values, e.g., Tsay’s example on page 131 seems to use number=24)

Code: Select all

----------------------------------
* Tsay, Analysis of Financial Time Series, 3rd edition
* ARCH example on exchange rate, page 131
*
open data exch-perc.txt
data(format=free,org=columns) 1 2487 exchperc
garch(p=0,q=3,resids=at,hseries=hvar) / exchperc
set resi = at/sqrt(hvar)
set resisq = resi^2
@RegCorrs(number=24) resi
@RegCorrs(number=24) resisq

2) I am also not completely certain about degrees of freedom correction in the Tsay example above, and the one in the notes:

Code: Select all

* ARCH(2) with Gaussian errors
*
garch(p=0,q=2,regressors,hseries=h) / y
# constant y{1}
graph(footer="ARCH(2) Residuals")
# %resids
set ustd = %resids/sqrt(h)
graph(footer="ARCH(2) Standardized Residuals")
# ustd
*
* Diagnostics
*
stats ustd
@regcorrs(number=10,dfc=1,nocrits,qstat,$
title="Standardized Residuals") ustd
disp "Q for Residual Serial Correlation" %qstat $
"significance level" %qsignif
set ustd2 = ustdˆ2
@regcorrs(number=10,dfc=2,nocrits,qstat,$
title="Standardized Squared Residuals") ustd2
disp "McLeod-Li for Residual ARCH=" %qstat $
"significance level" %qsignif

In the Tsay example, it appears that there are no degrees of freedom correction for the ARCH(3), but for the case in the workbook, there is a dfc=2 (since we are in an ARCH(2) case)? Should there be a correction in the Tsay example or am I missing something?

3) For extensions. When using an asymmetric Garch/Egarch, do we need to adjust the degrees of freedom? I think that Example 10.3 in the book would suggest that the answer is yes and that a EGARCH(1,1) with asymmetry would be have a dfc=3?
4) Finally a question about adding in the regressors and xregressors. Do we need any additional degrees of corrections for these?
Thanks for your help.

[TomDoan]

Hi,
I am new to working with ARCH/GARCH and have recently purchased the course notes you are selling. After going over the univariate models, I have a couple of questions about testing the standardized residuals from a GARCH(1,1) and EGARCH(1,1).
First, in the notes from the ARCH/Garch course, it seems that that for the standard GARCH(1,1) the notes have the test as being:

So the first test we have dfc=1 because of one lag, and in the test for the squared residuals McLeod-Li is 2 (the # of GARCH parameters (lagged variance/residual squared terms) for this case – in general I think this is just p+q)
I have four questions about the example/extensions:
1) How do we know that number=10 is the right value for the tests? Other examples use other values, e.g., Tsay’s example on page 131 seems to use number=24)

There's no "right" number of lags. I prefer smaller to larger, particularly for the squared residuals, simply because there is no easy fix for a problem correlation at lag 20.

2) I am also not completely certain about degrees of freedom correction in the Tsay example above, and the one in the notes:

In the Tsay example, it appears that there are no degrees of freedom correction for the ARCH(3), but for the case in the workbook, there is a dfc=2 (since we are in an ARCH(2) case)? Should there be a correction in the Tsay example or am I missing something?

One thing to remember about textbook examples is that we try to replicate the results from the textbook. In the Tsay example, there should have been a DFC=3 on the squared residuals, but the book didn't do that.

3) For extensions. When using an asymmetric Garch/Egarch, do we need to adjust the degrees of freedom? I think that Example 10.3 in the book
would suggest that the answer is yes and that a EGARCH(1,1) with asymmetry would be have a dfc=3?

I'm not sure there is any theorem which would show what the effects would be on the asymptotics in that case. I would probably do the correction just to be on the safe side.

4) Finally a question about adding in the regressors and xregressors. Do we need any additional degrees of corrections for these?
Thanks for your help.

There are unquestionably no theorems on the effects of those. Once you're outside of a "self-contained" DGP, it's not clear what the asymptotics are.

[jack]

Dear Tom,

I have estimated an EGARCH-X(1,1) and here is the results.

But there are some correlations between standardized squared residuals at beginning lags: Is my model correct? Do I have to correct for those correlations?

GARCH Model - Estimation by BFGS
Convergence in    42 Iterations. Final criterion was  0.0000000 <=  0.0000100

With Heteroscedasticity/Misspecification Adjusted Standard Errors
Dependent Variable RATE
Usable Observations                      2866
Log Likelihood                      8950.4912

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                     -0.000217006  0.000110089     -1.97119  0.04870240
2.  RATE{1}                       0.493057127  0.137295985      3.59120  0.00032916
3.  Mvg Avge{1}                  -0.521542744  0.144919568     -3.59884  0.00031964

4.  C                             0.000003442  0.000001254      2.74383  0.00607273
5.  A                             0.077563881  0.020159648      3.84748  0.00011934
6.  B                             0.912060238  0.019282479     47.29995  0.00000000
7.  D                            -0.037237231  0.022602389     -1.64749  0.09945718
8.  DUM                          -0.000000750  0.000000568     -1.32101  0.18649856
9.  DUMUS                         0.000003513  0.000001797      1.95450  0.05064176
10. Shape                         7.851643183  1.225672601      6.40599  0.00000000

Lag  Corr  Partial   LB Q    Q Signif
  1  0.015   0.015 0.6325204
  2  0.012   0.011 1.0224177
  3 -0.024  -0.025 2.7340092    0.0982
  4 -0.012  -0.011 3.1148165    0.2107
  5  0.015   0.016 3.7781657    0.2864
  6  0.008   0.007 3.9415181    0.4140
  7 -0.006  -0.007 4.0501342    0.5422
  8  0.021   0.022 5.3303209    0.5022
  9 -0.018  -0.018 6.2354396    0.5125
 10 -0.014  -0.014 6.7675090    0.5619

Q for Residual Serial Correlation       6.76751 significance level       0.56191
Lag  Corr  Partial   LB Q    Q Signif
  1 -0.013  -0.013  0.488438
  2  0.043   0.043  5.729153
  3 -0.004  -0.003  5.781101    0.0162
  4 -0.021  -0.023  7.031524    0.0297
  5  0.017   0.017  7.863805    0.0489
  6  0.000   0.002  7.863805    0.0967
  7  0.028   0.026 10.056839    0.0736
  8 -0.013  -0.013 10.562488    0.1029
  9  0.014   0.012 11.134970    0.1328
 10 -0.009  -0.008 11.373014    0.1814

McLeod-Li for Residual ARCH=      11.37301 significance level       0.18145

[TomDoan]

You are almost never going to get better than that with 2800 data points. You do, however, have what's almost certainly an unnecessary MA parameter (you're getting AR-MA cancellation), and your dummies don't seem to be necessary either.