MATLAB: Estimate GARCH(4,1) using estimate: Parameter GARCH{2} is missing

Econometrics Toolboxestimategarch

Hi,

I want to fit an AR(1) Model to a time series of returns (x) and the variance process follows a GARCH(4,1) model.

model = arima('ARLags',1,'Variance',garch(4,1))
fit = estimate(model,x)

returns:

    ARIMA(1,0,0) Model:
    --------------------
    Conditional Probability Distribution: Gaussian
                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant   -0.000754276   0.000531875       -1.41815
        AR{1}       0.223266     0.0279149        7.99808
    GARCH(4,1) Conditional Variance Model:
    ----------------------------------------
    Conditional Probability Distribution: Gaussian
                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant    1.77228e-05   3.59061e-06        4.93587
     GARCH{1}        0.55604      0.113503        4.89891
     GARCH{3}       0.193253      0.193158        1.00049
     GARCH{4}       0.065569     0.0953147       0.687922
      ARCH{1}       0.176294     0.0259334        6.79798

Why is the parameter GARCH{2} missing in the table? Is it because it may be very unsignificant? I mean the parameters GARCH{3} and GARCH{4} are pretty unsignificant as well. Or are there too few observations? The time series x is 1545 observations long. When I limit the estimation to a fit period of just 1000 observations via

model = arima('ARLags',1,'Variance',garch(4,1))
fit = estimate(model,x(1:1000))

it returns:

    ARIMA(1,0,0) Model:
    --------------------
    Conditional Probability Distribution: Gaussian
                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant    0.000184079   0.000783618       0.234909
        AR{1}       0.234824     0.0336974         6.9686
    GARCH(1,1) Conditional Variance Model:
    ----------------------------------------
    Conditional Probability Distribution: Gaussian
                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant      4.899e-05   1.53734e-05        3.18668
     GARCH{1}       0.796332      0.279077        2.85345
      ARCH{1}       0.157239     0.0455136        3.45478

Now the result is a GARCH(1,1) model, allthough I defined the model to be a GARCH(4,1).

Using Econometrics Toolbox for MATLAB R2014a.

Thanks in Advance!

Best Answer

Nevermind, I found the solution. In the 'garch.m' function of the Econometrics toolbox it is stated:

%   o The coefficients GARCH and ARCH are each associated with an
%     underlying lag operator polynomial and subject to a near-zero 
%     tolerance exclusion test. That is, each coefficient is compared to 
%     the default zero tolerance 1e-12, and is included in the model only 
%     if the magnitude is greater than 1e-12; if the coefficient magnitude 
%     is less than or equal to 1e-12, then it is sufficiently close to zero 
%     and excluded from the model. See LagOp for additional details.

So the parameters are too close to zero and therefore excluded from the model...

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In fact if you retain all of the data but just change the single, largest negative return at observation #449 from -0.1516 to -0.05 (which is something more in line with the rest of the dataset), the results match up much more closely.

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Mean: ARMAX(0,0,0); Variance: GARCH(1,1)

Conditional Probability Distribution: Gaussian

Number of Model Parameters Estimated: 4

                             Standard          T

Parameter Value Error Statistic

----------- ----------- ------------ -----------

         C    0.00060142     0.00076854       0.7825
         K    0.00026368     2.4286e-005     10.8571
GARCH(1)    0.01546        0.040954         0.3775
   ARCH(1)    0.51909        0.065912         7.8755

Here, notice that the t-stat for the Gaussian distribution GARCH(1) parameter was so low - meaning rejection of the GARCH(1,1) model.

This would imply mis-specification with the Gaussian GARCH(1,1) model and that, for some reason, the Student's t GARCH(1,1) model does a better job at handling what seems to be long-period lags in the data.

If you run "autocorr(series.^2)" on the data you can see that there are some really long lags in the data. It appears rather random, then a big correlation at the 20-th lag.

Incidentally, the DoF estimate is solidly a Student's t distribution so, in some sense, it is just a better fit and does not indicate a bug in the GARCHFIT function.

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