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

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!

[Karl-Martin]

Does someone has a clue?

Answer 1

[Karl-Martin]

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...