| GARCH Toolbox | |
Boundary Constraints and Statistical Inferences
The estimation process places stationarity and positivity constraints on the parameters (see Eq. (1-6) in the section Homoskedasticity of the Unconditional Variance).
Whenever garchfit actively imposes parameter constraints (other than user-specified equality constraints) during the estimation process, the statistical results based on the maximum likelihood parameter estimates are invalid (see Hamilton [10], page 142). This is because statistical inference relies on the log-likelihood function being approximately quadratic in the neighborhood of the maximum likelihood parameter estimates. This cannot be the case when the estimates fail to fall in the interior of the parameter space.
As an example of an actively imposed parameter constraint, fit a GARCH(1,2) model to the returns of the XYZ Corporation. This model is intentionally misspecified and estimations for such models often have difficulty converging. You can increase the likelihood of convergence by making the requirement for convergence less stringent. To do this increase the termination tolerance parameter TolCon from 1e-6 (the default) to 1e-5.
spec = garchset('P', 1, 'Q', 2, 'TolCon', 1e-5);
[coeff, errors, LLF, innovations, sigma, summary] = garchfit(spec, xyz);
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Diagnostic Information
Number of variables:5
Functions
Objective: garchllfn
Gradient: finite-differencing
Hessian: finite-differencing (or Quasi-Newton)
Nonlinear constraints: garchnlc
Gradient of nonlinear constraints: finite-differencing
Constraints
Number of nonlinear inequality constraints:0
Number of nonlinear equality constraints: 0
Number of linear inequality constraints: 1
Number of linear equality constraints: 0
Number of lower bound constraints: 5
Number of upper bound constraints: 0
Algorithm selected
medium-scale
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End diagnostic information
max Directional
Iter F-count f(x) constraint Step-size derivative Procedure
1 6 -5922.27 -1.684e-005 1 -3.34e+004
2 36 -5922.27 -1.684e-005 1.19e-007 -578
3 46 -5926.29 -1.474e-005 0.125 -60
4 60 -5926.45 -1.558e-005 0.00781 -51.6
5 68 -5952.6 -7.79e-006 0.5 -27.5
6 76 -5964.39 -3.895e-006 0.5 -12.4
7 84 -5964.42 -1.947e-006 0.5 -95.4
8 98 -5964.43 -2.084e-006 0.00781 -27.4
9 106 -5971.69 -1.552e-006 0.5 -7.6
10 114 -5974.09 -7.762e-007 0.5 -97.8
11 129 -5974.17 -9.254e-007 0.00391 -0.556
12 136 -5974.59 4.337e-019 1 -0.0767
13 145 -5974.6 5.421e-019 0.25 -0.0075
14 152 -5974.6 1.084e-018 1 -0.00322
15 159 -5974.6 2.168e-018 1 -0.00152
16 166 -5974.6 4.337e-018 1 -0.00084
17 173 -5974.6 8.674e-018 1 -0.000282
18 183 -5974.6 9.758e-018 0.125 -6.16e-005
19 191 -5974.6 1.464e-017 0.5 -0.000145 Hessian modified twice
20 205 -5974.6 1.475e-017 0.00781 -1.94e-006
Optimization terminated successfully:
Search direction less than 2*options.TolX and
maximum constraint violation is less than options.TolCon
Active Constraints:
5
Warning:Boundary Constraints Active; Standard Errors may be Inaccurate.
The warning message explicitly states that garchfit has imposed constraints. If you choose to suppress the estimations details (i.e., set the specification structure field Display to off), the same information is available from the constraints field of the summary output structure.
summary
summary =
warning:'No Warnings'
converge:'Function Converged to a Solution'
covMatrix:[5x5 double]
iterations:20
functionCalls:208
constraints:'Boundary Constraints Active; Errors may be Inaccurate'
Examine the estimation results to see exactly what happened.
garchdisp(coeff, errors)
Number of Parameters Estimated: 5
Standard T
Parameter Value Error Statistic
----------- ----------- ------------ -----------
C 0.00048993 0.00025674 1.9083
K 8.1018e-007 2.9827e-007 2.7163
GARCH(1) 0.96327 0.0062937 153.0524
ARCH(1) 0.031503 0.016075 1.9597
ARCH(2) 0 0.018615 0.0000
The 0 value of ARCH(2)reveals that garchfit has enforced the variance positivity constraint of the second ARCH parameter. It indicates that the estimated GARCH(1,2) model is in fact a GARCH(1,1) model, and further emphasizes that the default model is well suited for the returns of the XYZ Corporation.
Furthermore, since a parameter constraint has been actively imposed during the estimation process, the statistical results based on the maximum likelihood parameter estimates are invalid. These statistical results include the standard errors shown in column two, as well as any likelihood ratio tests based on the lratiotest function.
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