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Estimating Initial Parameters

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Computing User-Specified Initial Estimates Computing Automatically Generated Initial Estimates Working With Parameter Bounds

Computing User-Specified Initial Estimates

The constrained nonlinear optimizer, fmincon, requires a vector of initial parameter estimates. The garchfit function computes initial parameter estimates if you provide none. At times, however, it might be helpful to compute and specify your own initial guesses to avoid convergence problems. You can specify complete initial estimates for either or both the conditional mean equation and the conditional variance equation.

For the conditional mean estimates to be complete, specify the following parameters:

• C

• AR

• MA

These must be consistent with the orders you specified for R and M. The length of AR must be R, and the length of MA must be M. If you provide a regression matrix X, you must also specify the Regress parameter. C, AR, MA, and Regress correspond respectively to C, Φ_j, θ_i, and β_k in Equation 1-3.

Note   Set C = NaN (Not-a-Number) to remove the constant C from the conditional mean model. This fixes C = 0 without providing initial parameter estimates for the remaining parameters. In this case, the value of FixC has no effect.

For the conditional variance estimates to be complete, specify these specification structure parameters for all conditional variance models:

• K

• GARCH

• ARCH

These must be consistent with the orders you specified for P and Q. The length of GARCH must be P, and the length of ARCH must be Q. You must also specify the Leverage parameter for GJR and EGARCH conditional variance models. The parameters K, GARCH, ARCH, and Leverage correspond respectively to κ, G_i, A_j, and L_j in the conditional variance equations.

You can use garchset to create the necessary specification structure, Spec, or you can modify the Coeff structure returned by a previous call to garchfit.

If you provide initial parameter estimates for a model equation, you must provide all the estimated constants and coefficients consistent with the specified model orders. For example, for an ARMA(2,2) model with no regression matrix, you must specify the parameters C, AR, and MA. If you specify only MA, the specification is incomplete, and garchfit ignores the MA you specified and automatically generates all the requisite initial estimates.

The following specification structure provides C and AR as initial parameter estimates, but does not provide MA, even though M = 1. In this case, garchfit ignores the C and AR fields, computes initial parameter estimates, and overwrites existing parameters in the incomplete conditional mean specification.

spec = garchset('R',1,'M',1,'C',0,'AR',0.5,...
                'P',1,'Q',1,'K',0.0005,'GARCH',0.8,'ARCH',0.1)
spec = 

         Comment: 'Mean: ARMAX(1,1,?); Variance: GARCH(1,1)'
     Distribution: 'Gaussian'
                R: 1
                M: 1
                C: 0
               AR: 0.5000
               MA: []
    VarianceModel: 'GARCH'
                P: 1
                Q: 1
                K: 5.0000e-004
            GARCH: 0.8000
             ARCH: 0.1000

However, the structure explicitly sets all fields in the conditional variance model. Therefore, garchfit uses the specified values of K, GARCH, and ARCH as initial estimates, subject to further refinement.

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Computing Automatically Generated Initial Estimates

garchfit automatically generates initial estimates if you provide incomplete or no initial coefficient estimates for a conditional mean or variance model. It first estimates the conditional mean parameters as needed, and then estimates the conditional variance parameters as needed. Again, garchfit ignores incomplete initial estimates. It estimates initial conditional mean parameters using standard statistical time series techniques, dependent upon the parametric form of the conditional mean equation.

Estimating Conditional Mean Models Without a Regression Component

ARMA Models.   Initial parameter estimates of general ARMA(R,M) conditional mean models are estimated by the three-step method outlined in Box, Jenkins, and Reinsel [10], Appendix A6.2.

• garchfit estimates the autoregressive coefficients, Φ_j, by computing the sample autocovariance matrix and solving the Yule-Walker equations.

• Using these estimated coefficients, garchfit filters the observed Series to obtain a pure moving average process.

• garchfit computes the autocovariance sequence of the moving average process, and uses it to iteratively estimate the moving average coefficients, θ_i. This also provides an estimate of the unconditional variance of the innovations.

Estimating Conditional Mean Models with a Regression Component

ARX Models (No Moving Average Terms Allowed).   Ordinary least squares regression generates initial estimates of the autoregressive coefficients, Φ_j, and the regression coefficients, β_k, of the explanatory data matrix X.

For more information, see Regression.

ARMAX Models (Moving Average Terms Included).   Initial parameter estimation of the general ARMAX conditional mean models requires two steps:

• garchfit estimates an ARX model by ordinary least squares.

• garchfit estimates an MA(M) = ARMA(0,M) model, as described in Estimating Conditional Mean Models Without a Regression Component.

Estimating Conditional Variance Models

Unlike conditional mean parameters, initial estimates of conditional variance parameters are based on empirical analysis of time series. The approach is dependent upon the conditional variance model you select.

Estimating GARCH(P,Q) Models.   For GARCH models, garchfit assumes that the sum of the G_i, (i = 1, ...,P) and the A_j, (j = 1, ...,Q) is close to 1. Specifically, for a general GARCH(P,Q) model, garchfit assumes that

G₁ + ... + G_P + A₁ + ... + A_Q = 0.9

If P > 0 (lagged conditional variances are included), then garchfit equally allocates 0.85 out of the available 0.90 to the P GARCH coefficients. It allocates the remaining 0.05 equally among the Q ARCH coefficients.

P = 0 specifies an ARCH(Q) model in which garchfit allocates 0.90 equally to the Q ARCH terms.

The following examples clarify this approach.

Initial estimates of the GARCH(1,1) model are expressed as follows:

A GARCH(2,1) model is initially expressed as:

An ARCH(1) model is initially expressed as:

An ARCH(2) model is initially expressed as:

Finally, garchfit estimates the constant κ of the conditional variance model by first estimating the unconditional, or time-independent, variance of {ε_t}:

In terms of the parameters, this can also be expressed as:

and so

Estimating GJR(P,Q) Models.   garchfit treats a GJR(P,Q) model as an extension of an equivalent GARCH(P,Q) model with zero leverage terms. Thus, initial parameter estimates of GJR models are identical to those of equivalent order GARCH models (see Estimating GARCH(P,Q) Models), with the additional assumption that all leverage terms are zero: L_i = 0, 1 ≤ i ≤ Q.

Estimating EGARCH(P,Q) Models.   For EGARCH models, garchfit assumes that the sum of the G_i, (i = 1, ..., P) is 0.9, and the sum of A_j, (j = 1, ..., Q) is 0.2. Specifically, for a general EGARCH(P,Q) model, garchfit assumes that:

G₁ + G₂ + ... + G_P = 0.9

A₁ + A₂ + ... + A_Q = 0.2

and

L_i = 0, 1 ≤ i ≤ Q

If P > 0 (lagged conditional variances are included), then garchfit equally allocates the available weight of 0.9 to the P GARCH coefficients. It equally allocates the available weight of 0.2 to the Q ARCH coefficients.

In EGARCH models, the standardized innovation, z_t, serves as the forcing variable for both the conditional variance and the error. Thus, the G_i terms captured volatility clustering (that is, persistence). In other words, EGARCH models make no allowance for the relationship between the G_i and A_j coefficients regarding initial parameter estimates. Because of this, EGARCH(0,Q) models ignore the persistence effect commonly associated with financial returns, and are unusual. Some examples clarify the approach.

The EGARCH(1,1) model is by far the most common, and initial estimates are expressed as:

Initial estimates for an EGARCH(2,2) model are expressed as

An EGARCH(0,1) model would be initially expressed as

As you can see, initial parameter estimates for EGARCH models are most effective when P > 0.

Finally, you can estimate the constant κ of an EGARCH conditional variance model by noting the approximate relationship between the unconditional variance of the innovations process, σ², and the G_i parameters of an EGARCH(1,1) model:

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Working With Parameter Bounds

The garchfit function bounds some model parameters to provide stability in the optimization process. See the example Lower Bound Constraints for more information on overriding these bounds in the unlikely event they become active.

Setting Bounds for the Conditional Mean Model

For the conditional mean model, Equation 1-3, garchfit bounds the conditional mean constant C and the conditional mean regression coefficients β_k, if any, in the interval [–10, 10]. However, if the coefficient estimates that you specify or that garchfit generates are outside this interval, garchfit sets the appropriate lower or upper bound equal to the estimated coefficient.

Setting Bounds for the GARCH(P,Q) and GJR(P,Q) Conditional Variance Models

For GARCH(P,Q) and GJR(P,Q) conditional variance models, garchfit uses 5 as an upper bound for the conditional variance constant κ. If the initial estimate is greater than 5, garchfit uses the estimated value as the upper bound.

Setting Bounds for the EGARCH(P,Q) Conditional Variance Model

For EGARCH(P,Q) conditional variance models, garchfit places arbitrary bounds on the conditional variance constant κ, such that –5 ≤ κ ≤ 5. If the magnitude of the initial estimate is greater than 5, garchfit adjusts the bounds accordingly.

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Maximum Likelihood Estimation

Presample Data

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