garchfit - Estimate ARMAX/GARCH model parameters

Spec

GARCH specification structure containing the conditional mean and variance specifications. It also contains the optimization parameters needed for the estimation. Create this structure by calling garchset, or use the Coeff output structure returned by garchfit.

Series

Time series column vector of observations of the underlying univariate return series of interest. Series is the response variable representing the time series to be fitted to conditional mean and variance specifications. The last element of Series holds the most recent observation.

X

Time series regression matrix of observed explanatory data. Typically, X is a matrix of asset returns (for example, the return series of an equity index), and represents the past history of the explanatory data. Each column of X is an individual time series used as an explanatory variable in the regression component of the conditional mean. In each column, the first row contains the oldest observation and the last row the most recent.

The number of valid (non-NaN) most recent observations in each column of X must equal or exceed the number of valid most recent observations in Series. If the number of valid observations in a column of X exceeds that of Series, garchfit uses only the most recent observations of X. If X = [] or is unspecified, the conditional mean has no regression component.

PreInnovations

Time series column vector of presample innovations that garchfit uses to condition the recursive mean and variance models. This column vector can have any number of rows, provided it contains sufficient observations to initialize the mean and variance equations. That is, if M and Q are the number of lagged innovations required by the conditional mean and variance equations, respectively, then PreInnovations must have at least max(M,Q) rows. If the number of rows exceeds max(M,Q), then garchfit uses only the last (that is, most recent) max(M,Q) rows.

PreSigmas

Time series column vector of positive presample conditional standard deviations that garchfit uses to condition the recursive variance model. This vector can have any number of rows, as long as it contains sufficient observations to initialize the conditional variance equation. That is, if P and Q are the number of lagged conditional standard deviations and lagged innovations required by the conditional variance equation, respectively, then PreSigmas must have at least P rows for GARCH and GJR models, and at least max(P,Q) rows for EGARCH models. If the number of rows exceeds the requirement, then garchfit uses only the last (that is, most recent) rows.

PreSeries

Time series column vector of presample observations of the return series of interest that garchfit uses to condition the recursive mean model. This vector can have any number of rows, provided it contains sufficient observations to initialize the conditional mean equation. Thus, if R is the number of lagged observations of the return series required by the conditional mean equation, then PreSeries must have at least R rows. If the number of rows exceeds R, then garchfit uses only the last (that is, most recent) R rows.

Coeff

GARCH specification structure containing the estimated coefficients. Coeff is of the same form as the Spec input structure. Toolbox functions such as garchset, garchget, garchsim, garchinfer, and garchpred can accept either Spec or Coeff as input arguments.

Errors

Structure containing the estimation errors (that is, the standard errors) of the coefficients. Errors is of the same form as the Spec and Coeff structures. If an error occurs in the calculation of the standard errors, garchfit sets all fields associated with estimated coefficients to NaN.

LLF

Optimized loglikelihood objective function value associated with the parameter estimates found in Coeff. garchfit performs the optimization using the Optimization Toolbox fmincon function.

Innovations

Innovations (that is, residuals) time series column vector inferred from Series. The size of Innovations is the same as the size of Series. If an error occurs, garchfit returns Innovations as a vector of NaNs.

Sigmas

Conditional standard deviation vector corresponding to Innovations. The size of Sigmas is the same as the size of Series. If an error occurs, garchfit returns Sigmas as a vector of NaNs.

Summary

Structure of summary information about the optimization process. The fields and their possible values are as follows:

exitFlag

Describes the exit condition:

• Positive — Loglikelihood objective function converged to a solution.

• 0 — Maximum number of function evaluations or iterations was exceeded.

• Negative — Loglikelihood objective function did not converge to a solution.

warning

One of the following strings:

• No Warnings

• ARMA Model Is Not Stationary/Invertible

converge

One of the following strings:

• Function Converged to a Solution

• Function Did NOT Converge

• Maximum Function Evaluations or Iterations Reached

constraints

One of the following strings:

• No Boundary Constraints

• Boundary Constraints Active; Errors May Be Inaccurate

covMatrix

Covariance matrix of the parameter estimates

iterations

Number of iterations

functionCalls

Number of function evaluations

lambda

Structure, output by fmincon, containing the Lagrange multipliers at the solution x

                                Max     Line search  Directional  First-order 
 Iter F-count        f(x)   constraint   steplength   derivative   optimality Procedure 
    0      5     -1725.47   -9.98e-005                                         
    1     22     -1725.67  -9.978e-005     0.000244       -752    8.54e+003   
    2     35     -1725.94  -9.939e-005      0.00391       -184    1.08e+005   
    3     43     -1726.98   -0.0003173        0.125      -75.1    5.08e+003   
    4     56     -1727.28   -0.0003188      0.00391       -143     1.3e+004   
    5     65     -1728.25    -0.000345       0.0625      -44.1    6.56e+003   
    6     72     -1728.29   -0.0002615         0.25      -4.52    1.06e+004   
    7     80     -1728.34   -0.0002529        0.125      -10.3     1.2e+004   
    8     85     -1728.39   -0.0002914            1      -4.47    7.07e+003   
    9     90      -1728.4   -0.0002815            1      -1.77    5.65e+003   
   10     95      -1728.4    -0.000274            1      -1.79    2.31e+003   
   11    100      -1728.4   -0.0002747            1      -1.78          691   
   12    105      -1728.4   -0.0002763            1      -0.32         2.12  Hessian modified

Local minimum possible. Constraints satisfied.

fmincon stopped because the predicted change in the objective function
is less than the selected value of the function tolerance and constraints 
were satisfied to within the selected value of the constraint tolerance.

No active inequalities.