GARCH Toolbox™

garchfit - Estimate univariate GARCH process parameters

Syntax

[Coeff,Errors,LLF,Innovations,Sigmas,Summary] = ... garchfit(Series) [...] = garchfit(Spec,Series) [...] = garchfit(Spec,Series,X) [...] = garchfit(Spec,Series,X,... PreInnovations,PreSigmas,PreSeries) garchfit(...)

Description

Given an observed univariate return series, garchfit estimates the parameters of a conditional mean specification of ARMAX form, and conditional variance specification of GARCH, EGARCH, or GJR form. The estimation process infers the innovations (that is, residuals) from the return series. It then fits the model specification to the return series by maximum likelihood.

[Coeff,Errors,LLF,Innovations,Sigmas,Summary] = ... garchfit(Series) models an observed univariate return series as a constant, C, plus GARCH(1,1) conditionally Gaussian innovations. For models more complicated than this one, you must provide model parameters in the GARCH specification structure Spec.
[...] = garchfit(Spec,Series) infers the innovations from the return series and fits the model specification, contained in Spec, to the return series by maximum likelihood.
[...] = garchfit(Spec,Series,X) provides a regression component X for the conditional mean.
[...] = garchfit(Spec,Series,X,... PreInnovations,PreSigmas,PreSeries) uses presample observations, contained in the time-series column vectors PreInnovations, PreSigmas, and PreSeries, to infer the outputs Innovations and Sigmas. These vectors form the conditioning set used to initiate the inverse filtering, or inference, process. If you provide no explicit presample data, the necessary presample observations derive from conventional time-series techniques (see Automatically Minimizing Transient Effects).

If you specify at least one, but fewer than three, sets of presample data, garchsim does not attempt to derive presample observations for those you omit. When specifying your own presample data, be sure to specify all data required for the given conditional mean and variance models. See User-Specified Presample Observations.

garchfit(...) with input arguments as shown but with no output arguments, displays the final parameter estimates and standard errors. It also produces a tiered plot of the original return series, the inferred innovations, and the corresponding conditional standard deviations.

Input Arguments

`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.

Output Arguments

`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 log-likelihood 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 `NaN`s.
`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 `NaN`s.
`Summary`	Structure of summary information about the optimization process. The fields and their possible values are as follows:
	`exitFlag`	Describes the exit condition: `Positive` — Log-likelihood objective function converged to a solution. `0` — Maximum number of function evaluations or iterations was exceeded. `Negative` — Log-likelihood 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`

Note garchfit calculates the error covariance matrix of the parameter estimates Summary.covMatrix, and the corresponding standard errors found in the Errors output structure using finite difference approximation. In particular, it calculates the standard errors using the outer-product method. For more information, see Section 5.8 in Hamilton (References follow).

Examples

Example 1

The following example uses garchfit to estimate the parameters for a return series of 1000 simulated observations based on a GARCH(1,1) model. Because the 'Display' parameter defaults to 'on', garchfit displays diagnostic and iterative information:

spec = garchset('C',0,'K',0.0001,'GARCH',0.9,'ARCH',0.05);
randn('state', 0); rand('twister', 0);
[e,s,y]        = garchsim(spec,1000);
[Coeff,Errors] = garchfit(spec,y);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   Diagnostic Information 

Number of variables: 4

Functions 
 Objective:                            garchllfn
 Gradient:                             finite-differencing
 Hessian:                              finite-differencing (or Quasi-Newton)
 Nonlinear constraints:                armanlc
 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:          4
 Number of upper bound constraints:          4

Algorithm selected
   medium-scale


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 End diagnostic information

                                Max     Line search  Directional  First-order 
 Iter F-count        f(x)   constraint   steplength   derivative  optimality Procedure
    0      5     -1762.16   -9.98e-005                                          
    1     21     -1762.62  -9.975e-005     0.000488    1.32e+004    1.47e+004   
    2     33     -1763.04  -9.897e-005      0.00781          126    2.13e+005   
    3     40     -1764.68  -7.423e-005         0.25         4.18    1.28e+005   
    4     53     -1764.72  -7.477e-005      0.00391         6.92    1.12e+005   
    5     59     -1765.27  -4.128e-005          0.5        0.216     3.2e+003   
    6     72     -1765.28   -4.75e-005      0.00391          4.2    2.99e+004   
    7     82     -1765.28  -4.619e-005       0.0313        0.066    3.02e+004   
    8     93     -1765.29  -4.832e-005       0.0156          0.2    2.48e+003   
    9     98     -1765.29  -4.639e-005            1    -0.000171         14.6   
   10    117     -1765.29  -4.639e-005    -6.1e-005   -1.03e-005         14.6   
   11    124     -1765.29  -4.638e-005         0.25   -5.52e-006         2.34   
Hessian modified twice
   12    132     -1765.29  -4.638e-005        0.125    7.71e-006           38  
Hessian modified twice
   13    138     -1765.29  -4.638e-005          0.5    1.05e-006          126   
Hessian modified twice
Optimization terminated: magnitude of directional derivative in search
direction less than 2*options.TolFun and maximum constraint violation
is less than options.TolCon.
No active inequalities.

Example 2

Using the same data as in Example 1, the example sets 'Display' to 'off' and calls garchfit with no output arguments. garchfit then displays the final parameter estimates and standard errors, then produces a tiered plot:

spec = garchset(spec,'Display','off');
garchfit(spec, y);
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    -0.0024759     0.0012919       -1.9165
          K    4.6877e-005    5.3555e-005      0.8753
   GARCH(1)    0.93904        0.041604        22.5707
    ARCH(1)    0.035503       0.015123         2.3477

 Log Likelihood Value: 1765.29

References

Bollerslev, T., "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," Review of Economics and Statistics, Vol. 69, 1987, pp 542-547.

Bollerslev, T., "Generalized Autoregressive Conditional Heteroskedasticity," Journal of GARCH, Vol. 31, 1986, pp 307-327.

Box, G.E.P., G.M. Jenkins, and G.C. Reinsel, Time Series Analysis: Forecasting and Control, Third edition, Prentice Hall, 1994.

Enders, W., Applied Econometric Time Series, John Wiley & Sons, 1995.

Engle, Robert, "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Vol. 50, 1982, pp 987-1007.

Engle, R.F., D.M. Lilien, and R.P. Robins, "Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model," Econometrica, Vol. 59, 1987, pp 391-407.

Glosten, L.R., R. Jagannathan, and D.E. Runkle, "On the Relation Between Expected Value and the Volatility of the Nominal Excess Return on Stocks," The Journal of Finance, Vol.48, 1993, pp 1779-1801.

Hamilton, J.D., Time Series Analysis, Princeton University Press, 1994.

Nelson, D.B., "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Vol. 59, 1991, pp 347-370.

garchdisp

garchget