Fit GARCH Models to Time Series

Fit a Generalized Autoregressive Conditional Heteroscedastic GARCH(p, q) time series model to the data by computing the maximum-likelihood estimates of the conditionally normal model.

Usage

garch(x, order = c(1, 1), series = NULL, control = garch.control(...), ...)

garch.control(maxiter = 200, trace = TRUE, start = NULL,
  grad = c("analytical","numerical"), abstol = max(1e-20, .Machine$double.eps^2),
  reltol = max(1e-10, .Machine$double.eps^(2/3)), xtol = sqrt(.Machine$double.eps),
  falsetol = 1e2 * .Machine$double.eps, ...)

Arguments

x

a numeric vector or time series.

order

a two dimensional integer vector giving the orders of the model to fit. order[2] corresponds to the ARCH part and order[1] to the GARCH part.

series

name for the series. Defaults to deparse(substitute(x)).

control

a list of control parameters as set up by garch.control.

maxiter

gives the maximum number of log-likelihood function evaluations maxiter and the maximum number of iterations 2*maxiter the optimizer is allowed to compute.

trace

logical. Trace optimizer output?

start

If given this numeric vector is used as the initial estimate of the GARCH coefficients. Default initialization is to set the GARCH parameters to slightly positive values and to initialize the intercept such that the unconditional variance of the initial GARCH is equal to the variance of x.

grad

character indicating whether analytical gradients or a numerical approximation is used for the optimization.

abstol

absolute function convergence tolerance.

reltol

relative function convergence tolerance.

xtol

coefficient-convergence tolerance.

falsetol

false convergence tolerance.

...

additional arguments for qr when computing the asymptotic covariance matrix.

Details

garch uses a Quasi-Newton optimizer to find the maximum likelihood estimates of the conditionally normal model. The first max(p, q) values are assumed to be fixed. The optimizer uses a hessian approximation computed from the BFGS update. Only a Cholesky factor of the Hessian approximation is stored. For more details see Dennis et al. (1981), Dennis and Mei (1979), Dennis and More (1977), and Goldfarb (1976). The gradient is either computed analytically or using a numerical approximation.

Value

A list of class "garch" with the following elements:

order

the order of the fitted model.

coef

estimated GARCH coefficients for the fitted model.

n.likeli

the negative log-likelihood function evaluated at the coefficient estimates (apart from some constant).

n.used

the number of observations of x.

residuals

the series of residuals.

fitted.values

the bivariate series of conditional standard deviation predictions for x.

series

the name of the series x.

frequency

the frequency of the series x.

call

the call of the garch function.

vcov

outer product of gradient estimate of the asymptotic-theory covariance matrix for the coefficient estimates.

References

A. K. Bera and M. L. Higgins (1993): ARCH Models: Properties, Estimation and Testing. J. Economic Surveys 7 305–362. doi:10.1111/j.1467-6419.1993.tb00170.x .

T. Bollerslev (1986): Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics 31, 307–327. doi:10.1016/0304-4076(86)90063-1 .

R. F. Engle (1982): Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50, 987–1008. doi:10.1093/oso/9780198774310.003.0001 .

J. E. Dennis, D. M. Gay, and R. E. Welsch (1981): Algorithm 573 — An Adaptive Nonlinear Least-Squares Algorithm. ACM Transactions on Mathematical Software 7, 369–383. doi:10.1145/355958.355966 .

J. E. Dennis and H. H. W. Mei (1979): Two New Unconstrained Optimization Algorithms which use Function and Gradient Values. J. Optim. Theory Applic. 28, 453–482. doi:10.1007/BF00932218 .

J. E. Dennis and J. J. More (1977): Quasi-Newton Methods, Motivation and Theory. SIAM Rev. 19, 46–89. doi:10.1137/1019005 .

D. Goldfarb (1976): Factorized Variable Metric Methods for Unconstrained Optimization. Math. Comput. 30, 796–811. doi:10.2307/2005399 .

Author

A. Trapletti, the whole GARCH part; D. M. Gay, the FORTRAN optimizer

See also

summary.garch for summarizing GARCH model fits; garch-methods for further methods.

Examples

n <- 1100
a <- c(0.1, 0.5, 0.2)  # ARCH(2) coefficients
e <- rnorm(n)  
x <- double(n)
x[1:2] <- rnorm(2, sd = sqrt(a[1]/(1.0-a[2]-a[3]))) 
for(i in 3:n)  # Generate ARCH(2) process
{
  x[i] <- e[i]*sqrt(a[1]+a[2]*x[i-1]^2+a[3]*x[i-2]^2)
}
x <- ts(x[101:1100])
x.arch <- garch(x, order = c(0,2))  # Fit ARCH(2) 
#> 
#>  ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 
#> 
#> 
#>      I     INITIAL X(I)        D(I)
#> 
#>      1     2.363190e-01     1.000e+00
#>      2     5.000000e-02     1.000e+00
#>      3     5.000000e-02     1.000e+00
#> 
#>     IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
#>      0    1 -2.325e+02
#>      1    3 -2.767e+02  1.60e-01  1.95e-01  1.7e-01  5.4e+03  1.0e-01  5.25e+02
#>      2    4 -2.960e+02  6.50e-02  8.53e-02  2.3e-01  2.8e+00  1.0e-01  2.47e+00
#>      3    5 -2.987e+02  8.98e-03  1.04e-01  1.7e-01  2.0e+00  1.0e-01  3.66e+02
#>      4    6 -3.149e+02  5.15e-02  4.21e-02  8.5e-02  2.0e+00  5.0e-02  6.29e+01
#>      5    8 -3.161e+02  3.78e-03  5.49e-03  2.6e-02  3.4e+00  1.9e-02  2.03e+01
#>      6    9 -3.175e+02  4.50e-03  5.92e-03  2.6e-02  2.0e+00  1.9e-02  1.43e+01
#>      7   10 -3.199e+02  7.52e-03  9.82e-03  5.3e-02  2.0e+00  3.9e-02  7.96e+00
#>      8   11 -3.222e+02  7.07e-03  1.54e-02  9.6e-02  2.0e+00  7.8e-02  2.03e+00
#>      9   13 -3.229e+02  2.14e-03  6.49e-03  2.7e-02  2.0e+00  3.6e-02  2.13e-01
#>     10   14 -3.237e+02  2.36e-03  3.49e-03  4.0e-02  2.0e+00  3.6e-02  1.75e-01
#>     11   15 -3.239e+02  7.96e-04  1.20e-03  3.0e-02  1.9e+00  3.6e-02  3.78e-02
#>     12   17 -3.240e+02  2.36e-04  4.66e-04  8.1e-03  7.1e+00  1.0e-02  4.45e-02
#>     13   18 -3.241e+02  2.92e-04  6.62e-04  7.8e-03  2.0e+00  1.0e-02  1.02e-01
#>     14   19 -3.242e+02  2.59e-04  4.09e-04  8.6e-03  2.0e+00  1.0e-02  1.73e-02
#>     15   20 -3.242e+02  2.77e-05  1.40e-04  8.7e-03  1.8e+00  1.0e-02  1.57e-03
#>     16   21 -3.242e+02  1.08e-04  1.23e-04  7.6e-03  8.3e-01  1.0e-02  1.67e-04
#>     17   23 -3.242e+02  6.68e-06  2.42e-05  2.8e-03  1.5e+00  3.0e-03  3.36e-05
#>     18   24 -3.242e+02  1.02e-05  1.27e-05  2.9e-03  5.7e-01  3.0e-03  1.40e-05
#>     19   25 -3.242e+02  8.70e-07  1.36e-06  1.1e-03  0.0e+00  1.2e-03  1.36e-06
#>     20   26 -3.242e+02  1.33e-07  1.34e-07  3.3e-04  0.0e+00  3.5e-04  1.34e-07
#>     21   27 -3.242e+02  1.50e-12  1.51e-12  9.0e-07  0.0e+00  1.0e-06  1.51e-12
#> 
#>  ***** RELATIVE FUNCTION CONVERGENCE *****
#> 
#>  FUNCTION    -3.242267e+02   RELDX        8.952e-07
#>  FUNC. EVALS      27         GRAD. EVALS      22
#>  PRELDF       1.509e-12      NPRELDF      1.509e-12
#> 
#>      I      FINAL X(I)        D(I)          G(I)
#> 
#>      1    9.157780e-02     1.000e+00     1.712e-05
#>      2    5.183412e-01     1.000e+00     1.519e-06
#>      3    1.342126e-01     1.000e+00     9.057e-06
#> 
summary(x.arch)                     # Diagnostic tests
#> 
#> Call:
#> garch(x = x, order = c(0, 2))
#> 
#> Model:
#> GARCH(0,2)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -3.0820433 -0.6446973 -0.0002197  0.7076624  3.1693239 
#> 
#> Coefficient(s):
#>     Estimate  Std. Error  t value Pr(>|t|)    
#> a0  0.091578    0.008291   11.045  < 2e-16 ***
#> a1  0.518341    0.069238    7.486 7.08e-14 ***
#> a2  0.134213    0.045741    2.934  0.00334 ** 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostic Tests:
#> 	Jarque Bera Test
#> 
#> data:  Residuals
#> X-squared = 1.1825, df = 2, p-value = 0.5536
#> 
#> 
#> 	Box-Ljung test
#> 
#> data:  Squared.Residuals
#> X-squared = 0.3441, df = 1, p-value = 0.5575
#> 
plot(x.arch)                        





data(EuStockMarkets)  
dax <- diff(log(EuStockMarkets))[,"DAX"]
dax.garch <- garch(dax)  # Fit a GARCH(1,1) to DAX returns
#> 
#>  ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 
#> 
#> 
#>      I     INITIAL X(I)        D(I)
#> 
#>      1     9.549651e-05     1.000e+00
#>      2     5.000000e-02     1.000e+00
#>      3     5.000000e-02     1.000e+00
#> 
#>     IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
#>      0    1 -7.584e+03
#>      1    8 -7.585e+03  1.45e-05  2.60e-05  1.4e-05  1.0e+11  1.4e-06  1.35e+06
#>      2    9 -7.585e+03  1.88e-07  1.97e-07  1.3e-05  2.0e+00  1.4e-06  1.50e+00
#>      3   18 -7.589e+03  6.22e-04  1.10e-03  3.5e-01  2.0e+00  5.5e-02  1.50e+00
#>      4   21 -7.601e+03  1.58e-03  1.81e-03  6.2e-01  1.9e+00  2.2e-01  3.07e-01
#>      5   23 -7.634e+03  4.22e-03  3.55e-03  4.3e-01  9.6e-01  4.4e-01  3.06e-02
#>      6   25 -7.646e+03  1.61e-03  1.85e-03  2.9e-02  2.0e+00  4.4e-02  5.43e-02
#>      7   27 -7.646e+03  3.82e-05  5.23e-04  1.3e-02  2.0e+00  2.0e-02  1.46e-02
#>      8   28 -7.648e+03  1.86e-04  1.46e-04  6.5e-03  2.0e+00  9.9e-03  1.54e-03
#>      9   29 -7.648e+03  3.12e-05  4.83e-05  6.4e-03  2.0e+00  9.9e-03  3.34e-03
#>     10   30 -7.648e+03  1.39e-05  6.31e-05  6.2e-03  1.9e+00  9.9e-03  1.86e-03
#>     11   31 -7.650e+03  2.70e-04  3.24e-04  6.0e-03  1.9e+00  9.9e-03  4.99e-03
#>     12   34 -7.656e+03  8.42e-04  8.57e-04  2.2e-02  1.7e-01  3.9e-02  2.22e-03
#>     13   36 -7.661e+03  6.12e-04  6.40e-04  1.9e-02  4.2e-01  3.9e-02  2.09e-03
#>     14   38 -7.665e+03  4.87e-04  8.63e-04  4.9e-02  4.1e-01  9.6e-02  9.69e-04
#>     15   48 -7.666e+03  1.02e-04  1.86e-04  1.9e-07  4.5e+00  3.5e-07  3.94e-04
#>     16   49 -7.666e+03  1.12e-07  1.01e-07  1.9e-07  2.0e+00  3.5e-07  6.22e-05
#>     17   57 -7.666e+03  1.60e-05  2.70e-05  2.0e-03  9.3e-01  3.7e-03  6.10e-05
#>     18   59 -7.666e+03  5.23e-06  7.01e-06  3.7e-03  3.9e-01  8.0e-03  7.77e-06
#>     19   60 -7.666e+03  4.08e-08  3.74e-08  1.4e-04  0.0e+00  3.1e-04  3.74e-08
#>     20   61 -7.666e+03  2.31e-09  8.57e-10  8.6e-06  0.0e+00  2.0e-05  8.57e-10
#>     21   62 -7.666e+03  5.35e-11  2.25e-13  7.6e-07  0.0e+00  1.6e-06  2.25e-13
#>     22   63 -7.666e+03  1.81e-12  7.06e-16  1.7e-08  0.0e+00  3.4e-08  7.06e-16
#>     23   64 -7.666e+03  7.00e-14  1.69e-17  1.0e-09  0.0e+00  2.4e-09  1.69e-17
#>     24   65 -7.666e+03 -1.16e-14  1.76e-20  1.9e-10  0.0e+00  4.0e-10  1.76e-20
#> 
#>  ***** X- AND RELATIVE FUNCTION CONVERGENCE *****
#> 
#>  FUNCTION    -7.665775e+03   RELDX        1.874e-10
#>  FUNC. EVALS      65         GRAD. EVALS      24
#>  PRELDF       1.760e-20      NPRELDF      1.760e-20
#> 
#>      I      FINAL X(I)        D(I)          G(I)
#> 
#>      1    4.639289e-06     1.000e+00    -2.337e-02
#>      2    6.832875e-02     1.000e+00    -8.294e-07
#>      3    8.890666e-01     1.000e+00    -2.230e-06
#> 
summary(dax.garch)       # ARCH effects are filtered. However, 
#> 
#> Call:
#> garch(x = dax)
#> 
#> Model:
#> GARCH(1,1)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -12.18398  -0.47968   0.04949   0.65746   4.48048 
#> 
#> Coefficient(s):
#>     Estimate  Std. Error  t value Pr(>|t|)    
#> a0 4.639e-06   7.560e-07    6.137 8.42e-10 ***
#> a1 6.833e-02   1.125e-02    6.073 1.25e-09 ***
#> b1 8.891e-01   1.652e-02   53.817  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Diagnostic Tests:
#> 	Jarque Bera Test
#> 
#> data:  Residuals
#> X-squared = 12947, df = 2, p-value < 2.2e-16
#> 
#> 
#> 	Box-Ljung test
#> 
#> data:  Squared.Residuals
#> X-squared = 0.13566, df = 1, p-value = 0.7126
#> 
plot(dax.garch)          # conditional normality seems to be violated