imsl.timeseries.garch

garch(p, q, series, guess, max_sigma=10.0)

Compute parameter estimates of a GARCH model.

Parameters:

p : int

Number of GARCH parameters.

q : int

Number of ARCH parameters.

series : (N,) array_like

Array containing the observed time series data.

guess : (p+q+1,) array_like

Array containing the initial values for the parameter array x.

max_sigma : float, optional

Value of the upper bound on the first element (sigma) of the array x of returned estimated coefficients.

Default: max_sigma = 10.

Returns:

A named tuple with the following fields:

x : (p+q+1,) ndarray

Array containing the estimated values of sigma squared, followed by the q ARCH parameters and the p GARCH parameters.

log_likeli : float

Value of the log-likelihood function evaluated at the estimated parameter array x.

aic : float

Value of the Akaike Information Criterion (AIC) evaluated at the estimated parameter array x.

var : (p+q+1, p+q+1) ndarray

Array of size (p+q+1) \(\times\) (p+q+1) containing the variance-covariance matrix.

Notes

The Generalized Autoregressive Conditional Heteroskedastic (GARCH) model for a time series \({w_t}\) is defined as

\[w_t = z_t \sigma_t\]

\[\sigma_t^2 = \sigma^2 + \sum_{i=1}^p \beta_i \sigma_{t-i}^2+ \sum_{i=1}^q \alpha_i w_{t-i}^2 ,\]

where \(z_t\) is a sequence of independent and identically distributed standard normal random variables,

\[\begin{split}0<\sigma^2< \text{max_sigma}, \beta_i \ge 0, \alpha_i \ge 0, \; \text{and} \\ \sum_{i=2}^{p+q+1} x_i = \sum_{i=1}^p \beta_i + \sum_{i=1}^q \alpha_i < 1.\end{split}\]

The above model is denoted as GARCH(p, q). The \(\beta_i\) and \(\alpha_i\) coefficients are referred to as GARCH and ARCH coefficients, respectively. When \(\beta_i=0, i=1,2,\ldots,p\), the above model reduces to ARCH(q) which was proposed by Engle ([R36]). The nonnegativity conditions on the parameters imply a nonnegative variance and, the condition on the sum of the \(\beta_i\)‘s and \(\alpha_i\)‘s is required for wide sense stationarity.

In the empirical analysis of observed data, GARCH(1,1) or GARCH(1,2) models have often found to appropriately account for conditional heteroskedasticity ([R37]). This finding is similar to linear time series analysis based on ARMA models.

Note that for the above models, positive and negative past values have a symmetric impact on the conditional variance. In practice, many series may have a strong asymmetric influence on the conditional variance. To take into account this phenomena, Nelson ([R38]) put forward exponential GARCH (EGARCH). Lai ([R39], [R40], [R41]) proposed and studied some properties of a general class of models that extended the linear relationship of the conditional variance in ARCH and GARCH into a nonlinear relationship.

The maximum likelihood method is used in estimating the parameters in GARCH(p, q). The log-likelihood of the model for the observed series \({w_t}\) with length m is

\[\begin{split}\log(L) = -\frac{m}{2}\log(2\pi)-\frac{1}{2}\sum_{t=1}^my_t^2/ \sigma_t^2 -\frac{1}{2}\sum_{t=1}^m \log(\sigma_t^2), \\ \text{where}\quad \sigma_t^2 = \sigma^2+\sum_{i=1}^p\beta_i \sigma_{t-i}^2+\sum_{i=1}^q\alpha_iw_{t-i}^2\end{split}\]

Thus, \(\log(L)\) is maximized subject to the constraints on the \(\alpha_i, \beta_i\), and \(\sigma\).

In this model, if q = 0, the GARCH model is singular since the estimated Hessian matrix is singular.

The initial values of the parameter vector x entered in vector guess must satisfy certain constraints. The first element of guess refers to \(\sigma^2\) and must be greater than zero and less than max_sigma. The remaining p+q initial values must each be greater than or equal to zero and sum to a value less than one.

To guarantee stationarity in model fitting,

\[\begin{split}\sum_{i=2}^{p+q+1}x_i = \sum_{i=1}^p \beta_i + \sum_{i=1}^q \alpha_i < 1\end{split}\]

is checked internally. The initial values should be selected from values between zero and one.

AIC is computed by

\[-2 \log(L) + 2 (p+q+1),\]

where \(\log(L)\) is the value of the log-likelihood function.

Statistical inferences can be performed outside the function GARCH based on the output of the log-likelihood function (log_likeli), the Akaike Information Criterion (aic), and the variance-covariance matrix (var).

References

[R36]

(1, 2) Engle, C. (1982), Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation, Econometrica, 50, 987-1008.

[R37]

(1, 2) Palm, F. C. (1996), GARCH models of volatility. In Handbook of Statistics, Vol. 14, 209-240. Eds: Maddala and Rao. Elsevier, New York.

[R38]

(1, 2) Nelson, Peter (1989), Multiple Comparisons of Means Using Simultaneous Confidence Intervals, Journal of Quality Technology, 21, 232-241.

[R39]

(1, 2) Lai, D. (1998), Local Asymptotic Normality for Location-Scale Type Processes, Far East Journal of Theorectical Statistics, Vol. 2, 171-186.

[R40]

(1, 2) Lai, D. (1999), Asymptotic distributions of the correlation integral based statistics, Journal of Nonparametric Statistics, 10(2), 127-135.

[R41]

(1, 2) Lai, D. (2000), Asymptotic distribution of the estimated BDS statistic and residual analysis of AR Models on the Canadian lynx data, Journal of Biological Systems, Vol. 8, 95-114.

Examples

The data for this example are generated to follow a GARCH(p,q) process by using a random number generation function sgarch. The data set is analyzed and estimates of sigma, the ARCH and GARCH parameters, the log-likelihood and the AIC are returned.

>>> import numpy as np
>>> from imsl.timeseries.garch import garch
>>> def sgarch(p, q, m, x, y, z, y0, sigma):
...     k = max(1, p, q)
...     for i in range(0, k):
...         y0[i] = z[i] * x[0]
...     # Compute the initial value of sigma
...     s3 = 0.0
...     if max(p, q) >= 1:
...         for i in range(1, p+q+1):
...             s3 += x[i]
...     for i in range(k):
...         sigma[i] = x[0] / (1.0 - s3)
...     for i in range(k, m+1000):
...         s1 = 0.0
...         s2 = 0.0
...         if q >= 1:
...             for j in range(q):
...                 s1 += x[j + 1] * y0[i - j - 1] * y0[i - j - 1]
...         if p >= 1:
...             for j in range(p):
...                 s2 += x[q + 1 + j] * sigma[i - j - 1]
...         sigma[i] = x[0] + s1 + s2
...         y0[i] = z[i] * np.sqrt(sigma[i])
...     # Discard the first 1000 simulated observations
...     y[0:m] = y0[1000:1000+m]
>>>
>>> np.random.seed(182198625)
>>> m = 1000
>>> p = 2
>>> q = 1
>>> wk1 = np.random.normal(size=m + 1000)
>>> wk2 = np.empty(m+1000, dtype=np.float64)
>>> wk3 = np.empty(m+1000, dtype=np.float64)
>>> y = np.empty(m, dtype=np.float64)
>>> x = np.array([1.3, 0.2, 0.3, 0.4], dtype=np.float64)
>>> guess = np.array([1.0, 0.1, 0.2, 0.3], dtype=np.float64)
>>>
>>> sgarch(p, q, m, x, y, wk1, wk2, wk3)
>>>
>>> result = garch(p, q, y, guess)
>>>
>>> print("Sigma estimate is {0:11.4f}".format(result.x[0]))
... 
Sigma estimate is      1.3083
>>> print("ARCH(1) estimate is {0:11.4f}".format(result.x[1]))
... 
ARCH(1) estimate is      0.1754
>>> print("GARCH(1) estimate is {0:11.4f}".format(result.x[2]))
... 
GARCH(1) estimate is      0.3519
>>> print("GARCH(2) estimate is {0:11.4f}".format(result.x[3]))
... 
GARCH(2) estimate is      0.3477
>>> print("\nLog-likelihood function value is {0:11.4f}".format
...       (result.log_likeli)) 
Log-likelihood function value is  -2558.5405
>>> print("Akaike Information Criterion value is {0:11.4f}".format
...       (result.aic)) 
Akaike Information Criterion value is   5125.0810
>>> print("\n    Variance-covariance matrix")
... 
    Variance-covariance matrix
>>> np.set_printoptions(precision=2)
>>> print(str(result.var)) 
[[   73.91   481.41   657.06   663.43]
 [  481.55  4823.93  5019.15  5040.94]
 [  657.06  5018.17  6448.31  6523.42]
 [  663.44  5039.96  6523.44  6645.81]]