The assumption of constant variance in the traditional time series models of
ARMA is a major impediment to their applications in financial time series data
where heteroscedasticity is obvious and cannot be neglected. To solve the stated
problem, Engle (1982) proposed Autoregressive Conditional
Heteroscedascity (ARCH) model. However, Engle in his first application of ARCH
noted that a high order of ARCH is needed to satisfactorily model time varying
variances. It is noted that many parameters in ARCH will create convergence
problems for maximization routines. To avoid these problems, Bollerslev
(1986) extended Engles model to Generalized Autoregressive
Conditional. Heteroscedasticity models (GARCH).
This models time-varying variances as a linear function of past square residuals and of its past value. It has proved useful in interpreting volatility clustering effects and has wide acceptance in measuring the volatility of financial markets. The ARCH and GARCH models are known as symmetric models. Other extensions based on observed characteristic of financial time series data are:
||The asymmetric models of which the exponential GARCH (EGARCH)
model of Nelson (1991), the model of Gosten
et al. (1993) (GJR-GARCH) of as well as the threshold model (T
GARCH) of Zakoian (1994) are representatives models.
These modes and interpret leverage effect where volatility is negatively
correlated with returns
||The Fractionally Integrated GARCH model (FIGARCH) of Baillie
et al. (1996 ) introduced to model long memory via the fractional
||The GARCH in mean models that allows the mean to influence
These models are popularly estimated by the Quasi-maximum Likelihood Method
(QMLE) under the assumption that the distribution of one observation conditionally
to the past is normal. The asymptotic properties of the estimator are well established.
Weiss (1986) showed that the QMLE estimates are consistent
and asymptotically normal under the fourth moment conditions.
These were again proved by Ling and Mcaleer (2003),
under only the second moment conditions. If the assumption of normality is satisfied
by the data then the method will produce efficient estimates otherwise inefficient
estimates will be produced. Engle and Gonzalez-Revera (1991)
studied the loss of estimation efficiency inherent in QMLE and concluded it
may be severe if the distribution density is heavy tailed.
The QMLE estimator requires the use of numerical optimization procedure which
depends on different optimization techniques for implementation. This potentially
leads to different estimates. This is confirmed by recent studies by Brooks
et al. (2001) and McCullough and Renfro (1999).
Both reported different QMLE estimates across various packages using different
optimization routines. These techniques estimates time varying variances in
different ways and may result to different interpretations and predictions with
varying implications to the economy. It is therefore important to undertake
studies that would develop appropriate techniques for estimating parameters
of processes used in modeling time series data.
To solve the stated problems, Eni and Etuk (2006) developed
an Autocovariance Base Estimator (ABE) for estimating the parameters of GARCH
models through an ARMA transformation of the GARCH model equation.The thrust
of this study is to rate the performance of the Autocovariance Base Estimator
when the normality assumption is violated.
The Autocovariance Base Estimator (ABE): Consider the GARCH (p, q) equation:
Or its ARMA (Max (p, q),q) transform:
To obtain the autoregressive parameters, we take advantage of the fact that the variance, var (ε2t ε2t-1) for i>q in Eq. 2 will contain no moving average parameter B1. Hence we set i = q + 1
q + p to get the estimator:
Where Vi is the set of variances associated with Eq.
2. We can easily obtain the autoregressive parameters αi
+ Bi by solving Eq. 3. Eni
and Etuk (2006) have shown that the moving average parameters can be obtained
Note that the quantity
is already known. The variance V having been calculated from the data and the
autoregressive parameters having been calculated from Eq. 3.
We find the moving average parameters Bi by solving the system:
Eqution 5 is nonlinear and the solution can be found only through an iterative method. A ready procedure to consider is the one which depends on the Newton-Raphson algorithm. In this case, the Br+1 solution is obtained from the rth approximation according to:
where, f (Br) and f (Br) represent the vector Eq. 5 and its derivative evaluated at B = Br. We note that:
So that (4.12) becomes:
The starting point for the iteration Eq. 8 is σ2r = 1, B0 = V0, Bi = 0, i = 1
Having computed the Autoregressive parameters Φi = (αi + Bi) and the Moving average parameter Bi, it is easy to obtain the GARCH (p, q) parameters αi and the constant parameter w0 which is estimated using:
MATERIALS AND METHODS
In this study, the Data Generating Process (DGP) involves the simulation of
1,500 data points with 10 replications using the random number generator in
MATLAB 5. The random number generator in MATLAB 5 can generate all the floating
point numbers in the interval [2-53, 1-2-53]. Hence it
can generate 21492 values before repeating itself. We note that the
data points of 1,500 are equivalent to 210.55 and with 10 replications;
we will have a mere 213.87267 data points. Hence the 1,500 data points
with 10 replications were obtained without repetitions. Also, we used a program
implementation for ARMA due to McLeod and Sales (1983)
to find the QMLE. Although, we would assume Normality, we actually simulated
the data points using Log normal distribution and the t-distribution with degree
of freedom of 5,10 and 15. Of the 1,500 data points generated for each of the
process, the first 200 observations were discarded to avoid initialization effects,
yielding a sample size of 12000 observations. The results are reported in sample
sizes of 200, 500, 1000 and 1,200.
These sample presentations are to enable us keep track of consistency and efficiency
of the estimators. The relative efficiency of the Autocovariances Based Estimator
(ABE) and the Quasi-maximum Likelihood (QML) estimators were studied under this
misspecification of distribution function. The selection criteria used is the
Aikake Information Criteria (AIC). For simulating the data points, the conditional
variance equation for low persistence due to Engle and Ng
(1993) is adopted:
and Z2t is any of
where, N = normality, tV = t-distribution with V degree of freedom, LN = Log normal.
RESULTS AND DISCUSSION
Apart from the parameter setting in the DGP, selected studies of the paramete
settings (W, α, B) = (0.1, 0.15, 0.85) and (W, α, B) = (0.1, 0.25,
0.65) due to Lumsdain (1995) as well (W, α, B)
= (1, 0.3, 0.6) and (W, α, B) = (1, 0.05, 0.9) due to Yi-Ting
(2002) where also studied and the results obtained are in agreement with
the result obtained from detail studies of the DGP. The results obtained from
the DGP are shown in Table 1.
Table 1 shows the result under a sample size of 200 data points. The Table 1 reveals that the estimates are poor for QMLE and ABE. However, on the bases of the Aikate Information Criteria (AIC), the QMLE performed better than the ABE except under log normal distribution where ABE performed better than the QMLE.
A study of Table 2 showed that the estimates are better although
still poor. The performance bridge between QMLE and ABE is closing. This can
be seen from the AIC of QMLE and ABE under the different probability distribution
functions except in the case of the log normality. Here and surprisingly too
the QMLE method failed to show consistency.
|| Performance rating of QMLE and ABE for sample size of 200
|| Performance rating of QMLE and ABE Estimates under a sample
size of 500
|| Performance rating of QMLE and ABE estimates under sample
size of 1000
|| Performance rating of QMLE and ABE estimates under sample
size of 1,200
We also note that the performance of the two method are enhanced under the
t-distribution as the degree of freedom increases. An examination of Table
3 shows that both estimation models that is the QMLE and the ABE have equal
performance ratings. However, the ABE has an edge in its performance under t(5)
and LN (0,1) while QMLE has an edge under t(10) and t(15).
The estimates under t(15) and t(10) are close to their
true values for both estimation methods. Generally, the two methods gave consistent
The result in Table 3 is further confirmed by examining Table 4 where the two methods have nearly equal rating judging from the values of their AIC. This is inspite of its very poor performance at the sample size of 200.
The study in this section shows that the ABE method is adequate in estimating GARCH model parameters and can perform as well as the maximum likelihood estimate for reasonable large data point when the distribution assumption is miss-specified.