Question

B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0 and 2IV2 t-1't-2 ..

Answer 1

If an autoregressive moving average model (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.^[2]

In that case, the GARCH (p, q) model (where p is the order of the GARCH terms $~\sigma ^{2}$ and q is the order of the ARCH terms $~\epsilon ^{2}$ ), following the notation of the original paper, is given by

${\displaystyle y_{t}=x'_{t}b+\epsilon _{t}}$

${\displaystyle \epsilon _{t}|\psi _{t-1}\sim {\mathcal {N}}(0,\sigma _{t}^{2})}$

${\displaystyle \sigma _{t}^{2}=\omega +\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}+\beta _{1}\sigma _{t-1}^{2}+\cdots +\beta _{p}\sigma _{t-p}^{2}=\omega +\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}+\sum _{i=1}^{p}\beta _{i}\sigma _{t-i}^{2}}$

Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, this means to test for ARCH and GARCH errors.

Exponentially weighted moving average (EWMA) is an alternative model in a separate class of exponential smoothing models. As an alternative to GARCH modelling it has some attractive properties such as a greater weight upon more recent observations, but also drawbacks such as an arbitrary decay factor that introduces subjectivity into the estimation.

GARCH(p, q) model specification

The lag length p of a GARCH(p, q) process is established in three steps:

1. Estimate the best fitting AR(q) model

   $y_{t}=a_{0}+a_{1}y_{{t-1}}+\cdots +a_{q}y_{{t-q}}+\epsilon _{t}=a_{0}+\sum _{{i=1}}^{q}a_{i}y_{{t-i}}+\epsilon _{t}$ .

2. Compute and plot the autocorrelations of $\epsilon ^{2}$ by

   $\rho ={{\sum _{{t=i+1}}^{T}({\hat \epsilon }_{t}^{2}-{\hat \sigma }_{t}^{2})({\hat \epsilon }_{{t-1}}^{2}-{\hat \sigma }_{{t-1}}^{2})} \over {\sum _{{t=1}}^{T}({\hat \epsilon }_{t}^{2}-{\hat \sigma }_{t}^{2})^{2}}}$

3. The asymptotic, that is for large samples, standard deviation of $\rho (i)$ is $1/{\sqrt {T}}$ . Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use the Ljung-Box test until the value of these are less than, say, 10% significant. The Ljung-Box Q-statistic follows $\chi ^{2}$ distribution with n degrees of freedom if the squared residuals $\epsilon _{t}^{2}$ are uncorrelated. It is recommended to consider up to T/4 values of n. The null hypothesis states that there are no ARCH or GARCH errors. Rejecting the null thus means that such errors exist in the conditional variance.

2 GARCH(l,1) process Definition 2.1 Le Zn) be a sequence of i.i.d. random variables such that ZN(0,1 (Xi) is called the generalized autoregressive conditionally heteroskedastic or GARCH(g.p) process if whe re (a) is a nonmegative proces such that tEz t-p and ao > 0, a' > 0-1, . . . ,q 属-0-1, . . . , p. (2.3) The conditions on parameters ensure strong positivity of the conditional variance (22 If we write the equation (2.2) in terms of the lag-operator B we get (2.4) where and (2.5)

If the roots of the characteristic equation, i.e lie outside the unit circle and the process (Xt) is stationary, then we can write (2.2) as 1-B() 1- B(B) (2.6) a0 where ao1-1) and δί are coefficients of B' in the expansion of a(B)II-3(B)]-1 Note that the expression (2.6) tells us that the GARCH(q.p) process is an ARCH process of infinite order with a fractional structure of the coefficients. From (2.1) it is obvious that the GARCH(1,1) process is stationary if the process (d) is stationary. So if we want to study the properties and higher order moments of GARCH(L.1) process it is enough to do so for the process ( The following theorem gives us the main result for stochastic difference equations that we are going to use in order to establish the stationarity of the process ( .