For many financial and economic time series, valuable information about variance is contained in the past term implying that (Ψt | Ψt1 ). For many purposes, conditional forecasts therefore are preferable to unconditional forecasts (Enders 2010). In the following, the influential ARCH and its generalized equivalent will be presented.
3.3.1 The ARCH model
Engle (1982) introduced his autoregressive conditional heteroskedastic (ARCH) model by which the dynamic dependency in variance can be exploited19.
“For real processes one might expect better forecast intervals if additional information from the past were allowed to affect the forecast variance…”
Engle (1982)
In a forecasting perspective, the mean and variance of a volatility model at time t is given as (3.14) µt=E r(t Ψt−1)
and
(3.15) σt2=var(rtΨt−1)=E r((t−µt)2 Ψt−1)
meaning that the conditional expected return as well as the expected squared deviation from the mean are functions of the information set Ψt-1.
18 The issue of integration relates to a number of interesting topics in econometrics outside the scope of this thesis.
19 In Engle (1982) the class of models is introduced and applied to a time series of inflation demonstrating vast superiority over the unconditional measures.
By letting the conditional variance be parameterized by information in Ψ, the model turns heteroskedastic. Engle defined the properties of the ARCH model using the following notation,
(3.16) Yt Ψt−1N x h( tβ, )t
(3.17) ht= +α α ε0 1 t2−1...+α εp t p2−
(3.18) εt= −Y xt tβ
where Ψ denotes the information set available at time t and β is a vector of unknown para-meters constituting the mean of Yt so that εt expresses the shock or innovation at time t (Engle 1982). In its simplest multiplicative form, Engle proposed a specification for εt so that
(3.19) εt =υt ht
where υ is a white noise process at time t with βt2 = 1. Due to the effect of υ it follows that E(εt)=0 and that εt to εtp are serially uncorrelated while dependent in their second moment.
For the ARCH(1) process this implies that
(3.20) E r((t−µt)2Ψt−1)=E(ε ε εt2 t−1, t−2...)=α0+α ε1 t2−1
whereby a large realized shock in t-1 will be reflected in the conditional variance at t (Enders 2010). Whereas returns can be negative as well as positive, only positive values make sense for variances. This restricts the sum of the parameters α0, α1,… αp to be positive. The stationarity of the processes is assured by restricting the parameters so that 0 ≤ ∑qi=1 αi ≤ 1.
In effect, since α1,… αp cannot be negative, the minimum value of α0+α1ε2t1,…+αpε2tp is zero.
If αi = 0, the term is non-existing, which for the ARCH(1) model, means that no ARCH effects are present in the data.
The ARCH models can be estimated using varying laglengths denoted p so that the predicted volatility ht at time t depends on the parameterized squared shocks in α1 through αp in equa
tion 3.17. Inserting the ARCH(p) into the specification for εt in equation 3.19 it is found that
(3.21) εt= α0+
∑
iq=1α εi t i2−3.3.1.1 Weaknesses of the ARCH model
Despite the Nobel Prize awarded to Engle20, the ARCH model has some weaknesses with importance for practical application on financial data.
First, as the ARCH model is purely descriptive it provides no guidance as to the causes of the behavior of the data. Nwogugu (2006) note that as such, the ARCH class models are naïve as they assume that volatility can be explained solely through mechanical descriptive analysis, ignoring other sources of volatility such as liquidity, psychology or legal issues21.
Secondly, the ARCH models assume symmetry in reactions to positive and negative shocks.
This follows from the structure of the model by which it reacts to the square of the previous period’s realizations thereby analyzing the residuals as absolute figures. As described in Section 2.1.3 and 2.1.5, this may be of importance as a difference is expected ex ante.
Third, as the ARCH models is a short memory specification, a large number of estimators may be needed in practice, which gives rise to high data requirements, and fourth, there is a risk that the deviations may be over predicted due to the inertia in the models reaction to large isolated shocks (Tsay 2005).
3.3.2 The Generalized ARCH model
By letting the conditional variance process mimic an ARMA process, Bollerslev (1986) intro
duced the generalized ARCH (GARCH) model. This model has been shown to accommodate financial time series well especially volatility clustering and excess kurtosis.
The error process follows the definition from equation 3.21 where still υt represents a white noise process with σ2t = 1. Yet, the heteroskedastic variance process ht in the generalized ARCH is revised to encompass a moving average term. The conditional variance thus depends on lagged squared residuals as well as lagged estimates of variance so that
(3.22)
ht i t i i t ih
i q i
=α0+
∑
p=1α ε2− +∑
=1β −20 Robert F. Engle received the Nobel prize in Economics sciences 2003 “for methods of analyzing economic time series with time-varying volatility (ARCH)” (Web: nobelprize.org 2010)
21 While this is true, the incorporation of independent variables in the model specifications may provide a valid remedy to such objections.
Responding to the criticism of Engle´s model, the advantages of the generalized version are clear. While the generalized volatility model contain q + p + 1 parameters rather than p+1 in the ARCH model, the moving average terms p, allow for a longer memory in a more parsi-monious representation. Similarly, a simple version such as the well-known GARCH(1,1) can be shown to mimics the behavior of an ARCH(∞) model whereby all previous residuals contribute to the parameterization of the volatility at time t (Anderson et al 2009). This makes it easier to identify and estimate and often leads to a less restricted specification22 (Enders 2010).
To verify the existence and stability of the variance, the restrictions 0 ≤ αi + βi < 1 must be fulfilled. If αi + βi > 1 the variance is covariance nonstationary and our models may fail to correctly assess future period´s data23 (Bollerslev et al 1992). In general, αi + βi expresses the persistence of the model, that is, how long a shock to the conditional variance remains in the data. In the GARCH(1,1) model, it is clear that larger values of αi leads to greater volatility in the forecasted errors, while high values of βi indicate higher persistence.
That a low number of parameters are usually sufficient in explaining the second moment of financial time series has been empirically backed by a vast academic literature. Bollerslev et al. (1992) review this and find that in most research the simple GARCH(1,1), GARCH(1,2) and GARCH(2,1) were the most adopted specifications. This is also the case with large samples over large time scales. French et al (1987) successfully applied a GARCH(2,1) model using daily S&P returns to calculate monthly standard deviations in a time period from 1928 to 1984. For many purposes, the simple GARCH(1,1) have proven sufficient for modeling volatility in stock returns, interest rates and foreign exchanges (Bollerslev et al 1992, Anderson et al 2009). Hansen and Lunde (2001) tested 330 different volatility model specifications24 on daily returns in currencies and concluded that no specifications could be shown to significantly outperform the simple GARCH(1,1)25.
In accommodating the stylized facts described in Section 2, the GARCHtype models are ef
fective by allowing for the volatility clusters frequently observed. This is the case because a high value of ε2ti or hti will result in a high ht, whereby the pattern of high volatility following
22 Because it can be specified using less parameters.
23 That is, the conditional variance forecasts will not converge to their unconditional variance.
24 Formally 55 different specifications but tested using different error distributions and mean equations.
25 When the ARCH(1,1) model was used as benchmark it was clearly dominated by a wide range of models.
high volatility and low following low is generated. Of importance is also that the GARCH models have been shown to capture the leptokurtic characteristics as previously described.
This is the case for most GARCHtype models given that βi > 0 (Tsay 2005)26.
3.3.2.1 Weaknesses of the GARCH models
The GARCH models are criticized for imposing restrictions to the parameters that sub
sequently are violated during estimations. Especially the restrictions that αi ≥ 0 and βi ≥ 0 are often violated in practice leading to disqualification of the specification (Nelson 1991).
Like the ARCH model, the GARCH model is criticized for not giving any inside as to the sources of variance. The modeling process is still mechanical and purely descriptive or in other words, the models are statistical rather than economic. Also, similarly to the ARCH model, the pure GARCH model is unable to capture asymmetric effects.
