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Estimation and Inference with the Efficient Method of Moments: With
Applications to Stochastic Volatility Models and Option Pricing
van der Sluis, P.J.
Publication date
1999
Link to publication
Citation for published version (APA):
van der Sluis, P. J. (1999). Estimation and Inference with the Efficient Method of Moments:
With Applications to Stochastic Volatility Models and Option Pricing. Thela Thesis. TI
Research Series nr. 204.
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Chapter 7
Forecasting Volatility
This chapter evaluates the performance of volatility forecasting based on stochastic volatility (SV) models. We show that the choice of squared asset-return residuals as proxy of ex-post volatility directly leads to extremely low explanatory power in the common regression analysis of volatility forecasting. We argue that since the measure of volatility is always model dependent, the performance of volatility forecasting should be evaluated in a consistent model framework. This chapter is based on Jiang and van der Sluis (1998a).
The plan of this chapter is as follows. Section 7.1 contains an introduction. Section 7.2 successively considers: a description of the data, the structural mod-els, and results on estimation and reprojection. Section 7.3 evaluates forecasting performance of reprojected volatility when alternative measures of ex-post volati-lity are used. Section 7.4 concludes.
7.1 Introduction
As volatility plays so important a role in financial theory and financial markets, accurate measures and good forecasts of future volatility are critical for the imple-mentation of asset- and derivative-pricing theories as well as trading and hedging strategies. Empirical findings, dating back to Mandelbrot (1963) and Fama (1965), suggest that financial asset returns display pronounced volatility persistence and clustering, see also Section 2.1.3. Various recent studies based on standard time-series models, such as ARCH/GARCH and SV models, also report a very high de-gree of intertemporal volatility persistence in financial time series, see e.g. Boller-slev et al. (1992), BollerBoller-slev et al. (1994), Ghysels et al. (1996) and Shephard (1996a) for surveys. Such high degree of volatility persistence, coupled with sig-nificant parameter estimates of the model, suggests that the underlying volatility of asset returns is highly predictable. However, empirical studies based on standard
ARCH/GARCH models have reported results suggesting otherwise, i.e. the per-formance of volatility forecasting based on ARCH/GARCH models is very poor, measured by standard econometric techniques. Andersen and Bollerslev (1998) provide a rationale why these standard techniques are not adequate for measuring forecasting performance of volatility models.
In this chapter we argue that since the measures of underlying stochastic vola-tility are always model-dependent, the evaluation of volavola-tility forecasting per-formance should be based on a consistent model framework. We illustrate in a stochastic volatility model framework that the squared asset-return residuals are poor proxies of the underlying volatility. We further show that the squared asset-return residuals do not even provide a consistent estimator of the underlying vola-tility when the conditional mean of asset return is mis-specified. The analytical re-sults derived from the standard stochastic volatility models offer clear explanation to the well documented low explanatory power in the common regression analy-sis of volatility forecasting when squared asset return residuals are used as proxy of ex-post underlying volatility. This chapter further evaluates the performance of volatility forecasting based on reprojected underlying volatility series from esti-mated SV models. We illustrate that the reprojected underlying volatility series offer a much better measure of stochastic volatility and much better performance of volatility forecasting. Moreover, we show that the performance of volatility fore-casting based on multivariate SV (MSV) models can improve over the univariate SV models due to the co-movements of asset return volatility. Up till now, most of the commercially available risk management software assumes volatility to be purely random rather than clustered or persistent in any pattern. In this chapter we show that there is in fact a lot to be gained from using the reprojected volatility series as a measure of stochastic volatility and in forecasting future volatility.
7.2 Data, Models and Estimation
7.2.1 Data and Notation
The data consists of the daily returns of four technology stocks (3Com, Applied Material, Cisco, and Oracle which are all traded in Nasdaq) over the period from February 16,1990 to January 5,1997 with 1,821 return observations. Since part of the data will be treated as ex-post in volatility forecasting, we introduce the follow-ingnotation. We denote the whole sampling period as [1, T] where T = 1,821,and further partition the whole period into sub-periods [1, £i], [ti + 1 , t2] and [t2 +1, T] where ti = 1, 321, t2 = 1, 571. That is, the last two sub-periods both contain 250 observations, roughly over a one-year sample period. When the model is estimated based on observations over the sample period [l,h] and is used to project and
fore-1.2. DATA, MODELS AND ESTIMATION 163 cast future volatility, data observed during the period of [ti + 1 , t2] is viewed as
ex-post information. Similarly, when the model is estimated based on observations over sample period [1, t2\ and is used to project and forecast future volatility, data observed during the period of [t2 +1, T] is viewed as ex-post information. In other words, the SV model is only to be (re-)estimated once for every year in our appli-cation for the purpose of volatility forecasting.
7.2.2 Models
We will consider the univariate and multivariate SV that were discussed in Chapter 2 for modelling the de-meaned returns yt on the asset St
yt = 100xAlnSt-pt (7.1)
The univariate model will be the Gaussian ASARMAV(1,0) given in (2.13) to (2.15) and the multivariate model is the AMSV(l) model given in (2.26) to (2.28). The stochastic trend in this chapter is defined as /it = /J,0 + p^A In St-i, i.e. the return process is assumed to be an AR(1). The parameter estimates of HI are sig-nificantly different from zero for all four series. The pre-whitened returns are dis-played in Figure 7.1.
For estimation we use as a score generator the MEGARCH(1,1) model de-fined in (3.30) to (3.32) as motivated by Monte Carlo results in Section 3.4. From plots of the data we observe that there is probably some time-homogeneous non-Gaussianity in the data. In both the class of stochastic volatility models and in the auxiliary EGARCH model this aspect is not modelled. The consequence is that mis-specification tests based on the parametric MEGARCH model with Kx = Kz = 0 will have no power against such alternatives. The advantage is that the small sample properties of the estimation are improved under the null of a Gaus-sian SV model. This is shown in Monte Carlo results in Chapter 3.
In the application here we observe that the cross-coefficients p\2 and p2\ in the auxiliary MEGARCH model were not significantly different from zero. Therefore there is no hope to find such behaviour in the MSV models, since in essence these MEGARCH models are a reflection of the MSV models.
The estimation results based on observations over different sample periods for both univariate SV models and multivariate model of four stock return series are reported in Table 7.1 & 7.2. The Hansen J-tests for all models are reported in Ta-ble 7.3 & 7.4. All models are accepted at any reasonaTa-ble significance level, but note that the J-test only has power in the directions given by the elements of the score generator, so e.g. for Kz = 0 the J-test will have no power against time-homogeneous error structures beyond those captured by the Gaussian EGARCH model. This means that excess kurtosis beyond that of the Gaussian SV model will not be detected by the J-test, see Section 3.1.2.
25 r 0 500 1000 1500 2000 0 500 1000 1500 2000 20 r 10 :
o
: -10 : -20 : 0 500 1000 1500 2000 0 500 1000 1500 25 0 -25m*m»**m
0 500 1000 1500 2000 0 500 1000 1500 2000Figure 7.1 : On the right-hand side are the prewhitened returns and on the left-hand side are the reprojected volatilities from the AMSV(l) model.
1.2. DATA, MODELS AND ESTIMATION 165
Estimates, 1-1321 Estimates, 1-1571 Estimates 1-1821
LÜQ .046(1.42) .046(1.44) 074 (4.27) W i .052(1.51) .077 (4.48) .104(14.8) LÜ2 .055(1.52) .057(1.63) .059(1.89) U}3 .095 (20.3) .090(6.51) .077 (5.33) 7o .982(12.3) .982(12.6) .972(19.7) 7i .978(12.4) .969 (32.3) .958 (54.6) 72 .973(13.1) .973 (13.7) .971 (15.5) 73 .965 (75.3) .964 (26.6) .969 (26.4) C J J O .102(5.64) .106(6.20) .135(19.7)
<v
.101(3.68) .098 (5.03) .114(9.03) C772 .139(4.61) .136(4.66) .135(5.15) 0"»;3 .067 (5.93) .153(13.3) .140(12.8) 90 -.671 (-106) -.672 (-111) -.648 (-104) 9i -.555 (-88.4) -.482 (-116) -.385 (-125) 92 -.623 (-91.1) -.682 (-103) -.677 (-105) 93 -.596 (-180) -.757 (-162) -.793 (-200)Table 7.1: Estimates of four univariate SV models for different sample periods. t-values are between brackets using de-meaned and AR(l)-filtered data-set.
Estimates, 1-1321 Estimates, 1-1571 Estimates 1-1821
Wo .040(1.07) .044(1.31) 049 (2.06) W i .028 (.421) .055 (4.48) .068 (8.43) U>2 .048 (1.40) .046 (1.35) .046(1.46) OJ3 .076 (2.95) .088 (5.83) .085 (5.50) 7o .984(10.6) .982 (12.2) .980(16.5) 7i .988(6.10) .978(31.5) .973(48.1) 72 .978 (12.6) .979(13.3) .978(14.5) 73 .971(13.8) .965 (24.0) .965 (24.4) <7T?0 .076 (3.05) .086(4.14) .086(5.55) on\ .078 (2.08) .074 (3.66) .078 (5.34) oni .115(3.63) .107(3.71) .106(3.95) Ov3 .161(10.3) .163(13.7) .164(14.6) 9o -.715 (-119) -.730 (-125) -.724 (-140) 9i -.509 (-99.1) -.272 (-96.1) -.218 (-98.8) 92 -.549 (-108) -.591 (-133) -.599 (-145) 93 -.824 (-150) -.795 (-189) -.793 (-205) coi .237 (8.70) .276(10.9) .284(12.2) C02 .290(11.2) .327(13.7) .348 (16.4) C03 .307(12.3) .321(13.8) .317(14.7) C12 .347(12.6) .382(15.1) .386(16.1) C l 3 .288(10.8) .321 (13.0) .317(13.8) C23 .336(12.9) .369(15.3) .372(16.4)
Table 7.2: Estimates of four variate MSV model for different sample periods, t -values are between brackets using de-meaned and AR(l)-filtered data-set.
1-1321 1-1571 1-1821 Series 0 J-test (P-value) 1.61 (0.204) 1.61 (0.204) .483 (0.487) Series 1 J-test (P-value) 1.66(0.198) .609 (0.435) .858 (0.354) Series 2 J-test (P-value) .384 (0.535) 1.42(0.233) .342 (0.559) Series 3 J-test (P-value) .157(0.692) .679 (0.410) .968 (0.325)
Table 7.3: Hansen J-test for the univariate SV models for different sample periods. The number of degrees of freedom equals 1 for all sample sizes and models.
1-1321 1-1571 1-1821 J-test P-value 3.25 .517 3.74 .442 4.63 .327
Table 7.4: Hansen J-test for the MSV model for different sample periods. The number of degrees of freedom equals 4 for all sample sizes.
In Figure 7.1, the reprojected volatility series based on the multivariate SV model for each stock over the whole sample period are displayed. The reprojected volatility series based on univariate SV model and over other sample periods be-have similarly and are thus not reported. Since it is computationally quit demand-ing to determine Li and L2 in (3.40) by AIC, we set Lx = 30 and L2 = 30 based on past experience and experimentation.
7.3 Forecasting Volatility based on SV Models
In this chapter, we distinguish volatility estimated from asset returns, which we call the underlying or historical volatility, from volatility implied from observed option prices through certain option pricing formula, which we call the implicit or implied volatility. Since volatility is not directly observable, the exact measures of both un-derlying volatility and implied volatility are model dependent. For instance, when the underlying volatility is modelled using standard time-series models, the under-lying volatility can be computed directly from estimated ARCH/GARCH models or reprojected based on estimated SV processes. In the financial literature, studies on volatility forecasting have been focused on two aspects, namely volatility casting from historical volatility based on time-series models and volatility fore-casting from observed market option prices based on certain option pricing models. This study will focus on the first aspect while the second aspect will be examined in a related study.Judging the forecasting performance of any volatility model, amounts to com-paring the predictions with the subsequent volatility realizations. A common prac-tice is to use a model-independent measure of the ex-post volatility in such
anal-7.3. FORECASTING VOLATILITY BASED ON SV MODELS 167 ysis. Write the return innovation be written as
Etb&n] = Et[4+ 1]Et[<t + 1], V?,* (7.2)
When Zj ~ 7V(0,1), i.e. E[z2] = 1, then the squared rate of return y2+l is a consis-tent estimator of the expected future volatility. In particular, when a2+1 is perfectly predictable at time t, e.g. in a ARCH/GARCH model, then the squared rate of re-turn is an unbiased estimator of of+1. If the model for a2 is correctly specified then
the squared return innovation over the relevant horizon can be used as a proxy for the ex-post volatility.
Standard volatility models such as ARCH/GARCH and SV models have been applied with great success to the modelling of financial time series. The models have in general reported significant parameter estimates for in-sample fitting with desirable time-series properties, e.g. covariance stationarity. It naturally leads peo-ple to believe that such models can provide good forecasts of future volatility. In fact, the motivation of a ARCH/GARCH model setup comes directly from fore-casting. For instance, the following univariate GARCH(1,1) model
0?it = 7 + « î & - i + 0 < t - i . i = 1,2,..., N;t= 1,2,...,T (7.3)
is an AR(1) model. The optimal prediction of the volatility in the next period is a fraction of the current observed volatility, and in ARCH(l) case (i.e. ß = 0), it is a fraction of the current squared observation. The fundamental reason here is that the optimal forecast is constructed conditional on the current information. The general GARCH formulation introduces terms analogous to moving average terms in an ARMA model, thereby making forecasts a distributed-lag function of past squared observations.
However, despite highly significant in-sample parameter estimates, many stud-ies have reported that standard volatility models perform poorly in forecasting out-of-sample volatility. Similar to the common regression-procedure used in evaluat-ing forecasts for the conditional mean, the volatility forecast evaluation in the liter-ature typically relies on the following ex-post squared return-volatility regression
y2+1 = a + bâ2t+l + ut+1 (7.4) where t = 0,1,..., and y2+1 is used as a proxy for the ex-post volatility as it is an
unbiased estimator of <r2+1 as shown in (7.2). The coefficient of multiple
determi-nation, or R2, from the above regression provides a direct assessment of the vari-ability in the ex-post squared returns that is explained by the particular estimates of <7t2+1. The R2 is often interpreted as a simple measure of the degree of
volatility forecasts.1 Many empirical studies based on the above regression
eval-uation of the volatility forecast have universally reported disappointingly low R2 for various speculative returns and sample periods. For instance, Day and Lewis (1992) reported R2 = 0.039 for the predicative power of a GARCH(U) model of weekly returns on the S&P100 stock index from 1983-1989; Pagan and Schwert (1990) reported R2 = 0.067 for the predictive power of a GARCH(1,2) model of monthly aggregate US stock market returns from 1835 to 1925; Jorion (1996) re-ported R2 = 0.024 for the predictive power of a GARCH(1,1) model of the daily DM-$ exchange-rate returns from 1985 to 1992; Cumby, Figlewski and Hasbrouck (1993) reported R2's ranging from 0.003 to 0.106 for the predictive power of an EGARCH model of the weekly stock and bond market volatility in the US and Japan from 1977 to 1990; West and Cho (1995) reported R2's ranging from 0.001 to 0.045 for the predictive power of a GARCH(1,1) model of five different weekly US dollar exchange rates from 1973 to 1989. These systematically low R2's have led some critics to the conclusion that standard ARCH/GARCH models provide poor volatility forecasts due to possible severe mis-specification, and consequently are of limited practical use.
Andersen and Bollerslev (1998) argue that the documented poor performance of volatility forecasting is not due to the mis-specification of standard volatility models but due to the measure of ex-post volatility. Even though the squared inno-vation provides an unbiased estimator for the latent volatility factor, it may yield very noisy measurements due to the idiosyncratic error term z2. This component typically displays a large degree of observation-by-observation variation relative to a2. It is not surprising to see a low fraction of the squared-return variation at-tributable to the volatility process. Consequently, the poor predictive power of volatility models, when using y2+1 as a measure of ex-post volatility, is an in-evitable consequence of the inherent noise in the return generating process. An-dersen and Bollerslev (1998) found that based on an alternative ex-post volatility measure building on the continuous-time SV framework, the high-frequency data allow for the construction of vastly improved ex-post volatility measurements via cumulative squared intraday returns. The proposed volatility measures, based on high-frequency returns, provide a dramatic reduction in noise and a radical im-provement in temporal stability relative to measures based on daily returns. Fur-thermore, when evaluated against these improved volatility measurements, they found that daily GARCH models perform well, readily explaining about half of the variability in the volatility factor. That is, there is no contradiction between good volatility forecasts and poor predictive power for daily squared returns.
1 As Andersen and Bollerslev (1998) point out, it is not without problem to use R2 as a guide of
the accuracy of volatility forecasts since under this scenario the R2 simply measures the extent of
idiosyncratic noise in squared returns relative to the mean which is given by the (true) conditional return variance.
7.3. FORECASTING VOLATILITY BASED ON SV MODELS 169 The improved forecasting performance based on the continuous-time SV
model with the measure of volatility constructed from high frequency data, as proposed in Andersen and Bollerslev (1998), in our opinion can be attributed to the following two factors: namely (i) increase of sample size and (ii) accounting of asset-return predictability. To illustrate our points, consider the following two continuous-time diffusion models which are special cases of SV models:
din S(t) = ßdt + aabdW(t) (7.5)
and
dIn S(t) = (a- /?(ln S(t) - at))dt + aoudW{t) (7.6)
with ß > 0. Here we adopt the common notation to index continuous-time vari-ables with the time index in brackets. Model (7.5) is an arithmetic Brownian mo-tion process for l n 5 ( i ) and model (7.6) is a trending Ornstein-Uhlenbeck pro-cess which reduces to (7.5) when ß = 0. It can be shown that in model (7.5) asset returns r(t,r) = l n S ( t ) — \.nS(t — r ) over r-holding period are unpre-dictable as Cor(r(t, r),r(t + A, r ) ) = 0 for A > r ; while in model (7.6) asset returns r(t, T) over r-holding period are predictable as Cor(r(i, r),r(t + A, r ) ) = - | e_ / J(A~T) ( l — e~ßT) for A > r . The consistent estimator of a\b in (7.5) based
on a set of observations {r(ti,T),r(t2,T), ...,r(tN,r)} is
à2ab =
with
In other words, the performance of the estimator à\b depends solely on the number
of observations. Given the same sampling period, the more frequently the data are sampled, the larger number of observations is obtained, and therefore the sample variance is a better estimator of the instantaneous volatility o\h. For model (7.6),
it can be shown that a consistent estimator of a2ou based on the same data set is
aou = I T-Vab (7-8)
1 — e~f>T
since a\h = ^ ( 1 — e_/3T) , where ß is a consistent estimator of ß. Thus, it is
clear that when asset-return predictability is present, namely ß > 0, the conditional instantaneous volatility is different from the sample variance as l~e0T < 1 for
ß > 0, r > 0, and equivalently a\b < o2ou. However, since
1 '~NT N i=i r)- Ar)2,
A =
l N J V ' i=l E|:*ii
= <&(! • 1 ~N ), Varl»l)
N K * > ' a i = a2ou + 0{T) (7.9)it is clear that the difference between the conditional volatility and unconditional sample variance diminishes as the sampling interval decreases even the predictabil-ity of asset returns is present (i.e. ß > 0). This result can be extended to the general case under the diffusion framework when the asset returns are not independently distributed over time, the conditional volatility can be well approximated by the unconditional sample variance based on the observations sampled with high fre-quency.
Similarly, in the discrete-time model, predictability of asset returns also has strong impact on the volatility forecasting performance when squared returns are used as proxy of ex-post volatility. For instance, let the true underlying process of the unexpected asset returns be
yt = AlnSt-ß0-ßiAlnSt-i, \ßi\ < 1 (7.10) it can be shown that Efojf] = (1 - /^)Var[AlnSt] and Var[AlnSf] ± E[y*}, if
/Xx^O.
However, high frequency sampling observations are not always available, even in the financial markets. Even when they are, the quality of such data is likely to be questionable as such observations are typically subject to measurement error and various market microstructure issues. In this chapter, instead of computing vola-tility from high frequency data, we propose an alternative measure of stochastic volatility using reprojected volatility series in a consistent SV model framework. We show that there is in fact a lot to be gained from using the reprojected volatility series as a measure of stochastic volatility and in forecasting future volatility.
7.3.1 Volatility Forecasting based on Univariate SV Models
To evaluate the performance of volatility forecasting based on SV models, we first, for the purpose of comparison, evaluate the performance using the squared-return residuals as the measure of ex-post volatility as in previous studies and then use the model reprojected volatility in a consistent SV model framework. Instead of using (7.2), we rely our evaluation on the following relationship,
Et[ l n ^+ 1] = Et[ma2t+1] + Et[mz?t+1], Vt,t (7.11)
since Et [In z\\ = —1.27 and Var[mzt2] = 7r2/2, the logarithmic squared return is an
unbiased estimator of the expected future latent volatility factor with a constant ad-justment. Advantages of using the above relationship include the following. First,
it provides an exact gauge of upper limit of explanatory power for the following regression
7.3. FORECASTING VOLATILITY BASED ON SV MODELS 171 Since Var[ln of t+1] = jzts and Var[ln z?t+1] = \ , the population R2 can be
de-rived as R2 = ( 1 + '*~ 72 ) ~ *. Second, as shown above, that it is easier to examine
n
the relative variance of the signal and noise terms which determine the explanatory power of the predictor with an additive error than with a multiplicative error term. Third, with the above upper bound, it is easier to understand the small sample prop-erties of the regression with the guidance of the asymptotic propprop-erties2.
The evaluation of volatility forecasting performance based on reprojected vola-tility series is outlined as follows. In the first step, we estimate the AMSV model based on sampling observations over [1, <i], the underlying volatility Ht, 1 < t < ii, are obtained through reprojection based on return observations over [l,ii]. Such reprojected volatility is referred as the true underlying or historical volati-lity in our model framework. In the second step, we use the estimated model to reproject volatility over the period [ti + 1, t2] which is about one year ahead the estimation sample period. At each time of point at ti + i where 1 < i < t2 — ti,the underlying volatility Kfl+i is reprojected using the information up to time ti + i,
denoted by Itl+i, which include all the past and present return observations and re-projected volatility series; In the third step, the one-period ahead forecast of vola-tility Nt l + i + 1 ) 0 < i < <2 — h — 1, are obtained based on the volatility series and
asset-return observations up to time tx + i through the specified volatility process, i.e.
ln(diag(Kt)2) = Û + £ fitf ln[diag(Ht)2] + tnQzt^ (7.13) The last term is to take account of the asymmetry between volatility and asset
re-turns; In the final step, in order to evaluate the forecasting performance of our model, we estimate the AMSV model again using sampling observations over time period [1, t2], the reprojected volatility series Htl+j+1,0 < i < t2 — h — 1, which
are the ex-post volatility, are compared to the volatility forecasts Kfl+j+1,0 < i <
t2 — t\ — 1, through regression analysis. The above exercise is replicated with [1,^2] as estimation period and [t2 + 1, T] as volatility forecasting period. Doing so, we are able to increase the sample size in the regression analysis while keeping the forecasting span relatively short, i.e. roughly one year period. In other words, for the purpose of volatility forecasting, in our application the SV models are only (re-)estimated once every year.
2One may wonder what happens to the R2 in case zt is in fact non-normal as is very likely for
financial time series. For the above SV model with zt following a standardized Student-t
distri-bution as in (2.19) to (2.23) we can show that R2 = (1 + ^ - ^ H ^ + ^ ' W2» )-i where tp'(-)
denotes the trigamma function, which is a monotone decreasing non-negative function in v, with lim„|2 ii'{vj2) = 00 and l i m ^ o o t[>'(i//2) = 0. This indicates that here the R2 will be smaller
Thus, our evaluation is based on the following two regressions:
In y2+l = a + b In <5f+1 + ut (7.14)
In <j]+l = a + bln â2+1 + ut (7.15)
where In y2+1, In a2t+l denote respectively the squared filtered asset-return residu-als and the reprojected volatility based on observed asset returns which are used as alternative measures of ex-post volatility realizations. The above regressions are estimated using both ordinary least squares (OLS) and instrumental variables (IV) estimation procedures. The use of IV procedure is to remedy a possible flaw of the OLS estimates due to the fact that there is an inevitable error in the forecast of future volatility. The error in variables (EIV) will lead to inconsistent OLS esti-mates of the regression coefficients. Since for IV estimation the R2 is not properly defined and not useful for making comparisons with the R2 from OLS, we report the squared correlations between the predicted and observed dependent variable in the regression. This measure enables a fair comparison with the R2 in OLS. A nat-ural choice of the instrumental variable in the above regression is the past volatility forecast of which is strongly correlated with the true volatility at time t + 1 but is quite plausibly unrelated to its associated error.
The regression results are reported in Table 7.5. It is noted that when the squared-return residuals are used as proxy of ex-post volatility, the population Rlc's, as calculated from the estimated SV model, are all very low, namely 6.0%, 3.1%, 6.6% and 6.1% for four series respectively. It confirms that the squared return residual is a very poor proxy of the post volatility and is an ex-tremely noisy measure due to the idiosyncratic factor In z2 which accounts over 90% of the variations of the logarithmic squared return residuals. It is thus not surprising that the R2,s in the evaluation regression are all very low, namely 2.0%, 0.8%, 4.5% and 0.6% respectively for four return series. However, the sta-tistical tests of the parameter estimates indicate that in general the constant pa-rameter estimates are not significantly different from zero and the slope parame-ter estimates are not significantly different from one, suggesting the relationship in (7.2) is not rejected for any of the series. This further indicates no rejection of the SV model specification for each of the time series. When the reprojected volatility based on the reprojection technique is used as the measure of ex-post volatility, the i?2's in the evaluation regression all increase substantially, namely to
90.9%, 85.5%, 89.6% and 46.1% respectively for the four series. The IV estimates of the parameter b are all close to one, namely 0.938,0.706,0.988, and 0.788 re-spectively. Statistically they are significantly different from one and the constant terms are all significantly different from zero except for series 3. It appears to indi-cate that the volatility forecasts based on reprojected series fail to provide an unbi-ased estimator of the future stochastic volatility, underlining the difficulty of vola-tility forecasting. This finding is not surprising in some sense as we notice that
7.3. FORECASTING VOLATILITY BASED ON SV MODELS 173 a i cd • T - H u •—
1
<oC/J v _ .va3
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i—1 II t * ui * * * ( H L ; * * * l— c^ * ^ LN u ; » » * * O O tu c o£
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< c* w oo OO 0 0 •- = a E o • o > > e t Uthe parameters of the conditional mean of asset returns are changing over differ-ent sample periods. When we base on the past conditional mean of asset returns to reproject the future volatility, it may generate slightly different levels of fore-casted volatility than the observed ex-post volatility. A possible remedy is to in-crease the frequency of the model estimation to reduce the effect of time-changing conditional mean. To further gauge the magnitude of forecasting error, we also re-port the sum of squared residuals (SSR) for each regression. It is noted that the SSR's are minimal when the reprojected volatility series are used as ex-post tility compared to those when the squared return residuals are used as ex-post vola-tility.
7.3.2 Volatility Forecasting based on Multivariate SV Models
In this section, similar analysis are conducted based on the estimated AMSV(l) model and the subsequently reprojected volatility series. The evaluations based on regressions in (7.14) and (7.15) are replicated for the AMSV(l) model. The regression results are reported in Table 7.6 . Similar to the univariate SV model, when the squared return residuals are used as the measure of ex-post volatility, both population fi^'s and regression i?2's are very low, namely 3.6%, 2.3%, 4.9%, and
7.3% and 2.0%, 0.9%, 3.9%,and 0.3% for four series respectively. Again, the sta-tistical tests of both H0 : a = 0 and H0 : b — 1 are in general not rejected, suggesting the AMSV(l) model is not mis-specified. When the reprojected volati-lity series via reprojection technique from the estimated AMSV(l) model are used as the measure of ex-post volatility, the explanatory power of the regressions also increase substantially, to the levels of 93.0%, 86.5%, 92.2%, and 74.5%. Com-pared to the univariate model, the explanatory powers are further improved. More-over, the parameter estimates of both a and b based on IV estimation are in gen-eral improved in the sense that a is closer to zero and b is closer to one, notably b = 0.941,0.794,0.989,0.941 respectively. The statistical test H0 : a = 0 based on the IV estimation is rejected at 99% critical level for series 1 & 2, but not re-jected for series 3 & 4 at critical levels below 90%. The statistical test H0 : b = 1 based on the IV estimation is rejected at 99% critical level for series 2, rejected at 90% critical level for series 1, but not rejected for series 3 and 4 at critical levels be-low 90%. The findings further underline the difficulty of volatility forecasting in a stochastic volatility framework. However, it is noted that for all four series, the val-ues of the constant parameter are really small with the highest magnitude of 0.13%. Again, the reported SSR's are minimal when the reprojected volatility series are used as ex-post volatility compared to those when the squared asset-return resid-uals are used as ex-post volatility. It is also noted that the SSR's are also smaller than their counterparts in the univariate SV models.
fol-7.3. FORECASTING VOLATILITY BASED ON SV MODELS 175 CO O o. i X <ü
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£ ^> r - : zlowing two ideas can be explored. First, as the individual volatility series In of are correlated to each other as suggested by the AMSV(l) model, volatility forecasting based on VAR-type model instead of the AR(1) model with correlated error terms as specified in (2.26) for each volatility series should improve the forecasting per-formance; Second, to incorporate asymmetry of stochastic volatility with respect to asset returns, we can further include yt/crt type terms in the VAR-type model for In of as in the EGARCH model and reprojection procedure. To see whether the volatility forecast of one series has explanatory power to the future volatility of other series, the regression analysis based on the following multivariate model fori = 1,2,3,4
In of i + 1 = a + 6i In ô\t+l + b2 In à\t+l + b3 In âjt+1 + b4 In â\t+l + ut (7.16)
is also conducted and the results are reported in Table 7.6. It is noted that the re-gression R2,s are in general increased and the SSR's are further reduced as ex-pected. The statistical test of the hypothesis H0 : b = [62, b3, 64] = 0 for series
1, H0 : b = [bi,b3, 64] = 0 for series 2, H0 : b = [h,b2, b4] = 0 for series 3, H0 : b = [bi,b2, h] = 0 for series 4 are all rejected at 99% critical level based on OLS estimation and at 90% critical level based on IV estimation. It suggests that the cross terms of volatility forecasts have significant effect on the future volatility for all four series.
7.4 Conclusion
This chapter evaluates the performance of volatility forecasting based on stochas-tic volatility models. We show that the choice of squared asset return residuals as proxy of ex-post volatility directly leads to extremely low explanatory power in the common regression analysis of volatility forecasting. We argue that since the mea-sure of volatility is always model dependent, the performance of volatility forecast-ing should be evaluated in a consistent model framework. The main contributions of this chapter include: First, we apply the EMM estimation method proposed by Gallant and Tauchen (1996b) to estimate the multivariate SV model of asset re-turns; Secondly, we further implement the underlying volatility reprojection tech-nique proposed by Gallant and Tauchen (1998) to the estimated multivariate SV model; Finally, we illustrate that the performance of volatility forecasting based on reprojected volatility series can be substantially improved. Furthermore, we show that the volatility forecasting performance based on multivariate SV model improves over the univariate SV models due to the correlated movements of asset return volatility.
