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Adopting the rational perspective that asset prices reflect the discounted value of future expected cash flows, such prices should react almost continuously to the myriad of news that arrive on a given trading day. Assuming that the number of news arrival is large, one may expect a central limit theory to apply and financial returns should be well approximated by a conditional normal distribution with the conditioning variable corresponding to the number of relevant news events. More generally, a number of other variables associated with the overall activity of the financial market such as the daily number of trades, the daily cumulative trading volume or the number of quotes may well be similarly related to the information flow in the market. These considerations inspire the following type of representation,

, (4.1)

where yt is the market “activity” variable under consideration, st is the strictly positive process reflecting the intensity of relevant news arrivals, :y represents the mean response of the variable per news event, Fy is a scale parameter, and zt is i.i.d. N(0,1). Equivalently, this relationship may be written,

. (4.2) This formulation constitutes a normal mixture model. If the st process is time-varying it induces a fat-tailed unconditional distribution, consistent with stylized facts for most return and trading volume series. Intuitively, days with high information flow display more price fluctuations and activity than days with fewer news releases. Moreover, if the st process is positively correlated, then shocks to the conditional mean and variance process for yt will be persistent. This is consistent with the observed activity clustering in financial markets, where return volatility, trading volume, the number of transactions and quotes, the number of limit orders submitted to the market, etc., all display pronounced serial dependence.

The specification in (4.1) is analogous to the one-step-ahead decomposition given in equation (3.5). The critical difference is that the formulation is endowed with a structural interpretation, implying that the mean and variance components cannot be observed prior to the trading day as the number of news arrivals is inherently random. In fact, it is usually assumed that the s_t process is unobserved by the econometrician, even during period t, so that the true mean and variance series are both latent. From a technical perspective this implies that we must distinguish between the full information set (s_t 0öt ) and observable information (st óTt ). The latter property is a defining feature of the genuine volatility class. The inability to observe this important component of the MDH model complicates inference and forecasting procedures as discussed below.

In the case of short horizon return series, :y is close to negligible and can reasonably be ignored or simply fixed at a small constant value. Furthermore, if the mixing variable st is latent then the scaling parameter, Fy , is not separately identified and may be fixed at unity. This produces the following return (innovation) model,

, (4.3)

implying a simple normal-mixture representation,

. (4.4)

Both univariate models for returns of the form (4.4) or multivariate systems including a return variable along with other related market activity variables, such as trading volume or the number of transactions, are referred to as derived from the Mixture-of-Distributions Hypothesis (MDH). The representation in (4.3) is of course directly comparable to that for the return innovation in equation (3.5). It follows immediately that volatility forecasting is related to forecasts of the latent volatility factor given the observed information,

. (4.5)

(4.5) will generally not represent the actual conditional return variance, E(st+h |öt ). This point is readily seen through a specific example.

In particular, Taylor (1986) first introduced the log SV model by adopting an autoregressive parameterization of the latent log-volatility (or information flow) variable,

log st+1 = 00 + 01 log st + ut , ut - i.i.d.(0, Fu2 ), (4.6) where the disturbance term may be correlated with the innovation in the return equation, that is,

D = corr( ut , zt ) … 0. This particular representation, along with a Gaussian assumption on ut, has been so widely adopted that it has come to be known as the stochastic volatility model. Note that, if D is negative, there is an asymmetric return-volatility relationship present in the model, akin to the “leverage effect” in the GJR and EGARCH models discussed in Section 3.3, so that negative return shocks induce higher future volatility than similar positive shocks. In fact, it is readily seen that the log SV formulation in (4.6) generalizes the EGARCH(1,1) model by considering the case,

, (4.7)

where the parameters 00 and 01 correspond to T and $ in equation (3.15) respectively. Under the null hypothesis of EGARCH(1,1), the information set, Tt, includes past asset returns, and the idiosyncratic return innovation series, z_t , is effectively observable so likelihood based analysis is straightforward. However, if u_t is not (only) a function of z_t , i.e., equation (4.7) no longer holds, then there are two sources of error in the system. In this more general case it is no longer

possible to separately identify the underlying innovations to the return and volatility processes, nor the true underlying volatility state.

This above example illustrates both how any ARCH model may be seen as a special case of a corresponding SV model and how the defining feature of the genuine SV model may complicate forecasting, as the volatility state is unobserved. Obviously, in representations like (4.6), the current state of volatility is a critical ingredient for forecasts of future volatility. We expand on the tasks confronting estimation and volatility forecasting in this setting in Section 4.1.3. There are, of course, an unlimited number of alternative specifications that may be entertained for the latent volatility process. However, Stochastic Autoregressive Volatility (SARV) of Andersen (1994) has proven particular convenient. The representation is again autoregressive,

, (4.8) where ut denotes an i.i.d. sequence, and st = g(vt ) links the dynamic evolution of the state variable to the stochastic variance factor in equation (4.3). For example, for the log SV model, g(vt ) = exp(vt ). Likewise, SV generalizations of the GARCH(1,1) may be obtained via g(vt ) = vt and an SV extension of a GARCH model for the conditional standard deviation is produced by letting g(vt ) = vt1/2. Depending upon the specific transformation g(@) it may be necessary to

impose additional (positivity) constraints on the innovation sequence ut, or the parameters in (4.8). Even if inference on parameters can be done, moment based procedures do not produce estimates of the latent volatility process, so from a forecasting perspective the analysis must necessarily be supplemented with some method of approximating the sample path realization of the underlying state variables.