 A COMPANION TO Theoretical Econometrics

# The SVD model

This model represents dynamics of both the conditional mean and variance in duration data. In this way it allows for the presence of both conditional under - and overdispersion in the data. Technically, it shares some similarities with the stochastic volatility models used in finance. The main difference in SVD m

specification compared to ACD is that it relies on two latent factor variables which are assumed to follow autoregressive stochastic processes. Note that despite the fact that the conditional variance in the ACD model is stochastic it is entirely determined by past durations. The introduction of additional random terms enhances the structure of the model and improves significantly the fit. On the other hand it makes the model more complicated and requires more advanced estimation techniques.

The approach is based on an extension of the exponential duration model with gamma heterogeneity. In this model the duration variable Y is exponentially distributed with the pdf : X exp(-Xy), conditional on the hazard rate X. Therefore, the duration variable may be written as:

Y = U/X, (21.31)

where U ~ у(1, 1). The hazard rate depends on some heterogeneity component V:

X = aV, (21.32)

where V ~ у(b, b) is independent of U. The marginal distribution of this hetero­geneity component is such that: EV = 1 and Var(V) = 1/b, while the parameter a is a positive real number equal to the expected hazard rate.

Equations (21.31) and (21.32) yield the exponential model with gamma hetero­geneity, namely:

Y = —, (21.33)

aV

where U, V are independent, U ~ у(1, 1) and V ~ у(b, b). This equation may be considered as a two factor model formula, where Y is a function of U and V. Some suitable nonlinear transformations can yield normally distributed factors. More explicitly, we get:

Y = G(1, ф№)) = H(1, F) (2134)

aG(b, Ф(Е2)) aH(b, F2)'

where F1, F2 are iid standard normal variables, Ф is the cdf of the standard normal distribution and G(b, .) the quantile function of the у (b, b) distribution. We have: H(1, F1) = - log[1 - Ф^)]. On the contrary, the function H(b, F2) has no simple analytical expression in the general case, but admits a simple approximation in the neighborhood of the homogeneity hypothesis; namely if b ~ : H(b, F2) ~ 1 + (1/Vb )F2 ~ exp(F2/-Jb), where the latter follows by the Central Limit Theorem.

The dynamics is introduced into the model through the two underlying Gaussian factors which follow a bivariate VAR process Ft = (F1t, F2t)', where the marginal distribution of Ft is constrained to be N(0, Id) to ensure that the mar­ginal distribution of durations belongs to the class of exponential distributions with gamma heterogeneity.

This approach yields the class of Stochastic Volatility Duration (SVD) models (Ghysels, Gourieroux, and Jasiak, 1997). They are defined by the following specification:

Y = - = 1 &(b, Ft), (say) (21.35)

a H(b, F21) a

where

p

Ft = ^fi-j + e <, (21.36)

j=1

and et is a Gaussian white noise random variable with variance-covariance matrix £(¥) such that Var(Ft) = Id.

References

Cox, D. R. (1975). Partial likelihood. Biometrika 62, 269-76.

Cox, D. R., and D. Oakes (1984). Analysis of survival data. Chapman and Hall.

Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987-1007.

Engle, R., and J. R. Russell (1998). The autoregressive conditional duration model. Econometrica 66, 1127-63.

Ghysels, E., C. Gourieroux, and J. Jasiak (1997). The stochastic volatility duration model. D. P. CREST.

Gourieroux, C. (1989). Econometrie des variables qualitatives. Economica.

Gourieroux, C., J. Jasiak, and G. Le Fol (1999). Intra-day market activity. Journal of Financial Markets 2:3, 193-226.

Heckman, J. (1981). Heterogeneity and state dependence. In S. Rosen (ed.) Studies in Labor Markets. University of Chicago Press.

Heckman, J., and B. Singer (1974). Econometric duration analysis. Journal of Econometrics 24, 63-132.

Heckman, J., and B. Singer (1984). The identifiability of the proportional hazard model. Review of Economic Studies 231-45.

Jasiak, J. (1999). Persistence in intertrade durations. Finance 19, 166-95.

Kalbfleisch, J., and R. Prentice (1980). The Statistical Analysis of Failure Time Data. Wiley. Lancaster, T. (1985). Generalized residuals and heterogenous duration models, with appli­cation to the Weibull model. Journal of Econometrics 28, 155-69.

Lancaster, T. (1990). The Econometric Analysis of Transition Data. Cambridge: Cambridge University Press.

Lee, L. (1981). Maximum likelihood estimation and specification test for normal distribu­tional assumption for accelerated failure time models. Journal of Econometrics 24, 159-79.

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