4{)0 12. Nonlinearilies in Financial Data

Terence between the conditional log standard deviation of returns and its mean; it follows a /ero-mcan AR( 1) process.

We can rewrite this system by squaring the return equation and taking logs to gel

log(n2) = a, 4-log(<= ?), or, = </> ,., 4-<$(. (12.2.21)

This is in linear state-space form except that the first equation of (12.2.21) has an error with a log x2 distribution instead of a normal distribution. To appreciate the importance of the nonnormality, one need only consider the facl that when e, is very close to /.его (an inlier ), log( ,) is a very large negative outlier.

The system can be estimated in a variety of ways. Mclino and Turnbull (1990) and Wiggins (1987) use GMM estimators. While this is straightforward, il is not efficient. Harvey, Ruiz, and Shephard (1994) suggest a quasi-maxhnum-likelihood estimator which ignores the nonnormality olTog(e;f) and proceeds as if both equations in (12.2.21) had normal error terms. More recendy.Jacquier, Poison, and Rossi (1994) have suggested a Baycsian approach and Shephard and Kim (1994) have proposed a simulation-based exact maximum-likelihood estimator.

12.2.2 Multivariate Models

So far we have considered only the volatility of a single asscl relurn. More generally, wc may have a vector of asset relurns whose conditional covariance matrix evolves through time. Suppose we have N assets with return innovations i-\...N. Wc slack these innovations into a-vnctor T)i+i = [ t.i+l n.i+iY and define cr .i=Var,(n ,+ i) and сгу = Cpv,(n,/+i, n;-/+i); hence £,=[cr ] is the conditional covariance matrix of all the returns. It is often convenient lo stack the nonredundant elements ofE,-those on and below the main diagonal-into a vector. The operator which performs this stacking is known as the vech operator: vech(E,) is a vector with N(N+l)/2 elements.

Multivariate GAIICH Models

Many of the ideas we have considered in a univariate context translate nat-u ally to the multivariate setting. The simplest generalization of the uni-v; riatc GARCH(l.l) model (12.2.0) relates vcch(E,) to vcch(r/,r/() and lo vqch(E, i):

vech(E,) = w + *vech(£,-i) + Avcch(77,T/,). (12.2.22)

H ere wis a vector with N(N+\)/2 elements, and Ф and A arc N(N-fl)/2 x N(N+l)/2 matrices; hence the total number of parameters in this model

is N2(N+\)-/2 + N(N+\)/2 which grows with the fourth power of N. It is clear that this model becomes unmanageable very quickly; much of die literature on multivariate GARGII models therefore seeks to place plausible restrictions on (12.2.22) to reduce the number of parameters. Another important goal of the literature is to find restrictions which guarantee lhat the covariance matrix E, is positive definite. Such restrictions are comparatively straightforward in a univariate setting-for example, all the parameters in a univariate GARCH (1,1) model must be positive-but are much less obvious in a multivariate model.

Kroner and Ng (1993) provide a nice survey of the leading multivariate GARCH models. A first specification, the VEC11 model of Bollerslev, Knglc, ami Wooldridgc (1988) (named alter the vech operator), writes the covariance matrix as a set of univariate GARCH models. Each element til E, follows a univariate GARCH model driven by the corresponding clcmenl of the cross-product matrix rj,r/,. The (/, /) element of E, is given by

°ij.t - Wij; + А7(7 .,- + aijijuiij,. (12.2.23)

This model is obtained from (12.2.22) by making the matrices Л and Ф diagonal. The implied conditional covariance matrix is always positive definite if the man ices of parameters [a>,y], [fi,j), and [a,j] arc- all positive definite. The model has three parameters for each element of E, and thus ЗЛ/(Л/+1)/2 parameters in all.

A second specification, the BEKK model of Knglc and Kroner (1995) (named after an earlier working paper by Bollerslev, Knglc, Kraft, and Kroner), guarantees positive dcliniteness by working with quadratic forms rather than the individual elements of E,. The model is

E, = CC-f BE, ,B-f Atj,t>,A, (12.2.24)

where С is a lower triangular matrix with N(N+\)/2 parameters, and В and A are square matrices with N2 parameters each, for a total parameter count of {5N2+N)/2. Weak restrictions on В and A guarantee that E, is always positive definite.

A special case oflhe BEKK model is die single-factor GARCI 1(1,1) model of Knglc, Ng, and Rothschild (1990). In this model we define N-veclors A and w and scalars a and в, and then have

E, == CC + AA[pVE, ,w-Fa(wT;,)-. (12.2.25)

Here С is restricted as in the previous equation. We can impose one normalizing restriction on this model; it is convenient to set tw=l, where i is

12. Nonlinearities in Financial Data

a vector of ones. I lie vector w can then he thought of as a vector of porllolio weights. We define /tywrj, and o-WTl,. The model can now he restated as

na< ~ a + iiaif.i

M>.i = >/ . + fiaM,J ] +anj,r (12.2.20)

The covariances nfany two asset returns move through lime only with the variance of the portfolio return, which follows a univariate (.ARCH? I,I) model. The single-factor CARCI I (1,1) model is a special case of the BEKK model where the matrices A and В have rank one: A = JawX and В = vZ/bvA. It has (Л-ЧГ)Л+2)/,2 free parameters. The model can he extended straightforwardly to allow for multiple factors or a higher-order GARCH structure.

Finally, Bollerslev (1990) has proposed a constant-correlation mode! in wliich each asset return variance follows a univariate GARCH(1,1) model and the covariance between any two assets is given by a constant-correlation coefficient multiplying the conditional standard deviations of the returns:

<V< = loa + Pa on..-1 + i)l,

= Pi) -J n.i .i- (12.2.27)

This model has N(.N+b)/2 parameters. It gives a positive definite covariance matrix provided thai the correlations pn make up a well-defined correlation matrix and the parameters w . or , and fi are all positive.

To understand the differences between these models, it is instructive to consider what happens to the conditional covariance between two asset returns after large shocks of opposite signs bit the two assets. In the VF.CI I model with a positive a coellit ienl, the negative cross-product r; i/(, lowers the conditional covariance. In ihe constant correlation model, on the other hand, thesign olthe cross-product j; , isirrelevant; any event that increases the variances of i wo positively correlated assets raises the covariance between them. In the factor ARCII model n,h, only moves with aM, so the effect of a negative cross-product / depends on the weights in portfolio />.

As in the univariate case, return volatilities may be persistent in multivariate CARCI I models. Multivariate models allow for the possibility that some asset volatilities may share common persistent components; for example, there might be one persistent component in a set of volatilities, so that all changes in one volatility relative lo another are transitory. Bollerslev and F.ngle (I99.>) exploit this idea, which is analogous lo the concept of t oinlcgralion in lite lileraltire on linear unit-root processes.

12.2. Models of Changing Volatility

Multivariate Stochastic-Volatility Models

The univariate stochastic-volatility model given in (12.20) is also easily

extended to a multivariate setting. We have

a, = Фа,., (12.2.28)

where rj € a and £, are now (Ax 1) vectors and Ф is an (WxiV) matrix. This model has N parameters in the matrix Ф, N(N+\)/2 parameters in the covariance matrix of 6 and N(N+\)/2 parameters in the covariance matrix of tj(, so the total number of parameters is N(2N+l). There is no need lo restrict a, to be positive and it is straightforward to estimate the e, and tj, covariance parameters in square-root form to ensure that the implied covariance matrix is positive definite. Harvey, Ruiz, and Shephard (1994) suggest restricted versions of this model in which Ф is diagonal (reducing jhe number of parameters to N(N+2)) or is even the identity matrix (furtljier reducing the number of parameters to N(N+l)). j

Кven without such extra restrictions, it is important to understand фа! the.specification (12.2.28) imposes constant conditional correlations of asjset returns. In this respect il is as restrictive as Bollerslevs (1990) constani-corrclatiou GARCH model, and it has more parameters than that model whenever N>?>.

Л Conditional Market Model

Even the most restrictive of the models we have discussed so far are hard to apply to a large cross-sectional data set because the number of their parameters grows with the square of the number of assets A. The problem is trial these models take the whole conditional covariance matrix of returns as the object to be studied. An alternative approach, paralleling the much earlier development of static mean-variance analysis, is to work with a conditional market model. Continuing to ignore nonzero mean returns, we write

= Pn iV+i +C./+I, (12.2.29)

where fi = a .,/a , is the conditional beta of asset i with the market, and i is an idiosyncratic shock which is assumed to be uncorrelated across assets. Within this framework we might model a mJ, the conditional variance of the market return as a univariate GARCH(1,1) process; we might model Pim.i or equivalently <7;, as depending on cr, , pim and the returns i) and ij ; and we might model the conditional variance of the idiosyncratic shock to return as another univariate GARCH(1,1) process. The covariance matrix implied by a model of this sort is guaranteed to be positive definite, and the number ol parameters in the model grows at rate N rather than Ыг, which makes the model applicable to much larger numbers of assets. Braim,

Nelson, and Sunicr (1995) lake this approach, using EGARCH functional forms for the individual components of the model.

12.2.3 Links between First and Second Moments

We have reviewed some extremely sophisticated models of lime-varying second moments in time series whose first moments are assumed to be constant and zero. But the essence of finance theory is that it relates the first and second moments of asset returns. Accordingly we now discuss models in which conditional mean returns may change with the conditional variances and covariances.

The GARCH-M Model

Englc, Lilien, and Robins (1987) suggest adding a time-varying intercept to the basic univariate model (12.2.2). Writing r,+ 1 for a continuously compounded asscl return which is the time series of interest (since we no longer work with a mean-zero innovation), wc have

r(+i = Hi + 0,ei+\,

Hi = Xu + m,

(12.2.30)

I where £,+ i is an 11D random variable as before, and of can follow any GARCH process. This GARCH-in-mean or GARCH-M model makes the conditional mean of the return linear in the conditional variance. It can be straightforwardly estimated by maximum likelihood, although it is not known whether the model satisfies the regularity conditions for asymptotic normality of the maximum likelihood estimator.

The GARCH-M model can also be specified so that the conditional mean is linear in the conditional standard deviation rather than the conditional variance. It has been generalized to a multivariate setting by Bollerslev, Engle, and Wooltlridge (1988) and others, but the number of parameters increases rapidly with the number of returns and the model is typically applied to only a few assets.

The Instrumental Variables Approach

As an alternative to the GARCH-M model, Campbell (1987) and Harvey (1989, 1991) have suggested lhat one can estimate ihe parameters linking , first and second moments by GMM. These authors start with a model foi1 the market return that makes the expected market return linear in its own variance, conditional on some vector H, containing /. instruments or forecasting variables:

E[r, .,+,H,] = Y + Y\ Var[r ,+ iH,].

(12.2.31)

Campbell and Harvey assume that conditional expected relurns are linear in the instruments and define errors

../+1 = rml+] - H,b, ,

m.i+i з rml+t - Уи - Yi(rmJ+l - H,b, )-. (12.2.32)

Here b , is a vector of regression coefficients of the market return on the instruments. The error u, ./+i is the difference between the market return and a linear combination of the instruments, while the error emJ+i is the difference between the market return and a linear function of u2 ,. The model (12.2.31) implies that ihe errors u, + , and e, ,+ , are both orthogonal to the instruments H,. Willi /, instruments, there are 41. orthogonality conditions available to estimate /.4-2 parameters (y , yu and the /. coefficients in b, ). Thus GMM delivers both parameter estimates and a lest for the overidentifying restrictions of die model.

This approach can easily be generalized lo include other assets whose expected returns arc given by

E[r + , I H,] = Ku + KiGov[r ,+ ; u+i I H,]. (12.2.33)

11 there are N such assets, we define a vector r,){ s \ц M,.....лмиГ-

The conditional expectation of r,+1 is given by E)r,+1 H, = H,B, where В is a matrix with NL coefficients. We define errors

u,+ i = r,+ i - H,B,

e,+ , = r,+ i - y -Ki(r,+ - H,B)(r ,.,+ 1 - H,bM), (12.2.34)

and wc get 2NL extra orthogonality conditions to identify NL + 1 extra parameters. The total number of orthogonality conditions in (12.2.33) and (12.2.34) is 2(N + \)L and the total number of parameters is Л(/.+ 1) + /. + 2. Thus the model is identified whenever two or more instruments are available.

Harvey (1989) further generalizes the model to allow for a time-varying price of risk. He replaces (12.2.33) by

E[r,.,+ i I H,] = y + yi,Cov[ru+1, r, .,+ , j H,], (12.2.35)

where yu varies through time but is common to all assets. Since (12.2.35) holds for die market portfolio itself,

E[r , ,+ l I H,) - yu Yu = ~v7~l-ГИТ- 12.2.30)

and Harvey uses this to estimate the model. He substitutes (12.2.30) into (12.2.35), multiplies through by Var(r, ./+, H,], and uses E > ,.,+ ,H,J =