Quadratic Arch Model

The Review of Economic Studies Ltd.

Quadratic ARCH ModelsAuthor(s): Enrique SentanaSource: The Review of Economic Studies, Vol. 62, No. 4 (Oct., 1995), pp. 639-661Published by: The Review of Economic Studies Ltd.Stable URL: http://www.jstor.org/stable/2298081Accessed: 01/03/2010 21:40

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Review of Economic Studies (1995) 62, 639-661 0034-6527/95/00290639$02.00 ? 1995 The Review of Economic Studies Limited

Quadratic ARCH Models ENRIQUE SENTANA

CEMFI

First version received December 1991; final version accepted July 1995 (Eds.)

We introduce a new model for time-varying conditional variances as the most general quadratic version possible within the ARCH class. Hence, it encompasses all the existing restricted quadratic variance functions. Its properties are very similar to those of GARCH models, but avoids some of their criticisms. In univariate applications to daily U.S. and monthly U.K. stock market returns, QARCH adequately represents volatility and risk premia. QARCH is easy to incorporate in multivariate models to capture dynamic asymmetries that GARCH rules out. Such asymmetries are found in an empirical application of a conditional factor model to 26 U.K. sectorial stock returns.

1. INTRODUCTION

The analysis of economic and financial time-series data usually involves the study of the first and possibly second conditional moments of the series (given past behaviour) in order to characterize the dependence of future observations on past values. Besides, these two conditional moments are often identified with important economic concepts. For example, consider the univariate stochastic process for stock market excess returns, x,, t = 1, 2, . . .. whose first two conditional moments given the information set (generated by) X, 1, =

{Xt-I, Xt-2, .}are:

p t = E(xt I Xt- OO)=(Xt - OO,(la)

a 2 = V(X,tl X,t_, oo. ) 2(Xt_ 1, ,o ( lb)

In this context, p, is usually associated with the risk premium for the stock market as a whole, a 2 with its volatility, and p, /U2 with the market price of risk. From an empirical point of view, the first step often consists in the estimation of the conditional mean and variance. In practice, though, this is not a simple task as both p ( ) and a2( ) are generally unknown functions of the information set X,_, . The most common approach employed is to assume a particular functional form for p( ) and &2( ) characterized by certain unknown parameters (which have to be estimated), although non-parametric techniques (which obtain estimates of p, and U2 directly) and mixed approaches (semiparametric) have gained increased attention recently.

A fact that should be borne in mind when modelling conditional variances is that they are non-negative (or, in the multivariate case, positive semi-definite). Just as it is reasonable to use non-linear models for p( ) when x, is always positive (e.g. Wooldridge (1992)), it makes sense to use functional forms for &2( ) that ensure positivity. Here, an additive parametric functional form for &2( ) is proposed as a natural extension of many ARCH models already studied. Since it is clear that a first-order polynomial in X,_ 1, cannot be always positive (unless it is trivially constant or x, has bounded support), and this is also true of any odd-order one, an even-order polynomial is required. The purpose of this paper is to discuss how to model the conditional variance a2( ) as a quadratic function of X,_ . To stress this point we shall call this formulation the Quadratic ARCH

639

640 REVIEW OF ECONOMIC STUDIES

(or QARCH) model. As we shall see, it combines many attractive features of several existing ARCH models while ensuring that estimated conditional variances are non- negative.

In particular, the QARCH model nests the recently proposed Augmented ARCH (AARCH) model of Bera and Lee (1990), which in turn encompasses Engle's (1982) original ARCH model. It also nests the linear "standard deviation" model discussed by Robinson (1991), and the asymmetric ARCH model in Engle (1990). This nesting has at least three non-trivial advantages. First, many theoretical results derived for these models still hold with minor modifications for the QARCH model, including estimation and testing, stationarity conditions and persistence properties, autocorrelation structure, temporal and contemporaneous aggregation, forecasting, etc. Second, QARCH conditional variances can also be easily integrated in economic models, just as linear time-series models for the conditional mean are (see Campbell and Hentschel (1992) for an application to a stock returns model). Third, the QARCH model is capable of improving the empirical success of ARCH models, since it avoids some of its criticisms without departing significantly from the standard specification. In particular, it provides a very simple way of calibrating and testing for dynamic asymmetries in the conditional variance function of the kind postulated for some financial series (see e.g. Black (1976) and Nelson (1991)).

The QARCH formulation can also be interpreted as a second-order Taylor approximation to the unknown function &2( ), or, alternatively, as the quadratic projection of the square innovation on the information set. Actually, this projection coincides with the projection of the true conditional variance a2 on a "quadratic information set", so that we can also understand the QARCH model as producing smooth filtered "estimates" of or (see Nelson (1992) and Nelson and Foster (1994) for a related interpretation of ARCH models as filters). In addition, we can also relate QARCH to random coefficients models (cf. Tsay (1987) and Bera and Lee (1990)).

By and large, most of the existing literature on ARCH models is concerned with univariate models (see e.g. Bollerslev, Engle and Nelson (1994) for a recent survey). Given that many issues in finance, and in particular, asset pricing theories, are related to the variances and covariances of many assets, it is of the utmost practical importance to be able to extend univariate models so as to capture time-variation in the (conditional) mean vector and covariance matrix. An additional advantage of the QARCH formulation is that it is very easy to generalize to multivariate models, either directly, or more con- veniently, through the different covariance structures suggested in the literature. In particular, QARCH formulations are the only ones that can be easily adapted to the context of conditionally heteroskedastic latent factor models, and do not entail a large computational burden. Hence, it can also be used to capture potential dynamic asymmetries at a multiple assets level.

The paper is organized as follows. Section 2 introduces the QARCH(q) model, states under what parameter restrictions all the other proposed quadratic models can be obtained from it, discusses its positivity, and interprets it both as a least squares filter of the true conditional variance u2 based on a "quadratic" information set, and as a quadratic Taylor approximation to the unknown conditional variance function. It also discusses testing procedures and its generalization to GQARCH(p, q) along the lines of Bollerslev (1986). Section 3 obtains the stationarity condition as well as an expression for the unconditional variance. Fourth moments, the autocorrelation function for the squares of the series and the covariances between these and lagged levels are also discussed. Univariate empirical applications to daily U.S. and monthly U.K. stock returns are carried out in Section 4. Potential generalizations to the multivariate case are entertained in Section 5. In particular,

SENTANA QUADRATIC ARCH MODELS 641

a latent factor model with GQARCH factors is discussed in some detail. In Section 6, an illustrative empirical application of the model to monthly data on U.K. sectorial stock returns is carried out. Specifically, a multivariate factor model is estimated for stock returns of 26 industrial sectors. Finally Section 7 concludes.

2. THE QARCH MODEL

2. 1. Definition

Assume initially that p(X,-,,,) )=O and that &2(X,_,,) is a function of the past q values of x, only, i.e. &2(X,l) = &2(X, _ I,q). The following conditional variance parameterization:

orX,- I1,q) = V + Eq I YtiXt-i+ Zq= I aijx2t-j+2 Y_j=i+I

= 0+ 'Xt, -,q+X;_i,qAXt -,q (2)

where ,i is a q x 1 vector and A a symmetric q x q matrix, is the most general quadratic version possible of the parametric ARCH variance function h(Xt- ,q; rb) considered in Engle (1982), and for that reason we shall call it the Quadratic ARCH (QARCH herein- after) model. It therefore encompasses all the examples of quadratic variance functions proposed in the literature: the Augmented ARCH model of Bera and Lee (1990), the standard' ARCH model of Engle (1982), the linear "standard deviation" model considered by Robinson (1991), and the asymmetric ARCH model in Engle (1990). The AARCH model assumes that fi = 0, whereas Engle's ARCH assumes that, in addition, A is diagonal. The linear standard deviation model assumes that 2 = ( p + g'X,t 1,q)2 which implies 0 =

p2, i = 2pq and A = oo', a rank 1 matrix.2 An obvious fourth restricted parameterization, of which the asymmetric ARCH model is a special case, is also encompassed: a model with A (band-)diagonal but ,ip 0, which we shall term the (band-)diagonal QARCH process.

The cross-product terms give an indication of the extra effect of the interaction of lagged values of x, on the conditional variance. Hence, by allowing them to be non-zero, we can account for the possibility that the occurrence of, for example, two successive large values of x, of the same sign affects the conditional variance by more than an ARCH model would allow. But the substantive advantage of the QARCH formulation vs. the AARCH and ARCH models which it nests, is that by allowing yV to take any value, i.e. by not centring the quadratic polynomial for &2( ) at 0, a dynamic asymmetric effect of positive and negative lagged values of x, on u2 is allowed. As an example, let's take the QARCH(I) model, i.e. o2 = 0 + f X, - + a IIxt- I . If vI is negative, the conditional variance will be higher when x,tI is negative than when it is positive.3 In the context of stock market volatility, this could capture the leverage effect noted by Black (1976). Hence, the QARCH model provides an additive heteroskedastic alternative to the asymmetric multiplicative heteroskedastic EGARCH model of Nelson (1991). On the other hand, in

1. Engle's (1982) original model is sometimes referred to as linear ARCH. However, the term linear is unfortunate since as (2) shows it is only linear in the squares of the past values, not in the information set X, - I. .

2. It does not encompass, though, the models of Taylor (1986) and Schwert (1989), in which the variance is quadratic in the absolute value of innovations.

3. Besides, when I <0 the absolute value of the derivative of &2(x, -I) with respect to x, -_ (= , + 2a, ,x, , ) is also higher for negative than positive x, -I. Hence, the conditional variance function is not only asymmetric, but also steeper for x, -, negative. The rate of growth of this derivative, though, is assumed to be the same (=a,,).


ARCH and AARCH models (i.e. V = 0), only the magnitude, not the sign, of x,_ affect 2 i ca,. In contrast, in the linear standard deviation model symmetry is only achievable if

p = 0 (in which case we have AARCH with a2(0) = 0), or under homoskedasticity. As we mentioned before, one of the main reasons for using a quadratic polynomial

is to ensure that our parameterization implies a non-negative variance everywhere. To see under what conditions the right-hand side of (2) will be non-negative for any X,_ 1q, let's re-write (2) as:

ip'/2 0 ][I

This quadratic function will be non-negative if and only if the matrix [,, A

t"E] and hence A, are positive semidefinite. In that case, &2(X,_ 1,q) is a convex function of X,_ l,q which reaches its minimum at any X, I,q satisfying the first-order condition AXt l,q =

- V/2, with X ilq =argmin { &(Xt1q )}. In particular, if we choose Xt'L7I,q where A+ is the Moore-Penrose inverse of A, we obtain the non-negativity condition 0 - ly'A +#/4 ?0. But from the practical point of view we are interested in parameterizing (3) in such a way that the estimated conditional variance is always non-negative. This can be achieved by writing a as follows (cf. Schwallie (1982)):

't Mt' 1q, 1)[-b' I]0' c][' I ][1 (4)

where P is a q x q unit lower-triangular matrix, D a q x q non-negative diagonal matrix, b a q x 1 vector and c a non-negative scalar, so that PDP' is the Cholesky decomposition of A. We can also write (4) equivalently as = c+(X -I,q - b)'A(Xt 1,q - b), which is particularly convenient if A is (band-)diagonal (cf. Engle (1990) or Campbell and Hentschel (1992)).

The above reparameterization is certainly unique when A has full rank. But even when A is positive semidefinite (as in the linear "standard deviation" model), uniqueness is retained if we set to zero the elements of b' and the sub-diagonal elements of P in columns corresponding to the zero diagonal elements of D.4 It is then straightforward to recover the parameters of interest, 0, q/ and A, or some combinations of them such as tr (A) from the underlying parameters P, D, b and c, by using the relationships A = PDP', w = -2PDP'b and 0 = c+b'PDP'b. The delta method (see e.g. Goldberger (1991)) can be used to derive standard errors and carry out Wald tests on the parameters of interest. A word of caution is in order, though, since Wald tests are not usually invariant to the parameterization used. For instance, in the QARCH(1) case, the null hypothesis of no "leverage effects" can be equivalently expressed as vyr=0 or b1=0 when all?0. But whether we actually test one or the other results in a different numerical value for the Wald test. In a MLE framework, the likelihood ratio test is invariant, but is hampered by its non-robustness in quasi-MLE procedures. Robust LM tests a la Wooldridge (1990) offer an attractive alternative.

4. In order to optimize an objective function with respect to D, P, b, and c, when A is suspected to be singular, it is advisable to use a constrained algorithm which ensures that d and the elements of D are non- negative, and drops dj, bj and 1i, (i>j) from the current active set if the constraint dj>O is binding, instead of carrying out unconstrained optimization in terms of new "free" parameters v), with d>= vj (see Gill, Murray and Wright (1981)).


2.2. Potential reinterpretations of QARCH models

Just as AR(r) models can be understood as first-order Taylor approximations to the unknown conditional mean function p( ), the QARCH formulation can be intuitively interpreted as the two leading terms in a quadratic Taylor approximation to a2( ). There is a closely related interpretation in terms of projections. Provided that E(x, IX,_ l- ) = 0, a, is also the conditional mean of x24 E(X21X,_ ,"). Under the assumption that E(x4) < oo, it is well known that o2 is the (least squares) projection of x2 on B,1, where B,tI is the collection of all measurable random variables with finite second moments generated from X,_ O.

In this framework, the QARCH formulation may be understood as the least-squares projection of x2 on a "quadratic information set" Q, I B,t spanned by linear combinations of the vector t,,=-=vech [(, Xt,_o)', X,_ l o )] = {1, xt_ I, Xt-2*

2 2 .. sal t rjcin x_, X,XlXtX2,X,iX,-3,...; x,_2,xt-2x,_3,..-3 We shall refer to such a projection, U,Q, as the quadratic projection of x4 on X, Importantly, by the law of iterated projections (see e.g. Hansen and Sargent (1991)), 7UQ is also the projection of cr2 on Q,-t . Hence, E(U2) = E(a2Q) a2, projection and projection error (i.e. CrQ and a2 _- c2) are uncorrelated, and V(a2Q) < V(a2 ). Therefore, we can perhaps better understand ca2 as a smooth, filtered "estimate" of at.

As usual, the prediction error a2 _ a2Q will be orthogonal to any linear combination of lag values of x,, its squares or cross-products, but not independent because of the heteroskedasticity. Furthermore, all QARCH parameters can be interpreted as the (theoretical) regression coefficients of x2 (or a2) on X,o I For example, consider the QARCH(1) model where a2 =0+ Vx_1 +a, I2_. Since this can be re-written as X,2= 0+ flx,t I +a, x,2_ I + i,, with 1,-x,2 _- T2, under the additional assumptions that E(4) = 0 and all necessary moments exist and are bounded, a, I = cor ( x,4x) and 1= cov (x2, x,t )/V(x,). Hence, a,, is a measure of the correlation in the square series, while VI measures dynamic asymmetry. An important result is that if the joint distribution of X,, is symmetric and the conditional variance function is also symmetric (i.e. a 2(Xt_ I 0) = a2(-Xt_ 10,)), then the quadratic projection will be symmetric too. The reason for this is that cov (X2, x,j.) = E(cr2(X,_ X O )x,-j) + E([X2-_ a2(X,t )]x,j) = 0, so yrj must be 0 for all j. This actually suggests that cov (2, x,j) could be a useful tool to identify potential dynamic asymmetries even if QARCH is not the true model, since when &2(X,_ l) # a2(-X,1 this third moment may differ from 0.

Given that the QARCH formulation is linear in ,t-, its interpretation in terms of projections is rather obvious.6 What might be perhaps less obvious is that QARCH formulations understood in this weak sense are closed under both temporal and contemporaneous aggregation. In this respect, the result discussed in the previous paragraph confirms those of Drost and Nijman (1993) and Nijman and Sentana (1994) in the context of temporal and contemporaneous aggregation respectively of GARCH processes. These papers show that if such terms do not appear in the quadratic projections of the underlying series, then they do not appear in the projection of the aggregated one either. But it is easy to prove that if one series shows dynamic asymmetry, then so will the aggregated one (see Drost (1993)).

5. Drost and Nijman (1993) Weak ARCH concept is a particular case of what we call quadratic projections, in which linear and cross product terms are not included.

6. The same is true of AR(r) models, which may be understood as approximating the conditional mean p (X, l.r) by the Minimum Mean Square Error Linear Predictor of x,, say p,CL= a + y'X,_ I,. Again, this yields the familiar interpretation of yj as a (theoretical) regression coefficient.


2.3. Estimation and testing for QARCH effects

Given the interpretation of QARCH models as quadratic projections of x2 on X,_loc, a consistent method of estimating the parameters is provided by the OLS regression of 4 on the relevant elements of X, - (cf. Weiss (1986)). The preferred method of estimation for ARCH models, though, has been maximum likelihood, but since this involves a non- linear procedure, it is of some interest to have a simple preliminary test for the presence of QARCH effects. For the ARCH(q), Engle (1982) proposed an LM test which can be computed as TR2 of the OLS regression of x4 on a constant and its first q lags. This test is distributed as a x2 under the null of no ARCH even if x, is not conditionally Gaussian provided that the fourth conditional moment of x, is constant and finite (see Koenker (1981)). Bera, Higgins and Lee (1992) have extended this test to the case of AARCH(q) by including cross-product terms of the form x,-ix,-j in the above regression, resulting in a Xq(q+ 1)/2 null distribution. It is straightforward to check that if we also add the first q lags of x, as regressors, an LM test for QARCH(q) is obtained, which will be distributed as q(qq+3)/2 under homoskedasticity. It is worth noticing that this test is the dynamic analogue of White's (1980) general test for static heteroskedasticity, which suggests that it may also have good power against most other dynamic heteroskedastic alternatives. As White's test, it can also be derived as a test for random coefficients or as a selective dynamic information matrix test (cf. Bera and Lee (1993)).

But as in the ARCH(q) model (see Demos and Sentana (1994)), some of the true parameters lie at the boundary of the parameter space under the null of no QARCH. Although this does not affect the distribution of the LM test, intuition suggests that a more powerful test could be achieved by taking the (partially) one-sided alternative into account. For the sake of clarity let's consider testing against a QARCH(1) alternative. As we have seen, the two-sided LM test is based on the regression of x4 on a constant, 42_I and x,- 1. Notice that if x, is unconditionally symmetrically distributed, the regressors are orthogonal under the null, and we would expect each OLS coefficient to take any sign independently of the other. But under the alternative we would expect the coefficient of X4 to be positive. Hence, a partially one-sided test of Ho: = =a,, =0 vs. H,: V/, I0, a,, > 0 seems more appropriate. The test statistic will be the result of adding to the square of the t-ratio associated with x,_1, the square t-ratio for x2_1 when the coefficient is positive (cf. Yancey et al. (1980)). Under the null this statistic is a 50:50 mixture of x2 and X2

All the above tests may have the wrong asymptotic size if the assumption of conditional homokurtosis does not hold. Nevertheless, Wooldridge (1991) has recently suggested a simple modification that can be used in the two-sided LM tests for QARCH to allow for heterokurtosis under the null. His robust procedure would involve regressing a constant on the product of de-meaned x2's times de-meaned lagged values of x,, its squares and cross-products, and using T- SSR as asymptotically X2(q?3)/2, where SSR is the sum of squared residuals from the regression.

In some cases, more robust versions of one-sided LM tests are also possible. Let's consider again the QARCH(1) example above with an unconditionally symmetric distribution, and let's suppose for simplicity that the conditional kurtosis, K,, is a function of x, 1 only, say KC(Xt, 1). A partially one-sided LM test could be computed from the same regression as Wooldridge (1991) by adding to the square of the t-ratio associated with de- meaned x, 1 times de-meaned x2, the square t-ratio for de-meaned x2_ 1 times de-meaned

7. It can be proved that the distribution of the corresponding LR and Wald tests is the same. To do so it is more convenient to work with the one-to-one reparameterization a, = o + 20o0x,- I + (O,, + 4021 /4o)x,.1 If x, is not unconditionally symmetric, the mixing weights in all three tests will change.


4x when the coefficient is positive. Under the null of conditional homoskedasticity, this statistic is a 50:50 mixture of x2 and x2 even if ict is time-varying, provided that K( ) is an even function of x, 1. Another important case arises when we have an ARCH(l) null and want to test if V/1 is significantly negative. The robustified two-sided LM test would consist in the squared t-ratio from the regression of 1 on the regression residuals of (x,/&l)- 1 on a constant and (x0_l/u) times the regression residuals of (x, '/c) on a constant and (x '/at), where a iS the estimated ARCH(1) variance. If the conditional distribution of x, is symmetric, we would expect the regression coefficient to be negative, and hence a more powerful LM test could be computed as the squared t ratio when the coefficient is negative and zero otherwise. Such a test would have a 50:50 mixture of %

and x2 distribution under the null, and the relevant critical value would be 2 7 instead of 3-8. In both cases, though, there is a trade-off between the increase in power from using the one-sided test and the reduction in robustness derived from the (weaker) maintained assumptions about higher-order moments.

2.4. GQARCH(p, q) models

So far we have maintained the assumption that the relevant information set contains only a finite number of lags, q. But this may be too restrictive (for instance, Nelson's (1992) consistency results depend on q being unbounded, and the projection interpretation would also generally lead to infinite q). Besides, even if q were finite, estimating the model for large q will be difficult as the number of parameters in A is 0(q2), and we would need to impose some structure on this matrix. For example, one could introduce a rank k (k < q) structure in A (e.g. if k = 1, we would thus obtain the linear "standard deviation" model) but even in that case, the number of parameters could still be excessive for q large. Perhaps the most natural way to solve this problem is by introducing a declining structure on the coefficients. Just as in conditional mean models, this structure can be either rapidly decay- ing or more slowly so (as in Robinson (1991) or Baillie, Bollerslev and Mikkelsen (1993)). In particular, an exponentially declining lag structure can be obtained by including lagged values of o2 on the right-hand side of (3), as in Bollerslev (1986). That is:

( = 0+V11X,-1,,q+Xt1-1,qAX,_1,q+ff ij 15.72 (5)

By analogy with Bollerslev's (1986) GARCH and Bera and Lee's (1990) GAARCH models, we shall term these models Generalized QARCH models of orders p and q, or GQARCH(p, q) for short. As in the case of ARMA models, these GQARCH models will generally result in longer memory models with a flexible lag structure, which, at the same time, could offer a more parsimonious approximation to the conditional variance function. For instance, the general GQARCH(1, 2) model

(1 -SL)r2=O + yvx,t + yf2x,t2+al1x,, +a22x,_2+2a12x,tIx,t2 (6a)

is equivalent for S1 < 1 to the following tri-diagonal QARCH(oo) model:

o =0/(1-31)+ ViXt_I+a1, 0/2 (v 2+S1 ,)37l X,-j

+Z02(a22+SaII)2 +2+2 8aa2112x,_jx,1xj2. (6b)

For higher-order (invertible) models the pattern of coefficients of A,, and W will become more complex, but notice that A,, will be band-diagonal with 2q (off-)diagonals, where q is the order of the QARCH part. As Drost and Nijman (1993) and Nelson and


Cao (1992) point out for the standard GARCH(p, q) model, requiring the positivity of the finite QARCH part plus Sj>O for all j is unduly restrictive except in the GQARCH(1, 1) case. Conditions for the positivity of the conditional variance in (5), could be obtained as an extension to O., ip.O and A,,, of the methods for finite q in Section 2.1.8

3. PROPERTIES OF THE QARCH MODEL

3.1. Stationarity conditions and moments of the unconditional distribution

It is possible to prove (see Appendix) that a GQARCH(p, q) model is covariance stationary if E'7 aii + 8j is less than 1. The unconditional variance of x, is then given by

V(x,) = 0/(I -LE,= 1 ai i= - 8j). 7

Notice that the covariance stationarity of x, does not depend at all on the linear term in the conditional variance, #P'X,- I,q, only on the quadratic term associated with the matrix A. Loosely speaking, it is as if the quadratic term asymptotically dominates the linear one. Besides, the stationarity condition for GQARCH (and hence GAARCH) is the same no matter what the off-diagonal elements of A are, and so it coincides with that of the nested GARCH model. Notice also that the actual value of the unconditional variance does not depend on yi. Besides, as (7) does not depend on the off-diagonal elements aij, the unconditional variance of a GAARCH(p, q) process equals the unconditional variance of the GARCH(p, q) process obtained from it by setting the off-diagonal elements of A toO.

The formal proof of the above result is obtained by re-writing the GQARCH process as a random coefficients model as in Tsay (1987) and Bera and Lee (1990). But an heuristic proof of (7) may help us understand the reason for the similarity of the unconditional variances for GQARCH, GAARCH and GARCH models. Let's suppose that E(x,) is bounded and therefore equal to a2 = E(a2 ). Then, if we use the parameterization of the conditional variance given in (8) (i.e. _t= O+Z y,1 ,x, = aiix2_ i+ 2 E<. a,-x,_ x, +>j a2_j), take expectations at both sides and solve for a2, we get exactfy expression (7$ as both the linear terms and the cross-products vanish because xt is a zero-mean uncorrelated process.

As in standard GARCH models, the sum Eq aii + Ej=I 3Aj provides a measure of the persistence of shocks to the variance process (but see Nelson (1990)). Hence, we can analogously define Integrated GQARCH process as those for which this sum is 1 (cf. Engle and Bollerslev (1986)). For instance, the IGQARCH(1, 1) will be defined as

2 +_ - +6U a, = 0 + yt,x I + (1 - )X, + 3_ , which shares with the IGARCH( 1, 1) the property that E(2+j I X,, )=jo + a2 9

In order to discuss higher moments, let's define x* as the standardized variable associated with x, (i.e. x* = x,/at). So far we have mostly assumed that E(x* IX,_l =0, and E(x7*2 IX, 1 ,o ) = 1, but if we assume that x, X,_ is symmetric, so is the unconditional distribution of x,. To obtain unconditional fourth moments, assume for simplicity that x* is i.i.d. with finite fourth moment K. Provided that the appropriate moments are

8. See Nelson and Cao (1992) for some GARCH cases and Demos and Sentana (1991) for several GQARCH ones.

9. IGARCH processes are clearly not covariance stationary, but they are strictly stationary and ergodic (cf. Nelson (1990) and Bougerol and Picard (1992). Given that the behaviour of GQARCH processes is dominated by the quadratic terms, one would expect a similar result to be true of IGQARCH.


bounded, by Jensen inequality the degree of leptokurtosis of the unconditional distribution of any zero mean conditionally heteroskedastic model is higher than the degree of leptokurtosis of the assumed distribution for x*. In general, though, it is very tedious to derive closed-form expressions for the leptokurtosis of GQARCH models.

The GQARCH(1, 1) process is an important exception. Assuming symmetry for x,*, it is possible to show that, provided Ka21 + 2a1 1, +432< 1, the unconditional fourth moment of xt is given by:

(I

- 1cl, - 2a 1s,

-O- [0(1 + a, I +1 ) + 21 (8)

which reduces to the expression in Bollerslev (1986) for v' =0 and ic =3 (i.e. normality). However, if V,I #0, the GQARCH(1, 1) process is more leptokurtic than the GARCH(1, 1) model which it -nests, although the condition for boundedness of fourth moments is the same (see Bollerslev (1986)).

3.2. Dynamic correlation structure

QARCH processes are uncorrelated but not serially independent. As an example, Bollerslev (1986) proves that in the GARCH(p, q) case the autocorrelation functions for x2 corre- sponds to that of an ARMA [max (p, q), p], and suggests using the sample autocorrelations as a tool for tentatively selecting the orders p and q. In the general GQARCH(p, q) model, though, the process for x2 is more complicated. Nevertheless, under the assumptions that x* is symmetrically distributed and the unconditional fourth moment of x, is finite, it is easy to see that for k > max (p, q - 1) the autocorrelations, Pk, follow the same Yule- Walker recursion as the GARCH(p, q) model, whereas for k<max (p, q- 1), they will depend on all the parameters (0, Vi, A, 8).

In particular, the autocorrelations of the squares of the GQARCH(1, 1) process are exactly the same as those of the GARCH(1, 1) model. As a matter of fact, the similarity between GQARCH(1, 1) and GARCH(1, 1) is remarkable: they both have the same unconditional mean, variance and autocorrelation functions for both the series and its squares. Nevertheless, the GQARCH(1, 1) has the advantage that with a single extra parameter (Vyr) it allows for both an asymmetric effect on the conditional variance and higher kurtosis than the GARCH(1, 1) process which it nests, and therefore, it goes in the right direction towards capturing some of the stylized facts characterizing many financial time series.

As we saw earlier on, the asymmetry in the conditional variance can be captured more formally by dynamic third moments of the form cov (X2, x,_j) = pfoj , where

= ij=cov (xt, x,t , x,_j) denotes the bicovariance function of xt (see Hinich (1990)).?0 In the GQARCH(p, q) case, 3o,j also follow an analogue of the Yule-Walker recursions for j> q. This apparently surprising result is again due to the quadratic structure of the conditional variance function, in which the square terms eventually dominate all others. These moments, though, have to be understood as unconditional, and therefore unable to capture conditional asymmetries. The AARCH(2) model provides an interesting example in which flojj=0 for allj, but E(x x,1 IX,t-2,)=2aI2x,t-2t-i*1.

10. Notice that our assumptions imply that the remaining elements of the bi-covariance function are all zero.


4. UNIVARIATE APPLICATIONS

4.1. Daily U.S. stock market returns

The analysis of financial time series has turned out to be the most fruitful application of conditionally heteroskedastic models (see Bollerslev, Chou and Kroner (1992) for a recent survey). ARCH-type models have been used mainly as a reduced-form description of conditional variances, but more recently, they have also been used as building blocks for structural economic models with time-varying volatility. The asymmetric model of changing volatility in stock returns of Campbell and Hentschel (1992) provides an illustrative example in which the QARCH model discussed here is used to develop a formal model of volatility feedback.

As part of their empirical exercise, Campbell and Hentschel (1992) estimate a restricted version of the general GQARCH( 1, 2) model in (6a) for daily U.S. stock market returns over the period 1926-1988 (a total of 16,981 observations). For tractability reasons, they impose the restrictions a,2=0 and YV2/a22= =V, /a,l, so that their model can then be written as 52 = c+a11 (xt-, -b, )2 +a22(X,- 2-b, )2 + S1o-,2_ l. This restricted model gener- ates non-negative variances whenever c, a,,, 3,> 0 and a22 ?-3,a,i (see Demos and Sentana (1991)). Campbell and Hentschel (1992) find that such a model appears to capture most of the "leverage effect", and that unlike a symmetric GARCH model, it does not significantly overestimate average risk premia.12

In their application, the parameter b,, which measures dynamic asymmetries, is estimated to be positive and very significantly different from zero in a Gaussian MLE framework. But their results also suggest that the maintained assumption about conditional Gaussianity is rejected by the data. Although this does not make the normal pseudo-MLE inconsistent, the sizes of standard Wald, LM and likelihood ratio tests could be affected. For that reason we have computed a robust version of the Wald statistic (as in Bollerslev and Wooldridge (1992)), which shows that the significance of b, is not an artefact of a misspecified distribution.'3

We next test their restricted parameterization against our most general GQARCH(1, 2) model. A robust LM test of the two restrictions (see Wooldridge (1991)) produces a value of 7 32, which is significant at the 5% level (X 2o o5 = 5 99). The parameter estimates of the general model together with standard errors are reported in Table 1. Importantly, we have estimated the model both in terms of the free elements in P, D, b and c, as well as in terms of the parameters of interest 0, A and yr directly. The first procedure guarantees non-negative variances in and out of sample (see Demos and Sentana (1991) for details) while the second does not. In our case, both set of estimates coincide, which is reassuring.

11. Conditional asymmetries are important to distinguish how different ARCH models capture leverage effects or other dynamic asymmetries. For instance, in the conditionally normal GQARCH(l, 1) model cor (a2, , X,-2., ) approaches ?1 as a2- 1-*O, and 0 as -2 00oo. Hence, "leverage effects" are relatively more important in times of low volatility. In contrast, cor (in a2, x,I X,-2,, ) is constant for the EGARCH model, so these effects have the same importance regardless of the level of oJ

12. To understand why, consider a GARCH model with conditional mean p in which the conditional variance is modelled in terms of squared innovations (x, - _P)2. Here, p is also forced to capture dynamic asymmetries, and as a result, it is usually overestimated. Unlike GARCH and GAARCH models (cf. Bera and Lee (1990) and Tsay (1987)), a quadratic ARCH is obtained for linear conditional mean models whether we use innovations or the actual series to model volatility.

13. Our data set differs slightly from the one in Campbell and Hentschel (1992), in that we use arithmetic daily returns instead of geometric returns over the monthly average of the safe rate, but this should not affect our qualitative conclusions. Our results are also insensitive to relaxing the assumption of constant conditional mean by including the conditional variance or an MA(I) component.


TABLE I

Gaussian pseudo-maximum likelihlood parameter estimates of a GQARCH(1, 2) model

U.S. daily stock returns January 1926-December 1988 (16,981 obs.)

2 = y + 2_ -

at 6+ V16,-1 + V28,-2+a,,e,-1 +a22,--2+2a12e-19,-2+3,t-,

=0/(I -81i ) +a,l2_ tE,- a2 lll)12_j

+ yi,x,_ | +,2 (V2 + i' )3 2x,1 +52 VIr a124 X,jXt,-j1.-

Parameter Estimates

r 005352 (0 00543) [0 005731 {0-00541} 6 0 00859 (0 00075) [0-000611 {0.00095} VIl -0 09857 (0.01013) [0 006381 {0-02010} V2 0-06119 (0-00996) [0-006881 {0.01538} all 0-13472 (0-01026) [0-005201 {0.02088} a,2 -0-00029 (0.00263) [0-001821 (0 00387} a22 -0 05926 (0-01056) [0.017971 (0 02073} 6, 0 91790 (0 00362) [0-002541 {0.00532}

Log-likelihood -21,100 07

Note: Hessian-based standard errors in parenthesis, outer product-based standard errors in square brackets, robust standard errors in curly brackets.

Then we check whether an even more general GQARCH model is required. On the basis of robust LM tests, it seems that extra terms in ,-t3, Ei_26,.3 and s,_ I?t3 are required. Parameter estimates together with robust standard errors for such a GQARCH(1, 3) model are reported in Table 2. The "persistence" measure given by all +a22 +35l(=0 99389) is very close to 1, but a robust Wald test indicates that it is significantly smaller.'4 Note that the estimated model is effectively a 5-band-diagonal QARCH(oo) model, in which the coefficients for yfj, at,, aj, I and ajj12 decay exponentially at the rate 31 after just a few free lags. Interestingly, the effects of the first lags of levels, squares and cross-products are the largest, since the estimates of 3, a22 and a23 have the opposite signs to those of I/fI, a Il and a,2 . In the case of cross-product terms of the form E,_jc,_j_ the effect is dramatic: they practically disappear after the first lag. In fact, a robust Wald test confirms that a23 + 3Ia,2 is not significantly different from zero, which explains why the coefficients on c,_jct_j_ I were insignificant in the GQARCH(1, 2) model in Table 1. The cross-product terms 61_jE6_-_2 have also very small coefficients (000832, 000767, . . . ). The leverage effect is also short lived, since after taking the values -0' 09483 and -0-08911 for the first two lags, it drops to -002070, -0-01907, etc. In contrast, the ARCH effect decays less dramatically from 0 14286 to 0'06084, 0 05608, etc. Therefore it seems that there is an implicit ranking of effects: the standard ARCH effect is the most important, followed by the "leverage effect" and finally the cross-product terms.

To see if the estimated model adequately captures the autocorrelation in the square returns we have computed Ljung-Box statistics up to 20-th order for the standardized residuals. We also use robust LM tests to see whether non-quadratic terms such as I E? -I i, I s,-21 and I ?- l are required in the variance. Since we already have ?,_- , ?,-2 and 5t-3

in, this equivalent to testing an alternative model that imposes no restrictions in the way positive and negative lagged innovations affect the conditional variance (cf. Engle and Ng

14. Recent work by Lumsdaine (1993) and Lee and Hansen (1994) shows that the Wald test for IGARCH has an asymptotic normal distribution in the GARCH(1, 1) model. We assume here that the same is true for the QARCH model, although it would be non-trivial to prove.


TABLE 2

Gaussian pseudo-maximum likelihood parameter estinmates of a GQARCH(1, 3) model

U.S. daily stock returns January 1926-December 1988 (16,981 obs.)

r, = r + E,.

ta2= + a,,l 4, +a226,2-2+ VI S,-, + /26-2 + /3gt- 3

+ 2al26,_l,-,2 +2a238,_26t_3 +2a,3E,- ICt_3+ ,a_1

- 9/(1 - 3, ) + a,l c2,-1 + (a22 +3 IaII ) 7j=0 I t-2-j

+2a2e, -,e,-2+2(a23+3ala2) E.j 8 16,-2-j-,-3-j +2a13 jOO 5jl,'e-l-jt-3-j-

Parameter Estimates

r 0-05208 (000540) [0-005701 {0.00537} 0 0-00805 (0-00072) [0 00060] {0.00090} VI, -0-09483 (0-01028) [0 006391 (0.02036} V2 -0-00169 (0-01450) [0-009401 {0.02786} Y/3 0-06145 (0-00920) [0-006181 {0-01768} al, 0-14286 (0-01055) [0-006381 {0-01998} a,2 -0-02709 (0.00751) [0-005541 {001 104} a22 -0-07086 (0-01077) [0-006811 {0-01961) a,3 0-00832 (0-00252) [0?001951 (0 00355} a23 0-02842 (0 00731) [0-00546] (0-01058} 3, 0 92189 (0 00345) [0.002431 (0005101 Log-likelihood -21,061 18

Note: Hessian-based standard errors in parenthesis, outer product-based standard errors in square brackets, robust standard errors in curly brackets.

(1993) or Hentschel (1994)). Both sets of tests suggest that the purely quadratic formulation adequately describes the conditional variance.

4.2. Monthly U.K. stock market excess returns

We also look at the monthly FTA500 excess return series, rt, for the period 1971:2 to 1990: 10. Given the small number of observations (237), we initially fit a GARCH(1, 1)- M model on the basis of a Gaussian pseudo-log-likelihood function (see Table 3). Note that the price of risk coefficient is positive and significantly different from zero. However, as in the U.S., the average estimated risk premium is substantially higher than the mean excess return.

It is worth mentioning that any GARCH-M model already allows for correlation between r2 and r,_,, since a2 is symmetric only in the innovations. To check if there is evidence for an extra dynamic asymmetric effect, we have computed a robust LM test for the inclusion of Et- I in the conditional variance along the lines of Bollerslev and Woold- ridge (1992).'5 The x2 test statistic takes the value of 3-9, which suggests that it might be worth estimating a GQARCH(l, 1) model.

The parameter estimates together with robust standard errors are reported in Table 3.16 Note that the Wald test (t-ratio = -2-73) confirms that yr is significantly negative. Also

15. As the variance parameters also affect the mean, the computations are more complicated than in Section 4.1 because a matrix regression is necessary.

16. In the GQARCH(I, 1) case, positivity of the variance is achieved if a,,, 3, _0 and V1' <4a,1O. To impose these restrictions, we have estimated the likelihood function with o = c2+1,1( e, ,-b, )2+dlda,2,, so that O=c2+l12l,b2, a, =1,2;, 3a=d2 and =-2l1,2b,.


TABLE 3

Gaussian pseudo-maximum likelihood parameter estimates of GARCH(I, l)-M and GQARCH(l, l)-M models

U.K. monthly excess stock returns 1971:2-1990:10 (237 obs.) r,=pa2+ _e.

CF2-a,~+ VI E_ I+a?, 12 + 41,_ .2

GARCH GQARCH Parameter estimates estimates

p 024348 0-16669

(0.07728) (0 08091) [0 080671 [0-088151 {0* 10514} {0.09905}

0 0-08926 0-14682

(003797) (0 05159) [0-052051 [0-040561 {0.059281 (0-11980}

Y/t -0-23969

(0.06786) [0-081441 {0-087781

all 020752 0-09783 (0-07198) (0-05324) [0.099411 [0-057341 {0-09690} {0-06994}

31 0 67928 0-66351

(0 08757) (0-11312) [0146861 [0-094711 {0.07172} {0-22878}

Log-likelihood -275 147 -268-081

Note: Hessian-based standard errors in parenthesis, outer product- based standard errors in square brackets, robust standard errors in curly brackets.

note that the ARCH parameter a,, is smaller in this case, while the GARCH parameter S, is roughly the same, which taken together implies a smaller persistence of the volatility shocks. The difference between both models is even more obvious graphically. Figures 1(a) and I (b) plot the conditional standard deviation of returns implied by the GARCH and GQARCH models around the two most significant episodes in our sample: the October 1987 crash (a 26 3% drop in stock prices) and the January 1975 bounce back (a 51-6% surge). Both models capture the effect of the 1987 crash in a very similar manner, and the same happens for the bearish 1974. In contrast, the period immediately after January 1975 is noticeably different. The problem with the GARCH(1, 1) model is that it treats both episodes in the same manner, despite that the rest of 1975 was not particularly volatile.

The implications for risk premia are also different. The price of risk parameter is smaller in magnitude and less precisely estimated in the GQARCH(1, 1) model, but still significantly positive if we consider the relevant one-sided test. Again, the average estimated risk premia is much closer to the mean excess return. Figures 2(a) and 2(b) show the


3 (a)

2*5

GARCH(l,l)

2

1X5

05

0

1 73 1 74 1 75 1 76

Estimated Conditional Standard Deviation

3 (b)

25

2

1-5

I - GARCH(l1,1) | \

0.5

GQARCH(1,1)

86 i 87 1 88 1 89

Estimated Conditional Standard Deviation

FIGURE I


difference very clearly. Like in the rest of the sample, the GARCH model implies a higher required return than the GQARCH one around October 1987, but with a similar time profile. However, in 1975 the GARCH model implies a maximum required return of 200% on an annualized basis, and a much slower reversion to the mean, which seems rather implausible.17

5. MULTIVARIATE EXTENSIONS

Most of the existing literature on ARCH models (including the preceding sections) is concerned with univariate models. Given that many issues in finance, and in particular, asset-pricing theories, are related to the variances and covariances of many assets, it is of the utmost practical importance to be able to extend univariate models so as to capture time-variation in the (conditional) mean vector and covariance matrix.

Multivariate generalizations of the QARCH model are straightforward in theory. Let y, be a multivariate stochastic process of dimension m whose conditional mean is 4,=

E(y, t Yt- 1,)-p( Y,t ,,) and whose conditional covariance matrix is t-= V(y, Yt- l ,) =

I( Y, - loo ) where Y,- l= vec (y,- I, y,-2,. . . ). Again, for the sake of clarity, we shall deal initially with the case in which the dependence of the conditional moments on the past is limited to a finite number of lags of y,. Let Y,_ ,q=vec (Y,- l, Yt -2, .. ., y,-q) be the mq x 1 vector containing the values of the m series for those q lags, and let s, =

vech (I, ) = vech [(( Y,t- ) )] S( ft,- l,q ), be the vector-valued function which contains all the distinct elements of the conditional covariance matrix. Since I, contains m conditional variances and m(m- 1)/2 different conditional covariances, the dimension of s, is n =

m(m+ 1)/2. For simplicity let's assume that p(Y,- ,,,)=0 so that s,=E,-, [vech (y,ty'). It is again clear that only an even-order polynomial can guarantee the positive (semi-)definiteness of the conditional covariance matrix for all possible values in the condi- tioning set. On this basis, we can define a multivariate QARCH model by using second- order polynomials as follows:

S(Yt-1,q)= 0+ TYt - 1q +(In 0Yt'_1,q)A Yt -1,q (9)

where E) is a n x 1 vector, T a n x mq matrix and A a nmq x mq matrix of coefficients. By analogy, we shall call this formulation the Multivariate QARCH(q) model.'8 If we partition the matrix A in n square blocks of size mq, and if, in addition to T being 0, each one of these n blocks is in turn a block-diagonal matrix with q square blocks of size m, the right- hand side of equation (9) reduces to the multivariate ARCH(q) model introduced by Kraft and Engle (1983):

s,t= Bo+ ,q_ Bicot-i (10)

where Bo is a n x I vector and the Bi's are n x n matrices and c,= vech (y,y,). On the other hand, when T = 0 but the blocks of A are not block-diagonal, equation (9) constitutes a multivariate generalization of the AARCH model.

Sufficient conditions to guarantee the positive (semi-)definiteness of S, can be obtained by rewriting the multivariate QARCH(q) model above as a random coefficients VAR(q) model for y,. Just as in the univariate case, one can then reparameterize in terms of the

17. Again we have computed Ljung-Box statistics on the standardized residuals and tested for lagged absolute innovations in the conditional variance by means of robust LM tests without being able to reject the GQARCH(1, 1) specification.

18. Infinite dependence on the past can again be achieved by including lagged values of s, on the right- hand side of (9) as in Bollerslev, Engle and Wooldridge (1988).


2 (a)

1-75

1-5

1\25

1 GARCH(l,1)

075

0-5

0

0225

1 73 1 74 1 75 1 76

Estimated Risk Premia

2 (b)

1-75

1*5

1*25

0*75 -

GARCH(l,l) 05 -

025 -

0 GQARCH( l, l)

-0-25 I I 1 1 A I I I I I I I I I I I I I I I I I ,I I

I 86 1 87 1 88 1 89

Estimated Risk Premia

FIGURE 2


Cholesky decomposition of the covariance matrix of the random coefficients. The main practical problem, though, is the sheer number of parameters involved. Even the multivariate ARCH(q) model contains n + qn2 parameters, and in practice further restrictions have been imposed (e.g. the diagonal ARCH model used in Attanasio and Edey (1987), Bol- lerslev, Engle and Wooldridge (1988), or the BEKK model in Engle and Kroner (1995)).

But as in standard multivariate ARCH models, there are several alternatives to sim- plify the multivariate QARCH model above. One would be a k-factor QARCH model (as in Engle (1987)) in which k linear combinations of the y,'s follow univariate QARCH processes. Another would be a model in which the variances follow QARCH processes but the conditional correlation structure is held constant (as in Bollerslev (1990)). Finally, a conditionally heteroskedastic factor model of the type introduced by Diebold and Nerlove (1989), and extended by King, Sentana and Wadhwani (1994) but with QARCH-type effects on the underlying factors can also be entertained. Such a model is particularly appealing in financial applications, since the concept of factors plays a fundamental role in major asset pricing theories such as the Arbitrage Pricing Theory of Ross (1976). Besides, it automatically guarantees a positive (semi-)definite conditional covariance matrix for y, once we ensure that the conditional variances of the factors are non-negative.

However, much care has to be exercised when dealing with conditional variances that depend on past values of the underlying factors, ft, as the true value of the factors do not necessarily belong to the econometrician information set, Y, 1,". An argument similar to that in Harvey, Ruiz and Sentana (1992) shows that if the variance of each of the factors conditional on an information set that also contains past values of the factors is given by a GQARCH(p, q) model like (5), the variances of each of the factors conditional on Y,- ,oot, Rt-I= V(ft1,IY,1, ), will be given by:

At,t_-1 = OI+y I = l i t i -i + Ei#oj a1yfi,-iit-ift-ft-j

+ E. aziif fitr-il t - i + clt-it - + 4"lt li t flj (1

where fit - il t-j = E(fi,t- i Yt-j,,, ) and co_ 0l h-ii-j= V(fi,,- i I Yt -oo ). Notice that the only difference between expression (11) and a pure GQARCH(p, q)

variance like (5) is the inclusion of a correction in the standard ARCH terms which reflects the uncertainty in the factor estimates, but not in the AARCH or linear terms. Under some additional assumptions, this correction can be easily evaluated via the Kalman filter. In this respect, it is worth mentioning that the Kalman Filter prediction, updating and smoothing equations derived in Harvey, Ruiz and Sentana (1992) for the latent factor model under the assumption that the conditional distribution of the factors is proportional to a (standardized) multivariate t remain valid here since they do not actually depend on the particular functional form adopted for the conditional variances. But a non-trivial advantage of the QARCH formulation in the context of conditionally heteroskedastic factor models is that it would be very difficult to handle analytically any functional form other than the quadratic for the conditional variances of the underlying factors.

6. A MULTIVARIATE APPLICATION TO U.K. STOCK RETURNS

As we have seen, one of the strengths of the QARCH formulation is that it is also able to detect dynamic asymmetries at a multiple asset level which could not be captured by multivariate ARCH models. To gauge their empirical relevance, we have jointly estimated by Gaussian pseudo-maximum likelihood a conditionally heteroskedastic latent factor


model of the type discussed in Section 5 for excess stock returns on 26 U.K. sectors for the period 1971:2 to 1990:10 (see the Data Appendix for details).

Given the relatively small number of observations (just over 200), we have only considered GARCH(1, 1) and GQARCH(1, 1) parameterizations for the common factor, and constant conditional variances for the idiosyncratic terms.'9 To model the conditional mean, we use the dynamic version of the APT in King, Sentana and Wadhwani (1994).2? Specifically, if r, is the vector of 26 sectorial excess returns, the model that we estimate can be written in vector notation as:

rt= CA1,-% ,p+ cft+ wJtC(pt1t-,1 +f,) + wt (12)

where f, is the common unobservable factor, w, represents idiosyncratic risks, c is the vector of factor loadings, p is the price of risk parameter, , , = V(f,l r,t,...)= O+iv,E(f,lir,t,,.. .)+a,iE(fr llIrt-,,.. .)+361A,)-1t-2, V(w,Ir,_,,. . .)=F diagonal and E(ffw,j r, - I,. . 0.2

The novel result that we obtain at the multivariate level is that there appears to be a significant leverage effect in the sector returns through the common unobservable factor. The robust Wald statistic for VY/ = 0 is 6 02. Comparing the two estimates of the conditional variances of the unobservable factor, the "leverage effect" is again particularly noticeable around January 1975. The estimate of p is positive for both conditional variance parameterizations, but once more it is too large in the GARCH case, which leads to overestimated risk premia.

The estimates of the common component in the factor model (i.e. pAt,- +f,t) for the GQARCH(l, 1) and GARCH(1, 1) formulations, though, are remarkably close (r= 0.999). Importantly, the correlation of the common component with the FTA500 excess return series used in Section 4.2 is also very high for both GQARCH (0 984) and GARCH parameterizations (0 984), and the same is true of the conditional variances (0.992 and 0 996 respectively). This may not be that surprising, though, if we note from (12) that if the one factor model is realistic, a diversified linear combination of r, such as the FTA500 index will have almost no idiosyncratic component (see also Sentana and Shah (1994)). In fact, if the market return is well diversified, the asset pricing relation above coincides with that of a conditional CAPM with constant market betas. In this respect, we obtain very similar parameter estimates for c and F when we regress r, on the FTA500 return.

Our results seem to provide empirical support for the QARCH formulation of the conditional variance as a potential candidate to generalize GARCH models for time- varying variances at the multivariate level. In addition, they confirm that QARCH also provides a more adequate representation of risk premia.

7. CONCLUSIONS

In this paper a general quadratic model for the conditional variance of a time series which ensures positivity is introduced. It turns out that this model is the most general quadratic version possible of the class of Autoregressive Conditionally Heteroskedastic (ARCH) models introduced by Engle (1982) and for that reason we have called it Quadratic ARCH, or QARCH. Its main distinctive feature is that it allows an asymmetric effect of positive

19. Hence, we allow for possible co-persistence in variance as defined in Bollerslev and Engle (1993). 20. As far as the conditional variance is concerned, similar results are obtained if we assume a constant

conditional mean. 21. For scaling purposes, estimation of the factor model was carried out imposing the restriction O=

I - a,, - 3, so that the unconditional variance of the unobservable factor is 1.


and negative lagged values of the series. To allow for infinite dependence on the past, lagged values of the conditional variance have been included as in Bollerslev (1986).

In turn, the QARCH model may be nested into two non-parametric approaches to dynamic conditional heteroskedasticity. First, a quadratic polynomial constitutes the leading term in Gallant's (1981) Flexible Fourier Form approach, where extra trigonometric terms are added to the conditional variance function (see Pagan and Hong (1991) and Pagan and Schwert (1990)). Second, a piecewise-quadratic spline approximation to the unknown conditional variance function would also encompass QARCH as a trivial smooth example, as well as the models of Glosten, Jaganathan and Runkle (1993), Schwert (1989), Taylor (1986), and Zakoian (1990). Hence, QARCH may also provide a useful benchmark to compare the relative performance of these models. Our approach is also similar in spirit to the semi-non-parametric estimation technique based on a polynomial expansion put forward in Gallant and Tauchen (1989) (see also Gallant, Rossi and Tauchen (1992)).

The time-series properties of (G)QARCH processes are very similar to those of Bollerslev's (1986) GARCH. In particular, GQARCH(1, 1) and GARCH(1, 1) models are remarkably close: they both have the same mean, variance and autocorrelation functions for both the series and its squares, as well as the same forecasting recursion rule. Nevertheless, the GQARCH(1, 1) has the advantage that by adding a single parameter, it can allow for both an asymmetric effect on the conditional variance and higher (unconditional) kurtosis, which goes in the right direction towards capturing some of the stylized facts characterizing many financial time series.

An additional advantage of quadratic ARCH models is that they are easy to integrate in the analysis of structural economic models, just as linear models for the conditional mean are. The asymmetric model of changing volatility in stock returns of Campbell and Hentschel (1992) provides an illustrative example. In their model, a multiplicative parametrization would have been analytically intractable, while the QARCH model made their derivations straightforward.

Multivariate extensions of the QARCH model are straightforward in theory, but as in multivariate ARCH models, difficult to estimate given the number of parameters involved. However, QARCH formulations are the only ones that can be easily adapted to the context of conditionally heteroskedastic latent factor models, and do not entail a large computational burden.

Two univariate empirical applications to daily U.S. and monthly U.K. stock market returns provide support for the fact that the GQARCH conditional variance function represents the data substantially better than a standard GARCH model. The main reason is that it is able to capture the so-called leverage effect, which the other is ruling out a priori. An application of a factor model to 26 U.K. industrial sectors also shows empirical support for the GQARCH formulation vs. the GARCH one. In addition, all three applications show that it also provides a better representation of risk premia.

There is obviously no compelling theoretical reason why the conditional variance function should be literally quadratic, just as there is no reason why conditional means should be linear. If anything, one could expect non-linearities as well as "non-quadratities" to be the rule, rather than the exception, and not surprisingly, the literature on non-linear conditional mean and variance models is growing fast. At the same time, semi-parametric and non-parametric methods are becoming ever more popular. Hence no generality claim should be made about our approach. But equally, it seems sensible to use the quadratic conditional variance function discussed here as a benchmark, just as one would initially use linear models for conditional means for variables that can take on negative and positive values.


APPENDIX COVARIANCE STATIONARITY OF GQARCH MODELS

The GQARCH(p, q) process in (5) is equivalent up to conditional second moments to the following random coefficients model:

x qs+j, ,,+j (Al) X, = ?, + Ej= I I?tx tX_i + vEi'= I jtat _ Al

where (?,, IIIt,. tqt, .It, . , . (e, p= , ), are i.i.d. random coefficients with zero mean and covariance matrix

L p//2 01

v/2 A 01, 0 0 Al

provided that A and A are positive semi-definite, with A diagonal and 0- y,'A'ip1/4> 0. This equivalence is not unexpected, since the GQARCH model encompasses the GAARCH model of Bera and Lee (1990), which is introduced as a random coefficients model, and the original ARCH model of Engle (1982), which was expressed as a random coefficients model by Tsay (1987). The random coefficients interpretation also suggests potential generalizations of the GQARCH model. For example, one could allow for non-zero covariances between , and/or ?, with 4,, as well as non-zero covariances between different {,,'s. For instance, the asymmetric nonlinear GARCH model in Engle and Ng (1993) can be written as ? + tj I X, - I + ,Itat - I, with cov (ij ,, ,,) # 0. Similarly, a model with terms of the form oa,-ja,, could be achieved with a non-diagonal A. Neither example, though, results in a QARCH(oo) model.

To prove covariance stationarity, it is actually more convenient to work with the following alternative reparameterization:

x, =(, + iq_, ti,(x, -i-bi) + E,j=, {j,tf,-j (A2)

where the bi's are constant parameters and (4,, ,, ,) are i.i.d. random coefficients with zero mean and covariance matrix

cO 0 1 0 LL' Oj, [ 0 A

with (,, iq, and 4, mutually independent. Let s=max (p,q), and let z'=(x,, x, I,.., Xt-s+,), v,=(a,, a,,. a,-,+), and u'=(,,0,. ,0) be

s x I vectors. Also, define the following [I + (s - 1)] x s matrices:

[I ?] [? ? ] [? O ]; 0 - s-I I q s-q p s-P I s-I

so that the process can be represented in state-space form as:

Zt = ((4 + Tt)zt-- I + ltvt- I +ut- T,b. (A3)

Let V, = E(z,z,). Then, recalling the mutual and serial independence of i7,, {, and 4,, it follows that:

V, = E(z-z, -z,

,4'D) + E('P,z, , z, ,P) + E(Ht,v, v,' n,H) + (c + b'LL'b)G. (A4)

Apart from the inclusion of b'LL'b in the constant term, this expression is the same as the one derived by Bera and Lee (1990) for the GAARCH model (i.e. when b=0), and hence the stationarity condition for GQARCH models, stated in the following proposition, is the same as the stationarity condition for GAARCH models (see Bera and Lee (1990) for the proof):

Let

Ls0=[ o ]O ] and As = [0 ]S"P

q s-q p s-p

be lower triangular and positive semi-definite diagonal s x s matrices respectively.


Then x, as generated by equation (A2) is covariance stationary if and only if all the eigenvalues of the s x s matrix R are less than 1 in absolute value, where:

R=(4d?4 )+vec (G) vec' (LsL'+ A,). (A5)

Provided that this condition is satisfied, the unconditional variance of z,, V, will be:

vec (V) = (c + b'LL'b)[(Is(?s) - RI-' vec (G). (A6)

Given the shape of G, the unconditional variance of x, will be given by (c +b'LL'b) multiplied by the element 1, 1 of [(Is?s) - R5-', which is easily seen to be the reciprocal of the determinant of [(Il?ls) -RI-'. Tedious algebra shows that this determinant is equal to the familiar expression I - Eqa, aii- =, A>, where a11 is the i-th diagonal element of the matrix A = LL'. Therefore the unconditional variance of x, is indeed given by equation (7).

DATA APPENDIX

The following list refers to the definition of the 26 Financial Times Actuaries Sector Indices as of December 31, 1990, and includes the DATASTREAM four-letter sector mnemonics (all starting with FTA).

BANK Banks BDIS Brewers and Distillers BMAT Building Materials CHEM Chemicals CONC Contracting, construction ELEC Electricals ENGG Engineering General FDMG Food Manufacturing FDRT Food Retailing INBR Insurance (Brokers) INCM Insurance (Composite) INLF Insurance (life) INVT Investment Trusts LEIS Leisure MERB Merchant Banks METL Metals and Metal Forming MISC Miscellaneous MISF Other Financial MTRS Motors NWSP Publishing and Printing OILS Oil and Gas PAPA Packaging and Paper PROP Property SHPT Shipping and Transport STOR Stores TEXT Textiles

Of these sectors, seven (Banks, Life Insurance, Insurance General, Insurance Brokers, Merchant Banks, Property and Investment Trusts) are not in the FTA 500 share index (FTA500I). The beginning of the month 3-month Tbill rate was used as the safe interest rate (UKTRSBL%).

Acknowledgements. This is an extensively revised version of Sentana (1991). Three anonymous referees have helped me greatly improve the paper. The author is also grateful to Manuel Arellano, Anil Bera, John Campbell, Russell Davidson, Antonis Demos, Frank Diebold, Steve Durlauf, Rob Engle, Christian Gourieroux, Ludger Hentschel, Theo Nijman, Peter Robinson, Mushtaq Shah, Sushil Wadhwani and seminar participants at CEMFI, LSE, Universidad Complutense, University of Southampton, the ESRC Econometric Study Group Conference (Bristol, July 1991) and the European Meeting of the Econometric Society (Cambridge, September 1991) for very useful comments. Of course, the usual caveat applies. Ernst Maug collected the U.K. sector data. Financial support from the LSE Financial Markets Group, the Spanish Ministry of Education and Science and the ESRC is also gratefully acknowledged.


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