prediction in dynamic models with time-dependent …similarly to the innovation process for the...

Journal of Econometrics 52 (1992) 91-113. North-Holland

Prediction in dynamic models with time-dependent conditional variances*

Richard T. Baillie Michigan State Unil~ersity, East Lansing, MI 48824, USA

Tim Bollerslev Northwe.ytern Unirmsity, Ecanston, IL 60208, USA

This paper considers forecasting the conditional mean and variance from a single-equation dynamic model with autocorrelated disturbances following an ARMA process, and innovations with time-dependent conditional heteroskedasticity as represented by a linear GARCH process. Expressions for the minimum MSE predictor and the conditional MSE are presented. We also derive the formula for all the theoretical moments of the prediction error distribution from a general dynamic model with GARCHtl, 1) innovations. These results are then used in the construction of ex ante prediction confidence intervals by means of the Cornish-Fisher asymptotic expansion. An empirical example relating to the uncertainty of the expected depreciation of foreign exchange rates illustrates the usefulness of the results.

1. Introduction

The ARCH class of models was originally introduced by Engle (1982) as a convenient way of modeling time-dependent conditional heteroskedasticity; see Bollerslev, Chou, and Kroner (1992) for a recent survey. Despite the extensive literature on ARCH and related models, relatively little attention has been given to the issue of forecasting in models where time-dependent conditional heteroskedasticity is present. Bollerslev (19861, Diebold (19881, and Granger, White, and Kamstra (1989) all discuss the construction of one-step-ahead prediction error intervals with time-varying variances. Engle and Kraft (1983) derive expressions for the multi-step prediction error variance in ARMA models with ARCH errors, but do not further discuss the characteristics of the prediction error distribution. The prediction error

*The authors are very grateful to Rob Engle, two anonymous referees, and the participants at the conference on ‘Statistical Models of Volatility’ at the University of California, San Diego for helpful comments, and thank the NSF for financial support under Grant SES90-22807.

0304-4076/92/$05.00 0 1992-Elsevier Science Publishers B.V. All rights reserved

92 R. T. Baillie and T. BollersleL>, Prediction in dynamic models

distribution is also analyzed in Geweke (1989) within a Bayesian framework using extensive simulation methods. It turns out that the presence of ARCH effects can make a substantial difference to the conduct of inference, such as constructing ex ante forecast confidence intervals and out-of-sample structural stability tests.

This paper considers prediction from a fairly general single-equation model as represented by a nonlinear regression function with ARMA disturbances and innovations with time-dependent heteroskedasticity. After a brief section discussing notation, section 3 describes how the minimum mean square error (MSE) predictor of the future values of the conditional mean can be constructed. In the absence of ARCH in the mean effects, the actual form of the predictor is the same as in the homoskedastic case, but the presence of ARCH changes the MSE of the predictor and can make it larger or smaller than the value obtained under the assumption of conditional homoskedasticity. By expressing the Generalized ARCH (GARCH) model, as in Bollerslev (19861, in a companion form representation, section 4 derives the minimum MSE predictor of future values for the conditional variance. Some theoretical results for the corresponding MSE for the predictions of the conditional variance are given in section 5. Section 6 then derives explicit expressions for all the theoretical moments of the conditional prediction error distribution for the popular GARCH(l,l) model. This allows the percentiles of the forecast density to be approximated by means of the Cornish-Fisher asymptotic expansion as discussed in section 7. These results are extended in section 8 to the important case in practice where the disturbances from a model have ARMA errors and GARCH(1, 1) innovations. In section 9 the practical relevance of the techniques is illustrated through a simple empirical example relating to the uncertainty of the expected depreciation in the forward foreign exchange rate market. A brief conclusion and suggestions for future work are given in section 10.

2. Notation and assumptions

In many practical contexts it is important to derive multi-step predictions of the conditional mean from dynamic econometric models. To keep the setup as general as possible, let {y,) refer to the univariate discrete time real-valued stochastic process to be predicted and let

Et-ICY,) =~r (1)

denote the conditional mean given information through time t - 1. The innovation process, {E,}, for the conditional mean is then given by

E, = Y, - PLt > (2)

R. T. Baillie and T. Boliersle~: Prediction in dynamic models 93

with corresponding, possibly infinite, unconditional variance

Var( Et) = E( &:) = u2. (3)

While the unconditional variance is assumed to be time-invariant, the conditional variance of the process is allowed to depend nontrivially on the set of conditioning information, so that

Var,_,( y,) = E,-i(&,“> E a?. (4)

It is important to note that both pt and a, are measurable with respect to the time t - 1 information set and assumed to be finite with probability one. We also define

u,=&,2-q2. (5)

Similarly to the innovation process for the conditional mean given in (2), (v,} is serially uncorrelated through time with mean zero, and is readily inter- preted as the time t innovation for the conditional variance.

The above setup allows for a wide variety of dynamic econometric models with time-varying second-order moments. In order to simplify the exposition, in the following analysis we shall concentrate on predictions from the standard ARMA(k, 1) class of models, i.e.,

The extension to the case of exogenous explanatory variables allowing for the possibility of co-integration, as in Engle and Granger (1987), is in principle straightforward. It is well-known that if the innovation variance, a*, is finite, the ARMA(k,l) model in (6) is covariance-stationary and invertible if and onlyifalltherootsof1-~,z-~~~-~,zk=Oand1-~,z-~~~-~,z’=O lie outside the unit circle.

One important exclusion from this framework concerns the ARCH in mean model, originally due to Engle, Lilien, and Robins (1987). Processes with feedback from the conditional variance to the conditional mean will considerably complicate the form of the predictor and its associated MSE. Analysis of such models is consequently left for future research.

Recently several alternative parameterizations for the time-varying conditional variance, a,*, have been suggested in the econometrics and time series literature. In this study, we shall focus on the popular linear GARCH(p, q)

94 R. T Baillie and T. Bollersleu, Prediction in dynamic models

class of models,

at (7) i=l i=l

where o > 0, and (Y~ and pi are restricted so that the coefficients in the infinite distributed lag representation of a,* in terms of lagged values of &: are all positive; see Nelson and Cao (1992). If (hi + . . . +a, + p, + . . . + /3, < 1, {Ed} is covariance-stationary and

a2=w(l -(Y, - ..’ -cr,-p, - ... -p,) -1

Similarly to the ARMA(k, f) model for the conditional mean, the GARCH(p, 4) structure could easily be extended to allow for exogenous explanatory variables entering the conditional variance. The particular pa- rameterization in the GARCH(p, 4) model in (7) has a,* expressed as a function of lagged squared innovations. An alternative, although less widely used, representation that could be analyzed in a similar fashion involves a,* being expressed as a function of serially correlated lagged disturbances u, = yt -f<x,; b), where f(x,; b) denotes a possible nonlinear regression function; see Bera and Lee (1988) and Bera, Lee, and Higgins (1990).

In part of the subsequent analysis we shall make use of the higher-order conditional moments for the (F,) process. For simplicity, we assume that this conditional distribution is symmetric with all the existing even-ordered moments proportional to the corresponding powers of the conditional variance,

E,_,(E:‘+‘) = 0,

E,_ ,( $‘) = K,o;2’,

r=O,l,..., K-l,

r = 0, 1, . . . , K.

(8)

(9)

Here K, denotes the rth-order cumulant for the conditional density of Em, and by definition K() = K~ = 1.

For instance, under the assumption of conditional normality often invoked when conducting inference in ARCH-type models, all the moments of the conditional distribution of &, are finite and

K,= fi(2i-I), r= 1,2 )... . (10) i=l

With conditional t-distributed errors as in Bollerslev (1987),

~,=(n-2)~r(r+l/2)~(n/2-r)~(1/2)-’T(n/2)-’, (11)

r= 1,2 ,..., K,

R. T Baillie and T. Bollerslec, Prediction in dynamic models 95

where r(.) denotes the gamma function, n > 2 the degrees of freedom in the t-distribution standardized to have a unit variance, and K = int(n/2). Note, only the first IZ moments of the t-distribution are finite.

While the vast majority of empirical studies using ARCH models tend to rely on parametric specifications for the conditional density of E, given information through time t - 1 as in (10) and (ll), different nonparametric methods have also been suggested in the literature, including the polynom- ial expansion in Gallant and Tauchen (1989) and Gallant, Hsieh, and Tauchen (1990) and the nonparametric density estimation in Engle and Gonzalez-Rivera (1991). Although, explicit expressions for the higher-order conditional moments may not be directly available from these methods, the implied K,. coefficients can easily be evaluated using numerical techniques.r

3. Prediction of the mean in ARMACR, I) models

Many alternative expressions are available for the minimum MSE predictor from the ARMA(k, 0 class of models. However, in order to provide an analogy with subsequent material it is convenient to express the ARMA(k, I) model given by (11, (2), (4), and (6) in the companion form representation as

Yr -$I, bk 0, f-J-

Yl-I 1 0 0 0 0

Yr-ktl 0 . ..lO o... 0 =

El + 0 0

Et-1 0 0 1 ::: 0 0

El-l+1 0 0 0 1 0

or more compactly,

Y, =w, + @K_, + (e, +ektl)E,,

+

(12)

(13)

where ej refers to the compatible vector of zeros except for unity in the jth element, here a (k + II-dimensional unit vector. Following Baillie (19871, upon repeated substitution in (13) the optimal s-step-ahead predictor is readily seen to be

k-l I- I

Et( Yt+s) = L.~ + C T,,.~Yt-i + C *i,s’t-i,

i=O i=O (14)

‘The K, coefficients from the density estimation in Engle and Gonzalez-Rivera (1991) are time-invariant by assumption, but the higher-order standardized conditional moments from the seminonparametric method in Gallant and Tauchen (1989) may be time-varying.

96 R. i? Baillie and i? Bollerslec, Prediction in dynamic models

where

Ls=e;(I+@+ ... +@‘-l)e,p, (15)

ri,s = e;cDsei+,, i=O ,...,k- 1, (16)

hi,,=e;@sek+i+i, i=O ,...,l- 1. (17)

This is a different and more tractable expression than that given by Baillie (1980) and Yamamoto (1981), where the ARMA process is represented in terms of an infinite-order autoregression.

Furthermore, by direct substitution and iterated expectations the forecast error for the s-step-ahead predictor in (14) is given by

e -Y*+s f,S =

- Et( ~r+s) = i +r-iFt+i, (18) i=l

with conditional MSE

Et(&) = Var,( y,+,> = i +LiEt(uhi), i=l

where

ICI,=e;@(e, +ek+,), i=O ,...,s- 1. (20)

Note, the I,!J~‘s correspond directly to the coefficients in the infinite-order moving average representation of the model.

With conditionally homoskedastic errors the conditional MSE for the optimal s-step-ahead predictor is identical to the unconditional MSE,

Var( y,,,) =u2 i 4s2-i, i=l

where a2 is assumed to be finite. However, when conditional heteroskedasticity is present, the forecast error uncertainty is generally changing through time, and from (19) the conditional MSE takes the form

Hence, the conventional first term is appended by a second term reflecting the differences between the average or the unconditional variance of the future innovations and the conditional variance given information through time t. If the process is covariance-stationary and invertible, $2 goes to zero and E,(u,:~) to u2 for increasing values of i, and the conditional MSE converges to the unconditional variance of the process as the forecast horizon increases. However, with a time-varying variance this convergence is not necessarily monotone. As previously noted by Engle and Kraft (1983), over

R. T Baillie and T. BoNerslel,, Prediction in dynamic models 97

certain time periods the conditional MSE may exceed the unconditional variance provided E,(a,!+;) > u2 for some 1 5 i IS.

It is important to note, that the presence of conditional heteroskedasticity does not change the expression for the minimum MSE predictor as given by (14), but the forecast error uncertainty associated with the optimal predictor will be time-varying as reflected in (19). Of course, in order to evaluate the expression for the conditional MSE given by the formula in (19) it is necessary to calculate the conditional expectations for the future conditional variances of the innovation process. This is the subject of the next section.

4. Prediction of the variance in GARCH(p, 4) models

Following Bollerslev (19%X), the linear GARCH(p, q) model in (4) and (7) can be conveniently rewritten as

E, 2=W + ((Y, +P,)E:-, + ... +(a, +&)E:_,

-p,V,_, - ... -pPV,_P+V,, (21)

where m = max(p, q), cyi = 0 for i > q, and pi = 0 for i >p. From the definition in (5), {v,) is a serially uncorrelated process, and (21) corresponds to an ARMA(m,p) model for {E:). Thus, analogous to the ARMA(k, 1) representation in (12), the GARCH( p, q) model may be expressed in companion first-order form as

+

(Y, +p, ffw, + Pm -p,

1 0 0 0

0 1 0 0 0 0 0 1 0

0 0 0 1

1 r 3

-p/J 0

0 0 0

0

(22)

98 R. T. Baillie and T. Bollerslel: Prediction in dynamic models

or more compactly,

v,‘=we,+rV,Z,+(e,+e,+,)v,. (23)

Upon repeated substitution,

s-1

C = C We, + em+l)V,+.y--r + we,) + rY*. i=O

However, E:+, = e’,y:, and E,(v,+~) = 0 for i > 0, and analogously to the derivation of (14) it follows, from pre-multiplication with e’, and the use of iterated expectations, that the minimum MSE s-step-ahead predictor for the conditional variance from the GARCH(p, q) process is given by*

p-1 m-1

Er($+,) = E,(d+s) = w_s + C 6i,sat2-i + C Pi,s&:-i, (24)

i=ll i=O

where

w,=e;(l+r... +rs-l)e,w, (25)

a,,,= -e;rsem+i+l, i=o ,...,P- 1, (26)

Pi, s =4rs(ei+l +e,+i+,), i=O ,...,P-1, (27)

P 1,s =e;rser+,, i=p,...,m - 1. (28)

As an illustration consider the popular GARCH(1, 1) process, where

s-l

=ygl( Ql f/q’+ (a* +P,)s-‘&.

If the model is covariance-stationary with (Y, + pi < 1 and a2 = ~(1 -(Y, - pi>-‘, the optimal predictor for at+, becomes

E&q:,) =a*+ (aI +a,)‘-‘(q:, -a2),

‘An alternative expression for the optimal predictor for the conditional variance is given by the recursions E,(a(!+.) = w + P,E,(a,?+,_,) + ... +P,E,(o,?+,_,) + ‘YIE,(F:+,_,)

+ +(Y~E,(E,?+, _J, where by definition E,(E,?+,) = E,(a,?+,) for i > 0, while E,(a,!+,) = u,?,

and EI(~:+I) = E:+, for i 5 0.

R. T. Baillie and T. Bollersler~, Prediction in dynamic models 99

a result previously derived by Engle and Bollerslev (1986). Once again, as the forecast horizon increases current information becomes less important, and the optimal forecast converges monotonically to the unconditional variance. However, for the Integrated GARCH(1, l), or IGARCH(1, 11, model with II, + p, = 1, current information remains important for forecasts of all horizons.

E,(a,:,) =w(s - 1) +u;+,.

Shocks to the conditional variance are persistent in the sense of Bollerslev and Engle (1989), as E,(u,$,,) - E,_ ,(u,:~) = a,?+, - w - a,’ = a,vt is a nontrivial function of the information set at time t for all forecast horizons s > 0.

The conditional MSE associated with the optimal forecasts for the mean in the general ARMA(k, I)-GARCH(p, s> class of models is readily obtained by combining eqs. (19) and (241,

“=A Y,,,) = 2 v&w; + fl$,,‘_, p&p;_, + yp,,,E:_, . i= I i= I i j = 0 i = 0 I (29)

For example, for the covariance-stationary AR(l) model with GARCH(1, 1) errors the optimal predictor becomes

E,(Y,+,) =&Y,>

with associated MSE

Var,( y,,,) = i 4f(‘-‘)(c2 + (a, +p,)‘-‘(fl,!+, -v’)) i=l

x((q +P,)‘y-4:s)(q l tPI -4-‘7

where the last equality is only valid for 4: # (Y, + /3,. The inclusion of the second term in the above expression for the conditional MSE may lead to an increase or a decrease in the prediction error variance compared to the conventional MSE, but as the horizon increases the conditional MSE will converge to the unconditional variance, i.e., var(y,) = ~(1 -(Y, - p,)-‘(l - 4:)Y’.

100 R. T. Baillie and T. Bollerslec, Prediction in dynamic models

5. Uncertainty in predicting future conditional variances

The results in the previous section provide formulae for the calculation of the forecast error variance of the mean in general dynamic econometric models with GARCH( p, q) errors. However, in many applications in financial economics the primary interest centers on the forecast for the future conditional variance itself. Such instances include option pricing as discussed by Day and Lewis (1992) and Lamoureux and Lastrapes (1990), the efficient determination of the market rate of return as in Chou (19891, and the relationship between stock market volatility and the business cycle as analyzed by Schwert (1989). In these situations it is therefore of interest to be able to characterize the uncertainty associated with the forecasts for the future conditional variances also. Some potentially useful results for this purpose are given by Lemma 1.

Lemma 1. For the GARCH(p,q) model gicen by (4), (5), (7), (81, and (9) the forecast error for the s-step-ahead predictor for the conditional cariance a,!+, in (24) equals

and the conditional MSE is

El(&) = Var,(di,) = b2 - 1) c x_i-A(4k), i=l

(31)

where

xi=e;T’(e, +e,+,), i=l ,...,s- 1. (32)

Proof. Since a,!+, - E,(aF+,) = E:+, - E,(u~+,) - v,_~, (30) and (32) follow from the companion form in (23) and slight modifications to the derivation of (18) and (19). By (5) and (9) Et(~,+i~t+j ) = 0 for 1 I i <j I s, and E,(vf+, j = k2 - ljE,@=+, > for i > 0. Hence (31) is a result of iterated expectations.

Q.E.D.

To illustrate, consider the GARCH(1, 1) model. From (22) and (32) it follows that xi = (Y,((Y, + /3,>‘-‘, i = 1,. . . , s - 1. Hence, the forecast error uncertainty associated with predictions of the future conditional variance

R. T. Baillie and T. BollersleL,, Prediction in dynamic models 101

equals

s- 1

c ,,s = aI c (a, +P,)‘-L-i~ i=l

with corresponding conditional MSE

q &> = (K* - 1)a; c (a, +tp,)2~r~‘~E,(~p,,_,). i=l

The empirical evaluation of this conditional MSE requires an expression for the fourth conditional moment. Such an expression is derived in the next section. However, given the focus in the present paper on optimal predictions for the conditional mean and the distribution of the corresponding prediction errors, we shall not pursue the topic of optimally forecasting the conditional variance any further in this paper.

6. Prediction error distributions in GARCH(1, 1) models

When conducting inference in GARCH(p, 4) models, distributional assumptions are generally placed on the conditional distribution of E, given a,*. This implies specific values for the cumulants K, in (9) that characterize the conditional even-ordered moments. However, in the presence of ARCH the unconditional distribution of &t has fatter tails than this conditional one- step-ahead prediction error distribution. In particular, given finite fourth

moment E,_ ,(E:) - K*(E,_ ,<&f>>’ = 0 < E(e:> - K,(E(E~>)~. Similarly, the conditional distribution of cltF for s > 1 given information through time t differs from the conditional distribution for s = 1. Even if the distribution of

e,+.s/\lEr(&s > is time-invariant for s = 1 by assumption, for s > 1 and ut2 time-varying the standardized prediction error distribution generally depends nontrivially on the information set at time t. As opposed to the conventional framework with conditionally homoskedastic errors where standardizing with the prediction error variance leads to a time-invariant distribution, the dependence in the higher-order moments for the GARCH(p, 4) model substantially complicates conventional multi-step prediction exercises.

For instance, the quantile regression techniques discussed in Granger, White, and Kamstra (1989) as a method for estimating the time-varying one-step-ahead prediction error intervals do not easily extend to multi-step predictors. Similarly, the numerical methods developed by Geweke (1989) for calculating the exact predictive density would require extensive simulations of the prediction error distribution for each particular realization of a,: ,.

102 R. T. Baillie and T. BollersleL,, Prediction in dynamic models

The presence of heteroskedasticity also alters the structure of the post- sample structural stability tests proposed by Box and Tiao (1976) commonly used as a tool in model evaluation and diagnostic checking; see, e.g., Chong and Hendry (1986), Lahiri (1975), and Liitkepohl (1985, 1988). In particular, under the assumption of conditional normality of the one-step-ahead prediction errors, ~,/a,, it follows that the test statistic

will possess the conventional chi-squared distribution with s degrees of freedom. Ignoring the temporal variation in Us!+; in constructing the test statistic can seriously bias inference.”

To overcome this apparent indeterminancy regarding the properties of the prediction error distribution from GARCH models, simple recursive expressions for all the existing conditional moments of the prediction error density for the widely-used GARCH(1, 1) model are provided by Theorem 1.

Theorem 1. For the GARCH(1, 1) model defined by (4), (d), (9), and

ff, 2= w+a& +p,&,

the 2 Kth first conditional moments of elts, s > 1, are given by

Et( $;;‘) = 0, r=O,l ,...,K- 1, (33)

E,(e::,)=+,(q!,‘,)> ,...,K, r = 0,l (34)

where

(35)

Proof. Eqs. (33) and (34) follow from (8) and (9) by the law of iterated expectations. On repeated use of the binomial formula and from the law of

3For instance, for the GARCH(1.1) model with w = 0.1, (Y, = 0.2, /3, = 0.7, and v,:, fixed at the unconditional variance of one, the estimated rejection frequencies for s = 2, 5, and 10 based on the 0.05 and 0.01 fractiles in the chi-squared distribution with s degrees of freedom are 0.064, 0.091, 0.114 and 0.027, 0.046, 0.067, respectively.

R. T Baillie and T. Bollersler, Prediction in dynamic models 103

iterated expectations,

E,(q?,) = E,((w + d+,-, + P,d-dr) r r

z.T w 1 w’-‘E,((d+,-, +Pd-I)‘) i = 0

which reduces to (35) and (36). Q.E.D.

Given the recursive expressions for all the conditional moments in Theo- rem 1, several alternative asymptotic expansions are available for approximating prediction error distributions. 4 To set out the particular expansion used below in forming prediction error intervals, it is convenient to introduce the standardized cumulants for the s-step-ahead prediction error,5

Yr,r,.~~Kr,r,s(K2.r,s)~r, r=2 ,..‘, K, (37)

where K,, s , , denotes the 2rth cumulant for e,,:, conditional on the time t information set. For the one-step-ahead predictton error Y,,~, 1 will be time- invariant for all r by the assumptions in (8) and (91, but in general Y,,~,~ will be a nontrivial function of the time t information set for s > 1. For instance, from Theorem 1 the conditional excess kurtosis for the two-step-ahead

‘The conditional moment sequence uniquely determines the conditional distribution if the Carleman condition is satisfied; i.e., E,(E,?+,)~‘/* + E,(E:+,)-“~ + E,(E~=‘+,)-“’ + = m. See Serfling (1980) for sufficient conditions.

‘The ‘Y, ,.r’ s play an important role in many asymptotic expansions, including the Gramm-Ckarlier, the Edgeworth, and the Cornish-Fisher approximation used below.


prediction error in the GARCH(1, 1) model is given by

Yz.r.2 = ((4 - ++,4,, + (K2 - 3)

which under the assumption of conditional normality, i.e., ~~ = 3, reduces to

Obviousk y2, !, 2

values of a,:, is an increasing function of both a,‘-+, and LY,. For large

the conditional kurtosis for the two-step-ahead prediction error distribution may exceed the unconditional excess kurtosis for et; i.e., 6cu:(l - /3: - 2a,p, - 3at)-’ where the denominator is assumed to be positive in order to ensure a finite fourth moment. More complicated expressions for the higher-order cumulants and longer forecast horizons from the GARCH(1, 1) model are readily available from Theorem 1.

7. Comish-Fisher expansion

For the purpose of constructing prediction error intervals that remain valid in the presence of heteroskedasticity, the inverse of the Edgeworth expansion original developed by Cornish and Fisher (1937) is particularly useful; see Barndorff-Nielsen and Cox (1989) for a recent discussion. Thus, let z,,,(P) denote the Cornish-Fisher approximation to the time-varying pth quantile in the conditional distribution for the s-step-ahead prediction error e, s. For symmetric deviations from conditional normality the expression for ;,,,(p>

then simplifies to

z,,s( P> =d P~E,(e:s)"2~

where f sp < 1, and

P,.S( P) = @-‘(PI +P2(@-‘( P1)Yz.r.s

+P22(@-‘( P))Y;,tJ +tP,(W P))YW,s + . . . . (39)

The first term in (39), W’(p), refers to the pth quantile in the standard normal distribution, the second term adjusts for conditional excess kurtosis, while the third and fourth terms are due to adjustments for UP to the sixth

conditional moment, and terms involving eight or higher-order moments have

R. T. Baillie and T. Bollerslec, Prediction in dynamic models 105

been omitted. Also, three important functions are given by’

pz( z) = ( z3 - 3r)/24,

p4( z) = ( z5 - 10z3 + 15z)/720,

p2z( z) E - (32’ - 242” + 29z)/384.

Of course, the assumption of conditional normality for the s-step-ahead prediction errors corresponds to fixing all these functions in (39) to zero.

In order to check the accuracy of the Cornish-Fisher approximation in obtaining prediction error intervals in the present context a series of simulations were performed for various GARCH(1, 1) parameterizations with conditionally normal one-step-ahead prediction errors. For small values of (or and short forecast horizons the convenient assumption of conditional normality of the multi-step predictions appeared to work reasonably well on average, with the estimated coverage probabilities being close to the true size of the intervals. However, consistent with the discussion in the previous section, for longer forecast horizons and/or larger values of LY, the true conditional prediction error distribution was more peaked at the center and had fatter tails than the normal distribution, and the normal approximation tended to overestimate the 0.50 and 0.20 fractiles while underestimating the 0.01 fractile. Interestingly, the crossing point for the densities for the true prediction error distribution and the normal approximation were generally close to the 5% fractiles. However, the simple Cornish-Fisher expansion in (39) based on the adjustment for up to the fourth conditional moment only, i.e., neglecting p&@-‘(~)), pZ2(@-‘(p)), and higher-order terms, proved a better representation of the extreme fractiles. Also, for large realizations of a,:, resulting in more marked deviations from conditional normality in the multi-step-prediction error distributions the Cornish-Fisher expansion based on only pJ@‘-‘(~1) performed quite well. Full details of these simulation results are available in Baillie and Bollerslev (1990b).

8. Prediction error distributions in ARMAfk, l)-GARCH(1, 1) models

Eqs. (14) and (29) provided explicit expressions for the optimal s-step-ahead predictor and its associated conditional MSE in the ARMA(k, 1)- GARCH(p, 91 class of models. However, in order to make use of any asymptotic expansions in approximating the prediction error distribution, expressions for the higher-order conditional moments are called for. Theo- rem 2 provides such a formula for the fourth moment of the prediction errors from a general dynamic econometric model with GARCH(1, 1) innovations.

“See Abramowitz and Stegum (1972) and Kendall and Stuart (1967) for a definition of the importance functions in terms of Hermite polynomials.

106 R. T. Baillie and T. Bollersleq Prediction in dynamic models

Theorem 2. The conditional fourth moment for the prediction error for the minimum MSE predictor in the ARMA(k, O-GARCH(1, 11 model given by (11, (21, (4), and (6)-(9) equals

s-1

J%(ep,,) =K~ c &%(6!+s--i) i=O

s-2 s-l

+6 c c k%;[h +fl,)‘-i-l(‘%“, +i3,) i=O j=i+l

XEt(u:+s-j ) + li+jEt(uAs-j)] 2

(40)

where

j-l-i

5i,j= C Otal +PlIh2 h=O

(41)

and E,(UfY+i) and E,(u,‘!+~) for i > 0 are given by (35).

Proof. From (8) the conditional expectation of terms involving E~+~ to odd pow&s is zero, and therefore

Et(ep.3) =Et[( :c4i&z+s+ir]

+Et[( i<@i&t+s-i))

= . . .

s-l s-2 s-l

= i~o$~E,(~P+s-i) + 6 C C ~Z~~EI(&:+s-i&:+r-j). i=O j=i+l

For 0 2 i I s it follows directly from (9) that

Et(EP+s-i ) =Et(Er+s-i-I(EP+~-i)) =K~Et(u:+s-i).

R. 7: Baillie and T. Bollerslec, Prediction in dynamic models 107

Similarly, for 0 I i <j <s it can be shown that

Et(Ef+s-,Ef+s-j)

=Er(&12+Z-,[~fru,e:,,-,-, +Pd-,-I])

= Et(“f+s-j [w + (a1 +P,ML-11)

=E,($+s_j[o(l + ((Y, +/3,) + ... +(a1 +/3,)‘-2-i)

=w(I + ((~1 +Pl) + ... +(a, +~~)‘-‘-‘)E,(~~Y+~_j)

+(a, +tp,)j-I-‘(‘QN, +PdEr(d-j),

which reduces to (40) and (41) upon substitution. Q.E.D.

Note that if LY, + p, # 1 the expression for &, in (41) simplifies to

&j=w(l - (a1 +a,,j-j)(l -LX, -p,>-‘.

As for the formula for the conditional MSE in (19), the results for the conditional fourth moments provided in Theorem 2 apply more generally to all dynamic econometric models with GARCH(1, 1) innovations and prediction errors that can be expressed as in eq. (20). Of course, in the absence of any serial dependence in the conditional mean, i.e., I+?~ = $2 = . . . = $,V_, = 0, e ,,s = F,+,, and (40) is just a special case of the more general results for the conditional moments in the GARCH(1, 1) model given in Theorem 1.

To assess the practical importance of the results in Theorem 2 we also carried out several simulations for different AR(l)-GARCH(1, 1) formula- tions with conditionally normal one-step-ahead prediction error distributions. Not surprisingly, the presence of serial dependence in the disturbances generally led to an increase in the conditional excess kurtosis for the prediction errors, y2, f, s, due to the temporal dependence in Et(~:+,Y_ie:+,_,>. For instance, for LYE = 0.2, /3, = 0.7, and s = 2 the average conditional excess kurtosis increased from 0.228 for 4, = 0.0 to 0.516 for 4, = 0.5 and 0.642 for


4, = 1.0. This is also borne out by the simulation results obtained for the coverage probabilities for the prediction error intervals. The conditional normal approximations for the AR(l)-GARCH(1, 1) models are too peaked at the center and too thin in the tails, and the serial dependence in the conditional mean tend to enhance these departures from normality even further when compared to the results from the simple GARCH(1, 1) models. Fortunately, the Cornish-Fisher prediction error intervals based on correc- tions for up to the fourth conditional moment generally provided reasonably close approximations.

9. Empirical example

To illustrate the techniques discussed above we now consider a simple empirical example relating to the uncertainty of four different forward foreign exchange rates as predictors of the corresponding future spot rates. The data are opening bid prices from the New York Foreign Exchange Market from March 1, 1980 to February 2, 1989, and constitutes a total of 462 weekly observations on the UK pound (UK), the West German Deutschemark (WG), the Swiss franc (SW), and the French franc (FR), all vis-a-vis the US dollar. The one-month-forward rates are taken on Tuesdays and the corresponding future spot rates four weeks and two days later on Thursdays;’ for a more detailed description of the data see Baillie and Bollerslev (1990a). Following Hansen and Hodrick (1980) and Baillie (19891, if the forward rate is an unbiased predictor of the future spot rate, but the sampling time interval of the data is finer than the maturity time of the forward contract, the forecast errors will be serially correlated. To take account of this fourth-order moving average error structure induced by the one-month-forward contracts and overlapping weekly data plus the volatility clustering, an MA(4)-GARCH(l, 1) model was estimated for each of the four currencies,

E,- I( ~0 = or = P + el&,- I+ Q-~ + G&,-j + O-4 (42)

and

Var,_,(y,) =0;2= w + Ly+:_, + p,&, (43)

where yt = log s, - log fl_4. The estimates reported in table 1 are maximum likelihood estimates obtained under the assumption of conditional normality.8 In accordance with the results in Baillie and Bollerslev (1990a), the estimates

‘This generally matches the forward rate with the spot rate in the future that would be used to cover an open position. However, this alignment could be one or two days off around the beginning of a new month; see Hodrick (1987).

‘For comparison purposes the numbers have been converted to monthly percentage rates by multiplication with 100.

R. T Baillie and T. Bollerslec, Prediction in dynamic models

Table 1

Maximum likelihood estimates.”

109

UK WCi SW FR

w

- 0.322 (0.305)

0.906 (0.048)

0.796 (0.054)

0.768 (0.053)

0.310 (0.047)

0.158 (0.116)

0.060 (0.029)

0.987 (0.050)

- 0.765 (0.317)

0.925 (0.052)

0.819 (0.055)

0.754 CO.0531

0.298 CO.0461

1.118 (0.523)

0.199 (0.076)

0.528 (0.168)

-0.915 (0.344)

0.930 (0.048)

0.826 (0.053)

0.784 (0.053)

0.325 (0.047)

0.899 (0.528)

0.151 (0.062)

0.662 (0.143)

- 0.487 (0.307)

0.928 (0.050)

0.825 (0.056)

0.742 (0.053)

0.312 (0.045)

0.960 (0.420)

0.214 (0.074)

0.552 (0.137)

K2 4.087 4.367 3.643 3.778 Q(tO) 6.670 9.150 6.293 6.269 Q*(lO) 6.371 7.762 9.911 10.909

“Maximum likelihood estimates with asymptotic standard errors in parentheses. K~ gives the sample kurtosis for the standardized residuals. Q(lO) and Q*(lO) refer to the Ljung-Box portmanteau test for up to IOth-order serial correlation in the levels and the squares of the standardized residuals, respectively.

for the four MA coefficients in (42) are all reasonably close to the values implied by the unbiasedness hypothesis and a martingale spot price process, i.e., 0.837, 0.773, 0.686, and 0.258, respectively. In fact for none of the four rates are these implied parameter values rejected by a formal likelihood ratio test constructed under the assumption of conditional normality. Additional diagnostic tests, including the portmanteau tests for remaining serial correlation in the levels and the squares of the standardized residuals reported in table 1, also indicate a reasonably good fit of the models for all the four currencies.

Optimal predictions for the MA(4)-GARCH(l, l) model are readily available from (14), while the corresponding forecast errors and forecast error uncertainty are given by eqs. (18) and (191, respectively. Note, in this situation I+!I~ = 0; for i = 1,. . . ,4 and I/J; = 0 for i > 4. In order to succinctly summarize the empirical distribution of the forecast errors associated with the expected depreciation from each of the four models, the first three rows in table 2 report the average rejection frequencies across the one- through six-week forecast horizons obtained over the whole sample period using three

110 R. T. Baikie and T BoUersleu, Prediction in dynamic models

Table 2

One percent confidence intervals rejection frequencies.” ,-

UK WG SW FR

1980.3- Homoskedastic 1.52 1.56 1.30 1.52 1989.2 Heteroskedastic 1.41 1.96 1.70 1.93

Cornish-Fisher 1.63 1.00 1.04 0.93

1985.2- Homoskedastic 5.45 3.21 3.53 4.49 1986.2 Heteroskedastic 0.32 4.49 3.21 3.85

Cornish-Fisher 0.32 2.24 1.92 1.60

“Average rejection frequencies with 1% confidence in- tervas for one- through six-steps-ahead forecast horizons. Homoskedastic denotes the confidence interval constructed under the assumption of conditionally homoskedastic normal errors. Heteroskedastic gives the rejections with a conditional heteroskedastic normal confidence interval, while the Cornish-Fisher intervals adjust for up to the fourth conditional moment.

different 1% confidence intervals.’ In particular, the entries labelled ‘homoskedastic’ denote the rejections that occur with a homoskedastic normal confidence interval; i.e., p = 0.995, E,(u(f+;) fixed at the unconditional variance, and all the higher-order correction terms in (39) set equal to zero. Similarly, ‘heteroskedastic’ refers to the average rejection frequencies across the six horizons that obtain by allowing for the GARCH(l,l) conditional heteroskedastic error structure, but omitting any of the higher-order correction terms for deviations from conditional normality in (39). Finally, the ‘Cornish-Fisher’ approximation adjusts for up to the fourth conditional moment in the prediction error distribution based on the sample kurtosis for the standardized residuals, i.e., K* in table 1.

For all three methods and four currencies, the actual number of rejections are generally fairly close to the expected values. Interestingly, this is also true for the homoskedastic confidence intervals, since the unconditional sample distribution for the prediction errors are not markedly different from the normal in the present context. It is worth pointing out, that although table 2 only reports the average number of rejections, no systematic pattern across the six forecast horizons is apparent for the four currencies. To illustrate, for the UK the actual number of rejections that occur with the Cornish-Fisher expansion for the six horizons are 7, 5, 5, 8, 10, and 9, respectively, compared to 8, 8, 4, 3, 2, and 2 for West Germany. Also, the results for other sized confidence intervals are very much in line with the findings reported in table 2. For instance, for the UK the average rejection frequencies for the whole

“Six observations were excluded in the beginning and the end of the sample to allow for startup problems and predictions up to six steps ahead.

R. T. Baillie and T. Bollersleu, Prediction in dynamic models 111

sample with a 5% confidence interval for each of the three different methods equals 4.63, 4.93, and 5.15, respectively.

While the actual rejections for each of the three intervals are quite close over the entire sample period, the results for certain subsamples of the data are very different. To illustrate, consider the one-year period beginning February 26, 1985, corresponding to the peak of the US dollar against the deutschemark at 3.477 mark to the dollar. Over the following year, stimu- lated by the Plaza agreement on September 22, 1985, the dollar experienced a volatile but fairly systematic depreciation against most major currencies. From table 2 this increase in volatility resulted in far more rejections with the homoskedastic 1% confidence intervals over this one-year period than were to be expected. Whereas the heteroskedastic normal confidence intervals generally do somewhat better, it follows also that correcting for higher-order deviations from conditional normality as in the Cornish-Fisher asymptotic expansion, may be very important under high volatility scenarios. These results for the 1% confidence intervals are also in line with the findings pertaining to other sized tests. For instance for the UK over the 1985.2-1986.2 period the 5% intervals result in average rejection frequencies of 10.90, 3.85, and 4.17, respectively, for each of the three different methods.

10. Conclusion

This paper has considered predictions from a general dynamic time series model with ARMA disturbances and time-dependent conditional heteroskedasticity, as represented by a GARCH process. Tractable formulae for the minimum MSE predictor of both the future values of the conditional mean and conditional variance are presented. Expressions for all the exact moments of the multi-step forecast errors in the presence of GARCH(l,l> are also derived, and it is shown how the Cornish-Fisher expansion can be used in approximating the forecast densities. As illustrated by the empirical example concerning the depreciation of exchange rates, these adjustment formulae can be especially useful in very volatile periods.

One potentially important issue not addressed relates to the effect of parameter estimation. For the processes considered in this study, the information matrix is block-diagonal between the parameters in the conditional mean and variance equations. This implies that the estimation uncertainty for the conditional variance parameters is irrelevant to the asymptotic MSE for predicting the conditional mean. However, adjustment of higher-order moments for this effect may be important when using asymptotic approximations for the prediction density in small sample sizes. The practical importance of this is hard to ascertain without a detailed Monte Carlo experiment, and is left as an area for future research.

112 R.T. Baillie and T. Bollersleu, Prediction in dynamic models

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prediction in dynamic models with time-dependent …similarly to the innovation process for the...

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