Estimating EGARCH-M models: Science or art?

The Quarterly Review of Economics and Finance, Vol. 38, No. 2, 1998, pages 167-180 Copyright © 1998 Trustees of the University of Illinois All rights of reproduction in any form reserved. ISSN 1062-9769

ESTIMATING E G A R C H - M

MODELS: SCIENCE OR A R T ?

EILEEN F. ST. PIERRE Universityof Northern Iowa

This paper shows that the EGARCH-M model should be estimated with caution. Regardless of the assumption made regarding the conditional error distribution, the EGARCH-M model is sensitive to the choice of starting values and the degree of computer precision. However, a simple forecasting example shows that these issues only become a concern when the EGARCH-M model is estimated on a high-precision computer, under the assumption that the conditional errors follow a generalized error distribution. The instability of the risk aversion parameter in the mean equation is the greatest influence on these results.

Beginning with the introduction of the Autoregressive Conditional Heteroscedasticit,/ (ARCH) model of Engle (1982), return-generating models incorporating timevarying volatility have received considerable attention in the finance and economics literature. In these models, volatility is decomposed into predictable and unpredictable components. Most of the literature has centered around the modeling of the predictable, time-varying component. Why is it important to correctly model time-varying volatility in security returns? Understanding how volatility changes through time is crucial to understanding how assets are priced. Many recent tests of the Capital Asset Pricing Model (CAPM), originally developed by Sharpe (1964) and Lintner (1965), have focused on how heteroscedasticity in stock returns affects the CAPM's results (Bollerslev, Engle, and Wooldridge, 1988; Ng, 1991; Schwert and Seguin, 1990; Turtle, Buse, and Korkie, 1994). Researchers have also used time-varying models to predict future volatility (Akgiray, 1989; Pagan and Schwert, 1990). Consequently, it is important that timevarying models produce reliable parameter estimates. Given then the importance of models of conditional volatility in the finance and economics literature, it is vital that these models produce robust results. However, there are many different models of conditional volatility that have been introduced in the literature. 1 These models have ranged in complexity, with one distinguishing feature being the form of the conditional variance process chosen (Bollerslev, 1986; Nelson, 1991). Another distinguishing feature is the assumption made regarding the conditional error distribution. The most common assumption made by researchers is one of conditional normality (Akgiray, 1989; Engle, 1982; 167

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Pagan a n d Schwert, 1990). However, Bollerslev (1987) and Nelson (1991), a m o n g others, have assumed a conditional e r r o r distribution that allows for conditional kurtosis, such as the Student-t or Generalized Error Distribution. Given the wide variety o f models of conditional volatility, it would be impossible to d e t e r m i n e if each m o d e l p r o d u c e d robust results. This p a p e r focuses solely on the E x p o n e n t i a l Generalized Autoregressive Conditional Heteroscedasticity Model (EGARCH) m o d e l of Nelson (1991). Specifically, the p e r f o r m a n c e of the EGARCHin-mean (EGARCH-M) m o d e l is investigated. T h e r e are also m a n y estimation issues, such as the choice o f starting values, the choice of the maximization routine, and the choice o f computer, that researchers must face when using these models. H o w these issues are resolved is rarely m e n t i o n e d in published research. Consequently, unless s o m e o n e has personally estimated o n e of these models, the vast majority of researchers may never realize how a n d if these issues are addressed. T h e p u r p o s e o f this p a p e r is to d e t e r m i n e if the results of the EGARCH-M m o d e l are influenced by these estimation issues. T h e analysis is c o n d u c t e d assuming the conditional errors follow either a n o r m a l distribution or a generalized e r r o r distribution (GED). This p a p e r shows that regardless o f the assumption m a d e regarding the conditional e r r o r distribution, the p a r a m e t e r estimates f r o m the EGARCH-M m o d e l are affected by how these estimation issues are resolved. T h e EGARCH-M m o d e l is very sensitive to the choice o f starting values a n d the degree of c o m p u t e r precision. However, a forecasting example shows that these estimation issues only b e c o m e a c o n c e r n when the EGARCH-M m o d e l is estimated on a high-precision computer, u n d e r the assumption that the conditional errors are distributed GED. T h e instability of the risk aversion p a r a m e t e r in the m e a n equation appears to be driving these results. T h e general conclusions of this p a p e r are that the EGARCH-M m o d e l should be estimated with caution and the estimation issues e x a m i n e d in this p a p e r should be seriously considered. T h e rest o f this p a p e r is organized as follows. A description of the two different EGARCH-M models is presented in the next section. A discussion of the reliability of the models' results follows. Next, a forecasting example is constructed to d e t e r m i n e w h e t h e r the models' results are robust. A brief conclusion ends the paper.

I.

MODEL DESCRIPTION

T h e EGARCH-M m o d e l developed by Nelson (1991) is used to characterize the expected return a n d conditional variance processes. Specifically, daily returns are m o d e l e d as

Rt= a + bR~-i + cO~tt + Et where Rt = the daily return for day t, a = the intercept, b = the degree of first-order autocorrelation in daily returns,

(1)

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c = the degree of correlation between daily returns a n d conditional variance (c can also be t h o u g h t of as a risk aversion p a r a m e t e r ) , a n d o~ = the conditional variance on day t. T h e e r r o r term, el, is assumed to have the following properties: 1.

el = z/o/,

2.

zl is i n d e p e n d e n t and identically distributed (iid) with E(zL) = 0 a n d Var(zt) = 1, and 3. E(et/nt-1) = 0 and V a r ( e t / n ~ l ) = o~, where n ~ l is the set o f all information available at time t - 1.

In the EGARCH-M model, the natural logarithm o f the conditional variance of el, ln(o2), is m o d e l e d as follows: (1 + ~1 L +... + T q L q) In(or2) = c~, + (1 - A1 L -- A - - ~ gCzt-1),

(2)

where A/= • j= p = q= ¢tl =

the the the the the

A R ( 0 coefficient to be estimated for i = 1 . . . . . ~, MA(j) coefficient to be estimated f o r j = 1 . . . . . q; o r d e r of the AR process; o r d e r o f the MA process; and natural logarithm o f the unconditional variance at time t 2.

Nelson (1991) proposes that the f o r m ofg(z~ be a function of b o t h the m a g n i t u d e and sign ofzt in o r d e r to a c c o m m o d a t e the asymmetric relation between stock returns a n d volatility changes (i.e., the leverage effect), which was d o c u m e n t e d by Black (1976). This relationship is expressed as g(z,) = 0zt + v[Iz~ - ~zJ], where the sequence {g(zl)} is a zero-mean, iid r a n d o m sequence by construction, 0 represents the effect of the sign of the return shock, zl, on conditional volatility and y represents the effect of the m a g n i t u d e of zl o n conditional volatility. T h e natural logarithm of the unconditional variance, cxt, is defined as at = a + ln(1 + NtS), where Art is the n u m b e r of n o n t r a d i n g days between trading days t - 1 a n d t, and ct and 6 are the p a r a m e t e r s to be estimated. T h e p a r a m e t e r 6 d e t e r m i n e s the contribution o f n o n t r a d i n g periods to volatility. T h e n u m b e r of n o n t r a d i n g days is included in the unconditional variance process to control for changes in stock volatility that may be caused by information arriving during days the stock m a r k e t is closed d u e to either weekends or holidays. French a n d Roll (1986) find that n o n t r a d i n g periods

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c o n t r i b u t e m u c h less than do trading periods to m a r k e t variance, so it is e x p e c t e d t h a t O < 8 < < 1. T h e density o f a GED r a n d o m variable n o r m a l i z e d to have a m e a n a n d variance o f zero a n d one, respectively, is given by

f(z) =

vexp[-(1/2)lz /~[ v] [~.20%)F(1/v)] '

w h e r e --~ < z < oo, 0 < v < oo, F( ) is the g a m m a function, l a m b d a is d e f i n e d as = [2(-2/01-'(1/v)/1"(3/v)] 0/2),

(3)

a n d v is a tail-thickness parameter. 3 W h e n v = 2, zt has a s t a n d a r d n o r m a l distribution. W h e n v < 2, the distribution o f z t has thicker tails t h a n a n o r m a l distribution. T h e E G A R C H - M m o d e l s are estimated by m a x i m i z i n g the following log-likelih o o d f u n c t i o n with a n d w i t h o u t the restriction that v -- 2: T

LT

= E

[In(v/k0

- (1/2)[(zt/~,)[

v -

(1 + v-l)ln(2) - l n [ F ( 1 / v ) ] - 1/2 In ( ~ ) ] ,

w h e r e ~. is d e f i n e d in E q u a t i o n 3, l n ( a 2) is d e f i n e d in E q u a t i o n 2, a n d zt can be c o m p u t e d recursively as

zt = a71 ( R t - a - bR~l - ca~).

II.

(4)

RET,TABII,,ITY O F R E S U L T S

T h e data used in this analysis consists o f daily security returns f r o m the University o f Chicago's C e n t e r for Research in Securities Prices (CRSP) for firms t r a d i n g o n the New York Stock E x c h a n g e (NYSE) a n d the A m e r i c a n Stock E x c h a n g e (AMEX) for the p e r i o d J a n u a r y 1967 to D e c e m b e r 1991. A n equally-weighted i n d e x m a d e u p o f firms o n the N Y S E / A M E X is constructed. A n A R M A ( p , q ) process m u s t be specified f o r the c o n d i t i o n a l variance process in E q u a t i o n 9. F o r e a c h o f the two models, the Schwartz criterion (Schwartz, 1978) will be used to d e t e r m i n e the a p p r o p r i a t e A R M A ( p , q ) process. Specifically, p a n d q are c h o s e n so as to m a x i m i z e

SC(p,q) = L T - [n X log(XU~], where:

SC(p,q) = LT = n ---T--

the the the the

Schwartz Criterion, value o f the log-likelihood function, n u m b e r o f m o d e l parameters, a n d n u m b e r o f time-series observations.

For an application o f this criterion, see Tucker a n d P o n d (1988) o r Kim a n d Kon (1994). T h e search will be c o n d u c t e d for p = 0 . . . . . 3 a n d q = 0 . . . . . 3. I n o r d e r to

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d e t e r m i n e if the choice of the appropriate ARMA(p,q) process is sensitive to the selection o f different starting values, several sets o f starting values are used during this part o f the analysis. A~ Conditional Normal EGARCH-M

T h r e e sets of starting values, labeled Sets 1, 2, a n d 3, are used to estimate the EGARCH-M m o d e l u n d e r the assumption of conditional normality. T h e p a r a m e t e r vector, B, is (a, b, c, a, 8, 0, y, A1, A2, Aa, ~1, ~2, Wa). For the first set o f starting values, all p a r a m e t e r s are started at zero except ix, which is set to -8.00. W h e n 8 is started at zero, the unconditional variance can be f o u n d by taking exp(t~). In this case, the unconditional variance is started at 0.00034, or 0.34%. This is a daily variance, so on an annual basis, the unconditional variance is approximately 8.5%. I f a is started at zero, then the daily a n d annual unconditional variance would be started at 1% and approximately 250%, respectively. Consequently, a m u s t be started at a large negative number. For the second set o f starting values, Equation 1 is estimated, assuming c = 0. Estimates o f a a n d b are used as starting values. T h e natural logarithm of the residual variance is used as a starting value for ix. T h e rest of the p a r a m e t e r s are started at zero. For the final set of starting values, the first 7 p a r a m e t e r s in B are set at frequently estimated values. T h e ARMA coefficients are started at zero. 4 T h e EGARCH-M models are estimated using the subroutine D U M I N F f r o m the International Mathematical a n d Statistical Libraries (IMSL). Since this is a minimization routine, the negative of the log-likelihood function is minimized. T h e r e are m a n y o t h e r optimization packages or p r o g r a m s that can also be used. S Since DUMINF is a minimization routine, the negative of the log-likelihood function is minimized. T h e estimation of the EGARCH-M is a two-stage process. It is necessary for the estimators in the first stage to be consistent so the next stage can p r o d u c e consistent estimators. T h e first stage involves estimating the initial conditional variances. In this paper, the unconditional variances are used as initial starting values a n d are assumed to be consistent estimators of the conditional variances. If this is the case, the rest o f the p a r a m e t e r s will be consistent regardless o f where they are started, as long as they are started within their legal domains, since they are estimated via a single-stage process. To begin the optimization procedure, a starting value for the conditional variance m u s t first be chosen. T h e conditional variance is initially set at e x p ( a ) , the unconditional variance. Its value will d e p e n d on the choice o f starting values for a. O n c e this is obtained, the conditional variance, along with the starting values for a, b, a n d c, are inserted into Equation 4 and the initial value of zt is calculated. Using the values for the conditional variance and z~ along with the starting value o f v, an initial value for the likelihood function is obtained. All of the p a r a m e t e r s in the vector B are u p d a t e d within the subroutine according to the criteria specified by the user in the subroutine. A new conditional variance is calculated, followed by new values for zt a n d the likelihood function. T h r o u g h o u t this analysis, special attention will be paid to the consistency o f the p a r a m e t e r estimates between the two EGARCH-M models. Also, the models will be

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estimated on two types of computers, a CRAYY-MP/432 s u p e r c o m p u t e r a n d an IBM RS/6000 m o d e l 530 c o m p u t e r (denoted as the CRAYand IBM, hereafter). T h e CRAY has a 64-bit word a n d the IBM has a 32-bit word. By estimating the EGARCH-M models on these two different computers, the relationship between c o m p u t e r precision and the model's results can be examined. T h e results of estimating the EGARCH-M m o d e l u n d e r the assumption of conditional normality, using each of three sets o f starting values, on the CRAY are p r e s e n t e d in Table 1. T h e choice of the ARMA(p,q) process for the conditional variance does a p p e a r to be sensitive to the choice of starting values. Two different ARMA(p,q) models are chosen, the ARMA(1,1) and ARMA(2,1) models. Using Sets 1 a n d 3 as starting values lead to the same p a r a m e t e r estimates and choosing the ARMA(1,1) model. T h e only p a r a m e t e r that differs somewhat is the p a r a m e t e r c. Using Set 2 as starting values leads to an ARMA(2,1) m o d e l being chosen. For this m o d e l to p r o d u c e stable results, the optimal f o r m o f the model, as well as the p a r a m e t e r estimates, should not dramatically change when the 25-year sample period is shifted. In o r d e r to further test the stability of this model, the 25-year sample is also shifted forward and backward one year. In o t h e r words, the same tests are p e r f o r m e d using data f r o m J a n u a r y 1966 to D e c e m b e r 1990, as well as J a n u a r y 1968 to D e c e m b e r 1992. However, when the sample is shifted backward o n e year, an ARMA(2,1) process is chosen for the first two sets of starting values. T h e p a r a m e t e r

Tab/e 1.

Results from Estimating the Conditional Normal EGARCH-M Set 1 CRAY

ARMA Process LogL

(1,1) 22820.352

Set 2 CRAY (2,1) 22830.941

Set 3 CRAY (1,1) 22820.352

Set 2 IBM (1,1) 22819.500

a

0.0008

0.0007

0.0008

b

0.3458

0.3495

0.3458

0.0006 0.3495

c

-3.8382

-3.1068

-3.8410

-0.8828

0t

-9.9354

-9.7812

-9.9354

-9.9536

8

0.2126

0.2108

0.2126

0.2088

0

-0.1149

-0.1155

-0.1149

-0.1132

y

0.2680

0.2397

0.2680

0.2615

A1

0.9670

1.8773

0.9670

0.9658

A2

--

-0.8781

--

--

~1

-0.2792

-0.9654

-0.2792

-0.2596

Note~. The model is estimated over the period 1967 to 1991 using different sets of starting values. The optimal

ARMA(p,q) process is selection using the Schwartz Criterion. LogL is the log-likelihood function. The estimated variables are defined as follows: a = the intercept in the return equation b = the degree of first-order autocorrelation in daily portfolio returns c = a risk-aversion parameter that measures the degree of correlation between the daily portfolio return a n d the daily conditional variance ~x= the constant in the conditional variance equation; 6 = measures the effect o f n o n t r a d i n g days on unconditional volatility 0 a n d y represent the effects of the sign a n d magnitude of return shocks o n conditional volatility, respectively AI a n d A~ are the AR parameters in the conditional variance equation; a n d hul is the MA parameter in the conditional variance equation.

ESTIMATING EGARCH-M MODF.I-~; 173 estimates are similar to those f o u n d for the ARMA(2,1) models in Table 1. For the third set, an ARMA(3,1) process is chosen. W h e n the sample is shifted forward o n e year, an ARMA(3,2) process is selected using Sets 1 and 3 as starting values; an ARMA(2,I) process is selected using Set 2 as starting values. It appears that when the sample period is slightly changed, the optimal m o d e l can change dramatically d e p e n d i n g o n the starting values chosen. Starting values f r o m Set 2 are used to examine how c o m p u t e r precision effects the EGARCH-M model. 6 T h e EGARCH-M model, u n d e r the assumption of conditional normality, is estimated again using starting values f r o m Set 2 and the IBM computer. These results are also contained in Table 1. C o m p u t e r precision does a p p e a r to effect the EGARCH-M model. Now an ARMA(1,1) m o d e l is selected. T h e p a r a m e t e r s a and b arc not affected very m u c h by a change in c o m p u t e r precision. However, the next 5 p a r a m e t e r s are affected, especially the risk aversion p a r a m e t e r c.7 In conclusion, when estimating the EGARCH-M m o d e l o f Nelson (1991) u n d e r the assumption of conditional normality, over long time periods, the m o d e l is sensitive to the choice of starting values and the degree of c o m p u t e r precision. A different ARMA process may be chosen, using the Schwartz Criterion, for the conditional variance process, d e p e n d i n g on the choice of starting values and the sample period. This would affect conditional variance and return forecasts that arc c o m p u t e d using these p a r a m e t e r estimates. Beside the ARMA coefficients, the only o t h e r p a r a m e t e r that appears to be dramatically affected is the risk aversion parameter c. An unstable risk aversion p a r a m e t e r could also lead to p o o r forecasts. GED EGARCH-M

T h e analysis is r e p e a t e d for the EGARCH-M models whose conditional errors follow a GED distribution in o r d e r to examine how the assumption m a d e regarding the conditional e r r o r distribution influences the estimation issues m e n t i o n e d in the previous section. Four different sets of starting values are now used. T h e first two sets are the converged estimates f r o m the first two columns o f Table 1. T h e third and fourth sets of starting values are the same as those in Sets 1 a n d 2 that wcrc used to estimate the EGARCH-M m o d e l u n d e r the assumption of conditional normality. In all four sets of starting values, v is started at 2. Recall that when v = 2, then the conditional e r r o r distribution is normal. W h e n v < 2, the distribution is m o r e leptokurtic than normal. T h e results o f estimating the EGARCH-M m o d e l using each of the sets of starting values on the CRAY f r o m 1967 to 1991 arc presented in Table 2. Regardless of the starting values used, an ARMA(2,1) m o d e l is chosen. T h e 25-year sample is then shifted backwards and forwards o n e year and the EGARCH-M models are re-estimated on the CRAY For the sake of brevity, these results are not reported. The choice of starting values now makes a difference when the sample is shifted backwards one year. Using the two sets of converged estimates as starting values, an ARMA(2,1) model is selected. Using the other two sets o f starting values, an ARMA(1,0) model is selected. When the sample period is shifted forward one year, three different ARMA processes are selected, ARMA(1,0), ARMA(2,1), and ARMA(2,2). Regardless of the sample period, the parameter v is consistently less than 2.00.

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Tab/e 2.

Results from Estimating the GED EGARCH-M Set 1 CRAY

ARMA Process LogL a b c a 5 0 y A1 A2 ~1 v

(2,1) 22974.918 0.0010 0.3347 -7.4281 -10.2875 0.2344 -0.0998 0.2194 1.9064 -0.9068 -0.9767 1.3731

Set 2 CRAY (2,1) 22974.918 0.0010 0.3348 -7.4241 -10.2876 0.2344 -0.0998 0.2194 1.9064 -0.9067 -0.9767 1.3731

Set 3 CRAY (2,1) 22974.918 0.0010 0.3347 -7.4356 -10.2877 0.2344 -0.0998 0.2194 1.9064 -0.9067 -0.9767 1.3732

Set 4 CRAY (2,1) 22974.918 0.0010 0.3347 -7.4357 -10.2876 0.2344 -0.0998 0.2194 1.9064 -0.9068 -0.9767 1.3731

Set 4 IBM (1,0) 22959.342 0.0007 0.3362 -0.4391 -10.1870 0.1974 -0.0714 0.2078 0.9617 --1.3679

Noteg The model is estimated over the period 1967 to 1991 using different sets of starting values. The optimal

ARMA(p,q) process is selection using the Schwartz Criterion. LogL is the log-likelihoodfunction. The estimated variables are defined as follows a = the intercept in the return equation; b = the degree of first-order autocorrelation in daily portfolio returns; c = a risk-aversionparameter that measures the degree of correlation between the daily portfolio return and the dally conditional variance; (x is the constant in the conditional variance equation; fi measures the effect of nontrading days on unconditional volatility; 0 and y represent the effects of the sign and magnitude of return shocks on conditional volatility,respectively; Al and 52 are the AR parameters in the conditional variance equation; hul is the MA parameter in the conditional variance equation; and v is the tail-thickness parameter in the GED density function.

T h e G E D E G A R C H - M m o d e l is t h e n e s t i m a t e d o n t h e I B M . T h e s t a r t i n g v a l u e s f r o m t h e f o u r t h set, t h e r e g r e s s i o n e s t i m a t e s , a r e u s e d t o b e c o n s i s t e n t w i t h t h e s a m e analysis performed with the conditional normal EGARCH-M. These results are also p r e s e n t e d i n T a b l e 2. A n A R M A ( 1 , 0 ) p r o c e s s is n o w s e l e c t e d . E x c e p t f o r t h e p a r a m e ter c and the ARMA coefficients, the parameter estimates do not differ dramatically from those obtained from estimating the same model on the CRAY C.

Analysis of the Risk Aversion Parameter

E x c e p t f o r t h e p a r a m e t e r c, t h e e s t i m a t e d p a r a m e t e r s i n T a b l e s 1 a n d 2 all a p p e a r t o b e c o n s i s t e n t w i t h t h e o r y . H o w e v e r , t h e r i s k a v e r s i o n p a r a m e t e r c is c o n s i s t e n t l y n e g a t i v e o v e r t h e 1 9 6 7 t o 1991 p e r i o d . T h e m a g n i t u d e o f c v a r i e s , d e p e n d i n g u p o n the starting values used, the computer used, and the conditional error assumption. 8 Glosten, Jagannathan a n d R u n k l e ( 1 9 9 3 ) r e m a r k t h a t e v e n t h o u g h i t is g e n e r a l l y agreed upon that investors, within a given time period, would require a larger e x p e c t e d r e t u r n f r o m a s e c u r i t y t h a t is riskier, t h i s r e l a t i o n s h i p b e t w e e n r i s k a n d r e t u r n m a y n o t h o l d t h r o u g h t i m e . T h e y p o s t u l a t e t h a t a p o s i t i v e as w e l l as a n e g a t i v e r e l a t i o n s h i p b e t w e e n r i s k a n d r e t u r n is c o n s i s t e n t w i t h t h e o r y . B a c k u s a n d G r e g o r y

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(1993) show that the relationship between risk premiums and conditional variances can be increasing, decreasing, flat, or n o n m o n o t o n i c . T h e shape of the relationship depends on both the preferences o f the representative agent and the stochastic structure of the economy. They also find that the lack of a theoretical structure between risk premiums and conditional variances may explain why this class of models work well for some assets but not for others. Glosten,Jagannathan and Runkle (GJR) state that this may also account for the apparent fluctuations in the price of risk. This could explain why GJR, as well as French, Schwert and Stambaugh (1987), Nelson (1991), and Kim and Kon (1994), have f o u n d such varied results, both positive and negative, for the risk aversion parameter c. A likelihood ratio test (LRT) is p e r f o r m e d o n all models in Tables 1 and 2 to see if the p a r a m e t e r c is significantly different from zero. T h e LRT statistic is equal to 2[Lu - Lr], where L u is the unconstrained log-likelihood function and L r is the constrained log-likelihood function, where c is set equal to zero (Amemiya, 1985). This test statistic is asymptotically distributed X2(1). The LRT statistic is not statistically significant for the EGARCH-M model u n d e r the assumption of conditional normality, regardless of the starting values and c o m p u t e r used. Whitelaw (1994) finds that focusing o n the c o n t e m p o r a n e o u s relationship between conditional risk and return may not be valid. H e finds that the conditional m e a n and volatility exhibit an asymmetric relation. Specifically, he finds that the volatility leads the expected return. His results may explain why the p a r a m e t e r c is not statistically significant from zero for this version o f the EGARCH-M model. However, the LRT statistic is statistically significant at the 95% level for the EGARCH-M model u n d e r the GED assumption when it is estimated on the CRAY for all sets o f starting values. When estimating this version of the EGARCH-M model on the IBM, the LRT statistic is not statistically significant. This result may still be consistent with Whitelaw (1994). If volatility is clustered through time, and if there is an asymmetric relationship between volatility and the expected return, Equation 2 may not adequately capture all of this. Any remaining conditional heteroscedasficity can be controlled for by assuming the conditional errors follow a GED distribution. Furthermore, given the existence of a near unit root in the conditional variance, an increase in c o m p u t e r precision may be n e e d e d to uncover this effect. Whitelaw uncovers a n o t h e r reason to interpret this p a r a m e t e r with caution. H e finds that the c o n t e m p o r a n e o u s correlation between the conditional m e a n and conditional variance is nonstafionary. This is consistent with the results of (;JR. Whitelaw states that imposing a constant linear relationship between these two variables may lead to erroneous inferences. Consequently, using these models to forecast expected returns and conditional volatility could lead to misleading forecasts. In summary, the estimation problems that exist in estimating the EGARCH-M model are still present regardless of the assumption made regarding the conditional e r r o r distribution. T h e exception to this is the parameter c. T h e instability of this p a r a m e t e r is a concern, especially when all sources of excess kurtosis are adequately controlled for and a high precision c o m p u t e r is used to estimate the EGARCH-M model. T h e next question that needs to be answered is whether these problems affect the forecasting ability of the EGARCH-M model.

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O U T OF SAMPLE FORECASTS

In o r d e r to d e t e r m i n e if these estimation p r o b l e m s are really o f concern, daily conditional variance forecasts are calculated for 1992 using the p a r a m e t e r estimates f r o m the EGARCH-M models over the time period 1967 to 1991. Recall f r o m Tables 1 a n d 2 that for s o m e of the EGARCH-M models, the same p a r a m e t e r estimates are f o u n d using different sets of starting values. Despite this, different variance forecasts are o b t a i n e d because ending values of z t, 02, and the Elzl in 1991 are n e e d e d to start the forecasting procedure. These ending values are different for all models. T h e m e a n absolute p e r c e n t a g e e r r o r (MAPE) is analyzed here since it provides a m o r e robust estimate of the variance of the forecast errors than the root m e a n square error. It penalizes less for oufliers. Consequently, this is a g o o d m e a s u r e to use when analyzing leptokurtic distributions. T h e first step in the analysis is to isolate the effect of starting values on the daily conditional variance forecasts. Forecast errors are c o m p u t e d by taking the difference between conditional variance forecasts p r o d u c e d by each EGARCH-M, but using different sets of starting values. Since different starting values may have led to different ARMA models being chosen, it is inevitable that different EGARCH-M models m a y be c o m p a r e d . T h e MAPEs of these forecasts are p r e s e n t e d in Table 3. In Panel A, the pairwise comparisons are shown for the CN EGARCH-M. T h e MAPEs are very low for all pairs o f forecasts. In Panel B, the pairwise comparisons are shown for the GED EGARCH-M. Now some o f the MAPEs increase considerably. Starting values in Set 2 p r o d u c e m u c h different forecasts of conditional variances than those in Sets 3 a n d 4. It appears that the choice of starting values only has an impact on the forecasting ability of the GED EGARCH-M, not the CN EGARCH-M. Next, the effects o f c o m p u t e r precision a n d the conditional e r r o r assumption on the conditional variance forecasts are isolated. These results are p r e s e n t e d in Table 4. In Panel A, MAPEs are c o m p u t e d from forecast errors between conditional variances, where the only difference in the forecasts is the type of c o m p u t e r used to estimate the EGARCH-M models. W h e n estimating the CN EGARCH-M, b o t h computers p r o d u c e similar conditional variance forecasts; the MAPE is a very low 0.80. However, c o m p u t e r precision does a p p e a r to affect forecasting ability for the GED EGARCH-M; the MAPE rises to 11.42. In Panel B, MAPEs are c o m p u t e d from forecast errors between conditional variances, where the only difference in the forecasts is the assumption m a d e regarding the conditional e r r o r distribution. W h e n the IBM is used to estimate the different types of EGARCH-M models, the conditional variance forecasts are very similar. However, w h e n the CRAY is used, the CN EGARCH-M a n d the GED EGARCH-M p r o d u c e m u c h different conditional variance forecasts; the MAPEs are m u c h higher a n d the m a g n i t u d e of t h e m d e p e n d s on the starting values used. In s u m m a r i z i n g this forecasting exercise, it appears that using a m o r e precise c o m p u t e r a n d m a k i n g a m o r e complex assumption regarding the conditional e r r o r distribution lead to m o r e inaccurate conditional variance forecasts. T h e EGARCH-M appears to p r o d u c e m o r e robust results when a low precision c o m p u t e r is used and conditional normality is assumed. Recall also that the risk aversion p a r a m e t e r is only

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E G A R C H - M MODI~.L~

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Tab/e 3. M e a n A b s o l u t e P e r c e n t a g e E r r o r s , I s o l a t i n g t h e E f f e c t o f S t a r t i n g Values, On Daily Conditional Variance Forecasts Using the CRAY Panel A: Pairwise C o m p a r i s o n s Conditional Variance Forecasts Conditional Normal EGARCH-M FE = o 2(CN,i) - o 2(CN, j) Sets 1 & 2

Sets 1 & 3

Sets 2 & 3

0.78

1.07

0.26

Panel B: Pairwise C o m p a r i s o n s Conditional Variance Forecasts GED EGARCH-M FE = cr2(GED,i) - o'2(GED,j) Sets l & 2 Sets l & 3 0.28

1.69

Sets l & 4

Sets 2&3

Sets 2&4

Sets 3&4

5.62

80.07

11.71

0.51

Notes:. FE is the daily forecast error, o~(CN,i) and o~(CNj) are the daily conditional variance forecasts for the conditional

normal EGARCH-M using starting values from Sets i and j, respectively, and o2(GED,0 and o2(GED,fl are the daily conditional variance forecasts for the GED EGARCH-M using starting values from Sets i and j, respectively.

Tab/e 4. M e a n A b s o l u t e P e r c e n t a g e E r r o r s , I s o l a t i n g t h e E f f e c t o f C o m p u t e r Precision and the Effect of the Conditional Error Assumption, on Daily Conditional Variance Forecasts Panel A: C o m p a r i s o n o f Conditional Variance Forecasts Effect of C o m p u t e r Precision FE = a2(k,IBM) - a2(k,CRAY) CN EGARCH-M, Set 2 0.80 Panel B: Comparison o f Conditional Variance Forecasts Effect o f the Conditional Error A s s u m p t i o n CN EGARCH-M vs GED EGARCH-M FE = o2(CN,i) - o2(GED,i)

GED EGARCH-M, Set 4 11.42

Using the IBM

Using the CRAYa

0.05

3.63, 81.40

Notex.. FE is the daily forecast error, oa(k, IBM) and o~(k,CRAY) are the daily conditional variance forecasts for model k,

where k is the either the conditional normal (CN) EGARCH-M or the GED EGARCH-M, estimated on the IBM and CRAYcomputers, respectively, and o'~(CN,i) and o3(GED,i) are the daily conditional variance forecasts for the CN EGARCH-M and the GED EGARC,H-M, respectively, using starting values from Set L aln order to compare the CN EGARCH-M forecasts with the GED EGARCH-M forecasts, the same set of starting values must be used. The set of starting values containing the regression starting values and the set of starting values containing all zeros except for (x and v were both used to estimate the EGARCH-M models on the CRAY. The first MAPE, 3.63, is obtained using the regression starting values and the second MAPE, 81.40, is obtained using the latter set of starting values.

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statistically significant for the GED EGARCH-M model, when it is estimated on the CRAY. The influence of this parameter may be affecting the variance forecasts.

IV.

CONCLUSION

This paper demonstrates that conditional hetcrosccdastic models should bc estimated with caution. This paper focused solely on the estimation of the EGARCH-M m o d e l o f Nelson (1991) over a specific time period, with one particular datasct, using a particular search algorithm. This paper f o u n d that the EGARCH-M model is sensitive to the choice of starting valucs and that c o m p u t e r precision also influences thc parameter estimates. All of the problems f o u n d when estimating the EGARCH-M model exist regardless of the assumption made regarding the conditional error distribution. The exception to this is the risk aversion parameter in the m c a n equation. This parameter is very unstable, but its instability only becomes a concern when all sources of excess kurtosis are adequatcly controlled for and a high precision c o m p u t e r is used to estimate thc EGARCH-M model. A forecasting example is constructed to see if these estimation problems are really of concern. The forecast errors increase dramatically when the EGARCH-M model is estimated on a m o r e precise computer, u n d e r a m o r e theoretically correct conditional error assumption. The instability of the risk aversion p a r a m e t c r may also be influcncing the variance forecasts in this case. Given the popularity of these models and the ease to which c o m p u t e r programs for thcsc models can bc acquired, it is impcrative that researchers recognize and resolve these estimation issues. However, before any general conclusions can be made about the reliability of the results of the EGARCH-M model, a complete simulation study should be performed. This would eliminate the possibility that the composition of the datasct or the search algorithm influenced the issues discussed in this paper.

Acknowledgment: The

author would like to thank Paul Beaumont, Davc Peterson, seminar participants at the Fall 1995 Front Range Financial Seminar and the 1995 Financial Managemcnt Association Annual Mccting, and an anonymous referee for their helpful comments and suggcstions. Thc author assumcs responsibility for any remaining errors.

NOTES * Direct all correspondence to: Eileen E St. Pierre, Department of Finance, College of Business Administration, University of Northern Iowa, Cedar Falls, IA 50614 @uni.edu. 1. Bollerslev, Chou and Kroner (1999) provide an excellent summary of many of these models. 2. The above process must be stationary and invertible in order for ln(t~) to be written as an ARMA(p,q) process. All of the models estimated in this paper are stationary and invertible. However, some of the models are very close to being nonstationary, indicating high persistence in variance. This result has been documented by previous research.

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3. The GED is a family of distributions. Nelson (1991) uses the GED to obtain finite unconditional moments of o~t and et. For some distributions, like the Student-t with finite degrees of freedom, these variables typically do not have finite unconditional moments. The GED not only includes the normal as a special case, but is also includes more fat-tailed distributions like the double exponential and more thin-tailed distributions like the uniform. For more information on the GED, see Harvey (1981) and Box and Tiao (1973). 4. Specifically, B is started at (0.0005, 0.18, 5.50, -10.00, 0.12, 0.23, 0.17, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00). 5. This subroutine relies on numerical derivatives. In other words, the user does not have to input the first-order conditions. Standard errors can be computed through a process involving the inversion of the Hessian matrix. However, using numerical derivatives does not ensure that the Hessian matrix will be positive definite. This is indeed the case for many of the model estimations in this paper. Consequently, standard errors are not reported. Optimizing using the subroutine DUMING was also attempted. With this subroutine the user needs to input the first-order conditions. However, reprogramming the first-order conditions whenever the order of the model changed (i.e., whenever p and q changed) proved to be extremely tedious. Consequently, DUMINF was used throughout the paper. Changing the convergence criterion within DUMINF did not significantly affect the results. 6. This set ofstarting values is used in this part of the analysis since using Set 2 consistently leads to an ARMA(2,1) process being chosen, regardless of sample set. 7. When the sample period is shifted forward and backward one year, an ARMA(1,0) and an ARMA(1,1) process are selected, respectively. The risk aversion parameter c and the parameters in the conditional variance equation are all affected by a decrease in computer precision. 8. In some circumstances, shifting the time period forward and backward one year caused c to become positive. The parameter c may be sensitive to the length and type of stock return data used. Using an EGARCH(1,3)-M model, Kim and Kon (1994) estimated significantly positive values for all individual stocks in the DowJones Industrial Average and for CRSP equally-weighted portfolio returns over the period July 2, 1962 to December 31, 1990. Nelson (1991 ) finds c to be -3.3608 after estimating EGARCH (2,1 ) -M model using daily excess returns on the value-weighted CRSP index. French et al. (1987) estimated c to be 2.410 using monthly excess returns from 1928 to 1984.

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