Prediction in ARMA models with GARCH in mean effects

24 Pages·2001·0.27 MB·English

Prediction in ARMA models with GARCH in mean
e(cid:11)ects
Menelaos Karanasos?
University of York, Heslington, York, YO10 5DD,UK
Abstract
This paper considers forecasting the conditional mean and variance from an ARMA model with
GARCH in mean e(cid:11)ects. Expressionsfor the optimal predictorsand their conditional and unconditional
MSE’s are presented. We also derive the formula for the covariance structure of the process and its
conditional variance.
Key Words: ARMA Model, GARCH in Mean E(cid:11)ects, Optimal Predictor, Autocovariances.
JEL Classi(cid:12)cation: C22
?
IwouldliketothankMKaranassouandananonymousrefereeforhelpfulcommentsandsuggestions.
For correspodence: Email: [email protected], Tel: [phone], Fax: [phone]
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1 Introduction
The autoregressive conditional heteroscedasticity (ARCH) model introduced by Engle (1982)
and its generalisation, the GARCH model introduced by Bollerslev(1986) have become increas-
ingly popular in modelling (cid:12)nancial and economic variables (see for example, the surveys of
Berra and Higgins (1993), Bollerslev, Chou and Kroner (1992), Bollerslev, Engle and Nelson
(1994), and for a more detailed description the book by Gourieroux (1997)). Following Engle’s
pathbreaking idea, several formulations of conditionally heteroscedastic models (e.g. Expo-
nential GARCH, Fractional Integrated GARCH, Switching ARCH, Asymmetric Power ARCH,
ComponentGARCH)havebeenintroducedintheliterature,forminganimmenseARCHfamily.
AlthoughtheliteratureonGARCHtypemodelsisquiteextensive, relativefewerpapershave
examined the issue of forecasting in models where the conditional volatility is time-dependent.
EngleandKraft(1983) considerpredictionsfromanARMAprocesswithARCHerrors,whereas
Engle and Bollerslev (1986), hereafter EB, and Baillie and Bollerslev (1992), hereafter BB,
consider predictions from an ARMA model with GARCH errors.
OneimportantexclusionfromthisframeworkconcernstheARCHinmeanmodel,introduced
by Engle, Lilien and Robins (1987). This model was used to investigate the existence of time
varying term premia in the term structure of interest rates. Such time varying risk premia
have been strongly supported by a huge body of empirical research, in interest rates (Hurn,
McDonald and Moody, 1995), in forward and future prices of commodities (Hall , 1991, Moosa
andLoughani,1994), inGDP(Price,1994), inindustrialproduction(CaporaleandMcKierman,
1996) and especially in stock returns (see Campbell and Hentscel, 1992, Glosten, Jagannathan
and Runkle, 1993, Black and Fraser, 1995, Fraser, 1996, Hansson and Hordahl, 1997 , Elyasiani
and Mansur, 1998).
In this paper we focus our attention on predictions from a general ARMA model with
GARCH-in-mean e(cid:11)ects. In Section 2 we present a new method for obtaining multiperiod
predictions from the ARMA(r,s)-GARCH(p,q)-in-mean model. In particular, we derive the
optimal multistep predictor in terms of past observations and past errors. The coe(cid:14)cients in
our formula are expressed in terms of the roots of the autoregressive polynomials (AR) and the
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parameters of the moving average (MA) ones. We should mention that we only examine the
case where the roots of the AR polynomial are distinct (the case of equal roots is left for future
research). To point out the importance of our results we quote BB (1992): \Processes with
feedback from the conditionalvariance to the conditionalmean willconsiderably complicate the
form of the predictor and its associated mean square error (MSE). Analysis of such models is
consequently left for future research".
The goal of our method is theoretical purity rather than the production of expressions in-
tended for practical use. However, our method can be employed in the derivation of multistep
predictionsfroma more complicatedmodelwithsimultaneousfeedback between the conditional
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mean and variance, namely the GARCH-M-X model (see Christodoulakis, Hatgioannides and
Karanasos, 1999, hereafter CHK). Another value of our method is that it can easily be general-
ized to multivariate GARCH and GARCH in-mean models.
In addition to our method for obtaining a closed form expression for the optimal predictor
(and its associated MSE) of the conditional mean from the ARMA(r,s) -GARCH(p,q) in-mean
model, the following three substantive problems for which solutions do not exist in the GARCH
literature are solved in this paper: (a) We give the in(cid:12)nite moving average (ima) representa-
tions of the conditional mean and variance. These formulae can be used to obtain alternative
expressions for the minimum MSE (MMSE) predictors of the conditional mean and variance
in terms of in(cid:12)nite past observations and errors. (b) We give the canonical factorization (cf)
of the autocovariance generating function (agf) of the process and its conditional variance, the
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covariances between the squared errors and the conditional variance, and (cid:12)nally the autoco-
variances and cross covariances for the process and its conditional variance. (c) We give the
MMSE predictor of future values of the squared conditional variance. We subsequently use this
predictor to obtain the conditional MSE associated with the MMSE predictor of the futures
values of the conditional mean for the ARMA-GARCH in-mean model.
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The GARCH-M-X model was introduced by Longsta(cid:11) and Schwartz (1992) and includes the GARCH-X
modelof Brenner, Harjes andKroner (1996) as a special case.
2
Theautocovariance functionof thesquared errors for theGARCH(p,q)modelis giveninKaranasos (1999a)
and Heand Terasvirta (1997).
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2 ARMA-GARCH in-mean Model
To our knowledge, the analysis of the covariance structure and the multistep predictions from
a general ARMA model with GARCH errors and in-mean e(cid:11)ects has not been considered yet.
This paper attempts to (cid:12)ll this gap in the literature.
In what follows we will consider the ARMA(r,s)-GARCH(p,q)-M process:
(cid:8)(L)yt = (cid:30)+(cid:14)ht +(cid:2)(L)(cid:15)t; (2.1)
2
B(L)ht = !+A(L)(cid:15)t; (2.1a)
where (cid:8)(L), (cid:2)(L), B(L) and A(L) are given by
r r s
j j
(cid:8)(L) = (cid:0) (cid:30)jL = (1(cid:0)(cid:21)jL); (cid:2)(L) = (cid:0) (cid:18)jL ; (cid:30)0 = (cid:18)0 =(cid:0)1; (2.1b)
j=0 j=1 j=0
X Y X
p q
i i
B(L) = (cid:0) (cid:12)iL ; A(L) = aiL ; (cid:12)0 = (cid:0)1 (2.1c)
i=0 i=1
X X
Corollary 2.1. The univariate ARMA representations of ht and yt are given by
?
p
? ? ?
B (L)ht =!+A(L)vt; B (L) =B(L)(cid:0)A(L) = (1(cid:0)fiL) (2.2)
i=1
Y
? (cid:14) ? (cid:14) ?
B (L)(cid:8)(L)yt =(cid:30) +(cid:14)A(L)vt +(cid:2)(L)B (L)(cid:15)t; (cid:30) = (cid:30)B (1)+(cid:14)! (2.3)
? 2
p = max(p;q) and vt =(cid:15)t (cid:0)ht. The vt’s are uncorrelated but they are not independentand
have avery complicateddistribution. Itisnot di(cid:14)cultto showthatunderconditionalnormality
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we have that var(vt) = (2=3)E((cid:15)t) (see Karanasos, 1999a, hereafter K). The fourth moment of
the errors is given in Ling (1999), He and Terasvirta (1997), hereafter HT, and K (1999a). A
general condition for the existence of the 2mth moment is given by Ling (1999) (see also Ling
and Li, 1997).
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Proof. In (2.1a) we add and subtract A(L)ht and we get (2.2). Moreover, multiplication of
?
(2.1) by B (L) and substitution of (2.2) into (2.1) gives (2.3).
Assumption 1: All the roots of the autoregressive polynomial [(cid:8)(L)] and all the roots of
the moving average polynomial [(cid:2)(L)] lie outside the unit circle (Stationarity and Invertibility
conditions for the ARMA model).
Assumption 2: The polynomials (cid:8)(L) and (cid:2)(L) have no common left factors other than
unimodular ones, i.e, if (cid:8)(L) = U(L)(cid:8)1(L) and (cid:2)(L) = U(L)(cid:2)1(L), then the common factor
U(L) must be unimodular (Irreducibility condition for the ARMA model).
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Assumption 3: All the roots of the autoregressive polynomial [B (L)] and all the roots of
the MA polynomial [A(L)] lie outside the unit circle (Stationarity and Invertibility conditions
for the GARCH model).
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Assumption 4: The polynomials B (L) and A(L) are left coprime. In other words the
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B (L)
representation A(L) is irreducible.
Assumption 5. The polynomials(cid:8)(L) and A(L) are left coprime. In other words the repres-
A(L)
entation (cid:8)(L) is irreducible.
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InwhatfollowsweonlyexaminethecasewheretherootsoftheARpolynomials[(cid:8)(L);B (L)]
are distinct.
Corollary 2.2. Under assumptions 1-5, the canonical factorization (cf) of the autocovariance
generating function (agf) for yt is given by
(cid:0)1 2 (cid:0)1 2 1
(cid:14)A(z)A(z )(cid:27)v (cid:2)(z)(cid:2)(z )(cid:27)(cid:15) j (cid:0)j
gz(y) = ? (cid:0)1 ? (cid:0)1 + (cid:0)1 = fj(cid:13)j(z +z ) (2.4)
(cid:8)(z)B (z)(cid:8)(z )B (z ) (cid:8)(z)(cid:8)(z )
j=0
X
:5 if j =0
where fj = and the (cid:13)j’s are given in Theorem 2.1c.
(0 otherwise
Proof. The proof follows immediately from the univariate ARMA representation (2.3) and
the cf of the agf of an ARMA model given in Nerlove, Grether and Carvalho (1979, pp. 70-78),
hereafter NGC, and Sargent (1979, p. 228).
Under assumptions 1-5, the in(cid:12)nite-order ma representation of yt+i is given by
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(cid:14) 1 r+p?min(n;q)
(cid:30) (cid:14) n(cid:0)j
yt+i = ? + [(cid:14)gnvt+i(cid:0)n +sn(cid:15)t+i(cid:0)n]; gn = ul0((cid:21)l) aj;
B (1)(cid:8)(1)
n=0 l=1 j=1 (2.5)
X X X
(cid:16)l;t+p?
p? if l =1;2(cid:1)(cid:1)(cid:1) ;r
?
Q((cid:21)l(cid:0)fj)
t+r(cid:0)1 8j=1
(cid:16)l;t = r (cid:21)l ; ult =>>>> p? (fm?)t+r+rp?(cid:0)1 if l =r+m; 1 (cid:20)m (cid:20)p? ;
((cid:21)l (cid:0)(cid:21)j) >>< Q (fm?(cid:0)fj?)Q(fm?(cid:0)(cid:21)j) (2.5a)
j=1 j=1 j=1
jQ6=l >j6=m
>
>
r min(n;s) >>
n(cid:0)j>: (cid:14) (cid:21)l if l = 1;(cid:1)(cid:1)(cid:1) ;r
sn =(cid:0) (cid:16)l0(cid:21)l (cid:18)j; (cid:21)l = ? ? (2.5b)
l=1 j=0 (fm if l = r+m; 1 (cid:20)m(cid:20)p
X X
Proof. The proof follows directly from the univariate ARMA representation (2.3) and the
Wold representation of an ARMA model given in Pandit and Wu (1983, p. 105) and Pandit
(1973).
In the following Theorem we present closed form algebraic expressions for the optimal pre-
dictor(and its associated MSE) of futurevalues for the conditionalmean fromthe above model.
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We obtain the expression for the optimal predictor by using a new method for solving a linear
homogeneous di(cid:11)erence equation together with a technique for the manipulation of lag poly-
nomials used in Sargent (1987). We believe that our method gives a useful insight into the
treatment of linear stochastic di(cid:11)erence equations. Another advantage of our method is that it
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can be used to derive formulae for the multiperiod predictions from the GARCH-M-X model
(see CHK, 1999). In addition, our method can be applied to even more complicated GARCH
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modelslikethe Component GARCH, the Asymmetric Power ARCH andthe SwitchingARCH .
Finally, our method can easily be generalized to multivariate GARCH and GARCH-in-mean
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models .
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We express the rth order di(cid:11)erence equation as an AR(1) process with an error term which follows a r-1
di(cid:11)erenceequation. Weobtainyt asafunctionofrpastvaluesandtherootsoftheassociatedauxiliaryequation.
Foranexcellentdiscussion onsolutions of linear di(cid:11)erenceequationssee Wei (1989, pp. 27-30) orBrockwell and
Davis(1983, Sect. 3.6).
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The GARCH-M-Xmodel was introduced by Longsta(cid:11) and Schwartz (1992) as a discretization of their two-
factor short-terminterest rate model.
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The component GARCH model was introduced by Ding and Granger (1996),the Asymmetric Power ARCH
wasintroducedbyDing,Engle andGranger (1992). TheSwitchingARCHandGARCHmodelswere introduced
byHamiltonand Susmel(1996) and Dueker(1997), respectively.
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The statistical properties of the multivariate GARCH (MGARCH) model have been recently examined by
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Theorem 2.1a. Under assumptions 1-5 the i-step-ahead predictor of yt+i is readily seen to
be
?
q(cid:0)1 s(cid:0)1 r+p (cid:0)1
0 (cid:14) (cid:14)
Et(yt+i)=(cid:30) +(cid:14) zinvt(cid:0)n+ zin(cid:15)t(cid:0)n+ xinyt(cid:0)n; (2.6)
n=0 n=0 n=0
X X X
r+p?min(i+n;q) r+p?
(cid:14) (cid:14) i+n(cid:0)j (cid:14)
zin = ul0((cid:21)l) aj; xin = uli(cid:13)ln; (cid:13)l0 =1; (2.6a)
l=1 j=n+1 l=1
X X X
?
n r+p (cid:0)(n(cid:0)j) n
n (cid:14)
(cid:13)ln =((cid:0)1) [ ] ((cid:21)kj); k0 =0; (2.6b)
j=1 kj=kj(cid:0)1+1 j=1
Y X Y
kj6=l
r min(i+n;s)
i+n(cid:0)j
zin =(cid:0) (cid:16)l0(cid:21)l (cid:18)j; (2.6c)
l=1 j=n+1
X X
?
r+p
0 (cid:14) 1 uli
(cid:30) =(cid:30) [ ? (cid:0) u(cid:22)li]; u(cid:22)li = (cid:14)
B (1)(cid:8)(1) l=1 1(cid:0)(cid:21)i
X
(2.6d)
(cid:14)
where ult and (cid:21)l are given in (2.5a) and (2.5b) .
It is importantto note that inthe presence of GARCH inmean e(cid:11)ects the optimalpredictor
for the conditional mean is a function of past values not only of the observations and the errors
2
(yt(cid:0)n; (cid:15)t(cid:0)n) but of the conditional variances and the squared errors (ht(cid:0)n; (cid:15)t(cid:0)n) as well.
Proof. The proof follows from the univariate ARMA representation (2.3) and the methodo-
logy presented in Appendix A.
In many applications in (cid:12)nancial economics the primary interest centres on the forecast for
the future conditional variance. Such instances include option pricing as discussed by Day and
Lewis(1992) andLamourex andLastrapes(1990), the e(cid:14)cient determinationof the market rate
of return as examined in Chou (1988), and the relationshipbetween stock market volatility and
researchers. For example, Lin (1997) analyses the impulse response function for conditional volatility in various
MGARCHmodels. ForatheoreticalandempiricalanalysisofthemultivariateGARCH-in-meanmodelsseeSong
(1996).
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the businesscycle as analysed by Schwert (1989). In these situations itis therefore of interest to
be able to characterise the uncertainty associated with the forecasts for the future conditional
variances as well. Some potentially useful results for this purpose are given in Lemma 2.1 and
Theorem 2.1b.
Lemma 2.1. The forecast error for the above i-step-ahead predictor is given by
i(cid:0)1
FEt(yt+i) =yt+i(cid:0)Et(yt+i) = [(cid:14)gnvt+i(cid:0)n+sn(cid:15)t+i(cid:0)n] (2.7)
n=0
X
with unconditional and conditional MSE given by
i(cid:0)1 i(cid:0)1
2 4 2 2 2
V[FEt(yt+i)] = (cid:14) (2=3)E((cid:15)t) gn+E((cid:15)t) sn (2.8)
n=1 n=0
X X
i(cid:0)1 i(cid:0)1
2 2 2 2
Vt[FEt(yt+i)] = 2(cid:14) gnEt(ht+i(cid:0)n)+ snEt(ht+i(cid:0)n) (2.9)
n=1 n=0
X X
where sn is given in (2.5b), and gn is given in (2.5).
Note thatinthepresenceofGARCH inmeane(cid:11)ects theconditionalMSEisafunctionofthe
forecasts of the future values not only of the conditional variance but of the squared conditional
variance as well.
Observe that the above formulae when(cid:8)(L) =(cid:14) = 1 and(cid:2)(L) = 0 give the iperiodforecast
error (and its conditional and unconditional variances) of the conditional variance.
Proof. The proof follows immediately from the in(cid:12)nite-order ma representation (eq. 2.5).
2
Theorem 2.1b The i-period ahead forecast for the squared conditional variance ht is given by
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q(cid:0)1 p(cid:0)1 q(cid:0)1
2
2 (cid:15) 2 h (cid:15) 4
Et(ht+i) =!(cid:22) + j;i(cid:15)t(cid:0)j + j;iht(cid:0)j + j;i(cid:15)t(cid:0)j
j=0 j=0 j=0
X X X
q(cid:0)2 q(cid:0)1 p(cid:0)1
2 2
(cid:15) 2 2 h 2
+ k1k2;i(cid:15)t(cid:0)k1(cid:15)t(cid:0)k2 + j;iht(cid:0)j
k1=0k2=k1+1 j=0
X X X
q(cid:0)1p(cid:0)1 p(cid:0)2 p(cid:0)1
2
(cid:15)h 2 h
+ lj;i(cid:15)t(cid:0)lht(cid:0)j + k1k2;iht(cid:0)k1ht(cid:0)k2 (2.10)
l=0 j=0 k1=0k2=k1+1
XX X X
where the !(cid:22) and all the ’s are functions of the GARCH parameters which can be obtained
from a system of di(cid:11)erence equations given in Appendix B. The above expression is very useful
becausetheoptimalforecastsofthesquaredconditionalvarianceisneededinordertoobtainthe
conditionalvariance of the forecast error associated withthe optimalforecast forthe conditional
mean from the ARMA-GARCH in-mean model (see eq 2.9 ).
In the following Theorem we give a formula for the covariance structure of the ARMA-
GARCH in-mean model which includes several simpler models as special cases.
Theorem 2.1c. Under Assumptions 1-5 the autocovariance function of yt is given by
?
r r+p
(cid:13)j =covj(yt) = eijzi;jvar((cid:15)t)+ (cid:25)ijdi;jvar(vt); (2.11)
i=1 i=1
X X
j+r(cid:0)1
(cid:21)i (cid:16)ij
eij = r r = r ; (2.11a)
(1(cid:0)(cid:21)l(cid:21)i) ((cid:21)i(cid:0)(cid:21)k) (1(cid:0)(cid:21)l(cid:21)i)
l=1 k=1 l=1
k6=i
Q Q Q
j+r+p?(cid:0)1
(cid:21)i
r r p? p? if i=1;(cid:1)(cid:1)(cid:1) ;r
? ?
8Q(1(cid:0)(cid:21)i(cid:21)l) Q ((cid:21)i(cid:0)(cid:21)k)Q(1(cid:0)(cid:21)ifl)Q((cid:21)i(cid:0)fk)
l=1 l=1 k=1
> k=1
(cid:25)ij =>>>>> k6=i (fn?)j+r+p?(cid:0)1 ? ;
>><Qr (1(cid:0)fn?(cid:21)l)pQ?(1(cid:0)fn?fl?)Qr (fn?(cid:0)(cid:21)k) pQ? (fn?(cid:0)fk?) if i=r+n; 1 (cid:20)n(cid:20)p (2.11b)
l=1 l=1 k=1
k=1
>> k6=n
>
>
>
>
>
>
:
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s j s(cid:0)l s s(cid:0)l
2 l (cid:0)l l l(cid:0)2j
zi;j = (cid:18)k+ (cid:18)k(cid:18)k+l((cid:21)i+(cid:21)i )+ (cid:18)k(cid:18)k+l((cid:21)i+(cid:21)i ); (2.11c)
k=0 l=1 k=0 l=j+1k=0
X XX X X
q j q(cid:0)l q(cid:0)1 q(cid:0)l
2 (cid:14) l (cid:14) (cid:0)l (cid:14) l (cid:14) l(cid:0)2j
di;j = ak + akak+l[((cid:21)i) +((cid:21)i) ]+ akak+l[((cid:21)i) +((cid:21)i) ]
k=1 l=1 k=1 l=j+1k=1
X XX X X
(2.11d)
(cid:14) (cid:14) ? ?
and (cid:21)i =(cid:21)i , for i = 1;(cid:1)(cid:1)(cid:1) ;r, (cid:21)i =fn, for i= r+n, 1 (cid:20) n(cid:20) p .
Observe that the above general formula incorporates the following results as special cases:
(a) the acf for the white noise process with GARCH(1,1) in mean e(cid:11)ects given in Hong
(1991), (b) the acf for the ARMA(r,s) model given inZinde-Walsh (1988), and K (1998, 1999b),
and (c) the acf of the conditional variance for the GARCH(p,q) model.
Proof. The covariance structure can be derived by using the following three alternative
methods: (i) the one used in K (1999a), (ii) the one based on the cf of the agf (2.4) and (iii) the
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one based on the in(cid:12)nite-order ma representation (2.5) .
Theorem 2.1d. The cf of the cross covariance gf between yt and ht (gz(yh)) is given by
1 (cid:0)1
m A(z)A(z ) 2
gz(yh) = (cid:13)mz = ? ? (cid:0)1 (cid:14)(cid:27)v (2.12)
(cid:8)(z)B (z)B (z )
m=(cid:0)1
X
Proof. The prooffollows directlyfromthe univariate ARMA representations(2.2), (2.3) and
the cf of the agf of ARMA processes given in Sargent (1979, p. 228).
Moreover, the cross covariances ((cid:13)m) are given by
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See K (1999c) for the use of methods (ii) and (iii) in the context of univariate and multivariate GARCH
models.
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