class of nonlinear arch

8/7/2019 Class of Nonlinear Arch

1/23

Economics Department of the University of Pennsylvania

Institute of Social and Economic Research -- Osaka University

A Class of Nonlinear Arch ModelsAuthor(s): M. L. Higgins and A. K. BeraSource: International Economic Review, Vol. 33, No. 1 (Feb., 1992), pp. 137-158Published by: Blackwell Publishing for the Economics Department of the University ofPennsylvania and Institute of Social and Economic Research -- Osaka University

Stable URL: http://www.jstor.org/stable/2526988Accessed: 01/03/2010 21:37

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/action/showPublisher?publisherCode=black.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

Economics Department of the University of Pennsylvania, Blackwell Publishing, Institute of Social and

Economic Research -- Osaka University are collaborating with JSTOR to digitize, preserve and extend accessto International Economic Review.

http://www.jstor.org
http://www.jstor.org/stable/2526988?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=blackhttp://www.jstor.org/action/showPublisher?publisherCode=blackhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/2526988?origin=JSTOR-pdf


2/23

INTERNATIONAL ECONOMIC REVIEWVol. 33, No. 1, February1992

A CLASS OF NONLINEAR ARCH MODELS*BY M. L. HIGGINSAND A. K. BERA'

A class of nonlinear ARCH models is suggested. The proposed classencompassesseveralfunctionalformsfor ARCHwhich have been putforthinthe literature.A Lagrangemultipliertest is developed to test Engle's ARCHspecificationagainstthe wider class of models. This test provides an easilycomputed diagnosticcheck of the adequacyof an ARCH model afterit hasbeen estimated. The theory is appliedto a numberof weekly exchangerateseries andwe find strongevidence of nonlinearARCH.

1. INTRODUCTIONSince the introduction of the autoregressive conditional heteroskedasticity(ARCH) model in an-influentialpaper by Engle (1982), there has been considerableinterest in models in which the conditionalvariance of the current observation is afunction of past observations. Engle formally defined the ARCH regression modelfor a dependent variable yt by specifying its conditional distributionas

Yt (t - I-

N(xtp, ht)whereht = h(Et- I, Et-2, Et-p; a),

E t = Yt - XtP I

PDis the informationset at time t, x, is a vector of exogenous variablesand laggedvalues of the dependent variables, and 3 and a are parameter vectors. Englesuggested several functional forms for h(-), but concentrated primarilyon the linearARCH(p) model,

(l) ~~~~~ht=-- a o + a I Et_ I + * + Ot E- PTo ensure that the conditional variance is strictly positive for all realizations of 'E,the linear ARCH model (1) requires that the parameter space be restricted to a0 >Oand ai > 0, i = 1, ..., p.Engle showed that for the ARCH(p) process to be covariance stationary, the

* Manuscript received November 1989; final revision July 1991.1 We would like to thank three anonymous referees for detailed comments and many helpfulsuggestions. We are also grateful to Tim Bollerslev, Nichola Dyer, Rob Engle, Clive Granger, RogerKoenker, Dan Nelson, Paul Newbold and Pin Ng. Thanks are also due to W. Davis Dechert for supplyinghis software for calculating the BDS test. Earlier versions of this paper were presented at the UCSDConference on Statistical Models for Financial Volatility and the Sixth World Congress of theEconometric Society. We thank the participantsat these conferences for helpfulcomments. However, weretain the responsibility for any remainingerrors.

137


3/23

138 M. L. HIGGINS AND A. K. BERAroots of 1 - aL - aPLP must be larger than one in absolute value. Thiscondition implies that

pEai < ,

and the unconditional variance of Et is given byvar (E,) = (2 = ap

1- > aior

a0 = aiaio2.

Substituting this expression for a0 into (1)P

ht I - Ei o,2 a2_ l+ *l+ a 2-Pi~ ~~a EtII EEngle's specification of the conditional variance is a weighted average of the"global" variance o2 and the "local" variances Et-1, . EtP.In many empirical applications, it has been found that ARCH models capturesome important features of time series data, such as nonlinear dependence,nonnormality and over-dispersion. One fruitful way of looking at ARCH is throughits close link with the random coefficient autoregressive (RCAR) model. Asdiscussed in Tsay (1987)and Bera, Higginsand Lee (1988), if we startwith a RCARmodel for Et, under certain conditions the first two conditional moments of theprocess will be the same as those of an ARCH model. If we further assumeconditional normality, all the moments of these two processes will coincide. Thisgives an intuitively appealing explanation of the presence of ARCH in time seriesdata.Since Engle's paper, many extensions and generalizationsof the ARCH modelhave appeared (see Engle and Bollerslev 1986 for a general survey and Bollerslevet al. 1990 for applications). We believe that this research falls into three generalareas, each of which addresses one of the assumptions of Engle's original ARCHspecification.First, a majority of research has examined ARCH models in which the condi-tional variance ht is a function not only of the Et's but also of other elements of_t-I(for example, see Bollerslev 1986; Engle, Grangerand Kraft 1984; Granger,Robins and Engle 1984and Weiss 1984, 1986).A second area of research addressesthe assumption of conditional normality. The conditional distribution of data forwhich ARCH models are used, is frequently skewed and leptokurtic. To account


4/23

NONLINEAR ARCH MODELS 139for heavy tails of the conditional distribution, Engle and Bollerslev (1986) andBollerslev (1987) use Student's t, Hsieh (1989) uses the generalized error distribu-tion, while Lee and Tse (1991) suggest using a Gram-Charliertype distribution. Athird extension of the ARCH model, by far the one having received the leastamount of attention to date, considers functional forms for ht other than Engle'slinear ARCH specification. Engle (1982) in fact suggested several functional forms,but concentrated on the linearmodel for its analytic convenience and its plausiblityas a data generating mechanism. Engle and Bollerslev (1986), Geweke (1986) andPantula (1986) have suggested other functional forms. The use of alternatives to thelinear ARCH model is hindered by the lack of a specification test to determine theadequacy of the linear ARCH model and a procedure for selecting an alternativefunctional form if the linear ARCH model is rejected.The specification of the correct conditional variance function is important inseveral respects. The accuracy of forecast intervals depends upon selecting thefunction which correctly relates the future variances to the current informationset.Also, the test for detecting the presence of ARCH is partially determined by thefunctional form of the ARCH process. Further, Pagan and Sabau (1987) haveshown that an incorrect functional form of the ARCH process for the errors of aregression model can result in inconsistent maximum likelihood estimators of theregression parameters.In this paper, we propose a nonlinear ARCH model which encompasses severalof the models which have been proposed in the literature. Using this more generalARCH model, the specification testing of the functional form of the ARCH modelis addressed. Section 2 is a survey of functional forms which have been proposedfor ARCH. In Section 3, we introduce a nonlinear ARCH model and show that itencompasses some of the models discussed in Section 2. A Lagrange multiplier(LM) test is developed in Section 4 to test the linear ARCH model against a classof nonlinear ARCH models. In Section 5, the theory is applied to analyze weeklyexchange rates of six major currencies against the U.S. dollar covering the periodJanuary 1973to June 1985. Section 6 contains some concluding remarks.

2. FUNCTIONALFORMSFOR ARCHAlthough Engle (1982)focused on the linear ARCH model, he acknowledgedthat"it is likely that other formulations of the variance model may be more appropriatefor the particular applications" (p. 993). He suggested two alternatives, theexponential and absolute value models:

(2) ht = exp (ao + a l t _I + O t-P)(3) ht = ao + al Et - I I + + ap|Et-PIGThe exponential model (2) has the advantage that the variance is positive for allvalues of a; however, as stated by Engle, it has the unfortunate property that datagenerated by such a process will have infinite variance, making estimation andinference difficult. The absolute value model (3), like the linear model, requiresrestrictions on the parameter space to ensure that the variance is positive. For this


5/23

140 M. L. HIGGINS AND A. K. BERAmodel, however, the variance of the generated data will be finite for all positiveparametervalues, a property not shared by the linear ARCH model.In an empirical application, Engle and Bollerslev (1986)report having estimateda variety of functional forms for a GARCH model of the U.S. dollar/Swiss francexchange rate. They report results only for the two best models. The ARCHanalogues of these GARCH models are(4) ht = ao + all Et 1| + + apIEt-Plyand(5) ht = a0 + a1 (2F((Et1//) )-1) + + ap (2F(Et )-PI 1),where F is the cumulative normal distribution.Model (4) is a simple extension ofthe absolute value model (3). The model (5) is an attempt to provide a model inwhich the conditional variance ht remains bounded as the values of E2Q,i =1, , p, become arbitrarilylarge.

Concerned with the nonnegativity restrictions on the parameters of the linearARCH model, Geweke (1986) and Pantula(1986) suggests the functional form(6) log (ht) = ao + ca log (Et) + + caplog (Et-P),which ensures that the conditional variance is positive for all values of a. Gewekealso demonstrates that the log-likelihoodfor this ARCH model is globally concave,making maximumlikelihood estimation comparativelyeasy. Engle and Bollerslev(1986) criticize this specification, however, because the likelihood function be-comes undefinedif a residual of zero is encountered. We will refer to specification(6) as the log ARCH model.Each of the above models has its individualbenefits and limitations. Clearly anyone model cannot be chosen a priori as the most plausible specificationof an ARCHprocess. Their usefulness depends upon the particular empirical application.

3. A GENERAL FUNCTIONALFORM FOR ARCHIn this section we propose a general functional form for the ARCH model and

show that this more general model encompasses some of the models in Section 2.Consider the following ARCH model(7) =ht [? o(o-2)6 + .E.2_..... + + (E2.. 511/5where

,J2 > 0i4 ?0 for i =0, 1, , p5 > 0

and the 4i are such thatp>i=1.i = 0


6/23

NONLINEAR ARCH MODELS 141The conditional variance function (7) hasp + 3 parameters.The restriction that theXi's sum to one reduces the dimension of the parameter space by one; hence, themodel effectively has one more parameterthan Engle's model. The restrictionthat6 > 0 ensures that the conditional variance is defined for all Et's. We will refer toa model with conditional variancefunction (7) as a nonlinear ARCH model of orderp, or in short NARCH(p),The function (7) is familiar to economists as the constant elasticity of substitution(CES) production function of Arrow et al. (1961). If we heuristically view theconditional variance as output and "global" and "local" variances as inputs, thenthe ARCH model has the form of a linear productionfunction with infiniteelasticityof substitution. The NARCH specification is very flexible; the elasticity ofsubstitution 1/(1 - 6) can take values in the range (1, oc)for 0 < 6 ? 1. Equation(7) could also be written as

h - 1 0- 1 (E - 1 ( E 2_ - 1(8) 6 = ok +66 6, + *. + Up - 5which is a Box-Cox (1964) power transformationon both sides of the Engle (1982)specification. This transformationhas traditionallybeen used to linearize otherwisenonlinear models (see Carrolland Ruppert 1988, p. 118). In time series analysis,Grangerand Newbold (1976)and Hopwood, McKeown and Newbold (1984) havefound that incorporating a power transformation into the ARIMA model isbeneficial in terms of forecast accuracy. Therefore, the NARCH specification canbe expected to improve the estimates of the forecast intervals. Moreover, as we willsee below, the linear ARCH model is a special case of the NARCH specificationand therefore, ourgeneral formprovides a frameworkfor developing a specificationtest for Engle's linear ARCH model.It is possible to estimate the conditional variance function ht = V(yYt?t-1) bysome nonparametric method in a more general way than our proposed NARCHmodel. For example, some kernel function of the lagged squared residuals will beable to estimate ht quite easily. However, our method is superior,in that, as we justmentioned, it provides a simple way to test the null hypothesis of linear specifica-tion against a variety of nonlinear alternatives. Moreover, with our parametricapproach we will be able to interpretdifferent parametervalues; particularly, anestimate of 6 will provide some idea about the degree of nonlinearity. We shouldalso mention that using the NARCH model in Bera and Higgins (1989), we havedeveloped a simple but powerful test for the presence of conditional heteroskedas-ticity in time series models.We now show that some of the models of Section 2 are special cases of theNARCH model (7).

PROPOSITION.The conditional variancefunction (7) is equivalent to:(i) the linear conditional variancefunction (1) when 6 = 1(ii) the logarithmic conditional variancefunction (6) as 6 -> 0.


7/23


8/23

NONLINEAR ARCH MODELS 143

3.0

2.52. . . . .\.

1.5 . . . . . . . . . .

0.0... .25 .5 .751.01.2

* . . . . . . . . . IG R 12.0 . . . . . . . . . . . . . . . . . .x

. . . . . . . . . . . . . .

1 5 . . . . . . . . . . . . . . .

1.0 .25.5.7 .. . ....1.25..............

REGIONFORFINITENARCH(1)VARIANCE

where d(O) = &l/&Ois the score vector and 1(0) = -E[&21/&0&0'] is theinformation matrix and " - " denotes quantities evaluated at the maximum likeli-hood estimates subject to the restriction 6 = 1.The form of the LM test simplifies significantlywhen the informationmatrix isblock diagonal between the regression parameters and the variance parameters.The block diagonality can be shown in general; for simplicity, we show it forNARCH(1). Let v/ = (o-2, ki , 4~6p~ ) be the vector of variance parameters.THEOREM2. For the NARCH(1), the information matrix is block diagonalbetween the regression parameters /3and the variance parameters v, that is

- [ 21t]=

The proof is given in the Appendix. Under block diagonality the LM statisticsimplifies toLM = dv(6)'I v 6) 'd v(6)


9/23

144 M. L. HIGGINS AND A. K. BERAwhere d,(0) andI,(O) denote the score and information matrixwith respect to thevariance parameters v. Differentiatingthe log-likelihood

al I aht (Eta t 2ht av ht Jand

&a21 E2&ht a&ht [ 2 a Fa1 h-(9) 3 -a -aV,+ I- -I- -(9) avv, t2ht &v v [ht &v' &vJaThe information matrix Iv(0) is given by the negative of the expectation of thematrix(9). This expectation can be simplifiedby takingiteratedexpectations on theinformation set D,_1:

- [aaV aE{E[v 'F8I A [2h2 &Vh V,and hence, Iv(O) can be estimated by

I, (0)= >. [1 aht][_ ht]The LM test for Ho is then

1 /I IIahtILM= dv()'I v(6) I'dv(0) = 2 As -lh!L tht av]ht av' EO, ]|,8vLt JLJaVJ

The test statistic will be asymptoticallydistributedas XI under the null hypothesis.If we let ft = (E,2 ht) - 1, zt = fi,-' ahta v, f' = (f1, fT) and Z' =(Z,I , ZT), the statistic can be expressed as

2 ' ] L ' i L ' ]This statistic can be computed as 1/2 the explained sum of squares from theregression of f on Z. Furthermore, since plim(f'f/T) = 2, an asymptoticallyequivalent form of the statistic is

LM = T- fZ(Z'Z)-'Z'flf'fwhich can be computed as T times the uncentered coefficient of determinationfromthe regression off on Z.


10/23

NONLINEAR ARCH MODELS 145The test statistic is similar in form to Engle's original LM test for ARCH. Itdiffers in the function ht and the elements of aht/lav. In the above test, under thenull, ht is the conditional variance as estimated from Engle's model. The elements

of ahtla P, given in the Appendix, evaluated underthe null hypothesis 8 = 1simplifytoah, = _(J2) + Ati for i =I Oe paht -a 2= 40

a_ =, - ht log (ht)where

pa, = & (&2 log (&2) + E P7ig 2, log (?,-)-The test requires estimating Engle's ARCH model and computing the conditionalvariance for each observation. It does not, however, represent a significantcomputingburden. Presumably, a researcher has already concluded that ARCH ispresent in the data andwould naturallyproceed by estimating Engle's model. Oncethe MLE's of Engle's model are obtained, the test can be performed on anystandardregression package. The test, therefore, should be viewed as a diagnosticcheck of the adequacy of Engel's model after it has been estimated.

5. APPLICATIONTO FOREIGNEXCHANGERATESARCH is frequently used to model the volatility of exchange rates (see forexample, Domowitz and Hakkio 1985, Mih0j 1987,Hsieh 1988, 1989,McCurdyand

Morgan 1988, Baillie and Bollerslev 1989, and Diebold and Nerlove 1989). In thissection, we compare the performanceof ARCH and NARCH models of the weeklyexchange rates of six major currencies from January 1973 to June 1985. Thecurrencies are British pound (BP), Canadian dollar (CD), French franc (FF),German mark(GM), Japanese yen (JY), and Swiss franc (SF). The analyzed seriesfor each exchange rate is the first differences of the logarithmsof the spot price ofa specific currency in terms of the U.S. dollar. Hence, the data represent thecontinuously compounded percentage rate of return for holding the particularcurrency one week. For convenience, the rates of return for each series werecentered about the sample mean prior to analysis. The sample size is 651observations for all currencies. These particularseries were chosen because, asshown below, their conditional means can be represented by simple autoregressive(AR) processes and ARCH models of low orderare evident. Furthermore,some ofthese series provide good examples where the linear ARCH functional form may


11/23


12/23

NONLINEAR ARCH MODELS 147TABLE 1

ESTIMATED MODELS FOR THE US/FF EXCHANGE RATE*

AR: p (2 AIC = 1992.2.297 1.326

(.037) (.237)ARCH: 3 ao0 a1 a2 AIC = 1935.9

.343 .614 .330 .296(.043) (.070) (.075) (.062)NARCH: 38 0i2 01 02 8 AIC = 1929.7

.302 2.696 .301 .224 .302(.028) (.555) (.047) (.052) (.109)

LOG ARCH: 3 ao0 a1 a2 AIC = 1969.9.250 .528 .190 .244(.002) (.007) (.007) (.004)

*Standard errors of the estimates are shown in parenthesis.also reported for the estimated AR model. Standard over fitting tests wereconducted, but the AR(1) specification could not be rejected.Table 2 provides a number of diagnostic checks for the observed series andvarious estimated models. The Q(p) and Q2( p) are the Box-Pierce statistics for thedata and the squared data. The first one tests for linear dependence, while thesecond one is designed to pick up higher order dependence and the presence ofconditional heteroskedasticity. Both of these statistics are asymptotically distrib-uted as y,2. We also report the values of the BDS test statistic of Brock, Dechertand Scheinkman (1987) to test the null hypothesis of 1ID errors against a widevariety of nonlinear alternatives. In many recent applications, this test has beenfound to be very useful (see, for example, Hsieh 1989 and Scheinkman andLeBaron 1989). For the BDS(m, 8) statistic, m is the dimension and 8 is a smallquantity used to compute the "correlation integral." The correlation integralessentially measures the spatial correlation in the data (for theoretical details seeBrock et al. 1987 and for an exposition see Hsieh 1989).The choice of m and 8 isquite crucial for the test and we tried a number of differentvalues. To save space,in Table 2 we report the results for only m = 2, 3, 4 and 8 = .2, .3, .4, .5. Underthe IID assumption, the BDS statistic is asymptotically distributed as N(O, 1).First comparing the results for the observed series and the AR residuals, it isclear thatAR(1)explains a substantialpartof the lineardependence, although Q(12)remains slightly significant at the 5 percent level. However, the Q2 and BDSstatistics reveal the presence of strong nonlinearity in the data. Both Q2(12) andQ2(24) are highly significant. Also, there is only a slight reduction in the BDSstatistics after incorporatingthe AR conditionalmean. This implies that most of thedependency is of higher order and an AR model is not adequate to describe thedata.


13/23

148 M. L. HIGGINS AND A. K. BERATABLE 2

DIAGNOSTIC CHECKS OF MODELS FOR THE FRENCH FRANCARCH NARCH LOG ARCH

Observed AR Standarized Standardized StandardizedDiagnostic Series Residuals Residuals Residuals ResidualsSkewness -.10 -.06 -.05 -.04 -.05Kurtosis 5.06 5.76 4.75 4.80 7.32Q(12) 100.51** 23.83* 20.63 24.52* 28.09**Q(24) 117.48** 32.21 29.20 34.17 37.23*Q2(12) 91.14** 68.47** 18.82 16.34 12.94Q2(24) 98.06** 71.28** 26.04 22.87 19.46BDS(2,.2) 7.23** 4.40** - .18 -.15 .86BDS(2,.3) 6.19** 3.22** -.75 -.35 .14BDS(2,.4) 6.34** 1.80 1.80 -.60 -.15BDS(2,.5) 5.24** 0.55 .55 - .59 -.17BDS(3,.2) 8.80** 6.62** -.51 -.08 -.26BDS(3,.3) 7.94** 5.18** -1.17 -.54 -.58BDS(3,.4) 6.87** 2.66** -1.07 -.85 -.56BDS(3,.5) 5.23** 0.87 -1.01 -.85 -.27BDS(4,.2) 9.29** 7.50** -.02 .48 -.44BDS(4,.3) 8.55** 6.08** -.67 -.13 -.17BDS(4,.4) 7.28** 3.36** .21 - .24 - .14BDS(4,.5) 5.64** 5.64** 2.35* 1.74 - .17

* and ** indicate significance at the 5 percent and 1 percent levels, respectively.

The individual autocorrelations of the squared residuals of the estimated AR(1)model also reveal that some form of nonlinearityis present and helps to identify theorderof the ARCH process. Using the asymptotic standarderror I/T112= .039 (seeMcLeod and Li 1983), the first two autocorrelations of the squared residuals, .15and .19, are significant at the 1 percent level. Engle's LM test for ARCH wasperformedfor orders 1 through 10. The computed statistic for second order ARCHhas a value of 34.08 which is highly significant, but the value of the statisticsincrease very little as additional ARCH terms are included. Hence, an ARCH(2)model is identified andestimated. The results of the estimation, shown as L-ARCH,aregiven in Table 1. HigherorderARCH models were also fitted, but the additionalARCH parameters were insigificant. The AIC clearly indicates that the ARCHmodel is an improvement over the AR model. Additional strong support for theARCH model comes from the results of ARCH standardized residuals in Table 2.If the conditional variance is correctly specified, the residuals standardized by thesquare root of the conditional variance should behave as white noise. Therefore, asa furthercomparisonof the models, Table 2 presents the diagnostic statistics for thestandardized residuals -t/A/1/2. The Q2statistics are no longer significant.Further-more, all the BDS statistics, except for BDS (4, .5), do not detect any nonlinearityafter incorporatingARCH.At this point, although the kurtosis coefficient remains very large, it is temptingto be content with the linear ARCH model. To determine whether linear ARCH


14/23

NONLINEAR ARCH MODELS 149provides an adequate functional form for the ARCH process, the LM test forNARCH is performed. The computed value of the test statistic is 3.35, which issignificant at the 10 percent level. We estimate an NARCH(2) model where initialparameter values were obtained from the estimated linear ARCH model, with 8taken to be 1 as implied by the linear ARCH specification. The estimatedparameters are shown in Table 1. The likelihood ratio statistic for linear ARCHagainst NARCH is 8.257, which is highly significant. The AIC also indicates thatNARCH is an improvement over linear ARCH. The striking feature of theestimated NARCH model is that 8 falls from 1 to .302. An asymptotic t-test of thehypothesis that 8 = 1 is easily rejected. Although a formal test also rejects thehypothesis 8 = 0, it appears that the true model lies closer to the logarithmic ratherthan the linear model. The log ARCH model was estimated and the results areshown in Table 1 as LOG ARCH. The AIC indicates that although the log ARCHmodel is better than the simple AR(1) model, it is worse than the linear ARCHmodel. In terms of this criterion, the NARCH model is a clear choice. From Table2, we note that the BDS statistic can not distinguish very much between the threeARCH models. For NARCH, Q(12)is slighly significantat the 5 percent level whilefor log ARCH we observe deteriorationin terms of Q(12), Q(24) and the kurtosiscoefficient.Earlierwe indicated thatARCH models are inherently nonnormal, andtherefore,it is expected that an ARCH model will account for part of the nonnormalityin thedata. However, the kurtosis coefficients in Table 2 reveal that there is only a slightreduction in kurtosis for the linear ARCH and NARCH models, and for the logARCH model it is even higherthan those of the observed series and AR residuals.At firstglance, these results look somewhat unexpected and discouraging.To get aclear picture of the overall pattern of the distributions, in Figure 3, we plotnonparametrickernel estimates of the densities of the standardized residuals andsuperimpose the N(0, 1) density in the same figure. From the plots we observe thatit is the peakedness of the distributions that is driving the kurtosis coefficientshigher, and not so much the thickness of the tails. The plots for ARCH andNARCHalmost coincide throughoutthe whole range, whereas the log ARCH distributionismore peaked than the other two and considerably higher peaked than the N(0, 1)density. Therefore, the kurtosis coefficient of 7.32 for log ARCH is not thatsurprising.As a final comparison of the performance of linear ARCH, NARCH and logARCH, we plot the residuals and their 95 percent forecast intervals ? 1.96(kt) 1/2 inFigures 4 through 8. The plot for the log ARCH residuals is not presented since itwas very similar to the one for NARCH residuals. From Figures 4 and 5, theresiduals are seen to be almost indistinguishable, indicating that NARCH modelmay not provide a significantimprovement over ARCH (or log ARCH) in terms ofthe point forecast. The estimated AR parametersfor the three models do not differvery much from each other. The forecast intervals, however, are very different.Comparing Figures 6 and 7, the NARCH conditional variances seem to moreaccurately reflect the behavior of the series. During periods of low volatility, suchas from 1974throughthe firstquarterof 1978and 1979to 1981,the forecast intervalsof the ARCH model frequently decline to the lower bound ? 1.96(ao)1/2 = + 1.54.


15/23

150 M. L. HIGGINS AND A. K. BERA

45 ~~~~~~~~ARCH/7 \\ NARCH*45f0 At 6.; LOGARCHNORMAL

.35-.3 -

.25-.2/

.15

.05-0 -3 -113

FIGURE 3DENSITY ESTIMATES OF STANDARDIZED RESIDUALS

ComparingFigures4 and6, it appearsthatthis intervaloverstatesthe varianceofthe series. For the NARCHintervals,duringthese same periods,the forecastintervals become smallerthan ? 1.54. As should be expected from a goodconditionalvariancemodel,we canpredicttheserieswithhigherconfidenceduringless volatileperiods. Duringthe volatileperiods,the NARCHintervalswiden asrequired,yetthe transitionis smoothandnotsubjectto veryerraticvariationwhichoverstatesthevarianceas occursfor the ARCHintervals.The confidenceintervalsfor thelogARCHmodelis presentedinFigure8. Thesearesimilarto and inmanycases shorterthan those for the NARCHmodel.In this case the lower boundisvery small since Ait is calculated using

hj = exp [a0 + a' log (a _1) + 2( - 2)].Wheneversmalllaggedresidualsareencountered,A, takessmall values.Appar-ently, if we concentrateour attentiononly on the confidenceintervals,the logARCH model looks very attractive.However,takinginto considerationthe AICvalues and the diagnosticstatisticsin Table2, this superiorityof the log ARCHmodel is not very clear. In fact it might provideartificiallyshorterconfidenceintervalsanda false sense of highprecision.In Table3, we presentsome summaryresultsfor the other currenciesBP, CD,GM, JY and SF. Firstwe note that the ordersof AR andARCHwere identifiedusingtheprocedurewe followedfor theFF series,andarepresentedinrowone ofthe table. Except for SF, the LR tests for NARCH are highly significant.In


16/23

NONLINEAR ARCH MODELS 15130-

20 -

10 -

IN ~~r~Ir-n-J r-" rwl-10

-20 -

-30 - I I ,1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985FIGURE4

ARCH RESIDUALS

30 -

20-

10 -

-20

-30 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985FIGURE5

NARCH RESIDUALS


17/23

152 M. L. HIGGINS AND A. K. BERA

o-

30

-20-

?30

1973 1974 1975 1976 1977 1978 1979 1980 198 1 1982 1983 1984 1985FIGURE6

95 PERCENT CONFIDENCE INTERVALS FOR ARCH RESIDUALS

3 0 ........-.

20-

10

-30 -f t I I I f I - -I- - ----i!--1 -I------ I--- --I - - .1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985FIGURE 7

95 PERCENT CONFIDENCE INTERVALS FOR NARCH RESIDUALS


18/23

NONLINEAR ARCH MODELS 15330

20 -

10at10

-10

-20-

- 3 0 Ll I l1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985FIGURE 8

95 PERCENT CONFIDENCE INTERVALS FOR LOGARCH RESIDUALS

TABLE 3NONLINEARITY AND DIAGNOSTIC CHECKS OF OTHER EXCHANGE RATES

BP CD GM JY SFModel AR(4) + AR(1) + AR(1) + AR(1) + AR(1) +

ARCH(3) ARCH(1) ARCH(2) ARCH(1) ARCH(2)LM test 2.09 6.88** 2.10 15.10** 2.14for NARCHLR test 7.89** 13.21** 7.89** 26.40** 2.57for NARCH8 2.611 .148 .163 .039 .630(.919) (.167) (.257) (.074) (.500)ARCH Skewness -.42 -.91 .16 .41 .08NARCH Skewness -.37 -.84 .20 .30 .09ARCH Kurtosis 5.97 9.21 4.73 8.86 5.47NARCH Kurtosis 5.60 8.57 4.49 7.90 5.26ARCH BDS(2,.4) -.04 -.27 -1.17 -.26 -.56NARCH BDS(2,.4) .10 -.01 -1.06 -.31 -.30

* and ** indicate significance at the 5 percent and 1 percent levels, respectively.


19/23

154 M. L. HIGGINS AND A. K. BERAcomparison, the LM tests are significantonly in two cases. [This might indicate"low" power for the LM test for NARCH. However, our preliminary analysisshows that the LM test is also undersized, and therefore, the numerical values aresomewhat smaller than the LR statistics.] The most interesting result in this tableis the values of 8. For CD, GM and JY, 8 is not significantlydifferent from zero,indicatingsuitability of the log ARCH model. In the case of BP, the value is quitehigh and it appears that neither linear nor log ARCH will be appropriate for thisseries. Finally for SF, 8 lies in between zero and one, and is not significantlydifferent from either of them. The diagnostic checks againreveal that ARCH takesaccount of most of the nonlinearity.However the kurtosis coefficients still remainvery high, although for the NARCH models they are somewhat lower.

6. CONCLUSIONIn this paper we propose a nonlinearfunctional form for the conditional varianceof an ARCH model. This is useful given the current evidence that although thelinear ARCH model does take account of some nonlinearity, it is not nonlinearenough to model some financial time series data. For example Hsieh (1989, p. 366)found that the GARCH model cannot fit some exchange rates satisfactorily;Scheinkman and LeBaron (1989, p. 313) found evidence that volatility in stockmarket data can not be captured completely by linear ARCH models. Oursuggested nonlinearARCH model has an intuitiveappeal since it can be viewed asa generalization of Box-Cox transformationto the conditional variance. Since theNARCH model encompasses various functional forms, we arguethat it provides auseful framework for testing Engle's original specification against a wide class ofalternatives. The test we suggest is a simple one and can be calculated usingstandardregression packages. Our illustrationsof modeling exchange rates dem-onstrate that the NARCH model can provide improvement over the linear ARCHmodel.Before we conclude, let us mention some possible extensions of our work. Wehave noted that point forecast from the linear ARCH, NARCH and log ARCHmodels are similar. While the confidence intervals using NARCH and log ARCHmodels are much shorter. An implicationof this observation may be that out-of-sample forecasts from NARCH type models may be more precise relative to thosefrom linear ARCH specifications. Therefore, it would be interestingto evaluate theout-of-sample forecasting properties of NARCH relative to other models.Under our framework, it is easy to generalize the NARCH model to nonlinearGARCH (NGARCH) by including Box-Cox transformations of lagged hit's inexpression (8). NGARCH could be further extended by relaxing the conditionalnormality assumption and using Student's t or the generalized error distribution totake account of part of the excess kurtosis.One of the important questions in macroeconometrics is how a superiorspecification that incorporates the inherent nonlinearityin the data affects testingeconomic theory. Hamilton (1988) addresses the much debated issue of testing theexpectation hypothesis of the term structure of interest rates. His conclusion(p. 419) is that the cross equationrestrictions implied by the expectation hypothesis


20/23

NONLINEAR ARCH MODELS 155is rejected when a linear process for short rates is assumed, but is accepted if anonlinear specification is used. It would be interesting to investigate what happensto tests of expectational theory or CAPM models if the nonlinearity in the data iscaptured by NARCH type models.

University of Wisconsin-Milwaukee, U.S.A.University of California at San Diego, U.S.A.

APPENDIXPROOFOF THEOREM1. We require the following lemma.LEMMA. Assuming that a NARCH(1) process began infinitelyfar in the past

with all initial moments finite, thenE(lt 128) < X0

if01 r-11228F(8 + 1/2) < 1,

where F(r) is the gamma function.PROOFOF LEMMA. The process (7) can be expressed as the stochastic difference

equation? t =- Zt(ht) 12

where the Zt are independent N(O,1)and independent of past Et's. ThereforeElEt121 = Elzjt28[00(or2)8 + 01EIE 1128]

= 001,k*(0c2) +011.k*El~t_ 1 121,where A* = ElZt128 which is assured to exist by the normalityof Zt. This is a firstorder lineardifferenceequation in E| t 128. If initial moments are finite, E| at 128 willconverge to a finite value if

01y* = El Zt|21 = 1 7-1/228F(8 + 1/2) < 1,which completes the proof of the lemma.

The condition for the lemma is also sufficient for the finiteness of the variance. If8 > 1, then(A. 1) E( 2) = E[E( 21(pt_1)]

= E[40(o-2)8 + 01(E 2_ 1)8 1/8


21/23

156 M. L. HIGGINS AND A. K. BERAwhere the inequality follows from Jensen's inequality since (A. 1) is a concavefunction of IEt- I 28. The lemma assures that the expectation is finite, and hence thevariance is finite. For 8 < 1, we apply Minkowski's inequality (see White 1984,p. 34), to establish(A.2) E(82) = E[E(E 21 t I)

=E[(o~r2)8 + (Etl) ]? [4 o (or 2) 1 + pI[E(E 2_ )1 81 1/8.

The expression in (A.2) is monotonically increasing in E(E27 ). By repeatedlyapplying Minkowski's inequalityE( 2)- [40( 2)1 + Xl 0( 2)8 + [E( E2_)]8] /8

k- I_ 2+S)5 E 0li + xk [E( E2_ 1)] 8] 1/8

i = 0As k -? oo,the variance will be finite if Xl < 1. The restrictions on the parameterspace of (7) guarantee Xl < 1, which completes the proof.

PROOFOFTHEOREM2. From the log-likelihood of a NARCH(p) processdv d: (v -id 2h2 av1af3j [I, ]1

12 2Etx t I ahtaht h h+~~~~~~~~2h,avhf- 1 2hi

The elements of the information matrixbetween ,(3and v are given by the negativeof the expectation of (A.3). The expectation simplifies by first taking iteratedexpectations on the information set at time t - 1(A.4) > -El E Ea t- X E l2

~av~af~i~t [[av~a t ajap t ~2h'v avja1

If the expectation of the termin squarebrackets is zero, then the theorem is proved.The elements of ahofIA areaht hth

(A.6) E E= - t| 2 j


22/23

NONLINEAR ARCH MODELS 157aht ht8(A.7) = [-(o2)8 + (E?2t)81 for i = 19 ..., p

(A.8) a = [hh 1 ht log (ht)],a88where

p= o log ( 2)( -2)8 + > 1 log ( "-) (E 2

For the NARCH(1) case, we requirethe above derivatives for p = 1 only. From (7)and (A.6) through (A.8), it is evident that ht and aht/a vi are symmetric functionsof Et-I. While from(A.4), aht/afi is an antisymmetricfunction of Et-I. Hence, thewhole expression is antisymmetric in t- 1 and since t- 1 is symmetricallydistributed around zero, the expectation in (A.4) is zero.REFERENCES

ARROW, K. H., H. B. CHENERY, B. S. MINHAS AND R. M. SOLOW,"Capital-Labor Substitution andEconomic Efficiency," Review of Economics and Statistics 43 (1961), 225-250.

BAILLIE, R. T. AND T. BOLLERSLEV, "The Message in Daily Exchange Rates: A Conditional-VarianceTale," Journal of Business & Economic Statistics 7 (1989), 297-305.

BERA, A. K. AND M. L. HIGGINS, "A Test for ConditionalHeteroskedasticity in Time Series Models,"mimeo, University of Illinois at Urbana-Campaign,1989.AND S. LEE, "Interaction Between Autocorrelation and Conditional Heteroskedasticity:

A Random Coefficient Approach," mimeo, University of Illinois at Urbana-Champaign,1988.BOLLERSLEV, T., "Generalized Autoregressive ConditionalHeteroskedasticity," Journal of Economet-

rics 31 (1986), 307-327., "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates ofReturn," Review of Economics and Statistics 69 (1987), 542-547., R. Y. CHOU,AND K. F. KRONER, "ARCH Modeling in Finance: A Review of the Theory and

Empirical Evidence," Journal of Econometrics (forthcoming).Box, G. E. P. AND D. R. Cox, "An Analysis of Transformations,"Journal of Royal Statistical SocietyB, 26 (1964), 211-243.BROCK, W., W. D. DECHERT AND J. SCHEINKMAN, "A Test for Independence Based on the CorrelationDimension," mimeo, University of Wisconsin-Madison, 1987.

CARROLL, R. J. AND D. RUPPERT, Transformationand WeightinginRegression (New York:ChapmanandHall, 1988).

DIEBOLD, F. X., "Testing for Serial Correlationin the Presence of Heteroskedasticity," Proceedings ofthe American Statistical Association, Business and Economic Statistics Section (1986), 323-328.

AND M. NERLOVE, "The Dynamics of Exchange Rate Volatility: A Multivariate Latent FactorARCH Model," Journal of Applied Econometrics 4 (1989), 1-21.

DOMOWITZ,I. AND C. S. HAKKIO, "Conditional Variance and the Risk Premiumin the Foreign ExchangeMarket," Journal of International Economics 19 (1985), 47-66.

ENGLE, R. F., "Autoregressive ConditionalHeteroscedasticity with Estimates of the Variance of U. K.Inflation," Econometrica 50 (1982), 987-1008.

AND T. BOLLERSLEV, "Modellingthe Persistence of ConditionalVariances," EconometricReviews5 (1986), 1-50.

, C. W. J. GRANGER AND D. KRAFT, "CombiningCompetingForecasts of Inflationusinga BivariateARCH Model," Journal of Economic Dynamics and Control8 (1984), 151-165.


23/23

158 M. L. HIGGINS AND A. K. BERAGEWEKE,J., "Comment," Econometric Reviews 5 (1986), 57-61.GRANGER,C. W. J. ANDP. NEWBOLD,"Forecasting TransformedSeries," Journal of Royal Statistical

Society B, 38 (1976), 189-203.,R. P. ROBINSANDR. F. ENGLE, "Wholesale and Retail Prices: Bivariate Time Series Modelingwith Forecastable Error Variances," in D. A. Belsley and E. Kuh, eds., Model Reliability(Cambridge:MIT Press,1984), 1-17.

HAMILTON,J. D., "Rational-ExpectationsEconometric Analysis of Changes in Regime: An Investigationof the Term Structure of Interest Rates," Journal of Economic Dynamics and Control 12 (1988),385-423.

HSIEH, D. A., "The Statistical Properties of Daily Foreign Exchange Rates: 1974-1983," Journal ofInternationalEconomics 22 (1988), 129-145., "Testing Nonlinear Dependence in Daily Foreign Exchange Rates," Journal of Business 62

(1989), 339-368.HOPWOOD,W. S., J. C. McKEOWNANDP. NEWBOLD,"Time Series Forecasting Models Involving PowerTransformations,"Journal of Forecasting 3 (1984), 57-61.LEE, T. K. Y. ANDY. K. TSE, "Term Structureof Interest Rates in the Singapore Asian Dollar Market,"Journal of Applied Econometrics 6 (1991), 143-152.MCCURDY,T. H. ANDI. G. MORGAN,"Testing the MartingaleHypothesis in Deutsche Mark Futures with

Models Specifying the Form of Heteroscedasticity," Journal of Applied Econometrics 3 (1988),187-202.

McLEOD,A. I. ANDW. K. Li, "Diagnostic Checking ARMA Time Series Models using Squared-ResidualAutocorrelations," Journal of Time Series Analysis 4 (1983), 269-273.

MILH0J, A., "The Moment Structure of ARCH Models," The Scandinavian Journal of Statistics 12(1985), 281-292., "A Conditional VarianceModelfor Daily Deviations of an Exchange Rate," Journalof Businessand Economic Statistics 5 (1987), 99-103.

PAGAN,A. R. AND H. SABAU, "On the Inconsistency of the MLE in CertainHeteroskedastic RegressionModels," mimeo, University of Rochester, 1987.PANTULA,S. G., "Comment," Econometric Reviews 5 (1986), 71-73.

SCHEINKMAN,J. A. ANDB. LEBARON,"NonlinearDynamicsand Stock Returns," Journalof Business 62(1989), 311-337.TSAY, R. S., "Conditional Heteroskedastic Time Series Models," Journal of the American Statistical

Association 82 (1987), 590-604.WEISS, A. A., "ARMA Models with ARCH Errors,"Journal of Time Series Analysis 5 (1984), 129-143., "Asymptotic Theory for ARCH models: EstimationandTesting," Econometric Theory2 (1986),107-131.WHITE,H., Asymptotic Theoryfor Econometricians (Orlando:Academic Press, 1984).

class of nonlinear arch

Documents