A Capital Asset Pricing Model with TimeVarying Covariances Author(s): Tim Bollerslev, Robert F. Engle and Jeffrey M. Wooldridge Source: Journal of Political Economy, Vol. 96, No. 1 (Feb., 1988), pp. 116131 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/1830713 . Accessed: 15/06/2014 10:36 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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A Capital Asset PricingModel with TimevaryingCovariances
TimBollerslev Northwestern University
RobertF. Engle University of California,San Diego
M. Wooldridge Jeffrey Massachusetts Institute of Technology
The capital asset pricing model provides a theoretical structure for the pricing of assets with uncertain returns. The premium to induce riskaverse investors to bear risk is proportional to the nondiversifiable risk, which is measured by the covariance of the asset return with the market portfolio return. In this paper a multivariate generalized autoregressive conditional heteroscedastic process is estimated for returns to bills, bonds, and stocks where the expected return is proportional to the conditional covariance of each return with that of a fully diversified or market portfolio. It is found that the conditional covariances are quite variable over time and are a significant determinant of the timevarying risk premia. The implied betas are also timevarying and forecastable. However, there is evidence that other variables including innovations in consumption should also be considered in the investor's information set when estimating the conditional distribution of returns. We wish to thank Douglas Breeden, Thomas MaCurdy, Ieuan Morgan, Michael Rothschild,and Larry Weiss for helpful comments.We would also like to thank seminar participantsat the FifthWorld Congress of the EconometricSociety,Universityof Chicago, NorthwesternUniversity,New York University,Universityof Toronto, Universityof BritishColumbia, Universityof Californiaat Berkeley,Irvine,and San Diego, and Yale University.An anonymous referee provided many useful suggestions in improvingan earlier version of the paper. The research was supported by National Science Foundation grantsSES 8420680 and SES 8008580 awarded to Engle. [Journalof PoliticalEconomy,1988, vol. 96, no. 11 K 1988 by The Universityof Chicago. All rightsreserved. 00223808/88/96010002$01.50
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CAPITAL
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Introduction
The capital asset pricing model (CAPM), originally proposed by Sharpe (1964) and Lintner (1965) followingthe suggestionsof mean variance optimizationin Markowitz(1952), has provided a simple and compellingtheoryof asset marketpricingfor more than 20 years. In its simplestform the theorypredicts that the expected returnon an asset above the riskfreerate is proportional to the nondiversifiable risk,which is measured by the covariance of the asset returnwith a portfoliocomposed of all the available assets in the market.The assumptionsimplicitin the versionof the model discussed here are that (1) all investorschoose meanvarianceefficientportfolioswitha oneperiod horizon, although they need not have identical utilityfunctions; (2) all investorshave the same subjective expectations on the means, variances, and covariances of returns; and (3) the market is fullyefficientin that there are no transactioncosts, indivisibilities, taxes, or constraintson borrowingor lending at a riskfreerate. Empiricaltestsof the CAPM have tended to focus on assumption 1 while strengthening2 to include the assumption that the common distributionsare constantover time and that the entiremarketis the market for equities. These tests generally have found that the risk premium on individual assets can be explained by variables other than the estimatedcovariance. In particular,the own variance, firm size, and the month of January seem to be variables that help to explain expected returns.See, for example, Jensen (1972) for a survey of many of these early studies and Ross (1978), Roll and Ross (1980), Chen (1983), and Schwert(1983) for more recent surveys. One interpretationfor the failure of the CAPM to fullyexplain observed risk premia, due to Roll (1977), is that any empirical covariance is computed from an incomplete marketfor assets. Such an objection nearly makes the CAPM untestable. Another explanation is, of course, that alternativetheories of asset pricing may be supportable such as the arbitragepricingtheoryof Ross (1976) or the consumptionbeta formulationintroduced by Breeden (1979). In this paper we focus attentionon the possibilitythat agents may have common expectationson the momentsof futurereturnsbut that these are conditionalexpectations and therefore random variables rather than constants. For a discussion along these lines, see also Ferson (1985), Rothschild (1985), and Ferson, Kandel, and Stambaugh (1986). Let ytbe the vectorof (real) excessreturnsof all assets in the market measured as the nominal returnduring period t, minus the nominal returnon a riskfreeasset, and let tL,and H, be the conditionalmean vectorand conditionalcovariance matrixof these returnsgiveninfor
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mation available at time t  1. Also let tl be the vector of value weightsat the end of the previous period so thatthe excess returnon
themarketis definedas yM,
=
YtI( it I Then thevectorofcovariances
withthe marketis simplyHtwt_ and the CAPM requires lt = 8Htt&1
(1)
In thisformulation,as derived byJensen (1972), 5 is a scalar constant of proportionality,which in equilibrium is an aggregate measure of relativerisk aversion given by the harmonic mean of the agents' degree of relativerisk aversion weighted by the agents' share of aggregate wealth (cf. Bodie, Kane, and McDonald 1983, 1984). Throughout the paper we assume 8 to be constant. The conditional variance of the market excess return is CM, =  I Fpt,whichfrom I HtHt I and the conditionalmean is ItM = (1) can be writtenas FM, = 81CM1
(2)
so that6 is seen to be the slope of the markettradeoffbetweenmean and variance. Definingthe beta of an asset to be the covarianceof that asset withthe marketdivided by the variance of the marketportfolio, Pt = Htt  I/CM,, and substitutingin (1) and (2) yields the familiar expression Ft =
ft FM,
(3)
Because the covariance matrixof returnsvaries over time,the mean returnsand the betas will in general also be timevarying. We have stated the CAPM in termsof conditional momentssince these reflectthe informationset available to agents at the time the portfoliodecisions are made. But this model also implies a relation between unconditional moments. In the special case in which the value weightsare fixed,the unconditionalmeans are constantand are given by E(y,) = 6V(y,)w

83V(HtW)w.
Only if V(Hw) = 0 will the unconditional momentssatisfythe same CAPM relationsas the conditional moments. By a similarargument, if the econometricianuses only a subset of the relevantconditioning information,then the estimatedconditional momentswill not satisfy the CAPM. In this paper the conditional covariance matrix of a set of asset returnsis allowed to vary over time followingthe generalized autoregressiveconditional heteroscedastic(GARCH) process (see Engle 1982; Bollerslev 1986). This essentiallyassumes that agents update theirestimatesof the means and covariances of returnseach period
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MODEL
using the newlyrevealed surprisesin last period's asset returns.Thus agents learn about changes in the covariance matrixonly frominformation on returns.There may, of course, be additional information relevantto agents' expectationsthatwould lead to misspecificationas mentioned above. The approach is a multivariategeneralizationof Engle, Lilien, and Robins (1987), which treated a single asset, and thereforeestimates the timevaryingrisk premium as a functionof the conditional variance of that asset return alone. A similar idea was employed in the recent papers by French, Schwert, and Stambaugh (1986) and Poterba and Summers (1986). The approach can also be seen as a statisticalimplementationof the intertemporalCAPM of Bodie et al. (1983, 1984), in which they had no unknown parameters and no statisticaltest of the model performance. Finally,the paper can be viewed as a generalization of Frankel (1985), who assumed that W, may be timevaryingbut that H, is not, and of Friedman (1985a, 1985b), who allowed H, to be timevaryingonly because investors must learn about the unconditional variance V(y,). II.
Econometric Methods
According to the economic model (1), any explanation of timevaryingexpected excess holding yieldsshould be builtaround a structure witha timevarying conditionalcovariance matrix.As mentioned above, a model ideally suited for this purpose is the multivariate GARCH in mean (GARCHM) model. For yt N x 1, the GARCH (p, q)M model takes the general form y, = b + 6HMtowI
+ Ett
EtIqjtI

L B vech(H,1), p
q
vech(H,) = C +
A vech(E,  E')
ij=
+
I
(4)
AN(O, Ht),
where vech(*)denotes the column stackingoperator of the lower portion of a symmetricmatrix,b is an N x 1 vectorof constants,E1 is an N x I innovationvector,C is a ?12N(N + 1) x I vector,and A, i = 1, ... , q, and B1,j = 1,. p, are 1'2N(N + 1) x 1?2N(N+ 1) matrices.A nonzero b vector might reflecta preferred habitat phenomenon or differentialtax treatmentof the assets. Of course the GARCH specification does not arise directlyout of any economic theory,but as in the traditional autoregressive moving average timeseriesanalogue, it provides a close and parsimonious approximation to the form of heteroscedasticitytypicallyencountered with economic timeseries data (cf. Bollerslev 1986; Engle and Bollerslev 1986).
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The conditional log likelihood functionfor (4) for the single time period t can be expressed as log2T 2

LJ(O)
 1/2logIH,(O)j  V/2Et(0)'HT'(O)Et(O),
(5)
where all the parameters have been combined into O' = (b', 6, C', vec(A1)' . . . vec(Aq)', . . , vec(Bp)'), an m x 1 vector. Thus, conditional on the initial values, the log likelihood functionfor the sample 1,. T is given by' T
L (0)
L(0).
E
(6)
As is obvious from(4), (5), and (6), the log likelihood functionL(0) depends on the parameters 0 in a highlynonlinear fashion,and the maximizationof L (0) requires iterativemethods.The approach taken here is to use the Berndt et al. (1974) algorithmalong withnumerical firstorderderivatives to approximate aLt(O)h3O.These derivatives provide an added flexibility to changes in the specification. Given standard regularityconditions (see Crowder 1976; Wooldridge 1986), it followsthat the maximum likelihood (ML) estimate for 0 will be asymptoticallynormal and unbiased with covariance matrixequal to the inverseof Fisher'sinformationmatrix.Therefore, traditionalinferenceprocedures are immediatelyavailable. In particular, when equation (4) is tested versus a more general specification, the Lagrange multiplier(LM) teststatistictakes the wellknownform LM = T  R , where RO is the uncentered coefficientof multiple correlation in the firstBerndt et al. iteration for the augmented model startingat the ML estimatesunder the null (cf. Engle 1984). As it stands, (4) is very general and involves a total of (N + 1) + 1V2N(N+ 1) + 1/4N (N + 1)2(p + q) parameters. A natural simplificationis to assume that each covariance depends only on its own past values and surprises.Throughout thispaper we shall therefore take p = q = 1 and impose diagonalityon the matricesAl and B1. With these simplifications,the GARCH(1, 1)M model considered here becomes
=
yt hijt
tqlpt I
=

b'i
+ 8 +
L
OtijEit
+
{
+ 3,jhijt_, iI
j=
1,...N,
N(O, Ht),
In practicethe presample values are set equal to theirexpected value, zero.
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(7)
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where subscripti refersto the ith element of the correspondingvector and ij to the ijth element of the correspondingmatrix.Thus only the own lagged moments and cross products appear in each of the conditional covariance equations. Model (7) extends the univariateARCH model introducedin Engle (1982) in several directionsby allowingformultipletimeseries,conditionalcovariance termsin the mean, and own past conditionalcovariances in each of the covariance equations. III.
Data Description
The market portfoliostudied in the present paper is composed of bills (6month Treasury bills), bonds (20year Treasury bonds), and stocks.In broad terms,these three assets account fora good part,but certainlynot all, of the liquid investmentopportunitiesavailable. The data are quarterlypercentage returnsfromthe firstquarter of 1959 throughthe second quarter of 1984, for a total of 102 observations. The returnon 3monthTreasury bills is taken to representthe riskfree return. For a detailed descriptionof the data sources and data transformations, see the Appendix. Two data sets have been analyzed for these three returnsseries. In the previous draftof this paper the Standard and Poor's 500 equity series was used withCitibase interestrates. In thisversion New York StockExchange valueweightedequityreturnsare used withSalomon Brothersbilland bond yields.The resultsare quite similar,so onlythe second data set will be discussed here. The original resultsare available fromthe authors. The mean of the excess holding yield on 6month bills over the sample is 0.142 percent at a quarterlyrate while the standard deviation is 0.356. For bonds the average excess holding yield is 0.761 percentwithstandard deviation 6.255, and forstocksthe excess holding yield and the standard deviation are  0.995 percent and 2.225. All the excess holding yield series, however, tend to be somewhat erratic.The maximum return on a 3monthbalanced portfolioobtained by borrowingat the 3monthrate and lending at the 6month rate was 2.046 percent at a quarterlyrate in the second quarter of 1980. On the other hand, the threeworstreturnsoccurred in the first, third,and fourthquartersof 1980 with  0.462,  0.777, and  0.515 percent.For bonds, the best returnon a balanced portfoliowas also in the second quarter of 1980 with 22.274 percent, whereas the two worst returns were in the previous and subsequent quarters with  18.461 and  14.422 percent, respectively.Stocks did best in the firstquarter of 1975 with3.746 percent,but two quarters before,the returnwas as poor as  8.642 quarterlypercentage rates. This sug
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12 2
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geststhatnot only do the conditionalmean excess holding yieldsvary over time, but also the conditional variances seem to be changing throughtime. IV.
Model Estimates
In thissectionwe presentmodel estimatesfora trivariateCAPM. The econometricspecificationof the model is as in (7). The ML estimates (withcorresponding standard errors in parentheses) are y~tYIt Y2t
.070hiEl
4.342 (1.030)
=

Y3t
hiit
h2, h31
(
fl
0
(.160)
h2j,
.011 (.004)
.445 (.105)
.176
.233 (.092)
e2
~~.018 ;(.009)
.197 (.132)
Etl
5.143 (2.820)
.165 (.093)
eA11E311
2.083
.078 (.066)
(1.466)
.598
(1052)
e1/1e2t 1 2 E2/1
2
.441 (.215)
+
I3t
(8a)
E2,
t
.466 h t (.056)
1
.188 (.113)
+
+
h31j
(.710)
_13.305 h22t = i(6.372)
h131
+
3.117ht
(.062)
h]t1
h111
(.032)
1

362h
(.361)
h2 h22ti,
(8b)
13t
.348h21 (1338)
469
(.333)
where i = 1, 2, 3 refersto bills,bonds, and stocks,respectively. The estimatesforthe model are appealing. The estimatedvalue for 8 = .499 is reasonable and highlysignificant,lending some support for the theorypresented here. The interceptterms vary substantiallyacross the three assets. Although the theoreticalmodel does not include constanttermsin the riskpremia, these effectsare of some importance.The large negative interceptsforbonds and stocksare not surprisingsince reduced capital gains taxes on longtermassets provide incentivesto hold these assets even at otherwise unfavorable rates of return. It is also well knownthatbond and equityholders did consistentlyworse than other asset holders over the sample period. The intercepts 4.3 and  3.1 reflectthis fact.
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CAPITAL < I
ASSET
PRICING
MODEL
~Excess Holding   Risk Premium
12.3
Yield
I
Add~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I L
1959
1963
1967
1971
FIG.
1975
T ime
1979
1983
1987
1991
1.Risk prernia for bills
The dynamicstructurein the second momentsforbillsand bonds is apparent as reflectedby the significantvariance and covariance parameters.Even though none of the six variance or covariance parametersforstocksis individuallysignificantat the usual 5 percentlevel,it is interestingto note that the likelihood ratio teststatisticfor absence of dynamics in the secondorder moments for stocksequals 18.639, whichexceeds the .995 value of a x' distribution,thus soundlyrejecting the null hypothesis.Any correctlyspecified intertemporalasset pricing model ought to take this observed heteroscedasticnature of asset returnsinto account. The same point has also been made in the recentpapers by Ferson (1985) and Ferson et al. (1986). In particular, tests of the CAPM that treat the conditional covariance matrix as constantover time invariablyfalter. The estimated risk premia from the model, b + 81h,w1,i, are plottedin figures13 along withthe excess holding yieldsforeach of the three assets. Figures 1 and 2 show that the estimatesfor bills and bonds are fairlysimilar except for a differencein scale. Both assets have risingriskpremia during the volatilepostOctober 1979 period. It is reasonable to believe that, on average, investorswere paid a positivepremium for holding bills or bonds during this period. Note that the negative premia observed for bonds and equities in some
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Excess Holding Risk Prem i um
YTield
do
;
G
_~~~~~~~~~~~~~~~~~~I n a
1 IS96
5
1967
G
19S71
19S75
T i me
19S79
19939
1997
1983
1987
1991
FIG. 2.Risk premia for bonds
6i~~~~~~~~~~~~~~~~~~~j n 
Excess Holdinga  Ri 1sk P remisum
Tield
24
cn
I
1959
1963
1967
1971
1975
Ti
me
1979
FIG. 3.Risk premia for stocks
124
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1991
13
'1959
1967
1963
FIG.
'1959
19;63
.
.
.19'67
4.Beta
1971
FIG.
1979
1975
19,71
1997
1991
for bills
19'75
t I me
5.Beta
1993.
1979
19683
for bonds
125
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1967
1991
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'959
19~63
19,67
1971
FI G.
1975
t Ime
6.Beta
1979
1983
ECONOMY
1 98s7
1991
for stocks
periods could be due to the preferentialtax treatmentas previously mentioned. Plottedin figures46 are the estimatedbetas. Not surprisingly, the beta forstocksis close to one, thatforbonds is slightlyabove one, and thatforbillsis close to zero. There is, however,substantialmovement over the sample period. V.
Diagnostic Tests
Given the evidence above, the trivariateCAPM seems to fitthe data reasonablywell. However, in order to assess the general validityof the model, a series of LM testswere performed.We shall consider here only a verysmall subset of these. The firstLM test involves the inclusion of the own conditional variances in each of the three equations for the conditional expectation of the excess holding yields.The teststatisticequals 1.148, which is asymptoticallya Xrandom variable if the null hypothesisis true. Thus the null cannot be rejected at any level under 23 percent,lending furthersupport to the model. This test is of particularinterest since in testsof the timeinvariantCAPM the own variance is often found to be highlysignificant.This mightalso provide an explanation
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for the empirical findingsin French et al. (1986) and Poterba and Summers (1986), where a timevaryingmeasure of the own conditional variance or volatilityis found to have littleexplanatorypower forthe expected returnon the stockmarket.Our resultssuggestthat a better measure might be the nondiversifiablerisk as given by the conditional covariance with the market. The next testconsiders the lagged excess holding yieldsas explanatoryvariables for each of the three risk premia. This testrejects the formulationof the CAPM given in (8). The value of the teststatisticis 18.311 and is highlysignificantat any reasonable level in the corresponding X2 distribution.Thus agents may use informationin addition to past innovationsin formingtheirexpectations.This abilityof the lagged dependent variable to help forecastreturnsis not all that surprisingin view of other recent results in the literature(see, e.g., Campbell 1987). One of the competing theories of the intertemporalCAPM presented here is the consumption beta formulationmentioned in the Introduction.It is thereforeinterestingto note thatthe testforinclusion of innovationsin the logarithmof per capita consumptionin the conditionalmean equals 14.027, the value of a x2 random variable in the absence of any correlation.This surprisingcorrelationof innovations in consumption and innovationsin asset returnssuggests that reformulationalong the lines of a consumption beta model might deserve some consideration.2 However, the results in Hansen and Singleton(1982, 1983) do not lend much support to a strictformulation of that model. Also, Mankiw and Shapiro (1986) reported findings in favor of the traditionalCAPM versus the consumption beta formulation. VI.
Conclusions
In summary,the resultsreported in this paper support several conclusions.First,the conditionalcovariance matrixof the asset returnsis stronglyautoregressive.The data clearly reject the assumption that thismatrixis constantover time. The expected returnor riskpremia for the assets are significantly influenced by the conditional second momentsof returns.There is also some evidence thatthe riskpremia are betterrepresentedby covariances withthe implied marketthanby own variances. However, informationin addition to past innovations 2 The test probably has too small a size since, even if the CAPM were true, there mightbe consumption out of portfoliowealth, which would lead to a rejection.The rejectionoccurs onlywhen futureconsumptionis included, and thisis,of course, not in the agents' informationset. The LM teststatisticfor innovationsin currentconsumption takes the value 3.289 corresponding to the .65 fractilein a x' distribution.
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in asset returns is important in explaining premia and heteroscedasticity. In particular, lagged excess holding yields and innovations in consumption appear to have some explanatory power for the asset returns. Probably even better econometric models with a richer specification for the risk premia, not necessarily derived directly from any economic theory, can therefore be constructed. See Hansen and Hodrick (1983) for a discussion along these lines. Other interesting questions that remain are the sensitivityof the results to the choice of the "market portfolio" and a quarterly oneperiod horizon. It is possible that wider definitions of the market would allow the model to do better. We leave the answer to all these questions for future research. Data Appendix The yields on 3month Treasury bills, 6monthTreasury bills,and 20year Recordof Treasury bonds were all taken fromthe Salomon Brothers'Analytic Yieldsand YieldSpreads.The yields are percentages per annum for the first tradingday inJanuary,April,July,and October. The returnswere converted
~I
cBmu
  Bonds and Notes 
Stocks
J /
V
1959
I953
19587
1971
FIG.
1975
t I me
1979
1983
AlIMarket weights
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1987
1991
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129
to quarterlyrates rt, rt"1,and rio d, where (1 + Rt) = 1 + rt.From these rates the onequarter excess holding yields were calculated as bill=
ot
=
10oK + bond
100
1r
r(
bond
rl)2

+ rto
1

f(A)
 1 r]
(A2)
The stockmarketyieldswere based on the returnon the valueweightedNew York Stock Exchange index including dividends obtained from the Center for Research in SecurityPrices at the Universityof Chicago. From the quarterlyflowof returns,rt , th quarter excess holding yield was simply calculated by s0(rtockr).(3 stock I 00O(rt rt)(A3) Yt The maturitydistributionof the interestbearingpublic debt held by private investorswas taken fromthe FederalReserveBulletinand, for 1976 until the present, from the TreasuryBulletin.To get from par values to market values, we multipliedthe outstandingdebt in the differentmaturitycategories within1 year, 15 years,510 years,and 10 years and over by the price indices reported in Cox (1985). The categories were then added togetherto get the marketvalues within1 year,bills,and longer than 1 year,bonds. The total marketvalue of corporate equities was obtained froma special tabulation of the balance sheets of the flow of funds accounts by the Board of Governors of the Federal Reserve System. The relative market values are illustratedin figureAl. Finally,the data for personal consumptionexpenditure on nondurables in 1972 dollars, Ct,were obtained fromCitibankEconomicDatabaseand fromthe Surveyof CurrentBusiness.
References Berndt, E. K.; Hall, B. H.; Hall, Robert E.; and Hausman, JerryA. "Estimation and Inference in Nonlinear StructuralModels." Ann. Econ. and Soc. 3 (October 1974): 65365. Measurement Bodie, Zvi; Kane, Alex; and McDonald, Robert L. "Why Are Real Interest Rates So High?" WorkingPaper no. 1141. Cambridge, Mass.: NBER, June 1983. "Why Haven't Nominal Rates Declined?" Financial AnalystsJ. 40 (March/April1984): 1627. Bollerslev, Tim. "Generalized AutoregressiveConditional Heteroskedastic31 (April 1986): 30727. ity."J.Econometrics Breeden, Douglas T. "An IntertemporalAsset PricingModel withStochastic Consumptionand InvestmentOpportunities."J.FinancialEcon. 7 (September 1979): 26596. Campbell,John Y. "Stock Returnsand the Term Structure."J.FinancialEcon. (1987), in press. Chen, Naifu. "Some Empirical Tests of the Theory of ArbitragePricing."J. Finance 38 (December 1983): 13931414.
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