a multivariate garch in mean approach to testing uncovered interest parity: evidence from...

Focus: Foreign Exchange

A multivariate GARCH in mean approach to testinguncovered interest parity: Evidence from Asia-Pacific

foreign exchange markets

Chu-Sheng Tai*

Department of Economics and Finance, College of Business Administration, Texas A&M University-Kingsville, MSC 186, Kingsville, TX 78363-8203, USA

Abstract

The existence of time-varying risk premia in deviations from uncovered interest parity (UIP) isinvestigated based on a conditional capital asset pricing model (CAPM) using data from fourAsia-Pacific foreign exchange markets. A parsimonious multivariate generalized autoregressive con-ditional heteroskedasticity in mean (GARCH-M) parameterization is employed to model the condi-tional covariance matrix of excess returns. The empirical results indicate that when each currency isestimated separately with an univariate GARCH-M parameterization, no evidence of time-varying riskpremia is found except Malaysian ringgit. However, when all currencies are estimated simultaneouslywith the multivariate GARCH-M parameterization, strong evidence of time-varying risk premia isdetected. As a result, the evidence supports the idea that deviations from UIP are due to a risk premiumand not to irrationality among market participants. In addition, the empirical evidence found in thisstudy points out that simply modeling the conditional second moments is not sufficient enough toexplain the dynamics of the risk premia. A time-varying price of risk is still needed in addition to theconditional volatility. Finally, significant asymmetric world market volatility shocks are found inAsia-Pacific foreign exchange markets. © 2001 Board of Trustees of the University of Illinois. Allrights reserved.

JEL Classifications: C32; F31; G12

Keywords: UIP; Time-varying risk premium; CAPM; Multivariate GARCH-M

* Tel.: �1-361-593-2355; fax: �1-361-593-3912.E-mail address: [email protected] (C.-S. Tai).

The Quarterly Review of Economics and Finance 41 (2001) 441–460

1062-9769/01/$ – see front matter © 2001 Board of Trustees of the University of Illinois. All rights reserved.PII: S1062-9769(01)00105-3

1. Introduction

The uncovered interest parity (UIP) hypothesis states that the domestic nominal interestrate equals the foreign nominal rate on a comparable asset plus the expected change in theexchange rate over the period to maturity of the asset. Under the standard assumption ofrational expectations, and risk neutral agents, the ex post excess returns of holding foreigncurrency deposits just equal the market true expected excess returns plus a forecast error thatis unpredictable ex ante. Given this joint assumption, tests of UIP are essentially tests of theefficiency of the forward market for exchange rates if covered interest parity (CIP) holds.1

One important conclusion from this market efficiency study is that there exist predictablecomponents in excess returns from holding foreign currency deposits.2 This predictableexcess return is one of the puzzles in international finance literature.3 Although the hypoth-esis that forward exchange rates are unbiased predictor of future spot rates has usually beenrejected, most researchers are still inconclusive as to whether the forward bias is due tomarket inefficiency (irrationality) or to the presence of a time varying risk premium.4

Since the zero risk premium is hardly compatible with the existing applied financeliterature, this time-varying risk premium argument has led to an intensive search for properspecification of the risk premium in foreign exchange markets. Theoretical internationalfinance models developed by Solnik (1974), Roll and Solnik (1977), Hodrick (1981), Adlerand Dumas (1983), and Stulz (1981, 1984) consider the pricing of foreign currency depositsin much the same way as that of other financial assets. In these model, the nominal returnfrom holding a foreign currency deposit in excess of domestic risk-free rate results from arisk premium that has to be paid to risk averse speculators for taking the risk of futurechanges in exchange rates. If this foreign exchange risk cannot be diversified when forminga well-diversified portfolio, then standard portfolio theory tells us that this risk is systematicand should be priced in an asset market in equilibrium. However, if the foreign exchange riskis completely diversifiable, it should not command a risk premium. As a result, if currencyspeculation involves systematic risk, speculative returns should be nonzero and are predict-able. In this case, UIP will be violated even if rational expectations hold.

Most existing models of time-varying risk premia in foreign exchange markets do nothave much empirical success. For example, Mark (1985), Cumby (1988), Kaminsky andPeruga (1990), Backus, Gregory and Telmer (1993) use the intertemporal asset pricing model(IAPM) to test the existence of a time-varying risk premium in the foreign exchange market.In this model the risk premium is due to consumption risk measured by the covariancebetween returns and the marginal utility of money. The results from these studies aredisappointing because the observable ingredients in the risk premium models do not varysufficiently to explain the high degree of variability in asset returns without implausibly largeestimates of the coefficient of relative risk aversion.5

Instead of using the consumption-based IAPM, Mark (1988) uses a single-beta capitalasset pricing model (CAPM) to price the forward foreign exchange contracts from the pointof view of a U.S. investor. He specifies the betas as ARCH-like process and estimates themodel jointly for four currencies using a generalized method of moments (GMM) procedure.His results show significant time variation for the betas and tests of the overidentifyingrestrictions are not rejected. However, as pointed out by Mark (1988), the GMM estimator

442 C.-S. Tai / The Quarterly Review of Economics and Finance 41 (2001) 441–460

is robust, but, in general, is not asymptotically efficient. Consequently, instead of usingGMM estimation, McCurdy and Morgan (1991) employ the single-beta CAPM with a bivariateGARCH parameterization to price deviations from UIP for five European currencies. Theyestimate their model currency by currency, while Mark (1988) estimates his model jointly acrosscurrencies, so the efficiency might be sacrificed in McCurdy and Morgan’s (1991) study.

The purpose of this paper is to inquire further into the problem of whether the observedpredictable components found in deviations from UIP can be attributed to the time-varyingrisk premium that has to be paid to risk averse speculators for holding an uncovered positionin foreign currencies. To accomplish this, we extend previous studies by specifying andtesting a model of time-varying systematic risk in deviations from UIP in foreign exchangemarkets based on a conditional capital asset pricing model (CAPM).6 This paper differs fromprevious studies in several aspects. Firstly, unlike McCurdy and Morgan (1991) who applythe bivariate GARCH process in modeling time-varying conditional second moments, weemploy a parsimonious parameterization of the multivariate GARCH in Mean (GARCH-M)process proposed by Ding and Engle (1994) to model the conditional covariance matrix ofunforecastable components of deviations from UIP for four currencies and excess returns ona benchmark portfolio. With this parameterization, we cannot only retain the maximumefficiency gain in testing UIP, but also uncover some interesting statistics that are mostlyignored in previous studies.7 Secondly, two versions of conditional CAPM are estimated andtested: a constant-price-of-risk CAPM and a time-varying-price-of-risk CAPM. Finally,since most previous empirical studies focused on European currencies and due to the recentAsian currency crisis, it seems interesting to investigate the deviations from UIP using datafrom Asia-Pacific foreign exchange markets.

The empirical results indicate that when each currency is estimated separately with anunivariate GARCH-M parameterization, no evidence of time-varying risk premia is foundexcept Malaysian ringgit. However, when all currencies are estimated simultaneously withthe multivariate GARCH-M parameterization, strong evidence of time-varying risk premia isdetected. As a result, the evidence supports the idea that deviations from UIP are due to a riskpremium and not to irrationality among market participants. In addition, the empiricalevidence found in this study points out that simply modeling the conditional second momentsis not sufficient enough to explain the dynamics of the risk premia. A time-varying price ofrisk is still needed in addition to the conditional volatility. Finally, significant asymmetricworld market volatility shocks are found in Asia-Pacific foreign exchange markets.

This paper is organized as follows. The next section motivates the conditional single-betaCAPM specification for the risk premium. Section 3 discusses the multivariate GARCH-Mmodel employed to characterize time-variations in conditional second moments of excessreturns and presents the test equations. Section 4 discusses the data used. The empiricalresults are reported in Section 5. Conclusion is reserved for Section VI.

2. The theoretical motivation

The theoretical model of intertemporal asset pricing, such as Lucas (1982), assumes thatinvestors maximize expected discounted utility defined as future levels of consumption

443C.-S. Tai / The Quarterly Review of Economics and Finance 41 (2001) 441–460

subject to an intertemporal budget constraint. In equilibrium, asset prices are set so that themarginal utility of one unit of current consumption forgone equals the expected discountedutility of the return from investing that unit of the consumption good in a risky asset. Thatis:

U��Ct� � �E�U��Ct�1�rt�1j �It� (1)

where U� (•) is the marginal utility of consumption Ct at time t, 0 � � � 1 is a subjectivediscount factor. Agents are assumed to be ‘rational’, so E(•�It) is interpreted to be both themathematical expectation conditional on information available at t, as well as the expectationof economic agents. Lastly r t�1

j is the one-period real return on asset j received at time t �1 in terms of the consumption good. Since Ct is known at time t, the equilibrium conditioncan be rearranged as

1 � �E�U��Ct�1�r t�1j

U��Ct��It� (2)

Now, let r t�1j be the real return to a domestic resident from taking an uncovered,

one-period investment in a foreign currency denominated deposit. Thus, Eq. (2) becomes

1 � �E�U��Ct�1��1�i*j,t�1�

U��Ct�

Pt

Pt�1

Sj,t�1

Sj,t�It� (3)

where Pt is the domestic currency price of the consumption good in t, i*j,t�1 is theone-period, nominally risk-free interest rate denominated in currency j at t � 1, and Sj,t is thedomestic currency price of one unit of currency j. Similarly, Eq. (3) must hold for theone-period, nominally risk-free interest rate denominated in domestic currency at t �1, it�1

That is,

1 � �E�U��Ct�1��1�it�1�

U��Ct�

Pt

Pt�1�It� (4)

taking the difference between Eq. (3) and (4) yields the fundamental restrictions imposedby the model

E��U��Ct�1�

U��Ct�

Pt

Pt�1� �1 � i*j,t�1��Sj,t�1

Sj,t� � �1 � it�1�� It� � E�Mt�1Rj,t�1�It� � 0

(5)

where Mt�1 � �U��Ct�1�

U��Ct�

Pt

Pt�1is the intertemporal marginal rate of substitution (IMRS)

of money, and Rj,t�1 � �1 � i*j,t�1��Sj,t�1

Sj,t� � �1 � it�1� is the excess currency returns

or deviations from the UIP for currency j. Eq. (5) states that the conditional first cross-moment between the IMRS of domestic money and the nominal return from foreign currencyspeculation must be zero. Exploiting the decomposition of conditional covariance, Eq. (5)can be rewritten as


E�Mt�1Rj,t�1�It� � COV�Mt�1,Rj,t�1�It� � E�Mt�1�It�E�Rj,t�1�It� � 0

� E�Rj,t�1�It��COV�Mt�1,Rj,t�1�It�

E�Mt�1�It�

(6)

where COV(•�It) is the conditional covariance.Since the IMRS of money is always expected to be positive, the expected deviations from

UIP for currency j (or conditional risk premium) is seen to be proportional to the conditionalcovariance of the IMRS of money and the excess return from foreign currency speculation.More specifically, the conditionally expected excess returns or nominal risk premiumassociated with an uncovered position in the foreign currency is proportional to the condi-tional covariance of the spot price with the IMRS of domestic currency. Under risk aversion,this risk premium will be positive when the conditional covariance between Mt�1 and Sj,t�1

is negative. The intuition behind Eq. (6) is straightforward. For example, if payoff from anuncovered long position is low when the utility denominated value of a unit of homecurrency at t�1, U�(Ct�1)/Pt�1, is relatively high compared to U�(Ct)/Pt at t because ofeither a lower future consumption or a higher purchasing power of the domestic currency, thenrisk averse investors will expect the payoff from the uncovered currency position to be highbecause the foreign investment does not provide a hedge against adverse consumption outcomes.

Empirical implementation of asset pricing relations such as Eq. (6) depends on onechooses the form of Mt�1. As mentioned earlier most existing models of time-varying riskpremia in foreign exchange markets applying consumption-based asset pricing model do nothave much empirical success. Consequently, it is useful to re-express the consumption-basedasset pricing relation Eq. (6) in terms of a benchmark portfolio on the conditional mean-variance frontier [see Hansen and Richard (1987)]. Assume that there exists a minimum-variance portfolio whose nominal return �mt is perfectly conditionally correlated with Mt�1.Then, any linear combinations of �mt and the risk-free return (say �bt) will also beconditionally mean-variance efficient. Now, Eq. (6) can then be expressed as

E[Rj,t�1�It]�COV�Rb,t�1,Rj,t�1�It�

E�Rb,t�1�It��

COV��b,t�1 � �f,t�1,Rj,t�1�It�

E��b,t�1 � �f,t�1�It�(7)

where Rb,t�1 � �b,t�1 �f,t�1 is the nominal excess return on benchmark portfolio b,and �f,t�1 is the risk-free rate of return. Eq. (7) also holds for the benchmark portfolio itself.That is,

E�Rb,t�1�It��VAR��b,t�1 � �f,t�1�It�

E��b,t�1 � �f,t�1�It�(8)

we can then re-express Eq. (6) as a conditional capital asset pricing model by taking theratio of Eq. (7) and (8).

E�Rj,t�1�It� �E�Rb,t�1�It�

VAR�Rb,t�1�It�COV�Rb,t�1,Rj,t�1�It� (9)

where VAR[•�It] is the conditional variance.


Following Harvey (1991) and Chan, Karolyi, and Stulz (1992), the conditional CAPM inEq. (9) can also be rewritten as

E�Rj,t�1�It� �E�Rb,t�1�It�

VAR�Rb,t�1�It�COV�Rb,t�1,Rj,t�1�It� � �tCOV�Rb,t�1,Rj,t�1�It� (10)

where �t can be interpreted as the price of covariance risk. Because the same equation hasto hold for the market portfolio, �t is also referred to as the price of market risk. Eq. (10) willbe subject to empirical tests in Section V.

3. Econometric model and test equations

3.1. Modeling the conditional variances and covariances

The conditional CAPM requires Eq. (10) to hold for every asset, including the benchmarkportfolio. Assume that the realized excess returns are unbiased estimates of the conditionalexpected excess returns. Therefore, in an economy with N risky assets, the following systemof pricing restrictions has to be satisfied, at each point in time

Rt�1 � �thN,t�1 � �t�1 �t�1�It � N�0,Ht�1� (11)

where Rt�1 be the (N � 1) vector of excess returns which includes (N – 1) risky assets andthe benchmark portfolio, the Nth element of Rt�1, Ht�1 is the (N � N) conditional covariancematrix of asset returns, hN,t�1 is the Nth column of Ht�1 and contains the conditionalcovariance of each asset with the benchmark portfolio. Eq. (11) follows directly from theconditional CAPM in Eq. (10). However, the model does not impose any restrictions on thedynamics of the conditional second moments. Given the computational difficulties in esti-mating a larger system of asset returns, parsimony becomes an important factor in choosingdifferent parameterizations. A popular parameterization of the dynamics of the conditionalsecond moments is BEKK, proposed by Baba, Engle, Kraft, and Kroner (1989). The majorfeature of this parameterization is that it guarantees that the covariance matrices in the systemare positive definite. However, it still requires researchers to estimate a larger number ofparameters. In this paper, instead of using BEKK specification, a parsimonious GARCHprocess originally proposed by Ding and Engle (1994) is modified to accommodate theasymmetric volatility effects in variances and covariances, which has been documented inrecent papers by, among others, Kroner and Ng (1998), Bekaert and Wu (2000), andDe Santis, Gerard and Hillion (2000). Specifically, the dynamic process for the conditionalvariance-covariance matrix of asset returns is specified as:

Ht � C0 � A� � �t1��t1 � A � B� � Ht1 � B � G� � �t1��t1 � G (12)

where Ht is N � N time-varying variance-covariance matrix of asset returns. C0 is a N �N symmetric matrix of unknown parameters. A and B are N � N diagonal matrices ofunknown parameters, and G is N � N matrix. Nt1 is N � 1 vector such �i,t1 � �i,t1

if �i,t1 �0, 0 otherwise. In this model the conditional variance and covariance of each


return are related to the past squared residuals and cross residuals, past squared asymmetricshocks and cross asymmetric shocks of all returns while they are only related to their ownpast conditional variance and covariance. Previous studies such as Black (1976), Nelson(1990), Glosten, Jagannathan and Runkle (1993), Engle (1993), Kroner and Ng (1998), andBekaert and Wu (2000) all find that an increase in volatility is likely to be greater followinga large downward move than that following a large upward move for equity markets, aphenomenon which is called asymmetry in volatility. One of the arguments of asymmetricvolatility is the leverage hypothesis, but the same argument cannot be applied to foreignexchange markets. As a result, in this paper we do not include asymmetric volatility effectto all currency returns. In addition, if the asymmetric volatility effect is incorporated to allasset returns, the number of parameters to estimate will increase significantly, so to reducethe size of parameters space, similar to De Santis, Gerard and Hillion (2000) we assume thatonly the market asymmetric shocks affect the variance and covariance of currency returns.Kroner and Ng (1998) show that the volatility asymmetry of both small and large stockportfolios in the U.S. stems mostly from shocks in the large stock portfolio. De Santis,Gerard and Hillion (2000) impose market asymmetric shocks on the international equitymarkets, but conclude that there may not be a strong contagion of negative shocks in thosemarkets. Consequently, it will be interesting to see if the volatility asymmetry of worldmarket portfolio, a ‘large’ portfolio, has any effect on currency returns, a ‘small’ portfolio.

Finally, the conditional second moment process is assumed to be covariance stationary, sothe matrix C0 can be written as follows:

Vec�C0� � � IN2 � � A � A�� B � B�� 1

2�G � G�� Vec�H0� (13)

where H0 is the unconditional covariance matrix of the residuals. This covariance-stationary assumption further simplifies the estimation since it reduces the number ofparameters to estimate in the conditional second moment process by N��N�1�

2parameters.

Under the assumption of conditional normality, the log-likelihood to be maximized can bewritten as,

lnL� ��TN

2ln2 �

1

2�t�1

T

ln�Ht� �� 1

2��t� � Ht� �1�t� � (14)

where is the vector of unknown parameters in the model. Since the normality assumptionis often violated in financial time series, we use quasi-maximum likelihood estimation(QML) proposed by Bollerslev and Wooldridge (1992) which allows inference in thepresence of departures from conditional normality. Under standard regularity conditions, theQML estimator is consistent and asymptotically normal and statistical inferences can becarried out by computing robust Wald statistics. The QML estimates can be obtained bymaximizing Eq. (14), and calculating a robust estimate of the covariance of the parameterestimates using the matrix of second derivatives and the average of the period-by-periodouter products of the gradient. Optimization is performed using BFGS algorithm.


3.2. Test equations

To obtain testable equations, we decompose rational expectations in Eq. (10), and (12)into their forecastable and unforecastable components and categorize these test equationsinto two versions of conditional CAPM.

3.2.1. Constant price of risk model (CPR)The first version of conditional CAPM we consider is to assume the price of market risk,

�m, is time-invariant. This implies that, although both the conditional risk-free rate and theconditional mean-standard deviation frontier can change in each period, the slope of thecapital market line is fixed. Another feature of this model is the moving average error processMA(4), which is used to correct for any serial correlation, found in most the currency returnsin the sample.8 The system of equations for the CPR model is as follow:

Rj,t � �mhjm,t � �k�1

4

�jk�j,tk � �j,t � j (15)

where �t�It1 � N�0,Ht�

Ht � C0 � A� � �t1��t1 � A � B� � Ht1 � B � G� � �t1��t1 � G

with Vec�C0� � � IN2 � � A � A�� B � B�� 1

2�G � G�� Vec�H0�

3.2.2. Time-varying price of risk model (TPR)Many empirical studies have shown that the prices of risk are time-varying. (e.g., Harvey

(1991), Dumas and Solnik (1995), De Santis and Gerard (1997, 1998), Tai (1999a), andamong others.) This time-varying price of risk is economically appealing in the sense thatinvestors use all available information to form their expectations about future economicperformance, and when the information changes over time, they will adjust their expectationsand thus their expected risk premia when holding different risky assets. Therefore, in thesecond version of the conditional CAPM, we relax the constant-price-of-risk assumption toallow the price of market risk to change over time.

The dynamic of price of market risk is chosen according to the theoretical asset pricingmodel developed by Merton (1980). In his model, the price of market risk is the coefficientof risk aversion of risk averse investors, and thus should be positive. Consequently, similarto Bekaert and Harvey (1995) and De Santis and Gerard (1997, 1998) an exponentialfunction is used to model the dynamic of �m,t1.

�m,t1 � exp��mzm,t1� (16)

where zm,t1�{CONSTANT, FPJPY,FPHKD,FPSGD,FPMYR} is a vector of informationvariables observed at the end of time t 1 and �’s are time-invariant vectors of weights.Bekaert and Hodrick (1992) show that forward premia have significant explanatory for bothcurrency and equity returns, so the price of market risk is assumed to be a exponential


function of information variables in Zm,t1, including a constant, and four forward premia(FPJPY,FPHKD,FPSGD,FPMYR).

As a result, the system of equations for the TPR model is as follow:

Rj,t � �m,t1hjm,t � �k�1

4

�jk�j,tk � �j,t � j (17)

where �t�It1 � N�0,Ht�; �m,t1 � exp(��mzm,t1)

Ht � C0 � A� � �t1��t1 � A � B� � Ht1 � B � G� � �t1��t1 � G

with Vec�C0� � � IN2 � � A � A�� B � B�� 1

2�G � G�� Vec�H0�

4. Data and summary statistics

This paper analyzes weekly data for the currencies of the Japanese yen (JPY), Hong Kongdollar (HKD), Singapore dollar (SGD), and Malaysian ringgit (MYR), all in relation to theU.S. dollar. In particular, we consider four currency deposits which are 7-day Euroyendeposit rate, 1-week Hong Kong deposit, 1-week Singapore deposit, and Malaysia 1-monthdeposit rate.9 To approximate the benchmark portfolio, we use a value-weighted MorganStanley Capital International (MSCI) world index (MSWRLD). The 7-day Eurodollar depositrate is used as risk-free rate to compute the excess returns on the MSCI world index, and thefour currency deposits (or the deviations from UIP). Excess equity return is calculated as

�pt�1

pt�ln(1�it

us$) where pt is the MSCI world total return index at time t, and excesscurrency return (or the deviations from UIP) is calculated as ln(1�i*t) ln�st�1

st�ln(1�it

US$)where st is the spot rate at time t expressed as domestic price (the U.S dollar) of one unit offoreign currency; i*t is the nominal rate of 7-day Eurocurrency deposit known at time t anditUS$ is 7-day Eurodollar. All the data are extracted from Datastream and cover the period

from January 1, 1988 through February 27, 1998. However, we work with rates of return thatleaves 530 observations expanding from January 8, 1988 to February 27, 1998. Panel A ofTable 1 presents summary statistics for the excess return series. The first four columns arefor the currency returns and the last column is for the world equity returns. Notably, all theexcess currency returns are negative except SGD. The mean excess returns on the MSCIworld index is positive (0.1082%) but it also has highest standard deviation (1.87%) amongall the return series. All the excess returns exhibit significantly skewness and excess kurtosis,indicating nonnormality in returns. The Bera-Jarque‘s test confirms this formally. TheLjung-Box test statistic for raw returns, Q(12), are significant for JPY, SGD, and MYR,indicating linear dependencies in these returns. For squared returns, Q2(12) is significant forall series, indicating strong nonlinear dependencies. This is consistent with the volatilityclustering observed in foreign exchange markets: Large (small) changes in prices tend to befollowed by large (small) changes of either sign. The GARCH models used in this study arewell known to capture this property. The unconditional correlation coefficients for the


information variables are reported in Panel B of Table 1. All the correlation coefficients arebelow 0.5, indicating that the selected instruments contain sufficiently orthogonal informa-tion

5. Empirical results

In the empirical section, to compare with previous results by Domowitz and Hakkio(1985), we first present the estimation results from an univariate GARCH(1,1)-M modelwhere the time-varying risk premium is modeled as the conditional variance of excesscurrency returns. We then present the estimation results based on the multivariateGARCH(1,1)-M model with constant price of risk. Finally, we reports the empirical resultsfrom the multivariate GARCH(1,1)-M model with time-varying price of risk.

Table 1Summary statistics for excess currency and world equity returns

Panel A: Summary statistics

JPY HKD SGD MYR MSWRLD

Mean (%) 0.0485 0.0051 0.0014 0.0715 0.1082Std. Dev. (%) 1.4893 0.0895 0.6294 1.4202 1.8700Minimum (%) 6.1742 0.6032 4.8509 15.2110 11.2771Maximum (%) 5.1237 0.3888 3.7259 10.3761 10.5590Skewness 0.3187** 1.2695** 0.5386** 1.4469** 0.4846**Kurtosis 1.2962** 8.6934** 11.3213** 44.4195** 11.0262**B-J 46.0798** 1811.35** 2856.10** 43757.49** 2705.58**Q(12) 26.0956** 19.6896 24.1654* 77.7565** 4.7517Q2(12) 25.6240* 73.6350** 315.0137** 393.4163** 78.0372**

Panel B: Unconditional correlation

JPY HKD SGD MYR MSWRLD FPJPY FPHKD FPSGD FPMYR

JPY 1HKD 0.1100 1SGD 0.4367 0.1444 1MYR 0.1559 0.1082 0.6480 1MSWRLD 0.0532 0.0587 0.0439 0.0149 1FPJPY 0.2548 0.0335 0.1010 0.1371 0.0136 1FPIIKD 0.0136 0.3211 0.0059 0.0622 0.0323 0.1790 1FPSGD 0.0615 0.1261 0.1478 0.0369 0.0421 0.0353 0.2782 1FPMYR 0.1077 0.0750 0.2690 0.3791 0.0515 0.1629 0.0849 0.2870 1

The statistics are based on the Friday-to Friday returns data from 01/08/88 to 02/27/98 (530 observations). Theworld excess returns are the MSCI world index returns in excess of the seven-day Eurodollar interest rate. TheBera-Jarque (B-J) tests normality based on both skewness and excess kurtosis and is distributed 2 with twodegrees of freedom. Q(12) and Q2(12) denote the Ljung-Box test statistics for up to the twelfth-orderautocorrelation of the raw and squared returns, respectively. The information variables are a constant (CON-STANT), forward premium for the JPY (FPJPY), forward preminum for the HKD (FPHKD), forward premiumfor the SGD (FPSGD), and forward premium for the MYR (FPMYR). * and ** denote statistical significance atthe 5% and 1% level, respectively.


5.1. Univariate GARCH(1,1)-M

Since Domowitz and Hakkio (1985) apply the univariate GARCH model and fail to findsignificant time-varying risk premia in European currency returns, it is interesting to see ifwe can detect any time-varying risk premia by employing the same model and using theAsia-Pacific currency returns. Specifically, the following system of equations is estimated.

Rt � a0 � �ht � b1Rt1 � b2Rt2 � b3Rt3 � b4Rt4 � �t (18)

ht � c0 � c1�t12 � c2ht1 (19)

where �t�It1 � N(0,ht)

Table 2 contains QML estimates of the parameter for the univariate GARCH-M model. Ascan be seen in the table, the point estimate for the conditional variance, �, is only significantfor the MYR. As a result, similar to Domowitz and Hakkio (1985) we do not find muchevidence of time-varying risk premia presented in the data when the univariate GARCH-Mis employed.

Next, consider the parameter estimates for the conditional variance equations. All thecurrency returns exhibit significant ARCH effect, as evidenced by the significant pointestimates of c1. However, the GARCH effect is only significant in one case: SGD.

Table 2 also reports some diagnostic tests performed on the standardized residuals and the

Table 2Estimation of univariate GARCH(1,1)-M model

Parameter JPY HKD SGD MYR

a0 0.0021 (0.0043) 2.28E-05 (0.0001) 0.0002 (0.0004) 0.0012 (0.0008)b1 0.0116 (0.0549) 0.0325 (0.0490) 0.5838 (0.0584)**b2 0.1081 (0.0458)* 0.0686 (0.0507) 0.2075 (0.0656)**b3 0.1036 (0.0434)* 0.0963 (0.0433)* 0.2541 (0.0106)**b4 0.0362 (0.0413) 0.1030 (0.0502)* 0.1162 (0.0218)**� 11.9890 (20.2501) 92.955 (164.5005) 13.8065 (13.1916) 24.4492 (10.1559)*c0 0.0001 (0.0001) 3.19E-07 (0.0000)* 8.3E-07 (0.0000) 3.24E-05 (.0000)**c1 0.1382 (0.0667)* 0.5688 (0.2316)* 0.1020 (0.0315)** 0.1863 (0.0121)**c2 0.3409 (0.4212) 0.1097 (0.2231) 0.8838 (0.0394)** 0.1791 (0.1107)LIK 1486.40 3037.8361 2020.6635 1764.9516Kurtosis 4.2362** 10.2773 5.9579** 20.6567**B-J 58.4446** 1198.2341** 206.4744** 7545.39**Q(12) 15.7791* 11.9174 8.3016 176.8863**Q2(12) 11.1601 7.6621 9.6525 132.8714**

Rt � a0 � �ht � b1Rt1 � b2Rt2 � b3Rt3 � b4Rt4 � �t

ht � c0 � ct12 � c2ht1

�t � It1 � N(0, ht)Robust t-statistics are given in parentheses. LIK is the maximum log-likelihood value. Q(12) and Q2(12) are

the Ljung-Box test statistics of order 12 for serial correlation in the standardized residuals and standardizedresiduals squared. B-J is the Bera-Jarque test statistic for normality. * and ** denote statistical significance at the5% and 1% level, respectively.


standardized residuals squared for the purpose of assessing the robustness of the results andthe adequacy of the model. Both the index of excess kurtosis and the Bera-Jarque teststatistics for the standardized residuals are lower than the most of the corresponding indicesfor the original return series. However, the hypothesis of normality is still rejected for allcases. Such evidence against normality provides the reason for us to use the quasi-maximumlikelihood method to estimate the parameter values and use the robust standard errors for theinference [see Bollerslev and Wooldridge (1992)]. We also compute the Ljung-Box port-manteau statistics to test the null hypothesis of zero autocorrelation up to 12 lags in both thestandardized residuals and the standardized residuals squared. The results indicate that theGARCH(1,1) specification used in this study performs very well in capturing the dynamicsof the conditional second moments since most of the Q2(12) statistics is statisticallyinsignificant. However, there still exist some linear dependencies in the conditional meanprocess for JPY and MYR, as evidenced by the significant Q(12) statistics.

5.2. Multivariate GARCH(1,1)-M

Since many issues in finance can only be fully addressed within a multivariate framework,in this section, we consider the multivariate GARCH specification in which the conditionalCAPM is estimated as a system for all currencies. In particular, we estimate the model undertwo different scenarios. First, we assume that the price of market risk is constant. Second, weallow the price of market risk to be time varying.

5.2.1. Constant price of risk model (CPR)Although we constrain the price of market risk to be time-invariant, the model we estimate

is still a “conditional” model in the sense that the dynamics of asset returns depend on theconditional covariances between the underlying asset returns and the market portfolioreturns. That is, we want to see if using a conditional model that allows for the time-varyingsecond moments is sufficient enough to explain the conditional expected currency returns.Table 3 reports the QML estimates of the parameter for the CPR model. The point estimatefor �m is 3.1765, but the 2.4534 value for the robust standard error is relatively large,implying that it is not significant. Therefore, the CPR model also fails to identify thetime-varying-risk premia in excess currency returns.

Next, consider the parameter estimates for the conditional variance process. As can beseen in the table, all the elements in the vectors a and b are statistically significant atconventional level for all currency returns. The market asymmetric volatility shock param-eter, g, is statistically significant at 1% level for both JPY and HKD, implying that foreignexchange markets in Japan and Hong Kong respond the world market volatility shockasymmetrically.

Again, to assess the robustness of the results and the adequacy of the model, we subjectthe model to two kinds of diagnostic tests. First, the autocorrelation of the multivariatesystem is investigate by employing a portmanteau test for the significance of the standardizedresidual autocorrelations and the standardized residuals squared autocorrelations up to the 12th

lag. Under the multivariate framework, the standardized residuals is computed as Zt � Ht1/2�t,

where H t1/ 2 is the inverse of the Cholesky factor of the estimated variance-covariance


Table 3Conditional CAPM with constant Price of Risk (CPR): multivariate GARCH(1,1)-M

Conditional mean process


�1 0.0195 (0.0410) 0.0646 (0.0221)** 0.0469 (0.0242)�2 0.0317 (0.0344) 0.0671 (0.0269)* 0.1503 (0.0400)**�3 0.0914 (0.0429)* 0.0708 (0.0429) 0.0166 (0.0443)�4 0.0494 (0.0349) 0.0409 (.0229) 0.0019 (0.0309)�m 3.1765 (2.4534)

Conditional variance process

a 0.2066 (0.0794)** 0.1961 (0.0261)** 0.3433 (0.0025)** 0.4041 (0.0037)** 0.0937 (0.0432)*b 0.6210 (0.2388)** 0.9723 (0.0092)** 0.9188 (0.0063)** 0.9055 (0.0018)** 0.9851 (0.0038)**g 2.9890 (0.7000)** 0.0754 (0.0225)** 0.5877 (0.6700) 0.1741 (0.2659) 0.0133 (0.0384)

Residual diagnostics

Kurtosis 1.0767** 10.3959** 3.0864** 21.6278** 9.4803**B-J 41.7487** 2460.63** 250.5025** 10743.25** 2016.92**Q(12) 9.7565 14.9832 14.4141 9.5384 4.3510Q2(12) 19.5954 28.6547** 13.0203 2.7355 69.9929**

Log-likelihood function:12320.19

Return: Rj,t � �mhjm,t � �k�1

4

� jk�j,tk � �j,t; � j � JPY, SGD, MYR

Rj,t � �mhjm,t � �j,t; � j � HKD, MSWRLD

where �t� It1 � N�0, Ht�

GARCH: H1 � C0 � A� � �t1��t1 � A � B� � Ht1 � B � G� � �t1��t1 � G

With Vec�C0� � �IN2� A R A��B R B��1

2�G R G�� Vec�H0�

Robust standard errors are given in parentheses. Q(12) and Q2(12) are the Ljung-Box test statistics of order 12 for serial correlation in the standardizedresiduals and standardized residuals squared. B-J is the Bera-Jarque test statistic for normality. * and ** denote statistical significance at the 5% and 1% level,respectively. 453

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matrix Ht. Second, the assumption of conditional normality of the standardized residuals ischecked by using Bera-Jarque test statistics. These statistics, which are displayed at thebottom of Table 3, show that the null hypothesis of no autocorrelation up to and includinglag 12 in the standardized residuals (Q(12)) cannot be rejected in any case at 5% level. Inaddition, the null hypothesis of no autocorrelation up to and including lag 12 in thestandardized residuals squared (Q2(12)) cannot be rejected at 5% level in all but one case.As for the assumption of conditional normality, most of the indexes of excess kurtosis andthe Bera-Jarque test statistics for the standardized residuals are lower than the correspondingindex for the original return series. However, the hypothesis of normality is still rejected inall cases. Again, such evidence against normality warrants the use of the quasi-maximumlikelihood method to estimate the parameter values and use the robust standard errors for theinference. Overall the empirical results in this section indicate that the GARCH(1,1) spec-ification used in this study is flexible enough to capture the dynamics of the conditionalsecond moments.

5.2.2. Time-varying price of risk model (TPR)The poor performance of CPR model in detecting the statistically significant relationship

between excess returns and conditional market risk may be due to the assumption of constantprice of market risk imposed on the model. In this section, we relax this assumption byallowing �m to evolve based on a set of information variables. In doing so, we obtain strongevidence of time-varying risk premia for currency returns.

Table 4 contains the estimation results for TPR model. The parameter estimates for boththe conditional mean and variance processes are reported in Panel A. Summary statisticsconcerning the risk premia and diagnostic test statistics for the standardized residuals areshown in Panel B. Finally, the hypothesis tests regarding the prices of risk are presented inPanel C. The results are very encouraging. For example, the null hypothesis of zero price ofmarket risk is strong rejected by the Wald test statistic with a p-value of zero. The nullhypothesis of constant price of market risk is also significantly rejected with a p-value of0.0034. These results are consistent with the findings of De Santis and Gerard (1997), andimply that simply modeling the conditional CAPM by allowing the time-varying secondmoments is not sufficient to find market as a priced factor. Incorporating the feature oftime-varying price of risk into the model is also needed to detect these price factors. Thestrong predictability of forward premia found here shed a new light on the usefulness offorward premium in predicting the dynamic of risk price since De Santis and Gerard (1998)fail to reject the joint null hypothesis of zero predictability of forward premia from threedeveloped forward markets: Deutsche marks, Japanese yen, and British pounds.

With regard to the conditional variance processes, the point estimates are similar to thosepresented in Table 3. Again all the currency returns exhibit strong GARCH effect. Themarket asymmetric volatility shock parameter, g, is statistically significant at 5% level for allcases but MYR, implying that Asia-Pacific foreign exchange markets are strongly affectedby the asymmetric volatility shock from the world equity market.

Panel B of Table 4 contains diagnostic statistics on the standardized residuals and thestandardized residuals squared for the TPR model. Based on the Ljung-Box statisticsreported in the panel, no significant autocorrelation is left in the series except HKD, and


Table 4Conditional CAPM with time-varying price of risk (TPR): multivariate GARCH(1,1)-M

Panel A: Parameter Estimates

Conditional mean process

Price of risk

CONSTANT FPJPY FPHKD FPSGD FPMYR

�m 0.8192 (0.5827) 118.2698 (89.5109) 395.4245 (191.7212)* 169.6721 (106.4953) 11.2827 (14.9141)


�1 0.0083 (0.0369) 0.0588 (0.0255)* 0.0425 (0.0304)�2 0.0420**(0.0375) 0.0873 (0.0292)** 0.1570 (0.0461)**�3 0.0971 (0.0430)* 0.0700 (0.0322)* 0.0223 (0.0391)�4 0.0456 (0.0360) 0.0347 (0.0238) 0.0313 (0.0400)

Conditional variance process

a 0.2422 (0.0490)** 0.2145 (0.0407)** 0.2949 (.0024)** 0.3779 (0.0003)** 0.1005 (0.0101)**b 0.0017 (0.0041) 0.9689 (0.0175)** 0.9343 (0.0012)** 0.9317 (0.0001)** 0.9852 (0.0037)**g 2.7293 (0.5547)** 0.0863 (0.0433)* 0.1458 (0.0722)* 0.1388 (0.0834) 0.0245 (0.0341)

Panel B: Summary statistics and diagnostics for the residuals


Time-Varying RiskPremium (%)

0.0042 0.0005 0.0023 0.0133 0.1521

ConditionalVolatility (%)

1.4978 0.0757 0.5554 0.8492 1.8465

Kurtosis 1.0460** 11.2100** 3.7360** 32.9439** 9.3452**B-J 41.1188** 2872.40** 359.88** 24666.22** 1967.46**Q(12) 18.9498 19.2872 14.8675 11.2375 4.6930Q2(12) 18.6457 27.5088** 19.0745 1.6533 68.7585**

(continued on next page)

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Table 4 (Continued)

Log-likelihood function: 12334.29

Panel C: Hypothesis tests concerning price of risk

Null hypothesis Wald d.f. P-Value

1. Is the price of market risk equal to zero?H0:�m

Zm � 0; Zm � {CONSTANT, FPJPY, FPHKD, FPSGD, FPMYR} 77.7467 5 0.00002. Is the price of market risk constant?H0: �m

Zm � 0; Zm � {FPJPY, FPHKD, FPSGD, FPMYR} 15.6994 4 0.0034

Return: Rj,t � �m,t1hjm,t � �k�1

4

�jk�j,tk � �j,t; � j � JPY, SGD, MYR

Rj,t � �m,t1hjm,t � �j,t; � j � HKD, MSWRLD

where �t� It1 � N�0, Ht�; �m,t1 � exp ��mzm,t1�; zm,t1 � CONSTANT, FPJPY, FPHKD, FPSGD, FPMYR�

GARCH: H1 � C0 � A� � �t1��t1 � A � B’ � Ht1 � B � G� � ��t1 � G

With Vec�C0� � �IN2� A R A��B R B��1

2�G R G�� Vec�H0�

Robust standard errors are given in parentheses. Q(12) and Q2(12) are the Ljung-Box test statistics of order 12 for serial correlation in the standardizedresiduals and standardized residuals squared. B-J is the Bera-Jarque test statistic for normality. * and ** denote statistical significance at the 5% and 1% level,respectively.

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MSWRLD indicating that the MGARCH(1,1) is flexible enough to capture most of thedynamics of the conditional second moments.

This is an interesting finding in light of the work of Baillie and Bollerslev (1990). Baillieand Bollerslev (1990) model the conditional second moment matrix of forward rate forecasterrors as a multivariate GARCH process, and use this model to test the hypothesis that riskpremium is a linear function of the conditional variances and covariances. Unfortunately theyfail to find much of significant relationship. In contrast to their study, we are able to detectsignificant relationship between the conditional covariances and the risk premia based on thepricing restriction imposed by the conditional CAPM. This empirical finding reinforces theimportance of modeling the time-varying risk premia not only within a multivariate frame-work, but also within an asset pricing framework which allows for the time-varying price ofrisk.

Panel B also reports several interesting statistics, including the sample means of theestimated excess currency returns (or risk premia), and conditional volatilities associatedwith those returns. A useful complement to those statistics is to display the time series ofthose estimated risk premia, and conditional volatilities. The time-series plots of the esti-mated time-varying risk premia for four currency returns are displayed in Fig. 1. As can beseen from the plots, they exhibit a fair amount of time variation. For example, The Japaneseyen fluctuates within a range of –0.4820% to 0.6683% per week. The Hong Kong dollarfluctuates within a range of 0.0016% to 0.0127%. The Singapore dollar fluctuates within arange of –0.3890% to 0.5011%. The Malaysian ringgit fluctuates within a range of –1.9477%to 1.9407%. Fig. 2 displays the time-series plots of the conditional volatility for eachcurrency returns. All the plots also show a fair amount of time variation. For example, theconditional volatilities range from 1.4442% to 3.8314% for JPY, 0.0% to 0.1732% for HKD,0.3873% to 2.0857% for SGD, and 0.4358% to 8.1975% for MYR.

6. Conclusion

In this paper the existence of time-varying risk premia in four Asia-Pacific foreignexchange markets has been tested using the intertemporal asset pricing model. In order toestimate the conditional covariance matrix of excess returns, a parsimonious parameteriza-tion of the multivariate GARCH in mean process has been employed. With this parameter-ization, we retain the maximum efficiency gain in testing UIP which is ignored in most ofthe previous studies.

Estimation results indicate that similar to Domowitz and Hakkio (1985), an univariateGARCH process does not provide much evidence of time-varying risk premium in explain-ing the deviations from UIP when the conditional own variance is used to model the riskpremium. However, as we consider the multivariate GARCH specification where all thecurrency returns are simultaneously estimated with the world equity returns, we are able todetect significant evidence of time-varying risk premia in excess currency returns for allmarkets when the price of the market risk is allowed to be time varying. Therefore, the resultssupport the idea that the deviations from UIP are due to a risk premium and not to theirrationality among market participants. Moreover, the empirical evidence found in this study


points out that simply modeling the conditional second moments alone is not sufficientenough to explain the dynamics of the risk premia. A time-varying price of risk is still neededin addition to the conditional volatility. Finally, significant asymmetric world market vola-tility shocks are found in Asia-Pacific foreign exchange markets.

Notes

1. If CIP holds, the deviations from UIP can be expressed as the difference betweenexpected future spot rates and current forward rates (i.e., forward bias or forwardforecast error).

2. Hodrick (1987) provides a detailed survey on the empirical studies of marketefficiency of forward and futures markets.

3. See Hodrick (1987), Cumby (1988), Korajczyk and Viallet (1992), Bekaert andHodrick (1993) and Lewis (1994).

4. See for example, Hansen and Hodrick (1980, 1983), Hodrick and Srivastava (1984),Korajczyk (1985), Mark (1985, 1988), Hodrick (1987), Cumby (1988), and Kamin-sky and Peruga (1990).

5. Engel (1996) provides a detailed survey on this issue.6. By estimating and testing a conditional single-factor CAPM in an international

setting, we implicitly assume that either all investors have logarithmic utility orpurchasing power parity (PPP) holds. Because the focus of this paper is try to see towhat extent the multivariate GARCH approach will allow us to detect significanttime-varying risk premia within the framework of the single-factor CAPM. Forstudies considering more than one factor asset pricing models in the sprit ofintertemporal CAPM by Merton (1973), please see Dumas and Solnik (1995),De Santis and Gerard (1998), and Tai (1999a,b) where the additional state variablesare due to the foreign exchange risk, and McCurdy and Morgan (1992) in which theyconsider consumption risk as the second state variable.

7. Baillie and Bollerslev (1990) apply multivariate GARCH approach to model riskpremia in forward foreign exchange markets using weekly data from four Europeancurrencies. Unfortunately, they fail to find any significant relationship between theelements of conditional covariance matrix and a time-varying risk premium.

8. In the empirical estimation, MA(4) is included for all the currency returns except forthe Hong Kong dollar because there is little evidence of serial correlation found inthe data based on Q(12) statistic [see Table1].

9. The data on one-week currency deposit rate for Malaysia ringgit is not available, soa one-month deposit rate is used.

Acknowledgements

This paper is based on one of my dissertation essays at The Ohio State University. I wishto thank my advisor, Nelson C. Mark, for his guidance, Paul Evans, Zhiwu Chen, J. Huston


McCulloch, Pork-Sang Lam, Steven Yamarik, as well as workshop participants at The OhioState University and participants at the 1998 Annual Meeting of Ohio Association ofEconomists and Political Scientists (Columbus, OH), 1999 Midwest Economics Associationmeetings (Nashville, TN), and the 1999 Midwest Finance Association meetings (Nashville,TN) for their helpful comments and suggestions. The usual disclaimer applies.

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