Is the relative risk aversion parameter constant over time? A multi-country study

Empir Econ (2010) 38:605–617DOI 10.1007/s00181-009-0281-y

ORIGINAL PAPER

Is the relative risk aversion parameter constantover time? A multi-country study

Samarjit Das · Nityananda Sarkar

Received: 31 January 2006 / Accepted: 27 June 2008 / Published online: 5 March 2009© Springer-Verlag 2009

Abstract In this paper, an information matrix (IM)-based test is developed for test-ing the hypothesis of constant relative risk aversion parameter in the GARCH-M setup. A detailed Monte Carlo study is then carried out to evaluate the performance of thistest in terms of size and power. Further, a bootstrap technique is suggested to correctthe over-size problem found in small samples. The proposed test is then applied to thetime series of returns on stock markets of five important countries to examine whetherthis important hypothesis holds or not, and it is found that the relative risk aversionparameter is not time invariant for all the five time series.

Keywords Bootstrap · Information matrix test · Parameter constancy ·Relative risk aversion and ARCH-M

JEL Classification C10 · C15

1 Introduction

In the literature on financial economics, perhaps the most important relationship isthe one between risk and return. Over the last two decades, this relationship has beenmodified and reoriented in different manners and used in a wide range of financialapplications. Most of the earlier studies concerning this relationship assumed the risk

The authors are grateful to an anonymous referee for very insightful comments on an earlier draft of thispaper.

S. Das (B) · N. SarkarEconomic Research Unit, Indian Statistical Institute,203 B.T. Road, Kolkata 700108, Indiae-mail: [email protected]

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606 S. Das, N. Sarkar

premium i.e., the compensation required by a risk-averse economic agent for holdingrisky assets, to be constant. But, it is only reasonable to expect that the degree of uncer-tainty in asset returns should vary over time and consequently the risk premium mustalso be time varying. To incorporate this variation in the degree of uncertainty in suchmodels, Engle et al. (1987) introduced the autoregressive conditional heteroscedastic-in-mean (ARCH-M) model. Several studies by Baillie and Bollerslev (1990), Baillieand DeGennaro (1990), Bollerslev et al. (1988), Nelson (1990) and Das and Sarkar(2000) have used the ARCH-M or close relatives of the ARCH-M model whereinrisk premium has been considered to be a function of conditional variances and/orcovariances of asset returns.

One of the economic theoretic implications of ARCH-M model is that economicagents are constant relative risk averse. This assumption seems to be fairly strong andmay not be true in many practical situations. If a representative agent maximizes atime additive von Neumann-Morgenstern utility, the mean/variance ratio or the rela-tive risk aversion parameter may change due to either change in perception towardsrisk involved in the financial market or change in the distribution of wealth or both.This is likely to be more so for a longer horizon of time. Moreover, during the courseof time, the changes in consumption and investment patterns of different economicagents might also influence the relative risk aversion parameter.

Now, insofar as empirical evidence in support of this are concerned, French et al.(1987) found a striking result to the effect that relative risk aversion parameter is veryunstable across the sample periods. Based on the New York Stock Exchange (NYSE)monthly value-weighted index, they obtained the parameter estimates to be 1.693,1.510 and 7.220 for the periods 1928–1984,1928–1952 and 1953–1984, respectively.They also found, using the S & P daily composite index, that the two sub-sample(1928–1952 and 1953–1984) estimates are 0.598 and 7.809, respectively. Pindyck(1988) estimated the risk aversion parameter by using the NYSE index for the period1949–1983 and 1962–1983, and obtained the estimate as 3.447 and 1.672, respectively.With rolling regression technique, Chou et al. (1992) found that the mean/varianceratio of S & P daily composite index ranges from −0.4 to 15.6 with a mean of 5.4 andstandard deviation 4.1. All these findings, therefore, indicate that it should be possibleto generalize the ARCH-M model with an appropriately defined time varying riskaversion parameter.

The effect of neglecting the time varying nature of relative risk aversion parameter,λ (say), are of two folds. First, one of the inputs required by investors seeking to holdrisky assets is the ex ante measure of risk premium which, in turn, depends on therelative risk aversion parameter. Now, if a time invariant λ is estimated using sampleinformation, the estimated risk premium is liable to be erroneous in case λ is notstable. So, from investors’s point of view a test for stability of λ is crucial. Secondly,validity of inference and post-sample forecasting crucially depends on the underlyingparameter being stable. Neglecting parameter variation across observations results ina misspecified likelihood function, and this may lead to inconsistent maximum like-lihood estimates of the relevant parameters (White 1982). In actual data analysis, itis, therefore, important to know whether the relative risk aversion parameter is timeinvariant or not. To this end, we propose, in this paper, a test procedure for testing thetemporal stability of the relative risk aversion parameter. In this test, we essentially

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Is the relative risk aversion parameter constant over time? 607

adopt Chesher’s (1983) interpretation of White’s (1982) information matrix (IM) testbeing a test of parameter variation. Since this test is known to suffer from size problem,we suggest a bootstrap procedure for correcting the size of the test and then presentthe results of a simulation study which is undertaken to find the effect of this sizecorrection. Finally, we apply this test to study the potential temporal dependence ofthe relative risk aversion parameter associated with the stock markets of five majorcountries viz., the USA, the UK, Japan, Hong Kong and Germany. We also exam-ine whether risk, as measured by conditional variance, has any impact (positive ornegative) on returns.

The paper has been organized as follows. The proposed test of constancy of therelative risk aversion parameter is discussed in Sect. 2. In Sect. 3, the findings of theMonte Carlo study are summarized. Section 4 presents the results of the bootstrap-based size corrected test. The findings of the empirical example are discussed in Sect. 5.The paper concludes with some remarks in Sect. 6.

2 Test for constancy of the relative risk aversion parameter

In this section, we derive the explicit form of the test statistic for testing the constancyof the relative risk aversion parameter λ. A natural and accepted procedure for detect-ing parameter constancy is to specify any particular probability density, say, g(.), forthe underlying parameter and then test for the variance of its distribution being zero.

Following Chesher (1983) and Cox (1983), we derive Rao’s (1948) score (RS)test (also called the Lagrange multiplier (LM) test in econometrics) statistic for thistesting problem. It is noteworthy that the score test involves examination of the localbehaviour of the log-likelihood function close to the null hypothesis of no parametervariation, and hence it does not require the explicit specification of the alternativehypothesis in the form of any arbitrary distributional assumption on parameter vari-ations. Moreover, the asymptotic properties of the score test are not affected (Beraand Ullah 1991; Bera et al. 1998; Moran 1971; Self and Liang 1987) by the boundaryvalue problem as opposed to likelihood ratio or Wald tests. Following the notationsof Chesher (1983), and Bera and Kim (2002), let the pd f of the underlying randomvariable y be f (y; θ) and the l × 1 parameter vector θ has density, called the priordensity, g(θ; θ0,�), where θ0 = E(θ) and� = {ωrs}, the variance-covariance matrixof θ . Let h(y; θ0,�) be the marginal distribution of y, and this can be written as

h(y; θ0,�) = Eθ [ f (y; θ)] =∫

f (y; θ)g(θ; θ0,�)dθ. (2.1)

Since, generally we don’t have much information on g(.) and we need only localbehaviour of h(y; θ0,�), we consider a small � approximation to this density andthis approximation is given by

ha(y; θ0,�) = f (y|θ0)

[1 + 1/2

l∑r=1

l∑s=1

(Frs

2 + Fr1 Fs

1

)ωrs

], (2.2)

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where Frs2 , Fr

1 are the (r, s)th and r th element of F2(y; θ0) and F1(y; θ0), respectivelyand Fj (y; θ) = ∂ j log f (y; θ)/∂θ j j = 1, 2. Now, Chesher and Cox have shown thatfor � close to zero, ha(y; θ0,�) is non-negative1 and a proper density.

Note that a test for parameter variation in θ is equivalent to the test for � being azero matrix, and hence we can perform the score test on � considering ha(y; θ0,�)

as the appropriate density. However, as in our case, one may not be interested in thewhole parameter vector, but only in a sub-vector of the parameter vector θ . There-fore, let us now consider the score test of H0: ωrs = 0, (r, s) ∈ A (an index set)against the alternative H1: ωrs �= 02 given ωuv = 0 f or(u, v) �∈ A. Let θn be thelikelihood estimate of θ0 based on n observations and DA(θn) be the estimated scorevector which consists of the elements, Drs, (r, s) ∈ A, and which can be written asDrs = 1

n

∑ni=1

∂lnha(y;θ0,�)∂ωrs

evaluated at � = 0, and θ0 replaced by the maximum

likelihood estimate, θn . Let V (θn) be some consistent estimator of the asymptoticvariance-covariance matrix of

√n DA(θn). Then the RS test statistic is obtained as

d = nD′A(θn)(V (θn))

−1 DA(θn), (2.3)

which is asymptotically distributed as χ2(central) under H0 with degrees of freedomas the cardinality of A.

As noted by Bera and Kim (2002), the score function as obtained from Eq. (2.2)is nothing but the criterion function of the IM test which was originally suggestedas a general model missspecification test by White (1982). Now, there are mainlytwo versions of the IM test: the outer product of gradient (OPG) version (Chesher1984; Lancaster 1984) and the “efficient score” version, as proposed by Orme (1990).Between the two, the OPG version is very popular because of its computational sim-plicity, but it is also well-known that it suffers from size distortion problem (see, Taylor1987; Chesher and Spady 1991). In our case also, we use the OPG form of the IM test.However, we propose to overcome this size distortion problem by using an appropriatebootstrap technique.

Insofar as computation of the test statistic is concerned, we note that using White’s(1982) expression for V (θn) leads to White’s test statistic, but it may be difficult tocalculate this as it involves third derivatives of the log-density, F(y, θ), which appearsin Q0 = −E(G2), where G2 = ∂G1

∂ψ ′ ,G1 = ∂G∂ψ,G = lnha(y; θ0,�) and ψ is the

full parameter vector i.e., ψ = (θ ′

0

...�c ′)′, where the row order of �c = vech(�).

However, a computationally simpler estimator of V (θn) is available from Chesher.This is given by Vn which is nothing but the inverse of the lower right (ν × ν) subm-

atrix of Q0−1, Q0 = 1

n G1′G1, G1 is the n × (l + ν) matrix with i-th row equal to

G ′1 with y replaced by yt and θ0 replaced by the ML estimator, θn , under H0, G ′

1 =[F1(y; θ0)

′...d(y; θ0)′], d(y; θ0) = [

F2(y; θ)+ F1(y; θ)F1(y; θ)′]c, ν = l(l + 1)/2

1 Superscript ‘a’ denotes that the expression is an approximation one.2 Obviously, ωrs > 0 when r = s.

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is the number of distinct elements in �. The IM test statistic (in 2.3) thus simplifiedis given as

d = nD′n(θn)Vn

−1Dn(θn), (2.4)

which, in turn, approximates to the well-known expression n R2, where R2 is thesquared value of the coefficient of multiple determination from the least squares esti-mation of the dependent variable in , an n × 1 column vector of ones, on the matrix ofindependent variables G1.

Now, in order to obtain the RS test statistic for our case, we consider the GARCH(1,1)-M model, originally due to Engle et al. (1987), which is specified as:

yt = x ′tβ + λht + εt , (2.5)

ht = α0 + α1ε2t−1 + φ1ht−1, (2.6)

where εt | t−1 ∼ N (0, ht ), α0 > 0, α1 ≥ 0, φ1 ≥ 0, α1 + φ1 < 1, yt is thedependent variable (return) , xt is a (k × 1) vector of weakly exogenous and laggeddependent variables (if any), t−1 is the increasing sequence of σ -field generated by{xt−1, xt−2, . . . , εt−1, εt−2, . . .}. Note that while the innovations of the linear regres-sion equation are serially uncorrelated, they are not independent as they are relatedthrough higher order moments.

Now, suppose σ 2λ be the variance of the relative risk aversion parameter λ which

is now assumed to be a random variable and lt (θ) be the conditional log-density ofthe t-th observation, t = 1, 2, . . . , n. Then the conditional log-likelihood function ofy1, y2, . . . , yn , l(θ), is given by

l(θ) = 1

n

n∑t=1

lt (θ),

where lt (θ) = constant − 12 ln ht − ε2

t2ht

. Further, let l(θ0,�) be the log-likelihoodfunction for n independent realizations from ha(y; θ0,�). Then the average score forH0: σ 2

λ = 0, is

Dn(θn) = 1

n

n∑t=1

d(yt ; θn), (2.7)

where θn , as already defined, is the maximum likelihood (ML) estimator of θ0 underH0. Now, it is quite evident from (2.5) that

d(yt ; θn) = 1

2

[∂2lt (θ)

∂λ2

∣∣∣∣θn

+(∂

∂λlt (θ)

)2 ∣∣∣∣θn

]. (2.8)

Given the form of the score function , it is easy to find Vn and hence the test statisticunder the null hypothesis, H0 : σ 2

λ = 0. Since the expression of the IM test statistic

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(OPG version) involves ∂lt∂λ, ∂ht∂λ, ∂

2lt∂λ2 and ∂2ht

∂λ2 , their exact expressions are given below.

∂lt (θ)

∂λ= 1

2ht

(ε2

t

ht− 1 + 2λεt

)∂ht

∂λ+ εt , (2.9)

∂ht

∂λ= −2α1εt−1ht−1 + (φ1 − 2α1λεt−1)

∂ht−1

∂λ, (2.10)

∂2lt (θ)

∂λ2 = ∂2ht

∂λ2

[ε2

t

2h2t

− 1

2ht+ λεt

ht

]+ ∂ht

∂λ

[εt − λht

ht− λ

− λεt ht + ε2t

h3t

]

+(∂ht

∂λ

)2 [λht + λεt

h2t

]−

(εt

ht+ ht

), (2.11)

and

∂2ht

∂λ2 = 2α1h2t−1 + ∂ht−1

∂λ

[4α1λht−1 + εt−1 − 2α1εt−1

]

+∂2ht−1

∂λ2

[φ1 − 2α1λεt−1

] + 2

(∂ht−1

∂λ

)2

α1λ2. (2.12)

Obviously, d will follow a χ21 distribution under the null hypothesis, H0 : σ 2

λ = 0.

3 Monte Carlo results

In the last section, we have derived the IM test in OPG form for testing the null hypoth-esis of constancy of the relative risk aversion parameter, λ.We now examine the finitesample performance of this test statistic through a Monte Carlo experiment. The com-putations were carried out using a program written in GAUSS. For the purpose ofstudying the performance of the test in terms of its size, we generated yt ’s from thefollowing GARCH-M model:

yt = 0.25ht + εt , (3.1)

ht = 1 + 0.05ε2t−1 + 0.2ht−1, (3.2)

where εt/√

ht were obtained as standard Student’s t distribution with 10 degrees offreedom.3 Insofar as the choice of sample size is concerned, samples of sizes 100,500, 1,000, 5,000,and 10,000 have been considered, and for each of these sample

3 We considered smaller degrees of freedom as well. However, for such choices, the performance of thetest has been found to be quite poor even for sample of size as large as 1,000. We also conducted the samesimulation study based on normal variate in place of Student’s t . The findings have been more encouraging,as expected. For brevity of space, we are not reporting these results.

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Table 1 Estimated type-I errorprobabilities (λ = 0.25)

Nominal size (%) Sample size

100 500 1,000 5,000 10,000

1 0.591 0.270 0.023 0.012 0.013

5 0.794 0.326 0.108 0.056 0.046

10 0.822 0.405 0.248 0.108 0.103

Table 2 Estimated powers(λ0 = 0.25 and ut ∼ U (0, 1))


1,000 5,000 10,000

1 0.627 0.895 0.960

5 0.759 0.921 0.992

10 0.896 0.987 1.000

sizes, 10,000 replications have been taken.4 The empirical sizes presented in Table 1show (in percentage) the number of times (out of 10,000) the computed value of theproposed test statistic has exceeded the critical values corresponding to the usual nom-inal sizes (1%, 5% and 10%) based on χ2 distribution with one degree of freedom.5

It is observed from Table 1 that the test (OPG version) rejects the null hypothesisfar too often at all nominal levels considered, although its performance does improveas the sample size increases. We further note that the nominal sizes are more or lessattained when the sample size is the maximum i.e., 10,000. As it is, this sample sizeseems to be very large. But, in reality, this is not considered to be so in case of financialtime series like stock prices which are available at high frequency levels.

Since there is hardly any sense in studying the power performance of the test forsample sizes for which there are extreme size distortions, our investigation on thepower performance of the test has been confined to samples of higher sizes viz., 1000,5000, and 10,000, for which no major size distortion has occurred. For the purposeof power calculation, λt was taken to be generated as λt = λ0 + ut where ut wasassumed to follow a U (0, 1) distribution under the alternative hypothesis.6

The results of power analysis are summarized in Table 2. We find from this tablethat as sample size increases , the power also increases, as expected. Overall, the testseems to have good power when the sample size is reasonably large.

4 In order to study the robustness of the results, we also conducted a small simulation study with a modelwhere the relative risk aversion parameter, λ, in (3.1) was taken to be 0.5 while the values of all otherparameters in (3.1) and (3.2) were kept the same. However, overall findings have remained the same forthis choice of the value of λ as well.5 These computational figures correspond to the value of λ being 0.25.6 We also conducted a small simulation study with ut ∼ U (0, 0.8). Furthermore, we considered jumps atvarious time points ( 25th percentile, 50th percentile and 75th percentile points) with various other modelsincluding one containing a lag one term. The test was found to have reasonable power in all such cases.

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4 Bootstrapping the test for size correction

We have already discussed and noted that the IM test in OPG version suffers fromserious over-size problem resulting in rejection of the null hypothesis far too oftenthan those understood by the assumed nominal levels unless the sample size is large.Some attempts have been made to obtain a size corrected test. Chesher and Spady(1991) have suggested that the critical values be obtained from the Edgeworth expan-sion through O(n−1) of the finite sample distribution of the test statistic. Horowitz(1994), on the other hand, has proposed a bootstrap-based method to get the size-corrected test. Since derivations involving the Edgeworth expansion is quite tedious,we have followed, in our case, the computer-intensive method of Horowitz (1994).In this section, we report the results of a limited Monte Carlo study of the proposedIM test where critical values have been obtained by using bootstrap procedure as sug-gested by Horowitz. However, this procedure cannot be applied straightway for ourmodel since the underlying ARCH-M specification involves conditional heterosced-asticity. Therefore, the usual bootstrapping of the data would not be appropriate asbootstrap samples will not be representatives of the original sample simply becausethe original sample is time dependent, while bootstrap samples are independent. Inorder to take care of such problems of dependence in the data, especially in time seriesdata, residual bootstrapping has been suggested in the literature (see, Freedman 1981;Bickel and Freedman 1981; Bose 1988, for details). Following this procedure, we havebootstraped the standardized residuals.

As noted in the preceding section, both the sample sizes and the number of rep-lications considered for the Monte Carlo experiment are quite large. Further, com-putations required very long period of time to be completed. It is also a fact thatMonte Carlo experiments with bootstrapping are far more time-consuming because ateach replication of the estimation sample there are many inner replications for whichthe parameters of the model under the null hypothesis are reestimated from bootstrapsamples. We were thus left with a limited choice. Accordingly, we have consideredsamples of sizes, 100, 200 and 300 only, and in each case 1000 replications have beentaken. For each replication, the number of inner-replications, called the bootstrap sizeand denoted by B, was taken as 100. The data generation was carried out for the modelstated in Eqs. (3.1) and (3.2). Each experiment consisted of repeating the followingsteps 1000 times.

Step I : We generated an estimation data of size n by random sampling from themodel under consideration. This data set is, in fact, the same data set with λ = 0.25,which was used for the Monte Carlo study in the previous section. The parametersof the model were estimated and the value of the test statistic was computed. Let thisstatistic be denoted as d0.

Step II : The residual series εt was obtained. The standardized residuals, ε∗t = εt −¯ε√ht

,

where ¯ε is the simple arithmetic mean of the residuals and ht is the estimated condi-tional variance at time point t , were then computed. A bootstrap sample of size n wasgenerated by random sampling from the standardized residual series, and then the cor-

responding residuals were obtained by using the relation εt = ¯ε+ ε∗t

√ht . Finally, the

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Table 3 Size estimates basedon bootstrapping (λ = 0.25)

Nominal size (%) Sample Size

100 200 300

2.5 0.029 0.023 0.031

5 0.049 0.059 0.048

10 0.16 0.096 0.12

Table 4 Power estimates basedon bootstrapping (λ = 0.25) andut ∼ U (0, 1))


100 200 300

2.5 0.097 0.245 0.298

5 0.285 0.391 0.409

10 0.393 0.508 0.592

bootstrap samples of the data were obtained by using the model under considerationbut with the estimated parameter values obtained in Step I instead of the true values.

Step III : Using the bootstrap sample of data as obtained in Step II, the model wasagain estimated and the test statistic d was computed. The critical values of the IMtest statistic at 2.5%, 5% and 10% significance levels were then estimated from theempirical distribution of d , the latter being obtained by repeating this step 100 times.The model being tested at the nominal level α was rejected if d0 > ˆcn,α, where cn,α

denotes the estimated α-level critical value of the test.The estimated size figures corrected by bootstrap technique are presented in Table 3.

We find from the entries of this table that the over-size problem, especially in smalland medium sample sizes, of the proposed IM test in OPG version no longer exists.For instance, while for n = 100, the estimated size is 0.794 at nominal level of 5% (cf.Table 1), the size of the test with bootstrap correction is only 0.049. In fact, most ofthe entries in Table 3 corroborate this finding. Thus, we find that these corrected sizevalues are now reduced more or less to their respective nominal values.

In addition to size, the values of bootstrap size-adjusted estimated power of the testare presented in Table 4. We find from the entries of this table that the test performsquite well with respect to power, and that the power of the test, as expected, increasesas number of observations increases. Power calculation with any other value of λcould not be considered due to enormous amount of time required for such simulationexercises.

5 Empirical findings

For the purpose of application of the test with actual data, we have considered fivemajor stock price indices at weekly frequency from five countries viz., S & P-500composite index (the USA), FTSE-100 (the UK), Topix (Japan), Hang-Seng (HongKong) and DAX (Germany). The choice of these five countries has been made keeping

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Table 5 Descriptive statistics of returns for the five series

Statistic Hang-Seng S & P Topix FTSE DAX

Mean 0.23 0.12 0.16 0.17 0.10

Variance 17.18 4.66 5.98 6.79 6.85

Skewness 0.60 0.35 0.37 0.44 0.48

Kurtosis 5.85 3.26 3.72 5.58 4.77

Jarque-Bera 3034.75 932.88 1653.43 1803.79 1992.29

Q(5) 54.92 3.85 11.15 4.23 20.36

Q(25) 98.58 33.87 54.28 32.31 35.32

QA(5) 828.27 271.79 339.53 46.51 453.62

QA(25) 2527.61 872.99 975.17 127.38 1302.21

QS(5) 376.66 166.08 187.24 30.77 347.77

QS(25) 895.71 320.60 359.56 79.47 715.45

Q(m), Q A(m) and QS(m) are the Ljung-Box statistics based on returns, absolute returns, and squaredreturns, respectively, and each follows aχ2 distribution with m degrees of freedom under the null hypothesis

in mind the diversity of the capital markets of these countries. For instance, the capitalmarkets of both Japan and Hong Kong are very old and yet very developed; both arealso very important economies in Asia. Germany and the UK, on the other hand, areimportant European countries with well-developed stock markets. The stock marketof the USA is probably the most important because of the size, nature and importanceof the US economy. Weekly data covering more than 30 years have been taken for allthe five indices.7 Data have been collected from ‘Datastream’. Return at time pointt is defined as yt = (ln pt − ln pt−1)×100, where pt is the stock price index at t .Hence, the analyzed series represents the continuously compounded rates of returnfor holding the (aggregate) securities for one week.

We first present some descriptive statistics which may provide some insights intothe data generating processes. We observe from Table 5 that while Hang-Seng yieldsthe maximum value of mean return with maximum variance, the S & P-500 has theminimum mean return with minimum variance. All the return distributions are highlypeaked as kurtosis coefficients are found to be very high as compared to normal dis-tribution, with Hang-Seng having the maximum peak of 5.85. Insofar as skewnesscoefficient is concerned, we find from Table 5 that all return distributions are posi-tively skewed, which may be attributed to the presence of leverage effect and volatilityfeedback mechanism in the series. Stock returns are known to have volatility cluster-ing and thick tail non-normal distributions. Jarque-Bera (JB) normality test stronglycorroborates this hypothesis. Ljung-Box test with squared returns and absolute returns

7 The spans of the data sets considered in this study are the following : S & P-500 composite index: lastweek of December, 1964—first week of September, 2003 (a total of 2019 observations); FTSE-100 (theUK): second week of December, 1978—second week of December, 2004 (a total of 1358 observations);Topix (Japan): second week of January, 1951—first week of September, 2003 (a total of 2749 observations);Hang-Seng: first week of August, 1965—first week of September, 2003 (a total of 2040 observations); DAX(Germany): first week of January, 1965—first week of September, 2003 (a total of 2018 observations).

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Table 6 Estimates of the parameters and test statistic values

Estimate Hang-Seng S& P Topix FTSE DAX

λ 0.005 0.02 0.01 0.004 0.01

(1.86∗∗) (2.526∗) (2.08∗∗) (1.98∗∗) (2.02∗∗)

α0 0.85 0.19 0.22 0.49 0.18

(1.97∗∗) (7.16∗) (7.79∗) (2.40∗) (6.42∗)

α1 0.15 0.07 0.16 0.06 0.11

(4.14∗) (2.23∗∗) (4.24∗) (1.78∗∗) (4.01∗)

φ1 0.81 0.89 0.82 0.87 0.87

(10.86∗) (12.23∗) (8.41∗) (7.04∗) (9.25∗)

GARCH-M LL −5499.73 −4311.87 −6129.95 −3198.30 −4566.50

GARCH LL −5834.81 −4489.65 −6591.17 −3217.85 −4612.19

Q(5) 14.09∗∗ 1.04 30.60∗ 3.78 21.05∗Q(25) 37.83 31.78 66.25∗ 27.95 33.96

QS(5) 8.86 3.49 2.29 3.94 10.16

QS(25) 41.96∗∗ 31.32 19.42 32.24 21.02

IM-test 5353.59∗ 7.98∗ 123.59∗ 105.86∗ 92.05∗[76.98] [3.90] [7.25] [6.47] [10.75]

Figures in parentheses indicate t-ratio values. Figures in [.] in the last row show the bootstrap based t-ratiosat 5% level of significanceThe statistics Q(m) and QS(m) are the Ljung-Box statistics for standardized residuals and standardizedsquared residuals, respectively. Each of Q(m) and QS(m) follows a χ2 distribution with m degrees offreedom under the null hypothesis. GARCH-M LL and GARCH LL denote the maximized log-likelihoodvalues for the GARCH-M and GARCH models, respectively* Indicates significance at 1% level, ** at 5% level only

suggests that the returns are non-linearly dependent and these have volatility cluster-ings. All the five return series have been found to be stationary by the augmentedDickey-Fuller test.

Next, we have estimated the conditional mean part of the model by assumingan autoregressive AR(p) process with the value of p being determined by the AICcriterion. The presence of ARCH in the estimated residuals of any series has beentested by regressing the squared residual on its own lags upto length 25, and thenapplying Engle’s (1982) LM test. All the test statistic values have been found tobe significant at 1% level of significance suggesting thereby the presence of condi-tional heteroscedasticity in the residuals. 8 Therefore, the GARCH model was intro-duced to incorporate volatility in the data analysis. It was then found that while theGARCH(1,1)model turned out to be the best, the AR parameters were all insignificant.Finally, the GARCH-M model was estimated with an AR(p) specification in the mean.In this case also, all the AR parameters were found to be insignificant. However, it isevident from Table 6 that there is significant gain in terms of maximum log-likelihoodvalues if the GARCH-M model is considered instead of a model with conditional

8 These results are not reported for brevity of space.

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mean given by an AR(p)process and conditional variance by the GARCH(1,1) speci-fication.9 Now, looking at the estimated λ-value for each of the five indices in Table 6,we can easily conclude that the relative risk aversion parameter is significant for allthe five series. Thus, it may be easily concluded that the investors in these countriesare all risk averse—although of varying degrees; the λ-value for Hang-Seng is thelowest (0.005) while the largest (0.02) is for the S & P-500 of the USA. The val-ues of Ljung-Box statistic with the standardized squared residuals suggest that theconditional heteroscedasticity has been fully explained by the respective estimatedGARCH-M models for four countries with the sole exception of Japan. For Japan, wetried with higher order GARCH processes as well as with the exponential GARCHformulation by Nelson (1990), but failed to improve upon in terms of Ljung-Box test.

Further, insofar as testing the constancy of λ is concerned, we observe from Table 6that the IM-test statistic value is large for all the five indices, and hence the hypoth-esis of constancy of λ is rejected strongly for all five series.10 The IM-test statisticvalue is largest (5353.59) for Hang-Seng and lowest (7.98) for the USA’s S&P-500.Bootstrap-based critical values at 5% level also strongly suggest time variation in λ.11

Since the capital markets of these countries vary in different respects, this finding ofrejection of time invariant λ for all the countries clearly establishes the time dependentnature of the relative risk aversion parameter. Naturally, we can conclude, based onempirical findings, that the usual assumption of constant relative risk aversion is quiteinappropriate.

6 Conclusions

In this paper, we have proposed an IM based (OPG version) test for testing the con-stancy of the relative risk aversion parameter in the framework of GARCH-M model.We have also carried out a detailed simulation study to evaluate the performance ofthis test in terms of size and power. This test has been found to suffer from over-sizeproblem. To correct this size distortion, we have suggested application of the bootstraptechnique and found that bootstrap method substantially improves the over-size prob-lem. The proposed test has been applied on return data from five major different stockmarkets, and it has been found that there is sufficient evidence to conclude that therelative risk aversion parameter λ is not constant over time. While this finding is veryuseful as well as interesting, it opens up, at the same time, the issue of providing expla-nations for the same. The explanations may lie in the role of other relevant variableslike risky assets other than stock indices, or for that matter, the role of rate of inflationin stock price movements. Obviously, this calls for more detailed investigations andfuture work in this direction is likely to be very fruitful.

9 In all cases across all the models, intercepts turned out to be insignificant. Hence, we have not incorpo-rated any intercept in any of the five models. Further, we have estimated all the models with mean adjustedreturns.10 It can be stated that our test with these data sets should not suffer from significant size distortion since thelength of the time series is quite large for all the five countries—thus avoiding the need for bootstrapping.11 As stated in the text, bootstrap replication is 1000 for these computations.

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Is the relative risk aversion parameter constant over time? A multi-country study

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