Download VIOLATION OF THE IID-NORMAL ASSUMPTION

VIOLATION OF THE IID-NORMAL ASSUMPTION :
EFFECTS ON TESTS OF ASSET-PRICING MODELS USING
AUSTRALIAN DATA
by
Nicolaas Groenewold
and
Patricia Fraser
DISCUSSION PAPER 99.12
DEPARTMENT OF ECONOMICS
THE UNIVERSITY OF WESTERN AUSTRALIA
NEDLANDS, WESTERN AUSTRALIA 6907
VIOLATION OF THE IID-NORMAL ASSUMPTION :
EFFECTS ON TESTS OF ASSET-PRICING MODELS USING
AUSTRALIAN DATA
by
Nicolaas Groenewold*
Department of Economics
The University of Western Australia
and
Patricia Fraser
Department of Accountancy
University of Aberdeen
DISCUSSION PAPER 99.12
DEPARTMENT OF ECONOMICS
THE UNIVERSITY OF WESTERN AUSTRALIA
NEDLANDS, WESTERN AUSTRALIA 6907
ISSN 0811-6067
ISBN 0-86422-920-8
*Corresponding author. We are grateful to Gino Rossi for research assistance and to the Australian
Research Council for financial support through a small ARC grant.
Abstract:
Financial data are typically not iid-normal. Yet standard tests of asset-pricing models
are based on this assumption and we have little information on how sensitive the tests
are to violations of iid-nonnality. Recent evidence suggests that test outcomes may
be reversed with the use of tests that can accommodate these violations. In this paper
we use Australian data to compare the standard test results with those which do not
require iid-normality: the GMM-J test and bootstrap-based tests. We find that
different tests produce differences in prob values at least as large are those in US
studies hut that test outcomes are generally robust.
JEL classification codes: GO and G 1
2
I. Introduction
Asset-pricing models such as the CAPM and the APT are generally tested using
tests which depend for their validity on the assumption that model errors are
identically and independently normally distributed (iid-normal). Most empirical
analysis of asset returns and model errors strongly suggest that the iid-norrnal
assumption is violated in practice.
1
Yet, we know little about the effects of this
violation on the outcome of the test.
Several recent papers indicate that the outcome of tests of the CAPM may be
sensitive to whether account is taken of the violation of the iid-norrnal assumption.
MacKinlay and Richardson (1991), using a US data set compare the outcomes of
standard Wald and GRS tests of the CAPM with those obtained using the J test
associated with the GMM estimator which is robust to a number of violations of the
iid-norrnal assumption. Faff and Lau (1997) provide an application of the MacKinlay
and Richardson analysis to Australian share-price index data and, like MacKinlay and
Richardson, provide evidence that the use of the J test may change the inference
drawn from the data. In a third paper, Chou and Zhou (1997), using a data set similar
to that used by MacKinlay and Richardson, compare both the GMM-J test and
bootstrapped Wald and F tests to standard tests. They, too, find that, for some
samples, test outcomes are reversed when more appropriate tests are used, suggesting
that existing test results based on an invalid iid-norrnal assumption may be
misleading2 •
1
See, e.g., Richardson and Smith ( 1993) for US evidence, Mills and Coutts ( 1996) for the UK and
results presented in this paper for Australia.
2
ln an interesting earlier paper Affleck-Graves and McDonald ( 1989) present Monte Carlo evidence on
the effects of non-normalities on tests of asset-pricing models.
3
It is important to know whether test sensitivity is confined to isolated incidents
or whether it is a general phenomenon applying to a range of countries, time periods
and models. In this paper we contribute to the very limited literature on the subject
and provide further evidence on this question by using Australian share-market data to
compare the results obtained from standard tests to those obtained from tests which do
not depend on the iid-normal assumption. The overall aim of the study is to
contribute to the exploration of the question of when deviations from the iid-nmmal
assumption matter and when they do not.
3
To achieve our aim we begin by testing the CAPM (or the mean-variance
efficiency of our chosen market portfolio) using standard Wald and F tests and
compare the outcome of these to the results of the J-test associated with the
generalised methods of moments (GMM) estimator which is robust to a wide range of
depmtures from the iid-normal assumption. A second alternative to the standard
procedures which we use is to compare the standard W and F statistics to critical
values based on the actual (non-iid-normal) characteristics of the data using the
bootstrapping method. We go on to extend the literature by applying the tests to two
additional asset-pricing models, both of which involve the use of macroeconomic
instrumental variables. Snch an investigation allow us to assess whether or not our
conclusions are model-specific and hence the extent to which test outcomes are likely
to be influenced by issues concerning the testing of joint hypotheses.
Consistent with existing evidence, we find widespread departures from the iidnmmal property in both returns and in market-model eTI"ors. However, in contrast to
the US and Australian evidence cited above, we find no instance of sensitivity of the
results to the test used. Thus for all the results -- for all three models and for all three
3
In an earlier paper, Groenewold and Fraser ( 1998), we also addressed this question. The research
reported here extends that work by using an 18-sector data set (compared lo five sectors in the earlier
4
tests -- we found cousistent outcomes. We argue that the contrast between our results
and those reported in the existing literature is more apparent than real since, on the
whole, we find differences in prob values between standard and alternative tests
similar in magnitude to those found previously. The different test outcomes are the
result of the fact that in the US the CAPM restrictions are usually close to being
rejected so that small changes in the prob values may change the outcome of the test.
For our Australian data set, however, we find that the CAPM is very far from being
rejected in almost all cases so that prob values can change very substantially as a
result of using a different test without changing the outcome of the test at
conventional significance levels. We arrive at the conclusion that the use of tests
which take into account departures from the iid-normal assumption do indeed affect
prob values. However, changes prob values are likely to affect the test outcome only
if the standard test produces a test statistic that is close to the critical value.
The structure of the remainder of the paper is as follows. We set out the three
asset-pricing models to be tested in the next section. Section Ill presents the tests
used in the paper. The data are described and tests of the iid-normal assumption are
reported in section IV. The following three sections discuss the results: first for the
unconditional CAPM, then the conditional CAPM and finally the macro-factor
version of the APT. Conclusions are drawn in the final section.
2. The Asset-Pricing Models
We begin with the standard (unconditional) CAPM. Assume that there exists a
risk-free asset with return Rr and N risky assets with returns R; (i=l,2,. . .,N). Denote
the return to the market portfolio of risky assets by Rm. Then the model states that
paper) and by extending the bootstrapping procedure to account for intertemporul dependence.
5
(1)
ECRi) = Rr+ [ECRm)- RrJBi> i=l,2, ... ,N
where E(.) is the unconditional expectations operator and Bi = cov(Rm,Rj}/var(Rm).
Given that Rr is non-stochastic, the model can be written in terms of excess returns,
(2)
i=l,2 ... ,N
where ri
=Ri - Rr and rm= Rm - Rr.
Tests of the above model can be interpreted as
tests of CAPM only if data on the return to the market portfolio are available. If this
is not the case and a proxy such as the return to a stock market index is used, the test
is generally interpreted as the test of the mean-vaiiance efficiency (MVE) of the
market portfolio proxy. We continue to refer to the test as one of the CAPM but keep
this qualification in mind.
An alternative reaction to the unobservability of the return to the market
pmtfolio is to treat this as a latent variable and, following Gibbons and Ferson (1985)
and Campbell ( 1987), relate it linearly to a set of observable variables, say, z 1, •.• ,Zk.
In that case
(3)
where 11 is a random error term. Substituting this into equation (2), we define the
conditional CAPM as:
i=l,2, ... ,N
(4)
where the model restricts the way in which the observable variables zi, ... ,zk enter the
asset-pricing equations.
Equation (4) is not unlike a multi-factor asset-pricing equation such as might
be obtaiued from the Arbitrage Pricing Theory (APT) with macroeconomic or
aggregate factors of the type first put forwai·d by Chen, Roll and Ross (1986). It has
been applied to US data by Chen, Roll and Ross and to UK data by Beenstock and
Chan (1988), Clare et al. (1993) and Clare and Thomas (1994). In the APT
6
interpretation, the restrictions on the asset-pricing equations are somewhat different to
those implied by (4). In particular, if we use the z variables as the macroeconomic
factors in an APT model, the restrictions implied by the APT may be imposed on the
equation for the ith asset as: 4
E(ril
= YiO + Yi1Z1 + Yi2 z2 +... + Yik Zk.
i=l,2,. . .,N
(5)
Yet another rationalisation for the use of aggregate variables in the assetpricing equation of the form of (5) is the consumption-CAPM based on the
intertemporal CAPM of Cox, Ingersoll and Ross (1985) and Balvers et al. (1990). In
this model the z variables incorporated in (5) now have predictive power for excess
returns through their use as information variables in the fonnulation of forecasts of
the conditional covaiiance between asset returns and the growth of marginal utility.
Such forecasts may, in turn, be considered to be perfectly correlated with a
benchmark return as in Campbell (1987). In this interpretation the aggregate
variables typically enter equation (5) with a lag. 5 The predictability of the excess
returns is of interest in that it gives insight into the investor's decision-making process
as well as the functioning of financial markets.
3. Estimation and Testing
Consider the standai·d (unconditional) CAPM first. In terms of excess returns,
the model was written in equation (2) as:
i=l,2 .. .,N
(2)
This suggests a test based on the excess-return form of the market model:
4
See McElroy, Burmeister and Wall ( 1985) for a derivation. Applications using US data are by
McE!roy and Burmeister ( 1988), to Singapore by Ariff and Johnson (1990) and to Australian data by
Groenewold and Fraser ( 1997).
7
r;, =a;+ ~i rm,+ E;,,
(6)
i=l ,2, ... ,N and t=l ,2, ... ,T
where the CAPM imposes the restriction that a; = 0 for all i. If we make the standard
assumption that the E;,s are iid-normal, we can test H 0 :a;=O using a Wald test based on
the statistic
W = a'[var(aff 1a
(7)
where a is the vector of estimated a;s which has covariance matrix var(a). If var(a) is
replaced by a consistent estimator, Wis asymptotically
x2-distributed with N degrees
of freedom under H 0 . An alternative to the Wald test is the GRS test due to Gibbons,
Ross and Shanken (1989) who demonstrated that an adjusted version ofW has an
exact F distribution under Ho and the iid-normal assumption:
F
= [(T-N-1)/TN] W -
(8)
FtN, T-N-11
Since this test does not rely on large-sample prope11ies, a comparison of W and F test
outcomes will give us an indication of the effect of applying an asymptotic test to a
finite sample. This is particularly important in tests of asset-pricing models which are
typically tested over relatively short peiiods in order to minimise the effect of
parameter changes over time. Thus, it is common to estimate CAPM
~s
using
monthly data over five-year periods and Campbell, Lo and MacKinlay (1997) show
that there are likely to be considerable size distortions in the application of asymptotic
tests to sample of size 60, effects which largely disappear when sample size reaches
240 and larger. In typical applications monthly data are available for at least 240
months (our sample is 282 months long). However, the well-known problem of
paran1eter drift means that the parameters are usually estimated and the models tested
over five-year periods - a sample of 60 monthly observations.
5
See Keim and Siambaugh (1986), Campbell (1987), Fama and French (1988), Fama (1991), Clare and
Thomas ( 1992) and Clare et al. ( 1997).
8
Both the above tests rely on the iid assumption, and are therefore not valid in
the face of heteroskedasticity, some evidence for which was found in our analysis of
the data and the market-model residuals in the previous section. An increasingly
popular alternative is to use Hansen's (1982) J test associated with the GMM
estimator. MacKinlay and Richardson (1991), using US data, and Faff and Lau
(1997), using Australian data, found that test outcomes may change if the J statistic is
used instead of standard W and F statistics.
The GMM procedure estimates the parameters of the model by using the
sample counterpart to a set of population moment restrictions implied by the model.
In the present case this is simply that the error vector is uncorrelated at each t with the
excess return to the market portfolio and a constant. More formally, if we denote by
z, the vector (l,r11n)' and bye, the vector of errors for period t, the moment condition is
that E[z,®e,]=0. The sample counterparts to these conditions provide 2N restrictions
which are exactly enough to identify the 2N parameters O:; and ~; (i=l,2, ... ,N) and
will, in tl1e linear case, produce OLS estimates. If we impose the restriction that O:;=O
for all i=l,2, ... ,N not all the moment conditions can be satisfied and the GMM
estimator chooses the parameters to minimise the following quadratic crite1ion
function:
(9)
where Wis a weighting mauix which in practice takes various fonns. In the case
where there are more moment conditions than parameters, the resulting overidentifying restrictions can be tested using the optimised value of the critelion
function. Hansen shows that under quite weak conditions ( "that excess returns are
stationary and ergodic with finite fourth moments", Campbell, Lo and MacKinlay
1997, p.208) the resulting J statistic is asymptotically dismbuted
x2 with N degrees of
9
freedom. This therefore provides a method of testing the asset-pricing model in the
face of non-iid-normal errors and a comparison of the W and J tests will provide us
with an indication of the effect of the violation of the underlying iid assumption of
asymptotic tests.
An alternative to the use of the GMM-J test is to use the bootstrapping
procedure to derive critical values for standard test statistics. Bootstrapping proceeds
by drawing repeated samples with replacement from the residuals of the model. These
residuals are then used to generate new "data" which are used to re-estimate the
model. With a large number of repeated samples it is possible to build up an
empirical distribution of the estimated paran1eters or test statistics which are used for
inference. If there is intertemporal dependence in the errors a block-bootstrapping
procedure if generally used since the re-sampling procedure must preserve the
intertemporal structure of the data. This is a matter we tum to later in this section
after a discussion of the simpler case of bootstrapping based on independent errors.
Since the bootstrapping method is based on sampling from the actual
residuals, it overcomes both the small-sample problem and the non-normality of the
errors. Chou and Zhou ( 1997) report a recent application of bootstrapping to tests of
asset-pricing model and find that the bootstrapped test rejects mean-variance
efficiency more often than the standard GRS test. Further, their Monte Carlo
evidence suggests that in general the bootstrapping procedure is superior to the GRS
test in terms of size but not necessarily in terms of power.
There are several ways of proceeding to bootstrap tests of asset-pricing
models. They differ both as to resampling scheme used and statistic to be sampled.
Little is known about the theoretical properties of the various alternatives and Monte
10
Carlo evidence is only beginning to emerge. Based on the discussion in Efron and
Tibshirani (1993) and Li and Maddala (1996), we use the following scheme:
(1) Estimate the market model, equation (6), using OLS, reserve the residuals and
compute the Wald statistic and the F statistic
(2) Draw a sample with replacement from the residuals reserved in step (1) and
generate a new set of return series under the null hypothesis (i.e. assuming that
ai=O for all i).
(3) Estimate the market model based on the generated returns from step (2) and
calculate the Wald and F statistics.
(4) Repeat steps (2) and (3) 10,000 times.
(5) Build up an empirical distribution of the Wald and F statistics from the results in
(4) and calculate critical values to which the values of the test statistics from step
(1) are compared.
The bootstrapping procedure just outlined is based on the implicit assumption
tlmt the model errors which are sampled are independent. If this is not the case then
the basic bootstrap method is inappropriate since it fails to preserve the
interdependence present in the data. A common alternative is the block bootstrap in
which the residuals in step l above are divided into B sets of L consecutive
observations or blocks of residuals (B.L=T) and a sample ofB blocks is drawn with
replacement. The sampled blocks are then combined to create a new series of T
observations and the remainder of the steps are undertaken as before. Various more
complicated alternatives have been suggested; for a recent discussion see Berkowitz
and Kilian (1996) and Davison and Hinkley (1997, Ch.8). Clearly there is a trade-off
in choosing block length, L, between blocks long enough to preserve the dependence
11
in the data and having a sufficient number of blocks from which to sample so as to
generate sufficient variability in the generated series. There is little guidance in the
literature in the choice of block size and we experiment with various alternatives.
The above tests deal only with the unconditional CAPM but similar W, F and J
tests and bootstrapping procedures may be applied to the conditional CAPM and the
APT. Consider first the conditional CAPM in which we assume that the market
portfolio is not observed but is linearly related to a set of observable variables. Recall
that in this case the restrictions of the CAPM may be written as:
E(fi) = ~;(E(rm)),
where
(3)
Thus the equivalent to the market model is now 6
(10)
Not all of the parameters of this model can be identified and we follow Campbell
(1987) and set a;= 0 for all i and
~ 1=1.
Then the restrictions can be tested in the
framework:
(11)
by testing H 0 : YiiilYID=···='fik/y1k for i=2,3, ... ,N. We again use three tests: the Wald,
GRS and GMM-J tests, comparing the calculated test statistics to both theoretical and
bootstrapped critical values. It should be noted that the GRS results were not derived
in the framework of a model such as the conditional CAPM and therefore the
distJibutional result for the GRS F statistic is not strictly valid. We apply it
Note that ri 1refers to the return obtained from holding and asset fro1n period t tot+ Iwhile z, refers to
period t so that the z1s 1nay be considered kno\vn when the expectation ofr1111 is being forn1ed and so
they muy be thought of as Jugged instruments.
fi
l2
nevertheless and interpret the ORS statistic as a general small sample correction of the
W statistic.
In the case of the macro-factor version of the APT, the unrestricted model is
again given by equation (11) but the restrictions on the coefficients are
Ym = (:n:1yi1+:n:2y;2+ ... +7rkY1k) for all i = 1,2, ... ,N and for some constants, :n: 1, :n:2,.•. ,:n:k
implying N-k restrictions. The ORS test will again be interpreted as a small-sample
coITection of the W test.
4. TheData
We used monthly data for 18 Australian industrial sectors for the period January
1973 to June 1998 obtained from Datastream. For each sector we calculated the
continuously-compounded rate of return which includes both capital gains and
dividends. The sectors for which data are available for the entire sample period are
listed in Table 1. Dataslream has a further 10 sectors at this level of disaggregation
but for none of them are data available for the whole sample period. To calculate the
market return we used Datastream's Total Markel index, again using an index which
includes both capital gains and dividends.
Table 1 contains summary statistics for the returns for the 18 sectors. The
skewness and excess kurtosis statistics are both asymptotically standard-nornially
distributed under the null hypothesis of normality of returns. The normality statistic is
the goodness-of-fit version; it is
x,2271
distributed under the null of normality. The
next three columns in the table report the previous three statistics for the sample
omitting the October 1987 observation. It is often found that the events surrouhnding
the Crash have a disproportionate effect on the properties of the data. The final two
13
columns of the table report ARCH(6) and ADF statistics; the ARCH (6) statistic is
Xfoi distributed in the absence of heteroskedasticity.
The results in Table 1 show widespread departures from normality in the returns
- there is significant skewness in 16 of the 18 sectors and excess kurtosis in all 18
sectors. The goodness-of-fit test rejects normality for 10 of the 18 sectors. There is a
noticeable effect on these results of the October 1987 observation, however; the
incidence of skewness falls to 6/18 while the excess kurtosis statistics fall markedly in
magnitude although there is still evidence of excess kurtosis for all sectors. The
normality statistic is still significant in nine of the 18 sectors. There is some evidence
of heteroskedasticity, at least of the ARCH type, with six of the series exhibiting
significant ARCH(6) effects. The ADF results indicate that the null hypothesis of
non-stationarity in the returns can be rejected for all sectors.
Since the tests used in assessing asset-p1icing theories generally depend on the
iid-normal nature of the model errors, rather than the data as such, we also investigate
the properties of the residuals of the market model specified in terms of excess
returns. Further, since the choice of bootstrapping procedure depends on the presence
or otherwise of serial correlation in the model errors, we also repmt serial correlation
coefficients and the Box-Pierce statistic. These are reported in Table 2. The first six
columns of the table report the autocorrelation coefficients of order 1 to 6 and the next
column contains the Box-Pierce statistic for the null hypothesis that the first six serial
correlation coefficients are jointly zero. Approximately 10% of the autocorrelation
coefficients are significantly different from zero at the 5% level, indicating come
autocorrelation. This is confirmed by the Q statistic which is significant at 5% for 6
of the 18 sectors. There is, therefore, scattered but not widespread evidence of
autocorrelation.
14
The next column in Table 2 has Engel's statistic for sixth-order ARCH. This
shows that ARCH is more prevalent in the residuals than it was in the original data
and is somewhat more widespread than autocorrelation. The final four columns
report statistics relevant to the question of the normality of the errors: statistics for
skewness, excess kurtosis and for two tests of normality, the Jarque-Bera test and the
goodness of fit statistic. The results for the normality tests is that there is widespread
evidence of non-normality; indeed, only one sector passes all four tests. We may
conclude therefore that there is some evidence of intertemporal dependence and
widespread evidence of departures from normality in the residuals from the market
model.
In addition to share-price index and dividend-yield data, we also used macro
variables in our empirical work as instmments for the return on the market portfolio in
the conditional CAPM and as macro risk factors in the macro-factor version of the
APT. Our choice of macro variables for this purpose was based on the hypothesis that
at the aggregate level, risk is influenced by three classes of factors - real domestic
activity, nominal domestic factors and foreign variables. Changes in any of these
variables may conceivably influence agents' risk perceptions and therefore the betas.
Data limitations restricted the choice of macro variables since several obvious
choices (such as GDP, CPI and average weekly earnings) are not available at the
monthly frequency of our index data. Therefore, in the first group of macroeconomic
factors we experimented with an index of production, employment and the
unemployment rate; for the nominal domestic influences we used an index of
manufacturing prices, award wages, M3, M6 and the 90-day bank-accepted bill rate;
foreign influences were captured by three alternative exchange-rate measures (in
tern1s of the US dollar, the Japanese Yen and a trade-weighted basket of currencies)
15
and the deficit on the current account of the balance of payments. In our estimates of
beta based on non-overlapping sub-samples, we used two-year averages of macro
variables and extended our set to include real GDP and the CPI inflation rate. The
variables chosen were broadly similar to those used in other studies of the macrofactor APT such as Chen, Roll and Ross (1986) for the US, Clare and Thomas (1994)
for the UK, Ariff and Johnson (1990) for Singapore, Martikainen (1991) for Finland
and Groenewold and Fraser (l 997) for Australia.
5. Results: Unconditional CAPM
5.a Standard Tests
We begin by discussing the results obtained from standard tests of the
traditional (unconditional) CAPM or, alternatively, the MVE of our chosen market
index. The results are reported in Table 3. Panel A contains the estimated market
model equations for the 18 sectors in the data set and panel B has the test results.
All the betas are significant, of the correct sign and of a plausible magnitude.
None of the intercept coefficients is significant. In general, the R2s indicate that a
substantial proportion of the variation in individual sector returns can be explained by
movements in the market return.
In panel B of the table we report vmious test statistics and their corresponding
prob values. The Wald statistic is
x2(IB) distributed and the calculated value of
19.4487 has a prob value of0.3647, clearly indicating that the null hypothesis
(H0 :ai=O, i=l,2, ... ,18) cannot be rejected, i.e. that the CAPM restrictions comfortably
compatible with the data. If we malce the small-sa!11ple adjustment to the Wald to get
the GRS statistic, the prob value is noticeably higher producing a stronger inability to
reject the model. The conclusions are consistent with consistent with the unrestricted
16
estimates of the market model reported in panel B where all the a;s can be seen to be
insignificant.
These outcomes of the standard tests are in some contrast to those reported for
the US by MacKinlay and Richardson (1991) and Chou and Zhou (1997) where prob
values are generally close to conventional significance levels. Our results are,
however, consistent with earlier Australian results reported by Faff (1991) and Wood
(1991) for cases similar to ours. Faff cannot reject the CAPM restrictions for his full
sample nor for any sub-samples when he uses a value-weighted market index. This is
so whether he uses an F test or a likelihood-ratio test. Similarly, Wood cannot reject
the CAPM restrictions with an F test when using industry-based portfolios, a valueweighted market portfolio and continuously-compounded returns (as we do).
However, in both papers the results are sensitive to the data used. In particular,
Wood's results change if he uses discrete returns and/or individual size-based
portfolios (as MacKinlay and Richardson and Chou and Zhou both do). Further,
Faffs results are sensitive to the choice of market index: if he uses an equallyweighted index then the CAPM is widely rejected by the likelihood-ratio test,
although less often if a small-sample adjustment is made to this test.
The results in Faff and Lau (l 997) are in some contrast to ours (and, so, to
those in Wood and Faff). If they use the excess-return form of the market model,
industry p01ifolios and a value-weighted market index (which is the closest
combination to our own specification), both F and Wald tests result in rejection of the
CAPM restrictions for the full sample and for two of the three snb-samples
considered. Generally, therefore, our results differ from those rep01ied for the US but
are broadly consistent with the Australian resnlts although the latter do show evidence
17
of sensitivity to the method of portfolio construction and the choice of market
portfolio.
It is possible to formulate the CAPM restrictions on the market model
somewhat differently if we use gross rather than excess returns. Then the market
model has the form
(12)
Ri1 = Ui +pi Rm1 + Ei1
and equation (1) implies the restlictions that CXj=a1Cl-Pil/(1-P 1l for i=2,3, ... ,18.
Alternatively, equation (12) may be written as
(13)
Ril = YiCl-Pil +pi Rm,+ Ei1
so that the relevant null hypothesis under the CAPM is H 0 :yi:yfor i=2,3, ... ,18. This
is the fo1m tested by Gibbons (1982); Faff and Lau (1997) test it under the heading of
the 'zero-beta CAPM' although it is clearly not restricted to the 'zero-beta CAPM' as
long the lisle-free rate can be assumed constant. In contrast to the tests reported in
Table 3, the tests based on equations (12) and (13) do not require data for the lisle-free
rate but instead estimate it (or the expected return to the zero-beta portfolio) on the
assumption that it is constant. Tests of these hypotheses using a Wald test result in
x"
statistics of 14.2571and13.5697 respectively with corresponding prob values of
0.6488 and 0.6973. A J test of the restrictions implied by (13) and the accompanying
Ho produce a test statistic of 15.228 with a prob of 0.5791. So, our conclusions are
not specific to the type of test used.
All the above should not be taken to imply that the betas successfully expain
cross-section variation in sector mean returns. A standard two-step test of the CAPM
results in a second-stage cross-section regression:
Ri = 0.0121 + 0.0024 bi ,
(2.39)
R2=0.0087
(0.38)
18
where R; is the sample mean return for sector i and b; is the estimated beta for sector i.
As is often the case, the betas appear to have no explanatory power for cross-section
variation in mean returns despite the fact that the standard tests decisively fail to reject
the model in the single-stage tests reported in Table 3. 7
5.b Tests which account for the violation of the iid-normal condition
Consider now the effects of non-iid-normal errors. We report the J statistics
deiived from the GMM estimator of the restiicted system first. Like the Wald, it is
x2<181 distributed and is smaller than the corresponding Wald statistic producing a
correspondingly larger prob value than both the Wald and GRS statistics, making it
even Jess likely that the restiictions implied by the CAPM should be rejected in this
particular case. An alternative to the use of the J test is to bootstrap the Wald and
GRS tests. This should account both for violation of standard assumptions regarding
the nature of the error process as well as any small sample problems. The result is
prob values very close to the theoretical probs suggesting that the test outcomes are
essentially unaffected by the combination of the non-no1mal properties of the en-ors
and small-sample considerations.
At first sight, the conclusion that the test outcomes are unaffected by the
adjustment for non-iid-normal errors seems in sharp contrast to the US papers cited
above where the opposite was often the case. However, this contrast is more apparent
than real since both MacKinlay and Richardson and Chou and Zhou reported prob
values for standard tests close to conventional significance levels so that relatively
small changes in prob values could change test outcomes. In contrast, our results sow
large prob values for standard tests so that changes in prob values in excess of 20
7
See, e.g., the full sample results in Table 2 ofGroenewold and Fraser (1997).
19
percentage points (as observed when comparing the Wald and J tests) have no effects
on test outcomes even though they are larger than any reported by MacKinlay and
Richardson and greater than most reported by Chou and Zhou. Comparing to the
existing Australian literature, our GMM-J test statistics have magnitude comparable
to those in Faff and Lau (1997) regardless of whether e they test the excess-return
form or the gross-return form of the model.
We can conclude that the market model fits the data well for most sectors and
that the CAPM restrictions on the model are not rejected. The failure to reject is the
outcome of all the tests used. The use of tests which are robust in the face of
departures from the iid-normal assumption does not change the outcome.
As demonstrated in much of the literature, CAPM is more likely to hold over
short than long periods due to intertemporal beta instability (see, e.g., Groenewold
and Fraser, 1997, and references there). We therefore proceed to assess the
robustness of the conclusions we have drawn so far by testing the CAPM over shorter
sub-samples, following convention and choosing five-year periods for this purpose.
The results are reported in Table 4.
The results in Table 4 provide some support for the common finding that the
CAPM is more likely to hold in shorter periods. For three of the five sub-pe1iods the
prob values for the tests are higher than they are for the full sample. This is
particularly tme for the earliest two sub-periods where the probs are close to 1. For
the first two sub-periods there is very little difference between the probs for the Wald,
GRS and J tests, whether they are theoretical or bootstrapped although the theoretical
probs are somewhat larger for the GRS test.
The difference between the Wald and J on the one hand and the GRS probs on
the other is more pronounced for the 1988-92 sub-period where the GRS prob is
20
almost 30 percentage points greater than that for the J statistic. This is not sufficient,
however, to reverse the outcome of the test given the relatively small test statistics
and large prob values.
The remaining two sub-periods, 1983-87 and 1993-98, are Jess favourable to
the model. For the period 1993-98 the model is still rejected at conventional
significance levels but the GRS and J statistics have much larger probs than the Wald
test. The results for the 1983-87 period are the least similar to those for the rest of the
sample, not surprisingly since it includes the share market crash of October 1987. For
this sub-period the model is decisively rejected at the 5% level by the Wald test but
not by the GRS and J tests. Hence both the small sample transformation from the
Wald to the GRS and the use of the J tests to account for non-iid-normal errors
reverse the outcome of the test. However, bootstrapping the GRS lest reverses the
outcome again. Thus we find that, consistent with the results reported by MacKinlay
and Richardson (1992) and Chou and Zhou (1997), the test outcome is sensitive to the
tests used (and in particular to the adjustment for non-iid-normal errors) if the model's
restrictions are close to being rejected by the conventional tests.
The unusual nature of the results reported for the 1983-87 sub-period begs the
question of the influence of the October 1987 observation. Ifwe omit 1987 altogether
from this sub-period, the outcomes are not materially altered: the Wald strongly
rejects the model but the GRS and J tests do not. If we omit only the last 6 months
from the sample but add the last six months of 1982 to the beginning to preserve a 60
period length, the results are again similar- the probs for W, GRS and J are 0.0137,
0.2508 and 0.1094 respectively. Hence there is more that is unusual about this subperiod than simply the October 1987 observation.
21
In general, we can conclude that the CAPM cannot be rejected by
conventional Wald and GRS tests even though the betas explain little cross-section
variation in mean returns. For the full sample and for most sub-periods this
conclusion is not reversed by the use of tests which are robust to the presence of noniid-norrnal errors despite the fact that different tests often produce noticeably different
prob values. This is because, on the whole, the prob values for the standard tests are
very far from conventional significance levels so that substantial changes can occur in
prb values with no change in test outcome. Only for the 1983-87 sub-pe1iod is the
model rejected by the Wald test. This conclusion is reversed if the more appropiiate J
test is used but bootstrapping does not affect the outcome. The unusual results for the
1983-87 period do not seem to depend on the inclusion of the October I 987
observation.
5.c Bootstrapping dependent residuals
We return now to the matter of the bootstrapping procedure. The bootstrapped
results reported in Table 3 and 4 were all derived from a sampling procedure based on
the assumption that the model errors are independent. However, we found some
evidence of both autocoffelation and ARCH. Hence, it is appropriate to assess the
sensitivity of our results to the independence assumption. We do this by recomputing the bootstrapped prob values for the full sample using a block
bootstrapping procedure in which we sample blocks of consecutive residuals rather
than individual residuals.
We begin by considering block length. There is a clear trade-off in the choice
of block length: the longer the length of the block the more likely we are to be able to
preserve the interternporal dependence in the data but the fewer blocks we have and
22
so the less variation there will be in our sampled time-series. Since there is little
guidance in the literature on this choice, we experimented with block lengths of 3, 6,
12 and 24 observations (reducing the end-point of the sample to 1997 giving 288
rather than 306 observations). For each block length we sampled blocks, created
1000 new time-series and examined the AR and ARCH characteristics of these
artificial data. The results are reported in Table 5. The prob values for the Wald
statistics are reported in the last line of the table.
In Table 5 the first column of figures provides the AR and ARCH statistics for
the data. These differ somewhat from those reported in Table 2 reflecting the slightly
shorter sample period. There is evidence of AR in five of the 18 sectors and evidence
of ARCH in eight of the 18 sectors. The results show that longer block sizes are
needed for the replication of the ARCH than for the AR characteristics of the data. A
block size of six observations produces AR in five sectors, four of which also have
AR in the original data while with this block size only four sectors have a significant
ARCH statistic compared to eight in the original data. Even a block size of 24
produces ARCH characteristics in only seven of the 18 sectors but produces
significant autocorrelation in 13. Thus the choice of block size must be a
compromise. This does not present us with a serious dilemma, however, since the last
line of the table shows that the prob values for the Wald test are always considerably
larger than conventional significance levels so that the outcome of the test of the
CAPM restrictions is never affected by the choice of block length.
If we turn to the sub-sample results which we discussed previously and
reported in Table 4, it is clear that for only two of the sub-samples is it at all likely
that the use of an alternative bootstrapping procedure might result in a different test
outcome, viz. the 1983-87 and 1993-98 periods. We therefore re-computed the prob
23
values for the Wald test for these periods (reducing the 1993-98 period to 1993-97) by
using the block bootstrapping method. Given that there are only 60 observations in
each sub-period, we restrict block size to 12. In each of the sub-periods there is
considerably less evidence of AR and ARCH. AR is significant in only two and four
of the 18 sectors in the 1983-87 and 1993-97 periods respectively and ARCH is
present in three sectors in 1983-87 and not at all in 1993-97. With a block length of
12 observations, the bootstrapped prob for the Wald statistic is 0.0050 for 1983-87
and 0.4730 in 1993-97. In neither case, therefore is there a change in test outcome as
a result of sampling blocks rather than individual residuals.
We conclude that the failure of the bootstrapping procedure to reverse the
outcomes of any of the model tests was not because of ignored temporal dependence
in the model residuals.
6. Results: The Conditional CAPM
Consider now the results of similar tests performed on the conditional CAPM.
The estimation and test results are reported in Table 6. The first feature of the
estimated equations which stands out is their poor explanatory power. Compared to
an average R 1 of approximately 0.45 for the unconditional CAPM equations, the R 1s
in the present case are all below 0.1 and approximately half of them are under 5%.
Clearly, the two vruiables BB90 and RUS are poor predictors of the return to the
market portfolio. Nevertheless, the coefficient of BB90 is significant at the 5% level
for all but four of the sectors and RUS is significant in half of the equations.
The tests of the restrictions implied by the model set out below equation (10) are
reported in the lower part of the table. Both the Wald and GRS tests clearly fail to
reject the restrictions with prob values very close to 1. If we move to the GMM-J test
24
to allow for departures from the iid-normal assumption, we find a very dramatic fall
in the prob but given the very high prob values for the Wald and ORS tests, this fall is
not sufficient to change the test outcome. This conclusion is supported by the
bootstrapped test results. 8 Bootstrapping the Wald and ORS tests also reduces the
prob value by a considerable amount but, again, the change in the probs produced by
the use of the bootstrap is not sufficient to reverse the test outcome.
In contrast to the results for the unconditional CAPM, these results are largely
repeated for the sub-periods reported in Table 7. In all sub-periods the prob values for
the Wald and ORS tests are very close to 1. They are reduced somewhat by the use of
bootstrapping but the test outcome is not affected. Jn all sub-periods the value of the J
statistic is considerably larger than the Wald but the prob value does not approach
conventional significance levels for any of the sub-periods.
The results obtained for the conditional CAPM are in some contrast to those
reported for the unconditional CAPM reported in the previous section. The results for
the conditional CAPM are more uniform across sub-pe1iods but less consistent across
test statistics. However, despite considerable variation in the prob values across tests,
the test outcome was never reversed by the use of a test which accommodates
departures from the iid-nom1al assumption.
7. Results: The Macro-Factor APT
Consider, finally, an alternative interpretation of the conditional CAPM- the APT
with macro factors. The explanatory variables are the same for the two models but
8
Only bootstrapping based on !he sampling of individual residuals was used for tests of the conditional
CAPM and APT given the conclusion reached in the previous section that block-bootstrappping did not
affect test outcomes and our finding for both the conditional CAPM and the APT that standard Wald
and F statistics had prob values very close to I.
25
the interpretation is different, resulting in different restrictions. The results for the
APT are reported in Table 8.
The top panel in the table reports the model with the APT restrictions imposed.
The coefficients of the two factors are generally significant and are similar in
magnitude to those obtained from the unrestricted system reported in Table 6. The
estimates of the additional parameters, rc 1 and rc2, are insignificant which is not a
surprising result in light of the low I-ratios of the intercepts in the unrest1icted
estimates of the equations.
The lower panel of the table reports test statistics for tests of the APT restrictions.
The outcomes of the Wald and GRS tests are very similar: both have prob values in
excess of 90%, indicating that the restrictions do not violate the data. The effect of
using tests which account for departures from the iid-nomial assumption result in
considerably different prob values. The use of the J statistic produces a prob value
approximately 20 percentage points lower than the Wald and GRS statistics.
However, as in many previous cases, the original prob is so far from common
significance levels that even a substantial change in the prob value does not alter the
test outcome. Similar but larger effects are evident when we use bootstrapped probs
for the Wald and GRS tests: the bootstrapped probs are less than half the theoretical
ones but still large enough that the null hypothesis cannot be rejected at conventional
significance levels. Table 9 reports the sub-period results for the APT. They are
quite smilar to those reported for the full sample in Table 8. The J test probs are
always smaller than those for the Wald and GRS tests but not sufficiently so to
reverse the original outcomes and the bootstrapped probs are always smaller than their
theoretical counterparts but, again, not by a large enough margin to change the test
outcomes. In all cases reported the APT restrictions cannot be rejected.
26
8. Conclusions
In this paper we have been concerned with the effects on tests of asset-pricing
models of the violation of the assumptioo that model errors are independently and
identically normally distributed. We have explored these effects for three assetpricing models using an 18-portfolio Australian share price data set. We have
computed traditional tests, the Wald and GRS tests and their theoretical prob values
and then investigated the effects on the test outcomes of accommodating the non-iidnorrnal errors by using the J test associated with the GMM estimator and by
computing bootstrapped prob values for the standard tests. Our overall finding was
that the use of appropriate tests generally lead to substantial changes in prob values at least as large as those rep011ed in recent US studies by MacKinlay and Richardson
(1992) and Chou and Zhou (1997). However, because all three models were
generally comfortably accepted by the data, even large changes in prob values had no
effect on the test outcomes. This is in contrast to the two US studies which often
found prob values close to the significance level so that even small changes in probs
could reverse the test outcome. The moral is: use standard tests if they are the most
convenient but if the resulting prob values are close to the chosen significance level
check the result by using a more appropriate test (of course, only if the model errors
are non-iid-normal).
27
References
Affleck-Graves, J. and B. McDonald (1989), "Non-Normalities and Tests of Asset
Pricing Theories", Journal of Finance, 44, 889-908.
Ariff, M. and L.W. Johnson (1990), "Ex Ante Risk Premia on Macroeconomic
Factors in the Pricing of Stocks: An Analysis Using Arbitrage Pricing
Theory", Chapter 18 in M. Ariff and L.W. Johnson, Securities Markets and
Stock Pricing, Longman, Singapore, 194-204.
Balvers, R. J., T. F. Cosimano and B. Mc.Donald (1990), "Predicting Stock Returns
in an Efficient Market", Journal of Finance, 65, 1109-1128.
Beenstock, M. and K.Chan (1988),"Economic Forces in the London Stock Market",
Oxford Bulletin of Economics and Statistics, 50, 27-39.
Berkowitz, J. and L. Kilian (1996), "Recent Developments in Bootstrapping Time
Series", Finance and Economics Discussion Series No. 1996-45, Division of
Research and Statistics and Monetary Affairs, Federal Reserve Boards,
Washington, D.C.
Campbell, J. Y. (1987), "Stock Returns and the Term Structure", Journal of Financial
Economics, 18, 373-399.
Campbell, J.Y. (1993), "Intertemporal Asset Pricing without Consumption Data",
American Economic Review, 83, 487-512.
Campbell, J. Y., A. Lo, and C. MacKinlay (1997), The Econometrics of Financial
Markets, Princeton University Press.
Chen, N., R. Roll and S.A. Ross (1986), "Economic Forces and the Stock Market",
Journal of Business, 59, 383-403.
Chou, P. H. and G. Zhou (1997), "Using Bootstrap to Test Portfolio Efficiency",
paper presented to the Fourth Asia-Pacific Finance Association Conference,
28
Kuala Lumpur, July 1997.
Clare, A., R. O'Brien, S. Thomas and M. Wickens (1993), "Macroeconomic Shocks
and the Domestic CAPM: Evidence from the UK Stock Market", Centre for
Empirical Research in Finance, Brunel University , Discussion Paper 93-02.
Clare, AD., P.N. Smith and S.H. Thomas (1997),"UK Stock Returns and Robust Tests
of Mean Variance Efficiency", Journal of Banking and Finance, 21, 641-661.
Clare, A.D. and S.N. Thomas (1992), "International Evidence for the Predictability of
Bond and Stock Returns", Economics Letters, 40, 105-112.
Clare, A.D. and S.N. Thomas (1994),"Macroeconomic Factors, the APT and the UK
Stock Market", Journal of Business Finance and Accounting, 21, 309-330.
Cox, J., J. Ingersoll and S. A. Ross (1985), "An Intertemporal General Equilibrium
Model of asset Prices", Econometrica, 53, 363-384.
Davison, A. C. and D. V. Hinldey (1997), Bootstrap Methods and their Application,
Cambridge University Press, Cambridge, UK.
Efron, B. and R. Tibshirani (1993 ), An Introduction to the Bootstrap, Chapman and
Hall, New York.
Faff, R. W. (1991), "A Likelihood Ratio Test of the Zero-Beta CAPM in Australian
Equity Returns", Accounting and Finance, 31, 88-95.
Faff, R. W. and S. Lau (1997), "A Generalised Methods of Moments Test of Mean
Variance Efficiency in the Australian Stock Market", Pacific Accounting
Review, 9, 2-16.
Fama, E. (1991), "Efficient Capital Markets II", Journal of Finance, 46, 1575-1617.
Fama, E., and K. R. French (1988), "Dividend Yields and Expected Stock Returns",
Journal of Financial Economics, 22, 3-25.
29
Gibbons, M.R. (1982), "Multivariate Tests of Financial Models: A New Approach",
Journal of Financial Economics, 10, 3-27.
Gibbons, M. and W. Ferson (1985), "Tests of Asset Pricing Models with Changing
Expectations and an Unobservable Market Portfolio", Journal of Financial
Economics, 14, 217-236.
Gibbons, M., S. Ross and J. Shanken (1989), "Testing the Efficiency of a Given
Portfolio", Econometrica, 57, 1121-1152.
Groenewold, N. and P. Fraser (1997), "Share Prices and Macroeconomic Factors"
Journal of Business Finance and Accounting, 24, 1997, 1367-1383.
Groenewold, N. and P. Fraser (1998), "Tests of Asset-Pricing Models: How Important
is the iid-Normal Assumption?'', Discussion Paper No. 98-20, Department of
Economics, University of Western Australia.
Hansen, L. (l 982), "Large Sample Properties of Generalized Methods of Moments
Estimators", Econometrica, 50, 1269-1286.
Keirn, D. B., and R. F. Stambaugh (1986), "Predicting Returns in Stock and Bond
Markets", Journal of Financial Economics, 17, 357-390.
Li, H. and G. S. Maddala (1996), "Bootstrapping Time Series Models", Economenic
Reviews, 15, 115-158.
MacKinlay, A. C. and M. P. Richardson (1991), "Using Generalized Methods of
Moments to Test Mean-V miance Efficiency", Journal of Finance, 46, 511-527.
Martikainen, T. (1991), "On the significance of the Economic Detenninants of
Systematic Risk: Empirical Evidence with Finnish Data", Applied Financial
Economics, 1, 97-104.
McElroy, M.B. and E. Burmeister (1988), "Arbitrage Pricing Theory as a Restricted
Nonlinear Multivariate Regression Model: Iterated Nonlinear Seemingly
30
Unrelated Regression Estimates", Journal of Business and Economic
Statistics, 6(1), 29-42.
McE!roy, M.B., E. Burmeister and K.D. Wall (1985), "Two Estimators for the APT
when Factors are Measured", Economics Letters, 19, 271-275.
Mills, T.C. and J.A._Coutts (1996), "Misspecification Testing and Robust Estimation of
the Market Model: Estimating Betas for the FT-SE Industry Baskets", European
Finance Journal, 2, 319-331.
Richardson, M. P. and T. Smith (1993), "A Test for Multivariate Nonnality in Stock
Returns", Journal of Business, 66, 295-321.
Wood, J. (1991), "A Cross-Sectional Regression tests of the Mean Variance
Efficiency of an Australian Value Weighted Market Pmtfolio", Accounting
and Finance, 31, 96-107.
31
Table 1: Summary Statistics
No. Sector Name
Mean
Variance
Ske,vness
l(urtosis
Normality
Ske\vness
J(urtosis
Normality
(ex. Oct' 87)
(ex. Oct ' 87)
(ex. Oct '87)
ARCH (6)
ADF
1
Other Mining
0.0086
0.0062
-5.819
16.711
33.178
-1.004
3.327
46.993
22.134
-5.299
2
Building Materials & Merch.
0.0080
0.0046
-4.786
13.577
26.759
-0.546
2.469
26.923
5.217
-4.565
3
Chemicals
0.0132
0.0058
-4.382
15.557
38.912
0.466
2.864
51.049
24.530
-3.986
4
Diversified industries
0.0129
0.0043
-4.985
16.411
42.703
-0.227
3.858
27.062
0.607
-4.341
5
Electronic & Elect.
0.0169
0.0075
-5.597
23.634
41.764
1.122
4.477
47.856
8.748
-4.007
6
Engineering
0.0082
0.0056
-4.499
7.145
46.235
-2.680
3.463
32.010
5.340
-4.533
7
Paper & Print
0.0104
0.0046
-2.696
12.050
36.648
1.227
3.075
32.420
16.989
-4.036
8
Brewers
0.0117
0.0085
-5.543
30.303
81.023
-0.034
18.501
59.067
1.318
-3.714
9
Food Producers
0.0096
0.0044
-5.847
14.065
32.603
-2.052
4.103
35.933
9.202
-3.964
IO
Health Care
0.0158
0.0062
4.115
l 1.649
72.946
4.313
I 1.941
75.213
3.165
-4.199
11
Pharmaceuticals
0.0113
0.0053
-4.070
16.524
64.583
-0.217
8.075
60.368
6.842
-4.661
12
Tobacco
0.0164
0.0067
-8.164
20.504
31.219
-2.490
2.181
27.794
2.746
-4.017
13
Media
0.0180
0.0158
-8.398
21.745
69.429
-3.951
9.192
74.542
5.684
-4.519
14
ISupport Services
0.0144
0.0051
-1.000
5.219
40.309
0.288
3.517
34.634
5.880
-5.328
15
!Transport
0.0124
0.0044
-3.166
11.888
23.267
0.145
4.756
25.521
16.796
-3.849
16
!Banks Retail
0.0121
0.0048
-1.669
11.197
29.637
-0.227
9.739
25.982
56.112
-4.864
17
Other Financial
0.0101
0.0030
-27.430
148.592
77.524
0.903
5.449
61.300
2.511
-5.380
18
Property
0.0140
0.0082
-8.40 I
28.985
79.000
-2.222
11.053
70.014
19.344
-4.776
Critical Values (5%): skewness, kurtosis (N(0,1)): J.96; Normality (X.'-17 ): 40.1 l; ARCH (X.'6): 12.59; ADF(l 0%): -.2.57
32
Table 2: Properties of the residuals from the excess-returns market model
No.
Sector
P1
P2
p3
p4
Ps
P6
Q(6)
ARCH Sk
Ku
JB
GF
(6)
1
2
3
4
5
6
7
8
9
JO
JI
12
13
14
15
16
17
18
Other Mining
Building Materials & Merch.
Chemicals
Diversified industries
Electronic & Elect.
Engineering
Paper & Print
Brewers
Food Producers
Health Care
Pharmaceuticals
Tobacco
Media
Support Services
Transport
Banks Retail
Other Financial
Property
Notes:
0.16
-0.05
-0.15
0.00
-0.23
-0.JO
-0.12
-0.16
0.07
0.00
-0.09
0.01
0.00
-0.07
-0.01
0.00
-0.03
0.02
0.05
-0.JO
0.10
-0.07
0.01
-0.10
0.00
-0.01
0.02
-0.04
0.09
0.01
0.00
0.00
-0. JO
0.01
0.06
-0.07
0.03
0.08
0.04
0.09
0.08
0.02
-0.07
-0.01
-0.03
-0.02
0.08
0.03
0.12
0.05
-0.02
0.06
-0.09
0.06
-0.07
-0.07
0.10
0.06
0.00
0.00
0.06
0.03
0.03
-0.02
0.03
0.10
0.04
0.02
-0.02
-0.01
0.17
0.14
-0.02
-0. 14
0.04
0.03
0.01
0.03
-0.08
-0.08
0.02
-0.01
0.05
0.12
0.01
-0.01
0.07
-0.0J
-0.03
-0.04
-0.04
-0.03
0.04
0.09
0.10
-0.0l
0.16
0.12
0.06
-0.03
-0.04
0.01
0.03
0.05
-0.01
-0.04
0.01
-0.17
I 1.21
13.75
15.J 2
7.97
22
7.16
17.73
15. 19
3.2
1.2
8.73
7.51
5.31
3.19
5.06
1.64
13.07
17.78
18.662
l 7.902
6.805
6.987
18.116
19.922
22.715
9.801
10.335
3.967
3.388
5.477
9.877
16.766
4.059
37.632
6.725
21.234
-3.5437
1.64542
0.3861
2.77722
J.76289
0.29083
1.5573 I
-2.44413
0.05946
4.78223
0.78367
-3.18983
-2.87894
-0.91404
0.16977
0.9606
-10.28009
-0.13037
6.7438
1.81674
7.64211
7.83938
4.63061
3.44017
2.19332
19.62558
3.43119
13.6166
7.6396
2.06612
6.25835
3.3719
0.72188
3.12792
39.70248
5.659
54.7154
5.5570
54.7303
65.0342
22.8424
10.8896
6.6671
370.0231
10.7534
197.1667
55.1525
13.7676
44.5850
11.1928
0.4310
9.8007
1599.5212
29.7482
40.74110
31.40740
37.10450
26.37360
30.68430
36.42460
23.84770
58.16780
23.85430
47.19810
60.73400
36.23260
45.54430
36.88180
26.740
26.975
23.405
38.059
The Pi are autocorrelation coefficients and have standard error of 0.06.
Q(6) is the Box-Pierce-Ljung statistic for first- to sixth-order autocorrelation; it has a
distribution which has a 5% critical value of 12.5916.
ARCH(6) is Engle's test for sixth-order ARCH; it has a
distribution with a 5% critical value of 12.5916.
Sk and Ku are tests for skewness and excess-kurtosis and are both distributed N(O, I).
JB is the Jarque-Bera test for normality and is
-distributed with a 5% critical value of 5.9915.
GF is the goodness-of-fit test for normality based on 17 partitions; it is
-distributed with a 5% critical value of 27.5871.
x'cr.i
x'coi
x'm
x'mi
33
Table 3: Tests of Unconditional CAPM
Panel A: The Unrestricted Model:
fjt
= ai + Pi fmt + Eit
i= 1,2, ... ,18; t= 1,2, ... ,304
Industry
Other Mining
Building Materials & Merch.
Chemicals
Diversified industries
Electronic & Elect.
Engineering
Paper & Print
Brewers
Food Producers
Health Care
Pharmaceuticals
Tobacco
Media
Support Services
Transport
Banks Retail
Other Financial
Property
Panel B: Tests of CAPM: Ho:
0'.1
B1
R-
-0.0018
(1.13)
-0.2210
(1.15)
D.0032
(0.94)
0.0030
11.09)
0.0068
(1.70)
-0.0018
(0.54)
0.0004
(0.15)
0.0015
(0.36)
-0.0004
10.13)
0.0063
(1.46)
0.0017
(0.44)
0.0065
(1.63)
0.0074
(1.34)
0.0045
(1.33)
0.0023
(1.02)
0.0021
(0.79)
0.0003
(0.14)
0.0036
(1.09)
1.1156
(44.95)
0.9036
(31.07)
0.7445
(14.61)
0.6903
(16.76)
0.7751
(12.72)
0.7217
(14.23)
0.7385
(18.05)
0.8855
(14.21)
0.6836
115.99)
0.3358
(5.09)
0.4496
(7.78)
0.6607
(10.97)
1.2167
(14.43)
0.6166
(11.91)
0.8099
(23.48)
0.8037
(20.26)
0.5990
(18.26)
1.0660
(21.33)
0.8696
Test Statistic
Prob
w
19.4487
1.0132
15.2518
0.3647
0.4446
0.6446
J
0.4135
0.4810
0.3483
0.4006
0.5181
0.3999
0.4577
0.0788
0.1663
0.2841
0.4073
0.3188
0.6454
0.5754
0.5240
0.6003
0'.1=0'.2= •.. =0'.1s=O.
Test
GRSF
0.7611
Bootstrapped
Prob
0.3707
0.3707
0.6450
34
Table 4: Sub-Sample Tests of Unconditional CAPM
Test
Test Statistic
Prob
Bootstrapped Prob
w
19.4487
0.3647
0.3707
GRSF
1.0132
0.4446
0.3707
J
15.2518
0.6446
0.6450
w
5.3028
0.9983
0.9983
GRSF
0.1997
0.9997
0.9983
J
7.5567
0.9845
0.9950
w
9.5588
0.9454
0.9590
GRSF
0.3629
0.9884
0.9590
J
10.0999
0.9286
0.9670
w
30.5589
0.0324
0.0388
GRSF
1.1601
0.3359
0.0388
J
23.1667
0.1843
0.1670
w
15.168495
0.6504
0.7241
GRSF
0.575840999
0.8965
0.7241
J
15.64992
0.6170
0.7260
w
24.9055
0.1275
0.1502
GRSF
0.9853
0.4915
0.1502
J
17.6604
0.4782
0.5280
1973-1998(6)
1973-1977
1978-1982
1983-1987
1988-1992
1993-1998(6)
35
Table 5: Block-Bootstrapping the Wald test of the CAPM
Statistic
/Sector
Q(6):
Sector I
Sector 2
Sector 3
Sector 4
Sector 5
Sector 6
Sector 7
Sector 8
Sector 9
Sector 10
Sector 11
Sector 12
Sector 13
Sector 14
Sector 15
Sector 16
Sector 17
Sector 18
ARCH(6):
Sector 1
Sector 2
Sector 3
Sector4
Sector 5
Sector 6
Sector 7
Sector 8
Sector 9
Sector 10
Sector 11
Sector 12
Sector 13
Sector 14
Sector 15
Sector 16
Sector 17
Sector 18
Wald Prob
Block Le112th
3
6
()
1
10.72
10.75
15.67
6.38
22.60
8.62
18.77
15.51
5.91
1.31
8.37
8.54
5.00
2.85
5.89
2.08
11.65
17.52
6.09
6.12
5.86
6.08
5.99
5.67
6.21
5.87
6.05
6.10
5.95
5.98
5.85
5.79
6.08
5.82
5.73
5.91
7.61
8.82
16.37
7.94
11.76
10.29
10.76
16.81
7.48
7.15
9.94
6.05
6.96
6.45
6.46
7.08
6.27
7.21
12.41
17.02
6.19
6.64
15.74
18.35
22.61
9.31
10.63
3.70
3.21
5.46
7.76
14.94
3.31
33.88
5.87
16.74
0.3634
4.71
5.72
5.39
4.85
5.68
5.74
5.85
5.15
5.50
5.15
5.56
5.62
5.38
5.71
5.85
5.62
3.94
5.70
0.3780
7.54
6.47
4.99
7.86
11.19
10.20
11.25
6.12
5.75
4.69
8.05
6.61
12.59
7.95
7.35
8.99
4.52
11.27
0.2690
12
24
9.87
12.12
16.45
12.51
13.75
11.96
21.71
15.70
7.03
7.06
15.37
9.32
7.14
7.30
8.93
6.92
9.41
10.24
10.46
13.01
18.43
11.62
24.43
17.44
19.96
17.92
9.86
7.90
18.01
14.23
9.01
11.87
9.59
7.13
10.02
15.99
13.55
17.06
21.63
10.84
27.82
14.96
23.07
18.55
10.67
7.09
15.11
17.64
14.39
10.17
14.19
7.37
17.10
15.59
7.18
9.85
6.24
10. J J
10.30
16.97
22.37
5.63
6.06
6.62
7.63
7.75
10.74
11.24
8.27
17.03
7.57
25.81
0.2250
9.74
12.38
9.93
9.37
13.65
18.66
24.49
5.52
8.43
8.04
6.76
8.30
9.08
12.82
9.35
25.47
6.12
27.97
0.3980
12.16
17.19
8.56
10.18
18.38
20.65
24.90
6.54
11.83
6.60
7.78
9.80
10.63
15.69
9.02
23.90
8.74
27.34
0.511
36
Table 6: Tests of Conditional CAPM
Panel A: The Unrestricted Model: r;, = y;o + y;1 BB90, + Y;2 RUS, + E;,
i = 1,2, .. " 18; l = 1,2,. . .,304
Industry
Other Mining
Building Materials & Merch.
Chemicals
Diversified industries
Electronic & Elect.
Engineering
Paper & Print
Brewers
Food Producers
Health Care
Pharmaceuticals
Tobacco
Media
Support Services
Transport
Banks Retail
Other Financial
Property
'Ym
-0.0030
(0.66)
-0.0037
(0.95)
0.0031
(0.69)
0.0018
(0.46)
0.0062
(1.22)
-0.0023
(0.52)
0.0006
(0.14)
0.0004
(0.08\
-0.0015
(0.39)
0.0058
(1.23)
0.0009
(0.21)
0.0076
(1.57)
0.0036
(0.49)
0.0031
10.74)
0.0013
(0.34)
0.0013
(0.31)
0.0007
(0.24\
0.0020
(0.38)
Panel B: Tests of CAPM: Ho: Y;1IY11
'Yu
-16.6000
(3.31 l
-15.0470
(3.50)
-17.9200
-(3.66)
-15.4430
(3.71)
-9.7591
-(1.73)
-19.3450
(4.04)
-13.5200
(3.10)
-10.4290
(1.74)
-15.8840
(3.76)
-1.1833
(0.23)
-10.0010
(2.12)
-16.2840
(3.08)
-25.3540
(3.19)
-6.5619
(1.41)
-17.6670
(4.23)
-15.3250
(3.43)
-15.9440
(4.60)
-18.6510
(3.23)
y,,,_
R1
0.2460
(2.38)
0.3004
(3.39)
0.1214
(1.21)
0.2351
(2.74)
0.1757
(1.51)
0.1686
(1. 71)
0.0884
(0.98)
0.2248
(1.82)
0.2179
(2.50)
0.0953
(0.89)
0.1318
(1.36)
0.0028
(0.03)
0.5582
(3.42)
0.2015
(2.1 Q)
0.2403
(2.79)
0.1887
(2.05)
0.0495
(0.69)
0.3350
(2.82)
0.0537
Test Statistic
Prob
w
8.3762
0.4363
36.4430
1.0000
0.9792
0.3557
J
0.0478
0.0678
0.0177
0.0614
0.0345
0.0212
0.0650
0.0029
0.0213
0.0306
0.0696
0.0213
0.0806
0.0516
0.0677
0.0592
=Yi2IY12 =y;o/y1 0, i = 2,3,. . ., 18.
Test
GRSF
0.0752
Bootstrapped
Prob
0.8328
0.8328
0.3614
37
Table 7: Sub-Sample Tests of Conditional CAPM
Test
Test Statistic
Prob
Bootstrapped Prob
w
8.3762
1.0000
0.8328
GRSF
0.4363
0.9792
0.8328
J
36.4430
0.3557
0.3614
w
2.1004
1.0000
0.9299
GRSF
0.0791
1.0000
0.9299
J
21.9868
0.9443
0.9930
w
6.3192
1.0000
0.9631
GRSF
0.2399
0.9990
0.9631
J
30.7260
0.6289
0.6847
w
0.7777
1.0000
0.9331
GRSF
0.0295
1.0000
0.9331
]
30.7889
0.6258
0.7397
w
6.8134
1.0000
0.9978
GRSF
0.2587
0.9984
0.9978
J
40.1553
0.2161
0.0721
w
2.3586
1.0000
0.9966
GRSF
0.0927
1.0000
0.9966
J
30.5019
0.6398
0.4875
1973-1998(6)
1973-1977
1978-1982
1983-1987
1988-1992
1993-1998(6)
38
Table 8: Tests of APT
Panel A: The Restricted Model: r;, = 1t1Y;1+1t2Y;2 + y;1 BB90, + Y;2RUS,+ E;,
n:, = -0.3745, 1t2 = -9.8673
(0.01)
{0.01)
Industry
Yn
Y:?,_
Other Mining
-8.7824
(2.52)
-9.5734
(2.78)
-6.6015
12.09)
-8.3284
(2.57)
-5.7502
11.56)
-7.8828
(2.44)
-4.9179
(1.95)
-6.8644
(1.97)
-8.0648
(2.62)
-2.1166
(0.70)
-4.8903
11.83)
-3.8062
(1.16)
-17.1100
(2.62)
-5.5125
(1.86)
-8.9373
(2.67)
-7.3616
(2.39)
-4.6483
(2.16)
-11.0070
(2.53)
0.3337
(3.52)
0.3638
(4.35)
0.2504
(2.73)
0.3160
(3.92)
0.2177
12.07)
0.2996
(3.30)
0.1867
12.32)
0.2605
(2.38)
0.3063
(3.79)
0.0797
(0.84)
0.1856
(2.16)
0.1439
(1.46)
0.6491
(4.18)
0.2089
(2.42)
0.3392
(4.15)
0.2794
(3.31)
0.1765
(2.70)
0.4176
(3.77)
Building Materials & Merch.
Chemicals
Diversified industries
Electronic & Elect.
Engineering
Paper & Print
Brewers
Food Producers
Health Care
Pharmaceuticals
Tobacco
Media
Support Services
Transport
Banks Retail
Other Financial
Property
Panel B: Tests of APT: Ho: Y;o = n:1Y;1+7t2 Y;2, i = 1,2, ... ,18
Test
Test Statistic
Prob
Bootstrapped
Prob
w
8.8429
0.4606
12.6920
0.9197
0.9721
0.6591
0.4022
0.4022
ORS F
J
39
Table 9: Sub-Sample Tests of APT
Test
Test Statistic
Prob
Bootstrapped Prob
w
8.8429
0.9197
0.4022
GRSF
0.4606
0.9721
0.4022
J
12.6920
0.6951
w
0.8857
1.0000
0.8733
GRSF
0.0334
1.0000
0.8733
J
8.3927
0.9363
w
2.6176
0.9999
0.8052
GRSF
0.0994
1.0000
0.8052
J
8.4849
0.9331
w
5.4491
0.9930
0.6716
GRSF
0.2069
0.9996
0.6716
J
15.8210
0.4655
w
6.9788
0.9737
0.7263
GRSF
0.2649
0.9982
0.7263
J
12.6010
0.7017
w
5.7627
0.9905
0.7309
GRSF
0.2266
0.9994
0.7309
J
15.6490
0.4779
1973-1998(6)
1973-1977
1978-1982
1983-1987
1988-1992
1993-1998(6)
40