1.
The Asymmetric Performance of Greek Closed-End Funds:
An Empirical Examination
Panayiotis T. Theodossiou
and
Dimitrios V. Kousenidis
December 2003
2.
First Draft: Comments are Welcome
Corresponding Author: Dimitrios V. Kousenidis, Aristotle’s University of Thessaloniki, Department of Economics, Division of
Business Administration,
54006 Thessaloniki, Greece
Electronic Address: kousenides@hotmail.com
Telephone: +302310997098
+306937028874
The Asymmetric Performance of Greek Closed-End Funds:
3.
An Empirical Examination
Abstract
The present paper examines the selectivity and market timing performance of Greek Closed-End Funds. The paper uses five
alternative performance models that allow separating returns into selectivity and timing components. Two of the models
incorporate information asymmetry across upside (bull) and downside (bear) markets. One such model is the asymmetric
response model (ARM) and the other is a modified version of it developed in the paper. The results of the paper indicate that
some performance models suffer some degree of misspecification. On the other hand, the results depict that professional fund
managers in Greece are unable to outperform and sufficiently time the market. The poor performance of Closed-End Funds in
Greece shows that profitability does not consist one of the reasons that individual and institutional investors hold shares of
managed portfolios.
Keywords: Closed-End Funds, Selectivity, Market Timing, Quadratic CAPM, Dual Beta CAPM, Asymmetric Response
Model, Modified Asymmetric Response Model
5.
1. Introduction
The performance of managed portfolios, such as mutual funds (open-end funds) and investment trusts (closed-end funds) has been
a puzzling topic for finance researchers. The variety of performance models used along with the contradicting results of the
relevant studies clearly imply that no exhaustive answer has yet been given to the question as to whether managers of managed
portfolios possess superior selectivity and timing skills.1
Much of the research in the area has taken the form of identifying alternative performance models that can decompose
portfolio returns into selectivity and timing components. Although most of the models are variants of the CAPM, in several
contextual settings, some models are found to outperform the others. These results are evidenced in both closed-end funds and
mutual funds data sets. Wermers (2000) and Dimson and Minio-Kozerski (1998) provide a review and an analysis of mutual fund
and closed-end fund performance studies respectively.
Despite the fact that the majority of managed portfolios performance studies refer to institutional environments that share
many commonalities (i.e., the US, the UK and the Australian), they often report contradictory results. For example, Wermers
(2000) and Chance and Hemler (2001) find evidence that professional portfolio managers are able to outperform the market.
Moreover, Allen and Tan (1999) and Bers and Madura (2000) report that the good performance of managed portfolios persists in
the long run although the formers observe a mean reversal in performance in the short run.
On the other hand, Connor and Korajczyk (1991), Coggin, Fabozzi and Rahman (1993), Fletcher (1995), and Hallahan and
Faff (1999) report evidence on selectivity but not on timing managerial skills. These studies also report an inverse relation between
selectivity and timing return components. Patro (2001) examining the performance of 45 US based international funds is unable to
find evidence neither on selectivity nor on timing. Bangassa (1999) uses multifactor performance measures for a sample of UK
unit trusts and finds that some of the funds exhibit superior selectivity performance. However, he also fails to document any
superior timing abilities among fund managers. Bollen and Busse (2001) show that the inability of existing studies to observe
significant timing managerial skills is due to the use of monthly-return data sets. They support that when researchers analyze daily
returns professional portfolio managers are shown to possess significant timing abilities.
1
The term selectivity refers to the ability of the manager of the portfolio to pick successful shares; the term timing refers to the ability of the manager of the
portfolio to foresee the movements of the market and adjust the risk of the portfolio towards the right direction (see Fama, 1972). The existence of
managerial skills does not contradict the EMH. Grossman and Stiglitz (1980) provide a version of EMH, which assumes that information is not free.
Therefore, some managers may possess valuable private information that is not shared by the market as a whole. The excess returns of active managers are
then presumed to concur with the information-gathering costs incurred. Lang et.al. (1992), also observe that even if there exists noise in the market, it does
not completely dilute the information content of share prices. In this context they admit that their results describe an environment under which it is not
contradictory to assume both traders’ rationality and private incentives to acquire information.
6.
Dahlquist et al. (2000) and Sawicki and Finn (2002) observe that the performance of managed funds is not unconditional to the
size of their portfolios. In particular, both studies report a size effect in the sense that small-size funds are, on average, found to
outperform large-size funds.
Allen and Soucik (2000, 2002) test for the sensitivity of univariate and multivariate CAPM-based performance models to
different factors. They find that the use of different benchmarking portfolios (market proxies) is the major source of conflicting
results. Moreover, they observe that some univariate versions of the CAPM exhibit a nontrivial degree of misspecification. Finally,
they find that fund managers may or may not display selectivity and timing skills depending on the time horizon that the data set
covers.
Jagannathan and Korajczyk (1986) and Kothari and Warner (2001) test for model misspecification by showing that it is
possible to construct portfolios that display artificial timing ability when no true timing ability exists. They then use a variety of
both univariate and multivariate performance measures to test for selectivity and timing in managed funds and in artificially
constructed portfolios. Their results reveal abnormal performance even in cases of artificial portfolios where none is known to
exist. These results indicate that models of managed portfolio performance may be highly misspecified. Because of this
Jagannathan and Korajczyk advocate the use of specification tests while Kothari and Warner support that multivariate models
exhibit lesser degree of misspecification than univariate models.
Finally, Pedersen and Satchell (2000) argue that studies on the performance of closed-end funds are subject to the small-
sample bias. The small number of observations that is usually available to researchers violates the assumption of normality and
produces biased OLS regression estimates of selectivity and timing return components. What they believe is that accurate
performance rankings can be obtained if researchers employ the Asymmetric Response Model (ARM), initially developed by
Fabozzi and Francis (1977, 1979) and Bawa, Brown and Klein (1981). The ARM has the intuitive appeal that it captures non-
normality in unconditional returns without demanding the use of specific distributional assumptions that incorporate skewness and
kurtosis. Moreover, the ARM nests the one-factor mean-variance CAPM and the lower partial moment CAPM, and produces
positive and negative excess-return betas for up and down markets.
The present study examines the performance of closed-end funds in Greece over a six-year period from 1-1-1997 to
12-31-2002. The performance models used consist of the Jensen (1968) model, the Treynor and Mazuy (1966) model, the
Henriksson and Merton (1981) model, and the ARM. Moreover, the paper expands on the ARM and produces a modified version
of it, which captures differences in both selectivity and timing across bull and bear markets. The modified ARM exhibits many
similarities with the ARM but it has the intuitive property that it offers new insights in the explanation of the selectivity and timing
coefficients. In the present study, the modified ARM is used as an alternative performance model and its validity is tested on an
empirical context.
7.
The results of the present paper depict that the two versions of the ARM exhibit the lesser degree of misspecification among all
the models used to test for portfolio performance. Moreover, it is shown that on average active portfolio managers in Greece are
unable to outperform the market and they posses no superior selectivity and timing skills. On the contrary, it is found that closed-
end funds in Greece display significantly negative selectivity performance in downside markets and positive but insignificant
selectivity performance in upside markets. In addition, the downside systematic risk measure is found higher than the upside
systematic risk measure implying that managers do not switch portfolios’ systematic risk according to market times.
The results of the paper imply that profitability and superior management skills do not consist one of the reasons why investors
place their money in shares of closed-end funds in Greece. Institutional and individual investors hold shares of closed-end funds
for several other reasons, which however, are assumed but not empirically supported in the present paper. In the context of this
paper, their empirical proof remains an implication for future research.
The remainder of this paper is organized as follows. Section two describes the legal and institutional environment of closed-
end funds in Greece. Section three explains data selection and the methodological approach of the paper. Section four analyzes the
results of the empirical research. Finally, Section five summarizes conclusions and implications for further research.
1. Legal and Institutional Framework of Closed-End Funds in Greece
Closed-end funds in Greece are termed as Portfolio Investment Companies (PICs) and are companies with limited liability
(Societe Anonyme), which are subject to statutory controls like the rest of the incorporated businesses. Their primary purpose is to
manage a portfolio of transferable securities. The term transferable securities comprises among others of shares, debentures, bonds
and certificates of deposit. The PICs are governed by Law 1969/91 and must have a minimum amount of share capital of €
5,870,000 which must be totally paid in at the incorporation. The Hellenic Capital Market Commission (HCMC) is the competent
authority for providing a license for the establishment of a PIC.
The PICs must apply for listing to the Athens Stock Exchange (ASE) within six months after their incorporation. At the time of
listing 50% of the share capital of the PICs must have already been placed in transferable securities. The funds of PICs can be
invested in transferable securities listed on stock markets of any EU member state or in transferable securities listed on stock
markets of non-member states, on the condition that these markets are well regulated, recognized and open to the public. In order
to secure diversification, the HCMC requires that at the time of realizing its placements, a PIC is not allowed to invest more than
10% of its own funds in securities of the same issuer. In addition, PICs are not allowed to acquire shares of any company
representing more than 10% of the voting share capital of the latter. An exemption from this rule is granted when a PIC
participates in the share capital increase of a company of its holdings and acquires shares when exercising its rights of preference.
8.
If the shares acquired exceed the limit of 10%, then the PIC is obliged to sell the excess shares within one year from the date of
acquisition.
PICs are obliged to publish every three months a table with all their investments indicating their average acquisition cost and
their market value, and every six months their total net worth and per share internal value at current prices. With a special license
from the HCMC, a PIC can be transformed into a mutual (open-end) fund, following a relatively simple procedure. The HCMC is
the competent authority for imposing administrative penalties on directors, executives and certain employees of PICs when they
infringe the provisions of this law.
In terms of financing and dividend policy, PICs are allowed to borrow funds up to 10% of their own capital for investment on
transferable securities and for obtaining real estate properties. On the other hand, PICs are not allowed to retain earnings in order
to form reserves. They have to distribute to their shareholders, in the form of cash dividends, the total of their net profits.
Following a decision of the General Meeting (or Assembly?) of Shareholders only capital gains may be transferred to a special
reserve in order to cover possible future losses from sales of transferable securities. This possibility is suspended when such
special reserve exceeds 300% of the PICs own funds. If at the end of any fiscal year the company has realized accounting losses
arising from the evaluation of its transferable securities, it may form an equivalent provision from that year’s profits. Moreover,
PICs are not allowed to pay to the members of their Boards of Directors emoluments exceeding 1/10 th of distributed profits during
any fiscal year or 1/20th of capital gains (surplus value) of their portfolio at the end of that year.
An important aspect for PICs is the tax environment. PICs are legally bound to pay three per mille (3‰) tax on a yearly basis,
calculated on their average investments, plus the liquid funds in current prices as they appear in the quarterly investment
statements. The tax is paid to the appropriate tax authorities within the first two weeks of July and January of the semester
following the calculation. This tax payment essentially consists of the sole tax obligation of the PIC and its shareholders. In all
other cases, PICs are exempted from taxes, duties, stamp duties, contributions, rights or any other charge in favor of the state. An
exemption is the capital concentration tax (1% on issued share capital) and the value added tax. Interest received by PICs is
subjected to a deduction at source, as provided by law (15% on bank deposits and 10% on government securities). Dividends
received are in general exempted from income tax.
A final important institutional environment for PICs is the ownership (clientele) structure. The majority of PICs in Greece
are owned by institutional investors (banks and stock brokerage houses). Hardouvelis and Tsiritakis (1997) report that at the end of
year 1993 two-thirds (approximately 67%) of the shares of PICs in Greece were on average held by institutional investors. At that
time there were fifteen listed PICs in the ASE. By the end of year 2002 the number of listed PICs increased to twenty-four,
however, the percentage of institutionally owned PICs dropped only to 62.5%. These figures show that the ownership structure of
PICs does not present significant variations across years. Moreover, as compared to studies from the USA and the UK, these
figures depict that there are significant differences in the ownership structure of closed-end funds in Greece and in other countries.
9.
For example, Hardouvelis et al. (1994) report that, average institutional holdings of US-traded country funds is around 14% and
Bangassa (1999) reports that institutional investors hold 81% of the equity of UK-traded country funds.
2. Data and Methodology
2.1. The Data Set
The present study uses monthly data for a sample of Greek closed-end funds over a period of 72 months (6 years) from 1/1/1997 to
12/31/2002. The sample size varies in number of firms from month to month, and thus, the number of pooled firm-month
observations varies from year to year. Indeed the yearly pooled observations range from 154 to 260 and amount to a total of 1152-
pooled firm-month observations.
The use of panel data is motivated by a number of reasons. First, the present study does not intend to rank individual
closed-end funds according to their performance. The purpose of the study is to assess the average performance of the Greek
closed-end funds sector and to examine whether there are differences in average performance across bull and bear markets. In this
respect, the paper takes the individual investors’ point of view (as in Baks, Metrick and Wachter, 2001) and attempts to answer the
question as to whether superior managerial skills and performance consist of one of the reasons of investors’ preference to active
fund portfolios. A second purpose of this study is to assess the ability of alternative statistical models to explain selectivity and
timing performance. In particular, the paper tries to investigate whether models that capture information asymmetry across up and
down markets perform better than models that simply incorporate a time-dependent beta coefficient. Finally, the use of panel data
relaxes the assumption that the results are affected by a survivorship bias.2 The sample of firms used in the present study includes
all listed closed-end funds during the period examined. The number of firms varies from 16 on 1/1997 to 24 on 12/2002. It is quite
obvious that the sample of firms contains the entire new fund listings that occurred during this period. It is worthwhile noting,
however, that within this period there was no fund that discontinued operations.
The return metric used in the present study is the funds’ monthly net asset value (NAV) return. These data are available
through the website of the Association of Greek Institutional Investors (www.agii.gr) NAV returns offer two main advantages.
First, they refer to fund portfolio returns and thus, allow for a direct examination of selectivity and timing managerial performance.
2
Brown et al., (1992) and Elton et al., (1996) provide an extensive set of arguments on the importance of considering the impact of survivorship bias on
portfolio performance.
10.
Second, unlike stock returns, NAV returns are not affected by potential IPO underpricing or investor overreaction and thus, new
listings in the sample require no adjustment.
For the regression models run in the empirical part of the study, the dependent variable is excess NAV portfolio returns
(Rp,t). This is calculated by deducting the risk-free rate of return (Rf,t) from the funds’ raw NAV return (RN,t). In the context of the
present study the monthly risk-free rate of return is approximated by the monthly yield of the 3-months Treasury Bills of the Greek
Government. Since the regression models estimated are variants of the CAPM, the independent variables of the models are
moderations of the excess return on the market portfolio (Rm,t). The monthly raw return on the market portfolio is approximated by
the monthly percentage change in the General Price Index (GPI) of the ASE. Because NAV returns do not include any dividends
paid to fund shareholders, no adjustment has been made to the GPI for dividends. The excess market return is calculated by
deducting the risk-free rate of return from raw market returns. Data for the 3-month T-Bill rates and for the GPI have been
extracted from the monthly editions of the ASE.
2.2. Models of Portfolio Performance
The present study employs five alternative models of portfolio performance evaluation. These models are:
3.2.1. The Jensen (1968) Model
This model is actually the standard CAPM, which advocates a linear relation between the excess NAV returns and the excess
market returns:
R p ,t = α 0 + α1 Rm ,t + ε p ,t (1)
where: R p ,t is the excess NAV return on fund p in the month t; α0 is the intercept of the regression equation which, measures the
selectivity performance of the funds (see Fama, 1972); Rm ,t is the excess market return in month t; α1, is the slope of the regression
equation which, represents the average systematic risk of the funds; and ε p ,t is a white noise process assumed independent of
excess market returns.
3.2.2. The Treynor and Mazuy (1966) Model
11.
This model is also known as the Quadratic CAPM and relaxes the linearity assumption of the CAPM reflecting thus, time-
variations in the systematic risk of the portfolios. Specifically, it is assumed that fund managers who exhibit good timing skills are
able to increase the systematic risk of the portfolio during market up-times and reduce it when the market falters. Hence the time
dependent systematic risk coefficient can be expressed as follows:
α t = α1 + α 2 Rm,t (2)
A positive α2 shows a direct relation between portfolio systematic risk and excess market returns, displaying the managers’ timing
ability. Substituting equation 2 into equation 1 yields the quadratic version of the CAPM:
R p ,t = α 0 + α1 Rm ,t + α 2 Rm ,t + ε p ,t
2
(3)
In this model the intercept α0 indicates the return specifically derived from stock selectivity and the slope α2 reflects the return
component from market timing. If a fund manager increases (decreases) the portfolio’s market exposure prior to a market increase
(decrease), then the portfolio’s return will be a convex function of the market’s return, and the slope α2 will be positive.
3.2.3. The Henriksson and Merton (1981) Model
This model is also known as the Dual Beta CAPM and adopts the same attitude towards time-varying systematic risk as the
Treynor and Mazuy model. Merton (1981) and Henriksson and Merton (1981) opted to separate the constant systematic risk
component from the changing time-dependent systematic risk component in the following manner:
α t = α1 + α 2 Dt+ (4)
+
where; Dt is a dummy variable that takes the value of 1 when Rm ,t >0 and zero otherwise. Substituting equation 4 into equation 1,
leads to the definition of a CAPM with two target systematic risk factors known as the dual beta CAPM:
R p ,t = α 0 + α1 Rm ,t + α 2 Rm ,t + ε p ,t
+
(5)
12.
and
Rm ,t = Dt+ Rm ,t
+
(6)
+
where; Rm ,t equals Rm ,t when Rm ,t >0 and zero otherwise. Once again, the intercept α0 reflects the return derived from selectivity
while the slope α2 exposes the return from market timing. The magnitude of the α2 slope in equation (5) measures the difference
between the two target systematic risk factors, and is positive for a manager who successfully times the market.
3.2.4. The Asymmetric Response Model (ARM)
The ARM has been initiated by Fabozzi and Francis (1977) and Bawa, Brown and Klein (1981) and has been used by Fabozzi and
Francis (1977, 1979) and Pedersen and Satchell (2000) for testing managed-funds’ portfolio performance. The ARM nests the one-
factor mean-variance CAPM and the lower partial moment CAPM and can accommodate various degrees of non-normality in
portfolio returns quite simply. The model captures asymmetry by producing positive and negative intercepts and excess return
slopes for up and down markets respectively. The ARM, in its basic form, emerges by multiplying the intercept and the slope of
+ + − +
equation 1 by the terms (1 + Dt ) and ( Dt + Dt ) respectively, where, Dt is a dummy variable that takes the value of one when Rm ,t
− + −
>0 and zero otherwise; Dt is a dummy variable that takes the value of one when Rm ,t <0 and zero otherwise; and Dt + Dt = 1 . In
this sense, the model captures information asymmetry in both selectivity and timing return components in the following manner:
R p ,t = α 0 + α1 Rm ,t + α 2 Rm,t + α 3 Dt+ + ε p ,t
+ −
(7)
and
Rm ,t = Dt− Rm ,t
−
(8)
−
where Rm ,t equals Rm ,t when Rm ,t <0 and zero otherwise; and with all other variables being as previously defined. By
construction, the ARM captures the asymmetry in excess market returns through the slopes α1 and α2. The intuition of the model is
13.
that investors expect to receive a risk premium for downside risk (which is viewed as unfavorable) and pay a premium for upside
variation of returns (which is viewed as favorable). The intercept α0, reflects the selectivity performance in downside markets
whilst the slope α3, reflects the selectivity performance in upside markets. Positive values of α0 and α3 imply superior managerial
selectivity abilities, whilst positive values of α1 and α2 indicate the timing skills of the funds’ manager. Moreover, by splitting
market returns into upside and downside contributions, the ARM reflects if the fund manager responds better in bullish markets (if
α1 > α2) or in bearish markets (if α1 < α2).
3.2.5. The Modified Asymmetric Response Model (Modified ARM)
The modified version of the ARM is derived by considering the Fabozzi and Francis (1977) version of the ARM:
R p ,t = α 0 Dt− + α1 Rm,t + α 2 Rm ,t + α 3 Dt+ + ε p ,t
+ −
(9)
+
Adding and subtracting the terms α 0 Dt and a2 Rm,t , equation 9 yields on rearrangement:
+
R p ,t = α 0 ( Dt+ + Dt− ) + (α1 − α 2 ) Rm ,t + α 2 ( Rm ,t + Rm ,t ) + (α 3 − α 0 ) Dt+ + ε p ,t
+ + −
(10)
+ −
Knowing that Dt + Dt = 1 and that Rm ,t + Rm,t = Rm ,t and substituting back into equation 10 yields a modified version of the ARM
+ −
in the following manner:
R p ,t = α 0 + (α1 − α 2 ) Rm ,t + α 2 Rm ,t + (α 3 − α 0 ) Dt+ + ε p ,t
+
(11)
or equivalently:
R p ,t = α 0 + α1* Rm ,t + α 2 Rm ,t + α 3 Dt+ + ε p ,t
* + * *
(12)
14.
Equations 11 and 12 represent the modified version of the ARM. As in the original model, the intercept α 0 of equation 12 is a
*
measure of selectivity performance in market downtimes. The slope α1* is a direct measure of managerial timing skills. An active
manager who possesses superior timing skills must be able to foresee market uptimes (downtimes) and increase (reduce) the
systematic risk of the funds’ portfolio. In this case the slope α1* must be positive and significant. The slope α 2 represents a
*
measure of the funds downside risk which investors deem to be relevant. The slope α 3 is a measure of incremental selectivity and
*
reflects if the fund manager responds better in bullish markets (if α3 > α0, then α 3 is positive) or in bearish markets (if α3 < α0, then
*
α 3 is negative).
*
3.3. Specification Tests
The present paper tests for model misspecification in two ways. First the functional form of all regression models is examined by
performing Ramsey’s RESET test, which calculates the squares of the fitted values (Pagan and Hall, 1983). F-tests are performed
to test the validity of the null hypothesis, which states that the model does not fit the data well. If the F-values are found to be
significant then the model is potentially misspecified.
The second way to test for model misspecification is the application of exclusion-restriction specification tests (ERST)
proposed by Jagganathan and Korajczyk (1986) and applied also by Hallahan and Faff (1999) and by Allen and Soucik (2000).
These tests require that the selectivity-timing separation model is augmented by additional variable(s) of higher order. If the
original model is correctly specified then the additional variables should not produce significant regression coefficients. The
inclusion of a higher order variable to the Treynor and Mazuy and the Henriksson and Merton models, results to the following
ERST-adjusted versions:3
R p ,t = α 0 + α1 Rm ,t + α 2 Rm ,t + ϕ Rm ,t + ε p ,t
2 3
(13)
3
Equation 14 is a cubic market model that is consistent with the four-moment CAPM of Fang and Lai (1997). In this model expected excess returns are
related to systematic skewness and systematic kurtosis in addition to systematic variance. The coefficients α1, α2, and φ, measure systematic risk, skewness
risk, and kurtosis risk respectively. Positive significant α2, and φ, coefficients indicate that investors are willing to pay a premium for skewness and kurtosis
risk. This could be explained by investors who prefer stocks that provide an opportunity for unusually large positive returns (albeit with low probability)
much like the distribution of returns that are expected for a typical lottery.
15.
R p ,t = α 0 + α1 Rm ,t + α 2 Rm ,t + ϕ Rm ,t + ε p ,t
+ 2
(14)
Jagganathan and Korajczyk and the rest of the studies that apply ERST tests do not consider ARM as a portfolio performance
model. However, thinking of Rm ,t as the base variable of both the ARM and the modified ARM their ERST-adjusted versions
could be as follows:4
R p ,t = α 0 + α1 Rm ,t + α 2 Rm,t + α 3 Dt+ + ϕ Rm,t + ε p ,t
+ − 2
(15)
R p ,t = α 0 + α1* Rm ,t + α 2 Rm ,t + α 3 Dt+ + ϕ Rm ,t + ε p ,t
* + * * 2
(16)
In these adjusted versions of the four models the regression coefficient φ should be statistically insignificant. The appropriate
significance test proposed by Jagganathan and Korajczyk is the F-test although they admit that a heteroskedasticity-adjusted t-test
performs equally well.
3. Analysis of the Results
The empirical analysis of the paper consists of testing the performance of Greek closed-end funds using five alternative
performance models. In all cases, regression test statistics are based on standard errors corrected for heteroskedasticity using
White’s (1980) consistent covariance matrix. Moreover, the stability of the yearly regression coefficients is examined using the
test of Chow (Pagan and Hall, 1983). Finally, model misspecification is examined by applying Ramsey’s test and ERST tests. The
application of these tests reveals if some of the models result in misleading indications about the performance of closed-end funds
in Greece.
INSERT TABLE 1 APPROXIMATELY HERE
Table 1 contains some descriptive statistics for the variables used in the empirical tests. It appears that on average closed-end
funds are unable to outperform the market. Moreover, this result is evidenced in both bull and bear markets. However, what is
+ − + − + − + −
It can easily be shown that Rm ,t = ( Rm ,t ) + ( Rm ,t ) . This is because Rm ,t * Rm ,t = 0 and thus the term ( Rm ,t + Rm ,t ) reduces to ( Rm ,t ) + ( Rm ,t ) .
4 2 2 2 2 2 2
16.
more important is the standard deviation of the funds’ returns. Closed-end funds in Greece on average display a more volatile
pattern of returns than does the market. Moreover, return volatility is higher in upside markets than in downside markets. Finally,
the results of table 1 clearly indicate that the returns of both the funds and the market violate the assumption of normality. In
particular, fund returns are shown to be negatively skewed in both bull and bear markets, whilst market returns are positively
skewed in both market conditions. On the other hand fund returns display very high positive kurtosis implying that the distribution
of funds returns in Greece is highly leptokurtic. The presence of skewness and kurtosis in fund and market returns essentially
justifies the use of models, such as the ARM or the modified ARM, that do not require distributional assumptions about fund
returns.
INSERT TABLE 2 APPROXIMATELY HERE
Table 2 shows the results of evaluating the performance of Greek closed-end funds using the Jensen (1968) model.
Regressions are run separately for each one of the six years and for all years pooled together. The results indicate that in five out of
the six years the intercept of the regression is negative. Moreover, the intercept is found to be significant only in three out of these
five years. The intercept is positive but insignificant only in year 2002. When all data are pooled together the intercept is found to
be negative and significant. These results depict that on average fund managers in Greece do not display superior selectivity skills.
What is worth noting is that the systematic risk coefficient is found to be significant in all six yearly regressions and in the pooled
sample regression. In all cases the risk coefficient is less than unity implying that on average Greek funds show a tendency to
maintain low systematic risk securities in their portfolios. Moreover, the values of the R-square show that market returns play a
significant role in explaining average fund returns in Greece.
On the other hand, the results of the Chow test show significant F-values only in two out of the six years implying a
considerable instability of the regression coefficients over time. Moreover, by looking at the values of the F-test for model
misspecification, we can observe significant coefficients in two yearly regressions and in the panel-sample regression implying
that Jensen’s model is potentially misspecified.
INSERT TABLE 3 APPROXIMATELY HERE
Table 3 summarizes the results of evaluating Greek funds using the Treynor and Mazuy (1966) model. The average
selectivity performance of the funds (intercept) is found to be negative in all six yearly regressions and significant in only two of
them. The systematic risk coefficient is found significant at the 1% level in five out of the six years and in all cases, except in year
17.
2002, its value lies below unity. Moreover, in three out of the six yearly regressions, the timing component of returns is found to
be positive and significant implying that Greek professional fund managers may possess superior timing skills.
This results however, changes when we look at the panel-sample regression. The coefficients now demonstrate that both
the selectivity and the timing components of the returns are negative and significant at the 1% and 5% levels respectively
(although that the timing component is very close to zero). As with the Jensen model, market risk is found to be less than one and
significant at the 1% level. Moreover, the values of the R-square show that market returns and quadratic market returns explain to
a high extent the variation in average fund returns.
A quite interesting result is that the F-values of the Chow test display stability in the regression coefficients in four out of
the six years. However, the model misspecification tests depict significant F-values in one yearly regression and in the panel-
sample regression implying that the model may exhibit some degree of misspecification.
INSERT TABLE 4 APPROXIMATELY HERE
Table 4 tabulates the results of fund performance evaluation when the Henriksson and Merton (1981) model is employed.
The results now indicate that selectivity performance is negative and significant in four out of the six years and positive but
insignificant in only one year. On the other hand, the model exhibits significant positive timing return coefficients in three out of
the six years and significant negative timing return coefficient in another year. The systematic risk coefficient is again significant
at the 1% level in five out of the six years.
However, the results change dramatically when we look at the results of the panel-sample regression. Average selectivity
and timing return coefficients are found to be negative and only marginally significant at the 10% level. As with the previous
models, the market risk coefficient is found to be significant at the 1% level and again less than unity implying an average
tendency of fund managers to hold low risk securities into their portfolios.
The values of the R-square depict that the model exhibits a significant explanatory power for average fund returns, and the
results of the Chow test display stability of the regression coefficients in four out of the six years. However, what is important
noticing is the results of the test for model misspecification. The F-values are found significant in two out of the six years whilst
the panel sample regression exhibits no degree of model misspecification. Moreover, in one of the two yearly regressions, the null
hypothesis of model misspecification is accepted at the 10% level implying that the Henriksson and Merton model exhibits lesser
degree of misspecification than the previous two models.
INSERT TABLE 5 APPROXIMATELY HERE
18.
Table 5 exhibits the results of evaluating the performance of Greek closed-end funds when employing the Asymmetric
Response Model (ARM). The results now indicate that professional fund managers are on average unable to outperform the
market. In two out of the six years selectivity performance is significant in both downside and upside markets. However, these
selectivity performance coefficients are found to be negative in bearish markets and positive in bullish markets. This of course
implies that professional portfolio managers do not on average display consistent selectivity skills across bull and bear markets and
are unable to systematically outperform the market even in market uptimes.
On the other hand, the results indicate that systematic risk is a significant variable in explaining fund returns. The upside
risk coefficient is found to be significant at the 1% level in all six years while the downside risk coefficient is found significant at
least at the 5% level in five out of the six years. Again however, in almost all cases the systematic risk coefficients are found to be
less than unity supporting the assertion that Greek fund managers on average hold low risk securities into their portfolios.
The results of the panel sample regression simply verify the results of the yearly regressions. The intercept is found
negative and significant at the 1% level implying negative selectivity fund performance in market downtimes. On the other hand,
positive but insignificant selectivity performance coefficient is observed in market uptimes implying an inability of fund managers
to consistently and systematically outperform the market. The two systematic risk coefficients are found to be significant at the 1%
level and again are both less than unity. However, what is more important is the fact that the average systematic risk in up markets
is lower than the average systematic risk in down markets. This clearly implies that fund managers in Greece do not display
superior timing abilities since they are shown to switch the risk of their portfolios in the opposite way that market movements
require.
As regards the overall performance of the model, the results are quite encouraging. The values of the R-square show that
the decomposed market returns exhibit significant explanatory power for average fund returns, and the results of the Chow test
display stability of the yearly regression coefficients in four out of the six years. Finally, the ARM appears to exhibit the lesser
degree of model misspecification among all models examined thus far. The values of the F-test are found to be significant at the
5% in only one out of the six years and insignificant in the rest of the yearly regressions and in the panel-sample regression. Thus
the null hypothesis that the model is misspecified can be rejected at a high confidence interval.
INSERT TABLE 6 APPROXIMATELY HERE
The results of table 6 indicate that the modified ARM performs equally well as the original ARM. Both models exhibit exactly the
same test statistic values for the R-square, for the adjusted R-square, for the Chow test, and for the test for model misspecification.
However, the modified ARM offers clearer insights in both the selectivity and timing abilities of professional fund managers in
Greece. First, as with the ARM, the downside risk coefficient, which investors deem to be relevant, is found to be significant in
19.
five out of the six years. However, the timing return coefficient, which is the difference between up market and down market
systematic risk, is found to be significant only in three out of the six years. In two of these three years the timing return coefficient
is found to be negative and in the third year it is found to be positive. When looking at the panel-sample regression the timing
return coefficient is negative but insignificant implying that on average fund managers in Greece exhibit an inability to
consistently time the market.
As regards the selectivity return coefficients the results are about the same. The intercept, which, as in the original model,
represents the selectivity return coefficient, is found to be negative and significant in two out of the six yearly regressions and in
the panel-sample regression (exactly the same as with the ARM). However, the incremental selectivity coefficient (the difference
in the two selectivity return coefficients) is found to be negative and significant in two yearly regressions and positive but
insignificant in the panel sample regression. This result indicates that on average fund managers of Greek closed end funds are
unable to pick up shares that yield returns in excess of the market return, and that the unsuccessful choices of securities spreads
equally between bull and bear markets.
INSERT TABLE 7 APPROXIMATELY HERE
In a final illustration, the paper performs ERST tests as alternative way to test for model misspecification. These tests
assume that the original models are augmented by an explanatory variable of higher order. If this variable is found to be significant
then the model is potentially misspecified. This test is performed for the Treynor and Mazuy, the Henriksson and Merton, the
ARM, and the modified ARM models.5
The coefficients, t-statistics and p-values for the additional variables of higher order are exhibited in table 7. The prevailing
coefficient in all four models is negative and close to zero. The ARM and the modified ARM again share the same coefficient and
the respective t-statistics and p-values. It appears that the coefficient of the additional variable is only significant at the one percent
level in the Treynor and Mazuy model, implying a significant degree of model misspecification. In all other cases the coefficient is
found to be insignificant thus, the hypothesis of model misspecification cannot be sustained for the rest three models. This result is
consistent with the results of the tests for the functional form of the models illustrated previously. Thus, a conclusion of the paper
can be that the performance of Greek closed-end funds may be assessed by the Henriksson and Merton model and the ARM and
the modified ARM without worrying about model misspecification.
4. Summary and Implications
5
The ERST adjusted version of the Jensen model is the Treynor and Mazuy model. In the latter model the coefficient of the square of market returns is
found to be significant at the 10% level, thus it can be assumed that the Jensen model exhibits some degree of misspecification.
20.
The present paper examines the average performance of Greek closed-end funds over a six-year period from 1997 to 2002.
Monthly data are used for a varying number of funds to estimate yearly panel regressions and full-sample panel regressions. The
regression models include the Jensen (1968) model, the Treynor and Mazuy (1966) model, the Henriksson and Merton (1981)
model, the Asymmetric Response Model (ARM), and a modified version of the ARM developed in the paper.
The results of the paper can be summarized into three general sets of arguments. First, it appears that some models present
some degree of misspecification. The paper uses two different ways to test for the functional form of the models. These tests are
the Ramsey’s RESET test and the Jagganathan and Korajczyk Exclusion-Restriction Specification Tests (ERST). The application
of both tests reveals potential misspecification in the case of the Jensen and the Treynor and Mazuy models. The other three
models are found to be well specified with both kinds of tests. Moreover, these three models are found to exhibit significant
stability of the regression coefficients across years, thus it can be argue that they impound some degree of predictive ability.
Surprisingly, however all models, even those that are found to be misspecified, provide with the same insights regarding the
performance of Greek closed-end funds. Thus, an implication is that the results of fund-portfolio performance studies may not be
misguided even when the performance models used potentially suffer from misspecification.
Second, the modified version of the ARM developed in the paper, appears to perform equally well as the original ARM.
However, the modified ARM has the intuitive appeal that it offers better insights in the timing and selectivity performance of fund
managers. In particular, the model provides a direct way to test for timing performance by estimating the difference in systematic
risk between upside and downside markets. Moreover, the model offers the possibility to examine whether fund managers display
incremental selectivity skills across bull and bear markets.
Finally, it appears that professional fund managers in Greece do not on average display superior timing and selectivity
abilities and that their performance exhibits no significant differences across bull and bear markets. All, yearly regressions and
full-sample regressions estimated in the paper depict that fund managers are on average unable to systematically and consistently
outperform the market.
This result implies that profitability does not consist one of the reasons why investors place their funds in shares of managed
portfolios. A potential explanation for the increased preference of investors to shares of Greek closed-end funds may be the one
offered by Baks et.al. (2001) who claim that investors buy shares of managed portfolios because they do not possess the discipline,
time and technology to implement sophisticated trading strategies by themselves. On the other hand, this explanation may be
extended to include institutional investors who hold the majority of shares of Greek closed-end funds. Institutional investors
arguably possess the discipline, the time and the technology to implement sophisticated trading strategies. A reasonable
explanation why they are not particularly interested in the profitability of the closed-end funds they own is because they use these
funds as a way to facilitate their overall strategy of market making activities.
21.
An objection to these explanations could be that managers of closed-end funds in Greece are unable to display superior
timing and selectivity skills because of the low volume of transactions, which restricts them from engaging into timely buy and
sell activities. However, this argument cannot be sustained by the results of the present paper particularly in years 1998, 1999, and
2000 in which the volume of transactions in the ASE was quite high. As compared to the results obtained in years 1997, 2001, and
2002 when the volume of transactions was moderate or even shallow we observe no significant systematic differences in average
fund portfolio performance. The implication is that the volume of transactions does not explain average performance of Greek
closed-end funds. However, in the context of the present paper this issue remains simply an implication that guarantees further
empirical research.
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25.
Table 1: Summary Descriptive Statistics of Variables
Total Observations Bullish Market Bearish Market
Excess NAV Excess Market Excess NAV Excess Market Excess NAV Excess Market
Return Return Return Return Return Return
Mean -1.369 0.118 5.005 9.160 -6.240 -6.791
Median -1.089 -0.868 4.901 7.579 -5.137 -4.736
Standard Deviation 11.751 10.388 12.250 8.180 8.619 5.386
Skewness -1.407 0.840 -3.041 1.924 -1.504 -1.039
Kurtosis 11.756 2.001 23.454 4.440 14.157 0.577
Count 1152 1152 499 499 653 653
26.
Table 2: The Performance of Greek Closed-End Funds Evaluated by the Jensen Model
R p ,t = a0 + a1 Rm ,t + e p ,t
Adjusted Model Chow Number of
Year α0 α1 R-square F-test
R-square Specification Test Observations
Coefficient 221.070
-0.746** 0.632* 0.593 0.590 0.542 0.369
1997 *
154
t-statistic -2.100 15.302
p-value 0.037 0.000 0.000 0.463 0.692
Coefficient -0.875 0.508* 0.370 0.366 98.625* 1.361 9.766*
1998 t-statistic -1.205 8.959 170
p-value 0.230 0.000 0.000 0.245 0.000
Coefficient -2.231 0.971* 0.132 0.127 27.432* 4.882* 1.816
1999 t-statistic -1.327 6.518 182
p-value 0.186 0.000 0.000 0.028 0.163
Coefficient -4.125* 0.637* 0.293 0.289 78.743* 2.177 19.672*
2000 t-statistic -6.792 12.156 192
p-value 0.000 0.000 0.000 0.142 0.000
Coefficient 639.647
-0.582** 0.789* 0.769 0.768 0.789 0.836
2001 *
194
t-statistic -2.209 25.400
p-value 0.028 0.000 0.000 0.375 0.434
Coefficient 147.098
0.100 0.8335* 0.363 0.360 17.669* N/A
2002 *
260
t-statistic 0.188 9.991
p-value 0.851 0.000 0.000 0.000
Coefficient 578.499
-1.376* 0.653* 0.334 0.334 6.433** N/A
All *
1152
t-statistic -4.891 19.944
p-value 0.000 0.000 0.000 0.011
Notes: Descriptions of variables: R p ,t is the funds’ monthly NAV returns in excess of the risk free rate; Rm ,t is the monthly return on the market portfolio in
excess of the risk free rate. Regression test statistics are adjusted for heteroskedasticity as in White (1980).
27.
*Indicates significance at the 1% level; ** indicates significance at the 5% level; *** indicates significance at the 10% level.
Table 3: The Performance of Greek Closed-End Funds Evaluated by the Treynor and Mazuy Model
R p ,t = a0 + a1 Rm ,t + a2 Rm ,t + e p ,t
2
Adjusted Model Chow Number of
Year α0 α1 α2 R-square F-test
R-square Specification Test Observations
Coefficient 110.473
-1.010** 0.632* 0.003 0.594 0.588 0.762 0.505
1997 *
154
t-statistic -2.475 15.309 1.013
p-value 0.014 0.000 0.313 0.000 0.384 0.679
Coefficient -0.266 0.564* -0.003 0.375 0.367 50.099* 0.101 5.923*
1998 t-statistic -0.260 9.507 -1.092 170
p-value 0.794 0.000 0.276 0.000 0.751 0.001
Coefficient 0.049*
-2.388 0.331 0.155 0.145 16.452* 0.010 4.859*
1999 *
182
t-statistic -1.427 1.057 2.548
p-value 0.155 0.292 0.012 0.000 0.920 0.002
Coefficient 0.011*
-5.302* 0.637* 0.301 0.293 40.703* 10.160* 14.690*
2000 *
192
t-statistic -5.509 11.938 2.223
p-value 0.000 0.000 0.027 0.000 0.002 0.000
Coefficient 319.866
-0.429 0.771* -0.002 0.770 0.767 0.024 8.333*
2001 *
194
t-statistic -1.340 21.614 -0.881
p-value 0.182 0.000 0.379 0.000 0.877 0.000
Coefficient -0.148 1.231* 0.040* 0.404 0.399 87.136* 1.266 N/A
2002 t-statistic -0.296 8.557 4.011 260
p-value 0.767 0.000 0.000 0.000 0.262
28.
Coefficient 293.833
-0.985* 0.685* -0.003* 0.338 0.337 10.005* N/A
All *
1152
t-statistic -2.837 24.485 -1.886
p-value 0.005 0.000 0.060 0.000 0.002
Notes: Descriptions of variables: R p ,t is the funds’ monthly NAV returns in excess of the risk free rate; Rm ,t is the monthly return on the market portfolio in
2
excess of the risk free rate; Rm ,t is the square of Rm ,t .
Regression test statistics are adjusted for heteroskedasticity as in White (1980).
*Indicates significance at the 1% level; ** indicates significance at the 5% level; *** indicates significance at the 10% level.
Table 4: The Performance of Greek Closed-End Funds Evaluated by the Henriksson and Merton Model
+
R p ,t = a0 + a1 Rm ,t + a2 Rm ,t + e p ,t
Adjusted Model Chow Number of
Year α0 α1 α2 R-square F-test
R-square Specification Test Observations
Coefficient 110.366
-1.151** 0.578* 0.104* 0.593 0.588 0.290 0.522
1997 *
154
t-statistic -2.251 8.790 8.790
p-value 0.026 0.000 0.000 0.000 0.591 0.667
Coefficient 0.381 0.679* -0.237** 0.376 0.368 50.335* 0.119 6.335*
1998 t-statistic 0.364 8.023 -1.742 170
p-value 0.716 0.000 0.083 0.000 0.730 0.000
Coefficient -5.005*** -0.369 1.615 0.142 0.133 14.921* 2.981*** 3.056**
1999 t-statistic -1.831 -0.424 1.572 182
p-value 0.069 0.672 0.118 0.000 0.086 0.028
Coefficient -5.811* 0.455* 0.391** 0.300 0.293 40.628* 2.223 14.333*
2000 t-statistic -5.134 3.948 2.087 192
p-value 0.000 0.000 0.038 0.000 0.138 0.000
Coefficient 320.700
-0.206 0.834* -0.118 0.770 0.768 0.160 5.051*
2001 *
194
t-statistic -0.502 16.633 -1.160
p-value 0.616 0.000 0.248 0.000 0.689 0.002
29.
Coefficient -1.218** 0.652* 0.973* 0.386 0.381 80.793* 4.229** N/A
2002 t-statistic -2.461 8.075 3.014 260
p-value 0.014 0.000 0.003 0.000 0.041
Coefficient 291.513
-0.758*** 0.748* -0.158*** 0.336 0.335 1.899 N/A
All *
1152
t-statistic -1.659 15.122 -1.668
p-value 0.097 0.000 0.095 0.000 0.168
Notes: Descriptions of variables: R p ,t is the funds’ monthly NAV returns in excess of the risk free rate; Rm ,t is the monthly return on the market portfolio in
+
excess of the risk free rate; Rm ,t is equal to Rm ,t when Rm ,t > 0, and zero otherwise;
Regression test statistics are adjusted for heteroskedasticity as in White (1980).
*Indicates significance at the 1% level; ** indicates significance at the 5% level; *** indicates significance at the 10% level.
Table 5: The Performance of Greek Closed-End Funds Evaluated by the Asymmetric Response Model
R p ,t = a0 + a1 Rm ,t + a2 Rm ,t + a3 Dt+ + e p ,t
+ −
Adjusted Model Chow Number of
Year α0 α1 α2 α3 R-square F-test
R-square Specification Test Observations
Coefficient -0.796 0.745* 0.609* -1.034 0.595 0.586 73.471* 0.038 0.507
1997 t-statistic -1.250 7.618 8.193 -0.965 154
p-value 0.213 0.000 0.000 0.336 0.000 0.845 0.730
Coefficient 1.651 0.474* 0.762* -2.137 0.378 0.367 33.687* 0.255 4.181*
1998 t-statistic 1.269 5.170 8.161 -1.074 170
p-value 0.206 0.000 0.000 0.284 0.000 0.614 0.002
Coefficient 1.542 1.417* 1.260 -8.470 0.153 0.138 10.718* 1.205 5.568*
1999 t-statistic 0.249 6.069 0.801 -1.234 182
p-value 0.803 0.000 0.424 0.219 0.000 0.274 0.000
Coefficient -7.383* 0.491* 0.315* 6.088* 0.314 0.303 28.743* 0.672 9.965*
2000 t-statistic -5.073 5.877 2.313 3.006 192
p-value 0.000 0.000 0.022 0.003 0.000 0.413 0.000
30.
Coefficient 213.240
-0.346 0.656* 0.821* 0.666 0.771 0.767 0.006 2.510*
2001 *
194
t-statistic -0.697 8.468 14.773 0.892
p-value 0.486 0.000 0.000 0.373 0.000 0.935 0.041
Coefficient 1.541**
-1.548** 1.318* 0.615* 0.388 0.381 54.245* 4.392** N/A
2002 *
260
t-statistic -2.573 6.874 6.879 1.683
p-value 0.011 0.000 0.000 0.094 0.000 0.037
Coefficient 195.696
-1.475* 0.540* 0.683* 1.534 0.338 0.336 0.930 N/A
All *
1152
t-statistic -2.952 7.195 13.594 1.641
p-value 0.003 0.000 0.000 0.101 0.000 0.335
Notes: Descriptions of variables: R p ,t is the funds’ monthly NAV returns in excess of the risk free rate; Rm ,t is the monthly return on the market portfolio in
+ − +
excess of the risk free rate; Rm ,t is equal to Rm ,t when Rm ,t > 0, and zero otherwise; Rm ,t is equal to Rm ,t when Rm ,t <0 and zero otherwise; and Dt is a
dummy variable that takes the value of 1 when Rm ,t > 0, and zero otherwise.
Regression test statistics are adjusted for heteroskedasticity as in White (1980).
*Indicates significance at the 1% level; ** indicates significance at the 5% level; *** indicates significance at the 10% level.
Table 6: The Performance of Greek Closed-End Funds Evaluated by the Modified Asymmetric Response Model
R p ,t = a0 + a1 Rm ,t + a2 Rm ,t + a3 Dt+ + e p ,t
+
Adjusted Model Chow Number of
Year α0 α1 α2 α3 R-square F-test
R-square Specification Test Observations
Coefficient -0.796 0.136 0.609* -1.832* 0.595 0.586 73.471* 0.038 0.507
1997 t-statistic -1.250 1.110 8.193 -2.125 154
p-value 0.213 0.269 0.000 0.035 0.000 0.845 0.730
Coefficient 1.651 -0.288* 0.762* -0.487 0.378 0.367 33.687* 0.255 4.181*
1998 t-statistic 1.269 -2.204 8.161 -0.323 170
p-value 0.206 0.029 0.000 0.747 0.000 0.614 0.002
Coefficient 1.542 0.156 1.260 -6.928** 0.153 0.138 10.718* 1.205 5.568*
1999 t-statistic 0.249 0.098 0.801 -2.321 182
p-value 0.803 0.922 0.424 0.021 0.000 0.274 0.000
31.
Coefficient -7.383* 0.175 0.315** -1.295 0.314 0.303 28.743* 0.672 9.965*
2000 t-statistic -5.073 1.095 2.313 -0.920 192
p-value 0.000 0.275 0.022 0.359 0.000 0.413 0.000
Coefficient 213.240
-0.346 -0.165*** 0.821* 0.320 0.771 0.767 0.006 2.510**
2001 *
194
t-statistic -0.697 -1.730 14.773 0.574
p-value 0.486 0.085 0.000 0.566 0.000 0.935 0.041
Coefficient -1.548** 0.702* 0.615* -0.007 0.388 0.381 54.245* 4.392** N/A
2002 t-statistic -2.573 3.320 6.879 -0.011 260
p-value 0.011 0.001 0.000 0.911 0.000 0.037
Coefficient 195.696
-1.475* -0.143 0.683* 0.059 0.338 0.336 0.930 N/A
All *
1152
t-statistic -2.952 -1.586 13.594 0.074
p-value 0.003 0.113 0.000 0.941 0.000 0.335
Notes: Descriptions of variables: R p ,t is the funds’ monthly NAV returns in excess of the risk free rate; Rm ,t is the monthly return on the market portfolio in
+ +
excess of the risk free rate; Rm ,t is equal to Rm ,t when Rm ,t > 0, and zero otherwise; and Dt is a dummy variable that takes the value of 1 when Rm ,t > 0, and
zero otherwise.
Regression test statistics are adjusted for heteroskedasticity as in White (1980).
*Indicates significance at the 1% level; ** indicates significance at the 5% level; *** indicates significance at the 10% level.
Table 7: Regression Estimates of the φ coefficient (Exclusion-Restriction Specification Tests)
Model
Treynor - Mazuy Henriksson - Merton ARM Modified ARM
Coefficient -0.0002* -0.0069 -0.0054 -0.0054
t-statistic -2.6955 -1.4685 -1.1217 -1.1217
p-value 0.0070 0.1420 0.2620 0.2620
Notes: Regression test statistics are based on standard errors corrected for heteroskedasticity using the method in White (1980).
* indicates significance at the 1% level
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