The values of t can then be plotted over j s 1, 2, . . . , N, with the estimated
variances used to set confidence bands or AtunnelsB above and below. An immediate extension is to specify,
K
r s
u g q e ,
Ý
i k
i k i
k s0
Ž .
where the k th constructed regressor function g R ; R may utilise the entire set
k i
of available R values to form the values g of each regressor g at R . Section 6
i k k
i
contains an example where the regressors g are constructed as a set of orthogonal polynomials over the given sample, a technique which is superior to ordinary
Ž .
polynomial curve fitting. Again, Eq. 18 gives the variances of the fitted t ,
j
conditional on the sample R values. The proof of the variance formula in each case is just a special case of that for the nonlinear case, below, and will not be
presented separately. The procedure for nonlinear least squares, where the parameters u in enter the
regression in nonlinear fashion, is very similar to that just described. For the asymptotic variance, the data matrix G is replaced by the N = k Jacobian matrix
Ž .
ˆ
of the partial derivatives of e R , u with respect to u. A derivation of the t
i j
estimates and the resulting variance estimator is given in Appendix A.
6. Empirical illustration
The empirical work illustrates the COMD and related assessment techniques with the historical performance of three New Zealand funds using monthly returns
over the period December 1988 to November 1995, with data kindly supplied by FPG Research. The funds are of significant size and are among the relatively few
long established NZ funds. The market rate R is taken as the monthly percent
Ž .
change in the NZSE top 40 gross accumulation index. The respective Jensen’s alpha for funds A, B, C over the period, computed
using the 30-day bank bill rate as the risk-free rate were, y0.09, y0.03, and
y0.02, and all were statistically quite insignificant. Thus, there is little indica- tion on the basis of Jensen’s alpha of managerial superiority over the market
benchmark in any of the funds. None of the funds’ second order stochastically dominate the market, by either the classic cross over rule for the empirical
distribution functions or the reversed t procedure of Section 4 of the present paper, though funds B and C both come quite close.
Fig. 2a–c are fitted COMD functions for the three funds. In each case, the top diagram is a scatter plot of the fund return r against the market return R, the
middle diagram is the COMD function estimated by the non-parametric technique, while the bottom diagram is the parametric estimate with confidence bands of
ˆ
STD t , i.e. one standard deviation.
j
The parametric estimate is prepared by a variant of polynomial curve fitting Ž
that uses Forsyth sample orthogonal polynomials see Carnahan et al., 1969, Chap. .
8.22 . These simplify the task of choosing the order of the polynomial of best fit.
Ž . Ž .
Fig. 2. a COMD: fund A against market benchmark. b COMD: fund B against market benchmark. Ž .
c COMD: fund C against market benchmark.
Ž .
Fig. 2 continued .
The polynomials are designed so that the regression matrix of sums of squares and cross products G
X
G is diagonal. One can then choose the order of the polynomial
as that of the highest order Forsyth polynomial that remains significant, either in terms of ordinary Student’s t or else more automatically in terms of simple
Ž .
Fig. 2 continued .
stepwise regression inclusion rules. Appendix B gives the formulas for the Forsyth polynomials, while Table B1 of the appendix lists the results from a stepwise
regression procedure. In the case of fund A, Forsyth polynomials of degrees 1, 3 and 5 were stepped in and retained as significant. For fund B, only the polynomial
R.J. Bowden
r Journal
of Empirical
Finance 7
2000 195
– 223
214
Table B1 Stepwise regression results
Fund A Stepping sequence
F-value df
df F-probability
numerator denominator
Step number Variable label
Status 1
G1 stepped in
725.3896 1
82 0.000000
2 G5
stepped in 7.3258
1 81
0.008287 3
G3 stepped in
4.5410 1
80 0.036163
Summary for potential Õariables not entered into the reg. equation G2
if entered 0.0327
1 79
0.857016 G4
if entered 0.0129
1 79
0.909926 G6
if entered 2.4081
1 79
0.124706 G7
if entered 0.3868
1 79
0.535785 End of stepping sequence
Variable name Estimated
Student’s coefficient
t-ratio 80 df G1
0.83766 28.56
y3
G3 0.58580=10
2.131
y4
G5 0.14281=10
2.765 Constant
0.79498 4.714
R.J. Bowden
r Journal
of Empirical
Finance 7
2000 195
– 223
215 Ž
. Table B1 continued
Fund B Stepping sequence
F-value df
df F-probability
numerator denominator
Step number Variable label
Status 1
G1 stepped in
501.2995 1
82 Summary for potential Õariables not entered into the reg. equation
G2 if entered
0.1932 1
81 0.661414
G3 if entered
1.9815 1
81 0.163057
G4 if entered
0.0395 1
81 0.842938
G5 if entered
0.4165 1
81 0.520522
G6 if entered
2.4766 1
81 0.119453
G7 if entered
0.3339 1
81 0.564946
End of stepping sequence Variable name
Estimated Student’s
coefficient t-ratio 82 df
G1 0.74262
22.39 Constant
0.83854 4.398
continued on the next page
R.J. Bowden
r Journal
of Empirical
Finance 7
2000 195
– 223
216 Ž
. Table B1 continued
Fund C Stepping sequence
F-value df
df F-probability
numerator denominator
Step number Variable label
Status 1
G1 stepped in
80.1965 1
82 0.000000
2 G6
stepped in 5.6898
1 81
0.019398 Summary for potential Õariables not entered into the reg. equation
G2 if entered
1.9529 1
80 0.166138
G3 if entered
0.0011 1
80 0.973787
G4 if entered
2.3265 1
80 0.131129
G5 if entered
1.8825 1
80 0.173880
G7 if entered
0.4060 1
80 0.525838
End of stepping sequence Variable name
Estimated Student’s
coefficient t-ratio 81 df
G1 0.67122
9.208
y5
G6 0.44150=10
2.385 Constant
0.4191 2.013
Ž Choice of the order of the Forsyth polynomials was executed using stepping in regression option in the SHAZAM econometric package White, 1977 and
. updates .
Variables stepped in: Fund A — G1, G5, G3; fund B — G1; fund C — G1, G6. Ž
. Criteria for stepping in
rout: if a variable becomes significant at the 5 level, it is included probability values are based on the F values for each regression . If a variable becomes less significant than 5, then it is deleted from the equation.
2 2
2
Fund A: R s 0.9119, R s 0.9086; variance of the estimate s s 2.3887.
2 2
2
Fund B: R s 0.8594, R s 0.8577; variance of the estimate s s 3.054.
2 2
2
Fund C: R s 0.5276, R s 0.5160; variance of the estimate s s14.752.
of degree 1 was stepped in, so that for this fund, a linear regression is a very good fit. Fund C stepped in orders 1 and 6. All this means that one has to go to a
5th-order polynomial to adequately represent fund A against the market bench- mark, to order 1 for fund B, and to order 6 for fund C. There is no need to
transform back to the natural or monic polynomials, as the various estimates can
Ž .
Fig. 3. Equivalent margin as function of xi. A: Non-parametric actual r . B: Non-parametric
Ž w x
. estimated e r .
all be expressed directly in terms of the Forsyth polynomials.
6
In all cases, Ramsey reset tests indicated no residual regressor–residual correlation at 5
significance levels for the resulting x
2
tests. Both parametric and non-parametric techniques yield closely similar results for
the COMD functions, in each case. Funds B and C are virtually COMD dominant over the market, failing only over the large or very small range of returns. Since
the latter receives only a small empirical probability, it is evident that unless the degree of risk aversion was either very high or very low, investors would prefer
these funds over the market. On the other hand, the performance of fund A relative to the market is much less convincing.
We can check the possible influence of the degree of investor risk aversion by Ž .
choosing a family of utility functions and varying the risk aversion parameter s . A convenient family is the constant relative risk aversion family mentioned in
Ž Section 3. We varied the parameter j from 0 which is effectively the log utility
. function on down to
y5, moving from mild to progressively more stringent risk aversion. For each value of j , the equivalent margin t
was calculated for the
j
three funds, and the results plotted in Fig. 3 against j . It is clear that at least for the CRRA family, fund C generates most investor surplus relative to the market,
while for lower degrees of risk aversion, the poor performance of fund A is self-evident.
As indicated in Section 2, the equivalent margin measures are primarily designed for testing against the market as benchmark, but they do provide a
bilateral measure for comparing two funds if one is willing to accept the discount Ž .
factor p R as defined in terms of market returns. Thus, to compare fund B with fund C in terms of logarithmic utility one would compute:
y1
N
1 q R
Ž .
i
p R r
y r ;
p R s .
Ž . Ž
. Ž
.
Ý
i B i
C i i
N
y1
i s1
1 q R
Ž .
Ý
j
j s1
6
Ž . While OLS is the correct method to use in estimating the conditional expectation function e R , an
operational problem arises in determining the degree of the approximating polynomial. In most packages, stepping in is determined in terms of tests of significance that are based on independent
spherical Normal distributions for the regression residuals. Financial returns data is often not so Ž
. obliging, e.g. Engle and Ng, 1993; Glosten et al., 1993 , and indeed the core methodology of the
present paper itself makes no such assumption. Thus, it is advisable to test for departures form Normality etc in the residuals and if thought necessary to adapt the stepping in algorithm somehow. In
Ž . the present instance, none of the fund residuals resulting from the stepped in choices for e R were
Ž .
serially correlated DW statistics of 2.11, 2.00 and 1.91, respectively , but they did exhibit some positive skewness and kurtosis, also heteroscedascity, though only marginally significant in most
Ž .
standard tests except possibly for the ARCH test e.g. chi sq. of 4.56 with 1 df for fund C . The Ž
general problem here will be to adapt the hypothesis testing of the OLS fitting for such departures rather .
than adapt the estimation methodology, as OLS in itself remains correct . Thanks go to a referee for pointing out a potential problem here.
The resulting figure was y0.024, virtually zero, so that for such an investor
there would be little to choose between the funds, in spite of the fact that C does a little better relative to the market as benchmark than does B. This indicates if
nothing else that a clear purpose is necessary for the proposed comparison. In the present paper, we have emphasised the role of the market as benchmark, simply
because investors are so often concerned with the question of whether a passive holding of the market index would result in better returns than employing a fund
manager.
7. Concluding remarks
