# A nonparametric test of market timing

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A nonparametric test of market timing

Wei Jiang*

Finance and Economics Division, Columbia Business School, 3022 Broadway, New York, NY 10027, USA

Abstract

In this paper, we propose a nonparametric test for market timing ability and apply the analysis to

a large sample of mutual funds that have different benchmark indices. The test statistic is formed to

proxy the probability that a manager loads on more market risk when the market return is relatively

high. The test (i) only requires the ex post returns of funds and their benchmark portfolios; (ii)

separates the quality of timing information a money manager possesses from the aggressiveness with

which she reacts to such information; and (iii) is robust to different information and incentive

structures, as well as to underlying distributions. Overall, we do not find superior timing ability

among actively managed domestic equity funds for the period of 1980 – 1999. Further, it is difficult

to predict funds’ timing performance from their observable characteristics.

D 2003 Elsevier Science B.V. All rights reserved.

JEL classification: G1; C1

Keywords: Mutual funds; Market timing; Nonparametric test; U-statistics

1. Introduction

Based on the theory of market efficiency with costly information, there has been ample

research work on measuring professional money managers’ performance. The emphasis

has been on one of the two basic abilities: securities selectivity and market timing. The

former tests whether a fund manager’s portfolio outperforms the benchmark portfolio in

risk-adjusted terms (Jensen, 1972; Gruber, 1996; Ferson and Schadt, 1996; Kothari and

Warner, in press). The latter tests whether a fund manager can out-guess the market by

moving in and out of the market (Treynor and Mazuy, 1966; Henriksson and Merton,

1981; Admati et al., 1986; Bollen and Busse, 2001).

* Tel.: +1-212-854-9679; fax: +1-212-316-9180.

E-mail address: wj2006@columbia.edu (W. Jiang).

0927-5398/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0927-5398(02)00065-8

400

W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

Measures of market timing have fallen into one of the two categories: portfolio- and

return-based methods. The former tests whether money managers successfully allocate

monies among different classes of assets (e.g., equity versus cash) to capitalize on market

ascendancy and/or to avoid downturns. If we could observe the portfolio composition of

mutual funds at the same frequency as we observe the returns, we could infer funds’

market timing by testing whether the portfolio holdings anticipate market moves. Graham

and Harvey (1996) empirically test market timing using investment newsletters’ asset

allocation recommendations.

Holdings, however, are often not available (especially in academic studies), which

limits the market timing analysis to the returns of funds and benchmark portfolios only.

The return-based method, on the other hand, only requires data on the ex post returns of

funds and the relevant market indices. The two most popular methods along this line are

those proposed by Treynor and Mazuy (1966) (henceforth ‘‘TM’’) and Henriksson and

Merton (1981) (henceforth ‘‘HM’’).

Most of the work on mutual fund performance measurement extends the CAPM or

multi-factor analysis of securities and portfolios to mutual funds. There has been

controversy over using such a metric to evaluate mutual fund performance. The static

a – b analysis misses the diversified and dynamic aspects of managed portfolios (Admati et

al., 1986; Ferson and Schadt, 1996; Becker et al., 1999; Ferson and Khang, 2001). A fund

manager may vary her portfolio’s exposure to the market or other risk factors, or alter the

fund’s correlation to the benchmark index in response to the incentive she faces (Chevalier

and Ellison, 1997). Consequently, the systematic part of the fund’s risk can be mis-

estimated when its manager is trying to time the market, and existing measures may

incorrectly attribute performance to funds, or fail to attribute superior returns to an

informed manager (Grinblatt and Titman, 1989). To address these issues, there has been a

great deal of study on capturing the effect of conditioning information on timing

performance measures (Ferson and Schadt, 1996; Becker et al., 1999; Ferson and Khang,

2001), controlling for spurious timing arising from not holding the benchmark

(Jagannathan and Korazjczyk, 1986; Breen et al., 1986), decomposing abnormal perform-

ance into selectivity and timing (Admati et al., 1986; Grinblatt and Titman, 1989), and

minimizing the loss of test power due to sampling frequencies (Goetzmann et al., 2000;

Bollen and Busse, 2001).

In this paper, we develop an independent test to measure the market timing ability of

portfolio managers without resorting to the estimation of a’s or b’s. The test is based on

the simple idea that a successful market timer’s fund rises significantly when the market

rises and falls slightly when the market drops. The nonparametric test has the following

properties. First, it is easy to implement because it only requires the ex post returns of

funds and their benchmark portfolios. Second, the test statistic is not affected by the

manager’s risk aversion because it separates the quality of timing information a fund

manager possesses from the aggressiveness of the reaction to such information. Third, the

test is more robust to different information and incentive structures, as well as to timing

frequencies and underlying distributions, than existing timing measures. Finally, the

method developed in this paper is readily applicable to analyzing the market timing

ability of financial advisors or newsletters (Graham and Harvey, 1996), or the timing

behavior of individual investors (Odean, 1998; Barber and Odean, 2000).

W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

401

The rest of the paper is organized as follows: Section 2 presents the nonparametric

statistic of market timing and compares it with the TM and HM methods. Section 3 applies

the method to a data set of mutual funds with different benchmark indices. Section 4

concludes.

2. Model

2.1. Market timing test statistics

We assume that a money manager’s timing information is independent of her

information about individual securities. This is a fairly standard assumption in the

performance measurement literature (e.g., see Admati et al., 1986; Grinblatt and Titman,

1989).1 With independent selectivity and timing, we have the following market model of

fund returns (all returns are expressed in excess of the risk-free rate):

ri;tþ1 ¼ ai þ bi;trm;tþ1 þ ei;tþ1;

ð1Þ

where i is the subscript for individual funds throughout this paper. bi,t is a random variable

adapted to the information available to the manager at time t and rm represents the return of

the relevant market (which can be a subset of the total market) in which the mutual fund

invests. It is the benchmark portfolio return against which the fund is evaluated. In the

simplest case, a market timer decides on bt at date t and invests bt percent in the market

portfolio and the rest in bonds until date t + 1. Eq. (1) represents the return process from

such a timing strategy.

For a triplet {rm,t , rm,t , rm,t } sampled from any three periods such that rm,t < rm,t <

1

2

3

1

2

rm,t , an informed manager should, on average, maintain a higher average b in the trm,t ,

3

2

rm,t b range than in the trm,t , rm,t b range. The b estimates for both ranges (given two

3

1

2

observations for each range) are (ri,t À ri,t )/(rm,t À rm,t ) and (ri,t À ri,t )/(rm,t À rm,t ),

2

1

2

1

3

2

3

2

respectively. Accordingly, we propose using the probability

r

À r

r

À r

h ¼

i;t

i;t

i;t

i;t

2Pr

3

2

>

2

1

À 1;

ð2Þ

rm;t À r

r

À r

3

m;t2

m;t2

m;t1

as a statistic of market timing ability. We motivate this market timing measure as follows.

A manager’s timing ability is determined by the relevance and accuracy of her

information. Let rˆm,t + 1 = E(rm,t + 1 | It) be the manager’s prediction about the next-period

market return based on It, her information set (both public and private) at time t. If It is not

informative at all, then the conditional distribution equals the unconditional one, that is,

f(rm,t + 1jrˆm,t + 1) = f(rˆm,t + 1), where f(Á) stands for the probability density function. In this

case, the conditional forecast equals the unconditional one and the manager would not be

able to tell when the market will enjoy relatively high returns. More specifically, for two

1

Correlated timing and selectivity information would in general cause technical difficulties in separating

abnormal performance due to timing from that due to selectivity. For a detailed discussion, see Grinblatt and

Titman (1989).

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W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

periods, t1 p t2, the following parameter takes the value of zero in the absence of timing

information2:

m ¼ Prðˆrm;t1þ1 > ˆrm;t2þ1 j rm;t1þ1 > rm;t2þ1Þ À Prðˆrm;t1þ1 < ˆrm;t2þ1 j rm;t1þ1 > rm;t2þ1Þ

¼ 2Prðˆrm;t1þ1 > ˆrm;t2þ1 j rm;t1þ1 > rm;t2þ1Þ À 1:

ð3Þ

At the other extreme, if the forecast is always perfect, that is, rˆm,t + 1 u rm,t + 1, then m attains

its upper bound of one. Symmetrically, m = À 1 represents perfectly perverse market

timing. Therefore, the value of ma[ À 1,1] indicates the fund manager’s market timing

ability: the more accurate the information It the higher the value of m. The next step is to

find a relationship between the manager’s forecast (rˆm,t + 1) and her action (bt) so that h

defined in Eq. (2) is a valid proxy of m.

Suppose the manager receives a favorable signal that leads to a high rˆm,t + 1. How

much market exposure (bt) the manager would like to take apparently depends on two

factors: the precision of the forecast and the aggressiveness with which she uses her own

information. The first part concerns natural ability, while the latter can be affected by the

manager’s risk aversion. Grinblatt and Titman (1989) show that an investor who has

independent timing and selectivity information and non-increasing absolute risk aver-

sion3 would increase bt in Eq. (1) as information about the market becomes more

favorable, or

Bbt

B

> 0. Combining

Bbt > 0 with Eq. (3), we see that the following

ˆ

rm;tþ1

Bˆrm;tþ1

probability is greater than zero if and only if the manager possesses superior timing

information:

2Prðbt > b Arm;t

1

t2

1 þ1 > rm;t2 þ1 Þ À 1:

ð4Þ

From the analysis above, therefore, superior timing ability m>0 (defined in Eq. (2))

translates into h>0 (defined in Eq. (2)) if a manager loads on more market risk when signals

about future market returns are more favorable. Eq. (2) is testable because the sample

analogue of h can be formed. Under the null hypothesis of no timing ability, the b has no

correlation with the market return, in which case the statistic h assumes the neutral value of

zero. Intuitively, an uninformed manager would move the market exposure of her portfolio

in the right direction as often as she would do in the wrong direction. Note that a triplet {ri,t ,1

ri,t , ri,t } is convex vis-a`-vis the market return if and only if (ri,t À ri,t )/(rm,t À rm,t )>

2

3

3

2

3

2

(ri,t À ri,t )/(rm,t À rm,t ). Therefore, h measures the probability that the fund returns bear a

2

1

2

1

convex relation with the market returns in excess of that of a concave relation.

2

The HM method tests whether the probability Prðˆrm;tþ1 > 0 j rm;tþ1 > 0Þ þ Prðˆrm;tþ1 < 0 j rm;tþ1 < 0Þis

greater than one. When the HM model is the correct specification, our measure picks up the manager’s timing

ability among a subset of triplets where at least two observations of market returns are of opposite signs. In

general, our measure allows the manager to make finer forecasts and uses more information in the return data by

looking at all triplets frm;t1þ1; rm;t2þ1; rm;t3þ1g for t1 6¼ t2 6¼ t3:

3

Non-increasing absolute risk aversion requires that the investor’s risk aversion measured byÀ u00ðwÞ be non-

u0ðwÞ

increasing in the wealth level w. Commonly used utility functions, such as the exponential, power, and log

utilities, all meet this criterion.

W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

403

The sample analogue to h becomes a natural candidate as a statistic. It is a U-statistic

with kernel of order three:

À1

X

ˆ

n

r

À r

r

À r

h

i;t3

i;t2

i;t2

i;t1

n ¼

sign

>

;

ð5Þ

3

r

À r

r

À r

r

m;t

m;t

m;t

m;t

m;t <r

<r

3

2

2

1

1

m;t2

m;t3

where n is the sample size and sign(Á) is the sign function that assumes value 1 ( À 1) if the

argument is positive (negative) and equals zero if the argument is zero. By the property of

pﬃﬃﬃ

U-statistics, hˆn is a

n-consistent and asymptotically normal estimator for h (Serfling,

pﬃﬃﬃ

d

1980; Abrevaya and Jiang, 2001). That is,

nð ˆ

hn À hÞ ! N ð0; r2ˆ Þ when n ! l. Further

hn

hˆn, as defined in Eq. (5), is the least variance estimator among all unbiased estimators for

the population coefficient h.

Abrevaya and Jiang (2001) provide the asymptotic distribution of the hˆn statistic. Let

zt u (rt , rm,t ), j={1, 2, 3}, and denote the kernel function of hˆn by

j

j

j

ri;t À ri;t

rt À ri;t

hðz

3

2

2

1

t ; z ; z Þ ¼ sign

>

j r

< r

< r

:

1

t2

t3

m;t

m;t

m;t

r

1

2

3

m;t À r

r

À r

3

m;t2

m;t2

m;t1

A consistent estimator of the standard error of hˆn is derived in Abrevaya and Jiang (2001):

!2

9 X

n

À1

n

X

ˆ

r2ˆ ¼

hðz ; z ; z Þ À ˆ

h

:

ð6Þ

h

t1

t2

t3

n

n

n

2

t1¼1

t2;t3

Simulation results in Abrevaya and Jiang (2001) show that the size of the test is very

accurate4 if we use the bootstrap method in standard error estimation for sample sizes

below 50 and use the asymptotic formula for larger sample sizes.

2.2. Properties

The new market timing measure (h) has a ready interpretation as the probability that a

fund manager takes relatively more systematic risk in a higher return period than in a low

return one. Since the seminal work of Treynor and Mazuy (1966) and Henriksson and

Merton (1981), there has been much work extending these measures in order to relax their

restrictive behavioral and distribution assumptions while retaining their intuitive appeal,

ease of implementation, and minimal data requirements.5 In this subsection, we discuss the

4

Using 1000 simulations, rejection rates at 5% significance level are between 4.5% and 5.5% for all error

specifications.

5

Goetzmann et al. (2000) had an excellent review of the research that addresses the limitation of the TM and

HM timing measures.

404

W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

contribution of the nonparametric timing measure on these grounds and point out its

limitations. The fund subscript i will henceforth be omitted where there is no confusion.

2.2.1. Information structure and behavioral assumptions

The nonparametric measure allows a more flexible specification of a fund manager’s

response to information. We require bt to be a non-decreasing function of rˆm,t + 1, that is,

the manager sets a higher b for the fund when her forecast of the next-period market return

is more favorable. Grinblatt and Titman (1989) show that sufficient conditions for this to

hold are i.i.d. random noise in market returns, independent selectivity and timing

information, and non-increasing absolute risk aversion. This requirement is less stringent

than those of the TM and HM measures, which require linear or binary response function

by the manager. The i.i.d. assumption, however, rules out heteroscedasticity in returns and

hence volatility timing by money managers. We will relax this assumption and discuss the

possible impact of volatility timing in a later section.

In general, a fund manager’s reaction to information depends on her risk aversion

(which could be affected by the incentive she faces) as well as her natural ability. The

functional form of such a response is difficult to specify without being somewhat arbitrary.

For example, the TM measure uses the following quadratic regression of a fund’s returns:

rtþ1 ¼ a þ brm;tþ1 þ c½rm;tþ1Š2 þ etþ1;

ð7Þ

where superior timing shows up in a positive coefficient ci. As analyzed in Admati et al.

(1986), the return process of Eq. (7) comes out of a linear response by the fund manager in

the form of:

bt ¼ ¯b þ k½ˆrm;tþ1 À EðrmÞŠ:

ð8Þ

The linear response function is consistent with the manager’s acting as if she were

maximizing the expected utility of a CARA preference. However, such an assumption is

questionable if the fund manager maximizes the utility related to her own payoff under the

incentive she faces instead of the fund’s total return. The deviation from maximizing a

CARA preference is large when there is non-linearity in the incentive, explicitly or

implicitly, in the forms of benchmark evaluation (Admati and Pfleiderer, 1997), option

compensation (Carpenter, 2000), or non-linear flow-to-performance responses by fund

investors (Chevalier and Ellison, 1997).

The HM measure, on the other hand, assumes that a manager takes only two b values—

a high b when she expects the market return to exceed the risk-free rate and a low b when

otherwise. The binary-b strategy results in the following return model:

rtþ1 ¼ a þ brm;tþ1 þ c½rm;tþ1Šþ þ etþ1;

ð9Þ

where [rm,t + 1]+ = max(0, rm,t + 1). The coefficient on [rm,t + 1]+ represents the value added

by effective timing that is equivalent to a call option on the market portfolio where the

exercise price equals the risk-free rate. Such a specification, while intuitive, is highly

restrictive as well. After all, there is no reason to expect a uniform reaction to information

by all fund managers. In comparison, the nonparametric measure offers more flexibility. It

W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

405

only requires the reaction function to be non-decreasing in the manager’s forecast of

market return.

When a linear reaction function is the correct specification, the nonparametric measure

gives the same result as the TM measure. In the TM model, the manager’s private signal,

yt, is generated according to

yt ¼ rm;tþ1 þ gt;

ð10Þ

where gt is a normal random variable that is independent of rm,t + 1 and is i.i.d. across time.

Timing ability is represented by the inverse of the variance of the noise term. For any two

gt and gt from two periods t1p t2, we can calculate Eq. (3) as follows:

1

2

m ¼ 2Prðgt À g < rm;t

2

t1

1 þ1 À rm;t2 þ1 j rm;t1 þ1 > rm;t2 þ1 Š À 1

!

Ar

¼

m;t

2U

1 þ1 À rm;t2 þ1 A

pﬃﬃﬃ

À 1;

ð11Þ

2rg

where U(Á) stands for the cumulative probability function of the standard normal

distribution. It is easy to see that m is monotonically increasing in 1/rg, the precision of

the private signal. An infinitely noisy signal (rg = l) leads to m = 0 (no timing) and a

perfect signal (rg = 0) implies m = 1 (perfect timing). Therefore, the nonparametric measure

will identify a good timer who adopts the TM timing strategy.

2.2.2. Ability versus response

A fund manager’s market timing performance relies on both the quality of her private

information (ability) and the aggressiveness with which the manager reacts to her

information (response). This constitutes a dichotomy that is difficult to decompose.

Except for special cases, existing performance measures are not able to extract the

information-related component of performance. As Grinblatt and Titman (1989) point

out, it would be better if performance measures (in addition to detecting abnormal

performance) could ‘‘also select the more informed of two [managers]’’. An investor

should be more concerned with the quality of the manager’s information than with the

manager’s aggressiveness because the investor can choose the proportion of her wealth

invested in the fund in response to the manager’s ability.

The TM and HM measures reflect both aspects of market timing. We see that the

estimated cˆTM in the TM regression will pick up the coefficient in the linear reaction

function (the k term in Eq. (8)). Hence, more aggressive funds can show up with

higher cˆTM. The cˆTM coefficient in the HM model is an unbiased estimate for the

product D(bH À bL), where D is the probability defined in footnote 4, and bH(bL) is the

manager’s target b when the predicted market excess return is positive (negative).

Thus, both ability (the D term) and aggressiveness (the bH À bL term) are reflected in

the estimated timing. The nonparametric statistic, on the other hand, measures how

often a manager correctly ranks a market movement and appropriately acts on it,

instead of measuring how aggressively she acts on it. We see that, in the linear

406

W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

response case (as in Eq. (8)), the k coefficient cancels out in the nonparametric

measure because

ˆ

r

r

À ˆr

r

ˆ

r

r

À ˆr

r

h ¼

m;t

m;t

m;t

m;t

m;t

m;t

m;t

m;t

2Pr k

3

3

2

2 > k

2

2

1

1 j rm;t < rm;t < rm;t

À 1

r

1

2

3

m;t À r

r

À r

3

m;t2

m;t2

m;t1

ˆ

r

r

À ˆr

r

ˆr

r

À ˆr

r

¼

m;t

m;t

m;t

m;t

m;t

m;t

m;t

m;t

2Pr

3

3

2

2 >

2

2

1

1 j rm;t < rm;t < rm;t

À 1:

r

1

2

3

m;t À r

r

À r

3

m;t2

m;t2

m;t1

Thus, our measure largely reflects the information quality component of performance.

Based on this analysis, we also see that there is great complementarity between the

nonparametric method and the two other methods. Used together in empirical work,

they can offer a more complete picture of the market timing performance of fund

managers.

2.2.3. Conditional information

The nonparametric measure can be extended to the context of conditional market

timing. The literature on conditional performance evaluation stresses the importance of

distinguishing performance that merely reflects publicly available information (as captured

by a set of instrumental variables) from performance that can be attributed to better

information. The conditional market timing approach (see, e.g., Ferson and Schadt, 1996;

Graham and Harvey, 1996; Becker et al., 1999; Ferson and Khang, 2001) assumes that

investors can time the market on their own using readily available public information, or

that by trading on other accounts they can undo any perverse timing that is predicted from

the public information. Under such circumstances, the real contribution of a fund manager

would be successful timing on the residual part of market returns that is not predictable

from public information.

Let r˜m,t and r˜i,t , j = 1, 2, 3, be the residuals of market returns and the fund return that

j

cannot be explained by lagged instrumental variables. The following statistic then proxies

the probability that a fund manager loads on more market risk when the market return is

higher, controlled for public information in both market and fund returns:

À1

X

˜

n

˜

r

À ˜r

˜

r

À ˜r

h

i;t3

i;t2

i;t2

i;t1

n ¼

sign

>

:

ð12Þ

3

˜r

À ˜r

˜

r

À ˜r

˜

r

m;t3

m;t2

m;t2

m;t1

m;t <˜

r

<˜r

1

m;t2

m;t3

Theoretically, h in Eq. (2) and h˜ in Eq. (12) can have different magnitudes or even

different signs because the probabilities are conditional on different states. That is, a

manager who successfully times the unpredicted part of the market return can show up as a

mis-timer on the gross market return if we do not control for public information. Both

public and private information can be used to enhance portfolio returns, but a truly

informed manager should have superior market timing based on information beyond that

which is readily available to the public.

W. Jiang / Journal of Empirical Finance 10 (2003) 399–425

407

2.2.4. Statistical robustness

Breen et al. (1986) point out that heteroscedasticity can significantly affect the

conclusions of the HM tests. Jagannathan and Korajczyk (1986) and Goetzmann et al.

(2000) demonstrate the bias of the HM measure due to skewness. The asymptotic

distribution of the hˆn statistic, on the other hand, is unaffected by heteroscedasticity or

skewness. Further, hˆn in Eq. (5) is the least variance estimator among all unbiased

estimators of h in Eq. (2). The simulation results shown in Abrevaya and Jiang (2001)

demonstrate that the nonparametric test has accurate size even for small samples and is

robust (in terms of both the value of the statistic and its standard error) to outliers, non-

normality, and heteroscedasticity that are common in financial data.6 However, we do

require the errors in Eq. (1) to be serially uncorrelated. As we will be using monthly return

data for our empirical test, this assumption is not a serious concern. However, the statistic

can be biased when applied to high-frequency data.7

The nonparametric method also offers a timing measure that has little correlation with the

estimation error in the standard selectivity measures. TM or HM type regression models

would produce a spurious negative correlation between estimated selectivity and timing

because of the negatively correlated sampling errors between the two estimates (Jagannathan

and Korajczyk, 1986; Coggin, 1993; Kothari and Warner, in press). Our simulation shows

that a significant negative correlation between the two estimated abilities will occur in the

TM or HM models (or between the selectivity measure from one model and the timing

measure from the other) even when the correlation is non-existent. Coggin et al. (1993) and

Goetzmann et al. (2000) have similar results. On the other hand, the correlation between hˆn

and the selectivity measures from standard regression models is close to the truth.

2.2.5. Model specification and potential bias

In this section, we discuss three specification issues that can affect the consistency and

power of market timing tests: the separability of timing from selectivity; the difference

between the frequencies at which data are sampled and at which the manager times the

market; the relationship between market timing and volatility timing. The nonparametric

measure is more robust to model specifications than the TM and HM measures, though it

does not overcome all the biases.

A manager can enhance portfolio returns by selecting securities and by timing the

market. Decomposing returns in this fashion, however, is empirically difficult (Admati et

al., 1986; Grinblatt and Titman, 1989; Coggin et al., 1993; Kothari and Warner, in press).

Our measure relies on two common assumptions to avoid detecting spurious timing

because of selectivity issues. The first assumption is that a portfolio manager’s information

on the selectivity side (movement of individual securities) is independent of her

6

For example, Bollen and Busse (2001) test the hypothesis that fund returns are normally distributed and

reject normality at the 1% level. They also conjecture that the relative skewness of market and fund returns is

driven by the crash of 1987 and other smaller crashes in the sample.

7

When applying the measure to high-frequency data, we would recommend the following modification in

forming hˆn: use only triplet observations {rm,t + 1, rm,t + 1, rm,t + 1} that are at least k periods apart, where k is the

1

2

3

lag of possible serial correlation, and rescale the statistics by the number of triplets actually used, denote it m. For

À1

n

any finite k, m !

when n ! l.

3

408

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information on the timing side (market movement). In practice, this requires that each

individual security constitutes only a small portion of a diversified portfolio and has a

negligible impact on the whole market (the manager does not select ‘‘too many’’ stocks at

one time, either); or the fund manager must act on selectivity at a much lower frequency

than on market timing (so that the manager keeps roughly constant the composition of her

risky portfolio when trying to time the market). The second assumption is that the portfolio

does not contain derivatives. Jagannanthan and Korajczk (1986) show that buying call

options, for example, can induce spurious timing ability. Kosik and Pontiff (1999) find that

21% of the 679 domestic equity funds in their sample hold derivative securities, but

detailed information about their derivative holdings is not available. Our measure, like the

TM and HM measures, cannot distinguish market timing from option-related spurious

timing.

For most timing measures, biases arise when the econometrician observes return data at

a frequency different from the frequency at which the manager times the market.

Goetzmann et al. (2000) show that monthly evaluation of daily timers using the HM

measure is biased severely downward. At the same time, a major component of timing

skill would show up as security-selection skill. Bollen and Busse (2001) show that the

results of standard timing tests are sensitive to the frequency of data used. Ferson and

Khang (2001) point out that an ‘‘interim trading bias’’ can arise when expected returns are

time varying and managers trade between return observation dates. The major source of

bias is the mis-specification of the regressor [rm]+ in the HM equation that should take

different values depending on the actual timing frequency rather than uniform frequencies

(such as monthly). Goetzmann et al. (2000) suggest replacing the monthly option value

[rm]+ with its accumulated daily option value when daily data of fund returns are not

readily available. Simulations show that the nonparametric measure is more robust to the

difference between timing frequency and sampling frequency because it does not rely on a

regression involving a potentially unknown regressor [rm]+ measured at the ‘‘right’’

frequency. Ferson and Khang (2001) use conditional portfolio weights to control for

interim trading bias as well as for trading on public information. Since our measure does

not use portfolio weights, it can potentially be subject to such bias.

The third model specification issue comes from the fact that the manager might be

timing market volatilities as well as market returns. Busse (1999) shows that funds attempt

to decrease market exposure when market volatility is high. Laplante (2001) shows that

observed mutual fund positions are not informative about future market volatility. If

volatility and expected return are uncorrelated, then our market timing measure remains

consistent in the presence of volatility timing. If the correlation is positive, the market

timing measure would underestimate the information quality of a successful volatility

timing manager.8 The opposite is true when the relation is negative. Research on the

relationship between the expected return and volatility (see, e.g., Breen et al., 1989;

Glosten et al., 1993; Busse, 1999) finds that the relation between return and volatility is

weak, both conditionally and unconditionally. If this is the case, the manager’s timing on

return and volatility likely to be weakly related.

8

If the manager tries to time the volatility, she may reduce market exposure even when the expected return is

high, if high-expected return tends to go with high volatility.

# Document Outline

- A nonparametric test of market timing
- Introduction
- Model
- Market timing test statistics
- Properties
- Information structure and behavioral assumptions
- Ability versus response
- Conditional information
- Statistical robustness
- Model specification and potential bias

- Simulations

- Testing the market timing of mutual funds
- Data
- Do funds out-guess the market?
- Some related questions
- Does experience matter?
- Do small funds fare better?
- Is high turnover rate justified as timing?
- Do investor flows affect market timing?

- Conclusion
- Acknowledgements
- References