Asymmetric Information and the Predictability ofReal Estate Returns*

Michael CooperKrannert School of Management, Purdue University

David H. DownsTerry College of Business, University of Georgia

Gary A. PattersonSchool of Management, SUNY-New Paltz

Abstract

This paper examines the relation between systematic price changes and the heterogeneity ofinvestors’ information sets in real estate asset markets. The empirical implications rely on atheoretical economy in which information asymmetry alters the dynamic relation between returnsand trading volume. We employ a filter-rule methodology to determine predictability in returnsand augment the return-based conditioning set with trading volume. The additional conditioninginformation is necessary since the model is underspecified when predictability is based onreturns alone. Our results provide new insight into the co-existence of informational and non-informational exchange in the speculative markets for real estate assets. Specifically, we findthat the predictability of real estate returns is generally more indicative of portfolio rebalancingeffects than an adverse selection problem. Importantly, these results are unique in addressingthe time-variation in information asymmetry.

Forthcoming Journal of Real Estate Finance and Economics

Final version: October 1998

JEL Classification: G10, G14, R33Key Words: Information, Predictability, Real Estate

*Address Correspondence to: David H. Downs, Terry College of Business, University of Georgia, AthensGA 30602-6255 or email to ddowns@cba.uga.edu

We wish to thank Crocker Liu and the participants of the AREUEA and FMA meetings for their helpfulsuggestions. We are especially grateful to Chinmoy Ghosh.

1

Asymmetric Information and the Predictability of Real Estate Returns

1. Introduction

The predictability of asset returns has been the focus of a large body of academic research with studies

attributing this apparent phenomenon to informational inefficiency, investor irrationality, time variation in

risk premia, and other market-specific effects. Recently, Mei and Gao (1995) examine whether the short-

term predictability of real estate assets is exploitable in an economically meaningful sense.1 They find that

the real estate security market is efficient with respect to trading profits, and thus, the real estate security

markets are not accessible to competent arbitrageurs. Their study portrays a general condition of efficiency

in the real estate markets without considering specific market conditions that may alter the behavior of

asset prices. In contrast, other studies suggest that market conditions may influence informational

efficiency and, consequently, asset prices. Damodaran and Liu (1993) conduct a study that focuses upon

events producing periods of asymmetric information that affect price movements

in real estate assets. They examine a sample of REITs that choose to reappraise themselves, an action that

endows the REIT insiders with private information. By identifying an event that heightens information

asymmetry, their study demonstrates that the trading activity of informed investors could influence the price

formation process. However, the extent to which informed trading might influence the predictability of real

estate assets is largely an empirical question.

In this paper we study the predictability of real estate returns for evidence of information-based

trading in a speculative market for real estate assets. We conduct our research of speculative markets as

presented by Wang (1994) and assume the existence of heterogeneous traders and asymmetric information.

In this economy, investors trade for informational and non-informational purposes. The informed investors

possess heterogeneous endowments of (1) private information about the future cash flows of the underlying

asset and (2) private investment opportunities. In this market, the uninformed investor rationally extracts

information from prices and other public signals to estimate expected returns. Consequently, the two

2

classes of investors trade competitively based on (a) the non-informational motives of the uninformed, (b)

the informational motives of the informed (i.e., trading based on private information), and (c) the non-

informational motives of the informed (i.e., portfolio rebalancing to accommodate private investment

opportunities). Thus, our paper extends the prior research of Mei and Gao (1995) and Damodaran and Liu

(1993) by examining predictability in short-term returns in a market containing information-based trading

in real estate securities and where investors are heterogeneous in their information and private investment

opportunities (Wang, 1994).2

The heterogeneities in Wang’s model serve to characterize the popular idea of REIT insiders as

private-market participants capitalizing on financing opportunities in the publicly traded markets. Wang

shows that heterogeneity among investors gives rise to different dynamic relations between trading volume

and returns.3 In essence, a high return accompanied by high volume implies low future returns (price

reversals) if informed investors are trading for changes in their private investment opportunities and not

because of private information. Yet when informed investors condition their trades upon private

information, then high future returns (price continuations) are expected when high returns are accompanied

by high trading volume. The model demonstrates that the underlying motivation behind investor behavior

produces different volume-return interactions that affect the pattern of return behavior. Wang’s model

allows us to characterize the nature of investor heterogeneity by examining the pattern of expected returns

that emerges from the interaction between returns and trading volume.

A testable implication of the dynamic relation between returns and trading volume is that the return

reversals documented by Mei and Gao (1995) will be correlated with volume. To test this possibility we

form contrarian portfolios with the aid of a filter rule methodology (Cooper, 1998). This approach avoids

the criticism levied against previous short-horizon contrarian papers that base their portfolio weights on the

cross-sectional distribution of lagged returns.4 More importantly, the filter method offers flexibility for the

detection of nonlinearities in the predictability of price changes. We test the effect of volume on the

3

autocorrelation of weekly returns with the realized portfolio returns acting as proxies for the expected

returns in the Wang model.

We find strong evidence of nonlinearities in the predictability of real estate returns when we

introduce volume into the trading rule. Specifically, the price-volume dynamic differs between high and low

volume periods, where the high volume periods reflect the exchange of real estate assets motivated by

private information. We also observe predictable patterns of return reversals when we form portfolios using

a filter rule based only upon lagged prices, which is consistent with earlier papers. Importantly, the study

highlights the adverse selection problem faced by investors who trade in public real estate markets where

representative insiders may have both private information and private investment opportunities.

The remainder of this paper is organized as follows. Section 2 describes the application of a filter

rule to determine return predictability. In Section 3 we describe the data and then present our empirical

results in Section 4; the analysis focuses on the price-volume dynamic for speculative trading of real estate

assets. We conclude in Section 5.

2. Empirical methods

2.1 Improving signal quality in short-term predictability

To address the testable implication concerning predictability (i.e., that return reversals are

correlated with volume), we employ two significant modifications to the overreaction portfolio formation

methodology used by Mei and Gao (1995). These modifications are designed to boost the “signal-to-noise”

ratio of the security selection process used to form contrarian portfolios. Specifically, our modifications to

the signal extraction process include (1) the use of filters, and (2) the use of a conditioning variable other

than price changes, namely volume.

The filter-rule method allows us to screen on the magnitude of lagged returns and percentage

changes in volume when forming loser and winner portfolios. In contrast, prior short-term contrarian

papers’ portfolio formation methodology (Lehmann, 1990, and Mei and Gao, 1995) typically emphasizes

4

forming portfolios by investing in all securities in their sample, giving greater weight to securities with

larger relative lagged cross-sectional returns. Including stocks regardless of lagged return magnitudes

results in inclusion of securities into the overreaction portfolios that may not be subject to investor

overreaction.5 In contrast, the filter portfolio formation method includes an asset in a loser (winner)

portfolio only if its lagged weekly return moved down (up) by a threshold amount. Hence, our method will

provide a more sensitive measure of predictability for analysis of the price-volume dynamic. Other papers

that use variations of the filter-rule method to boost the sensitivity of their analysis include Alexander

(1961), Fama and Blume (1966), Sweeney (1986, 1988), Brown and Harlow (1988), Lakonishok and

Vermaelen (1990), Bremer and Sweeney (1991), Corrado and Lee (1992), Cox and Peterson (1994), and

Fabozzi, Ma, Chittenden, and Pace (1995).

Our second method to improve the signal-to-noise ratio in a weekly overreaction portfolio strategy

is to utilize variables not directly derived from a security’s price. Because of the scarcity of

macroeconomic and microeconomic time series variables at shorter time intervals, a natural choice would

be to examine the time series properties of volume as it relates to subsequent weeks’ return behavior.

Theoretical papers that have taken this approach include Blume, Easley and O’Hara (1994), Campbell,

Grossman and Wang (1993), and Wang (1994) who present models suggesting there is valuable in-

formation in the time series of lagged volume for predicting a security’s price movement. Conrad, Hameed,

and Niden (1994) examine the interaction between lagged percentage changes in transactions, lagged

returns, and subsequent weekly returns to individual NASDAQ securities. They employ an overreaction

portfolio weighting scheme that produces returns from negative autocorrelation. Motivated by these results,

we incorporate a lagged, individual security volume measure into the overreaction portfolio formation rules.

Additionally, the joint use of volume and return filters allows this paper to examine the heterogeneity of

investor behavior.

5

2.2. Filter-rule methodology

The methodology we use is a first-order filter rule where lagged information from one week (i.e.,

returns, or returns and volume) is used to predict future returns. In all, six strategies are examined. The

first two strategies are price-only strategies. For example, if last week’s return is negative, it falls into the

strategy of “loser-price” filter. Hence, the two price-only strategies are “loser-price” and “winner-price,”

and they form portfolios that provide a baseline for interpreting the price-volume results. The remaining

four strategies incorporate both price and volume information. For example, if last week’s return and

percentage change in volume for a security are each negative, the security is assigned to a “loser-price |

low-volume” filter strategy. Likewise, the four price and volume strategies are “loser-price | low-volume”,

“loser-price | high-volume”, “winner-price | low-volume”, and “winner-price | high-volume.”

Past week’s returns are classified as winners or losers using the following criteria:

Return states = (1)

* if winner

if loser :5For

*)1( if winner

*)1( if loser :4 ,...,1 ,0For

1,

1,

1,

1,

*

*

*

*

≥

−<=

+<≤

+−≥>−=

−

−

−

−

AkR

k*ARk

AkRk*A

AkRk*Ak

ti

ti

ti

ti

Ak

Ak

Ak

Ak

where:Ri t, is the non-market adjusted return for security i in week t

k is the filter counter that ranges from 0, 1, …5.A is a parameter equal to 2 percent.

The low and high states for percentage change in individual security weekly volume (termed "volume

returns”) are defined to be:

Volume return states = (2)

* if high

* if low :5For

*)1(* if high

*)1(* if low :4,...,1 ,0For

1,*

1,*

1,*

1,*

≥

−<=

+<≤

+−≥>−=

−

−

−

−

CkVR

BkVRk

CkVRCk

BkVRBkk

tiCk

tiBk

tiCk

tiBk

where:VRi t, is the volume return for security i in week t

k is the filter counter that ranges from 0, 1, …5.B is a parameter equal to 15 percent.

6

C is a parameter equal to 50 percent.

The percentage change in individual security weekly volume, termed “volume returns,” adjusted for the

number of outstanding shares of a security, are defined as:

VRV

S

V

S

V

Si t

i t

i t

i t

i t

i t

i t,

,

,

,

,

,

,

= −

−

−

−

−

1

1

1

1

(3)

where:Si t, is the number of outstanding shares for security i in week t

Vi t, is the weekly volume for security i in week t .

Thus, k*A, k*B, and k*C are the grid increments for the price filters, low volume filters, and high volume

filters, respectively. For each of the strategies, the applicable price and volume filters are varied over their

respective domains, resulting in thirty-six sets of price and volume filter combinations for the price and

volume strategies.

The specific filter breakpoints are determined by examining the overall sample distributions of the

weekly price return and volume return from our sample of REITS and then choosing appropriate

filter values to span the distributions.6 Specifically, the return filters start at zero percent and increment in

steps of two percent, to a maximum (minimum) of positive (negative) ten percent for winner (loser) filters.

The low-volume return filters begin at zero percent and increment in steps of 15 percent to a minimum of

negative 75 percent. The high-volume return filters start out at zero percent and rise to a maximum of 250

percent in increments of fifty percent. The skewness in the volume return distribution produces the

asymmetry in the filters for volume.

For each combination of filter values, the securities whose lagged weekly returns (or returns and

volume) meet the filter constraints are formed into equally-weighted portfolios during week t. All portfolios

are held for a period of one week and then liquidated. The resulting portfolio’s mean return is calculated for

weeks in which non-zero positions are held. If mean returns of the portfolios are significantly different from

7

zero, this is taken as evidence in favor of return predictability. Thus, the null hypothesis of no predictability

is that the mean return of a portfolio equals zero.7

We use moment conditions to calculate test statistics for the mean returns since there may be some

dependence in the time series of portfolio returns, both contemporaneously and across periods. Specifically,

moment conditions are estimated with a generalized method of moments estimator (Hansen, 1982), and

Newey and West (1987) weights are employed on the variance/covariance matrix to compute the mean and

standard errors of the time series of trades for each portfolio and to perform comparisons between the

means of different strategies. Comparing the mean returns in a GMM framework has the advantage of

controlling for contemporaneous and time series correlations in the portfolio returns.

3. Data

To examine the interactions between lagged returns and volume, we construct a data set of Wednesday-

close to Wednesday-close weekly returns and weekly volume for 301 Real Estate Investment Trusts

(REITs) in the CRSP file between 1973 and 1995. Securities are included in the sample for week t if they

have daily volume in each of the previous ten trading days. Since the weights placed on individual securities

to form portfolios are based on non-market adjusted returns, the portfolio returns associated with our filter-

based strategy should not emanate from index autocorrelation.8

Table 1 reports sample statistics for the data set. The mean market equity over the entire sample

period is 119 million dollars and the average share price is approximately 15.4 dollars. REITs with a share

price less than 5 dollars are screened out of the sample as a precaution against bid-ask bounce effects. The

cross-sectional average of individual security weekly autocorrelation coefficients is –7.07 percent at the

first lag and –2.77 percent at the second lag. The negative autocorrelation is consistent with overreaction

for individual stocks or it may indicate the existence of a bid-ask spread effect. For this reason we also

report the four day return’s (e.g., the skip-day returns) first-order autocorrelation of -3.24 percent. The

8

magnitude of this statistic strongly suggests that negative autocorrelation induced by the bid-ask spread is

not driving the negative autocorrelations exhibited in the full weekly returns.

In addition, Table 1 presents descriptive statistics for the volume return (percentage change in

volume) measure used with the filter strategies. The measure of volume we use, itVR as defined in

equation 3, is the average percentage change in weekly volume, and over the 1294 week sample period,

VRi,t averages 67.4 percent.

4. Strategies that condition on price and volume

The empirical analysis in this paper relies on the use of information from trading volume to augment a

simple, lagged price filter rule. In turn, we attempt to determine whether predictability in real estate returns

is related to volume and interpret our findings in the context of the return-volume dynamics of Wang

(1994). Theoretical and empirical works suggest there is valuable information in the time series of lagged

volume for predicting a security’s price movements (Blume, Easley and O’Hara (1994), Campbell,

Grossman and Wang (1993) and Conrad, Hameed, and Niden (1994)). The inclusion of volume, which

represents the trading activity of investors, as well as an analysis of the behavior of portfolio returns, may

also provide important information about the level of heterogeneity of investors in the real estate market.

By concurrently examining return behavior and the investors’ information sets, Wang (1994) argues that

one may more accurately identify those periods where the predictability of returns is attributable to the

information heterogeneity in asset markets.

Figure 1 illustrates the general pattern in weekly portfolio returns when portfolio construction is

conditioned on different values of lagged returns and lagged volume. We find that conditioning on negative

percentage changes in volume results in increased negative return-autocorrelations for the more extreme

winner and loser filters. In contrast, conditioning on positive changes in volume results in decreased

negative return-autocorrelations for the more extreme winners and losers. Thus, we observe an inverse

9

relation between volume and the autocorrelation of returns, which supports critical tenets of the Wang

model. Namely, that prices alone are not sufficient to resolve the identification problem of investor

heterogeneity.

Table 2 presents detailed results for the graphical relations shown in figure 1 and allows us to

focus on a major element of our results. Specifically, we find the behavior of portfolio returns in week t

reveals two distinct interactions between price and volume based upon the level of lagged volume. Low

volume portfolios (i.e., stocks with a negative percentage change in volume from week t-2 to week t-1)

experience considerably greater reversals (i.e., stocks with positive (negative) returns in week t-1

experience larger magnitude negative (positive) returns in week t) than the high volume portfolios (i.e.,

stocks with a positive percentage change in volume from week t-2 to week t-1). Additionally, portfolios at

the extreme price | low volume filters consistently have greater reversals than do the portfolios with

comparable price-only filters. Panel A of Table 2 shows that portfolios from the loser-price | low-volume

strategy yield larger weekly portfolio returns than those produced by the price-only strategies shown in the

“No volume filter” row. For example, the average weekly returns approach 3 to 4 percent when we jointly

condition on extreme losers and low volume. When interpreted in the context of Wang’s model, an

environment with a greater proportion of trades motivated by private investment opportunities will produce

the observed reversals in portfolio returns.

The returns from winner portfolios also experience greater reversals when volume is added to the

portfolio formation process. Panel C shows the results of the winner-price | low-volume strategy where a -

0.376 percent (t-statistic = -1.676) weekly return from the “greater than 10%” filter for the price-only

strategy decreases to –3.330 percent (t-statistic = -3.21) when low-volume (<-75%) REITs are considered.

The return pattern associated with low volume is particularly evident at higher absolute magnitude price

filters in both winner (>10%) and loser (<-10%) portfolios. Overall, we observe large increases in the level

of reversals from incorporating volume information into the more extreme price filter rules. One

interpretation of this result is that transitory shifts in noninformational demand are more pronounced and

10

persistent when volume is low and returns are either very high or very low. This finding is consistent with

the argument that less active stocks are problematic not because there are too many informed traders, but

because there are too few uninformed ones (Easley, Kiefer, O’Hara, and Paperman, 1996). Downs and

Güner (1998) document a similar phenomenon among publicly traded real estate firms. Their paper shows

that the higher levels of information asymmetry contribute to a less-liquid, less-active REIT market,

perhaps due to the adverse selection problem faced by uninformed investors.

In contrast to the preceding discussion, conditioning on high volume lowers the magnitude of return

reversals, though the autocorrelation pattern remains strongly negative. The loser-price | high-volume re-

sults in Panel B of Table 2 reveal a trend of decreasing return reversals across the price filters for increas-

ing levels of volume. For example, loser portfolios formed by the “-10 percent or less” price-only filter

generated weekly returns of 2.20 percent (t-statistic = 6.34). At the same price filter, but also conditioning

on a weekly volume filter of greater than 250 percent, weekly returns diminish to 1.47 percent (t-statistic =

1.87). The same pattern of decreased reversals in subsequent weekly portfolio returns is seen in Panel D of

Table 2 when a winner-price | high-volume strategy is used to form portfolios. For both losers and winners,

the increased return reversals found in portfolios that condition on low volume are more evident at higher

price filters.9 These results suggest that the information content during periods of high volume reflects a

market that is responding to private information trades as well as trades motivated by changing investment

opportunities.

The empirical results show that the magnitude of return predictability varies considerably between

high and low volume periods. These results, with different price-volume dynamics, are consistent with an

economic model in which the relative proportion of trades motivated by private information and by

heterogeneous investment opportunities will affect the behavior of expected returns. By interpreting these

test results in the context of Wang’s model, the periods with high volume contain a greater proportion of

private information which leads to less predictable reversals in portfolio returns; the reverse is observed in

periods of low trading volume.

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4.1. Robustness

As a summary measure of the price-volume dynamic relation, we report the correlation of volume and

subsequent returns. This statistic allows us to formally test the relation between volume and future returns

and, as proposed by Wang (1994), to identify a dominant trading behavior in the speculative exchange of

real estate assets. Though the correlation analysis can not separate the data into high and low volume

periods, it measures the general price-volume relation that may help identify whether private information or

investment opportunities is, on average, the primary motivator of trading activity. For the entire sample

period, the correlation between the absolute value of weekly portfolio returns and the lagged volume filters

is negative and significant (-0.17 with a p-value of 0.038). This finding, interpreted in the context of

Wang’s model, supports the dominance of non-informationally motivated trading over informationally

motivated trading, which our earlier tests more clearly identify to be strongest during periods of low

volume.

We also consider the robustness of our results by drawing attention to the volume measure, itVR , used

in the filter-based method. Our findings show that the incorporation of volume improves the predictability

of returns, but in a manner not entirely consistent with a model in which investors trade because of their

differences. In other words, Wang (1994) emphasizes price changes accompanied by high volume when

identifying the type of information that produces price reversals or continuations. We find that large

absolute magnitude price changes (in week t-1) accompanied by high volume (e.g., the percentage change

in volume from week t-2 to week t-1 is positive) will reverse (in week t) but this pattern is stronger during

low volume periods. To ensure that our volume measure is not biasing the test results, we construct volume

measures using other time horizons. Specifically, we examine returns to strategies that condition on longer

horizon volume measures, so we construct two volume measures that employ an average of the last 4 and

20 weeks of volume to form trading shocks relative to longer term volume expectations. The substitute

volume measures are defined as:

12

VRit m

Vit

m Vi t j

j

m

m Vi t j

j

m,

( / ),

( / ),

=

− −=∑

−=∑

11

11

(4)

where m is equal to 4 or 20, the number of weeks used to form the volume average for security i in week t.

Similar to the calculations for one-week volume returns, weekly portfolio returns are calculated

using the four “price | volume” strategies (e.g., loser-price | low-volume). Subsequently, we construct the

summary measure of the price-volume dynamic relation as above. Recall this is the correlation of the abso-

lute value of the weekly portfolio returns and these alternate volume measures.

The correlation coefficients between weekly portfolio returns and the lagged 4-week and 20-week

volume measures are both negative and significant at -0.16 (p=0.05) and -0.20 (p=0.02), respectively.

These results, with their negative relationships between volume and expected returns, support our earlier

test results which suggest that speculative trading in real estate assets is dominated by portfolio rebalancing

and not private information. However, this strict test precludes the dynamic nature of information

asymmetry and the existence of a weak dominance in informational trading over other motives. For this

reason, we turn to our final set of results.

4.2 Additional analysis conditioning on price and volume

To examine the association between expected returns and the return-volume dynamic, we run a cross-

sectional regression of the average of all week t portfolio returns (RET) on week t-1 returns (RET_LAG):

RET = 0.48 + -0.11 ⋅ RET_LAG Adj. R2 = .56, N = 144. (5)(8.79) (-13.64) t-statistics in parentheses

To the extent that RET measures the expectation of future returns conditioning on current returns

for each price-volume filter, the significant negative parameter estimate is consistent with earlier studies

that find return reversals for real estate securities (Mei and Gao, 1995). Additionally, we use information

13

within trading volume to resolve the identification problem associated with investor heterogeneity. We

conduct an alternative test where we regress the average of all week t portfolio returns (RET) on week t-1

returns with an emphasis on high volume periods by the use of an interactive dummy variable. The dummy

variable HI_VOL has a value of 1 for all filter levels where volume in week t-1 is greater than or equal to

50 percent over the prior week’s volume. HI_VOL assumes a value of 0 otherwise.

RET = 0.48 + -0.12 ⋅ RET_LAG + 0.04 ⋅ RET_LAG ⋅ HI_VOL Adj. R2 = .58, N = 144. (6)(8.79) (-12.31) (2.61) t-statistics in parentheses

The coefficients on the RET_LAG and the interactive term are both significant at the 99 percent

level, but the point estimates have opposite signs.10 The test shows that an environment exists where low

(high) returns typically imply high (low) expected returns for real estate securities. Yet the positive

coefficient on the interactive term suggests that information in high trading volume periods dampens the

reversal effect that is found in periods with lower volume. This mitigating effect on return reversals is

consistent with the trading behavior of heterogeneously informed investors.

Finally, we examine the robustness of the price volume relationship across high and low vacancy

periods.11 This analysis allows us to assess the influence of the economic condition associated with the

underlying property market on the price-volume dynamic. As such, we construct an aggregate vacancy rate

measure for income-producing properties in the U.S. since 1980. The data are obtained from the 1997 U.S.

Bureau of the Census, Abstracts of the United States. A simple method to gauge the effect of occupancy

rates on the lagged return, lagged volume, subsequent reversal relationship is to form equally-weighted

portfolios of stocks in the bottom or top half of the price filters (i.e., week t-1 returns less than –6 percent

(loser-price) or greater than 6 percent (winner-price)) and the bottom half of the volume return filters (i.e.,

week t-1 volume returns less than 0 percent (low volume)). We name these portfolios loser-low and

winner-low, respectively, as they are formed by averaging the returns in the three most right columns of

Table 2, Panel A (loser-low) and the three most right columns of Table 2, Panel C (winner-low).

14

The pattern that emerges is one of greater reversals in high vacancy rate years relative to low

vacancy years for both the loser-low and winner-low portfolios. For example, the loser-low's average

weekly return is 3.85 percent (1.83 percent) in high (low) vacancy years (paired t-statisitic = 2.42). The

winner-low portfolio has average weekly returns of -1.44 percent in high vacancy periods versus returns of

-0.818 percent in the low vacancy periods (paired t-statistic = 0.95). Thus, the dampening of return

reversals during periods of low vacancy suggests that informed investors are trading on their private

information to extract what profits are available from public-market real estate. Just as our previous tests

found a relative increase in asymmetric information during high volume periods, the greater activity

occurring during strong real estate markets may provide opportunities for insiders to exploit their private

information more easily.

The greater return reversals observed during high vacancy years suggest there is a relative decrease

in asymmetric information during weak real estate markets. In the context of Wang’s model, the heightened

return reversals evident in high vacancy periods suggest that the trading of insiders in the real estate

securities market is driven by their private investment opportunities. In other words, the information

advantage of REIT insiders is less prominent in their trades than the need to rebalance portfolios in pursuit

of the private investment (e.g., vulture) opportunities often associated with a depressed real estate market.

5. Conclusion

Our study concludes that non-informational trading activity strictly dominates trading that is motivated by

the private information of informed investors. We arrive at this assessment by examining the predictability

of real estate returns in the context of the model in Wang (1994) where investor heterogeneity, in terms of

investment opportunities and information, leads to alternative specifications of a price-volume dynamic.

Our observance of strict dominance is based on a price-volume relation that exists across all periods and, in

this sense, is consistent with the Mei and Gao (1995) approach to documenting real estate market

overreaction. However, Damodaran and Liu (1993) provide compelling evidence that information

15

asymmetry, a principal determinant of the price-volume dynamic, changes across periods. Consequently,

our study attempts to reconcile some of the apparent contradictions in the real estate literature.

Our analysis of the predictability of real estate returns, conditioning on volume, demonstrates that

reversals are less pronounced during periods of high volume. This result is consistent with a weak-form

dominance of informationally motivated trading, which may explain why Mei and Gao do not find

economically significant price reversals. Most importantly, our findings contribute to the understanding of

the time-variation in the adverse selection problem of real estate investors.

Intuition suggests that a sophisticated investor with private information about publicly-traded real

estate assets might also have competing private-investment opportunities. Our research suggests that the

predictability in real estate returns is more a function of the rebalancing effects associated with the latter

endowment opportunities than the market corrections generated by the former asymmetric information (i.e.,

private information) opportunities. Importantly, our research also suggests that the risk of trading with a

more-informed investor is higher during periods of active trading as well as during periods when occupancy

rates are high.

16

Acknowledgements

This paper is an extended version of the second half of an earlier working paper with the same title. The

first half of the earlier working paper focuses specifically on trading strategies (Cooper, Downs and

Patterson, 1998). We wish to thank Crocker Liu, participants of the AREUEA and FMA meetings and two

anonymous reviewers for their helpful suggestions. We are especially grateful to Chinmoy Ghosh.

17

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Brown, K., and W. Harlow. (1988). “Market Overreaction: Magnitude and Intensity,” The Journal ofPortfolio Management 14 (Winter), 6-13.

Campbell, J.Y., S.J. Grossman and J. Wang. (1993). “Trading Volume and Serial Correlations in StockReturns,” The Quarterly Journal of Economics 108, 905-939.

Conrad, J., M. N. Gultekin, and G. Kaul. (1997). “Profitability and Riskiness of ContrarianPortfolio Strategies,” Journal of Business and Economic Statistics 15, 379-386.

Conrad, J., A. Hameed, and C. M. Niden. (1994). “Volume and Autocovariances in Short-HorizonIndividual Security Returns,” Journal of Finance 49, 1305-1329.

Cooper, M. (1998). “The Use of Filter Rules and Volume in Uncovering Short-Term Overreaction,”Review of Financial Studies, forthcoming.

Cooper, M., D.H. Downs, and G.A. Patterson. (1998). “Real Estate Securities and a Filter-based, Short-term Trading Strategy,” Journal of Real Estate Research, forthcoming.

Corrado, C., and S. Lee. (1992). “Filter Rule Tests of the Economic Significance of Serial Dependenciesin Daily Stock Returns,” The Journal of Financial Research 15, 369-387.

Cox, D., and D. Peterson. (1994). “Stock Returns following Large One-Day Declines: Evidence on Short-Term Reversals and Longer-Term Performance,” Journal of Finance 49, 255-267.

Damodaran, A., and C. Liu. (1993). “Insider Trading as a Signal of Private Information,” The Review ofFinancial Studies 6, 79-119.

Downs, D.H., and Z.N. Güner. (1998). “Is the Information Deficiency in Real Estate Evident in PublicMarket Trading?” Real Estate Economics, forthcoming.

Easley, D., N. Kiefer, M. O’Hara, and J. Paperman. (1996). “Liquidity, Information, and InfrequentlyTraded Stocks,” Journal of Finance 51, 1405-1436.

Fabozzi F., C. Ma, W. Chittenden, and R. Pace. (1995). “Predicting Intraday Price Reversals,” TheJournal of Portfolio Management 21 (Winter), 42-53.

18

Fama, E.F., and M.E. Blume. (1966). “Filter Rules and Stock-Market Trading,” Journal of Business 39,226-241.

Hansen, L. P. (1982). “Large Sample Properties of Generalized Method Of Moments Estimators,”Econometrica 50, 1029-1054.

Keim, D. B., and A. Madhavan. (1997). “Transaction Costs and Investment Style: An Inter-exchangeAnalysis of Institutional Equity Trades,” Journal of Financial Economics 46, 265-292.

Lakonishok, J., and T. Vermaelen. (1990). “Anomalous Price Behavior around Repurchase TenderOffers,” Journal of Finance 45, 455-477.

Lehmann, B. (1990). “Fads, Martingales, and Market Efficiency,” Quarterly Journal of Economics 105,1-28.

Ling, D., A. Naranjo, and M. Ryngaert. (1998). “The Predictability of Equity REIT Returns: TimeVariation and Economic Significance,” Journal of Real Estate Finance and Economics, forthcoming.

Ling, D., and M. Ryngaert. (1997). “Valuation Uncertainty, Institutional Involvement, and theUnderpricing of IPOs: The Case of REITs,” Journal of Financial Economics 43, 433-456.

Liu, C., and J. Mei. (1992). “Predictability of Returns on Equity REITs and Their Comovement withOther Assets,” Journal of Real Estate Finance and Economics 5, 401-418.

Lo, A.W., and A.C. MacKinlay. (1990). “When are Contrarian Profits Due to Stock MarketOverreaction?,” Review of Financial Studies 3, 175-205.

Mei, J., and C. Liu. (1994). “The Predictability of Real Estate Returns and Market Timing,” Journal ofReal Estate Finance and Economics 8, 115-135.

Mei, J., and B. Gao. (1995). “Price Reversal, Transaction Costs, and Arbitrage Profits in the Real EstateSecurities Market,” Journal of Real Estate Finance and Economics 11, 153-165.

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Sweeney, R.J. (1986). “Beating the Foreign Exchange Market,” Journal of Finance 41, 163-182.

Sweeney, R.J. (1988). “Some New Filter Rule Tests: Methods and Results,” Journal of Financial andQuantitative Analysis 23, 285-300.

Wang, J. (1994). “A Model of Competitive Stock Trading Volume,” Journal of Political Economy 102,127-168.

Wang, K., S. Chan and G. Gau. (1992). “Initial Public Offerings of Equity Offerings: AnomalousEvidence using REITs,” Journal of Financial Economics 31, 381-410.

19

Table 1. Summary statistics for the REIT sample (N=301), period from 1973 through 1995.

Mean Median Std. Dev. Min. Max. 1ρ (s.d.)

5 day return (%) 0.267 0.0 3.914 -46.078 112.00 -7.073(13.647)

4 day return (%) 0.205 0.0 3.532 -40.298 96.296 -3.236(11.402)

VRi t, (%) 67.363 -2.152 643.521 -100.00 1049.00 -17.810(12.226)

Capitalization ($, millions)

119 57.2 168 0.018 1760

Price ($ per share)

15.430 13.75 8.947 5.00 132

Five day return is a Wednesday-to-Wednesday close weekly holding period return. Four day return is a “skip-day” Wednesday-to-Tuesday close four day holding period return. REITs with prices less than $5 per shareare excluded from the sample. The mean, median, and standard deviation of capitalization and price arecalculated across time and across securities. The statistic ρ1 is the average first-order autocorrelationcoefficient of weekly returns of individual stocks. The population standard deviation is given in parentheses.Since the autocorrelation coefficients are not cross-sectionally independent, the reported standard deviationscannot be used to draw the usual inferences; they are presented as a measure of cross-sectional variation inthe autocorrelation coefficients.

20

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Lagged weekly volume return (%)

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Weekly Portfolio Returns

Figure 1. Weekly portfolio returns conditioning on percentage changes in price and volume.

This figure presents the average weekly portfolio returns based upon a contrarian trading strategy usingprice-volume filters. The greatest return reversals occur with the loser-price | low-volume portfolios in theupper right quadrant. The upper left quadrant presents the performance of the loser-price | high-volumeportfolios. The lower left and right quadrants respectively reflect the average weekly returns of the winner-price | high-volume and winner-price | low-volume portfolios.

21

Table 2. Weekly portfolio returns to price and volume strategies_______________________________________________________________________________

Panel A: Loser-price | Low-volume

Lagged weekly return filter (%)

<0 and≥≥-2

<-2 and≥≥-4

<-4 and≥≥-6

<-6 and≥≥-8

<-8 and≥≥-10

<-10

No Mean (%) 0.202 0.561 0.835 1.212 1.354 2.200volume Stand. dev. 1.944 2.519 3.342 4.236 5.499 7.008

filter N 1279 1260 1097 786 496 477t-Statistic 2.677 6.624 7.843 7.579 5.271 6.338

<0 and Mean (%) 0.157 0.650 0.911 1.251 0.889 3.476≥≥-15 Stand. dev. 2.645 4.068 4.369 5.426 5.394 5.764

N 833 602 285 111 50 41t-Statistic 1.567 4.008 3.892 2.727 0.890 3.953

<-15 and Mean (%) 0.201 0.503 0.866 1.426 2.188 1.240≥≥-30 Stand. dev. 2.958 3.660 3.874 4.684 7.734 7.727

N 878 641 273 121 49 44t-Statistic 1.874 3.455 3.065 3.234 1.696 1.840

Volume <-30 and Mean (%) 0.225 0.415 0.449 0.955 2.271 4.655Filter ≥≥-45 Stand. dev. 3.003 3.852 4.892 4.726 7.166 10.640(%) N 879 632 332 109 48 38

t-Statistic 2.004 2.757 1.796 2.402 2.494 2.570<-45 and Mean (%) 0.265 0.361 0.860 1.342 -0.738 3.039

≥≥-60 Stand. dev. 2.900 3.400 4.304 5.275 7.668 9.187N 834 579 279 121 42 42

t-Statistic 2.687 2.586 3.425 3.453 -0.531 2.433<-60 and Mean (%) 0.158 0.415 0.858 1.106 1.200 3.292

≥≥-75 Stand. dev. 2.608 4.181 5.393 5.303 4.582 7.596N 713 440 203 86 33 19

t-Statistic 1.690 2.274 2.188 2.011 1.206 1.558<-75 Mean (%) 0.014 0.190 1.156 2.118 1.451 3.784

Stand. dev. 2.790 3.533 4.514 6.590 6.923 10.699N 682 370 164 73 29 18

t-Statistic 0.127 1.080 2.937 2.800 1.411 3.085

22

Table 2. (continued)_________________________________________________________________________________

Panel B: Loser-price | High-volume

Lagged weekly return filter (%)

<0 and≥≥-2

<-2 and≥≥-4

<-4 and≥≥-6

<-6 and≥≥-8

<-8 and≥≥-10

<-10

No Mean (%) 0.202 0.561 0.835 1.212 1.354 2.200volume Stand. dev. 1.944 2.519 3.342 4.236 5.499 7.008

filter N 1279 1260 1097 786 496 477t-Statistic 2.677 6.624 7.843 7.579 5.271 6.338

≥≥0 and Mean (%) 0.118 0.664 0.823 1.100 0.706 1.892<50 Stand. dev. 2.724 3.741 4.709 4.724 5.499 7.887

N 1095 909 523 243 113 96t-Statistic 1.262 5.394 3.672 3.182 1.309 2.871

≥≥50 and Mean (%) 0.398 0.458 0.615 1.413 2.353 1.659<100 Stand. dev. 2.911 4.174 3.926 4.515 8.555 6.493

N 834 613 329 152 87 89t-Statistic 3.827 2.448 2.651 3.835 2.238 2.582

Volume ≥≥100 and Mean (%) 0.382 0.376 0.943 1.744 0.420 1.345Filter <150 Stand. dev. 3.012 3.430 5.413 4.155 5.198 7.841(%) N 604 410 206 106 56 54

t-Statistic 2.684 2.187 2.564 4.668 1.063 1.355≥≥150 and Mean (%) 0.436 0.366 1.024 1.499 0.933 0.289

<200 Stand. dev. 2.701 3.666 6.504 6.423 7.696 6.726N 401 243 142 65 45 50

t-Statistic 3.777 1.173 1.692 2.496 0.804 0.057≥≥200 and Mean (%) 0.169 0.507 1.359 1.225 2.102 1.505

<250 Stand. dev. 3.218 3.512 4.446 4.881 6.567 5.611N 316 179 94 44 23 32

t-Statistic 1.012 2.005 3.363 0.910 1.555 2.144≥≥250 Mean (%) 0.461 0.351 0.811 1.182 1.475 1.470

Stand. dev. 3.309 3.733 4.108 4.993 5.452 7.941N 621 460 274 169 99 148

t-Statistic 3.355 1.976 3.548 2.746 2.902 1.869

23

Table 2. (continued)_________________________________________________________________________________

Panel C: Winner-price | Low-volume

Lagged weekly return filter (%)

≥≥0 and<2

≥≥2 and<4

≥≥4 and<6

≥≥6 and<8

≥≥8 and<10

≥≥ 10

No Mean (%) 0.192 0.121 0.063 -0.086 -0.021 -0.376volume Stand. dev. 1.499 2.165 3.177 4.003 4.612 5.687

filter N 1278 1256 1151 925 662 748t-Statistic 3.597 1.665 0.643 -0.643 -0.064 -1.676

<0 and Mean (%) 0.221 0.216 0.207 -0.301 0.841 0.025≥≥-15 Stand. dev. 2.537 3.626 3.826 4.635 5.548 5.796

N 865 606 336 154 71 78t-Statistic 2.659 1.274 0.994 -0.941 1.470 -0.130

<-15 and Mean (%) 0.334 0.302 0.149 -0.841 -1.238 0.529≥≥-30 Stand. dev. 2.960 3.688 4.077 4.371 4.647 7.124

N 894 649 313 171 59 89t-Statistic 3.123 2.057 0.679 -2.648 -2.536 0.542

Volume <-30 and Mean (%) 0.266 0.087 -0.086 -0.029 -0.169 -0.524Filter ≥≥-45 Stand. dev. 2.616 3.203 3.813 4.107 4.840 6.245(%) N 879 606 338 165 69 72

t-Statistic 2.922 0.660 -0.335 0.024 -0.294 -0.764<-45 and Mean (%) 0.179 0.305 -0.465 0.726 -0.675 0.457

≥≥-60 Stand. dev. 2.706 3.483 3.760 6.019 4.357 8.986N 876 555 261 119 73 67

t-Statistic 1.964 1.981 -1.682 1.209 -1.176 0.374<-60 and Mean (%) 0.114 0.037 -0.417 -0.706 -1.876 -0.359

≥≥-75 Stand. dev. 2.575 3.756 3.883 4.312 5.956 6.257N 777 442 208 95 34 40

t-Statistic 1.057 0.068 -1.383 -1.420 -1.084 -0.890<-75 Mean (%) 0.205 -0.026 -0.532 -1.035 -0.978 -3.330

Stand. dev. 2.954 3.588 3.879 4.907 5.326 6.681N 780 385 149 101 44 34

t-Statistic 1.887 -0.235 -1.611 -1.807 -1.318 -3.208

24

Table 2. (continued)______________________________________________________________________________

Panel D: Winner-price | High-volume

Lagged weekly return filter (%)

≥≥0 and<2

≥≥2 and<4

≥≥4 and<6

≥≥6 and<8

≥≥8 and<10

≥≥ 10

No Mean (%) 0.192 0.121 0.063 -0.086 -0.021 -0.376volume Stand. dev. 1.499 2.165 3.177 4.003 4.612 5.687

filter N 1278 1256 1151 925 662 748t-Statistic 3.597 1.665 0.643 -0.643 -0.064 -1.676

≥≥0 and Mean (%) 0.183 0.040 0.154 -0.177 0.386 -0.438<50 Stand. dev. 2.529 2.774 6.095 4.426 6.198 7.158

N 1098 919 648 359 180 196t-Statistic 2.265 0.410 0.635 -0.807 1.215 -1.154

≥≥50 and Mean (%) 0.406 0.129 0.213 0.273 -0.337 -0.752<100 Stand. dev. 2.636 3.170 4.388 5.045 4.961 5.313

N 845 644 403 238 131 145t-Statistic 4.150 1.084 0.792 0.820 -0.732 -1.889

Volume ≥≥100 and Mean (%) 0.273 0.104 0.311 -0.041 0.075 -0.786Filter <150 Stand. dev. 2.991 3.426 4.366 4.675 4.264 6.872(%) N 629 445 259 143 87 119

t-Statistic 2.136 0.607 1.183 0.139 0.406 -0.966≥≥150 and Mean (%) 0.345 0.020 -0.496 -0.439 0.303 -0.575

<200 Stand. dev. 3.113 3.422 3.453 3.795 6.059 6.256N 421 260 164 80 47 77

t-Statistic 2.240 0.110 -2.012 -0.706 0.562 -0.728≥≥200 and Mean (%) 0.250 0.353 -0.431 0.294 -0.005 0.375

<250 Stand. dev. 3.623 4.582 4.027 4.439 6.408 5.879N 321 174 98 53 37 52

t-Statistic 1.324 0.907 -0.789 0.553 -0.025 0.105≥≥250 Mean (%) 0.331 0.376 -0.156 -0.148 -0.096 -0.639

Stand. dev. 3.130 3.540 3.933 4.445 4.912 5.669N 655 471 321 182 97 229

t-Statistic 2.778 1.999 -0.653 -0.377 -0.267 -1.666

Panels A, B, C, and D give the corresponding portfolio’s means, standard deviations, and t-statistics for amean equal to zero null hypothesis for the four joint price and volume strategies for weeks in which equitypositions are held. Securities are included in a given portfolio if the lagged weekly return and lagged volumereturn (percentage changes in volume) meet the filter conditions for both lagged return and lagged volume.Four price-volume strategies are examined: loser-price | low-volume, loser-price | high-volume, winner-price| low-volume, and winner-price | high-volume in panels A, B, C, and D, respectively. A "No volume filter"corresponds to a price-only strategy and is included for comparison purposes with the volume strategies.The sample consists of REITs for the period from January 1973 to December 31, 1995. N is the number ofportfolio weeks the strategy traded at the respective price and volume filter levels out of a possible 1294weeks. The t-statistics are robust to heteroskedasticity and autocorrelation.

25

Notes

1 In contrast to Mei and Gao (1995), Cooper, Downs and Patterson (1998) use a filter-based trading strategy and

find relatively strong evidence of short-term predictability for real estate securities. See also Liu and Mei (1992),

Mei and Liu (1994), and Ling, Naranjo and Ryngaert (1998) for evidence on the predictability of real estate returns

based on macro economic forecasting factors such as yield spreads, dividend yields, and capitalization rates on

equity REITS.

2 Whether a particular case of return predictability is attributable to market inefficiencies or time-varying risk

premia is often a contentious point, especially in longer horizon predictability. Lehmann (1990) and others suggest

that this disagreement may be resolved by examining the predictability of short-term (weekly) stock returns based

on the assumption that expected returns are not likely to change over a week. Specifically, Lehmann cites Sims

(1984) who hypothesizes that as time intervals shorten, prices should follow a random walk because there should

be few systematic changes in valuation over daily and weekly periods if information arrival is unpredictable. Thus,

we examine weekly return horizons. Furthermore, our study differs from previous research as we (1) employ a

filter-based portfolio construction methodology, (2) include volume as an additional forecast variable, and (3)

interpret our finding based on a theoretical framework in which the heterogeneity across investors gives rise to

different price-volume dynamics.

3 Wang (1995) derives a relation between current period returns (Rt) and volume (Vt), and expected returns (Rt+1),

ttttt RVVRRE )(],|[ 2101 φφ −≅+ . Here, φ0 and φ1 are constants and the sign of φ1 depends on the information

asymmetry between the two types of investors. Specifically, if φ1 < 0 then trading is dominated by informational

motives and φ1 > 0 indicates that trading is dominated by non-informational motives. An inherent feature of the

Wang model is the emphasis on trading motives of the informed (i.e., competitive trading to benefit from private

information or competitive trading to re-balance portfolios due to a shift in private investment technology.) For

this reason, the implications of the model tend to highlight high volume. One might expect the dynamic between

current and expected returns to be similar for low volume scenarios, although the magnitude of the reversal or

26

continuation may differ from the case where volume is high. See Wang (1995) for a complete discussion of the

implications.

4 Lehmann (1990) was the first to examine short-horizon reversals using a relative cross-sectional weighting

method. Mei and Gao apply a similar method to examine reversals in the real estate securities market. Several

papers subsequent to Lehmann provide alternative explanations for the profits found by employing cross-sectional

weighting methods. Lo and MacKinlay (1990), for example, show that up to 50% of Lehmann’s contrarian profits

are due to lagged forecastability across large and small securities. Other important citations include Ball, Kothari,

and Wasley (1995) and Conrad, Gultekin, and Kaul (1997).

5 The filter method may more closely correspond with the academic evidence on the psychology of overreaction.

Related studies (see DeBondt (1989) for a review) show that individuals tend to overreact to a greater degree when

confronted with a large information shock relative to their prior base-rate expectations. This realization leads

DeBondt and Thaler (1985) to postulate an overreaction hypothesis that states; “(1) Extreme movements in stock

prices will be followed by extreme movements in the opposite direction; (2) The more extreme the initial

movement, the greater will be the subsequent adjustment.” This hypothesized predictable behavior, manifested in

extreme price movements, forms the basis of the filter rules. In these rules, a security is included in a portfolio only

if its lagged return is within the filter level. Thus, by employing filters on lagged returns, we are able to screen

stocks for “large” past price movements which may likely be investor overreaction, and subsequently eliminate

securities that experienced smaller lagged returns (or those that may be noise to a contrarian strategy). Cooper

(1998) examines large capitalized NYSE and AMEX stocks and finds that weekly contrarian strategies based on

filter rules generally earn greater weekly profits than do portfolios formed from relative cross-sectional weighting

rules.

6 The price and volume filter breakpoints are determined by using each variable’s overall sample distribution

percentiles of approximately 1, 2.5, 5, 10, 25, 50, 75, 90, 95, 97.5, and 99 percent. As with all filter-based

methods, the primary goal in setting the breakpoints is to generate maximum dispersion in the return and volume

27

distributions. As such, the filter breakpoints are chosen to span the distribution of return and volume conditioning

variables and, therefore, are independent of the results.

7 We follow the practice of other short-horizon contrarian papers and report mean equal to zero t-statistics. We

also calculate t-statistics (not reported in the paper) by subtracting the unconditional weekly mean return of the

sample from the return of each filter portfolio and find that this measure of excess returns produces little variation

in the reported t-statistics.

8 Our method employs weights conditioning on raw returns. As such, the profits from the filter strategies will be

based upon individual security autocovariances and individual security unconditional mean weekly returns. This is

an important point as Conrad, Gultekin and Kaul (1997) and Lo and MacKinlay (1990), using a profit

decomposition originally derived in Lehmann (1990), show that contrarian strategies that base their weights on a

security’s deviation from an equally-weighted index of those securities result in a large percentage of profits

attributable to positive autocovariances of the returns of an equally-weighted portfolio of the component assets.

As we will report, the average weekly unconditional return of the REIT sample is 0.267 percent. This

mean return is relatively small compared to the magnitude of the profits from many of this paper’s filters

strategies, suggesting that the primary source of predictability is individual security autocovariance.

9 Determining whether there are “arbitrage” opportunities in the publicly-traded real estate markets is an

interesting topic which takes us away from our main objective of studying the characteristics of predictability in the

context of informational and non-informational motives for trading real estate. However, a casual observation of

the more extreme filters suggests the possibility for profitable trades. Keim and Madhavan (1997) report round-

trip total execution costs of 0.96 percent (price impact, bid-ask spreads, and commission costs) calculated from

actual trades placed by 21 institutional investors on the smallest quintile of NYSE securities over the 1991 to 1993

period for medium sized trades. The issue of trading costs is more fully explored in Mei and Gao (1995) and

Cooper, Downs and Patterson (1998).

28

10 We investigate alternative specifications of the relation between expected returns and returns, which include

omitting the 36 extreme portfolios (i.e., price filters < -10%, >=10%, and volume filters < -75%, >= 250%). The

results of this regression test do not change.

11 Liu and Mei (1992) show that REITs are a hybrid security in that returns are influenced not only by stock market

conditions but by conditions in the property markets, as well. Wang, Chan, and Gau (1992), and Ling and

Ryngaert (1997) find evidence supporting the influence of the changing real estate market environment on real

estate returns. We are indebted to Crocker Liu for suggesting this test.