differentiate between islamic banks and conventional banks

7/30/2019 differentiate between islamic banks and conventional banks

1/16

Cross-correlations and Predictability ofStock Returns

D. OLSON1

AND C. MOSSMAN2

1American University of Sharjah, United Arab Emirates2University of Manitoba, Canada

ABSTRACT

Studies have shown that small stock returns can be partially predicted by thepast returns of large stocks (cross-correlations), while a larger body ofliterature has shown that macroeconomic variables can predict future stockreturns. This paper assesses the marginal contribution of cross-correlationsafter controlling for predictability inherent in lagged macroeconomicvariables. Macroeconomic forecasting models generate trading rule protso f u p t o 0431% per month, while the inclusion of cross-correlationsincreases returns to 0516% per month. Such results suggest that cross-correlations may serve as a proxy for omitted macroeconomic variables instudies of stock market predictability. Macroeconomic variables are moreimportant than cross-correlations in forecasting small stock returns andencompassing tests suggest that the small marginal contribution of cross-correlations is not statistically signicant. Copyright # 2001 John Wiley &Sons, Ltd.

INTRODUCTION

Recent studies, such as Lo and MacKinlay (1990), have shown that small stock returns can be

predicted, in part, by the past returns of larger stocks. The cross-correlations are asymmetric in

the sense that returns to small stocks are correlated with lagged returns on large stocks, but

lagged returns for small stocks do not help predict returns to large stocks. The existence of this

leadlag relationship between large and small stocks raises questions about market eciency and

to date, two studies have examined whether trading rules can exploit the predictability inherent in

cross-correlations. McQueen, Pinegar, and Thorley (1996) devise a trading rule that yields

annualized abnormal returns of 68%, while Knez and Ready's (1996) non-parametricforecasting technique generates trading rule prots of up to 21% per year. However, Knez and

Ready (1996) argue that the inclusion of realistic transaction costs eectively eliminates trading

rule prots.

In addition to the predictability arising from past stock returns, macroeconomic variables have

been shown to predict the time series of stock returns, while stock market fundamentals help

Copyright # 2001 John Wiley & Sons, Ltd.

Journal of Forecasting

J. Forecast. 20, 145160 (2001)

* Correspondence to: Dennis Olson, School of Business, PO Box 26666, American University of Sharjah, Sharjah,United Arab Emirates.


2/16

explain the cross-section of stock returns. Connor (1995) categorizes models designed to capture

these sources of predictability as statistical factor models, macroeconomic factor models, and

fundamental factor models. For a pooled cross-sectional time series of US stock returns for

19851993, he nds that macroeconomic variables contain no marginal explanatory power when

added to either fundamental or statistical factor models. In contrast, Lo and MacKinlay (1990)

hypothesize that macroeconomic information impacts large companies rst and is transmitted

with a lag to smaller companies. If this hypothesis is correct, with the `right' set of

macroeconomic variables as predictors, the proper lag structure, and functional form, any

economically signicant prediction from cross-correlations should be eliminated. Following this

argument, one would expect macroeconomic variables to forecast small stock returns better than

statistical models involving cross-correlations, which is the opposite of Connor's (1995) ndings.

This study examines the relative importance of cross-correlations versus macroeconomic

variables in models that forecast returns for portfolios of US small stocks. Unlike previous

studies that examine predictability within-sample, comparisons between these two sources of

predictability are made using out-of-sample tests.1

Following an approach developed byPesaran and Timmermann (1995), various models are tted within-sample and tested for one-

month-ahead out-of-sample predictability. The models are updated monthly using a rolling

120-month estimation window. Small stocks are purchased and held as long as one-month-

ahead portfolio returns are predicted to be positive, while the risk-free asset is held whenever

the forecast for excess stock returns (returns above the risk-free rate) is negative. Base-case

forecasting models are developed for both macroeconomic variables and cross-correlations.

Then, lagged large stock returns and macroeconomic variables are included in the same model

to determine the marginal contribution of each source of predictability. The models are judged

on the basis of directional forecast accuracy and trading rule prots before and after the

inclusion of trading costs.

LITERATURE REVIEW

Cross-correlations

Cross-correlations are perhaps the least researched of the many sources of predictability in stock

returns that are now well documented in the nance literature. Badrinath, Kale, and Noe (1995)

suggest that cross-autocorrelations between large and small stocks arise primarily from levels of

institutional ownership, rather than stock market value. Institutionally favoured stocks tend to

be larger than institutionally unfavoured rms, so that the leadlag eect in size portfolios may

be caused more by the level of institutional ownership than rm size. In contrast, McQueen et al.

(1996) document that observed leadlag relationships between large and small stocks are more

size related than the result of institutional ownership. They also discovered a directional

asymmetry in cross-correlations. Small stocks respond quickly to bad macroeconomic news, but

respond with a delay to common good news. Hence, the observed leadlag relationship applies

only to positive returns to large stocks.In an attempt to understand why small stock returns can be predicted using cross correlations,

Boudoukh, Richardson, and Whitelaw (1994) categorize possible explanations into three

1 For example, Ferson and Korajczyk (1995) use factor models to determine which variables are most responsible forwithin-sample returns predictability. Similarly, Connor's (1995) analysis of three types of factor models involves in-sample comparisons.

146 D. Olson and C. Mossman

Copyright # 2001 John Wiley & Sons, Ltd. J. Forecast. 20, 145160 (2001)


3/16

groups loyalists, revisionists, and heretics according to their relationship with the ecient

market hypothesis. The loyalist group looks to data mismeasurement or specic institutional

features such as dierential bidask spreads to defend market eciency. Non-synchronous

trading is consistent with this explanation, but Lo and MacKinley (1990) argue that the

frequency of non-trading is not sucient to be the primary source of observed stock cross-

correlations.

Revisionist arguments expressed by Conrad, Gultekin, and Kaul (1991) suggest that

predictability arises from time-varying expected returns and does not violate market eciency.

More recently, Hameed (1997) has shown that predictability from cross-correlations can likely be

attributable to dierences in the level of time variation in expected returns. However, McQueen et

al. (1996) note that such an explanation does not indicate why returns to large stocks can not be

predicted in the same way. Also, their formal test for this theory fails to support the time-varying

risk premium argument.

The heretic explanations for predictability rest on over-reaction, under-reaction, noise trader

response, or feedback strategies that lead to a form of market ineciency. Over-reaction couldlead to contrarian prots, and under-reaction to protability of momentum strategies. For

example, Grinblatt, Titman, and Wermers (1995) argue that mutual fund managers follow each

other in buying winners, but make independent decisions about selling losers. Since less

information is available for small stocks, herding occurs once managers have observed a rather

imprecise signal, such as a positive return on large stocks in the previous period. This behaviour is

consistent with the directional asymmetry in cross-correlations, as identied by McQueen et al.

(1996).

Macroeconomic variables

Studies such as Fama and French (1989) demonstrate that macroeconomic variables

representing general business conditions can help predict the time series of stock returns.Perhaps the most important of these variables are the levels and changes in interest rates.

Short-term rates (yields on T-bills or commercial paper), term spreads (yields on long-term

government bonds less short-term yields), and default spreads (yields on high-risk corporate

bonds versus low-risk corporate or government bonds) have been shown to have predictive

power in numerous studies. For example, Kairys (1993) shows that changes in commercial

paper rates help explain excess stock returns in the USA from the 1830s to the present. Lo and

MacKinlay (1990) show that large stocks respond to macroeconomic news in the same month

that the news is received, while the response of small stocks can take up to eight weeks (based

upon the signicance of lagged autocorrelations). Jegadeesh and Titman (1995) nd that stock

prices over-react to rm-specic information, but react with a delay to common factors or

macroeconomic information.

Pesaran and Timmermann (1995) show that information about industrial production,ination, monetary growth, dividend yields, and earningsprice ratios improve upon the

predictability discovered by interest rate variables alone. Using a methodology that serves as

the base case for this study, they update the parameters and variables in their forecasting

model each month. Before paying transactions costs, this technique provides annual returns of

344375% above the return obtained from a buy-and-hold strategy during the years 19601992.

Cross-correlations and Predictability of Stock Returns 147



4/16

DATA AND METHODOLOGY

The data set consists of 540 monthly observations for US stock returns and macroeconomicvariables for the years 19501994. Stock returns data are obtained from the Center for Research

in Security prices (CRSP). For each company, end-of-year market capitalization is used to sort

stocks into ve size portfolios where company stock returns are equally weighted within quintile

groups. Macroeconomic data on US interest rates, dividend yields, earningsprice ratios, and

ination rates are from Pinnacle Data Corporation. Data from the Federal Reserve include

various monthly interest rate series, as well as money supply, industrial production, the index of

leading indicators, and quarterly gross domestic product. The ination rate is measured by

changes in the consumer price index. Short-term interest rates are represented by yields on 90-day

Treasury bills and monthly rates for commercial paper. The dierence in yield between the 30-

year government bond and the risk-free rate on 90-day Treasury bills is termed the risk premium,

while the yield dierential between low- and high-grade corporate bonds is referred to as the

default premium. Finally, stock market fundamentals are reected by the dividend and earnings

yield for stocks in the Standard and Poor's 500 index.

The data set includes all of the macroeconomic variables used by Pesaran and Timmermann

(1995) to forecast stock returns, as well as a variety of variables used in other studies. The data

examined include current and past levels, as well as annual and monthly changes in each of the

variables. Although any type of lag structure may be possible, only lags of one and two months

and annual changes in the variables had signicant explanatory power. Also, higher-order

Almon lags and various complicated functional forms failed to improve upon the forecasting

abilities of simple lagged variables.

The data are initially divided into two time periods. The ìn-sample' period consists of ten years

of monthly observations (120 observations) where the rst thirteen months of data are used only

for the purposes of dening annual changes in interest rates, dividend yields, and earningsprice

ratios. The òut-of-sample' period extends 34 years from February 1961 through December of

1994, for a total of 407 observations. Several dierent models are developed to forecast thedirection of one-month-ahead small stock portfolio excess returns, RST rSt rft, where RStrepresents an excess portfolio return, rSt is the actual return to small stocks in period t, and rft is

the risk-free rate in period t. Information available in periods t1 or earlier is used to forecast

current excess returns for the smallest quintile of stocks, RSt. Cross-correlations are considered to

arise from excess returns for the largest quintile of stocks in the previous period, RLt1, whichthen aects small stock returns in the next period RSt. For each of the models, the relevant stock

portfolio is purchased and held as long as one-month-ahead portfolio excess returns are

predicted to be positive. The risk-free asset is held if excess returns are forecasted to be negative.

As in other studies, we also consider purchasing the large stock portfolio whenever excess returns

are forecasted to be negative, but results are not as strong as for holding the risk-free assessment.

Following Pesaran and Timmermann (1995), macroeconomic variables (excluding cross-

correlations) are tted for the initial 10 year ìn-sample' estimation period.

2

The variables areselected using stepwise regression and cuto signicance level of 10%. This technique

approximately maximizes the within-sample adjusted R-squared statistic and it is used to

2 A 10-year period was used rather than the 6-year period used by Pesaran and Timmermann (1995). Previous work byBrockman, Mossman, nad Olson (1997) using macroeconomic variable to forecast Canadian stock index returns foundthat a 10-year estimation period led to more stable and accurate one-month-ahead forecasts of stock returns than arolling 5- or 6-year estimation period.




5/16

forecast one-month-ahead excess returns to the small stock portfolio. The estimating model is

then rolled forward one month by successively dropping the oldest observation and adding the

most recent monthly observation. The rolling regression window is always maintained at 120

months and the macroeconomic variables and model parameters are updated each month to

generate one-month-ahead forecasted returns for the entire 407-month out-of-sample period.3

A series of ordinary least squares (OLS) regression models compare the forecasting ability of

macroeconomic variables versus cross-correlations. For each of the models, the forecasts are

obtained using information available at time t1 to forecast period t. Model A, which is the base-

case macroeconomic variable model is estimated as:

RSt N

i1alMVi l Model A

This indicates that small stock returns in any period are a function of the N macroeconomic

variables (MVi) lagged one or more periods and an error term (t). Each of the lags of t1, t 2,and annual changes in the macroeconomic variables are each considered as separate variables.

The regression parameter ai takes on a non-zero value whenever MVi is included in the regression

model.

The variables selected in the `best' within-sample specications of Model A during any of the

407 estimation periods are listed below (i 1,2 month lag):

TBYti yield on 3-month T-bills lagged i periods (i 1,2)DYLDti dividend yield on the S&P 500 lagged i monthsEPti earningsprice ratio for the S&P 500 lagged i monthsCPAPti interest rate on commercial paper lagged i monthsRISKP

tirisk premium lagged i months (yield on 30-year government bonds in

t1 minus TBYt1)DEFPti default premium lagged imonths (yield on lower-grade BAA corporate

bond in period t1 minus the yield on hight-grade AAA corporate

bonds in t1)

INFLti ination rate lagged i monthsIPti monthly change in industrial production lagged i monthsJANtY SEPTtDECt dummy variable set equal to one when the month to be forecasted is

January, September, or October

In addition, changes in the macroeconomic variables over the past year are signicant in many

periods. For example, the variable CTBY12 TBYt1 TBYt13 is the change in yield on 90-day T-bills over the past twelve months. It is signicant in about one-quarter of the estimation

periods. Among all the variables considered, only the January seasonal dummy is signicant inall estimation periods. The September and October seasonal dummies are signicant in about

3 This continual updating of both parameters and model specication mitigates the lookback bias. It arises if researchers,with the benet of hindsight, select a single set of factors that best `predicts' historical returns over an entire data set. Asnoted by Pesaran and Timmermann (1995. p. 1202), the updating methodology using rolling regressions makes it possibleto `simulate investors' decisions in real time using publicly available information on a set of factors thought a priori tohave been relevant to forecasting stock returns'.




6/16

half of the estimation periods. The remaining macroeconomic variables are not as

important individually, but some combination of them is signicant in every estimation

period. For example, one or more measures of short-term interest rates

(TBYt1Y TBYt2Y CPAPt1Y CPAPt2), or annual changes in these Treasury bill orcommercial paper rates (CTBY12 or CCPAP12) are signicant in every estimation period.

Similarly, some combination of lagged dividend and/or earnings yields (DYLDti or EPti) issignicant in most periods, while some measure of long-term interest rates (RISKPti orDEFPti) or annual changes in these variables enters each of the estimation models. In contrast,various lags of the ination and industrial production variables enter only a few of the estimation

periods. Other variables not listed, such as changes in the index of leading indicators or changes

in the money supply, have predictive power if considered individually, but are subsumed by

various combinations of the listed variables.

We can also address the importance of updating the model specication and regression

parameters monthly, as done in Pesaran and Timmermann (1995). The functional form and

included variables selected do not vary signicantly from month to month. Nevertheless, it is

important to periodically review model specication because over a period of years the variables

selected change and the regression parameters also evolve over time.

Models BE add cross-correlation information to the macroeconomic variables. Model B adds

one period of lagged large stock returns to Model A as follows:

RSt N

i1aiMVi b1RLt1 l Model B

where bi is a regression coecient showing the signicance of the cross-correlation variable

RLt1. Returns with lags of up to six periods were examined, but only the rst two lags provedsignicant in any of the estimation periods if macroeconomic variables were also included in themodel. Model B could be estimated by simply adding lagged large stock returns to the variables

already selected by Model A. However, then cross-correlations are only signicant in about half

of the estimation periods. An alternative way of modelling is to force cross-correlations into each

of the models and then select the macroeconomic variables using step-wise regression. Using this

technique, cross-correlations are signicant in all estimation periods; but as seen in the next

section, they do not capture as much information as various lagged macroeconomic variables.

Model C replaces lagged large stock returns by Model B by one period of lagged asymmetric

returns as shown below:

RSt N

i1 alMVi b2RLt1up b3RLt1down l Model C

This type of model is attributable to McQueen et al. (1996). It allows cross-correlations to aect

small stock returns dierently, depending on whether last period's large stock excess returns were

positive RLt1up, or negative RLt1down, and b2 and b3 are regression coecients showingthe signicance of cross-correlations.




7/16

Model D adds a variable dened as lagged small stock minus large stock returns to the

macroeconomic variables (signicance measured by b4

) in the following manner:

RSt N

i1aiMVi b4RSt1 RLt1 l Model D

This variable used by Knez and Ready (1996), adds information about small stock's own

autocorrelations to the cross-correlation information contained in lagged large stock returns.

Finally, Model E incudes any signicant cross-correlation variables (CCVj) involving large

and small stocks lagged one or two periods as follows:

RSt N

i1atMVi

M

j1bjCCVj t Model E

During about one-fourth of the estimation periods there were no signicant cross-correlationterms (all bj 0) and Models A and E were identical. In other periods, usually one of the cross-correlation variables was signicant, but the included variable often frequently changed between

RLt1Y RLt2Y RSt1 RLt1, RLt1(up), and RLt1(down). Stock return lags of longer thantwo months were not signicant in any estimation periods.

Models FH correspond to Models BD, respectively, except that only cross-correlation

information is used to forecast small stock returns. Macroeconomic variables and seasonal

dummies are not included. These models provide information about the role of cross-correlations

versus macroeconomic variables in forecasting small stock returns.

RESULTS

The results for one-month-ahead out-of-sample forecasts of small stock returns are presented inTable I. For the years 19611994, a buy-and-hold strategy that always forecasts an up market

would have been correct 5294% of the time. The abnormal return for a buy-and-hold strategywould have been, by denition, 0% per month. The directional forecast accuracy for Models A

H exceeds that of a buy-and-hold strategy, and positive abnormal returns exist for all models,

given zero transaction costs.4

The base case involving only macroeconomic variables, Model A, provides 5381% directionalaccuracy. It holds stock for 6462% of the 407 months in the sample and only trades in 2015% ofall months. In the absence of transaction costs it provides abnormal returns of 0431% permonth, which is only slightly lower than for Models B, D, and E, which also include information

about cross-correlations. Model E allows one or more signicant lagged returns from large stocks

or small stocks to enter the model. It provides the largest abnormal returns of any of the

models 0516% per month, in absence of transaction costs. Models BD force cross-

correlations into the estimating equation, even if the relationships are not signicant in all the in-

sample estimation periods. As a result, these models do little to improve upon the results from

4 Returns for large stocks, as other studies have shown, cannot be forecasted as well as small stock returns. A buy-and-hold strategy would have forecasted market direction correctly for 5539% of the months in the sample, while the bestforecasting model has a directional forecasting accuracy of only 5515% and provides abnormal returns (beforetransaction costs) of 0174% per month.




8/16

macroeconomic variables alone. In fact, the asymmetric version of lagged returns in Model C

yields lower abnormal and Model B provides a lower directional forecasting accuracy than Model

A. Focusing on the theoretically preferred model, Model E, we note that addition of cross-

correlations adds a 147 percentage point improvement in directional forecast accuracy and a0085 percentage point increase in abnormal returns (at zero trading cost) over Model A.

Models FH show the forecasting ability of cross-correlations alone. Directional forecast

accuracy is generally as high or higher than the results obtained for Models AE, but trading rule

prots are much lower than for the models involving macroeconomic variables. Such results are

consistent with previous studies showing that cross-correlations are a source of predictability, but

it appears that this information is not readily exploitable in terms of protability. For example,

monthly abnormal returns for zero trading cost are 0215% for Model F (cross-correlations),versus 0431% for Model A (macroeconomic variables) and 0516% for Model E(macroeconomic variables plus cross-correlations). Comparing these models, the marginal

contribution of macroeconomic variables is 0321% (05160215) versus 0085% (05160431)for cross-correlations. While cross-correlations add little to protability generated by

macroeconomic variable models, macroeconomic variables can add signicantly to the abnormal

returns generated by cross-correlations alone. This evidence is indicative of the possibility that

cross-correlations may serve as a proxy for omitted lagged macroeconomic variables.

Encompassing tests are another way to judge the relative out-of-sample importance of

macroeconomic variables models versus statistical models with cross correlations. Donaldsonand Kamstra (1996, p. 57) note that a model, such as our Model A, should be preferred to

another model, such as Model F, if A explains what F cannot explain and F cannot explain what

A cannot explain. They show that a formal test for encompassing between any two models, such

as A and F, involves regressing the forecast error from Model A on the forecast from Model F to

see if Model F can explain what Model A cannot explain. Then the forecast error from Model F




9/16

is regressed on the forecast from Model A to see if Model A can explain what Model F cannot

explain.5 For models A, E, and F, we nd that both models A and E encompass Model F (cross-

correlations alone) at the 5% signicance level. Model A is not encompassed by either Model E

or Model F and similarly Model E is not encompassed by either Model A or Model F. This test

conrms earlier results that macroeconomic variables are more important for out-of-sample

forecasts than cross-correlations. In fact, it suggests that the marginal contribution of cross-

correlations is not statistically important in distinguishing between Models A (macroeconomic

variables alone) and Model E (macroeconomic variables cross-correlations).Adding trading costs of 025% per trade, or 05% roundturn, which Berkowitz, Logue, and

Noser (1988) found to be the average trading costs faced by large institutional investors reduces

abnormal returns for the best model, Model E, to 0 457% per month. In the absence of tradingcosts, the annualized abnormal return is 637%, which is nearly double the return found byPesaran and Timmermann (1995) for trading S&P 500 stocks. Although trading rule prots

appear to be substantial, small stocks have much larger transaction costs than the 05%roundturn costs identied for trading large stocks. Knez and Ready (1996) suggest thatroundturn costs could approach 6%, or trading costs of 3% per trade in our framework. The last

column of Table I shows that none of the models provide positive abnormal returns given such

costs. However, Keim and Madhavan (1995) calculate that trading costs of small stocks for large

investors are 135%286% (27%536% roundturn costs). For 1% trading costs, Models AEall provide positive abnormal returns. Monthly abnormal returns for the best models are 0230%(2795% annualized) for Model A and 0280% (3412% annualized) for Model E. For 135%trading costs, Models A and E provide monthly abnormal returns of 0159% and 0197%, whichdecline to 0028% and 0044% for 2% trading costs. Whether such returns are exploitabledepends upon the proper identication of trading costs. Also, even if trading costs are as low as

135%, our results may not constitute a violation of market eciency. Trading costs were higherand the technology needed to exploit this predictability may not have been available during the

earlier years of our data set.

ALTERNATIVE TRADING RULES AND FORECASTING TECHNIQUES

Table II considers the abnormal returns that could have been obtained using simple trading rule

strategies examined in previous studies. Model I assumes that an investor buys small stocks

whenever monthly returns to large stocks have been positive. The strategy provides a directional

accuracy of 5497% and an abnormal return of 0281% per month, in the absence of transactioncosts. Model J buys small stocks if their own most recent excess return has been positive. The

abnormal returns obtainable from this strategy are0241% per month, or just slightly below that ofModel I. Model K requires an investor to buy and hold small stocks if the excess return to either the

5 Chong and Hendry (1986) rst introduced encompassing tests to make model comparisons within-sample. Donaldsonand Kamstra (1996) provide a clear explanation of how to use the tests to judge forecasting performance out-of-sample.Their notation applied to our models yields the rst regression equation: eAt a0 a1fFt t, where eAt is the forecasterror from Model A in period t, a0 and a1 are regression parameters, fFt is the forecast from Model F, and t is an errorterm. The second regression equation is: eFt b0 b1fAt Zt, where eFt is the forecast error from Model F in period t, b0and b1 are regression parameters, fAt is the forecast from Model A, and Zt is an error term. If a1 is signicant at the 5%level and b1 is not, then Model F encompasses Model A. If b1 is signicant at the 5% level and a1 is not, then Model Aencompasses Model F. If both a1 and b1 are signicant or if neither is signicant, then neither model encompasses theother.




10/16

small or large stock portfolio has been positive in the most recent month, while Model L buys small

stocks only if returns to both portfolios have been positive in the last month. Model K provides the

largest abnormal return among these simple strategies. The abnormal return of 0341% per month(in the absence of transaction costs) is smaller than the abnormal returns generated by Models A or

E, although the directional forecasting accuracy is similar to that obtained with anyof the previous

models. Such dierences in abnormal returns illustrate the importance of including macro-

economic variables within a trading strategy designed to exploit predictability generated by cross-

correlations. However, the similarities in directional forecasting accuracy and the dierences in

abnormal returns between macroeconomic variable models and the cross-correlation models may

be due to forecasting techniques employed. Perhaps cross-correlations inuence future returns in a

more complex non-linear pattern than is captured by OLS regression.

Several previous studies have assumed that an investor would hold the large stock portfolio,

instead of T-bills, whenever the forecasted return for small stocks is negative. This doubles thenumber of stock trades made, and would add at least 025% trading costs to the costs of buyingand selling small stocks. Even in the zero transaction cost environment, this strategy provides

smaller abnormal returns for Models AH than in the case where the risk-free asset is held in

forecasted down months. For the simple trading rules embodied in Models IL, abnormal

returns in the absence of transaction costs are 002 to 012 percentage points higher per monththan those presented in Table II. For example, Model I provides abnormal monthly returns of

0366% for this case, versus 0281% in Table II. Once trading costs of even 025% are included,abnormal returns are lower than in the case where the investor holds T-bills during forecasted

down markets.

Non-parametric techniques, such as those employed by Knez and Ready ( 1996), may improve

upon the results from ordinary least squares forecasts if the relationship between returns and

fundamental variables or cross-correlations is non-linear in nature.6 Perhaps the most popular of

6 An Almon lag structure could be used to introduce some degree of non-linearity into the models without going to theadded complexity of non-parametric estimation. Almon lags of up to twelve months and rst-, second-, and third-degreepolynomials for the lag structure were examined, but results are not reported. The Almon lag models forecasted nobetter, and generally performed worse out of sample, than the models estimated by ordinary least squares (OLS).Similarly, a two-equation vector autoregression (VAR) model that uses the same macroeconomic variables to estimatereturns for small and large stocks simultaneously failed to forecast as well as the OLS models for the 19611994 period.




11/16

the non-parametric regression (NPR) techniques is the kernel density estimator. It is used in this

paper as an alternative to OLS estimation and forecasting techniques. Further description of

non-parametric estimation is provided in the Appendix.

The rolling regression window of 120 observations used for OLS estimation is maintained for

non-parametric regression estimation (NPR). For each model, the corresponding OLS variable

set was used to estimate each month's model. This was necessary because there is no step-wise

NPR estimator. Table III provides evidence about the one-step-ahead forecasting performance of

some representative non-parametric forecasting models for the period 19611994. Model A,

which includes only macroeconomic variables, provides abnormal returns of 0134% per monthand forecasts the sign of one-month-ahead returns in 5479% of all cases. Model B, which addslagged large stock returns gives a directional forecasting accuracy of 55

28% and monthly

abnormal returns of 0294%. Model D, which adds lagged small and large stock returns to themacroeconomic variables, proves a directional accuracy of 5503% and abnormal monthlyreturns of 0302% per month. Since non-parametric techniques are being used, the asymmetricversion of the forecasting models does not oer any theoretical improvement over Model B.

(Actual regressions conrmed that in practice, Model C also generates lower abnormal returns

and has a lower directional accuracy than Model B.) The contribution of cross-correlations in

moving from Model A to Models B or D is 0160 to 0168 percentage points per month, which issomewhat larger than the previous results from OLS forecasts. However, we note that NPR

abnormal returns are consistently lower than those generated by comparable OLS models, even

though again directional forecast accuracy is similar. The apparent inferiority of non-parametric

techniques may be due to the diculties, in practice, with estimating various complicated models

involving macroeconomic variables. Problems may also arise from the well-known over-tting

problem that plagues non-parametric and neural network forecasting.7 Other studies have found

7 Backpropagation neural networks were also constructed to forecast for Models AH with results generally better thanfor the non-parametric models, but worse than for the OLS forecasting models each of the models. For example, monthlyabnormal returns (in the absence of trading costs) over the 19611994 period are 0 176% for Model A and 0407 forModel D for neural networks. Comparable results are 0134% and 0302% for non-parametric regression forecasts and0431% and 0447% for OLS forecasting techniques.




12/16

non-linearities in stock return series, but our results suggest that the non-linearities are not easily

exploitable, perhaps because the relationships between variables changes over time.

VARIABILITY OF FORECAST ACCURACY BETWEEN TIME PERIODS

We now explore the accuracy and protability of the forecasting models over various subperiods

of our data set. Forecasting results for the years 19831994 are shown in Table IV. This period

corresponds to the period of analysis used by McQueen et al. (1996). Most of the models perform

better over this subperiod than over the entire 19611994 period. Model E provides the largest

abnormal returns 0689% per month in the absence of transaction costs, while Model C givesthe best directional forecasting accuracy 5903%. Cross-correlations add very little to thepredictability achieved by macroeconomic variables, but macroeconomic variables seem to

improve upon predictability from cross-correlations. In this period non-parametric techniques

perform as well as OLS forecasts and the NPR estimation of Model D actually generates slightly

larger trading rule abnormal returns than the OLS model.

McQueen et al. (1996) found abnormal returns of 055% per month over the 19831994period using a strategy that is represented by their Model I. In contrast, our Model I gives

abnormal returns of 0397% per month by holding T-bills during forecasted down months(instead of the large stock portfolio). We can replicate McQueen et al.'s (1996) results, but fortrading costs of 05% or higher it is better to adopt our Model I. Also, the strategy ofswitching between large and small stocks does not work as well as holding T-bills during the

earlier 19611982 subperiod. Comparison of Models AE with Models H and I shows the

importance of macroeconomic variables relative to cross-correlations. The small marginal

contribution of cross-correlations (0086%) is virtually identical to the contribution over theentire data set (0085%).




13/16

Table V presents results of selected forecasting models for the 19881992 period analysed by

Knez and Ready (1996). Directional accuracy and abnormal returns are generally higher for this

period than for the whole data set or in other subperiods. Model A has a directional accuracy of

5333% and abnormal returns of 0

984% before transaction costs. Model E again gives the best

results with a 6333% directional accuracy and an abnormal monthly return of 1230%. Knezand Ready (1996) used a non-parametric model similar to Model H to forecast weekly changes in

stock prices during this same period. Their strategy generated annual abnormal returns (before

transaction costs) that ranged from 203% down to 528% depending upon the switching pointand whether the trade occurred at the last trading price or the estimated execution price. In

contrast, Models A and E, which use monthly data, provide annualized abnormal returns of

1247% and 1580%. Model H, which most closely approximates the Knez and Readymethodology, provides annualized abnormal returns of 395%.

Given the large dierences in forecasting performance over the full period versus the

subperiods considered in other studies, we also compare forecasting performance of Models A

and E over 5-year subperiods. Results for other models follow a similar pattern and are not

presented. In the absence of transaction costs, mean abnormal monthly returns for Model A for19611965, 19661970, 19711975, 19761980, 19811985, 19861990, and 19911994 are

0384%, 0770%, 1420%, 0445%, 0106%, 1400%, and 0065%. For Model E abnormalreturns are 0523%, 0652%, 1563%, 0045%, 0100%, 1624%, and 0039% for the sameyears. The models have the best forecasting accuracy and largest abnormal returns during the

years with the greatest number of down movements in stock prices. They do not perform as well

when markets are trending upward. Perhaps most importantly, the results show considerable




14/16

variability in protability of macroeconomic variable and cross-correlation trading strategies

over time. In fact the entire success of the trading rules rests upon returns in three of the

subperiods: 19661970, 19711975, and 19861990. In other periods the abnormal returns are

either negative or negligible. This variability of success over time is a type of risk that along

with transaction costs may explain why abnormal returns persist over 34 years of out-of-sample

tests.

SUMMARY AND CONCLUSIONS

This paper has examined two sources of predictability in small stock returns: macroeconomic

variables and cross-correlations. They both yield similar out-of-sample directional forecast

accuracy, macroeconomic variables are preferred using encompassing tests and on the basis of

trading rule prots. Cross-correlations marginally improve upon the forecast accuracy and

trading rule prots generated by models using macroeconomic variables alone, while adding

macroeconomic variables to cross-correlation models substantially increases abnormal returns.

Cross-correlations seem to serve as a proxy for omitted lagged macroeconomic variables in

studies of small stock predictability. The reason they may be signicant in some periods and add

marginally to trading rule prots is because the macroeconomic variables included in the best

model are generally stable from month to month, but they do change slowly over time. During

periods when the macroeconomic variables included are changing, cross-correlations may pick

up changing market conditions faster than lagged macroeconomic variables.

APPENDIX

To illustrate the use of non-parametric techniques and the kernel density estimator, we denote the

time series of returns to small stocks by Yt, where Yt y1Yy2Yy3Yyn for t 1Y 2Y 3Y F F F n. A groupof fundamental variables and cross-correlation terms, denoted by the vector Xt, helps to explain

Yt. The return series may be represented by Yt mXt t, where mXt is an arbitrary xedfunction of unknown form, and t is the error term. When mXt is linear, Yt can be estimated bylinear regression, but ifmXt is of an unknown non-linear form, Yt should be estimated by non-parametric regression techniques. Assuming sucient smoothness, the time series observations

of Xt near a point of evaluation x0 should be close to x0 and the corresponding values for Ytshould be near mx0. All observations within some neighbourhood of x0 (denoted by h) areassumed to inuence Yt, but values ofXt closer to x0 are assumed to have greater inuence. mx0can then be estimated by a weighted average of the values of Yt, where the weights depend upon

the distances of any Xt from x0.A common way to assign weights within the neighbourhood about x0 is to use a non-

parametric kernel density function, Khx, which is probability density function with nite meanand variance satisfying the conditions that Khx5 0 and

II Khxdx 1. The parameter, h,

which can be estimated from the data or selected a priori, controls the size of the neighbourhood

about x0. It is also called the bandwidth, window width, or the smoothing parameter since it

controls the smoothness of the kernel and essentially determines which observations are used in




15/16

local averaging. The statistics literature has shown that many dierent kernel functions can be

used for Khx

and that results are not very sensitive to choice of kernels. Since the Gaussian

kernel is the most popular and since it is readily available in the GAUSSX statistical package, it is

used in this paper. The NadarayaWatson kernel estimator mx of mx, or the conditionalmean of Yt, is given by:

mx n

j1 Khx0 XtYtnj1 Khx0 Xt

Assuming a Gaussian kernel, the weighting scheme is based upon Euclidean distance away from

x0 and the kernel simplies to:

Khx 1

h2p

p ex2a2h2

REFERENCES

Badrinath S, Kale J, Noe T. 1995. Of shepherds, sheep, and the cross-autocorrelations in equity returns.Review of Financial Studies 8: 401430.

Berkowitz S, Logue D, Noser E Jr. 1988. The total costs of transactions on theNYSE. Journal of Finance 43:97112.

Boudoukh J, Richardson M, Whitelaw R. 1994. A tale of three schools: Insights on autocorrelations ofshort-horizon stock returns. Review of Financial Studies 7: 539573.

Brockman P, Mossman C, Olson D. 1997. Predictability of Canadian stock returns and choice of equityportfolios. Working paper, University of Manitoba.

Chong Y, Hendry D. 1986. Econometric evaluation of linear macroeconomic models. Review of EconomicStudies 53: 671690.

Connor G. 1995. The three types of factor models: A comparison of their explanatory power. Financial

Analysts Journal 51: 4246.Conrad J, Gultekin M, Kaul G. 1991. Asymmetric predictability of conditional variances. Review of

Financial Studies 4: 597622.Donaldson R, Kaamstra M. 1996. Forecast combining with neural networks. Journal of Forecasting 15: 49

61.Fama E, French K. 1989. Business conditions and expected returns on stocks and bonds. Journal of

Financial Economics 25: 2349.Ferson W, Korajczyk R. 1995. Do arbitrage pricing models explain the predictability of stock returns?

Journal of Business 68: 309349.Grinblatt M, Titman S, Wermers R. 1995. Momentum investment strategies, portfolio performance, and

herding: A study of mutual fund behavior. American Economic Review 85: 10881105.Hameed A. 1997. Time-varying factors and cross-autocorrelations in short-horizon stock returns. Journal of

Financial Research 20: 435458.Jegadeesh N, Titman S. 1995. Overreaction, delayed reaction, and contrarian prots. Review of Financial

Studies 8: 972993.

Kairys J. 1993. Predicting sign changes in the equity risk premium using commercial paper rates. Journal ofPortfolio Management 19: 4151.

Keim D, Madhavan A. 1995. Execution costs and investment performance: An empirical analysis ofinstitutional equity trades. Working paper 9-95, Wharton School, University of Pennsylvania.

Knez P, Ready M. 1996. Estimating the prots from trading strategies. Review of Financial Studies 9: 11211163.

Lo A, MacKinlay C. 1990. When are contrarian prots due to stock market overreaction? Review ofFinancial Studies 3: 175205.




16/16

McQueen G, Pinegar M, Thorley S. 1996. Delayed reaction to good news and the cross-autocorrelation ofportfolio returns. Journal of Fiance 51: 889919.

Pesaran M, Timmermann A. 1995. Predictability of stock returns: Robustness and economic signicance.Journal of Finance 50: 12011228.



Documents

differentiate between islamic banks and conventional banks