The conditional beta and the crosssection of expected returns.
We examine the crosssectional relation between conditional Subject to change; dependent upon or granted based on the occurrence of a future, uncertain event. A conditional payment is the payment of a debt or obligation contingent upon the performance of a certain specified act. betas and expected stock returns for a sample period of July July: see month. 1963 to December December: see month. 2004. Our portfoliolevel analyses and the firmlevel crosssectional regressions indicate a positive, significant relation between conditional betas and the crosssection of expected returns. The average return difference between high and lowbeta portfolios ranges between 0.89% and 1.01% per month, depending on the timevarying specification of conditional beta. After controlling for size, booktomarket, liquidity, and momentum, the positive relation between market beta and expected returns remains economically ec·o·nom·i·cal adj. 1. Prudent and thrifty in management; not wasteful or extravagant. See Synonyms at sparing. 2. Intended to save money, as by efficient operation or elimination of unnecessary features; economic: and statistically significant. ********** The Sharpe (1964), Lintner (1965), and Black (1972) capital asset pricing model Capital asset pricing model (CAPM) An economic theory that describes the relationship between risk and expected return, and serves as a model for the pricing of risky securities. (CAPM CAPM See: Capital asset pricing model CAPM See capitalasset pricing model (CAPM). ) implies (logic) implies  (=> or a thin right arrow) A binary Boolean function and logical connective. A => B is true unless A is true and B is false. The truth table is A B  A => B + F F  T F T  T T F  F T T  T It is surprising at first that A => the meanvariance efficiency of the market portfolio in the sense of Markowitz Markowitz  The author of the original Simscript language. (1959). The primary implication implication In logic, a relation that holds between two propositions when they are linked as antecedent and consequent of a true conditional proposition. Logicians distinguish two main types of implication, material and strict. of the CAPM is that there is a positive linear relation between expected returns on securities and their market betas, and that variables other than beta should not capture the crosssectional variation in expected returns. However, over the last three decades, many studies have tested the empirical em·pir·i·cal adj. 1. Relying on or derived from observation or experiment. 2. Verifiable or provable by means of observation or experiment. 3. performance of the static (or unconditional HEIR, UNCONDITIONAL. A term used in the civil law, adopted by the Civil Code of Louisiana. Unconditional heirs are those who inherit without any reservation, or without making an inventory, whether their acceptance be express or tacit. Civ. Code of Lo. art. 878. UNCONDITIONAL. ) CAPM in explaining the crosssection of realized average returns. The findings of these earlier studies indicate that firm size, booktomarket ratio BookToMarket Ratio A ratio used to find the value of a company by comparing the book value of a firm to its market value. Book value is calculated by looking at the firm's historical cost, or accounting value. , earningstoprice ratio, liquidity, and momentum have significant explanatory ex·plan·a·to·ry adj. Serving or intended to explain: an explanatory paragraph. ex·plan power for average stock returns, but that market beta has little or no power. Early tests of the CAPM are based on the crosssectional regressions of average stock returns on estimates of individual stock betas. Two obvious problems with these tests are errorsinvariables and residual Residual See:Residual value correlations. First, beta estimates for individual stocks are imprecise im·pre·cise adj. Not precise. impre·cisely adv. and generate a measurement error problem when they are used to explain average returns. To improve the accuracy of estimated betas, Blume Blume , Judy Born 1938. American novelist best known for depicting the everyday problems of adolescence. Her works include Are You There God? It's Me, Margaret (1970). (1970), Friend and Blume (1970), and Black, Jensen Noun 1. Jensen  modernistic Danish writer (18731950) Johannes Vilhelm Jensen , and Seholes (1972) use portfolios instead of individual stocks in their crosssectional tests. Since estimates of betas for diversified diversified (di·verˑ·s portfolios are more precise than estimates for individual stocks, using portfolios in the crosssection regressions of average returns on betas diminishes the errorsinvariables problem. Second, the regression regression, in psychology: see defense mechanism. regression In statistics, a process for determining a line or curve that best represents the general trend of a data set. residuals Residuals (1) Part of stock returns not explained by the explanatory variable (the market index return). Residuals measure the impact of firmspecific events during a particular period. have common sources of variation. Positive correlation Noun 1. positive correlation  a correlation in which large values of one variable are associated with large values of the other and small with small; the correlation coefficient is between 0 and +1 direct correlation in the residuals yields downward bias in the usual ordinary least squares (OLS OLS Ordinary Least Squares OLS Online Library System OLS Ottawa Linux Symposium OLS Operation Lifeline Sudan OLS Operational Linescan System OLS Online Service OLS Organizational Leadership and Supervision OLS On Line Support OLS Online System ) estimates of the standard errors of the crosssectional regression slopes. Fama and MacBeth (1973) introduce a method for addressing the inference (logic) inference  The logical process by which new facts are derived from known facts by the application of inference rules. See also symbolic inference, type inference. problem caused by correlation correlation In statistics, the degree of association between two random variables. The correlation between the graphs of two data sets is the degree to which they resemble each other. of the residuals in crosssectional regressions. Rather than estimating a single crosssection regression of average monthly returns on betas, they estimate monthbymonth crosssection of regressions of monthly returns on betas. The timeseries averages of the monthly slopes and intercepts and their standard errors are used to test whether the average market risk premium is positive and the average intercept is equal to the riskfree rate Riskfree rate The rate earned on a riskless asset. . In crosssectional tests, Douglas (1969), Black, Jensen, and Scholes Scholes(/skowlz/ or /šowlz/) could refer to the following places: United Kingdom:
1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities. 2. on beta is less than the average excess market return and the intercept is greater than the average riskfree interest rate RiskFree Interest Rate Describes return available to an investor in a security somehow guaranteed to produce that return. The riskfree interest rate compensataes the investor for the temporary sacrifice of consumption. . In their widely cited study, Fama and French (1992) examine the static version of the CAPM and find both at the firm and portfolio level that the crosssectional relation between market beta and average return is flat. (1) They interpret To run a program one line at a time. Each line of source language is translated into machine language and then executed. this fiat [Latin, Let it be done.] In old English practice, a short order or warrant of a judge or magistrate directing some act to be done; an authority issuing from some competent source for the doing of some legal act. relation as strong empirical evidence against the CAPM. As indicated by Jagannathan and Wang (Wang Laboratories, Inc., Lowell, MA) A computer services and network integration company. Wang was one of the major early contributors to the computing industry from its founder's invention that made core memory possible, to leadership in desktop calculators and word processors. (1996), although a flat relation between the unconditional expected return Expected Return The average of a probability distribution of possible returns, calculated by using the following formula: and the unconditional market beta may be evidence against the static CAPM, it is not necessarily evidence against the conditional CAPM. The CAPM was originally developed within the framework of a hypothetical Hypothetical is an adjective, meaning of or pertaining to a hypothesis. See:
There is substantial empirical evidence that conditional betas and expected returns depend on the nature of the information available at any given point in time and vary over time.3 In this paper, we investigate whether timevarying conditional betas can explain the crosssection of expected returns at the firm and portfolio level. There is substantial empirical evidence that conditional betas and expected returns depend on the nature of the information available at any given point in time and vary over time. Earlier studies use either a single or rolling long sample of monthly data in estimating beta. Instead, we use daily returns within a month to compute To perform mathematical operations or general computer processing. For an explanation of "The 3 C's," or how the computer processes data, see computer. realized beta for each stock trading at the New York Stock Exchange New York Stock Exchange (NYSE) World's largest marketplace for securities. The exchange began as an informal meeting of 24 men in 1792 on what is now Wall Street in New York City. (NYSE NYSE See: New York Stock Exchange ), American Stock Exchange (Amex), and Nasdaq for our sample period of July 1963 to December 2004. We propose three alternative specifications of expected future beta based on the past information on realized beta using autoregressive Autoregressive Using past data to predict future data. Notes: Essentially it's forecasting, similar to the weather... Sometimes even the weatherman can be caught in an unexpected downpour. , moving average, and generalized gen·er·al·ized adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. autoregressive conditional heteroskedasticity Autoregressive Conditional Heteroskedasticity (ARCH) A nonlinear stochastic process, where the variance is timevarying, and a function of the past variance. ARCH processes have frequency distributions which have high peaks at the mean and fattails, much like fractal distributions. (GARCH GARCH Generalized Autoregressive Conditional Heteroskedasticity )inmean models to obtain timevarying conditional betas for each stock. We estimate conditional betas by using the entire history of returns on a stock. Hence, the high and lowconditional beta portfolios we form cannot be exactly replicated by an investor at any given point in time. Our focus is more in the nature of a hypothesis An assumption or theory. During a criminal trial, a hypothesis is a theory set forth by either the prosecution or the defense for the purpose of explaining the facts in evidence. test that has asymptotic validity. Thus, our approach is somewhat different from standard practice, which identifies ex ante measures of risk based on information available at a given point in time that a particular portfolio will earn a higher return on average than another portfolio. For each specification of conditional beta, we find that stocks with high (low) market betas have, on average, high (low) average returns. Our portfoliolevel analyses and the firmlevel crosssectional regressions indicate that the positive relation between the conditional betas and the crosssection of average returns is economically and statistically significant. Average portfolio returns increase almost monotonically when moving from low to highbeta portfolios. The [R.sup.2] values from the regression of average portfolio returns on average portfolio betas are in the range of 82% to 98% for 10, 20, 50, and 100 beta portfolios. When we form the equalweighted decile decile one of the groups when a series of ranked data is divided into ten equal parts, or dividing points between such groups. See also quartile. portfolios by sorting the NYSE/Amex/Nasdaq stocks based on conditional beta, we find that the average return difference between decile 10 (high beta) and decile 1 (low beta) portfolios ranges between 0.89% and 1.01% per month, depending on the timevarying specification of conditional beta. For 20, 50, and 100 beta portfolios, the average return difference ranges from 1.01% to 1.23% per month. To check whether our findings are driven by small, illiquid Illiquid An asset or security that cannot be converted into cash very quickly (or near prevailing market prices). Notes: A house is a good example of an illiquid asset. See also: Cash, Liquidity Illiquid In the context of finance. , and lowprice stocks, we exclude the Amex and Nasdaq stocks and form the beta portfolios by sorting only the NYSE stocks based on the conditional betas. The results indicate that excluding the Amex and Nasdaq sample has almost no effect on our original findings. We also control for the wellknown crosssectional effects, including size and booktomarket (Fama and French, 1993, 1995, 1996), liquidity (Amihud, 2002; Pastor and Stambaugh Stambaugh is a city and a township in Iron County, Michigan
n. New England & Upstate New York 1. A runt, especially one of a litter of pigs. 2. A small person. See Regional Note at tit^{1}. , 1993). After controlling for these effects, we estimate the crosssectional beta premium as being in the range of 0.86% to 1.46% per month. The paper is organized as follows. Section I contains the data and variable definitions. In Section II, we discuss the average raw returns and the average riskadjusted returns on beta portfolios, and in Section III we present the firmlevel crosssectional regression results. Section IV provides a battery of robustness checks, including portfoliolevel crosssectional regressions, testing the longterm predictive power The predictive power of a scientific theory refers to its ability to generate testable predictions. Theories with strong predictive power are highly valued, because the predictions can often encourage the falsification of the theory. of conditional betas, and some additional tests after controlling for liquidity and momentum, after excluding the Amex and Nasdaq sample, and after controlling for microstructure mi·cro·struc·ture n. The structure of an organism or object as revealed through microscopic examination. microstructure Noun a structure on a microscopic scale, such as that of a metal or a cell effects. In Section V, we investigate whether our main findings are robust for size/BM/beta portfolios. In Section VI, we discuss the crosssectional implications of the conditional CAPM approach. Section VII concludes the paper. I. Data and Variable Definitions Our first data set comprises all NYSE, Amex, and Nasdaq financial and nonfinancial Adj. 1. nonfinancial  not involving financial matters financial, fiscal  involving financial matters; "fiscal responsibility" firms. We obtain this information from the Center for Research in Security Prices This article or section needs sources or references that appear in reliable, thirdparty publications. Alone, primary sources and sources affiliated with the subject of this article are not sufficient for an accurate encyclopedia article. (CRSP CRSP Collaborative Research Support Program (USA) CRSP Collaborative Research Support Program CRSP Center for Research in Security Prices CRSP Center for Research in Security Prices ) for the period from July 1963 through December 2004. We use the daily stock returns to generate the conditional beta measures. Our second data set is Compustat '''Standard & Poor's Compustat^{®} is a database of financial, statistical and market information on active and inactive companies throughout the world. Compustat^{®} data has a reputation for extensive coverage, standardization, expertise and timeliness. , which we use primarily to obtain the book values for individual stocks. For each month from July 1963 to December 2004, we compute the following variables for each firm in the sample. A. Size Following other studies, we measure firm size (ME) by the natural logarithm Natural logarithm Logarithm to the base e (approximately 2.7183). of the market value of equity (a stock's price times shares outstanding in millions of dollars) for each stock. B. BooktoMarket Following Fama and French (1992), we compute a firm's booktomarket ratio (BE/ME) by using its market equity at the end of June of year t  1 and the book value of common equity plus balancesheetdeferred taxes for the firm's latest fiscal year ending in calendar year t  1. To avoid giving extreme observations heavy weight in our analysis, like Fama and French (1992), we set the smallest and largest 0.5% of the observations on booktomarket ratio equal to the next largest and smallest values of the ratio (the 0.005 and 0.995 fractiles). C. Realized Beta To estimate the monthly beta for an individual stock, we assume a singlefactor returngenerating process in the form of a market model: [R.sub.i,d,t] = [[alpha].sub.i,t] + [[beta].sub.i,t] [R.sub.m,d,t] + [[epsilon].sub.i,d.t], (1) where [R.sub.i,d,t] is the daily return on stock i on day d of month t, [R.sub.m,d,t] is the daily market return on day d of month t, [[epsilon].sub.i,d.t] is the residual term, (4) [[alpha].sub.i,t] is the intercept, and [[beta].sub.i,t] is the realized beta of stock i in month t. We define the realized beta as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], (2) where [[bar.R].sub.i,t] is the average daily return on stock i in month t, [[bar.R].sub.m,t] is the average daily market return in month t, and n is the number of daily return observations in month t. In our empirical analysis, we measure [R.sub.m,d,t] by the CRSP daily valueweighted index, that is, the daily valueweighted average returns of all stocks trading at the NYSE, Amex, and Nasdaq. Although earlier studies generally use monthly returns to estimate beta and test the CAPM and other factor models, we use daily returns because in principle, we believe we can estimate betas more precisely with higherfrequency data, just as Merton Merton, outer borough (1991 pop. 161,800) of Greater London, SE England. The area is largely residential with some industry, including tanning and the manufacture of silk and calico prints, varnish and paint, and toys. (1980) observed for variances. In practice, using daily returns creates microstructure issues caused by nonsynchronous Adj. 1. nonsynchronous  not occurring together unsynchronised, unsynchronized, unsynchronous asynchronous  not synchronous; not occurring or existing at the same time or having the same period or phase trading. Nonsynchronous prices can have a big impact on shorthorizon betas. Lo and MacKinlay (1990) show that small stocks react with a significant (i.e., a week or more) delay to common news, so a daily beta will miss much of the smallstock covariance Covariance A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns vary inversely. with market returns. To mitigate mit·i·gate v. To moderate in force or intensity. miti·gation n. the problem, we exclude the Amex and Nasdaq stocks and control for the size effect using twodimensional size/beta portfolios based on the NYSE sample. Also, following Dimson (1979), we use both current and lagged market returns in the regressions. In Equation (3) we estimate the realized beta as the sum of the slopes ([[??].sup.1.sub.i,t] and [[??].sup.2.sub.i,t]): [R.sub.i,d,t] = [[alpha].sub.i,t] + [[beta].sup.1.sub.i,t] [R.sub.m,d,t] + [[beta].sup.2.sub.i,t] [R.sub.m,d1,t] + [[epsilon].sub.i,d,t], (3) where the sum of the slopes, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], adjusts for nonsynchronous trading (see Scholes and Williams, 1977; Dimson, 1979). We estimate the timevarying conditional betas based on the following autoregressive of order one AR(1), moving average of order one MA(1), and GARCH(1,1)inmean models: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6) We drop the i subscript (1) In word processing and scientific notation, a digit or symbol that appears below the line; for example, H_{2}O, the symbol for water. Contrast with superscript. (2) In programming, a method for referencing data in a table. in Equations (4) to (6) to save space. Here, E([[beta].sub.t]  [[OMEGA 1. (programming) Omega  A prototypebased objectoriented language from Austria. ["TypeSafe ObjectOriented Programming with Prototypes  The Concept of Omega", G. Blaschek, Structured Programming 12:217225, 1991]. 2. ].sub.t1]) denotes the current conditional mean of realized beta estimated with the information set at time t  1, [[OMEGA].sub.t1]. In AR(1) and MA(1) models, we assume that the conditional variance In statistics, conditional variance is a special form of the variance. If we have a conditional distribution YX the conditional variance is defined as where of realized beta, denoted by E([[epsilon].sup.2.sub.t]  [[OMEGA].sub.t1]), is constant. We use the GARCHinmean model, which was originally introduced by Engle En´gle n. 1. A favorite; a paramour; an ingle. v. t. 1. To cajole or coax, as favorite. I 'll presently go and engle some broker.  B. Jonson. , Lilien, and Robins (1987), to model the conditional mean of asset returns as a function of the conditional volatility Volatility 1. A statistical measure of the tendency of a market or security to rise or fall sharply within a period of time. 2. A variable in option pricing formulas that denotes the extent to which the return of the underlying asset will fluctuate between now and the . In the GARCHinmean model, we assume that the conditional variance of realized beta follows the GARCH(1,1) model of Bollerslev (1986). We compare the conditional betas ([[beta].sup.AR.sub.tt1]], [[beta].sup.MA.sub.tt1], [[beta].sup.GARCH.sub.tt1] with the lagged realized beta [[beta].sup.realized.sub.t1] in terms of their power to predict the crosssection of onemonthahead average stock returns. Earlier studies find significant persistence (1) In a CRT, the time a phosphor dot remains illuminated after being energized. Longpersistence phosphors reduce flicker, but generate ghostlike images that linger on screen for a fraction of a second. in the conditional beta estimates for industry, size, or booktomarket portfolios (e.g., Braun Braun , Eva 19121945. German lover and later wife of Adolf Hitler. They began living together in 1936, but the liaison was kept secret, and she was never seen in public with him. They were married hours before their double suicide on April 30, 1945. , Nelson, and Sunier, 1995; Ang and Chen, 2007). However, these studies do not estimate conditional betas at the firm level. Ang, Chen, and Xing Xing Crossing (2006) compute realized beta at the firm level using daily returns over the past 12 months and propose alternative measures of downside risk Downside Risk An estimation of a security's potential to suffer a decline in price if the market conditions turn bad. Notes: You can think of this as an estimate of the amount that you could lose on a stock or other investment. based on the unconditional realized betas. According to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. their descriptive statistics descriptive statistics see statistics. , the average AR(1) coefficient of realized betas is in the range of 0.077 to 0.675 depending on their specification of downside risk. We generate conditional beta estimates for each stock using AR(1) and MA(1) models given in Equations (4) and (5). Jagannathan and Wang (1996) examine the relation between unconditional betas and the crosssection of unconditional expected returns by assuming that the conditional CAPM holds period by period. As described in Jagannathan and Wang (1996), when the conditional CAPM is assumed to hold for each period, crosssectionally, the unconditional expected return on any asset is a linear function of its expected beta and its betapremium sensitivity. In other words Adv. 1. in other words  otherwise stated; "in other words, we are broke" put differently , the standard static (or unconditional) CAPM leads to a twofactor unconditional asset pricing model Asset pricing model A model for determining the required or expected rate of return on an asset. Related: Capital asset pricing model and arbitrage pricing theory. , where the first factor is the unconditional market beta that measures average market risk and the second factor is the unconditional premium beta that measures betainstability risk. According to this model, stocks with higher expected betas should have higher unconditional expected returns. Similarly, stocks with betas that are correlated cor·re·late v. cor·re·lat·ed, cor·re·lat·ing, cor·re·lates v.tr. 1. To put or bring into causal, complementary, parallel, or reciprocal relation. 2. with the market risk premium and hence are less stable over the business cycle should also have higher unconditional expected returns. Jagannathan and Wang (1996) indicate that the betapremium sensitivity of an asset measures the instability instability /in·sta·bil·i·ty/ (stahbil´ite) lack of steadiness or stability. detrusor instability of the asset's beta over the business cycle. We model the conditional mean of market beta as a function of its conditional variance as in the GARCHinmean specification. Equation (6) models the current conditional mean and conditional variance of realized betas as a function of the information set at time t  1. To provide an alternative justification justification In Christian theology, the passage of an individual from sin to a state of grace. Some theologians use the term to refer to the act of God in extending grace to the sinner, while others use it to define the change in the condition of a sinner who has received for our use of the GARCHinmean model, we compute the correlations between the realized beta ([[beta].sub.t]) and the conditional standard deviation of realized beta ([[sigma].sub.t]), the realized beta ([[beta].sub.t]) and the conditional variance of realized beta ([[alpha].sub.t]), the conditional mean of realized beta (E([[beta].sub.t]  [[OMEGA].sub.t1])) and the conditional standard deviation of realized beta ([[sigma].sub.t]), and the conditional mean of realized beta (E([[beta].sub.t]  [[OMEGA].sub.t1])) and the conditional variance of realized beta ([[sigma].sub.t]). Table I presents the percentiles of the correlation measures for all stocks trading at the NYSE, Amex, and Nasdaq. The correlation statistics indicate a strong relation between the monthly realized betas and their conditional volatility, and a strong relation between the conditional mean of monthly realized betas and their conditional volatility. We also find that the estimated slope coefficients ([[??].sub.1]) in [[beta].sub.t] = [c.sub.0] + [c.sub.1][[sigma].sub.2tt1]] + [[epsilon].sub.t] are statistically significant. This result provides further justification of our use of the GARCHinmean model. Table II presents percentiles of the timeseries mean and standard deviation of realized and expected conditional betas. The statistics presented in Panel A of Table II are based on realized betas that we compute by using daily returns over the previous month without lagged market return. The statistics shown in Panel B of Table II are based on realized betas that we compute by using daily returns over the previous month with the lagged market return. In both panels, we only report the sample mean of the realized betas because theoretically, the mean of conditional betas is the same as the mean of realized beta. Theoretically, the means should be the same, but the discrepancies are caused by the filtration filtration: see sewerage; water supply. Filtration The separation of solid particles from a fluidsolids suspension of which they are a part by passage of most of the fluid through a septum or membrane that retains most of the solids of conditional beta. The standard deviation of realized betas is greater than the standard deviation of expected betas. In both panels, the standard deviation of conditional betas obtained from the GARCHinmean model is somewhat greater than the standard deviation of conditional betas that we obtain from the AR(1) and MA(1) models. We compare the conditional betas ([[beta].sub.ARs.sub.tt1]], [[beta].sub.MA.sub.tt1]], [[beta].sup.GARCH.sub.tt 1]) with the lagged realized beta [[beta].sub.realized.sub.t1] in terms of their power to predict the onemonthahead realized beta, [[beta].sup.realized.sub.t]. Table III shows the percentiles of [R.sup.2] values from the regression of onemonthahead realized betas on the lagged realized beta and conditional betas. The performance of conditional betas in predicting the onemonthahead realized beta is much higher than the lagged realized beta. The 1 percentile of [R.sup.2] is 0.01% for [[beta].sup.realized.sub.t1] and the 99 percentile of [R.sup.2] is 26.82%. The corresponding figures are 1.14% and 33.87% for [[beta].sup.GARCH.sub.tt1]. These results provide some explanation for why the earlier studies that use lagged realized beta or unconditional beta could not identify a positive and significant relation between market beta and expected stock returns. We think that to generate more accurate measures of expected futures betas and to explain the crosssectional variation in stock returns, one needs to use conditional betas. II. Average Returns and FF3 Alphas on Beta Portfolios This section presents univariate univariate adjective Determined, produced, or caused by only one variable and bivariate bi·var·i·ate adj. Mathematics Having two variables: bivariate binomial distribution. Adj. 1. portfoliolevel analysis after controlling for size and booktomarket. A. Univariate PortfolioLevel Analysis Table IV presents the equalweighted average returns of decile portfolios that are formed by sorting the NYSE/Amex/Nasdaq stocks based on the lagged realized beta, and the conditional AR(1), MA(1), and GARCHinmean betas. We base the results in Panel A on the realized betas that we compute using daily returns over the previous month without lagged market return. When we sort portfolios based on the lagged realized beta, [[beta].sup.realized.sub.t1], the average return difference between decile 10 (high beta) and decile 1 (low beta) is about 0.49% per month with the NeweyWest (1987) tstatistic of 2.53. Although this result does not support the empirical validity of CAPM, it is not conclusive Determinative; beyond dispute or question. That which is conclusive is manifest, clear, or obvious. It is a legal inference made so peremptorily that it cannot be overthrown or contradicted. , because as discussed earlier, [[beta].sup.realized.sub.t1] is not a precise estimator of [[beta].sup.realized.sub.t]. The static CAPM predicts a contemporaneous con·tem·po·ra·ne·ous adj. Originating, existing, or happening during the same period of time: the contemporaneous reigns of two monarchs. See Synonyms at contemporary. positive relation between expected stock returns and market betas. However, we cannot use the current realized beta in empirical tests because of the statistical problems indicated by Miller and Scholes (1972) and Fama and MacBeth (1973). When we examine the crosssectional predictive power of the conditional beta measures, we see that they are more accurate estimators of [[beta].sup.realized.sub.t].When we sort decile portfolios based on [[beta].sup.AR.sub.tt1], [[beta].sup.MA.sub.tt1] and [[beta].sup.GARCH.sub.tt1], the average return difference between decile 10 (high beta) and decile 1 (low beta) is in the range of 0.74% to 0.92% per month and highly significant. In addition to the average raw returns, Panel A of Table IV also shows the magnitude magnitude, in astronomy, measure of the brightness of a star or other celestial object. The stars cataloged by Ptolemy (2d cent. A.D.), all visible with the unaided eye, were ranked on a brightness scale such that the brightest stars were of 1st magnitude and the and statistical significance of the intercepts (FF3 alphas) from the regression of the equalweighted portfolio returns on a constant, excess market return, small minus big (SMB (1) (Small to Mediumsized Business) Also called "SME" (small to mediumsized enterprise), it refers to companies that are larger than the small office/home office (SOHO), but not huge. ), and high minus low (HML HML Hämeenlinna (Finland) HML Hawaii Medical Library HML High Minus Low (Book to Market Value ratio) HML Hard Money Lender (real estate) HML Human Media Lab ) factors. If the conditional CAPM is right and FF3 alphas do not adequately capture time variations in betas, then conditionalbetasorted portfolios will have alphas different from zero. Panel A shows that the 101 difference in the FF3 alphas is negative for [[beta].sup.realized.sub.t1], but it is positive and highly significant for the AR(1), MA(1), and GARCHinmean betas. Our results in Panel B of Table IV are based on the realized betas that we compute using daily returns over the previous month with the lagged market return. When we sort portfolios based on [[beta].sup.realized.sub.t1] the average return difference between high and lowbeta portfolios is negative but marginally mar·gin·al adj. 1. Of, relating to, located at, or constituting a margin, a border, or an edge: the marginal strip of beach; a marginal issue that had no bearing on the election results. 2. significant. When decile portfolios are sorted based [[beta].sup.AR.sub.tt1], [[beta].sup.MA.sub.tt1], and [[beta].sup.GARCH.sub.tt1], the average return difference between high and lowbeta portfolios is positive, in the range of 0.89% to 1.01% per month, and highly significant. Panel B also shows that the 101 difference in the FF3 alphas is negative for [[beta].sup.realized.sub.t1] but positive and highly significant for the AR(1), MA(1), and GARCHinmean betas. Overall, the results in Table IV indicate that the strong positive relation between the conditional betas and expected returns is robust to the measurement of realized betas. (5) Due to space considerations, in the following sections we report results only from the realized beta measures estimated with the lagged market return. B. Controlling for Size and BooktoMarket We test whether there is a positive relation between conditional beta and expected returns after we control for size and booktomarket. We control for size by first forming decile portfolios ranked based on market capitalization Market Capitalization A measure of a public company's size. Market capitalization is the total dollar value of all outstanding shares. It's calculated by multiplying the number of shares times the current market price. This term is often referred to as market cap. . Then, within each size decile, we sort stocks into decile portfolios, which we rank based on GARCHinmean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. Panel A of Table V shows that in each size decile, the highest (lowest) beta decile has a higher (lower) average returns. The column labeled "Average Returns" averages across the 10 size deciles to produce decile portfolios with dispersion dispersion, in chemistry dispersion, in chemistry, mixture in which fine particles of one substance are scattered throughout another substance. A dispersion is classed as a suspension, colloid, or solution. in market beta but containing all sizes of firms. This procedure creates a set of decile beta portfolios with nearidentical levels of firm size, and thus these decile beta portfolios control for differences in size. After controlling for size, the average return difference between high and lowbeta portfolios is 1.41% per month with the NeweyWest (1987) tstatistic of 3.48. Thus, market capitalization does not explain the high (low) returns to high (low) beta stocks. We also control for booktomarket (BM) by first forming decile portfolios ranked based on the ratio of book value of equity to market value of equity. Then, within each BM decile, we sort stocks into decile portfolios, which we rank based on GARCHinmean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. Panel B of Table V shows that in each BM decile, the highest (lowest) beta decile has a higher (lower) average returns. The last two columns report the average returns and NeweyWest (1987) tstatistics of 10 beta portfolios after controlling for BM. The average return difference between high and lowbeta portfolios is 1.16% per month with the NeweyWest tstatistic of 3.69. Thus, booktomarket ratio does not explain the high (low) returns to high (low) beta stocks. (6) Table VI shows the average return differences and FF3 alphas on highbeta minus lowbeta portfolios within each size and booktomarket decile. As shown in Panel A of Table VI, for all specifications of conditional beta, the average return differences and FF3 alphas are positive and economically significant within each size decile. Except for the two biggest size portfolios (sizes 9 and 10), the average return differences and FF3 alphas are also statistically significant at the 5% level or better. For example, for the smallest size decile, the average return difference between decile 10 (high beta) and decile 1 (low beta) is 2.49% per month for [[beta].sup.AR.sub.tt1], 2.57% per month for [[beta].sup.MA.sub.tt1] and 2.50% per month for [[beta].sup.GARCH.sub.tt1]. The corresponding FF3 alphas are 2.02% per month for [[beta].sup.AR.sub.tt1], 2.06% per month for [[beta].sup.MA.sub.tt1], and 2.02% per month for [[beta].sup.GARCH.sub.tt1]. This strong positive relation between market beta and expected return is present for size 1 to size 9 portfolios, and the relation becomes somewhat weaker for the largest size portfolio (size 10). Panel B of Table VI shows that for all specifications of conditional beta, the average return differences and FF3 alphas are positive and economically significant within each BM decile. The average return differences and FF3 alphas are also statistically significant at the 5% level or better. For example, for the lowest BM decile, the average return difference between decile 10 (high beta) and decile 1 (low beta) is 1.50% per month for [[beta].sup.AR.sub.tt1], 1.49% per month for [[beta].sup.MA.sub.tt1], and 1.29% per month for [[beta].sup.GARCH.sub.tt1]. The corresponding FF3 alphas are 1.60% per month for [[beta].sup.AR.sub.tt1], 1.54% per month for [[beta].sup.MA.sub.tt1], and 1.36% per month for [[beta].sup.GARCH.sub.tt1]. Although there is no obvious pattern, the strong positive relation between market beta and expected return is more pronounced for BM I to BM 9 portfolios, and the relation becomes weaker for the highest BM portfolio. III. FirmLevel CrossSectional Regressions Here, we present the timeseries averages of the slope coefficients from the crosssection of average stock returns on the lagged realized beta, conditional beta, size, and BM. The average slopes provide standard FamaMacBeth (1973) tests for determining, on average, which explanatory variables have nonzero non·ze·ro adj. Not equal to zero. nonzero Not equal to zero. expected premiums. We run monthly crosssectional regressions for the following econometric e·con·o·met·rics n. (used with a sing. verb) Application of mathematical and statistical techniques to economics in the study of problems, the analysis of data, and the development and testing of theories and models. specifications: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10) In Equations (7) to (10) [R.sub.i,t] is the realized return Realized return The return that is actually earned over a given time period. on stock i in month t, [logME.sub.i,t1] is the natural logarithm of market equity for firm i in month t  1, log([BE.sub.i,t1]/[ME.sub.i,t1]) is the natural logarithm of the ratio of book value of equity to market value of equity for firm i in month t  1, [[beta].sup.realized.sub.i,t1] is the lagged realized beta of stock i in month t  1, and [[beta].sup.AR.sub.i,tt1], [[beta].sup.MA.sub.i,tt1], and [[beta].sup.GARCH.sub.i,tt1] are the conditional expected beta of stock i in month t estimated with the information set at month t  1. Table VII reports the timeseries averages of the slope coefficients [[gamma].sub.i,t] (i = 1, 2, 3) over the 498 months from July 1963 to December 2004. The NeweyWest (1987) adjusted tstatistics are given in parentheses. The results show a negative but insignificant relation between the lagged realized beta and the crosssection of average stock returns. The average slope, [[gamma].sub.1,t], from the monthly regressions of realized returns on [[beta].sup.realized.sub.i,t1] alone is about 0.07% with a tstatistic of 1.19. The univariate regression results indicate a significant positive relation between average stock returns and conditional betas. The average slope, [[gamma].sub.1,t], from the monthly regressions of realized returns on [[beta].sup.AR.sub.i,tt1], [[beta].sup.MA.sub.i,tt1], or [[beta].sup.GARCH.sub.i,tt1] in the range of 0.44% to 0.50% and statistically significant at the 5% or better. These values imply a reasonable expected market risk premium of 5.33% to 5.87% per annum Per annum Yearly. . The univariate regression results also indicate a significant negative relation between average stock returns and firm size. The average slope, [[gamma].sub.2,t], from the monthly regressions of realized returns on [logME.sub.i,t1] alone is about 0.24% with a tstatistic of4.75. The parameter (1) Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind. estimates show a significant positive relation between average stock returns and BM ratio. The average slope, [[gamma].sub.3,t], from the monthly regressions of realized returns on log([BE.sub.i,t1]/[ME.sub.i,t1]) alone is about 0.42% with a tstatistic of 5.98. The findings of negative size and positive BM effect in FamaMacBeth (1973) regressions are consistent with Fama and French (1992) and related studies. We find that the strong positive relation between conditional beta and expected stock returns is robust across different econometric specifications. When we add size to the univariate regressions, the average slope coefficient on [[beta].sup.AR.sub.i,tt1], [[beta].sup.MA.sub.i,tt1], or [[beta].sup.GARCH.sub.i,tt1] is about 0.8% and statistically significant at the 1% level. When we add BM to the univariate regressions, the average slope coefficient on conditional betas is in the range of 0.65% to 0.68% and statistically significant at the 1% level. When we include both size and BM in the univariate regressions, the average slope coefficient on the conditional betas is about 0.9% and statistically significant at the 1% level. The [R.sup.2] values from the univariate regressions of realized returns on conditional beta are in the range of 2.02% to 2.13%. When we add size and BM to these univariate regressions, the [R.sup.2] values increase to 4.7% to 4.87%. Although the [R.sup.2] values from univariate and multivariate crosssectional regressions are small, they are consistent with the earlier studies that report [R.sup.2] for the firmlevel crosssectional regressions. IV. Robustness Check This section presents results from a battery of robustness checks. A. Alternative Portfolio Partitions We compute the equalweighted average returns of 20, 50, and 100 portfolios that we form by sorting the NYSE/Amex/Nasdaq stocks based on the conditional AR(1), MA(1), and GARCHinmean betas. Although not presented in the paper, the average return difference between high and lowbeta portfolios is in the range of 0.83% to 1% per month [[beta].sup.AR.sub.i,tt1], 0.89% to 1.11% per month for [[beta].sup.MA.sub.tt1], and 1.06% to 1.31% per month [[beta].sup.GARCH.sub.tt1] All these average return differences are statistically significant at the 5% level or better. In addition to the average raw returns, we also find the magnitude and statistical significance of the FF3 alphas. The 101 difference in the FF3 alphas is positive and highly significant for the AR(1), MA(1), and GARCHinmean betas. In addition to the firmlevel FamaMacBeth (1973) regressions, we examine the crosssectional relation between conditional beta and expected returns at the portfolio level. We present the timeseries averages of the slope coefficients from the crosssection of average portfolio returns on the conditional portfolio beta: [R.sub.p,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[beta].sup.AR.sub.p,tt1] + [[epsilon].sub.i,t], (11) [R.sub.p,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[beta].sup.MA.sub.p,tt1] + [[epsilon].sub.i,t], (12) [R.sub.p,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[beta].sup.GARCH.sub.p,tt1] + [[epsilon].sub.i,t]. (13) In Equations (11) to (13), [R.sub.p,t] is the realized return on portfolio p in month t calculated as the equalweighted average returns of all stocks in portfolio p, and [[beta].sup.AR.sub.p,tt1], [[beta].sup.MA.sub.p,tt1], and [[beta].sup.GARCH.sub.p,tt1] are the conditional expected betas of portfolio p that we obtain from the equalweighted average conditional beta of all stocks in portfolio p in month t estimated with the information set at month t  1. First, we form 10, 20, 50, and 100 portfolios by sorting the NYSE/Amex/Nasdaq stocks based on the conditional AR(1), MA(1), and GARCHinmean betas. Then, for each month from July 1963 to December 2004, we compute each portfolio's return as the equalweighted average return of all stocks in the portfolio, and we calculate the portfolio's conditional beta as the equalweighted average conditional beta of all stocks in the portfolio. We run the univariate regressions of average portfolio returns on the average conditional portfolio beta for each month from July 1963 to December 2004. We calculate the timeseries averages of the slope coefficients and the NeweyWest (1987) adjusted tstatistics. The univariate regression results indicate a significant positive relation between average portfolio returns and average portfolio betas. Although not presented here, for 10 beta portfolios, the average slopes from the monthly regressions of average portfolio returns on [[beta].sup.AR.sub.p,tt1], [[beta].sup.MA.sub.p,tt1] and [[beta].sup.GARCH.sub.p,tt1] are about 0.44%, 0.47%, and 0.53%, respectively. These average slope coefficients have NeweyWest tstatistics of 2.25, 2.37, and 2.61, respectively. A notable point is that for 20, 50, and 100 beta portfolios, the average slope coefficients are similar. In other words, the results are robust across different portfolio formations. These slope coefficients, which imply an expected market risk premium of 5.28% to 6.36% per annum, are also similar to our earlier findings from the firmlevel crosssectional regressions. The [R.sup.2] values are much higher for the portfoliolevel regressions: about 58% to 59% for 10 beta portfolios, 48% to 49% for 20 beta portfolios, 35% to 36% for 50 beta portfolios, and 25% to 26% for 100 beta portfolios. In addition to the monthbymonth FamaMacBeth (1973) regressions, we take the timeseries average of the monthly portfolio returns and the monthly portfolio betas and compute overall average portfolio return and overall average portfolio beta for 10, 20, 50, and 100 portfolios. We plot the average portfolio return against the average portfolio beta and find a strong positive relation between market beta and expected returns for all portfolio partitions. The [R.sup.2] value is 97.94% for 10 beta portfolios, 95.97% for 20 beta portfolios, 88.47% for 50 beta portfolios, and 81.5% for 100 beta portfolios. We also find the slope coefficients for each portfolio partition A reserved part of disk or memory that is set aside for some purpose. On a PC, new hard disks must be partitioned before they can be formatted for the operating system, and the Fdisk utility is used for this task. . The results are similar to our earlier findings from the monthbymonth firmlevel and portfoliolevel FamaMacBeth regressions. The slope on average portfolio beta is almost identical for different portfolio partitions: 0.53% for 10, 50, and 100 beta portfolios, and 0.52% for 20 beta portfolios. B. LongTerm Predictive Power of Conditional Betas Table VIII presents the equalweighted returns of decile portfolios that we form by sorting the NYSE/Amex/Nasdaq stocks based on the conditional GARCHinmean betas. (7) The column labeled "[[beta].sup.GARCH.sub.tt1]" repeats our earlier result for onemonthahead predictability: when we sort decile portfolios based on [[beta].sup.GARCH.sub.tt1] the average return difference between decile 10 (high beta) and decile 1 (low beta) is 1.01% per month with the NeweyWest (1987) tstatistic of 2.83. To test threemonthahead predictability, we form decile portfolios by sorting stocks based on their conditional betas at time t  2 obtained from the information set at time t  3, [[beta].sup.GARCH.sub.t2r3] average return difference between high and lowbeta portfolios is 0.93% per month with the tstatistic of 2.6. As shown in Table VIII, the conditional beta can predict up to 12 months ahead because the average return difference between high and lowbeta portfolios is 0.75% per month with a tstatistic of 2.09. Table VIII also presents the magnitude and statistical significance of the FF3 alphas from the regression of the equalweighted portfolio returns on a constant, excess market return, SMB and HML factors. The 101 difference in the FF3 alphas is positive and significant at the 5% level or better up to ninemonthahead predictability. However, the economic and statistical significance of FF3 alpha gradually grad·u·al adj. Advancing or progressing by regular or continuous degrees: gradual erosion; a gradual slope. n. Roman Catholic Church 1. reduce to 0.43% per month with the tstatistic of 1.88 for 12monthahead returns. C. Controlling for Liquidity and Momentum Following Amihud (2002), we measure stock illiquidity as the ratio of absolute stock return to its dollar volume: [ILLIQ.sub.i,t] = [absolute value of [R.sub.i,t]] / [VOLD.sub.i,t], (14) where [R.sub.i,t] is the return on stock i in month t, and [VOLD.sub.i,t] is the respective monthly volume in dollars. This ratio gives the absolute percentage price change per dollar of monthly trading volume Trading volume The number of shares transacted every day. As there is a seller for every buyer, one can think of the trading volume as half of the number of shares transacted. That is, if A sells 100 shares to B, the volume is 100 shares. . As in Amihud (2002), [ILLIQ.sub.i.t] follows the Kyle's (1985) concept of illiquidity, that is, the response of price to the associated order flow or trading volume. The measure of stock illiquidity given in Equation (14) represents the price response associated with one dollar of trading volume. Thus, it serves as a rough measure of price impact. We control for liquidity by first forming decile portfolios ranked based on Amihud's (2002) measure of illiquidity. Then, within each illiquidity decile, we sort stocks into decile portfolios, which we rank based on the GARCHinmean betas so that decile 1 (10) contains stocks with the lowest (highest) market beta. In each illiquidity decile, the highest (lowest) beta decile has a higher (lower) average returns. The column labeled "Illiquidity" in Table IX presents the average returns across the 10 illiquidity deciles to produce decile portfolios with dispersion in market beta. This procedure creates a set of decile beta portfolios with nearidentical levels of illiquidity. Thus, these decile beta portfolios control for differences in illiquidity. After controlling for illiquidity, we find that the average return difference between high and lowbeta portfolios is 1.20% per month with the NeweyWest (1987) tstatistic of 3.05. Thus, liquidity does not explain the high (low) returns to high (low) beta stocks. When we measure liquidity of individual stocks using dollar trading volume, we obtain similar results. The column labeled "Volume" presents the average returns across the 10 volume deciles to produce decile portfolios with dispersion in Table IX market beta. After controlling for dollar trading volume, we find that the average return increase monotonically from 0.92% to 2.38% when moving from low to highbeta portfolios. The average return difference between high and lowbeta portfolios is 1.46% per month with the NeweyWest (1987) tstatistic of 3.68. Thus, trading volume does not explain the high (low) returns to high (low) beta stocks either. We control momentum by first forming decile loserwinner portfolios ranked based on the past sixmonth average returns of individual stocks. Then, within each sixmonth momentum portfolio, we sort stocks into decile portfolios ranked based on GARCHinmean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The column labeled "MOM6" in Table IX presents the average returns across the 10 momentum deciles to produce decile portfolios with dispersion in market beta. This procedure creates a set of decile beta portfolios with nearidentical levels of past average sixmonth returns. Thus, these decile beta portfolios control for differences in momentum. After controlling for momentum, the average return difference between highand lowbeta portfolios is 0.99% per month with the NeweyWest (1987) tstatistic of 2.89. Thus, momentum does not explain the high (low) returns to high (low) beta stocks. We obtain similar results when we form loserwinner portfolios based on the past 12month average returns (MOM12). The average return difference between high and lowbeta portfolios is 0.86% per month with a tstatistic of 2.74. After controlling for liquidity, momentum, size, and BM, we investigate whether the positive relation between conditional beta and the crosssection of expected returns holds in the firmlevel FamaMacBeth (1973) regressions. Table X presents the timeseries averages of the slope coefficients and the NeweyWest (1987) adjusted tstatistics in parentheses. The regression results indicate a significant positive relation between average stock returns and the conditional GARCHinmean betas after controlling for illiquidity, trading volume, past average 6 and 12month returns with and without size, and BM. The average slope coefficient on [[beta].sup.GARCH.sub.tt1] is about 0.51% with [ILLIQ.sub.t1] and 0.52% with [VOL VOL Volume VOL Volunteer VOL Volcano VOL Volvo (stock symbol) VOL Verdingungsordnung für Leistungen (German) VOL Volatile Organic Liquid Vol Volscan (linguistics) .sub.t1], and both coefficients are highly significant. The average slope coefficient on [[beta].sup.GARCH.sub.tt1] is about 0.41% and significant at the 5% level when we add [MOM6.sub.t1] or [MOM12.sub.t 1] along with [[beta].sup.GARCH.sub.tt1] in the crosssectional regressions. When we include alternative measures of liquidity and momentum along with [[beta].sup.GARCH.sub.tt1], size, and BM, the average slope coefficient on [[beta].sup.GARCH.sub.tt1] becomes stable in the range of 0.62% to 0.66% for different specifications. As shown in for all these multivariate regressions with liquidity, momentum, size, and BM, the average slope coefficient on [[beta].sup.GARCH.sub.tt1] is statistically significant at the 1% level. At an earlier stage of the study, we replicated our results presented in using the conditional beta estimates obtained from the AR(1) and MA(1) specifications. The results turn out to be similar to those from [[beta].sup.GARCH.sub.tt1]. We do not present our findings here from [[beta].sup.AR.sub.i,tt1] and [[beta].sup.MA.sub.i,tt1]. They are available on request. D. Results from the NYSE Sample To check the robustness of our findings, we exclude the Amex and Nasdaq stocks from our sample and form the beta portfolios by sorting only the NYSE stocks based on the conditional GARCHinmean betas. Table XI shows that for the univariate sort of NYSE stocks, the average return difference between high and lowbeta portfolios is about 0.86% with the NeweyWest (1987) tstatistic of 2.79. The 101 difference in the FF3 alphas is 0.37% with a tstatistic of 2.44. We further examine the crosssectional relation by forming the beta portfolios within each size and BM decile. Table XI shows that the average return difference between high and lowbeta portfolios is 0.84% after we control for size and 0.78% after we control for BM. Both return differences are statistically significant at the 1% level. The 101 differences in the FF3 alphas are also positive and highly significant. These results indicate that excluding the Amex and Nasdaq sample has almost no effect on our previous findings. These results remain the same for alternative specifications of conditional beta ([[beta].sup.AR.sub.tt1] and [[beta].sup.MA.sub.tt1]). E. Controlling for Microstructure Effects and NYSE Breakpoint The location in a program used to temporarily halt the program for testing and debugging. Lines of code in a source program are marked for breakpoints. When those instructions are about to be executed, the program stops, allowing the programmer to examine the status of the program Above, we excluded the Amex and Nasdaq stocks and presented the return/beta estimates from the portfolios of NYSE stocks formed based on the NYSE breakpoints. However, these results may be contaminated contaminated, v 1. made radioactive by the addition of small quantities of radioactive material. 2. made contaminated by adding infective or radiographic materials. 3. an infective surface or object. by microstructure effects because there is only a onemonth gap between the conditional beta estimates and portfolio returns. Here, we follow Fama and French (1992) by skipping skip v. skipped, skip·ping, skips v.intr. 1. a. To move by hopping on one foot and then the other. b. To leap lightly about. 2. the month following portfolio formation to avoid microstructure effects and use the NYSE breakpoints to generate beta portfolios of NYSE/Amex/Nasdaq stocks with a relatively more balanced average market share. Since there are so many smallcap Smallcap A stock with a small capitalization, meaning a total equity value of less than $500 million. smallcap 1. Of or relating to the common stock of a relatively small firm having little equity and few shares of common stock Nasdaq stocks, we determine portfolio breakdowns by using only NYSE stocks. Doing so enables us to avoid the beta portfolios that contain small stocks from being too small in terms of average market share. Table XII presents the average returns on the beta portfolios of NYSE/Amex/Nasdaq stocks with NYSE breakpoints after we skip the month following portfolio formation. When we sort portfolios based on the lagged realized beta, the average return difference between high and lowbeta portfolios is economically and statistically nonsignificant non·sig·nif·i·cant adj. 1. Not significant. 2. Having, producing, or being a value obtained from a statistical test that lies within the limits for being of random occurrence. . When we form portfolios based on the AR(1), MA(1), and GARCHinmean beta estimates, the average return difference between deciles 10 and 1 is about 0.7%, 0.72%, and 0.92% per month, respectively. Similar to our earlier findings, for all conditional beta estimates, the 101 differences in average returns and FF3 alphas are positive and highly significant. Overall, the results in Table XII indicate that forming portfolios with CRSP or NYSE breakpoints and skipping the month following portfolio formation does not affect our main conclusions. V. Results from Size/BM/Beta Portfolios When we construct beta portfolios, we control for size or BM ratio, but not both. Here, we test whether the significantly positive relation between conditional beta and expected returns remains intact after we control simultaneously si·mul·ta·ne·ous adj. 1. Happening, existing, or done at the same time. See Synonyms at contemporary. 2. Mathematics for size and BM. Table XIII presents the average returns and FF3 alphas on the quintile quin·tile n. 1. The astrological aspect of planets distant from each other by 72° or one fifth of the zodiac. 2. Statistics The portion of a frequency distribution containing one fifth of the total sample. portfolios of realized and conditional betas after we control for size and BM. At the beginning of each month t from July 1963 to December 2004, we first sort all NYSE/Amex/Nasdaq stocks into five size (market equity) portfolios. Then within each size portfolio, stocks are sorted into five BM (booktomarket equity ratio) portfolios. Finally, within each portfolio formed based on the intersections of five size and five BM portfolios, we sort stocks into five beta portfolios based on their realized and conditional betas in month t  1. Table XIII shows that when we sort stocks in the 5 x 5 size/BM portfolios into five realized beta ([[beta].sup.realized.sub.t1]) portfolios, the average return difference between high and lowbeta portfolios is about 0.5% per month with a tstatistic of 0.04. Similar to our earlier findings from the univariate and bivariate sorts, there is no significant relation between lagged realized beta and the crosssection of expected returns from trivariate sorts. When we sort the stocks in the 5 x 5 size/BM portfolios into five AR(1), MA(1), and GARCHinmean beta portfolios, the average return differences between high and lowbeta portfolios are about 0.97%, 1.01%, and 1.06% per month, respectively. These return differences are statistically significant at the 1% level. Moreover, for all conditional beta estimates, the 51 differences in FF3 alphas are positive and highly significant. Overall, the results in Table XIII indicate that the significant positive relation between conditional beta and the crosssection of expected returns remains the same after we control simultaneously for both size and BM. To provide further evidence for the significant positive link between conditional beta and expected returns on size/BM/beta portfolios, we run the FamaMacBeth (1973) regressions using the 125 (5 x 5 x 5) portfolios of size, BM, and beta. First, we compute the monthly realized beta for each of the 125 portfolios, using daily returns within a month. Then, we generate the conditional beta estimates for each of the 125 portfolios using the AR(1), MA(1), and GARCHinmean specifications. We use the average firm size and average BM ratio of each portfolio as additional controls in FamaMacBeth regressions. Table XIV shows that the average slope coefficients on conditional betas are positive and highly significant with and without controlling for the portfolios' size and BM. Confirming the earlier findings from firmlevel regressions, the average slopes on size and BM turn out to be significantly negative and positive, respectively. Overall, we can conclude that the FamaMacBeth regressions at the firm level and at the portfolio level yield similar results on the relation between market beta and expected returns. To check whether the crosssectional relation still holds after we control for the timeseries relation between conditional betas and expected returns, we run the pooled panel regressions using both the crosssection and time series of 125 portfolio returns and betas. Table XV presents the parameter estimates and the tstatistics that are corrected for heteroskedasticity, firstorder firstorder  Not higherorder. autocorrelation Autocorrelation The correlation of a variable with itself over successive time intervals. Sometimes called serial correlation. , and contemporaneous crosscorrelations in the error terms. Similar to our earlier findings, the pooled panel regressions indicate a positive and highly significant relation between conditional beta and expected returns, but the relation between lagged realized beta and expected returns is not significant. These results hold after controlling for size and BM in crosssection and timeseries setting. VI. CrossSectional Implications of the Conditional CAPM The static (or unconditional) CAPM of Sharpe (1964), Lintner (1965), and Black (1972) indicates that there is a positive linear relation between expected returns on securities and their market betas: E([R.sub.i,t]) = [[beta].sub.i] E ([R.sub.m,t]), (15) where E([R.sub.i,t]) is the unconditional expected excess return of asset i, E([R.sub.m,t]) is the unconditional expected excess return of the market portfolio, and [[beta].sub.i] = Cov([R.sub.i,t], [R.sub.m,t])/Var([R.sub.m,t]) is the unconditional beta of asset i. Fama and French (1992) and related studies find that the unconditional market beta cannot explain the crosssectional variation in expected stock returns. The unconditional CAPM was derived de·rive v. de·rived, de·riv·ing, de·rives v.tr. 1. To obtain or receive from a source. 2. by examining the behavior of investors in a hypothetical model in which they live for only one period, but in the real world, investors live for many periods. Hence, in an empirical examination of the CAPM that uses data from the real world, it is necessary to make certain assumptions. One of the most common assumptions in the static CAPM framework is that the betas of the assets remain constant over time. However, this assumption is not reasonable, because the relative risk of a firm's cash flow is likely to vary over the business cycle. As indicated by Harvey Harvey, city (1990 pop. 29,771), Cook co., NE Ill., a suburb S of Chicago; inc. 1895. Its manufactures include steel castings, metal products, chemicals, machinery, and electronic equipment. Harvey has an oil research center. The city was founded by Turlington W. (1989), Shanken (1990), Jagannathan and Wang (1996), Ferson and Harvey (1991, 1999), and Lettau and Ludvigson (2001), betas and expected returns generally depend on the nature of the information available at any given point in time, and thus will vary over time. The conditional version of the CAPM imposes the restriction restriction  A bug or design error that limits a program's capabilities, and which is sufficiently egregious that nobody can quite work up enough nerve to describe it as a feature. that conditionally con·di·tion·al adj. 1. Imposing, depending on, or containing a condition. See Synonyms at dependent. 2. Grammar Stating, containing, or implying a condition. 3. expected returns on assets are linearly related to the conditionally expected return on the market portfolio in excess of the riskfree rate. The coefficient in the linear relation is the asset's conditional beta or the ratio of the conditional covariance of the asset's return with the market to the conditional variance of the market: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16) where E([R.sub.i,t+1]  [[OMEGA].sub.t]) is the conditional expected excess return of asset i, E([R.sub.m,t+1]  [[OMEGA].sub.t]) is the conditional expected excess return of the market portfolio, [[beta].sub.i,t+1] = Cov([R.sub.i,t+1], [R.sub.m,t+1]  [[OMEGA].sub.t]) / Var ([R.sub.m,t+1]  [[OMEGA].sub.t]) is the conditional market beta of asset i, and [OMEGA].sub.t] denotes the information set at time t. We can rewrite re·write v. re·wrote , re·writ·ten , re·writ·ing, re·writes v.tr. 1. To write again, especially in a different or improved form; revise. 2. Equation (16) to simplify the followup followup, n the process of monitoring the progress of a patient after a period of active treatment. followup subsequent. followup plan expressions: E ([R.sub.i,t+1]  [[OMEGA].sub.t] = [A.sub.m,t+l] x [[beta].sub.i,t+1], (17) where [A.sub.m,t+1] : E([R.sub.m,t+1]  [OMEGA].sub.t]) is the time t + 1 conditional expected market risk premium. Taking the unconditional expectation of both sides of Equation (17), we obtain the unconditional implication of the conditional CAPM: E[[R.sub.i,t+l]] = [[bar.A].sub.m] x [[bar.[beta]].sub.i] + Cov([A.sub.m,t+l], [[beta].sub.i,,t+l]), (18) where Cov([A.sub.m,t+1], [[beta].sub.i,t+1]) denotes the unconditional covariance, and E[[A.sub.m,t+1] = [[bar.A].sub.m] and E[[beta].sub.i,t+1] = [[bar.[beta]].sub.i] are the unconditional means of the corresponding conditional estimates. We note that the last term in Equation (18) depends only on the part of the conditional beta that is in the linear span of the market risk premium, which motivates Jagannathan and Wang (1996) to decompose de·com·pose v. de·com·posed, de·com·pos·ing, de·com·pos·es v.tr. 1. To separate into components or basic elements. 2. To cause to rot. v.intr. 1. the conditional beta of any asset i into two orthogonal components by regressing the conditional beta on the market risk premium. For each asset i, we run the following regression: [[beta].sub.i,t+1] = [[bar.[beta]].sub.i] + [[lambda].sub.i] ([A.sub.m,t+1]  [[bar.A].sub.m]) + [u.sub.i,t+1], (19) where [[lambda].sub.i] = Cov([A.sub.m,t+l], [[beta].sub.i,t+1])/Var([A.sub.m,t+1]) is the unconditional market betapremium sensitivity that measures the sensitivity of conditional beta to the market risk premium. Substituting (19) into (18) gives: E[[R.sub.i,t+1]] = [[bar.A].sub.m] x [[bar.[beta]].sub.i] + [[lambda].sub.i] x Var([A.sub.m.t+l]). (20) Hence, crosssectionally, the unconditional expected excess return on any asset i is a linear function of the unconditional average of its conditional market beta ([[bar.[beta]].sub.i]) and its unconditional market betapremium sensitivity ([[lambda].sub.i]). Equation (20) implies that stocks with higher expected betas have higher unconditional expected returns, as do stocks with betas that are prone to vary with the market risk premium and hence less stable over the business cycle. Hence, the onefactor conditional CAPM leads to a twofactor model Twofactor model Usually, Fischer Black's zerobeta version of the capital asset pricing model. It may also refer to another type of model whereby expected returns are generated by any two factors. for unconditional expected returns. A complete test of the conditional CAPM specification requires that we estimate the expected beta ([[bar.[beta]].sub.i]) and betapremium sensitivity ([[lambda].sub.i]). Here, we use the average conditional beta estimates obtained from AR(1), MA(1), and GARCHinmean specifications as a proxy See proxy server. (networking) proxy  A process that accepts requests for some service and passes them on to the real server. A proxy may run on dedicated hardware or may be purely software. for [[bar.[beta]].sub.i]. We estimate betapremium sensitivity [[lambda].sub.i] using the lagged market return as a proxy for the expected market risk premium, that is, [[lambda].sub.i] = Cov([R.sub.m,t], [[beta].sub.i,t+1])/Var([R.sub.m,t]), where we use the lagged market return, [R.sub.m,t], as a proxy for the time t + 1 conditional expected market risk premium, [A.sub.m,t+1] = E([R.sub.m,t+1]  [[OMEGA].sub.t]) = [R.sub.m,t]. For each month, we run the following crosssectional FamaMacBeth (1973) regressions: [R.sub.i,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[bar.[beta]].sup.AR.sub.i] + [[gamma].sub.2,t] x [[lambda].sup.AR.sub.i] + [[epsilon.sub.i,t], (21) [R.sub.i,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[bar.[beta]].sup.MA.sub.i] + [[gamma].sub.2,t] x [[lambda].sup.MA.sub.i] + [[epsilon.sub.i,t], (22) [R.sub.i,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[bar.[beta]].sup.GARCH.sub.i] + [[gamma].sub.2,t] x [[lambda].sup.GARCH.sub.i] + [[epsilon.sub.i,t], (23) In Equations (21) to (23), [[bar.[beta]].sup.AR.sub.i] , [[bar.[beta]].sup.MA.sub.i], and [[bar.[beta]].sup.GARCH.sub.i] are the timeseries average [[bar.[beta]].sup.AR.sub.i,tt1,], [[bar.[beta]].sup.MA.sub.i,tt1], and [[beta]].sup.GARCH.sub.i,tt1] respectively. Here, [[lambda].sup.MA.sub.i], [[lambda].sup.GARCH.sub.i] are obtained from the regression of [[beta]].sup.AR.sub.i,tt1] [[beta]].sup.MA.sub.i,tt1] and [[beta]].sup.GARCH.sub.i,tt1] on [R.sub.m,t], respectively. The lagged return on the CRSP valueweighted index is our proxy for [R.sub.m,t]. We compute the timeseries averages of the slope coefficients and their NeweyWest (1987) tstatistics from the monthly crosssectional FamaMacBeth (1973) regressions of stock returns on their average conditional beta and betapremium sensitivity. The average slopes on [[bar.[beta]].sup.AR.sub.i], [[bar.[beta]].sup.MA.sub.i] and [[bar.[beta]].sup.GARCH.sub.i]are about 0.622, 0.6189, and 0.5705 with the NeweyWest tstatistic of 2.89, 2.85, and 2.67, respectively. However, the average slope coefficients on betapremium sensitivity are economically and statistically nonsignificant for all specifications of the conditional beta measures. The results indicate a significant positive relation between average conditional beta and the crosssection of expected returns within the conditional CAPM framework. To provide further evidence on the correlation between conditional beta and market risk premium, we investigate the correlations between the conditional betas and the Chicago Chicago, city, United States Chicago (shĭkä`gō, shĭkô`gō), city (1990 pop. 2,783,726), seat of Cook co., NE Ill., on Lake Michigan; inc. 1837. Fed National Activity Index (CFNAI CFNAI Chicago Fed National Activity Index ), which is a weighted average of 85 existing monthly indicators of national economic activity constructed to have an average value of zero and a standard deviation of one. Since economic activity tends toward trend growth rate over time, a positive index reading corresponds to growth above trend, and a negative index reading corresponds to growth below trend. We expect to find a positive relation between expected stock returns and innovations in output (or growth above trend). Actual increases in real economic activity, if greater than expected (or greater than the trend), may increase agents' expectations of future growth. Forecasts of higher economic growth should make stocks more attractive and thus cause an immediate jump in share prices. That is, the positive relation between expected returns and the CFNAI makes economic sense. Since there is a positive relation between conditional betas and expected returns, we also expect to find a positive link between the conditional beta and the CFNAI. Figure 1 shows that the sample correlations for almost all of the 125 portfolios are positive for the AR(1), MA(1), and GARCHinmean beta estimates. These results provide further evidence on the capability of conditional betas to predict the timeseries and crosssectional variation in stock returns. [FIGURE 1 OMITTED] VII. Conclusion In this paper, we investigate the crosssectional relation between conditional betas and expected stock returns for the sample period of July 1963 to December 2004. First, we use daily returns within a month to compute realized beta for each stock trading at the NYSE, Amex, and Nasdaq and then use autoregressive, moving average, and GARCHinmean models to obtain timevarying conditional betas for each stock. For each specification of conditional beta, we find that the average portfolio returns increase almost monotonically when moving from lowbeta to highbeta portfolios. The portfoliolevel analyses and the firmlevel crosssectional regressions indicate that the positive relation between the conditional betas and the crosssection of average returns is economically and statistically significant. For the NYSE/Amex/Nasdaq sample, the average return difference between high and lowbeta portfolios is in the range of 0.89% to 1.01% per month, depending on the timevarying specification of conditional beta. To check whether our findings are driven by small, illiquid, and lowprice stocks, we exclude the Amex and Nasdaq stocks and form the beta portfolios by sorting only the NYSE stocks based on the conditional betas. The results indicate that excluding the Amex and Nasdaq sample has almost no effect on our original findings. We also control for the crosssectional effects of size, BM, liquidity, and momentum. After controlling for these effects, we estimate the crosssectional beta premium to be in the range of 0.86% to 1.46% per month. These results are robust across different measures of conditional beta. We thank Bill Christie (the Editor) and two other anonymous Nameless. See anonymous post and anonymous Web surfing. referees for their extremely helpful comments and suggestions. We also benefited from discussions with Hadiye Asian, Ozgur Demirtas, Armen Armen may refer to:
American army engineer and parliamentary authority. He designed the defenses for Washington, D.C., during the Civil War and later wrote Robert's Rules of Order (1876). Noun 1. Whitelaw, and seminar participants at Barueh College, Graduate School, and University Center of the University of New York There is no institution of higher education in the State of New York or the United States of America that bears the name University of New York. However, in confusion, it is possible that such a reference may regard the following: References Amihud, Y., 2002, "Illiquidity and Stock Returns: CrossSection and TimeSeries Effects," Journal of Financial Markets 5, 3156. Ang, A. and J. Chen, 2007, "CAPM over the LongRun: 19262001," Journal of Empirical Finance 14, 140. Ang, A.J., J. Chen, and Y. Xing, 2006, "Downside Risk," Review of Financial Studies 19, 11911239. 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(2) This is because an asset that is on the conditional meanvariance frontier frontier, in U.S. history, the border area of settlement of Europeans and their descendants; it was vital in the conquest of the land between the Atlantic and the Pacific. need not be on the unconditional frontier, as Dybvig and Ross (1985) and Hansen and Richard (1987) point out. Also see Chan and Chen (1988) who indicate that even when betas vary over time, unconditional CAPM can hold. (3) An incomplete list includes Bollerslev, Engle, and Wooldridge (1988), Harvey (1989, 2001 ), Shanken (1990, 1992), Ferson and Harvey (1991, 1999), Fama and French (1997), Lettau and Ludvigson (2001), Campbell and Vuolteenaho (2004), Jostova and Philipov (2005), Petkova and Zhang (2005), Ang and Chen (2007), Lewellen and Nagel (2006), and Bali (2008). (4) French, Schwert, and Stambaugh (1987), Campbell, Lettau, Malkiel, and Xu (2001), Goyal and SantaClara (2003), and Bali, Cakici. Yan, and Zhang (2005) use withinmonth daily returns to estimate the monthly market variance The discrepancy between what a party to a lawsuit alleges will be proved in pleadings and what the party actually proves at trial. In Zoning law, an official permit to use property in a manner that departs from the way in which other property in the same locality or the monthly idiosyncratic id·i·o·syn·cra·sy n. pl. id·i·o·syn·cra·sies 1. A structural or behavioral characteristic peculiar to an individual or group. 2. A physiological or temperamental peculiarity. 3. or total volatility of each stock trading at the NYSE, Amex, and Nasdaq. (5) Brav, Lehavy, and Michaely (2005) use analysts' expected rates of return instead of realized rates of return as a proxy for expected return and identify a positive, robust relation between expected return and market beta. Based on their experimental study, Bloomfield and Michaely (2004) find that market professionals expect firms with higher betas to be riskier investments and to generate higher returns. Harris, Marston, Mishra, and O'Brien (2003) estimate the relation between market beta and expected returns for S&P 500 stocks, and their findings indicate that for estimating the cost of equity, the choice between the domestic and global CAPM may not be a material issue for many large US firms. (6) In Table V, we form decile portfolios based on the GARCHinmean beta estimates. The results from the AR(I) and MA(1) models are similar to those in Table V and are available from the authors on request. (7) We do not present the results from [[beta].sup.AR.sub.i,tt1] and [[beta].sup.MA.sub.i,tt1] which are similar to those in the table. They are available on request. Turan Turan (trän`), desert lowland, shared by Kazakhstan, Uzbekistan, and Turkmenistan, S and E of the Aral Sea. G. Bali, Nusret This article is about the Turkish minelayer. For the Pakistani musician, see Nusrat Fateh Ali Khan Nusret also known as Nusrat is a Turkish minelayer that strongly influenced the course of the 1915 Battle of Gallipoli. Cakici, and Yi Tang tang, in zoology tang: see butterfly fish. * * Turan G. Bali is the David Krell David Krell is a founder and President & CEO of ISE, LLC and ISE Holdings. From 1997 to 1998, he was Chairman and cofounder of KSquared Research, LLC, a financial services consulting firm. Chair Professor of Finance at the University of New York in New York NE and Visiting Professor of Finance at Koc University, Turkey. Nusret Cala'ci is a Professor of Finance at Fordham University Fordham University (fôr`dəm), in New York City; Jesuit; coeducational; founded as St. John's College 1841, chartered as a university 1846; renamed 1907. Fordham College for men and Thomas More College for women merged in 1974. in New York, NY. Yi Tang is an Assistant Professor of Finance at Fordham University in New York, NE Table I. Correlation between Realized Beta and Conditional Volatility This table presents the percentiles of the correlation measures for all stocks trading at the NYSE, Amex, and Nasdaq. The correlation statistics indicate a strong relation between the monthly realized betas and their conditional volatility, and a strong relation between the conditional mean of monthly realized betas and their conditional volatility. [[beta].sub.t], [[beta].sub.t], Correlation [[sigma].sub.t] [[sigma].sup.2.sub.t] 1% 0.625 0.650 5% 0.390 0.422 10% 0.272 0.301 20% 0.150 0.168 30% 0.069 0.080 40% 0.002 0.004 50% 0.067 0.067 60% 0.130 0.135 70% 0.196 0.204 80% 0.273 0.287 90% 0.390 0.412 95% 0.493 0.520 99% 0.696 0.711 E([[beta].sub.t] E([[beta].sub.t] [[OMEGA].sub.t1]), [[OMEGA].sub.t1]), Correlation [[sigma].sub.t] [[sigma].sup.2.sub.t] 1% 0.988 0.989 5% 0.971 0.973 10% 0.944 0.946 20% 0.838 0.826 30% 0.505 0.474 40% 0.073 0.066 50% 0.583 0.549 60% 0.791 0.773 70% 0.882 0.877 80% 0.929 0.931 90% 0.963 0.967 95% 0.979 0.981 99% 0.992 0.993 Table II. TimeSeries Mean and Standard Deviation of Realized and Conditional Betas This table presents the percentiles of the timeseries mean and standard deviation of realized and conditional betas for the sample period of July 1963 to December 2004. In Panel A, we compute the realized beta for each stock trading at NYSE\Amex\Nasdaq by using daily returns over the previous month without lagged market return. In Panel B, we compute the realized beta by using daily returns over the previous month with the lagged market return. We use the CRSP valueweighted index as our proxy for the market portfolio. We estimate conditional betas based on the AR(1), MA(1), and GARCHinmean models: AR(1): [[beta].sub.t] = [a.sub.0] + [a.sub.1][[beta].sub.t1] + [[epsilon].sub.t], E([[beta].sub.t]  [[OMEGA].sub.t1]) = [[beta].sup.AR.sub.t\t1] = [??.sub.0] + [??.sub.1] [[beta].sub.t1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], MA(1): [[beta].sub.t] = [b.sub.0] + [b.sub.1][[epsilon].sub.t1] + [[epsilon].sub.t], E([[beta].sub.t]  [[OMEGA].sub.t1]) = [[beta].sup.MA.sub.t\t1] = [??.sub.0] + [??.sub.1] [[epsilon].sub.t1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], GARCHinmean: [[beta].sub.t] = [c.sub.0] + [c.sub.1] [[sigma].sub.t\t1] + [[epsilon].sub.t], E([[beta].sub.t]  [[OMEGA].sub.t1]) = [[beta].sup.GARCH.sub.t\t1] = [??.sub.0] + [??.sub.1][[sigma].sup.2.sub.t\t1], E([[epsilon].sup.2.sub.t] [[OMEGA].sub.t1]) = [[sigma].sup.2.sub.t\t1] = [[gamma].sub.0] + [[gamma].sub.1][[epsilon].sup.2.sub.t1] + [[gamma].sub.2] [[sigma].sup.2.sub.t1]. Panel A. Realized Beta Is Estimated without the Lagged Market Return Mean 1% 5% 10% 20% [[beta].sup.realized.sub.t] 0.1426 0.0347 0.1112 0.2325 SD 1% 5% 10% 20% [[beta].sup.realized.sub.t] 0.3929 0.5348 0.6360 0.7841 [[beta].sup.AR.sub.t\t1] 0.0020 0.0109 0.0220 0.0442 [[beta].sup.GARCH.sub.t\t1] 0.0014 0.0101 0.0164 0.0454 30% 40% 50% [[beta].sup.realized.sub.t] 0.3503 0.4610 0.5642 SD 30% 40% 50% [[beta].sup.realized.sub.t] 0.9236 1.0641 1.2175 [[beta].sup.AR.sub.t\t1] 0.0657 0.0891 0.1154 [[beta].sup.GARCH.sub.t\t1] 0.0747 0.1071 0.1431 60% 70% 80% [[beta].sup.realized.sub.t] 0.6862 0.8163 0.9777 SD 60% 70% 80% [[beta].sup.realized.sub.t] 1.3879 1.5970 1.8609 [[beta].sup.AR.sub.t\t1] 0.1439 0.1830 0.2369 [[beta].sup.GARCH.sub.t\t1] 0.1880 0.2435 0.3251 90% 95% 99% [[beta].sup.realized.sub.t] 1.2420 1.4954 2.0480 SD 90% 95% 99% [[beta].sup.realized.sub.t] 2.3190 2.8373 4.2165 [[beta].sup.AR.sub.t\t1] 0.3378 0.4527 0.8169 [[beta].sup.GARCH.sub.t\t1] 0.4693 0.6275 1.1644 Panel B. Realized Beta Is Estimated with the Lagged Market Return Mean 1% 5% 10% 20% [[beta].sup.realized.sub.t] 0.1450 0.0692 0.1744 0.3217 SD 1% 5% 10% 20% [[beta].sup.realized.sub.t] 0.5428 0.7126 0.8439 1.0454 [[beta].sup.AR.sub.t\t1] 0.0026 0.0117 0.0234 0.0472 [[beta].sup.GARCH.sub.t\t1] 0.0059 0.0101 0.0221 0.0461 Mean 30% 40% 50% [[beta].sup.realized.sub.t] 0.4587 0.5885 0.7111 SD 30% 40% 50% [[beta].sup.realized.sub.t] 1.2325 1.4229 1.6222 [[beta].sup.AR.sub.t\t1] 0.0723 0.0976 0.1260 [[beta].sup.GARCH.sub.t\t1] 0.0802 0.1189 0.1635 Mean 60% 70% 80% [[beta].sup.realized.sub.t] 0.8392 0.9726 1.1352 SD 60% 70% 80% [[beta].sup.realized.sub.t] 1.8551 2.1360 2.4868 [[beta].sup.AR.sub.t\t1] 0.1609 0.2071 0.2737 [[beta].sup.GARCH.sub.t\t1] 0.2201 0.2877 0.3853 Mean 90% 95% 99% [[beta].sup.realized.sub.t] 1.3826 1.6126 2.2350 SD 90% 95% 99% [[beta].sup.realized.sub.t] 3.0675 3.6365 5.4468 [[beta].sup.AR.sub.t\t1] 0.3971 0.5455 1.0561 [[beta].sup.GARCH.sub.t\t1] 0.5551 0.7714 1.5448 Table III. Performance of Lagged Realized and Conditional Betas in Predicting Future Realized Beta This table presents the percentiles of the [R.sup.2] values from the regression of onemonthahead realized betas on the lagged realized and conditional betas for our sample period of July 1963 to December 2004. In Panel A, we compute the realized beta for each stock trading at NYSE/Amex/Nasdaq by using daily returns over the previous month without lagged market return. In Panel B, we compute the realized beta by using daily returns over the previous month with the lagged market return. We use the CRSP valueweighted index as our proxy for the market portfolio. We estimate conditional betas based on the AR(1), MA(1), and GARCHinmean models. We run the following OLS tests to obtain the [R.sup.2] values: [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.realized.sub.t1] + [[epsilon].sub.t], [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.AR.sub.t\t1] + [[epsilon].sub.t], [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.MA.sub.t\t1] + [[epsilon].sub.t], [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.GARCH.sub.t\t1] + [[epsilon].sub.t], [R.sup.2] 1% 5% 10% 20% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t1] 0.01% 0.11% 0.22% 0.69% [[beta].sup.AR.sub.t\t1] 0.74% 2.04% 2.87% 4.58% [[beta].sup.MA.sub.t\t1] 0.63% 1.94% 2.67% 4.48% [[beta].sup.GARCH.sub.t\t1] 1.14% 2.46% 3.33% 4.93% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t1] 0.01% 0.03% 0.08% 0.22% [[beta].sup.AR.sub.t\t1] 1.29% 2.28% 3.12% 4.22% [[beta].sup.MA.sub.t\t1] 1.29% 2.35% 3.14% 4.15% [[beta].sup.GARCH.sub.t\t1] 1.17% 2.12% 3.09% 4.23% [R.sup.2] 30% 40% 50% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t1] 1.18% 1.94% 3.22% [[beta].sup.AR.sub.t\t1] 6.16% 7.88% 10.31% [[beta].sup.MA.sub.t\t1] 5.95% 7.65% 9.98% [[beta].sup.GARCH.sub.t\t1] 6.75% 8.90% 11.01% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t1] 0.46% 0.81% 1.21% [[beta].sup.AR.sub.t\t1] 5.29% 6.10% 7.13% [[beta].sup.MA.sub.t\t1] 5.19% 6.04% 7.13% [[beta].sup.GARCH.sub.t\t1] 5.04% 6.02% 7.10% [R.sup.2] 60% 70% 80% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t1] 4.66% 6.89% 9.94% [[beta].sup.AR.sub.t\t1] 13.12% 15.77% 19.24% [[beta].sup.MA.sub.t\t1] 12.54% 15.39% 18.72% [[beta].sup.GARCH.sub.t\t1] 13.80% 16.70% 20.67% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t1] 1.73% 2.65% 4.12% [[beta].sup.AR.sub.t\t1] 8.38% 10.20% 12.66% [[beta].sup.MA.sub.t\t1] 8.40% 10.19% 12.50% [[beta].sup.GARCH.sub.t\t1] 8.31% 10.50% 12.58% [R.sup.2] 90% 95% 99% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t1] 13.26% 17.67% 26.82% [[beta].sup.AR.sub.t\t1] 23.90% 27.76% 33.82% [[beta].sup.MA.sub.t\t1] 23.26% 26.70% 32.32% [[beta].sup.GARCH.sub.t\t1] 24.88% 27.88% 33.87% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t1] 6.63% 9.54% 16.76% [[beta].sup.AR.sub.t\t1] 16.20% 19.60% 24.77% [[beta].sup.MA.sub.t\t1] 15.92% 19.16% 24.00% [[beta].sup.GARCH.sub.t\t1] 16.99% 19.73% 24.62% Table IV. EqualWeighted Portfolios Sorted by Realized and Conditional Beta We form equalweighted decile portfolios every month from July 1963 to December 2004 by sorting the NYSE/Amex/Nasdaq stocks based on realized and conditional beta. In Panel A, we compute the realized beta for each stock by using daily returns over the previous month without lagged market return. In Panel B, we compute the realized beta by using daily returns over the previous month with the lagged market return. We use the CRSP valueweighted index as our proxy for the market portfolio. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or conditional betas. The row "HighLow" refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the FamaFrench (1993) model. NeweyWest (1987) adjusted tstatistics appear in parentheses. AR(1): [[beta].sub.t] = [a.sub.0] + [a.sub.1][[beta].sub.t1] + [[epsilon].sub.t], E([[beta].sub.t]  [[OMEGA].sub.t1]) = [[beta].sup.AR.sub.t\t1] = [??.sub.0] + [??.sub.1] [[beta].sub.t1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], MA(1): [[beta].sub.t] = [b.sub.0] + [b.sub.1][[epsilon].sub.t1] + [[epsilon].sub.t], E([[beta].sub.t]  [[OMEGA].sub.t1]) = [[beta].sup.MA.sub.t\t1] = [??.sub.0] + [??.sub.1] [[epsilon].sub.t1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], GARCHinmean: [[beta].sub.t] = [c.sub.0] + [c.sub.1] [[sigma].sup.2.sub.t\t1]  [[epsilon].sub.t], E([[beta].sub.t]  [[OMEGA].sub.t1]) = [[beta].sup.GARCH.sub.t\t1] = [??.sub.0] + [??.sub.1][[sigma].sup.2.sub.t\t1], E([[epsilon].sup.2.sub.t]  [[OMEGA].sub.t1]) = [[sigma].sup.2.sub.t\t1] = [[gamma].sub.0] + [[gamma].sub.1][[epsilon].sup.2.sub.t1] + [[gamma].sub.2] [[sigma].sup.2.sub.t1]. Panel A. Realized Beta Estimated without the Lagged Market Return [[beta].sup.realized [[beta].sup.AR .sub.t1] .sub.t\t1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.56 1.65 1.09 0.08 2 1.36 1.37 1.21 0.24 3 1.38 0.01 1.30 0.37 4 1.38 0.23 1.41 0.48 5 1.35 0.47 1.37 0.58 6 1.41 0.72 1.44 0.69 7 1.34 1.01 1.51 0.81 8 1.27 1.37 1.53 0.96 9 1.25 1.91 1.58 1.16 10 High [beta] 1.07 3.40 1.83 1.55 High to low 0.49 ** 0.74 ** (2.53) (2.33) Alpha 0.48 *** 0.50 ** (2.85) (2.17) [[beta].sup.MA [[beta].sup.GARCH .sub.t\t1] .sub.t\t1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.07 0.02 1.11 0.00 2 1.21 0.25 1.15 0.23 3 1.31 0.37 1.27 0.36 4 1.36 0.48 1.29 0.47 5 1.41 0.59 1.38 0.58 6 1.46 0.70 1.41 0.69 7 1.46 0.82 1.49 0.82 8 1.54 0.96 1.51 0.97 9 1.60 1.15 1.67 1.18 10 High [beta] 1.85 1.62 2.03 1.71 High to low 0.78 ** 0.92 *** (2.47) (2.65) Alpha 0.53 ** 0.60 ** (2.33) (2.60) Panel B. Realized Beta Estimated with the Lagged Market Return [[beta].sup.realized [[beta].sup.AR .sub.t1] .sub.t\t1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.44 2.25 1.13 0.03 2 1.32 0.55 1.15 0.33 3 1.33 0.08 1.27 0.47 4 1.39 0.24 1.38 0.60 5 1.42 0.53 1.36 0.71 6 1.41 0.85 1.41 0.83 7 1.36 1.21 1.49 0.95 8 1.38 1.65 1.57 1.08 9 1.25 2.32 1.62 1.27 10 High [beta] 1.08 4.25 2.02 1.74 High to low 0.33 * 0.89 *** (1.92) (2.66) Alpha 0.35 ** 0.63 *** (2.36) (2.76) [[beta].sup.MA [[beta].sup.GARCH .sub.t\t1] .sub.t\t1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.13 0.05 1.10 0.02 2 1.15 0.33 1.19 0.32 3 1.25 0.48 1.25 0.46 4 1.38 0.60 1.28 0.59 5 1.37 0.72 1.39 0.71 6 1.42 0.83 1.40 0.83 7 1.49 0.95 1.44 0.95 8 1.58 1.08 1.55 1.09 9 1.64 1.26 1.69 1.29 10 High [beta] 2.00 1.70 2.11 1.83 High to low 0.87 ** 1.01 *** (2.54) (2.83) Alpha 0.60 ** 0.71 *** (2.62) (3.06) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table V. EqualWeighted Portfolios Sorted by GARCHinMean Beta after Controlling for Size and BM In Panel A, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their market capitalizations. Then, within each size decile, we sort stocks into decile portfolios ranked based on GARCHinmean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The column labeled "Average Returns" averages across the 10 size deciles to produce decile portfolios with dispersion in market beta and with nearidentical levels of firm size, and thus these decile beta portfolios control for differences in size. In Panel B, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their booktomarket ratios (BM). Then, within each BM decile, we sort stocks into decile portfolios ranked based on GARCHinmean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The column labeled "Average Returns" averages across the 10 BM deciles to produce decile portfolios with dispersion in market beta and with nearidentical levels of BM. Thus, these decile beta portfolios control for differences in BM. Panel A. EqualWeighted Returns on Beta Portfolios after Controlling for Size Small Size Size 2 Size 3 Size 4 1 Low [beta] 2.85 0.61 0.48 0.63 2 2.12 0.98 1.03 0.83 3 2.63 0.98 1.06 0.86 4 2.40 1.31 1.13 1.05 5 2.93 1.29 1.20 1.19 6 3.18 1.51 1.17 1.18 7 3.39 1.79 1.55 1.29 8 3.49 1.83 1.62 1.65 9 4.46 2.18 1.88 1.63 10 High [beta] 5.36 3.04 2.31 2.23 Size 5 Size 6 Size 7 Size 8 1 Low [beta] 0.68 0.78 0.88 0.83 2 0.88 0.86 1.11 1.00 3 0.81 0.84 0.99 1.10 4 0.96 0.98 1.12 1.07 5 1.13 1.12 1.24 1.24 6 1.15 1.18 1.16 1.13 7 1.17 1.17 1.19 1.11 8 1.53 1.40 1.27 1.34 9 1.72 1.69 1.28 1.26 10 High [beta] 2.05 2.13 2.03 1.75 Big Average t Size 9 Size Returns statistic 1 Low [beta] 0.84 0.80 0.94 *** 5.07 2 1.05 0.88 1.07 *** 5.26 3 0.96 0.93 1.12 *** 4.76 4 1.03 0.99 1.20 *** 4.67 5 1.01 1.02 1.34 *** 4.72 6 1.06 0.97 1.37 *** 4.58 7 1.28 0.99 1.49 *** 4.67 8 1.11 1.01 1.63 *** 4.54 9 1.28 1.00 1.84 *** 4.56 10 High [beta] 1.51 1.08 2.35 *** 4.55 High [beta] 1.41 *** 3.48 to low [beta] Panel B. EqualWeighted Returns on Beta Portfolios after Controlling for BM Low BM BM 2 BM 3 BM 4 1 Low [beta] 0.35 0.40 0.77 0.86 2 0.19 0.65 0.72 0.90 3 0.51 0.79 0.93 1.04 4 0.59 0.70 1.11 1.10 5 0.69 1.01 1.19 1.11 6 0.45 0.92 0.90 1.20 7 0.85 1.09 1.39 1.31 8 1.00 1.19 1.34 1.40 9 1.03 1.39 1.34 1.58 10 High [beta] 1.64 1.62 1.68 1.90 BM 5 BM 6 BM 7 BM 8 1 Low [beta] 0.82 1.15 1.21 1.42 2 1.03 1.05 1.22 1.49 3 1.02 1.16 1.19 1.32 4 1.05 1.22 1.47 1.49 5 1.13 1.27 1.45 1.57 6 1.20 1.36 1.57 1.64 7 1.27 1.37 1.52 1.68 8 1.17 1.67 1.54 1.70 9 1.63 1.73 1.89 2.05 10 High [beta] 2.04 2.07 2.44 2.31 High Average t BM 9 BM Returns statistic 1 Low [beta] 1.35 1.91 1.02 *** 4.55 2 1.31 1.95 1.05 *** 4.62 3 1.60 1.83 1.14 *** 4.79 4 1.90 2.28 1.29 *** 5.14 5 1.84 2.13 1.34 *** 5.04 6 1.71 1.85 1.28 *** 4.69 7 1.79 2.22 1.45 *** 4.87 8 1.91 2.26 1.52 *** 4.76 9 2.43 2.43 1.75 *** 4.93 10 High [beta] 2.94 3.16 2.18 *** 4.80 High [beta] 1.16 *** 3.69 to low [beta] *** Significant at the 0.01 level. Table VI. Average Return Differences and FF3 Alphas within Each Size and BM Deciles In Panel A, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their market capitalizations. Then, within each size decile, we sort stocks into decile portfolios ranked based on conditional beta so that decile 1 (10) contains stocks with the (lowest) highest market beta. We report the average return differences and alphas along with their NeweyWest (1987) adjusted tstatistics in parentheses for each size decile. In Panel B, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their booktomarket ratios (BM). Then, within each BM decile, we sort stocks into decile portfolios ranked based on GARCHinmean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. We report the average return differences and alphas along with their NeweyWest (1987) adjusted tstatistics in parentheses for each BM decile. Panel A. Average Return Differences and FF3 Alphas within Size Deciles [[beta].sup.AR.sub.t\t1] Decile High [beta] to Low [beta] Alpha Small size 2.49 *** 2.02 *** (5.79) (5.54) Size 2 2.17 *** 1.65 *** (4.94) (4.55) Size 3 1.92 *** 1.40 *** (4.00) (3.55) Size 4 1.67 *** 1.25 *** (3.65) (3.48) Size 5 1.54 *** 1.14 *** (3.46) (3.56) Size 6 1.49 *** 1.17 *** (3.25) (3.59) Size 7 1.19 *** 1.01 *** (2.68) (3.26) Size 8 0.72 * 0.64 ** (1.67) (2.34) Size 9 0.60 0.63 ** (1.33) (2.09) Big size 0.24 0.42 (0.60) (1.58) [[beta].sup.MA.sub.t\t1] Decile High [beta] to Low [beta] Alpha Small size 2.57 *** 2.06 *** (6.09) (6.05) Size 2 2.22 *** 1.69 *** (5.01) (4.67) Size 3 1.88 *** 1.33 *** (3.88) (3.36) Size 4 1.68 *** 1.23 *** (3.57) (3.44) Size 5 1.52 *** 1.13 *** (3.40) (3.57) Size 6 1.51 *** 1.16 *** (3.21) (3.57) Size 7 1.08 ** 0.88 *** (2.39) (2.87) Size 8 0.82 * 0.73 *** (1.87) (2.64) Size 9 0.65 0.67 ** (1.39) (2.13) Big size 0.24 0.43 (0.60) (1.64) [[beta].sup.GARCH.sub.t\t1] Decile High [beta] to Low [beta] Alpha Small size 2.50 *** 2.02 *** (5.89) (5.35) Size 2 2.43 *** 1.83 *** (5.34) (5.14) Size 3 1.83 *** 1.27 *** (3.82) (3.40) Size 4 1.60 *** 1.05 *** (3.46) (3.44) Size 5 1.37 *** 0.94 *** (3.06) (3.05) Size 6 1.35 *** 0.93 *** (2.83) (2.69) Size 7 1.15 ** 0.88 *** (2.56) (2.91) Size 8 0.92 ** 0.87 *** (2.09) (3.05) Size 9 0.67 0.65 ** (1.45) (2.13) Big size 0.28 0.50 * (0.68) (1.91) Panel B. Average Return Differences and FF3 Alphas within BooktoMarket Deciles [[beta].sup.AR.sub.t\t1] Decile High [beta] to Low [beta] Alpha Low BM 1.50 *** 1.60 *** (3.75) (4.59) BM 2 1.27 *** 1.27 *** (3.44) (3.83) BM 3 0.89 ** 0.80 *** (2.58) (2.71) BM 4 0.98 *** 0.68 *** (2.90) (2.59) BM 5 1.18 *** 0.71 *** (3.48) (2.71) BM 6 0.90 *** 0.44 * (2.90) (1.73) BM 7 0.90 ** 0.47 (2.46) (1.53) BM 8 0.82 ** 0.41 (2.27) (1.39) BM 9 1.51 *** 1.01 *** (4.09) (3.25) High BM 0.60 (0.15) (1.62) (0.46) [[beta].sup.MA.sub.t\t1] High [beta] to Low [beta] Alpha Low BM 1.49 *** 1.54 *** (3.74) (4.50) BM 2 1.31 *** 1.33 *** (3.66) (4.12) BM 3 0.82 ** 0.70 ** (2.37) (2.37) BM 4 0.93 *** 0.65 ** (2.76) (2.41) BM 5 1.22 *** 0.75 *** (3.48) (2.73) BM 6 0.84 *** 0.39 (2.66) (1.56) BM 7 0.90 ** 0.46 (2.39) (1.45) BM 8 0.74 ** 0.30 (2.15) (1.07) BM 9 1.41 *** 0.94 *** (3.86) (2.98) High BM 0.49 (0.24) (1.32) (0.73) [[beta].sup.GARCH.sub.t\t1] High [beta] to Low [beta] Alpha Low BM 1.29 *** 1.36 *** (3.27) (4.21) BM 2 1.22 *** 1.19 *** (3.28) (3.57) BM 3 0.91 ** 0.73 *** (2.61) (2.67) BM 4 1.03 *** 0.67 ** (3.06) (2.52) BM 5 1.22 *** 0.79 *** (3.22) (2.65) BM 6 0.92 *** 0.37 (2.83) (1.53) BM 7 1.24 *** 0.81 ** (3.08) (2.38) BM 8 0.89 ** 0.37 (2.38) (1.25) BM 9 1.59 *** 1.14 *** (4.10) (3.52) High BM 1.25 *** 0.59 * (3.21) (1.75) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table VII. FirmLevel CrossSectional Regressions Panel A presents the firmlevel crosssectional regression results for the NYSE/Amex/Nasdaq stocks for our sample period of 23193 to December 2004. We estimate the monthly conditional betas based on the AR(1), MA(1), and GARCHinmean specification of the realized beta measures. We calculate the realized beta of each stock by using daily data over the previous month with the lagged market return. Here, log[ME.sub.t1] is the last month's log market capitalization (size), and log([BE.sub.t1]/[ME.sub.t1]) is the last fiscal year's log booktomarket ratio. The timeseries average slope coefficients are reported in each row. NeweyWest (1987) adjusted tstatistics appear in parentheses. The last column presents the average [R.sup.2] values. [[beta].sub. [[beta].sub.AR [[beta].sub.MA [[beta].sub. realized.sub. .sub.t\t1] .sub.t\t1] GARCH.sub. t1] t\t1] 0.0678 (1.19) 0.4443 ** (2.09) 0.4890 ** (2.26) 0.4565 ** (2.32) 0.0124 (0.21) 0.7985 *** (3.07) 0.8063 *** (3.09) 0.8010 *** (3.05) 0.0357 (0.68) 0.6694 *** (3.34) 0.6796 *** (3.36) 0.6470 *** (3.41) 0.0045 (0.08) 0.8967 *** (3.65) 0.9041 *** (3.66) 0.8929 *** (3.60) log[ME.sub.t1] log [R.sup.2] ([BE.sub.t1]/ [ME.sub.t1]) 1.04% 2.06% 2.02% 2.13% 0.2373 *** 1.81% (4.75) 0.4185 *** 1.00% (5.98) 0.2029 *** 0.2396 *** 2.69% (3.83) (3.21) 0.2229 *** 2.79% (4.18) 0.2907 *** 4.34% (4.69) 0.2901 *** 4.31% (4.68) 0.2869 *** 4.27% (4.52) 0.4328 *** 1.85% (6.67) 0.4721 *** 2.83% 7.39 0.4718 *** 2.79% 7.37 0.4771 *** 2.89% (7.44) 0.1861 *** 0.2765 *** 3.50% (3.31) (4.03) 0.2587 *** 0.2745 *** 4.70% (4.01) (4.29) 0.2579 *** 0.2740 *** 4.70% (4.00) (4.28) 0.2526 *** 0.2872 *** 4.87% (3.86) (4.62) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table VIII. LongTerm Predictive Power of Conditional Beta This table presents the equalweighted average returns, average return differences, and alphas from the 1 to 12monthahead predictability of stock returns. We form equalweighted decile portfolios for every month from July 1963 to December 2004 by sorting the NYSE/Amex/Nasdaq stocks based on the GARCHin mean beta estimates conditional on time t  1 to t  12. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or expected betas. The row HighLow refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the FamaFrench (1993) model. NeweyWest (1987) adjusted tstatistics appear in parentheses. [[beta].sup. [[beta].sup. [[beta].sup. [[beta].sup. Decile GARCH.sub. GARCH.sub. GARCH.sub. GARCH.sub. t\t1] t1\t2] t2\t3] t3\t4] 1 Low P 1.10 1.11 1.12 1.12 2 1.19 1.16 1.18 1.22 3 1.25 1.24 1.22 1.26 4 1.28 1.29 1.29 1.23 5 1.39 1.41 1.38 1.41 6 1.40 1.37 1.41 1.43 7 1.44 1.46 1.48 1.42 8 1.55 1.55 1.52 1.56 9 1.69 1.69 1.69 1.66 10 High [beta] 2.11 2.06 2.05 2.04 High to low 1.01 *** 0.95 *** 0.93 *** 0.92 ** (2.83) (2.69) (2.60) (2.55) Alpha 0.71 *** 0.64 *** 0.62 *** 0.60 *** (3.06) (2.82) (2.69) (2.58) [[beta].sup. [[beta].sup. [[beta].sup. [[beta].sup. Decile GARCH.sub. GARCH.sub. GARCH.sub. GARCH.sub. t4\t5] t5\t6] t6\t7] t7\t8] 1 Low P 1.12 1.16 1.16 1.17 2 1.25 1.20 1.20 1.20 3 1.22 1.23 1.25 1.27 4 1.31 1.30 1.31 1.29 5 1.37 1.42 1.33 1.37 6 1.44 1.41 1.48 1.44 7 1.44 1.43 1.42 1.44 8 1.56 1.58 1.57 1.53 9 1.67 1.64 1.64 1.65 10 High [beta] 2.04 2.03 2.00 1.98 High to low 0.92 ** 0.87 ** 0.84 ** 0.81 ** (2.54) (2.42) (2.33) (2.23) Alpha 0.61 *** 0.56 ** 0.53 ** 0.49 ** (2.63) (2.4l) (2.28) (2.10) [[beta].sup. [[beta].sup. [[beta].sup. [[beta].sup. Decile GARCH.sub. GARCH.sub. GARCH.sub. GARCH.sub. t8\t9] t9\t10] t10\t11] t11\t12] 1 Low P 1.19 1.21 1.21 1.22 2 1.21 1.23 1.24 1.25 3 1.25 1.24 1.27 1.27 4 1.30 1.28 1.33 1.32 5 1.37 1.40 1.36 1.39 6 1.43 1.43 1.44 1.44 7 1.42 1.42 1.40 1.38 8 1.53 1.55 1.56 1.54 9 1.67 1.64 1.66 1.67 10 High [beta] 1.96 1.98 1.97 1.97 High to low 0.77 ** 0.77 ** 0.76 ** 0.75 ** (2.l3) (2.l6) (2.l2) (2.09) Alpha 0.46 ** 0.44 * 0.43 * 0.43 * (l.96) (l.93) (l.88) (l.88) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table IX. Average Returns on Beta Portfolios after Controlling for Liquidity and Momentum This table presents the equalweighted average returns and average return differences on beta portfolios after controlling for liquidity and momentum. We first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked according to their illiquidity, dollar trading volume, past average 6month (MOM6), and past average 12month (MOM12) returns. Then, within each illiquidity, volume, MOM6, and MOM12 decile, we sort stocks into decile portfolios ranked based on GARCHinmean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The average returns reported below are the averages across the 10 illiquidity, volume, MOM6, and MOM12 deciles to produce decile portfolios with dispersion in market beta and with nearidentical levels of illiquidity, volume, MOM6, and MOM12. NeweyWest (1987) adjusted tstatistics appear in parentheses. Decile Illiquidity Volume MOM6 MOM12 1 Low [beta] 0.99 *** 0.92 *** 0.86 *** 1.05 *** (5.23) (4.92) (3.89) (4.82) 2 1.13 *** 1.05 *** 0.94 *** 1.18 *** (5.30) (5.13) (3.99) (5.10) 3 1.18 *** 1.09 *** 1.02 *** 1.22 *** (5.06) (4.72) (4.19) (5.17) 4 1.19 *** 1.19 *** 1.16 *** 1.34 *** (4.63) (4.70) (4.53) (5.34) 5 1.29 *** 1.32 *** 1.30 *** 1.53 *** (4.74) (4.82) (5.02) (6.01) 6 1.38 *** 1.33 *** 1.36 *** 1.55 *** (4.61) (4.51) (5.00) (5.71) 7 1.37 *** 1.45 *** 1.39 *** 1.52 *** (4.44) (4.51) (4.82) (5.43) 8 1.57 *** 1.58 *** 1.45 *** 1.60 *** (4.44) (4.43) (4.61) (5.21) 9 1.75 *** 1.83 *** 1.57 *** 1.67 *** (4.36) (4.51) (4.47) (4.83) 10 High [beta] 2.20 *** 2.38 *** 1.85 *** 1.91 *** (4.33) (4.58) (4.11) (4.34) High to low 1.20 *** 1.46 *** 0.99 *** 0.86 *** (3.05) (3.68) (2.89) (2.74) *** Significant at the 0.01 level. Table X. FirmLevel CrossSectional Regressions with Size, BM, Liquidity, and Momentum This table presents the firmlevel crosssectional regression results for the NYSE/Amex/Nasdaq stocks for the sample period of July 1963 to December 2004. We estimate the monthly conditional beta based on the GARCHinmean specification. Here, [ILLIQ.sub.t1] is the last month's illiquidity measure of each stock, [VOL.sub.t1] is the last month's dollar trading volume, [MOM6.sub.t1] is the past 6month average return, [MOM12.sub.t1] is the past 12month average return, log[ME.sub.t1] is the last month's log market capitalization (size), and log([BE.sub.t1]/[ME.sub.t1]) is the last fiscal year's log booktomarket ratio. The timeseries average slope coefficients are reported in each row. NeweyWest (1987) adjusted tstatistics appear in parentheses. The [R.sup.2] column presents the average [R.sup.2] values. [[beta].sup.GARCH.sub.tt1] [ILLIQ.sub.t1] [VOL.sub.t1] 0.5092 *** 6.4938 *** (2.45) (3.41) 0.5183 ***  (2.46) 6.3239 *** (2.57) 0.4123 *** (2.06) 0.4058 *** (2.12) 0.6453 *** 3.1506 ** (3.00) (2.20) 0.6609 *** 3.2395 ** (3.l9) (2.22) 0.6235 *** (0.00) (2.92) (0.01) 0.6361 *** 0.11 (3.09) (0.08) [[beta].sup.GARCH.sub.tt1] [MOM6.sub.t1] [MOM12.sub.t1] 0.5092 *** (2.45) 0.5183 *** (2.46) 0.4123 *** 3.6474** (2.06) (2.33) 0.4058 *** 1.31 (2.12) (0.61) 0.6453 *** 4.6308 *** (3.00) (3.25) 0.6609 *** (0.09) (3.l9) (0.05) 0.6235 *** 4.7820 *** (2.92) (3.41) 0.6361 *** (0.31) (3.09) (0.16) [[beta].sup.GARCH.sub.tt1] log[ME.sub.t1] log([BE.sub.t1]/ [ME.sub.t1]) 0.5092 *** (2.45) 0.5183 *** (2.46) 0.4123 *** (2.06) 0.4058 *** (2.12) 0.6453 *** 0.1480 ** 0.3141 *** (3.00) (2.51) (4.46) 0.6609 *** 0.1659 *** 0.3065 *** (3.l9) (2.84) (4.55) 0.6235 *** 0.1893 *** 0.3215 *** (2.92) (2.98) (4.58) 0.6361 *** 0.2087 *** 0.3136 *** (3.09) (3.31) (4.67) [[beta].sup.GARCH.sub.tt1] [R.sup.2] 0.5092 *** 2.87% (2.45) 0.5183 *** 2.64% (2.46) 0.4123 *** 3.45% (2.06) 0.4058 *** 3.53% (2.12) 0.6453 *** 6.27% (3.00) 0.6609 *** 6.33% (3.l9) 0.6235 *** 6.14% (2.92) 0.6361 *** 6.20% (3.09) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table XI. EqualWeighted Portfolios of NYSE Stocks Sorted by Conditional Beta We form equalweighted decile portfolios for every month from July 1963 to December 2004 by sorting the NYSE stocks based on the conditional GARCHinmean beta. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or conditional betas. The row "HighLow" refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the FamaFrench (1993) model. NeweyWest (1987) adjusted tstatistics appear in parentheses.Decile Average Return Average Return Average Return (Univariate Sort) (After (After Controlling Controlling for Size) for BM) 1 Low [beta] 1.11 1.06 1.11 2 1.21 1.20 1.15 3 1.34 1.24 1.22 4 1.36 1.32 1.25 5 1.35 1.37 1.33 6 1.35 1.37 1.37 7 1.45 1.45 1.30 8 1.51 1.56 1.46 9 1.54 1.65 1.58 10 High [beta] 1.97 1.90 1.89 High to low 0.86 *** 0.84 *** 0.78 *** (2.79) (2.81) (2.94) Alpha 0.37 ** 0.43 *** 0.34 ** (2.44) (2.87) (2.23) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table XII. EqualWeighted Portfolios Using NYSE Breakpoints and Controlling for Microstructure Effects We form equalweighted decile portfolios for every month from July 1963 to December 2004 by sorting the NYSE/Amex/Nasdaq stocks based on the realized and conditional beta. We compute the realized beta by using daily returns over the previous month with the lagged market return. We generate portfolios based on the NYSE breakpoints and skipping the month following portfolio formation. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or conditional betas. The row "HighLow" refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the FamaFrench (1993) model. NeweyWest (1987) adjusted tstatistics appear in parentheses. Decile [[beta].sup.realized. [[beta].sup.AR. sub.t2] sub.t1t2] Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.31 1.56 1.17 0.21 2 1.31 0.17 1.30 0.52 3 1.36 0.18 1.40 0.65 4 1.36 0.45 1.39 0.76 5 1.38 0.70 1.44 0.86 6 1.40 0.96 1.45 0.95 7 1.38 1.24 1.58 1.04 8 1.35 1.59 1.57 1.14 9 1.31 2.09 1.65 1.29 10 High [beta] 1.23 3.82 1.87 1.73 High to low (0.08) 0.70 ** (0.51) (2.24) Alpha 0.09 0.48 ** (0.82) (2.32) Decile [[beta].sup.MA.sub. [[beta].sup.GARCH.sub. t1t2] t1t2] Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.15 0.22 1.14 0.19 2 1.32 0.53 1.24 0.50 3 1.38 0.66 1.33 0.64 4 1.41 0.77 1.43 0.75 5 1.43 0.86 1.36 0.85 6 1.48 0.95 1.43 0.94 7 1.53 1.04 1.49 1.03 8 1.59 1.14 1.57 1.14 9 1.67 1.28 1.68 1.30 10 High [beta] 1.87 1.69 2.06 1.80 High to low 0.72 ** 0.92 *** (2.29) (2.76) Alpha 0.49 ** 0.63 *** (2.38) (3.07) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table XIII. EqualWeighted Beta Portfolios after Controlling for Size and BooktoMarket Simultaneously This table presents average returns for each beta quintile, the average return differences between high and lowbeta portfolios, and the FF3 alpha differences between high and lowbeta portfolios. We report the results for realized and conditional betas after controlling for size and booktomarket. At the beginning of month t, we first sort the NYSE/Amex/Nasdaq stocks into five size (market equity) portfolios. Then within each size portfolio, we sort the stocks into five BM (booktomarket equity ratio) portfolios. Finally, within each portfolio formed from the intersections of five size and five BM portfolios, we sort the stocks into five beta portfolios, based on their realized and conditional betas in month t1. NeweyWest (1987) adjusted tstatistics appear in parentheses. [[beta].sup. [[beta].sup.AR. Quintile real?zed.sub.t1] sub.tt1] Low [beta] 1.3016 0.9565 (4.79) (4.70) 2 1.3720 *** 1.1579 *** (5.44) (4.76) 3.00 1.3938 *** 1.3315 *** (5.23) (4.87) 4.00 1.4153 *** 1.5181 *** (4.83) (4.91) High [beta] 1.3066 *** 1.9297 *** (3.69) (4.81) Return dif. 0.0050 *** 0.9731 *** High [beta] to low [beta] (0.04) (3.75) Alpha diff. 0.1138 *** 0.6903 *** High [beta] to low [beta] (1.02) (3.92) [[beta].sup.MA. [[beta].sup.GARCH. Quintile sub.tt1] sub.tt1] Low [beta] 0.9380 0.9351 (4.59) (4.69) 2 1.1623 *** 1.1771 *** (4.79) (4.88) 3.00 1.3190 *** 1.2629 *** (4.80) (4.65) 4.00 1.5217 *** 1.5126 *** (4.95) (4.89) High [beta] 1.9492 *** 1.9944 *** (4.83) (4.84) Return dif. 1.0112 *** 1.0593 *** High [beta] to low [beta] (3.84) (3.83) Alpha diff. 0.7272 *** 0.7552 *** High [beta] to low [beta] (4.07) (4.08) *** Significant at the 0.01 level. Table XIV. FamaMacBeth (1973) CrossSectional Regressions Using 125 Size/BM/Beta Portfolios This table presents the crosssectional regression results from the 125 size/BM/beta portfolios for the sample period of July 1963 to December 2004. We estimate the monthly conditional betas based on the AR(1), MA(1), and GARCHinmean specification of the realized beta measures. We calculate the realized beta of each portfolio by using daily data over the previous month with the lagged market return. Here, [logME.sub.t1] is the last month's log market capitalization (size) of each portfolio, and [log(BE.sub.t1]/[ME.sub.t1]) is the last fiscal year's log booktomarket ratio of each portfolio. We report the timeseries average slope coefficients in each row. NeweyWest (1987) adjusted tstatistics appear in parentheses. The [R.sup.2] column presents the average [R.sup.2] values. [[beta].sup. [[beta].sup. [[beta].sup. realized.sub. AR.sub.tt1] MA.sub.tt1] t1] 0.0759 (1.25) 0.3507 * (l.82) 0.3876 ** (l.96) 0.0183 (0.28) 0.8091 *** (2.83) 0.8580 *** (2.94) 0.0389 (0.69) 0.5709 *** (2.97) 0.610l *** (3.10) 0.0074 (0.12) 0.8472 *** (3.04) 0.8942 *** (3.l5) [[beta].sup. (log(Be.sub.t1]/ GARCH.sub.tt1] [logMe.sub.t1] [ME.sub.t1]) 0.4359 ** (2.l3) 0.1864 *** (3.43) 0.2673 *** (3.97) 0.2698 *** (3.99) 0.8093 *** 0.2583 *** (2.78) (3.88) 0.7236 *** (6.64) 0.7860 *** (7.24) 0.7891 *** (7.25) 0.6356 *** 0.7856 *** (3.12) (7.22) 0.1222 ** 0.5424 *** (2.17) (4.9l) 0.2081 *** 0.4918 *** (2.99) (4.82) 0.2106 *** 0.4914 *** (3.01) (4.8l) 0.8451 *** 0.1973 *** 0.5045 *** (2.98) (2.88) (5.l1) [R.sup.2] 7.98% 13.88% 13.72% 14.41% 23.05% 30.33% 30.26% 30.60% 15.51% 20.38% 20.28% 20.91% 28.98% 35.27% 35.23% 35.36% *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table XV. Pooled Panel Regressions Using 125 Size/BM/Beta Portfolios This table presents the pooled panel regression results from the 125 size/BM/beta portfolios for the sample period of July 1963 to December 2004. We estimate the monthly conditional betas based on the AR(1), MA(1), and GARCHinmean specification of the realized beta measures. We calculate the realized beta of each portfolio by using daily data over the previous month with the lagged market return. Here, [logME.sub.t1] is the last month's log market capitalization (size) of each portfolio, and [log(BE.sub.t1] /[ME.sub.t1]) is the last fiscal year's log booktomarket ratio of each portfolio. We report the slope coefficients in each row. We adjust the tstatistics given in parentheses for heteroskedasticity, firstorder autocorrelation, and contemporaneous crosscorrelation in the error term. [[beta].sup. realized.sub. [[beta].sup. [[beta].sup. t1] AR.sub.tt1] MA.sub.tt1] 0.0505 ** (2.44) 0.3067 *** (3.8l) 0.3250 *** (4.03) (0.02) (0.95) 0.3061 *** (6.65) 0.6251 *** (6.68) (0.03) (1.30) 0.4372 *** (5.49) 0.4615 *** (5.79) (0.01) (0.56) 0.6194 *** (6.83) 0.6435 *** (6.89) [[beta].sup. (log(Be.sub.t1]/ GARCH.sub.tt1] [logMe.sub.t1] [ME.sub.t1]) 0.2906 *** (3.74) 0.2003 *** (8.33) 0.1960 *** (8.01) 0.1887 *** (7.60) 0.5147 *** 0.1781 *** (5.92) (7.13) 0.5279 *** (11.77) 0.5546 *** (12.16) 0.5310 *** (11.46) 0.4103 *** 0.5378 *** (5.32) (12.05) 0.1368 *** 0.4582 *** (5.66) (10.15) 0.1283 *** 0.4704 *** (5.24) (10.41) 0.1227 *** 0.4436 *** (4.92) (9.60) 0.5360 *** 0.1098 *** 0.4578 *** (6.l8) (4.36) (l0.31) *** Significant at the 0.01 level. ** Significant at the 0.05 level. 

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