The presence of traders with superior information leads to a positive bid-ask spread even when the specialist is risk-neutral and makes zero expected profits. The resulting transaction prices convey information, and the expectation of the average spread squared times volume is bounded by a number that is independent of insider activity. The serial correlation of transaction price differences is a function of the proportion of the spread due to adverse selection. A bid-ask spread implies a divergence between observed returns and realizable returns. Observed returns are approximately realizable returns plus what the uninformed anticipate losing to the insiders.

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We find support for a negative relation between conditional expected monthly return and conditional variance of monthly return, using a GARCH‐M model modified by allowing (1) seasonal patterns in volatility, (2) positive and negative innovations to returns having different impacts on conditional volatility, and (3) nominal interest rates to predict conditional variance. Using the modified GARCH‐M model, we also show that monthly conditional volatility may not be as persistent as was thought. Positive unanticipated returns appear to result in a downward revision of the conditional volatility whereas negative unanticipated returns result in an upward revision of conditional volatility.

We find support for a negative relation between conditional expected monthly return and conditional variance of monthly return, using a GARCH-M model modified by allowing (1) seasonal patterns in volatility, (2) positive and negative innovations to returns having different impacts on conditional volatility, and (3) nominal interest rates to predict conditional variance. Using the modified GARCH-M model, we also show that monthly conditional volatility may not be as persistent as was thought. Positive unanticipated returns appear to result in a downward revision of the conditional volatility whereas negative unanticipated returns result in an upward revision of conditional volatility. There is general agreement that investors, within a given time period, require a larger expected return from a security that is riskier. However, there is no such agreement about the relation between risk and return across time. Whether or not investors require a larger risk premium on average for investing in a security during times when the security is more risky remains an open question. At first blush, it may appear that rational risk-averse Most of the support for a zero or positive relation has come from studies that use the standard GARCH-M model of stochastic volatility.2 Other studies, using alternative techniques, have documented a negative relation between expected return and conditional variance. In order to resolve this conflict we examine the possibility that the standard GARCH-M model may not be rich enough to capture the time series properties of the monthly excess return on stocks. We consider a more general specification of the GARCH-M model. In particular, (1) we incorporate dummy variables in the GARCH-M model to capture seasonal effects using the procedure first suggested by Glosten, Jagannathan, and Runkle ( THE TRADEOFF BETWEEN RISK and return has long been an important topic in II. Estimating the Model A. Econometric IssuesThe parameter p in the model given by (2) cannot be estimated without specifying how variances change over time, since Var(xt + 1? Ft) is not directly observed by the econometrician. To appreciate the difficulties involved, project both sides of (2) on Gt, the econometrician's information set, which is a strict subset of the agents' information set Ft. The term on the left in equation (4) Since the estimated slope coefficient cl is a consistent estimate of /3 bl, and d1 provides a consistent estimate of b1, the ratio of any two corresponding elements of cl and d1 provides a consistent estimate of /8. If zt-, is not a scalar, then we may impose the constraint that the slope coefficients in (5) and the slope coefficients in (6) differ only by the scale factor, /3. Such a restriction also provides a natural test for the validity of the model specification. We call this approach Campbell's Instrumental Variable Model. Another approach, the GARCH-M model, assumes that Var(vtllGt_-) is identically zero, and that zt-1 consists of innovations and variances that, while unobservable, can be...

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