where R,, is the continuously compounded rate of return for the common
stock of firm j on day t, and R,, is the continuously compounded rate of
return for the CRSP equally weighted index on day t. The coefficients a, and
pj are ordinary least squares estimates of firm j’s market model parameters.
The estimation period is 140 trading days, beginning 21 trading days after the
issuance or cancellation.
The estimation period follows the issuance or cancellation because many
types of security offerings follow a period of statistically significant abnormal
returns. Market efficiency implies that abnormal returns following the issuance
should not differ systematically from zero. We avoid a bias in estimation of
market model parameters due to stock returns that are systematically non-zero
in the estimation period. However, parameter estimates derived from a period
that follows the offering reflect any shift in the parameters due to a change in
the firm’s financial leverage. Thus, our estimation period may induce a bias in
the measurement of prediction errors prior to the issuance.
Prediction errors are calculated for each day in the event period that begins
60 trading days before the announcement and ends 20 trading days after the
issuance or cancellation. The length of the event period differs among the
security offerings due to the varying number of days between the announcement and the issuance or cancellation.
The average prediction error on event day t for a sample of size N is