354 B.E. Eckbo et al.
5.4.2. Time-varying factor loadings
Nonstationary factor loadings may produce (i) significant performance in subperiods,
(ii) predictable changes in factor loadings which affect the alpha estimates, and (iii) sig-
nificant effect of using value-weighted instead of equal-weighted issuer portfolios.
Nonstationarities. Eckbo, Masulis, and Norli (2000)examine holding periods of be-
tween one and five years. For example, with a two-year holding period, firms enter the
SEO issuer portfolio as before, but exit after only two years (or at a subsequent security
offer or delisting, whichever occurs earlier). This serves to check whether any subpe-
riod abnormal performance is washed out in the averaging of returns over the five-year
holding period. The conclusion emerging from the analysis of one-to-five-year holding
periods remain the same: none of the alphas are significantly different from zero.
Eckbo, Masulis, and Norli (2000)also reestimate alphas using factor-mimicking port-
folios that are continuously updated. That is, the portfolio weights are constructed using
a rolling estimation period where the factor loadings are reestimated every month.
This rolling estimation procedure relaxes the stationarity assumption on the factor-
mimicking weights. The alphas are again all insignificant.
Predictable changes in factor loadings. Eckbo, Masulis, and Norli (2000)andEckbo
and Norli (2005)reexamine the null hypothesis of zero abnormal performance using
a conditional factor model framework.^59 They followFerson and Schadt (1996)and
assume that factor loadings are linearly related to a set ofLknown information variables
Zt− 1 :
β 1 pt− 1 =bp 0 +Bp 1 Zt− 1. (13)Here,bp 0 is aK-vector of “average” factor loadings that are time-invariant,Bp 1 is
a(K×L)coefficient matrix, andZt− 1 is anL-vector of information variables (ob-
servables) at timet−1. The productBp 1 Zt− 1 captures the predictable time variation
in the factor loadings. After substituting equation(13)into equation(10), the return-
generating process becomes
rpt=bp′ 0 rFt+b′p 1 (Zt− 1 ⊗rFt)+ept, (14)where theKL-vectorbp 1 is vec(Bp 1 )and the symbol⊗denotes the Kronecker prod-
uct.^60 This factor model is estimated after adding a constant termαp, which equals zero
under the null hypothesis of zero abnormal returns. The information variables inZt− 1
include the lagged dividend yield on the CRSP value-weighted market index, the lagged
30-day Treasury bill rate, and the lagged values of the credit and yield curve spreads,
BAA–AAA and TBILLspr, respectively. The alpha estimates all remain insignificantly
different from zero.
(^59) A survey of conditional factor model econometrics is found inFerson (1995).
(^60) The operator vec(·)vectorizes the matrix argument by stacking each column starting with the first column
of the matrix.