财务危机预警文献,仅供学习研究
where yi is a dummy variable, which is set to 1 only in the year in which a bankruptcy filing occurred. Shumway (2001) indicates that multi-period Logit model is estimated with the data from each firm year as if it were a separate observation. The likelihood function of the multi-period Logit model can be written as: L=Π(F(ti,xi;θ)yiΠ[1 F(j,xi;θ)]) (6)
i=1
j<ti
n
The cumulative density function,F(t,x;θ), has a value between 0 and 1. F(t,x;θ) can also be written as hazard function, φ(t,x;θ). Replacing F(t,x;θ) with the hazard function,φ(t,x;θ) in equation (6), the likelihood function is written as L=Π(φ(ti,xi;θ)yiΠ[1 φ(j,xi;θ)]) (7)
i=1
j<ti
n
According to Cox and Oakes (1984), survival function of discrete-time hazard model satisfies
S(t,x;θ)=Π[1 φ(j,xi;θ)] (8)
j<ti
Substituting equation (8) into equation (5) verifies that the likelihood function of a multi-period logit model is equivalent to that of a discrete-time hazard model. Different from single-period logit model, the multi-period logit model (discrete-time hazard model) incorporates time-varying covariates by making x depend on time, and therefore provides more consistent and unbiased parameters estimation (Shumway 2001).
We define hazard function as logit function, defined as:
