4 Estimation Methodology
As discussed in Section 2, the bankruptcy cost parameter μ determines the magnitude
of financial market frictions in the BGG framework. Recognizing that this parameter
may exhibit substantial temporal variation, we use the cross-section of firm-level
observations for each time period t, and apply non-linear least squares (NLLS) to
obtain the value of μt that minimizes the sum of squared deviations between observed
credit spreads and those predicted by the model. In implementing this procedure,
we allow the idiosyncratic shock variance, σit , to vary across firms as well as time
periods.
For any given value of the bankruptcy cost parameter, the predicted credit spread
of a given firm i at time period t is computed as follows. First, using the observed
leverage ratio and expected default probability, the following two equations can be
solved to obtain values for the default threshhold ω,tt and the idiosyncratic shock
variance σit :
|
I---------------------------------------------------------1 ≥d to I_________________________________________________________I |
_ ψ,(ωit; σit)ξ(ωit; σa,μt). i,t-1 Ψ (ωit ; σit ) ξ (ωit ; σtt,, μt ) ’ |
(9) |
|
EDFi |
flnωtt - 0.5σit∖ ,t-1 ∣t = φ ------------------- , σit |
(10) |
where [B/N]i,t-1 denotes the firm’s leverage at the end of the previous quarter, and
EDFi,t-1|t denotes the probability (as of the end of quarter t - 1) that firm i will
default in quarter t. Under the assumption of log-normality of the idiosyncratic
shocks, the functions ψ and ξ are given by equations 7 and 8, respectively, while Φ
denotes the standard normal cumulative distribution function.19
Next, the predicted credit spread [Rbit/Rit]* is obtained by evaluating the follow-
ing expression using these values of ω*t and σ*t together with the observed leverage
r Rb 1 ∙ = ω*tψ' (ω*t ; σ*t ) (1+ [ N ] it-1)
(11)
_ r J it ψ(ω*t; σ*)ξ(ω*t; σit, μt) - ψ(ω*t; σ*t)ξ(ω*t; σ*t, μt)
19 To match the one-period nature of the BGG framework, we convert the MKMV year-ahead
expected default frequency (EDFit|t+1...t+4) to a quarterly basis using the simplifying assumption of
a constant hazard rate over each four-quarter horizon; that is, EDFit|t+1 = (1+EDFit|t+1...t+4)1/4-1.
21