Encuesta aplicada a los estudiantes de la Escuela Fiscal Mixta “24 de

We now turn to our estimation results. Table 3.3 shows the parameter esti- mates for the Cox model and the log-logistic model.29 _{The results refer to}

lifetimes which have been (additionally) censored at 60 months as described at the end of section 3.2.2. All coefficients have the expected sign and are highly significant. Higher profitability (NITA), higher stock market returns (RET) and larger firm sizes (SIZE) are associated with lower default risk. On the opposite side, higher debt levels (TLTA), extreme growth (GRO) and more volatile returns (VOLA) correspond to higher hazard rates. As we estimate the Cox model with the partial likelihood approach, we do not estimate parameters for the baseline hazard (and thus no intercept) here. The results from the Cox model and the log-logistic model turn out to be quite similar. The goodness-of-fit can not be directly compared due to the different estimation procedures but if we compare the log likelihood values of the log-logistic and the Weibull model (-68249.16 vs. -69558.03) we find that the PO approach has a better fit than the PH approach.

Since the absolute parameter values are hard to interpret in nonlinear mod- els we calculated the marginal effects of the covariates in Table 3.4. The marginal effects are evaluated at the means of the covariates and refer to the ceteris paribus effect of a one-unit increase of a covariate on the 5-year default probability. For instance, in the log-logistic model increasing prof- itability (NITA) by 1% is estimated to lower the 5-year default probability by 0.2782% and a 10% increase in leverage (TLTA) is estimated to raise the 5-year default probability by 0.945%. The results for the log-logistic model and the Cox model are relatively close to each other while the marginal effects are somewhat lower for the Cox model.

While the better goodness-of-fit of the log-logistic model gives us first evi- dence on its appropriateness we will now do some further analyses. Figure 3.2 shows on the left hand side the course of the median firm’s hazard rate

29_{Using the Weibull instead of the Cox model makes almost no difference. The coeffi-} cients are very close and Spearman’s rank correlation for the predictions from the Weibull and the Cox model is as high as 0.99999883.

Table 3.3: Results from hazard regressions

Cox model (PH) Log-logistic model (PO)

Coef. Std. error Coef. Std. error

NITA -5.598 (1.358) -6.804 (1.271) TLTA 2.426 (0.296) 2.311 (0.254) GRO 0.212 (0.054) 0.184 (0.053) RET -0.826 (0.056) -0.813 (0.053) VOLA 6.142 (0.526) 6.062 (0.461) SIZE -0.374 (0.031) -0.336 (0.027) const. -11.992 (0.278) α 1.255 (0.023) firm-months 339 222 339 222 logL -214084.69 -68249.16 Wald χ2 2351.88 2415.62

Table 3.4: Marginal effects on 5-year default probability

Log-logistic model (PO) Cox model (PH)

NITA -0.2782 -0.1971 TLTA 0.0945 0.0854 GRO 0.0075 0.0075 RET -0.0332 -0.0291 VOLA 0.2478 0.2161 SIZE -0.0137 -0.0132

over the forecast time according to the log-logistic model.30 _{The hazard rate}

is monotonically increasing in the first 60 months which makes sense since the median firm is not supposed to be close to default in the beginning.31

The right-hand side of Figure 3.2 provides some support to our conjecture

30_{The median firm refers to the median of the predictions from the log-logistic model,} i.e. it refers to the observation where 50% of all firm-months are estimated to be less risky. 31_{Note that the log-logistic model implies, given that the shape parameterα is larger} than 1, that the hazard rate rises until (α−1)1/α

exp(β′xit) and decreases thereafter (Kalbfleisch & Prentice, 2002, Ch. 2.2.6).

Figure 3.2: Hazard rate curves 0 10 20 30 40 50 60 0.0000 0.0002 0.0004 0.0006 0 10 20 30 40 50 60 0.0000 0.0002 0.0004 0.0006

The plot shows the continuous-time hazard rate of the median firm according to the log-logistic model (left hand side) and the nonparametrically estimated discrete-time hazard rate of a BBB firm (right hand side) plotted against forecast time (in months). Continuous-time and discrete-time hazard rates are comparable if hazard rates are small and periods are short sinceλd_{(s) = 1−exp−}Rs

s−1λ

c_{(u)du= 1−exp(−λ}c_(s))≈λc_(s).

that the hazard curve of the log-logistic model is a realistic one. We see that the hazard curve of a BBB rated firm,32 _{estimated completely nonparametri-}

cally with the life-table estimator (see section 4.1), exhibits a similar pattern and does not seem to have any important characteristics that are smoothed away by the parametric structure of the log-logistic model.

Our primary motivation to estimate the log-logistic model was its property of declining hazard ratios. To study this aspect we plotted in Figure 3.3 the evolution of the hazard ratios of selected pairs of firms over the forecast time. The left hand side shows the hazard ratios for the upper and lower quartile firm while the right hand side refers to the upper and lower decile firm.33

The decline of the hazard ratios is evident but happens at a moderate pace in our model. This does not surprise as we do not expect that the hazard rates of high-risk and low-risk firms approach each other very quickly. By comparing both curves we further see that more extreme hazard ratios decline more quickly. Note that in the Cox model, the hazard ratios for the same

32_{In our sample, the median S&P rating is BBB.}

Figure 3.3: Evolution of hazard ratios 0 10 20 30 40 50 60 5.0 6.0 7.0 8.0 0 10 20 30 40 50 60 35 40 45 50

Hazard ratios of the upper and lower quartile firm (left-hand side) and upper and lower decile firm (right-hand side) derived from the log-logistic model. The abscissa represents forecast time (s) in months.

quantiles are constant at 5.30 and 24.28, respectively.