4. The Role of Accounting, Market and Macroeconomic Variables for the
4.5. Methods: Panel Logit Model Specification
The sample is divided into two groups, failed firms and healthy or non-failed firms.
The outcome is a binary dependent variable. Our approach is to model the outcome within a panel logit framework, (Altman et al. 2010 and Altman and Sabato 2007), and follow Shumway (2001) and Nan et al. (2008) who show that a panel logit model, that corrects for period at risk and allows for time varying covariates95, is equivalent to a hazard model.
Other influential studies that have used the logit methodology for the development of default prediction models are Keasey and Watson (1987), Peel and Peel (1987), Storey et al.
(1987). Among the studies concerned with large firms using the logit methodology we can cite: Martin (1977), Ohlson (1980), Mensah (1984), Gentry et al. (1985), Zavgren (1985, 1988), Platt and Platt (1991), Charitou and Trigeorgis (2000), Becchetti and Sierra (2003), Altman et al. (2010).
The logistic regression model used in this study is based on the following mathematical definition. Let be a random sample from a conditional logit distribution. Next, let be a collection of independent variables denoted by the vector . Assuming that each of these variables is at least interval scaled and that the conditional probability of the outcome is present is denoted by | then the logit of the logistic regression model is denoted by:
and
then
[ | ] [ | ]
therefore
[ | ]
∑
95 Shumway (2010), p. 123.
In addition to the estimates computed through this statistical methodology, marginal effects for each of the variables are also presented. This calculation, despite its usefulness in the interpretation of individual variables on the performance of the model, has been overlooked in previous default/distress prediction models. The present study fills this gap in the literature by computing marginal effects for all variables in the models. The calculations are intended to measure the expected instantaneous change in the response variable as a function of a change in a specific predictor variable while keeping all other covariates constant. The marginal effect of a predictor is defined by the SAS Institute as the partial derivative of the event probability with respect to the predictor of interest96. The marginal effects measurement is therefore very useful in order to interpret the effects of the regressors on the dependent variable for discrete dependent variable models, in this case, a logit binary choice model. Marginal effects are therefore mathematically expressed as follows:
For simplicity, consider now the same model but with only one regressor. It is called logit because:
[ | ] )
where is the explanatory variable and and are unknown parametersto be estimated, and
is the distribution function for the logistic (logit) distribution.
If then [ | ] ) is an increasing function of : [ | ]
where is the derivative of ;
96 Usage Note 22604: Marginal effects estimation for predictors in logistic and probit models.
http://support.sas.com/kb/22/604.html
Thus, the marginal effect of on [ | ] depends on :
[ | ]
( )
In practice, there are two frequently used approaches to estimate either average or overall marginal effects. According to the SAS Institute, one of them is to calculate the marginal effect at the sample means of the data and the other is to estimate marginal effect at each observation and then to compute the sample average of individual marginal effects to obtain the overall marginal effect97. For large samples, both approaches yield similar results, however, for the purposes of this analysis; the average of the individual marginal effects is preferred. The present study outputs the marginal effects estimated for each observation in the dataset and then computes the sample average of individual marginal effects in order to obtain the overall marginal effects. SAS software code was employed to get the estimated marginal effects.
97 See SAS/ETS Web Examples. Computing Marginal Effects for Discrete Dependent Variable Models.
Table 4-3 Summary Statistics for Model 1
This table presents summary statistics for Model 1, which includes only financial statement variables. It covers the Mean, Standard Deviation, Minimum and Maximum Values and the number of observations that were used in the logistic regression for the ratios Total Funds from Operation to Total Liabilities (TFOTL), Total Liabilities to Total Assets (TLTA), the No Credit Interval (NOCREDINT), and Interest Coverage (COVERAGE). Panel A contains summary statistics for the entire dataset; Panel B for financially healthy firms, and Panel C for failed firms.
Variable TFOTL TLTA NOCREDINT COVERAGE
Panel A: Entire Data Set
Mean 0.069322 0.485737 -0.120307 0.532898
Std. Dev. 0.337566 0.188089 0.986233 0.818508
Min -1 -0.432123 -1 -1
Max 1 1 1 1
Observations 18,158
Panel B: Financially Healthy Firms
Mean 0.073826 0.48373 -0.112936 0.545882
Std. Dev. 0.335902 0.187087 0.987091 0.810734
Min -1 -0.432123 -1 -1
Max 1 1 1 1
Observations 17,843
Panel C: Failed Firms
Mean -0.185767 0.599386 -0.537879 -0.202545
Std. Dev. 0.33396 0.208933 0.837612 0.916257
Min -1 0.005761 -1 -1
Max 0.796339 1 1 1
Observations 315
Table 4-4 Summary Statistics for Model 2.
This table presents summary statistics for Model 2, which includes financial statement ratios as well as macroeconomic variables. It covers the Mean, Standard Deviation, Minimum and Maximum Values and the number of observations that were used in the logistic regression for the ratios Total Funds from Operation to Total Liabilities (TFOTL), Total Liabilities to Total Assets (TLTA), the No Credit Interval (NOCREDINT), Interest Coverage (COVERAGE) the Retail Price Index (RPI), and the proxy for interest rates, the 3-month Short Term Bill Rate adjusted for inflation (SHTBRDEF). Panel A contains summary statistics for the entire dataset; Panel B for financially healthy firms, and Panel C for failed firms.
Variable TFOTL TLTA NOCREDINT COVERAGE RPI SHTBRDEF
Panel A: Entire Data Set
Mean 0.068245 0.485506 -0.116483 0.528139 178.27559 2.061429 Std. Dev. 0.339132 0.188782 0.986674 0.821606 32.152036 2.412037
Min -1 -0.432123 -1 -1 94.59 -4.69551
Max 1 1 1 1 235.18 7.7407
Observations 17,952
Panel B: Financially Healthy Firms
Mean 0.072782 0.483472 -0.108957 0.541189 178.15488 2.060159 Std. Dev. 0.337499 0.187782 0.987519 0.813904 32.242693 2.419031
Min -1 -0.432123 -1 -1 94.59 -4.69551
Max 1 1 1 1 235.18 7.7407
Observations 17,637
Panel C: Failed Firms
Mean -0.185767 0.599386 -0.537879 -0.202545 185.03432 2.132532 Std. Dev. 0.33396 0.208933 0.837612 0.916257 25.739411 1.983302
Min -1 0.005761 -1 -1 115.21 -4.69551
Max 0.796339 1 1 1 235.18 7.1745
Observations 315
Chapter 4: Financial Distress and Bankruptcy Prediction using Accounting, Market and Macroeconomic Variables83
Table 4-5 Summary Statistics for Model 3
This table presents summary statistics for the full model, or Model 3, which includes financial statement ratios, macroeconomic indicators and market variables. It covers the Mean, Standard Deviation, Minimum and Maximum Values and the number of observations that were used in the logistic regression for the ratios Total Funds from Operation to Total Liabilities (TFOTL), Total Liabilities to Total Assets (TLTA), the No Credit Interval (NOCREDINT), Interest Coverage (COVERAGE) the Retail Price Index (RPI), and a proxy for interest rates, the 3-month Short Term Bill Rate adjusted for inflation (SHTBRDEF), the firm’s Equity Price (PRICE), the firm’s annual Abnormal Returns (ABNRET ), the firm’s Relative Size (SIZE), and the ratio Market Capital to Total Debt (MCTD). Panel A contains summary statistics for the entire dataset; Panel B for financially healthy firms, and Panel C for failed firms.
Variable TFOTL TLTA NOCREDINT COVERAGE RPI SHTBRDEF PRICE ABNRET IDYRISK SIZE MCTD
Panel A: Entire Data Set
Mean 0.089856 0.495185 -0.18222 0.582094 178.41205 2.057968 4.408757 -0.098878 0.122309 -10.083135 0.912645 Std. Dev. 0.28928 0.171132 0.976069 0.783736 32.300452 2.474547 1.719708 0.401444 0.080925 2.212375 0.18944
Min -1 -0.162029 -1 -1 97.82 -4.69551 -4.60517 -0.999987 0.012299 -16.602146 0.002877
Max 1 1 1 1 235.18 7.7407 13.785052 0.997029 0.787179 -2.374161 1
Observations 14,203
Panel B: Financially Healthy Firms
Mean 0.094928 0.492868 -0.173282 0.597459 178.26942 2.057772 4.445545 -0.092124 0.120757 -10.044873 0.91643 Std. Dev. 0.286516 0.169794 0.977685 0.772992 32.396525 2.483145 1.691239 0.397685 0.079442 2.206104 0.183715
Min -1 -0.162029 -1 -1 97.82 -4.69551 -3.912023 -0.999987 0.012299 -16.602146 0.002877
Max 1 1 1 1 235.18 7.7407 13.785052 0.996455 0.751478 -2.374161 1
Observations 13,929
Panel C: Failed Firms
Mean -0.167974 0.612982 -0.636597 -0.198991 185.6627 2.067917 2.538613 -0.442228 0.201189 -12.028228 0.720245 Std. Dev. 0.311704 0.196133 0.764131 0.919496 26.004402 1.992703 2.084033 0.440864 0.111096 1.566622 0.327313
Min -1 0.052458 -1 -1 115.21 -4.69551 -4.60517 -0.99966 0.015639 -15.922758 0.00588
Max 0.49607 1 1 1 235.18 7.1745 10.96388 0.997029 0.787179 -5.641377 1
Observations 274