Predicting Recurrent Financial Distresses with Autocorrelation
Structure: An Empirical Analysis from an Emerging Market
Ruey-Ching Hwang&Huimin Chung&Jiun-Yi Ku
Received: 4 September 2011 / Revised: 19 March 2012 / Accepted: 27 March 2012 / Published online: 5 May 2012
# Springer Science+Business Media, LLC 2012
Abstract The dynamic logit model (DLM) with autocorrelation structure (Liang and Zeger Biometrika 73:13–22, 1986) is proposed as a model for predicting recurrent financial distresses. This model has been applied in many examples to analyze repeated binary data due to its simplicity in computation and formulation. We illustrate the proposed model using three different panel datasets of Taiwan industrial firms. These datasets are based on the well-known predictors in Altman (J Financ 23:589–609,1968), Campbell et al. (J Financ 62:2899–2939,2008), and Shumway (J Bus 74:101–124,2001). To account for the corre-lations among the observations from the same firm, we consider two different autocorrela-tion structures: exchangeable and first-order autoregressive (AR1). The predicautocorrela-tion models including the DLM with independent structure, the DLM with exchangeable structure, and the DLM with AR1 structure are separately applied to each of these datasets. Using an expanding rolling window approach, the empirical results show that for each of the three datasets, the DLM with AR1 structure yields the most accurate firm-by-firm financial-distress probabilities in out-of-sample analysis among the three models. Thus, it is a useful alternative for studying credit losses in portfolios.
Keywords Autocorrelation structure . Dynamic logit model . Expanding rolling window approach . Predictive interval . Predicted number of financial distresses . Recurrent financial distresses
JEL Classification G20 . G33 . C33 DOI 10.1007/s10693-012-0136-0
R.-C. Hwang (*)
Department of Finance, National Dong Hwa University, No. 1, Sec. 2, Da Hsueh Road, Hualien, Taiwan 974, Republic of China
e-mail: rchwang@mail.ndhu.edu.tw
H. Chung
Graduate Institute of Finance, National Chiao Tung University, No. 1001 University Road, Hsinchu, Taiwan 300, Republic of China
J.-Y. Ku
Department of Applied Mathematics, National Dong Hwa University, No. 1, Sec. 2, Da Hsueh Road, Hualien, Taiwan 974, Republic of China
1 Introduction
The prediction of financial distress is an important tool for credit risk management. The well-known models for prediction can be separated into two types: static and dynamic. The static model uses only single-period data of firms. The model includes the following types of methods: the discriminant analysis model (Altman 1968), the Merton model (Merton 1974; Vassalou and Xing 2004), the logit model (Ohlson 1980), the probit model (Zmijewski 1984), and the mixed effect logit model (Alfo et al. 2005). However, the static model might suffer from a loss of predictive power (Shumway 2001) because it ignores the changing characteristics of firms over time. On the other hand, the dynamic model uses multiple-period data of firms so that the prediction involves the effects of time-varying firm characteristics. The dynamic model has proved over time to be more powerful than the static model (Hillegeist et al. 2004; Campbell et al. 2008; Hwang et al. 2011; Glennon and Nigro 2005). Examples of dynamic models are the discrete-time hazard model (Shumway 2001; Chava and Jarrow 2004), the default-intensity model (Duffie et al.2007), the quantile-regression approach (Li and Miu 2010), and the contingent-claim approach (Tang and Yan 2006). However, these dynamic models solely focus on the point in time when the financial distress first happens to a firm and ignores the possibility that subsequent financial distresses still might happen to that firm. Thus, these models do not generate predictions for firms with financial-distress experiences.
To avoid the above restriction on the dynamic model, we suggest using the dynamic logit model (DLM) to predict recurrent financial distresses. The DLM applies the logit model to the panel data containing not only the first financial distresses but also subsequent financial distresses of firms. Thus, this model has the advantage of using all available information to predict a firm’s financial distress at any point in time whether or not that firm has financial-distress experience. Under the independence assumption, the important parameters in DLM can be simply determined by maximizing the log-likelihood function of the panel data under study. However, this inde-pendence assumption might not be proper in practice, because repeated observations from the same firm tend to be correlated with one another. If one imposes an improper independence assumption on DLM, then one might suffer from a loss of predictive power.
For more accurately predicting recurrent financial distresses, we consider the DLM with autocorrelation structure. This model is more robust in predicting financial distresses for firms than the DLM with independent structure. Under the autocorrela-tion structure, the unknown parameters in DLM are estimated using the generalized estimating equations (GEE) approach (Liang and Zeger1986; Lipsitz et al. 1994). The consistency and asymptotic normality of the resulting estimators are given in Sec-tion 2. There are many software packages having the capabilities to implement GEE analyses, for example, SAS, S-Plus, and STATA. Thus, the computation required for the DLM with autocorrelation structure is as simple as that for the DLM with independent structure. There are many examples in the literature in which the GEE approach has proved to be more powerful for analyzing repeated data than the approach using independence estimating equations (IEE). For a detailed introduction to GEE, see for example, the monograph by Hardin and Hilbe (2002).
To implement the proposed model, exchangeable and first-order autoregressive (AR1) autocorrelation structures are used to account for the correlations among the observations from the same firm at different points in time. These two autocorrelation structures assume respectively that the magnitude of the correlation remains unchanged and decreases
dramatically as the number of time lags increases. They have the advantage of being formulated using only one nuisance parameter. See Lee and Geisser (1975) and Geisser (1981) for the importance of parsimonious correlation structures in prediction problems based on repeated measurements.
In Section 3, we illustrate the proposed model using three different panel datasets of industrial firms listed on the two major Taiwan stock exchanges: Taiwan Stock Exchange (TWSE) and GreTai Securities Market (GTSM). These panel datasets are based on the well-known predictors suggested by Altman (1968), Campbell et al. (2008), and Shumway (2001). The three prediction models comprise the following: the DLM with independent structure, the DLM with exchangeable structure, and the DLM with AR1 structure. These models are separately applied to each of the three panel datasets. We measure the perfor-mance of the proposed models through out-of-sample analysis with two perforperfor-mance metrics. These two performance metrics are the absolute difference (AD) between the actual number of financial distresses (ANFD) and the predicted number of financial distresses (PNFD) as well as the predictive interval (PI) of ANFD. These metrics are based on the actual magnitudes of financial-distress probabilities of firms. Using an expanding rolling window approach (Hillegeist et al.2004; Chava et al.2011), the empirical results in Section3 show that for each of the three panel datasets, the DLM with AR1 structure has the best performance among the three prediction models. Thus, this model has the potential to be a powerful model for studying credit losses in portfolios.
The remainder of this paper is organized as follows. In Section2, we develop the method for predicting recurrent financial distresses based on DLM. Section3presents the empirical results. Section4contains the concluding remarks and future research topics. Theappendix gives a computational procedure based on the GEE approach for estimating unknown parameters in the DLM with autocorrelation structure.
2 Method
In this section, we first describe the formulation of the DLM with independent structure and estimate its unknown parameters using a maximum likelihood method. Then we give the idea of the DLM with autocorrelation structure and estimate its unknown parameters using the GEE approach. Afterwards, we introduce two out-of-sample performance metrics to measure the performance of the discussed prediction methods.
2.1 The DLM with independent structure
The DLM has the advantage of using all available information to predict each firm’s financial distress at any point in time. In the following, we describe the structure of the panel data used in the prediction methods based on the DLM.
Two factors determine the panel data: the sampling criteria and the sampling period. Our sampling criterion is that all industrial firms that are listed on the TWSE or GTSM during the sampling period are included in the sample. All information occurring at the discrete points in time during the sampling period is collected from the Taiwan Economic Journal (TEJ) database. Let the sampling period be [ξ1,ξ2]. Suppose that there are n selected firms under the particular sampling scheme. We denote the panel data by
Yi;j; xi;j
: j ¼ si; ; ti; i ¼ 1; ; n
Here sidenotes the first observation time and tithe last observation time for the i-th firm in the sampling period [ξ1,ξ2] The value of Yi,j01 indicates that the financial status of the i-th firm at time j is in distress and Yi,j00 otherwise. Therefore, for the i-th firm, the results of
Pti j¼si Yi;j¼ 0; P ti j¼si Yi;j¼ 1; and P ti j¼si
Yi;j> 1 indicate respectively that the firm either has no, one, or repeated financial-distress experiences during the sampling period. Section3provides the definition of financial distress for the sampled firms. Further, we let xi,jbe the value of the d×1 predictor X collected from the i-th firm at time j.
Under the independence assumption, the likelihood function of the panel data is expressed as:
L ¼Y n i¼1 Yti j¼si pYi; j i; j 1 pi; j 1Yi; j: ð1Þ
Here pi; j¼ p Yi; j ¼ 1jxi; j p j; xi; j stands for the probability of financial distress happening to the i-th firm at time j. Thus, the probability function pi,j can be of any functional form with values in the interval (0, 1). The DLM considers a linear logistic function for pi,j, that is:
pi; j¼ exp a þ bxi; j
1þ exp a þ bxi; j ; ð2Þ
whereα and β are 1× 1 and 1× d vectors of parameters respectively. Plugging Eq. (2) into Eq. (1), the resulting log-likelihood of the panel data becomes:
‘ ¼Xn i¼1
Xti
j¼si
Yi; ja þ bxi; j log 1 þ exp a þ bxi; j
: ð3Þ
The maximum likelihood estimate baI; bbI
of (α, β) in Eq. (3) can be obtained by maximizing ‘ with respect to (α, β), or by solving the normal equations:
0¼ @ ‘ @ a; bð ÞT ¼ Xn i¼1 Xti j¼si
Yi; j exp a þ bxi; j
1þ exp a þ bxi; j ( ) 1 xi; j : ð4Þ
Using Eq. (2) and replacing the unknown parameters α and β with their maximum likelihood estimates baI and bbI , if a firm has the predictor value x0 at time t0 then its predicted financial-distress probability based on the DLM with independent structure is:
bpIðt0; x0Þ ¼ exp baIþ bbIx0
1þ exp baIþ bbIx0
: ð5Þ
Under the independence assumption, Liang and Zeger (1986) show that the maximum likelihood estimate baI; bbI
is consistent for (α, β). Thus, the predicted financial-distress probability bpIðt0; x0Þ converges to the true financial-distress probability p t0; x0ð Þ ¼
expðaþbx0Þ
1þexp aþbxð 0Þ . This result shows that the DLM with independent structure is an efficient
Further, through a straightforward calculation, the normal equations in Eq. (4) can also be equivalently expressed as the IEE:
0¼ @ ‘
@ a; bð ÞT ¼ Xn
i¼1
DTiVi1ðYi piÞ: ð6Þ
We use this result in subsection2.2to develop the GEE for the DLM with autocorrelation structure. The notations of Yi, pi, Viand Diin Eq. (6) are defined by
Yi¼ Yi;si Yi;siþ1 .. . Yi;ti 2 6 6 6 4 3 7 7 7 5; pi¼ pi;si pi;siþ1 .. . pi;ti 2 6 6 6 4 3 7 7 7 5; Vi¼ Vi;si 0 0 0 Vi;siþ1 0 .. . .. . . . . .. . 0 0 Vi;ti 2 6 6 6 4 3 7 7 7 5; Di¼ Di;si Di;siþ1 .. . Di;ti 2 6 6 6 4 3 7 7 7 5; where
Vi; j¼ pi; j1 pi; j Var Yi; jjxi; j ; Di; j¼ pi; j1 pi; j 1; xT i; j
@=@ a; bf ð Þgpi; j: Under the independence assumption, the quantities pi, Vi, and Distand for E Yijxið Þ , Cov Yijxið Þ , andf@=@ a; bð Þgpi, respectively, for each i ¼ 1; ; n; where xi¼ xi;si; ; xi;ti
.
2.2 The DLM with autocorrelation structure
The DLM with autocorrelation structure is developed by assuming that the time series observations Yi,jfrom the same firm are correlated with one another, but those from different firms are not correlated. Under the autocorrelation assumption, the value of (α, β) in DLM is estimated using the GEE approach. To formulate the GEE, let ρj,k be the correlation coefficient between Yi,jand Yi,k, where j; k 2 si; ; tif g , for each i ¼ 1; ; n . Thus, the covariance matrix of Yican be expressed as:
Cov Yijxið Þ ¼ Vi1=2AiVi1=2 Gi; ð7Þ
for each i ¼ 1; ; n . Subsection2.1provides the definitions of Yi, xi, and Vi, and
Ai¼ 1 ρsi;siþ1 ρsi;ti ρsiþ1;si 1 ρsiþ1;ti .. . .. . . . . .. . ρti;si ρti;siþ1 1 2 6 6 6 4 3 7 7 7 5:
Note that Ai stands for the correlation matrix of YiUsing the result from Eq. (7), the associated GEE are similarly formulated as the IEE in Eq. (6) with Vi replaced by Gi Specifically, the GEE are:
0¼X
n i¼1
DTi G1i ðYi piÞ: ð8Þ
By way of generation, the GEE alleviate the need to correctly specify the joint distribu-tion of Yi. But the price paid by the GEE for simple formulation is that the corresponding
log-likelihood function of the panel data is not available, and thus the GEE approach is not a maximum likelihood method. A computational procedure for finding the solution to the GEE in Eq. (8) is in theappendix.
In this paper, we consider two types of autocorrelation structures for the time series observations Yi,jfrom the same firm. One is the exchangeable structure withρj,k0 ρ, and thus the corresponding correlation matrix of Yiis:
Ai¼ 1 ρ ρ ρ 1 ρ .. . .. . . . . .. . ρ ρ 1 2 6 6 6 4 3 7 7 7 5;
for each i ¼ 1; ; n . The other is the AR1 structure with ρj;k¼ ρjjkj , and thus the corresponding correlation matrix of Yiis:
Ai¼ 1 ρ ρtisi ρ 1 ρtisi1 .. . .. . . . . .. . ρtisi ρtisi1 1 2 6 6 6 4 3 7 7 7 5;
for each i ¼ 1; ; n . Given each of these two autocorrelation structures, if ρ00 then Gi0 Vi, for each i ¼ 1; ; n , and the resulting GEE become the IEE in Eq. (6).
We also consider other types of autocorrelation structures such as the m-dependence structure withρj,k00 for jj kj > m , the banded correlation structure with ρj;k¼ ρjjkj forjj kj 1 , and the unstructured correlation without constraints on ρj, k. However, their corresponding estimates of the regression parametersα and β provided by our computational algorithm are not of normal convergence, and thus we do not report the results. The poor computational results might be due to there being too many nuisance parameters ρj,k involved. The numbers of nuisance parameters in these autocorrelation structures are
Pm k¼1
x2 x1þ 1 k
ð Þ , ξ2−ξ1, and ðx2 x1Þ x2ð x1þ 1Þ=2 , respectively. Here ξ1and ξ2 are the start and the end points in time for the sampling period of the panel data respectively.
Given each of exchangeable and AR1 structures, set baG; bbG
as the solution of the corresponding GEE in Eq. (8). Regardless of whether the imposed covariance matrix Giis correctly specified for Cov Yijxið Þ ,baG; bbG is consistent for (α, β) and has an asymptotic normal distribution with covariance matrix:
V1¼ lim n!1 Xn i¼1 DT iG1i Di !1 Xn i¼1 DT iG1i Cov Yijxið ÞG1i Di ( ) Xn i¼1 DT iG1i Di !1 : If Giis correctly specified so that Cov Yijxið Þ ¼ Gi, then V1reduces to
V2¼ lim n!1 Xn i¼1 DTiG 1 i Di !1 :
Consistent estimators bV1 and bV2 for V1 and V2 can be constructed by replacing the unknown quantities α, β, ρ, and Cov Yijxið Þ with their estimates baG; bbG; bρ;
and Yið bpiÞ Yið bpiÞ
T , respectively in each case. Herebρ ¼ Pn i¼1 ti si ð Þ tið siþ 1Þ=2 d 1g1P n i¼1 P ti1 j¼si Pti k¼jþ1bei; jbei;k
for the exchangeable structure,bρ ¼ P n i¼1 ti si ð Þ d 1g1 Pn i¼1 P ti1 j¼si
bei; jbei; jþ1 for the AR1 structure, bei; j¼ bpi; j 1bpi; j
n o1=2 Yi; jbpi; j ; bpi; j¼ exp baGþbbGxi; j 1þexpbaGþbbGxi; j
; bpi¼ bpi;si; ; bpi;ti
T
.
The estimator bV1 is called the robust covariance estimator because it is consistent for V1 regardless of whether or not Giis correctly specified for Cov Yijxið Þ . In contrast, bV2 is called the naive covariance estimator because it is based on the assumption that Giis correctly specified for Cov Yijxið Þ . See Liang and Zeger (1986) and Lipsitz et al. (1994) for a detailed introduction of the asymptotic properties of baG; bbG
.
Using Eq. (2) and replacingα and β with their estimates baG and bbG, if a firm has the predictor value x0at time t0, then its predicted financial-distress probability based on the DLM with autocorrelation structure is:
bpGðt0; x0Þ ¼ exp baGþ bbGx0
1þ exp baGþ bbGx0
: ð9Þ
By the consistency of baG and bbG , bpGðt0; x0Þ is also consistent for the true financial-distress probability p(t0, x0). Thus, the DLM with autocorrelation structure is a reliable prediction model for a firm’s financial distress whether or not the autocorrelation structure imposed on the time series observations Yi,jfrom the same firm is correctly specified.
The same developments for baG; bbG
, V1, V2, bρ; bV1; bV2; and bpGðt0; x0Þ based on the DLM with autocorrelation structure can also be applied to the DLM with independent structure by taking the estimate of the nuisance parameter ρ as zero. Thus, both the consistency and the asymptotic normality of baG; bbG
are shared with baI; bbI
. The principle disadvantage of baI; bbI
is that it might not have high efficiency in cases where the autocorrelation is large (Liang and Zeger1986).
2.3 Measuring prediction performance
For assessing the prediction models introduced in subsections 2.1and2.2, there are some standard performance metrics, for example, the out-of-sample type I and type II error rates (Cheng et al.2010) and the out-of-sample accuracy ratio obtained from the cumulative accuracy profile curve (BCBS2005). However, these standard performance metrics are only based on the relative ordinal rankings of financial-distress probabilities and not on the actual magnitudes of those probabilities. Thus, they are not suitable to assess whether a prediction model generates financial-distress probabilities that are adequate in absolute terms. Due to the fact that the credit losses of portfolios depend on the actual magnitudes of financial-distress probabilities, a proper prediction model must be able to provide accurate firm-by-firm financial-distress probabilities. Accordingly, we use the out-of-sample AD (ADout) between ANFD and PNFD and the out-of-sample PI (PIout) of ANFD as performance metrics for assessing the prediction models. Both
ADoutand PIoutare developed using the actual magnitudes of financial-distress probabilities rather than the relative firm-riskiness rankings. Korablev and Dwyer (2007), Duffie et al. (2009), and Chava et al. (2011) also consider similar performance metrics.
To compute ADoutand PIout, the out-of-sample data are selected. In contrast, the panel data that are used to build the prediction models in subsections2.1–2.2are considered as the “in-sample” data. The out-of-sample data are generated in a similar fashion to the in-sample data. In this paper, the out-of-sample period isðx2; x2þ 1 , where ξ2is the end time of the sampling period of in-sample data. The out-of-sample firms comprise all industrial firms that are listed on the TWSE or GTSM during the out-of-sample period. Suppose that there are n0out-of-sample firms. All values of the d×1 predictor X for the n0out-of-sample firms occurring at timeξ2+1 are collected from the TEJ database. The out-of-sample data are denoted by
eYk;x2þ1;exk;x2þ1
: k ¼ 1; ; n0
n o
:
Here eYk;x2þ1¼ 1 indicates that the financial status of the k-th out-of-sample firm is in
distress at timeξ2+1, and eYk;x2þ1¼ 0 otherwise. Further,exk;x2þ1 is the value of X collected
from the k-th out-of-sample firm at time ξ2+1, for each k ¼ 1; ; n0. Using the out-of-sample data eYk;x2þ1;exk;x2þ1
: k ¼ 1; ; n0
n o
and the result from Eq. (9), we first evaluate the predicted financial-distress probabilities bpG x2þ 1;exk;x2þ1
, for each k ¼ 1; ; n0. Then we apply the convolution calculation technique (Duan2010) to these predicted financial-distress probabilities, so that the distribution of the number of financial distresses based on the DLM with autocorrelation structure for the n0out-of-sample firms at timeξ2+1 can be obtained. The statistical characteristics of this distribution such as the mean, variance, and quantile can be evaluated. For the n0 out-of-sample firms, their PNFD at timeξ2+1, denoted by PNFDout(ξ2+1), is taken to be the mean of this distribution. Further, the value of PNFDout(ξ2+ 1) can also be equivalently obtained using PNFDoutðx2þ 1Þ ¼P
n0
k¼1bpG
x2þ 1;exk;x2þ1
. Thus the value of ADoutat timeξ2+1 is defined by
ADoutðx2þ 1Þ ¼ ANFDoutj ðx2þ 1Þ PNFDoutðx2þ 1Þj; where ANFDoutðx2þ 1Þ ¼ Pn
0
k¼1eYk;x2
þ1 and PNFDoutðx2þ 1Þ stands for a prediction of the value of ANFDoutðx2þ 1Þ which has a mean of P
n0
k¼1
p x2þ 1;exk;x2þ1
. On the other hand, for the n0out-of-sample firms, the 95 % PI of their ANFDout(ξ2+1) can be taken as [p0.025, p0.975] Here, p0.025and p0.975denote respectively the 2.5-th and 97.5-th percentiles of the distribution of the number of financial distresses. This 95 % PI of ANFDout(ξ2+1) is denoted by PIout(ξ2+1). The out-of-sample performance metrics ADout(ξ2+1) and PIout(ξ2+1) based on the DLM with autocorrelation structure can be similarly defined for the DLM with independent structure by replacingbpGx2þ 1;exk;x2þ1 withbpI x2þ 1;exk;x2þ1
.
3 Empirical studies
In this section, we conduct empirical studies to compare the performance of the three prediction models: the DLM with independent structure, the DLM with exchangeable structure, and the DLM with AR1 structure.
3.1 Data and estimation results
To investigate the performance of the three prediction models, we collect three different panel datasets based on the well-known predictors from Altman (1968), Campbell et al. (2008), and Shumway (2001). For simplicity of presentation, these predictors are called the Altman, Campbell, and Shumway predictors respectively. Table 1gives their definitions. Further, the panel data used to build the three prediction models are also called the in-sample data in this paper.
The sampling period for each of the three panel datasets is 1986 to 2008. The in-sample firms consist of all industrial firms listed on the TWSE or GTSM during the sampling period. We exclude financial firms from the sample due to the unique capital requirements and regulatory structure of that industry group. The predictor values come from the calendar year-end data collected from the TEJ database. In order to eliminate outliers, the values of each predictor (except the predictor PRICE) are winsorized using a 5/95 percentile interval (Campbell et al.2008). The resulting predictor values measure the risk of financial distress
Table 1 The definitions of the Altman, Campbell, and Shumway predictors. The results are given respec-tively in Panels A, B, and C
Variable Definition
Panel A: Altman predictors
WCTA Working capital divided by total assets
RETA Retained earnings divided by total assets
EBITTA Earnings before interest and taxes divided by total assets
METL Market equity divided by total liabilities
STA Sales divided by total assets
Panel B: Campbell predictors
NIMTAAVG NIMTAAVG¼ 1 wð 3Þ 1 wð 12Þ1NIMTA
4þ w3NIMTA3þ w6NIMTA2þ w9NIMTA1
ð Þ ,
the weighted average of four quarterly NIMTA with ω02−1/3and NIMTA as net income
divided by market-valued total assetsa
TLMTA Total liabilities divided by market-valued total assetsa
EXRETAVG EXRETAVG¼ 1 wð Þ 1 wð 12Þ1EXRET
12þ þ w11EXRET1
ð Þ , the weighted average
of twelve monthly excess returns (EXRET) with ω02−1/3and EXRET as the return on the
firm minus the TWSE capitalization weighted stock index (TAIEX) return
SIGMA Annualized square root of the average of the squared deviations in the firm’s daily stock
returns from zero over the past 3 months
RSIZE Logarithm of each firm’s market equity value divided by the total TWSE market equity value
CASHMTA Cash and short-term investments divided by market-valued total assetsa
MB Market equity value divided by book equity value
PRICE Logarithm of stock price if the price is below NT$22, and logarithm of NT$22 otherwise
Panel C: Shumway predictors
SIGMA Annualized square root of the average of the squared deviations in the firm’s daily stock
returns from zero over the past 3 months
RSIZE Logarithm of each firm’s market equity value divided by the total TWSE market equity value
NITA Net income divided by total assets
TLTA Total liabilities divided by total assets
EXRET The return on the firm minus the TAIEX return
a
over the 12-month period beginning 4 months after the calendar year end (Hillegeist et al. 2004). Using the TEJ definitions for characteristics such as negative net worth and bank-ruptcy, we identify the financial-distress filings covering the period from May, 1987 to April, 2010. Table2presents the frequency distribution of the in-sample firms in each of the three panel datasets according to the number of financial distresses that a firm experiences during the sampling period. The results in Table2show that for each of the three panel datasets, more than 12 % of the in-sample firms have financial-distress experience. Table3gives the summary statistics of the predictor values in each of the three panel datasets.
Given the in-sample data, Table4reports the estimation results of the three prediction models using each set of the Altman, Campbell, and Shumway predictors. Table 4shows that the values of the estimated coefficients in each of the three models all agree with their expected signs. Also, the table indicates that the robust standard errors for most of coeffi-cient estimates are of larger magnitude than the associated naive standard errors.
3.2 Prediction results
To compare the prediction performance of the models based on each set of the Altman, Campbell, and Shumway predictors, we collect the corresponding out-of-sample data with the predictor values in 2009 and financial statuses in 2010 (covering the period from May, 2010 to April, 2011) from the TEJ database. The out-of-sample data based on the Altman predictors consist of 24 financial distresses and 1,387 firm-year observations, the Campbell predictors consist of 22 financial distresses and 1,242 firm-year observations, and the Shumway predictors consist of 23 financial distresses and 1,368 firm-year observations.
Using the out-of-sample data and the estimation results in Table4, Fig.1and Table 5 present the out-of-sample prediction performance of the three models. Figure1 gives the distributions of the number of financial distresses in 2010. Panel (a) of Fig.1presents the results for the three models based on the Altman predictors. The panel shows that the distributions generated from the three models are quite different. From Panel (a) of Fig.1, we see that the mean of the distribution generated by the DLM with AR1 structure is the closest to the ANFD among the models. Also, the distribution generated by the DLM with AR1 structure shifts to the left relative to the distribution produced by each of the other two
Table 2 The frequency and percent frequency (in parentheses) distributions of the in-sample firms in each of the three panel datasets according to the number N of financial distresses that a firm has experienced during the sampling period. These panel datasets are collected from the TEJ database for the period of 1986 to 2008
N Altman predictors Campbell predictors Shumway predictors
0 1,495 (87.12 %) 1,209 (84.84 %) 1,435 (86.65 %) 1 66 (3.85 %) 79 (5.54 %) 68 (4.11 %) 2 73 (4.25 %) 71 (4.98 %) 73 (4.41 %) 3 29 (1.69 %) 29 (2.04 %) 32 (1.93 %) 4 28 (1.63 %) 19 (1.33 %) 25 (1.51 %) 5 9 (0.52 %) 9 (0.63 %) 14 (0.85 %) 6 7 (0.41 %) 3 (0.21 %) 3 (0.18 %) 7 4 (0.23 %) 5 (0.35 %) 3 (0.18 %) 8 1 (0.06 %) 0 (0.00 %) 2 (0.12 %) 9 4 (0.23 %) 1 (0.07 %) 1 (0.06 %) Total firms 1,716 (100 %) 1,425 (100 %) 1,656 (100 %)
models, and the distribution has a smaller mean. The results in Panel (a) of Fig.1are the same as those in Panels (b) and (c) of Fig.1for the distributions generated by the models based on the Campbell and Shumway predictors respectively.
Table5gives the values of PNFD, AD, and 95 % PI produced from the distributions in Fig.1. Among the three models, Table5shows that the value of AD from the DLM with AR1 structure is the smallest one for each set of the Altman, Campbell, and Shumway predictors. Also, the 95 % PI from the DLM with AR1 structure is the only one containing the ANFD for the Altman and Shumway predictors. By the results presented in Fig.1and Table5, we conclude that the DLM with AR1 structure has the best out-of-sample prediction performance in 2010 among the three models.
3.3 Robustness performance
In this section, we use an expanding rolling window approach to assess the robustness of the advantage of the DLM with AR1 structure, that is, it has the best out-of-sample prediction performance among the three models. For simplicity, we use the same predictors to generate the in-sample and out-of-sample data. Also, we use the same computational procedures as in Table 3 Summary statistics of predictors in each of the three panel datasets. These panel datasets are collected from the TEJ database for the period of 1986 to 2008. Panels A, B, and C give the results for the Altman, Campbell, and Shumway predictors respectively
Variable Mean Median Standard deviation Minimum Maximum
Panel A: Altman predictors
570 financial distresses, 14,269 firm-year observations, and 1,716 firms
WCTA 0.167 0.160 0.170 −0.141 0.490
RETA 0.060 0.082 0.143 −0.325 0.272
EBITTA 0.046 0.053 0.077 −0.138 0.175
METL 3.575 2.112 3.830 0.301 14.890
STA 0.758 0.664 0.440 0.164 1.827
Panel B: Campbell predictors
491 financial distresses, 12,858 firm-year observations, and 1,425 firms
NIMTAAVG 0.003 0.008 0.018 −0.049 0.025 TLMTA 0.346 0.312 0.202 0.063 0.749 EXRETAVG −0.006 −0.007 0.045 −0.090 0.087 SIGMA 0.447 0.432 0.155 0.205 0.755 RSIZE −8.149 −8.231 1.480 −10.567 −5.348 CASHMTA 0.084 0.060 0.074 0.007 0.270 MB 1.514 1.340 0.799 0.473 3.315 PRICE 2.642 2.996 0.648 −2.408 3.091
Panel C: Shumway predictors
544 financial distresses, 13,655 firm-year observations, and 1,656 firms
SIGMA 0.459 0.439 0.166 0.207 0.809
RSIZE −8.231 −8.310 1.523 −10.769 −5.389
NITA 0.030 0.038 0.075 −0.155 0.149
TLTA 0.394 0.388 0.166 0.122 0.716
T able 4 Es timation res ults of the three mode ls. Panels A , B, and C present the results base d o n the Altman, Campb ell, and Shumw ay predictors respectively . The se panel data are collected for the period of 1986 to 2008 to study recurrent financial distresses. The three models are the D L M with indepen dent struc ture, the D L M with excha ngeable struc ture, and the D L M with AR1 structure V ariable Independent Exchangeable AR1 Coef ficient estimate Naive standard error Robust standard error Coef ficient estimate Naive standard error Robust standard error Coef ficient estimate Naive standard error Robust standard error Panel A: Altman predictors 570 financial distresses, 14,269 firm-year observations, and 1,716 firms Intercept − 3.507 0.126*** 0.177*** − 3.493 0.142*** 0.176*** − 3.749 0.170*** 0.236*** WCT A − 0.878 0.333*** 0.469* − 0.981 0.357*** 0.453** − 0.672 0.431 0.561 RET A − 10.455 0.436*** 0.575*** − 9.640 0.464*** 0.549*** − 8.917 0.556*** 0.712*** EBITT A − 1.762 0.800** 1.092 − 1.879 0.823** 1.016* − 0.782 0.953 1.249 METL − 0.109 0.026*** 0.039*** − 0.090 0.026*** 0.036** − 0.1 18 0.036*** 0.056** ST A − 0.304 0.123** 0.181* − 0.267 0.133** 0.178 − 0.295 0.163* 0.218 Panel B: Campbell predictors 491 financial distresses, 12,858 firm-year observations, and 1,425 firms Intercept − 8.841 0.701*** 0.982*** − 8.172 0.768*** 0.950*** − 8.197 0.771*** 0.889*** NIMT AA VG − 29.127 2.922*** 3.877*** − 27.021 2.899*** 3.661*** − 21.638 2.961*** 3.509*** TLMT A 2.807 0.432*** 0.659*** 2.497 0.443*** 0.602*** 2.489 0.461*** 0.603*** EXRET A V G − 4.492 1.264*** 1.254*** − 3.669 1.218*** 1.126*** − 3.174 1.194*** 1.031*** SIGMA 2.069 0.425*** 0.459*** 1.564 0.414*** 0.400*** 1.457 0.423*** 0.388*** RSIZE − 0.346 0.055*** 0.073*** − 0.339 0.062*** 0.076*** − 0.361 0.064*** 0.071*** CASHMT A − 1.660 0.955* 1.189 − 1.462 0.953 1.075 − 2.172 1.088** 1.061** MB 1.341 0.099*** 0.155*** 1.140 0.097*** 0.127*** 1.061 0.099*** 0.124*** PRICE − 0.862 0.099*** 0.143*** − 0.783 0.100*** 0.133*** − 0.749 0.103*** 0.128*** Panel C: Shumway predictors 544 financial distresses, 13,655 firm-year observations, and 1,656 firms Intercept − 9.250 0.413*** 0.596*** − 9.275 0.545*** 0.815*** − 8.869 0.576*** 0.657***
T able 4 (continu ed) V ariable Independent Exchangeable AR1 Coef ficient estimate Naive standard error Robust standard error Coef ficient estimate Naive standard error Robust standard error Coef ficient estimate Naive standard error Robust standard error SIGMA 1.692 0.331*** 0.381*** 1.041 0.350*** 0.390*** 0.156 0.342 0.345 RSIZE − 0.226 0.042*** 0.057*** − 0.292 0.055*** 0.080*** − 0.304 0.060*** 0.067*** NIT A − 12.278 0.778*** 1.053*** − 9.794 0.860*** 1.268*** − 5.862 0.835*** 1.128*** TL T A 5.585 0.378*** 0.669*** 4.964 0.459*** 0.775*** 5.160 0.510*** 0.783*** EXRET − 0.134 0.124 0.122 − 0.134 0.126 0.135 − 0.045 0.1 14 0.125 The nota tions ***, **, and * indicate the significa nce of the parameter based on the W ald chi-squared test at the 1 % , 5 %, and 10 % levels res pective ly
Fig. 1 Plot of the distributions of the number of financial distresses in 2010. The distributions are generated
by applying Duan’s (2010) convolution calculation technique to the predicted financial-distress probabilities
of the out-of-sample firms based on the three models. These predicted financial-distress probabilities are computed using the predictor values of the out-of-sample firms in 2009. In each panel, the blue dashed, the green short dashed, and the red solid curves denote the distributions produced using the DLM with independent structure, the DLM with exchangeable structure, and the DLM with AR1 structure respectively. Also, the location of the black vertical solid curve stands for the value of ANFD. Panels (a), (b), and (c) show the results based on the Altman, Campbell, and Shumway predictors respectively
Table 5 The numerical values of PNFD, AD, and 95 % PI for the out-of-sample firms in 2010. The results are
produced from the distributions of the number of financial distresses in Fig.1. These distributions are obtained
by applying Duan’s (2010) convolution calculation technique to the predicted financial-distress probabilities
of the out-of-sample firms in 2010 based on the three models. The three models are the DLM with independent structure, the DLM with exchangeable structure, and the DLM with AR1 structure. Panels A, B, and C present the numerical results based on the Altman, Campbell, and Shumway predictors respectively
Independent Exchangeable AR1
Panel A: Altman predictors
1,387 out-of-sample firms, ANFD024
PNFD 53.404 50.234 32.484
AD 29.404 26.234 8.484
PI [41.057, 65.262] [37.980, 62.035] [22.074, 42.604]a
Panel B: Campbell predictors
1,242 out-of-sample firms, ANFD022
PNFD 37.220 37.247 35.507
AD 15.220 15.247 13.507
PI [27.742, 46.259] [27.257, 46.823] [25.497, 45.107]
Panel C: Shumway predictors
1,368 out-of-sample firms, ANFD023
PNFD 43.082 35.221 32.922
AD 20.082 12.221 9.922
PI [31.770 53.951] [24.450, 45.662] [22.169, 43.458]a
subsections3.1and3.2to obtain the distributions of the number of financial distresses based on the three models in each year during the period of 2001 to 2010.
For the first window, we estimate the updated coefficients for each of the three models using the in-sample data comprising the predictor values from 1986 to 1999 and financial statuses from 1987 to 2000. The updated coefficients are combined with the predictor values of out-of-sample firms in 2000 to predict the financial-distress probabilities and to generate the distribution of the number of financial distresses for out-of-sample firms in 2001. The out-of-sample performance based on AD and 95 % PI during 2001 is measured for each of the three models. For the second window, we estimate the updated coefficients for each of the three models using the in-sample data comprising the predictor values from 1986 to 2000 and financial statuses from 1987 to 2001. The updated coefficients are combined with the predictor values of out-of-sample firms in 2001 to predict the financial-distress probabilities and to generate the distribution of the number of financial distresses for out-of-sample firms in 2002. The out-of-sample performance based on AD and 95 % PI during 2002 is measured for each of the three models. The process is continued so that the updated coefficients for each of the three models in the last window are based on the in-sample data comprising the predictor values from 1986 to 2008 and financial statuses from 1987 to 2009. The last set of updated coefficients is combined with the predictor values of out-of-sample firms in 2009 to predict the financial-distress probabilities and to generate the distribution of the number of financial distresses for out-of-sample firms in 2010. Thus, the out-of-sample performance based on AD and 95 % PI during 2010 is measured for each of the three models. The process carried out in the last window is the same as that performed in subsections 3.1and 3.2. Table6gives the numbers of firm-year observations in the in-sample and out-of-sample data for each of the ten windows.
Figures2and3and Table7present the out-of-sample prediction performance of the three models based on each set of the Altman, Campbell, and Shumway predictors. Figures2and 3give the out-of-sample prediction comparisons in terms of AD and 95 % PI respectively. From Fig.2, we see that the DLM with AR1 structure has the best performance among the three models for most of the windows, in the sense of yielding the smallest values of AD. From Fig. 3, we see that the 95 % PI from the DLM with AR1 structure is of the best performance among the three models in terms of the number of times that the 95 % PI contains ANFD over the ten windows.
Table7 summarizes the out-of-sample prediction performance in Figs.2 and3for the three models. Table7gives the sample average and standard deviation of the values of AD as well as the number of times that the 95 % PI contains ANFD over the ten windows. For each set of the Altman, Campbell, and Shumway predictors, Table7shows that the values of AD generated by the DLM with AR1 structure over the ten windows not only have the smallest sample average but also have the lowest volatility, and the number of times that the 95 % PI contains ANFD over the ten windows is the largest. These results confirm the robustness of the advantage of the DLM with AR1 structure having the best out-of-sample prediction performance among the three models.
4 Concluding remarks and future research topics
In this paper, we propose to use the DLM with autocorrelation structure to predict recurrent financial distresses. We construct the model by applying the logit model to the panel data containing not only the first financial distresses but also subsequent financial distresses of firms. Thus the proposed model has the advantage of using all of the available information to
predict a firm’s financial distress at any point in time whether or not the firm has financial-distress experience. Further, the model assumes that the financial statuses collected for the same firm at different time points are correlated with one another, but those obtained from different firms are not correlated. Such an autocorrelation assumption is more appropriate than the usual independence assumption, because repeated observations from the same firm tend to be correlated with one another. We estimate the unknown parameters in the proposed model using the GEE approach. From the theoretical results in Liang and Zeger (1986) and Lipsitz et al. (1994), the estimated financial-distress probability based on the proposed Table 6 The numbers of firm-year observations in the in-sample and out-of-sample data. The data are collected in each of the ten windows for investigating the robustness performance of the three models based on each set of the Altman, Campbell, and Shumway predictors. Panels A and B present the results for the in-sample and out-of-in-sample data respectively
Window (predictor sampling period)
Altman predictors Campbell predictors Shumway predictors
Financial distresses Total firm-years Financial distresses Total firm-years Financial distresses Total firm-years
Panel A: In-sample data First window (1986–1999) 139 4,071 87 3,641 121 3,771 Second window (1986–2000) 173 4,812 119 4,359 154 4,507 Third window (1986–2001) 212 5,641 156 5,161 192 5,330 Fourth window (1986–2002) 252 6,632 196 6,098 232 6,289 Fifth window (1986–2003) 304 7,752 247 7,120 284 7,354 Sixth window (1986–2004) 366 9,012 304 8,209 345 8,539 Seventh window (1986–2005) 435 10,290 369 9,330 414 9,752 Eighth window (1986–2006) 486 11,579 414 10,477 464 11,018 Ninth window (1986–2007) 537 12,924 462 11,664 514 12,341 Tenth window (1986–2008) 570 14,269 491 12,858 544 13,655
Panel B: Out-of-sample data
First window (2000) 34 741 32 718 33 736 Second window (2001) 39 829 37 802 38 823 Third window (2002) 40 991 40 937 40 959 Fourth window (2003) 52 1,120 51 1,022 52 1,065 Fifth window (2004) 62 1,260 57 1,089 61 1,185 Sixth window (2005) 69 1,278 65 1,121 69 1,213 Seventh window(2006) 51 1,289 45 1,147 50 1,266 Eighth window (2007) 51 1,345 48 1,187 50 1,323 Ninth window (2008) 33 1,345 29 1,194 30 1,314 Tenth window (2009) 24 1,387 22 1,242 23 1,368
model is consistent for the true financial-distress probability whether or not the imposed autocorrelation structure is correct. Thus, the DLM with autocorrelation structure is a reliable prediction model.
The performance of the proposed model in predicting recurrent financial distresses is illustrated using three panel datasets based on the well-known predictors from Altman (1968), Campbell et al. (2008), and Shumway (2001). Its robustness assessment is also investigated using different numbers of years of data. To implement the proposed method, we use two different autocorrelation structures: exchangeable and AR1, to account for the correlations among the observations from the same firm at different points in time. The data collected from the TEJ database are based on all industrial firms listed on the two major Taiwan stock exchanges: TWSE and GTSM. Using an expanding rolling window approach, our empirical results show that for each of the three panel datasets, the DLM with AR1 structure has better performance than the other two discussed models: the DLM with independent structure and the DLM with exchangeable structure. The DLM with AR1 structure yields more accurate PNFD and PI in out-of-sample analysis. Thus, from the empirical results, the DLM with AR1 structure has the potential to be a powerful model for studying credit losses in portfolios.
There are some possible extensions for the prediction models considered in this paper. First, a measure of aggregate systemic risk can be generated using the predicted financial-distress probabilities based on each of the three models. For example, using the expanding rolling window approach in subsection 3.3, systemic risk can be the equal-weighted (or value-weighted) average of predicted financial-distress probabilities for out-of-sample firms collected in each window. It is of interest to examine whether this type of measure of aggregate systemic risk can predict the future market return, economic downturn, TED spread, or loss-given-default rate. Caselli et al. (2008), Acharya et al. (2010), Allen et al. (2010), Adrian and Brunnermeier (2011), and Kelly (2011) all consider similar studies on systemic risk. Second, each of the three models can be extended to predict financial Fig. 2 The out-of-sample AD between ANFD and PNFD produced by the three models using an expanding rolling window approach for the period of 2001 to 2010. In each panel, the blue dashed, the green short dashed, and the red solid curves denote the values of AD produced using the DLM with independent structure, the DLM with exchangeable structure, and the DLM with AR1 structure respectively. Panels (a), (b), and (c) show the results based on the Altman, Campbell, and Shumway predictors respectively
distresses over different future periods (Campbell et al.2008; Duan et al.2011). It is of interest to study the term structure of forward financial-distress probabilities based on each of the three models. Third, we study the performance of the three models in this paper only using firm-specific variables. The macroeconomic variables such as the real GDP growth rate and interest rate have been considered in Salas and Saurina (2002) for assessing credit risk. Researchers might study the effects of macroeconomic variables on the discussed models for predicting recurrent financial distresses. Finally, each of the three models assumes that the firm-specific effects on financial distress prediction are constant. However, in practice, these firm-specific effects should depend on business cycles (Pesaran et al. 2006), especially in cases of severe economic downturns. Thus, it would be more sensible to allow the parameters of logistic function to evolve with the effect of changes in macroeco-nomic dynamics. To do so, the idea of the varying coefficient model (Fan and Zhang2008) can be considered. Specifically, the logistic function in Eq. (2) is replaced by
pi; j¼ exp a zj
þ b zj xi; j
1þ exp a zj þ b zj xi; j ;
Fig. 3 The out-of-sample 95 % PI of ANFD produced by the three models using an expanding rolling window approach for the period of 2001 to 2010. The blue, the green, and the red solid curves denote the out-of-sample 95 % PI of ANFD produced using the DLM with independent structure, the DLM with exchange-able structure, and the DLM with AR1 structure respectively. In each panel, the black stars and the solid circles stand for PNFD and ANFD respectively. Panels (a)–(c) show the results based on the Altman
where b zj ¼ b1 zj ; ; bd zj , and each ofa zj ; b1 zj ; ; bd zj is an unknown but smooth function of the value zjcollected at time j from the k×1 macroeconomic variable Z. Thus, the resulting prediction models would allow the effects of firm-specific predictors on credit risk to change with observable macroeconomic dynamics factors.
Acknowledgments The authors thank the reviewers for their valuable comments and suggestions that have greatly improved the presentation of this paper. This research is supported by the National Science Council, Taiwan, Republic of China.
Appendix: A computational procedure for finding the solution baG; bbG
of GEE
The value of baG; bbG
can be computed using the Fisher-scoring algorithm. Setθ ¼ a; bð ÞT . Given a starting value bθ0 forθ, iterate
bθmþ1¼ bθmþ Xn i¼1 Di bθm T Gi bθm; bρm 1 Di bθm ( )1 Xn i¼1 Di bθm T Gi bθm; bρm 1 Yi pi bθm n o " # ; until bθmþ1¼ bθm baG; bbG T
andbρmþ1¼bρmbρ . Here the nuisance parameter ρ in the m-th iteration is estimated by bρm¼ Pn i¼1 ti si ð Þ tið siþ 1Þ=2 d 1 1Pn i¼1 P ti1 j¼si Pti k¼jþ1be ðmÞ i;j be ðmÞ i;k
for the exchangeable structure,bρm¼ Pn i¼1 ti si ð Þ d 1 1Pn i¼1 P ti1 j¼si beðmÞi; jbe ðmÞ
i; jþ1 for the AR1
structure,beðmÞi; j ¼ bpðmÞi; j 1 bpðmÞi; j n o h i1=2 Yi; j bpðmÞi; j , andbpðmÞi; j ¼ pi; j bθm
. Liang and Zeger
Table 7 The sample average and standard deviation of the values of AD and the numberN of times that the
95 % PI contains ANFD over the ten windows for the three models based on each set of the Altman, Campbell, and Shumway predictors. The three models are the DLM with independent structure, the DLM with exchangeable structure, and the DLM with AR1 structure
Independent Exchangeable AR1
Panel A: Altman predictors
Average 11.071 7.010 3.847
Standard deviation 11.575 10.922 7.411
N 8 8 9
Panel B: Campbell predictors
Average 17.413 15.934 9.096
Standard deviation 14.084 13.595 11.468
N 4 4 7
Panel C: Shumway predictors
Average 15.568 11.442 7.399
Standard deviation 13.786 10.414 8.928
(1986) suggest taking the starting value bθ0 as the maximum likelihood estimate ofθ produced in subsection2.1under the independence assumption.
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