Sketches of the Proofs

In this section, sketches of the proofs for Theorem 2.1 will be given. The following notations will be used. Let ⁽¹⁾_j and ⁽²⁾_j be the gradient vector and the Hessian matrix of j, for each j = 1,2,3, given in (22)-(24), respectively. Also, H⁽²⁾(x) is the Hessian matrix of H(x). Define P0 as the event that the number of control data points falling into the neighborhood N (x; b/2) of x is less than ρ₀n0

R

N (x; b/2) f0(t) dt, where ρ₀ is a positive constant satisfying ρ₀ ≤ 1/4, and Q⁰ the event that the number of control data

points falling into the neighborhood N (x; b) of x is greater than ϕ₀n0

R

N (x; b)f0(t) dt, where ϕ₀ is a positive constant satisfying ϕ₀ ≥ exp(1). The definition of neighborhood N (x; b) of the given point x has been given in Section 1.4. The events P1 and Q1 are similarly defined for case data points with n0, f0, ρ₀, and ϕ₀ replaced respectively by n1, f1, ρ₁, and ϕ₁.

The proofs of the asymptotic bias and variance for each of the estimators ˆH1(x), ˆθ, and ˆH(x)are given below in sequence.

Proof of the asymptotic bias and variance for ˆH1(x). Set η = (α, β, θ)^T and ˆη = (ˆα, β, ˆˆ θ)^T, the maximizer of 1(α, β, θ; x) in (22). By the first order Taylor expansion, we have

0 = ⁽¹⁾₁ (ˆη; x) = ⁽¹⁾₁ (η; x) + ⁽²⁾₁ (η^∗; x) (ˆη− η), (31) for each x ∈ [0, 1]^d, where η^∗ lies in the line segment connecting η and ˆη.

Using conditions (C1)-(C5), (21), and the large deviation theorem in Section 10.3.1 of Serfling (1980), a straightforward calculation leads to the following asymptotic re-sults: as n → ∞,

P (P0∪ Q⁰∪ P¹∪ Q¹) = O{exp(−n b^θ)}, (32) E{ ⁽¹⁾1 (η; x)} = (1/2) n b²θ ζ(1− ζ) A¹+ O(n b³_θ+ b^−d_θ ), (33) E{ ⁽²⁾1 (η; x)} = (−1) n ζ(1 − ζ) B¹+ O(n bθ), (34) V ar{ ⁽¹⁾1 (η; x)} = n b^−dθ ζ(1− ζ) C¹+ O(n b^−d+1_θ ), (35) for each η. Here

A1 ={u⁰(x) D0(x), u^T₁(x) D0(x), u0(x) D₁^T(x)}^T,

B1 = Z k1

m1

· · · Z kd

m_d

Λ1(x, t) K^#(t) dt,

C1 = Z k1

· · · Z k_d

Λ1(x, t) K^#(t)² dt,

where

Using condition (C3) and the results of (32)-(35) and comparing the magnitudes of

(1)

1 (η; x) = Op(n b²_θ+ n^1/2 b^−d/2_θ )and ⁽²⁾₁ (η^∗; x) = Op(n) in (31), we have

ˆ

η− η = o^p(1). (36)

Using (31)-(36) and approximations to the standard errors of functions of random variables in Section 10.5 of Stuart and Ord (1987), the results of the asymptotic bias and variance of ˆH1(x) in (25) and (26) follow, respectively.

Proof of the asymptotic bias and variance for ˆθ. Set φ = (α0, θ) and ˆφ = (ˆα0, ˆθ), the maximizer of 2(α0, θ)in (23). Using the fact that α0is a normalizing constant for f1(x, z) = f0(x, z) exp{α⁰+ ˆH1(x) + θ z}, the results of the asymptotic bias and variance of Hˆ1(x) in (25) and (26), (C1)-(C5), (21), and approximations to the standard errors of functions of random variables, through a straightforward calculation, we have

exp(α0) = (1 + b²_θ c1+ c2)⁻¹ {1 + o^p(1)},

Next, using this result and (C1)-(C5), through a straightforward calculation, we have,

as n → ∞,

Following the same arguments as those of (31) and (36) and using (37)-(39), we have ˆφ− φ = o^p(1). Combining this result and using approximations to the standard errors of functions of random variables in Section 10.5 of Stuart and Ord (1987), the results of the asymptotic bias and variance of ˆθ in (27) and (28) follow, respectively.

Proof of the asymptotic bias and variance for ˆH(x). Set ξ = (α^∗, β)^T and ˆξ = (ˆα^∗, ˆβ)^T, the maximizer of 3(α^∗, β; x) in (24), where α^∗ = H^∗(x) + α1 and α1 is a normalizing constant. Using (C1)-(C5), (21), the asymptotic bias and variance of ˆθin (27) and (28), and approximations to the standard errors of functions of random variables, we have, as n → ∞,

Λ3(x, t) =

⎛

⎜⎝ 1, t^T t, t t^T

⎞

⎟⎠ D⁰(x).

Following the same arguments as those of (31) and (36) and using (32) and (40)-(42), we have ˆξ− ξ = o^p(1). Combining this result with (40)-(42) and using approximations to the standard errors of functions of random variables, the results of the asymptotic bias and variance of ˆH(x) in (29) and (30) follow, respectively. Hence the proof of Theorem 2.1 is completed.

CHAPTER III

DYNAMIC PREDICTION METHODS FOR BANKRUPTCY AND FINANCIAL DISTRESS

3.1 Introduction

As introduced in Sections 1.1 and 1.7, the DSM (19) is a dynamic forecasting method against the static forecasting method LLM (1). The DSM (19) has the advantage of using all available historical information to determine each firm’s bankruptcy risk at each point in time; but the LLM (1) uses only one set of predictor values collected at a specific time point for each firm.

In this chapter, we shall first study asymptotic properties of maximum likelihood estimators ˆψ = (ˆα0, ˆα1, ˆβ, ˆθ)^T given in Section 1.5 for the parameters ψ = (α0, α1, β, θ)^T in DSM (19) with discrete-time survival data. Then the practical performance of the bankruptcy prediction method based on the DSM (19) with discrete-time survival data is studied through a real data example. For these, the composition of the discrete-time survival data and the formulation of the DSM (19) are recalled.

According to the discrete-time survival sampling introduced in Section 1.3, the discrete-time survival data are expressed as

(ti, Yi, xi,1, · ··, x^i,ti, zi,1, · ··, z^i,ti), for i = 1, · ··, n.

Here ti ∈ {1, 2, · · ·, m} denotes the duration time of the i-th company in the sampling period, and m is a positive integer standing for the length of the sampling period. Also, at the duration time ti, Yi = 0 indicates that the i-th company is nonbankrupt, and 1 the i-th company is bankrupt. Further, xi,j and zi,j are values of the d-dimensional continuous and q-dimensional discrete explanatory variables X and Z collected at the duration time j, respectively in each case, for each j = 1, · · ·, tⁱ and for the i-th company.

Combining the discrete-time survival data, the logistic hazard function, and the natural logarithm function of the duration time, the DSM (19) is expressed by the log-likelihood function for the discrete-time survival data as

DSM(ψ) =

Using the log-likelihood function DSM(ψ), maximum likelihood estimators ˆψ = (ˆα0, ˆ

By the functional form of the normal equations, we can not derive a closed-form solution for ψ = (α0, α1, β, θ)^T from the normal equations. But, practically, there are many software packages including, for example, S-plus, Gauss, and SAS providing available procedures to solve the normal equations. In this dissertation, we use the Gauss software to process the computational work.

This chapter is organized as follows. Section 3.2 presents asymptotic properties of maximum likelihood estimators ˆψ = (ˆα0, ˆα1, ˆβ, ˆθ)^T for the DSM (19) with the discrete-time survival data. To illustrate the bankruptcy prediction method based on the DSM (19) with the discrete-time survival data, a real data set is analyzed in Section 3.3.

Finally, the concluding remarks and future research topics are given in Section 3.4.

3.2 Theoretical Results

In this section, we shall study the asymptotic properties of maximum likelihood

estimators ˆψ = (ˆα0, ˆα1, ˆβ, ˆθ)^T for the parameters ψ = (α0, α1, β, θ)^T in the DSM (19).

By the properties of maximum likelihood estimators given in Section 3.3 of Cox and Oakes (1984), ˆψ = (ˆα0, ˆα1, ˆβ, ˆθ)^T are asymptotically normally distributed as

to estimate In(ψ). Replacing the quantity In(ψ)⁻¹ in the above asymptotic normal distribution of ˆψ with its estimate In(ˆψ)⁻¹, the resulting distribution can be used to derive the confidence interval estimate and test the values of ψ.

3.3 A Real Data Example

In this section, the DSM (19) with the discrete-time survival data was applied to the data occurred in Taiwan. Our discrete-time survival data were drawn by three

the year 1981 to December of the year 1999, and the sampling criterion was defined as those firms starting to be listed in Taiwan Stock Exchange during the sampling period. By characteristics of industries, the financial institutions (with industry code M2800: Banking and Insurance) were eliminated from the sample due to the unique capital requirements and regulatory structure in that industry group. Also, electronic companies (with industry code M2300: Electron) were not considered because such companies provide much less historical data and have much smaller financial failure rate, compared to traditional companies. Further, the companies providing incomplete values of explanatory variables were excluded. Hence, our discrete-time survival data were selected from the traditional companies providing complete values of explanatory variables.

In the second step, 249 companies satisfying the above sampling considerations were selected, and called the in-sample companies. Finally, in the third step, the historical data of the 249 selected in-sample companies were drawn from the financial database provided by Taiwan Economic Journal Co. Ltd.

Note that since there are only few bankrupt companies among the 249 selected ones, it is difficult to predict bankruptcy well. In this case, to provide more failure companies to proceed our research, we replaced our target on bankruptcy prediction with financial distress prediction. According to the definition of financial distress given by Taiwan Stock Exchange, financial distress companies are those whose stocks were delisted, stopped trading, or traded by cash.

For predicting financial distress, two different sets of predictors were considered in the DSM (19). The first set of predictors was that used in Altman (1968), and the second was that employed in Zmijewski (1984). The two sets of predictors are given as follows.

Altman’s predictors:

1. WC/TA = Working capital / Total assets.

2. RE/TA = Retained earnings / Total assets.

Table 5: The information about industry and financial status of in-sample companies.

3. EBIT/TA = Earnings before interest and taxes / Total assets.

4. ME/TL = Market value of equity / Book value of total debts.

5. S/TA = Sales / Total assets.

Zmijewski’s predictors:

1. NI/TA = Net income / Total assets.

2. TL/TA = Total debts / Total assets.

3. CA/CL = Current assets / Current liabilities.

Since there is no discrete predictor considered by Altman and Zmijewski, our discrete-time survival data only contain continuous predictors. To perform the financial distress prediction, the duration time t required by the DSM (19) was taken as the firm’s trading age (Shumway 2001).

The values of Altman’s and Zmijewski’s predictors were collected for our selected 249 in-sample companies from the Taiwan Economic Journal database. For each

se-Table 6: Summary statistics of variables in our discrete-time survival data.

mean median std min max

log(t) 1.3867 1.6094 0.7535 0.0000 2.8332 WC/TA 0.1486 0.1395 0.1803 −1.0713 1.1979 RE/TA 0.0533 0.0593 0.1232 −1.6818 0.6752 EBIT/TA 0.0592 0.0602 0.0854 −1.0061 0.6526 ME/TL 5.4051 3.3237 6.4633 0.0355 78.7321 S/TA 0.7228 0.6318 0.4568 −0.1665 4.1400 NI/TA 0.0347 0.0407 0.0984 −1.6825 0.6407 TL/TA 0.4000 0.3942 0.1690 0.0485 1.5139 CA/CL 1.9803 1.4929 1.6762 0.0129 21.1094

lected company, the annual values of Altman’s and Zmijewski’s predictors were collected during the sampling period. The information about industry and financial status of our selected 249 in-sample companies are given in Table 5. The summary statistics of log(t) and the predictors considered by Altman and Zmijewski in our selected discrete-time survival data are given in Table 6.

In this dissertation, in order to predict financial distress, the data used in the LLM (1) and DSM (19) were standardized so that the prediction results are unaffected by range, outliers, and other factors. The maximum likelihood estimates for parameters in the DSM (19) using the discrete-time survival data with Altman’s predictors, and those in the LLM (1) using the last annual data of the discrete-time survival data with Altman’s predictors are presented in Table 7. By characteristics of Altman’s predictors, the larger the values of Altman’s predictors, the smaller the probability of financial distress. Combining the result and the fact that the logistic function is strictly increasing, the coefficient estimates of Altman’s predictors in both models LLM (1) and DSM (19) should be negative. From Table 7, the coefficient estimates of the three predictors, WC/TA, RE/TA, and S/TA, are negative in both the DSM (19) and LLM (1). But those of the two predictors, EBIT/TA and ME/TL, are positive in both models. The tests of the latter two coefficient estimates are not significant at 0.05 level in the DSM (19). Therefore, the resulting coefficient estimates of Altman’s predictors in the DSM (19) are reasonable. On the other hand, the tests of the latter two coefficient

Table 7: The estimated values of parameters in each of the DSM and the LLM using our discrete-time survival data with Altman’s predictors. A z statistic was given to test the significance of the value of each parameter. The value given in the parenthesis stands for the p-value of the corresponding z test. Each value marked by^∗ denotes that the corresponding coefficient is significant at 0.05 level.

prediction

model log(t) WC/TA RE/TA EBIT/TA ME/TL S/TA DSM −0.0151 −0.0291 −0.3219 0.0879 0.0486 −0.2000

(0.747) (0.548) (0.000)^∗ (0.078) (0.300) (0.670) LLM −0.1383 −0.1561 −3.9517 1.8612 0.4580 −0.2876

(0.326) (0.405) (0.000)^∗ (0.000)^∗ (0.001)^∗ (0.080)

estimates are significant at 0.05 level in the LLM (1). Hence, it is unreasonable in this situation. The coefficient test of the firm’s trading age is not significant in each of the DSM (19) and LLM (1).

Table 8 shows the maximum likelihood estimates of the parameters in the DSM (19) and the LLM (1), similar to Table 7, but with Zmijewski’s predictors instead of Altman’s predictors. By characteristics of Zmijewski’s predictors, the larger the values of NI/TA and CA/CL, and the smaller the value of TL/TA, the smaller the probability of financial distress. Combining the result and the fact that the logistic function is strictly increasing, the coefficient estimates of the two predictors, NI/TA and CA/CL, should be negative and that of TL/TA should be positive. From Table 8, the coefficient estimates of the predictors, NI/TA and TL/TA , are negative and positive, respectively, in the DSM (19). The same remark is also made for the LLM. Those of the predictors CA/CL are positive in both the DSM (19) and LLM (1). The test of coefficient estimate of the predictor CA/CL is not significant at 0.05 level in the DSM (19). Therefore, the resulting coefficient estimates of Zmijewski’s predictors in the DSM (19) are reasonable.

On the other hand, the test of coefficient estimate of the predictor CA/CL is significant at 0.05 level in the LLM (1). Hence, it is unreasonable in this situation. The coefficient test of the firm’s trading age is not significant in each of the DSM (19) and LLM (1).

In this illustration, the selected optimal cutoff value ˆp^∗was taken as the minimizer of

Table 8: The estimated values of parameters in each of the DSM and the LLM using our discrete-time survival data with Zmijewski’s predictors. A z statistic was given to test the significance of the value of each parameter. The value given in the parenthesis stands for the p-value of the corresponding z test. Each value marked by^∗ denotes that the corresponding coefficient is significant at 0.05 level.

prediction

model log(t) NI/TA TL/TA CA/CL DSM 0.0009 −0.2421 0.0704 0.0522

(0.985) (0.000)^∗ (0.146) (0.267) LLM −0.0223 −1.5164 0.6676 0.3827 (0.874) (0.000)^∗ (0.000)^∗ (0.006)^∗

Table 9: The selected optimal cutoff values ˆp^∗ obtained by applying each of the DSM and the LLM to our discrete-time survival data with Altman’s predictors, and those with Zmijewski’s predictors.

predictors DSM LLM Altman 0.5281 0.4808 Zmijewski 0.5476 0.5621

of in-sample type I error rate. The type I and II error rates have been introduced in Section 1.8. For each of the DSM (19) and LLM (1), Table 9 shows the selected optimal cutoff values ˆp^∗ using the discrete-time survival data with each set of Altman’s and Zmijewski’s predictors.

In order to compare the financial distress prediction performance of LLM (1) and DSM (19), the 220 healthy companies in the sampling period were used to predict their financial status in the sample period. These companies were called the out-of-sample companies. The out-of-out-of-sample period was taken as the one during January of the year 2000 to December of the year 2002. Table 10 presents their industry and financial status in the out-of-sample period.

The last annual values of predictive variables in the out-of-sample period were col-lected for the out-of-sample companies. These data was called the out-of-sample data.

Table 10: The information about industry and financial status of out-of-sample

The out-of-sample data is expressed as

(tk, ˜Yk, xk,tk), for k = 1, ..., n0.

Here n0 = Pn i=1

(1− Yⁱ) is the number of the out-of-sample companies, n is the number of the in-sample companies, and tk denotes the duration time of the k-th out-of-sample company in the out-of-sample period. Also, at the duration time tk, ˜Yk = 0 indicates that the k-th out-of-sample company is healthy, and 1 the k-th out-of-sample company is of financial distress. Further, xk,tk denotes the values of Altman’s or Zmijewski’s predictors collected at the duration time tk.

Given each set of Altman’s and Zmijewski’s predictors, the financial distress pre-diction performance of the LLM (1) and that of the DSM (19) were compared on their out-of-sample error rates. Given each set of Altman’s and Zmijewski’s predictors, the

out-of-sample error rates of the DSM were calculated as follows. First, use the maxi-mum likelihood estimates ˆψof parameters ψ in Tables 7 and 8 for the DSM to calculate the predicted probability of financial distress for the company with predictor values (tk, xk,tk), for each k = 1, ..., n0. Second, use the resulting predicted probability of financial distress to compare with the selected optimal cutoff value ˆp^∗ in Table 9 for the DSM.

The k-th company with predictor values (tk, xk,tk) is classified as a financial distress company if

ˆhk = h(tk, xi,tk; ˆψ) = exp{ˆα0+ ˆα1 log(tk) + ˆβ xi,tk}

1 + exp{ˆα0+ ˆα1 log(tk) + ˆβ xi,tk} > ˆp^∗,

otherwise a healthy company. Finally, the three out-of-sample error rates of the DSM corresponding to each set of Altman’s and Zmijewski’s predictors defined by

αDSM =

were computed. Here αDSM is the error rate of misclassifying the financial distress companies to healthy ones, and β_DSM is the error rate of misclassifying the healthy companies to financial distress ones for the prediction rule DSM.

The same computational procedures were also applied to the prediction rule based on the LLM (1). See Section 1.3 for a detailed introduction of the computational procedures for the LLM (1). Let αLLM, β_LLM, and τLLM be similarly defined as the out-of-sample error rates for the LLM (1) corresponding to each set of Altman’s and Zmijewski’s predictors.

Given each set of Altman’s and Zmijewski’s predictors, Table 11 presents the finan-cial distress prediction performance of the DSM (19) and that of the LLM (1). Using Zmijewski’s predictors, Table 11 shows that the performance of the DSM (19) is nearly

Table 11: The out-of-sample error rates obtained by applying each of the DSM and the LLM to our discrete-time survival data with Altman’s predictors, and those with Zmijewski’s predictors.

equal to that of the LLM (1). But, using Altman’s predictors, Table 11 shows that the performance of the DSM (19) is much better than that of the LLM (1). Hence, from Table 11, to predict the financial distress for the traditional companies listed in Taiwan Stock Exchange, we suggest using the DSM (19) with Altman’s predictors.

3.4 Discussion

In this chapter, the prediction of financial distress based on the DSM (19) are proposed for the discrete-time survival data collected in Taiwan. The DSM (19) has the advantage of using all available historical information to determine each firm’s bankruptcy risk at each point in time, and it is a dynamic forecasting model.

The maximum likelihood method is employed to estimate the values of parameters ψ of the DSM (19), and the resulting estimators ˆψ are asymptotically normally dis-tributed as ˆψ ≈ N{ψ, Iⁿ(ψ)⁻¹}, where Iⁿ(ψ) = (−1)En the asymptotic normal distribution of ˆψ with its estimate In(ˆψ)⁻¹, the resulting dis-tribution can be used to derive the confidence interval estimate and test the values of ψ.

To decide the optimal prediction rule, we propose to define the optimal cutoff value ˆ

p^∗ as the minimizer of the sum of in-sample type I and type II error rates without restriction on the magnitude of the in-sample type I error rate. Based on our real data example, our DSM (19) performs better than LLM (1), in the sense of having much

smaller and having almost equal out-of-sample total error rates using Altman’s and Zmijewski’s predictors, respectively. In conclusion, it is better for the financial distress prediction of the traditional companies listed in Taiwan Stock Exchange to use the DSM (19) with Altman’s predictors.

Although the DSM (19) has the above advantages, it has some practical drawbacks.

Firstly, it needs to collect each firm’s time-series data. However, such time-series data may be incomplete and some ad hoc imputation methods are frequently employed.

For example, Shumway (2001) suggested substituting variable values from past years for the missing values in the cases when the explanatory variables are not completely observed. Secondly, there will be a problem encountered in economical structure change.

Grice and Dugan (2001) showed that using the prediction models to time periods other than those used to develop the models may result in significant decline in prediction accuracies. Finally, the hazard function employed by the DSM (19) is a linear logistic function, which is not robust with respect to the misspecification of the linear relation.

This is the same problem happened to the LLM (1). Hence, to avoid such drawback to the DSM (19), one remedy is to consider the hazard function as a semiparametric logistic function

h(t, x, z) = exp{H(t, x) + θ z}

1 + exp{H(t, x) + θ z},

as discussed in Section 1.4 and Chapter 2. Here H(t, x) is an unknown, but smooth function of (t, x). It will be interesting to study the financial distress prediction per-formance for the model combining both the SLM and DSM.

CHAPTER IV CONCLUSIONS

Ohlson (1980) proposed the LLM to predict bankruptcy. The LLM postulates that the logit function of bankruptcy probability is a linear function of the predictors. The

Ohlson (1980) proposed the LLM to predict bankruptcy. The LLM postulates that the logit function of bankruptcy probability is a linear function of the predictors. The