Methodology

3.2.1 Proxy for the firm’s financial condition: Default Likelihood Indicators

Vassalou and Xing (2004) is the first research of using Merton’s model to calculate default measures for individual firms and appraise the effect of default risk on equity returns. The conception of Merton (1974) is that the equity of a firm is thought of as a call option on the firm’s assets and then one can estimate the value of equity by using Black and Scholes (1973) formula. Our procedure in calculating default risk measures using Merton’s option pricing model is similar to the one used by KMV.

8 Because little information for 423 firms is not obtained on COMPUSTAT we lose a small number of stocks each month when merging the TAQ database with the variables calculated from

COMPUSTAT.

9 Because little information for 423 firms is not obtained on CRSP we lose a small number of stocks each month when merging the TAQ database with the variables calculated from CRSP.

We assume the capital structure of the firm including equity and debt. The market value of a firm’s underlying assets follows a Geometric Brownian Motion (GBM) of the form:

dV_A=µV_Adt+σ_AV_AdW (1)

Where V_A is the firm’s value with an instantaneous drift µ and an instantaneous volatility σ_A. A standard Wiener process is W.

Let be the book value of the debt at the time t, that has maturity equal to T.

is used as the strike price of a option, since the market value of equity, , can be regard as a call option on with time to expiration equal to T. By the Black and Scholes model for call options, the market value of equity will be estimated:

Xt

Xt V_E

VA

V_E =V_AN(d₁)− Xe^−rTN(d₂) (2) where

d d T

T

T r

X V

d _A

A

A

A σ

σ

σ = −

+

= + _{2 1}

2

1 ,

2 ) ( 1 ) / ln(

(3) r is the risk-free rate, and N (．) is the cumulative density function of the standard normal distribution.

An iterative procedure is adopted to estimate σ_A and then to estimate daily value of V_A for each sample month. As follow steps:

Step1) Daily data from the past 12 months will be using to secure an estimate of the volatility of equityσ_E, which is then viewed as an initial value for the estimation of σ_A, σˆ_A(1).

Step2) Using the Black and Scholes formula Eq.(2), we obtain using as the market value of equity of that day for each trading day of the past 12

VA V_E

months. In this mode, we capture daily values of for the past 12 months.

VA

Step3) We then compute the standard deviation of those from the last step, which is used as the value of

VA

σA,σˆ_A(2), for the next iteration.

Step4) Step2) to Step3) are repeated to obtain σˆ_A(3),σˆ_A(4),…et al., until the values of σ_A from two consecutive iterations converge. Our tolerance level for convergence is 10E-4.

Step5) Once the converged value of σ_A is obtained, we use it to back out daily values of V_A for month t through Eq.(2).

The above procedure is repeated at the end of every month, resulting in the estimation of monthly values of σ_A. The estimation window is always kept equal to 12-months. The risk-free rate used for each monthly iterative process is the 1-year T-bill rate observed at the end of the month. Once daily values of for month t are estimated, we can compute the drift µ, by calculating the annual compound return rate of asset value, . The default measure is the probability that the market value of firm’s assets will be less than the book value of the firm’s liabilities as maturity T

VA

) / log(V_A,t−1 V_A,t−q

10. In other words,

P_{def ,t} = Pr(V_A,t+T ≤ X_tV_A,t) = Pr(ln( V_A,t+T) ≤ ln( X_t)V_A,t) (4)

Because the value of the assets follows the GBM of Eq.(1), the value of the assets at any time t is given by:

10 We use the “Debt in One Year” plus the half of “Long-Term Debt” as book value of debt.

)

Hence, we can get the default probability as follows:

)

Distance to default (DD) could be defined as follows:

[

^ln( , ^{) (} 12 ²⁾T

]

T ⁽⁸⁾ DD_t = ^VAt_Xt +

µ

−

σ

_A

σ

_A

When the ratio of the value of assets to debt is less than 1 or its log is negative, default occurs. The DD tells us by how many standard deviations the log of this ratio needs to deviate from its mean in order for default to occur. Pay careful attention to although the value of the call option in Eq.(2) does not depend on µ, DD does, resulting from DD depends on the future value of assets which is given in Eq.(3).

Using the normal distribution implied by Merton’s model, the theoretical¹¹ probability of default, called Default Likelihood Indicators (DLI), will be given by:

)

11 Strictly speaking, Pdef is not a default probability because it does not correspond to the true probability of default in large samples. Thus Vassalou and Xing (2004) do not call that measure default probability, but rather DLI.

DLI increases as (1) the value of debt raises, (2) the market value of equity as well as assets goes down, and (3) the assets volatility increases. Comparatively, DLI is superior to other approaches using accounting financial information to measure default risk. DLI has a range of strengths as follows: (1) it has strong theoretical underpinnings; (2) default likelihood indicator takes into account the volatility of a firm’s assets; (3) it is forward looking based on stock market data rather than historic book value accounting data.

3.2.2 Regressions with control variables

The important factors of effect percentage bid-ask spreads have been approved by numerous earlier papers (e.g. Benston and Hagerman (1974), Tinic and West (1972) and Stoll (1978)). Besides stock price, the spread is influenced by some factors such as trading volume, variance of stock returns, market value of equity, and even the structure of exchange market. The declining prices of poorly performing firms will make percentage bid-ask spreads to rise. In particular, volatility of stock returns, which proxies for informed trading (Black, 1986), may rise and increase in spreads. It is also possible that although trading by informed investors rises, trading volume in general will decline because of fewer investors in the market, hence higher spreads ensued. For the purpose of examining the influence of the firm’s default risk on the bid-ask spreads clearly, we recommend controlling for these determinants. Under controlling variables in our regressions and find that the increase in spreads is still directly linked to the firm’s financial condition. Every variable is monthly average data except for DLI. DLI is default probability of the first trading day for each month. That implies that DLI forecasts PSP in ex ante base.

Following most of the literature, we employ log regression. Consequently, our

cross-sectional regression has the following form:

) 10 (

5 4

3 2

1 it it it it it it

it a b logCP b logNT b SIG b logMV bDLI

PSP = + + + + + +ε

Based on the results of past literature, we suppose b1, b2 and b4 should be negative, and b3 be positive. Because higher default risk implies financial condition exacerbation, b5 should be predictable positive according to Agrawal, et al., (2004).

Considering the heteroscedasticity of data, we use generalized least squares estimation (GLS) to estimate the coefficients of these explanatory variables in Eq.

(10) and test statistical significance.

3.2.3 Panel data models

The important motivation for using panel data is to solve the omitted variables problem. If the omitted variable is correlated with explanatory variables, we cannot consistently estimate parameters without additional information. It is uncertain whether the explanatory variable, DLI, is exogenous. If the test result shows that an omitted variable is presence, unobserved effects panel data model is one of ways to address the problem.

First of all, suppose that the omitted variable is not change over time. That means the omitted variable has the same effect on the mean response in each time period. An unobserved, time-constant variable is called an unobserved effect in panel data analysis. Moreover there are two different estimation methods, including random effects estimation and fixed effects estimation which be adopted in this

study. In modern econometric parlance, differing from traditional notion¹² to panel data models, “random effect” is synonymous with zero correlation between the observed explanatory variables and the unobserved effect. However, “fixed effect”

means that one is allowing for arbitrary correlation between the observed explanatory variables and the unobserved effect. The random effects approach estimates the coefficients under the assumption that the unobserved effect is orthogonal to explanatory variables. But this assumption is too strong to satisfy easily. On the other hand, fixed effects approach releases the strict exogeneity in addition to orthogonality between the observed explanatory variables and the unobserved effect in random effects approach. Thus fixed effects approach will more make sense for our nonexperimental panel data. More detailed discussion concerning fixed effects methods are as follows.

The linear unobserved effects model for T time periods will be written as:

y_it = x_itβ + c_i + u_it (11)

where the subscript i is cross section observation indexing individuals such as firms and subscript t indexes time. is 1 × k and can comprise observable variables that change across t as well as i, variables that change across t but not i, and variables that change across i but not t. presents unobserved effects that change across i but not t, thus also named individual effect. As what we have mentioned, the individual effect be viewed as correlation with the in fixed effect models.

The called idiosyncratic errors or idiosyncratic disturbances change across not only t but also i.

xit

ci

ci x_it

uit

12 In the traditional approach, “random effect” means unobserved component is treated as a random variable. In contrast, an unobserved component is treated as parameter to be estimated for each cross section observation I, called “fixed effect”.

The individual effect is better to be removed if the pooled OLS will be applied to estimate . The within transformation is one of those transformations that are able to accomplish the purpose. By first averaging Eq.(11) over t = 1,…,T the within transformation is obtained to get cross section eqation:

ci

β

y_i = x_iβ + c_i + u_i (12) .

T and

, T

, T

where y_i = ^-1Σ^T_t=1 y_it x_i = ^-1Σ^T_t=1 x_it u_i = ^-1Σ^T_t=1 u_it Next step, we take difference between Eq.(12) and Eq.(11) for each t to get the within transformation equation as Eq.(13). As a result, the time demeaning of original Eq.

(11) has eliminated the individual effect and the OLS estimator of fixed effected method in Eq.(13) can be obtained.

ci

y_it^∗ = x^∗_itβ_i + u_it^∗ (13) .

and ), (

,

where y_it^∗ = y_it − y_i x_it^∗ = x_it - x_i u_it^∗ = u_it − u_i Our sample is designed for balanced panels, so we took the subset of 276 stocks which are observed for the period 2001/02-2002/05.

3.2.4 Appropriate econometric techniques for threshold regression model with panel data

As so far, a regression equation estimates the identical slopes across all observations in a sample for each regressor. In reality, regression functions probably fall into discrete classes, called non-linear regression. This question may be addressed using threshold regression techniques. Threshold regression models express that individual observations can be divided into distinct classes by the value of an observed variable. This paper treats DLI as a threshold variable and our single

threshold regression equation has the following form:

where I(．) is the indicator function and γ₁ is value of threshold. PDLI is equal to time DLI by 100. In Eq.(14) the logPDLI is introduced in place of DLI in Eq.(10) because of the necessity to prevent the matrix from singular one. In the similar way, the double threshold regression equation is

it

where γ₂ is the other threshold value. The observations are dividend into two and three regimes respectively in Eq.(14) and Eq.(15). Both models have only the slope coefficient on logPDLI switch between regimes, because we can focus attention on this key variable of interest. Based on the results of past literature, b1, b2 and b4

should be negative. On the other hand, b3, b5, b6 and b7 should be positive.

In contrast to decide threshold levels arbitrarily, econometric techniques developed in Hansen (1999) appropriate for threshold regression with panel data are applied here. Hansen’s threshold regression method is suitable for non-dynamic panels with individual specific fixed effects and shows that the model is rather straightforward to estimate using a fixed-effects transformation. An asymptotic distribution theory is derived which is used to construct confidence intervals for the parameters. A bootstrap method to assess the statistical significance of the threshold effect is also described in Hansen (1999). GAUSS programs are modified to fit our

sample and regression model.

What we will continue to introduce is how to estimate thresholds in econometric techniques and test statistical significance of the threshold effect. We put threshold variable qit in one-rgressor regression model and structure equation of interest is given by

function. Similar to the procedure of fixed effects transformation introduced in previous section, within transformation is used to eliminate the individual effect

and now let Y

ci ^＊, X^＊ and u^＊ denote data stacked over all individuals. The threshold equation with fixed effects transformation could be written as

Y^∗ = β X^∗(γ) + u^∗ (17) The parameter β can be estimated by OLS for any given γ .

βˆ(γ )= (X^∗(γ )'X^∗(γ ))^-1X^∗(γ )'Y ^∗ (18) The regression residuals is

uˆ^∗(γ ) = Y^∗ - X^∗(γ )βˆ(γ )

Using least square to estimate γ is suggested by Chan(1993) and Hansen(1999). It is easiest to achieve by minimizing the sum of squared errors Eq.(17). Thus the least squares estimator of γ is

When the threshold γˆ is available, the coefficient of the regressor is , the residual is , and the residual variance is

) γˆ ˆ(

ˆ β

β= )

γˆ ˆ ( ˆ^∗= u^∗

u σˆ² = S₁(γˆ) n(T -1) .

It is very important to identify whether the threshold effect has observable influence on coefficient estimator. In other words, it is necessary to test the hypothesis of no threshold effect in Eq.(16). The null hypothesis is set up as

2 1 0 :

H β =β .

The null hypothesis is tested by likelihood ratio test. Firstly, if the null hypothesis is hold, the model is

it i it

it x β c u

y = ₁ + +

And then the equation with fixed effects transformation could be written as

∗

∗

∗ = _{it 1} + _it

it x β u

y

The parameter is able to be estimated by OLS, generating coefficient β₁ β^~₁, residuals u~_it^∗ and SSE S₀ = uˆ^∗'uˆ^∗. The likelihood ratio test of H₀ is based on

2 1

0

1 (S S (γˆ)) /σˆ

F = −

Hansen(1996) advanced a bootstrap to simulate the asymptotic distribution of likelihood ratio test. The bootstrap estimates the asymptotic p-value for F1 under

. If p-value is less than the desired critical value, the null hypothesis of no threshold effect is rejected.

H0

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