Financial disclosure, investor protection and stock market behavior: an international comparison

Abstract

The main purpose of this study is to examine the relationship between financial opacity, investor protection and stock market behavior for sixteen countries. We use the 1995 CIFAR corporate disclosure ratings and the 2006 World Bank investor protection index to measure a country’s relative level of financial transparency and legal protection for investors. The return behavior of each country is examined using numerous time series tests such as serial correlation, Markov chain, runs, duration dependence and variance ratio tests. We found that the results show no significant differences between high and low disclosure countries. However, high disclosure countries appear to be associated with a lower level of stock market volatility. Cox proportional hazard test results indicate that extreme returns (positive and negative) are more likely in low disclosure countries.

Technical appendix

1.1 Non-parametric runs test

The runs test examines the likelihood of sequential dependence among series of the same type, such as an increase or decrease (run up and down) without requiring normality of the return distribution. For a “run up”, a positive sign is assigned if the return in a sequence is greater than the preceding return, whereas a negative sign is assigned if the return in the sequence is smaller than the preceding return.

The Z-statistic for the null hypothesis that the real return for a particular country is random, implying weak-form efficiency, is computed as follows:

$$ Z = {\frac{M - E(M)}{{\sigma^{2} (M)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} }}}, $$

(1)

where M is the actual numbers of runs and E(M) is the expected number of runs which is calculated as:

$$ E(M) = {\frac{(2N - 1)}{3}}, $$

(2)

where N is the number of observation and σ ²(M) is the variance of M runs which is computed as:

$$ \sigma^{2} (M) = {\frac{(16N - 29)}{90}}. $$

(3)

The Z test statistic has a standard normal distribution.

1.2 Duration dependence test

The duration dependence methodology developed by McQueen and Thorley (1994) examines the hazard rate (h _i) for positive and negative runs. The hazard rate is defined as the probability of obtaining a negative return (ε _t  < 0) given a sequence of i prior positive returns (ε _t−i  > 0). The hazard rate, in the presence of negative duration dependence,

$$ h_{i} = {\text{Prob}}(\varepsilon_{{t}} < \left. 0 \right|\varepsilon_{{{{t}} - 1}} > 0, \, \varepsilon_{{{{t}} - 2}}> 0 \ldots ,\varepsilon_{{{{t}} - {{i}}}} > 0, \,\varepsilon_{{{{t}} - {{i}} - 1}} < 0),{\text{ decreases with }}i.$$

(4)

To apply the duration dependence test, the stock returns of each country are transformed into run lengths of positive and negative returns. The hazard function (h _i ), defined as h _i  ≡ Prob(I = i∣I ≥ i), represents the probability that a specific run ends at length i, provided that it lasts until length i. Specifically, the presence of a bubble indicates that the hazard rates for runs of positive return should decrease as the length of the run increases, h _i+1 < h _i for all i.¹⁶ The log likelihood of the hazard function is defined as follows:

$$ {\text{L(}}\left. \theta \right|{\text{S}}_{\text{T}} )= \sum\limits_{i = 1}^{\alpha } {N_{i} {\text{Ln }}h_{i} + M_{i} {\text{Ln}}\left( { 1- h_{i} } \right) + Q_{i} {\text{Ln}}\left( { 1- h_{i} } \right)} . $$

(5)

The functional form for the hazard function is based on the logistic transformation of the log of i.

$$ h_{i} = {\frac{1}{{1 + e^{ - (\alpha + \beta \,Lni)} }}}, $$

(6)

where N _i is the number of completed runs of length i in the sample. M _i and Q _i are the numbers of completed and partial runs with length greater than i. The sample estimated hazard rate$(\hat{h}_{i} )$ for length i is derived from maximizing the log likelihood function in (5) which yields

$$ \hat{h}_{i} = {\frac{{N_{i} }}{{(M_{i} + N_{i} )}}}. $$

(7)

The null hypothesis of no bubble or a constant hazard rate requires that β = 0. The alternative hypothesis of a bubble suggests that the probability of a run of positive return ending should decrease with the length of the run (i.e. a decreasing hazard rate, β < 0).

Duration dependence tests are performed by maximizing the log likelihood function with respect to α and β. The hazard function is estimated as a Logit regression in which the independent variable is the log of the current run length and the dependent variable is 1 if the run ends in the next period and 0 if it does not. Under the null hypothesis of no bubble (β = 0), the likelihood ratio test (LRT) is asymptotically distributed χ² with one degree of freedom.

$$ {\text{LRT}} = 2\left[ {{\text{Log Unrestricted}} - {\text{Log Restricted}}} \right]\sim \chi_{1}^{2} . $$

(8)

1.3 Third-order Markov chain

Under the Markov chain technique, the random walk process requires that transition probabilities from one state to another are equal, regardless of the prior return sequence. If the returns have unequal transition probabilities then it is likely that the return series follow a non-random walk pattern.

To test the random walk hypothesis, a two-state Markov chain process {I _t } can be formulated by letting I = 1 if the returns are positive (R _t  > 0) and I = 0 if the returns are negative (R _t  < 0). Therefore, the finite Markov process {I _t } can be defined as follows:

$$ I_{t} = \left\{ \begin{gathered} 1\quad if \quad R_{t} > 0 \\ 0\quad if \quad R_{t} \le 0 \\ \end{gathered} \right. $$

(9)

If the positive and negative returns are random, the probability of observing positive or negative returns should not depend on the prior sequence. To test for the randomness of the returns series, the transition counts and transition probabilities are formed based on the behavior of process {I _t }. For a two-state third-order Markov chain, the transition counts and transition probabilities vary depending on the returns of three prior states. The transition counts (N _ijk and M _ijk ) and probabilities (λ _ijk ) are defined as N ₀₀₀ represents the number of observations in state 0 0 0 0. M ₀₀₀ represents the number of observations in state 0 0 0 1. For example, the transition probabilities (λ ₀₀₀ ) can be defined as follows:

$$ \lambda_{000} = Prob\left[ {I_{\text{t}} = \left. 0 \right|I_{{{\text{t}} - 3}} = 0, \, I_{{{\text{t}} - 2}} = 0, \, I_{{{\text{t}} - 1}} = 0} \right], $$

(10)

where λ ₀₀₀ is the probability that a negative return will persist in the current period given a sequence of three negative returns and therefore (1 − λ ₀₀₀ ) is the probability that a sequence of three negative returns will revert to a positive return in the current period. If the returns series are random or symmetric, the probability of observing a negative or positive return in the current period should be similar and independent of what happened in the prior states. The null hypothesis of a random walk is tested by imposing the restriction on transition probabilities:

$$ {\text{Null Hypothesis 1: }}\lambda_{000} = \lambda_{111} , $$

(11)

$$ {\text{Null Hypothesis 2: }}\lambda_{000} = \lambda_{001} = \lambda_{010} = \lambda_{011} = \lambda_{100} = \lambda_{101} = \lambda_{110} = \lambda_{111} . $$

(12)

The maximum likelihood estimators λ _ijk ,

$$ \hat{\lambda }_{ijk} = {\frac{{N_{ijk} }}{{(N_{ijk} + M_{ijk} )}}}, $$

(13)

and their asymptotic variances,

$$ \sigma^{2} ( \hat{\lambda }_{ijk} ) = {\frac{{\hat{\lambda }_{ijk} (1 - \hat{\lambda }_{ijk} )}}{{N_{ijk} + M_{ijk} }}}. $$

(14)

are derived from the following log likelihood function

$$ L\left( {S_{T} ,{\Uplambda}^{\prime} ,\pi_{0} } \right) = \log \pi_{0} + \sum\limits_{ijk = ooo}^{111} {N_{ijk} \log \lambda_{ijk} + M_{ijk} \log \left( {1 - \lambda_{ijk} } \right)} . $$

(15)

Given the estimates of the transition probabilities, the likelihood ratio test (LRT) is used to test the null hypothesis of equal transition probabilities. The LRT is

$$ {\text{LRT}} = 2\left[ {{\text{Log Unrestricted}}-{\text{Log Restricted}}} \right]\sim \chi_{n}^{2} . $$

(16)

1.4 Variance ratio test

The variance ratio test developed by Cochrane (1988) has been employed in numerous studies as an alternative test of the random walk hypothesis (see for example, Lo and MacKinlay 1988; Huang 1995; Long et al. 1999). Cochrane (1988) showed that if the data generating process is a random walk, the variance of k-period return should be equal to k times the variance of the one-period return. Therefore, under the null hypothesis of a random walk process, the expected value of the variance ratio (VR(k)) for all lags k should be unity. The variance ratio is defined as:

$$ VR(k) = 1 + 2\sum\limits_{j = 1}^{k} {\left[ {1 - {\frac{j}{(k + 1)}}} \right]\rho (j).} $$

(17)

where ρ(j) is the sample autocorrelation coefficient of the one period return at lag j and k is the number of autocorrelations used in the calculation of VR(k). A VR(k) significantly lower (higher) than one indicates the presence of negative (positive) serial correlation in the return series.

Under the assumption of homoscedasticity, the asymptotic variance of VR(k) statistic can be defined as follows:

$$ \Upphi_{k} = [2(2k - 1)(k - 1)]/3kT, $$

(18)

where T is the sample size. With the assumption of homoskedasticity, the VR(k) statistics under the null hypothesis of a random walk have a normal distribution. The Z(k) test statistic is estimated as:

$$ Z(k) = {\frac{VR(k) - 1}{{[\Upphi (k)]^{1/2} }}}\sim N(0,1). $$

(19)