Predicting Recurrent Financial Distresses with Autocorrelation Structure: An Empirical Analysis from an Emerging Market

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 correlations among the observations from the same firm, we consider two different autocorrelation structures: exchangeable and first-order autoregressive (AR1). The prediction 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.

Appendix: A computational procedure for finding the solution \( \left( {{{\hat{\alpha }}_G},{{\hat{\beta }}_G}} \right) \) of GEE

The value of $\left( {{{\hat{\alpha }}_G},{{\hat{\beta }}_G}} \right)$ can be computed using the Fisher-scoring algorithm. Set $\theta = {\left( {\alpha, \beta } \right)^T}$. Given a starting value ${\hat{\theta }_0}$ for θ, iterate

$$ {\hat{\theta }_{m + 1}} = {\hat{\theta }_m} + {\left\{ {\sum\limits_{i = 1}^n {{D_i}{{\left( {{{\hat{\theta }}_m}} \right)}^T}{G_i}{{\left( {{{\hat{\theta }}_m},{{\hat{\rho }}_m}} \right)}^{ - 1}}{D_i}\left( {{{\hat{\theta }}_m}} \right)} } \right\}^{ - 1}}\left[ {\sum\limits_{i = 1}^n {{D_i}{{\left( {{{\hat{\theta }}_m}} \right)}^T}{G_i}{{\left( {{{\hat{\theta }}_m},{{\hat{\rho }}_m}} \right)}^{ - 1}}\left\{ {{Y_i} - {p_i}\left( {{{\hat{\theta }}_m}} \right)} \right\}} } \right], $$

until ${\hat{\theta }_{m + 1}} = {\hat{\theta }_m} \equiv {\left( {{{\hat{\alpha }}_G},{{\hat{\beta }}_G}} \right)^T}$ and ${\hat{\rho }_{m + 1}} = {\hat{\rho }_m} \equiv \hat{\rho }$. Here the nuisance parameter ρ in the m-th iteration is estimated by ${\hat{\rho }_m} = {\left\{ {\sum\limits_{i = 1}^n {\left( {{t_i} - {s_i}} \right)\left( {{t_i} - {s_i} + 1} \right)/2} - d - 1} \right\}^{ - 1}}\sum\limits_{i = 1}^n {\sum\limits_{j = {s_i}}^{{t_i} - 1} {\sum\limits_{k = j + 1}^{{t_i}} {\hat{e}_{i,j}^{(m)}\hat{e}_{i,k}^{(m)}} } }$ for the exchangeable structure, ${\hat{\rho }_m} = {\left\{ {\sum\limits_{i = 1}^n {\left( {{t_i} - {s_i}} \right)} - d - 1} \right\}^{ - 1}}\sum\limits_{i = 1}^n {\sum\limits_{j = {s_i}}^{{t_i} - 1} {\hat{e}_{i,j}^{(m)}\hat{e}_{i,j + 1}^{(m)}} }$ for the AR1 structure, $\hat{e}_{i,j}^{(m)} = {\left[ {\hat{p}_{i,j}^{(m)}\left\{ {1 - \hat{p}_{i,j}^{(m)}} \right\}} \right]^{ - 1/2}}\left( {{Y_{i,j}} - \hat{p}_{i,j}^{(m)}} \right)$, and $\hat{p}_{i,j}^{(m)} = {p_{i,j}}\left( {{{\hat{\theta }}_m}} \right)$. Liang and Zeger (1986) suggest taking the starting value ${\hat{\theta }_0}$ as the maximum likelihood estimate of θ produced in subsection 2.1 under the independence assumption.