Motivation

Want to model the loss distribution of b banks.

Motivation

Want to model the loss distribution of b banks.

Motivation

Want to model the loss distribution of b banks.

Focus on \(Y_{it}\).

First Idea

Assume conditional independence and e.g. let the default intensity be

\[\log\lambda_{it} = \vect\beta^\top \vect x_{it} + \vect\gamma^\top \vect z_t\]

Add Frailty

Duffie et al. (2009) suggest to generalize to

\[ \begin{aligned} \log\lambda_{it} &= \vect\beta^\top \vect x_{it} + \vect\gamma^\top \vect z_t + A_t \\ A_t &\sim \theta A_{t-1} + \epsilon_t \\ \epsilon_t&\sim N(0,\sigma^2) \end{aligned} \]

The auto-regressive frailty, \(A_k\), captures excess clustering.

Remarks

Generalize

\[ \begin{aligned} \log\lambda_{it} &= \vect\beta^{(1)\top}\vect x_{it}^{(1)} + \vect\gamma^\top \vect z_t + \vect\beta^{(2)\top} \vect f(\vect x_{it}^{(2)}) + \vect A_t^\top\vect u_{it} \\ \vect A_t &\sim F\vect A_{t-1} + \vect \epsilon_t \\ \vect \epsilon_t&\sim \vect N(\vect 0, Q) \\ \vect x_{it} &= \left(\vect x_{it}^{(1)\top}, \vect x_{it}^{(2)\top}\right)^\top \end{aligned} \]

\(\vect A_t \in \mathbb{R}^p\) is low dimensional and some elements in \(\vect u_{it}\) and \(\vect x_{it}\) may match.

Generalize

\[ \begin{aligned} \log\lambda_{it} &= \color{gray}{\vect\beta^{(1)\top}\vect x_{it}^{(1)} + \vect\gamma^\top \vect z_t} + \vect\beta^{(2)\top} \vect f(\vect x_{it}^{(2)}) + \vect A_t^\top\vect u_{it} \\ \color{gray}{\vect A_t} &\sim F \color{gray}{\vect A_{t-1} + \vect \epsilon_t} \\ \color{gray}{\vect \epsilon_t} &\color{gray}\sim \color{gray}{\vect N(\vect 0, }Q\color{gray}) \\ \color{gray}{\vect x_{it}} & \color{gray}= \color{gray}{ \left(\vect x_{it}^{(1)\top}, \vect x_{it}^{(2)\top}\right)^\top} \end{aligned} \]

\(\vect A_t \in \mathbb{R}^p\) is low dimensional and some elements in \(\vect u_{it}\) and \(\vect x_{it}\) may match.

Marginal Likelihood

\[ \begin{aligned} L &= \int_{\mathbb R^{pd}} \mu_0(\vect A_1)g_1\Cond{\vect y_1}{\vect A_1} \\ &\hspace{30pt}\cdot\prod_{t=2}^d g_t\Cond{\vect y_t}{\vect A_t} f\Cond{\vect A_t}{\vect A_{t-1}}\mathrm{d}A_{1:d} \\ \vect y_t &= \{y_{it}\}_{i\in \mathcal{O}_t} \end{aligned} \]

\(\mathcal{O}_t\) is the risk set and \(g_t\) is a conditional density.

Monte Carlo Method

Use Monte Carlo expectation maximization (EM).

Approximate E-step with a particle smoother.

Get arbitrary precision.

Particle Smoother

Implemented the generalized two-filter smoother suggested by Briers, Doucet, and Maskell (2009).

Features

Implemented in C++ and supports computation in parallel.

More Software

rollRegres package: fast rolling regression.

DtD package: fast estimation of the Merton model.

Summary

Add covariates, non-linear effects, and a random slope to model in Duffie, Saita, and Wang (2007) and Duffie et al. (2009).

Provide evidence of time-varying size slope.

Show improved firm-level performance and industry-level performance.

Data Source

Default data from Moody’s Default Risk Service Database.

Covariates from Compustat and CRSP.

Additions

• Working capital to size.
• Operating income / size.
• Market value / total liabilities.
• Net income / size.
• Total liabilities / size.
• Current ratio.
• Log relative market size.
• Idiosyncratic volatility.

Adding Time-Varying Coefficient

\[\begin{aligned} \vec z_{it} &= (\vec x_{it}^\top, \vec m_t^\top, u_{it}, \alpha_t, b_t)^\top \\ g(P(Y_{it} = 1 \mid \vec z_{it})) &= \vec \beta^\top\vec f(\vec x_{it}) + \vec\gamma^\top\vec m_t + \alpha_t + b_tu_{it} \\ \begin{pmatrix}\alpha_t \\ b_t \end{pmatrix} &= \begin{pmatrix}\theta_1 & 0 \\ 0 & \theta_2 \end{pmatrix} \begin{pmatrix}\alpha_{t-1} \\ b_{t-1} \end{pmatrix} + \vec\epsilon_t \\ \vec\epsilon_t & \sim N\left(\vec 0, \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}\right) \end{aligned}\]

Adding Time-Varying Coefficient

\[\begin{aligned} \color{gray}{\vec z_{it}} & \color{gray}= \color{gray}{(\vec x_{it}^\top, \vec m_t^\top, u_{it}, \alpha_t, b_t)^\top} \\ \color{gray}{g(P(Y_{it} = 1 \mid \vec z_{it}))} & \color{gray}= \color{gray}{\vec \beta^\top\vec f(\vec x_{it}) + \vec\gamma^\top\vec m_t} + \alpha_t + b_tu_{it} \\ \begin{pmatrix}\alpha_t \\ b_t \end{pmatrix} &= \begin{pmatrix}\theta_1 & 0 \\ 0 & \theta_2 \end{pmatrix} \begin{pmatrix}\alpha_{t-1} \\ b_{t-1} \end{pmatrix} + \vec\epsilon_t \\ \vec\epsilon_t & \sim N\left(\vec 0, \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}\right) \end{aligned}\]

Summary

Argued for random effects and non-linear effects.