Spatial Dependence in Subprime Mortgage Defaults

Abstract

We analyze the spatial default dependence between pairs of nonconforming securitized mortgages, originated in Los Angeles between 2000 and 2011 and clustered by zip code. Our approach allows us to estimate the range and shape of the spatial dependence function, which relates zip-code center-to-center distance between mortgages to the dependence parameter of a number of different copulas. We find significant evidence for the presence of spatial dependence, which decays to zero within 40km and can be well characterized by a squared exponential function, a special case of the Matérn spatial correlation function.

Appendix: Loglikelihood, Score and Hessian

This appendix develops the loglikelihood function, score and Hessian for the pairwise likelihood estimation of the bivariate default probability with copula. Define Y_i, i = 1, 2 to be Bernoulli variables with probability π_i of default for mortgage i, and $C(\bar {\pi }_{1},\bar {\pi }_{2};\theta )$ the copula for joint default, where $\bar {\pi }_{i}=1-{\pi }_{i}$. The loglikelihood is the sum of the contributions of all four possible outcomes:

$$ L=Y_{1} Y_{2} \log(p_{11})+(1-Y_{1}) Y_{2} \log(p_{01})+Y_{1} (1-Y_{2}) \log(p_{10})+(1-Y_{1}) (1-Y_{2}) \log(p_{00}), $$

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where we leave out the arguments $(\bar {\pi }_{1},\bar {\pi }_{2};\theta )$ and write $C \equiv C(\bar {\pi }_{1},\bar {\pi }_{2};\theta )$. In order to compute standard errors, we need to evaluate the score and Hessian. The score can be written as

$$ \frac{\partial L}{\partial \theta}=C_{\theta} \left( Y_{1} Y_{2} \frac{1}{p_{11}} -(1-Y_{1}) Y_{2} \frac{1}{p_{01}}- Y_{1} (1 - Y_{2}) \frac{1}{p_{10}}+ (1-Y_{1}) (1-Y_{2}) \frac{1}{p_{00}}\right), $$

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where $C_{\theta }=\frac {\partial C}{\partial \theta }$. Using the chain rule, the Hessian can be written as

$$ \begin{array}{cc} \frac{\partial^{2} L}{\partial \theta^{2}}= C_{\theta\theta}\left( Y_{1} Y_{2} \frac{1}{p_{11}}-(1-Y_{1}) Y_{2} \frac{1}{p_{01}}-Y_{1} (1-Y_{2}) \frac{1}{p_{10}}+ (1-Y_{1}) (1-Y_{2}) \frac{1}{p_{00}}\right)\\ -\left( C_{\theta}\right)^{2} \left( Y_{1} Y_{2} \frac{1}{p_{11}^{2}}+(1-Y_{1}) Y_{2} \frac{1}{p_{01}^{2}}+Y_{1} (1-Y_{2}) \frac{1}{p_{10}^{2}}+(1-Y_{1}) (1-Y_{2}) \frac{1}{p_{00}^{2}}\right), \end{array} $$

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where $C_{\theta \theta }=\frac {\partial ^{2} C}{\partial \theta ^{2}}$. We now provide expressions for C_𝜃 and C_𝜃𝜃 for each copula.

a. :

Farlie Gumbel Morgenstern (FGM)

$$ \begin{array}{lc} C_{\theta}=\bar{\pi}_{1}\bar{\pi}_{2} (1-\bar{\pi}_{1})(1-\bar{\pi}_{2})\\ C_{\theta\theta}=0 \end{array} $$

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All Archimedian copulas such as the Gumbel, Frank and Clayton, can be defined in terms of a generator function ϕ, as follows:

$$ \begin{array}{lc} C(\bar{\pi}_{1},\bar{\pi}_{2};\theta)=\phi^{-1}(\phi(\bar{\pi}_{1};\theta)+\phi(\bar{\pi}_{2};\theta)). \end{array} $$

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As a result, for all Archimedean copulas, the score and Hessian depend on their generator function:¹

$$ \begin{array}{lll} C_{\theta}&=& \frac{1}{\phi_{C}(C;\theta)}\left[\phi_{\theta}(\bar{\pi}_{1};\theta)+\phi_{\theta}(\bar{\pi}_{2};\theta)-\phi_{\theta}(C;\theta)\right]\\ C_{\theta\theta}&=&\frac{1}{\phi_{C}(C;\theta)}\left[\phi_{\theta\theta}(\bar{\pi}_{1};\theta)+\phi_{\theta\theta}(\bar{\pi}_{2};\theta)-\phi_{\theta\theta}(C;\theta)-\left[\phi_{CC}(C;\theta)C_{\theta}+2\phi_{C\theta}(C;\theta)\right]C_{\theta}\right] \end{array} $$

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We now provide expressions for the partial derivatives of the generator ϕ, that are needed for computation of the score and Hessian for each of our remaining copulas.

b. :

Gumbel

$$ \begin{array}{lll} \phi(t;\theta)&=& (-\log t)^{\theta}\\ \phi_{C}(t;\theta)&=& -\frac{\theta (-\log(t))^{{\theta} - 1}}t\\ \phi_{\theta}(t;\theta)&=& \log(-\log(t)) (-\log(t))^{\theta}\\ \phi_{CC}(t;\theta)&=& -\frac{\theta (-\log(t))^{\theta} (\log(t) - {\theta} + 1)}{t^{2} \log(t)^{2}}\\ \phi_{C \theta}(t;\theta)&=& -((-\log(t))^{{\theta} - t} (\theta \log(-\log(t)) + 1))/t\\ \phi_{\theta \theta}(t;\theta)&=& \log(-\log(t))^{2} (-\log(t))^{\theta} \end{array} $$

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c. :

Frank

$$ \begin{array}{lll} \phi(t;\theta)&=& -\log\frac{e^{\theta t}-1}{e^{\theta} - 1}\\ \phi_{C}(t;\theta)&=& -\frac{\theta}{e^{\theta t} - 1}\\ \phi_{\theta}(t;\theta)&=& \frac{e^{\theta t}-e^{\theta}}{(e^{\theta} - 1)(e^{\theta t} - 1)}\\ \phi_{CC}(t;\theta)&=& \frac{\theta^{2}}{4 \sinh^{2}(\theta t/2)}\\ \phi_{C \theta}(t;\theta)&=& \frac{e^{\theta t} (\theta t - 1) + 1}{(e^{\theta t} - 1)^{2}}\\ \phi_{\theta \theta}(t;\theta)&=& -\frac{e^{\theta} (e^{2 \theta t} + 1) - e^{\theta t} (t^{2} e^{2 \theta} - e^{\theta} (2 t^{2} - 2) + t^{2})}{(e^{\theta t} - 1)^{2} (e^{\theta} - 1)^{2}} \end{array} $$

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d. :

Clayton

$$ \begin{array}{lll} \phi(t;\theta)&=& \frac{1}{\theta}(t^{-\theta} - 1)\\ \phi_{C}(t;\theta)&=& -t^{-(\theta + 1)}\\ \phi_{\theta}(t;\theta)&=& - \frac{t^{-\theta}-1}{\theta^{2}}-\frac{1}{\theta}t^{-\theta}\log(t)\\ \phi_{CC}(t;\theta)&=& (\theta + 1)t^{-(\theta + 2)}\\ \phi_{C \theta}(t;\theta)&=& \log(t)t^{-(\theta + 1)}\\ \phi_{\theta \theta}(t;\theta)&=& \frac{\theta^{2} \log(t)^{2} - 2 t^{\theta} + 2 \theta \log(t) + 2}{t^{\theta} \theta^{3}} \end{array} $$

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