Multivariate t-Distribution Estimation

The MODEL Procedure

Multivariate t-Distribution Estimation

The multivariate t-distribution is specified using the ERRORMODEL statement with the T option. Other method specifications ( FIML and OLS, for example ) are ignored when the ERRORMODEL statement is used for a distribution other than normal.

The probability density function for the multivariate t-distribution is

$P_q=\frac{\Gamma( \frac{df +m}2 ) }{ (\pi* df)^{\frac{m}2} * \Gamma( \frac{df... ...(y_{t} \hspace*{1pt}, x_{t}\hspace*{1pt}, {{\theta}})}{df} )^{-\frac{df+m}2 }$

where m is the number of equations and df is the degrees of freedom.

The maximum likelihood estimators of ${{\theta}}$ and ${{\sigma}}$ are the ${\hat{{\theta}}}$ and ${\hat{{\sigma}}}$ that minimize the negative log-likelihood function:

$l_{n}({{\theta}}, {{\sigma}}) &=& -\sum_{t=1}^n \ln (\frac{ \Gamma( \frac{df +... ...}2*\ln(|\Sigma|) - \sum_{t=1}^n \ln(|\frac{\partial q_t}{ \partial y'_t} | )$

The ERRORMODEL statement is used to request the t-distribution maximum likelihood estimation. An OLS estimation is done to obtain initial parameter estimates and MSE.var estimates. Use NOOLS to turn off this initial estimation. If the errors are distributed normally, t-distribution estimation will produce results similar to FIML.

The multivariate model has a single shared degrees of freedom parameter, which is estimated. The degrees of freedom parameter can also be set to a fixed value. The negative log-likelihood value and the l₂ norm of the gradient of the negative log-likelihood function are shown in the estimation summary.

t-Distribution Details

Since a variance term is explicitly specified using the ERRORMODEL statement, ${\Sigma}({\theta})$ is estimated as a correlation matrix and $q(y_{t}\hspace*{1pt}, x_{t}\hspace*{1pt}, {{\theta}})$ is normalized by the variance.

The gradient of the negative log-likelihood function with respect to the degrees of freedom is

$\frac{\partial l_n}{\partial df} &=& \frac{n m}{2 df} - \frac{n}2 \Gamma'( \fr... ...rac{0.5 (df + m)}{(1 +\frac{q'\Sigma^{-1} q}{df})} \frac{q'\Sigma^{-1} q}{df^2}$

The gradient of the negative log-likelihood function with respect to the parameters is

$\frac{\partial l_n}{\partial \theta_i}=\frac{0.5 (df + m)}{(1 +q'\Sigma^{-1}... ... - \frac{n}2 {\rm trace}( \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_i})$

where

$\frac{{\partial}{\Sigma}({\theta})}{{\partial} {\theta}_{i}}=\frac{2}n \sum_{... ...t}\hspace*{1pt}, x_{t}\hspace*{1pt}, {{\theta}})'} {{\partial} {\theta}_{i}} }$

and

$q(y_{t}\hspace*{1pt},x_{t}\hspace*{1pt}, {{\theta}})=\frac{ \epsilon({\theta}) }{\sqrt{h({\theta})}} \in R^{m x n}$

The estimator of the variance-covariance of ${\hat{{\theta}}}$ (COVB) for the t-distribution is the inverse of the likelihood Hessian. The gradient is computed analytically and the Hessian is computed numerically.

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