GHK Algorithm
The GHK algorithm (Geweke, Hajivassiliou and Keane) is an importance sampling method for simulating choice probabilities in the multivariate probit model. These simulated probabilities can be used to recover parameter estimates from the maximized likelihood equation using any one of the usual well known maximization methods (Newton's method, BFGS, etc.). Train has well documented steps for implementing this algorithm for a multinomial probit model. What follows here will applies to the binary multivariate probit model. Consider the case where one is attempting to evaluate the choice probability of \Pr(\mathbf , \mathbf, \Sigma) where \mathbf = (y_1, ..., y_J), \ (i = 1,...,N) and where we can take j as choices and i as individuals or observations, \mathbf is the mean and \Sigma is the covariance matrix of the model. The probability of observing choice \mathbf is : \begin \Pr(\mathbf, \mathbf, \Sigma) = & \int_\cdots\int_f_N(\mathbf^*_i, \mathbf, \Sigma) dy^*_1\do ...
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Importance Sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally attributed to a paper by Teun Kloek and Herman K. van Dijk in 1978, but its precursors can be found in statistical physics as early as 1949. Importance sampling is also related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both. Basic theory Let X\colon \Omega\to \mathbb be a random variable in some probability space (\Omega,\mathcal,P). We wish to estimate the expected value of ''X'' under ''P'', denoted E 'X;P'' If we have statistically independent random samples x_1, \ldots, x_n, generated according to ''P'', then an empirical estimate of E 'X;P''is : \widehat_ ;P= \frac \sum_^n x_i \quad \m ...
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Multivariate Probit Model
In statistics and econometrics, the multivariate probit model is a generalization of the probit model used to estimate several correlated binary outcomes jointly. For example, if it is believed that the decisions of sending at least one child to public school and that of voting in favor of a school budget are correlated (both decisions are binary), then the multivariate probit model would be appropriate for jointly predicting these two choices on an individual-specific basis. J.R. Ashford and R.R. Sowden initially proposed an approach for multivariate probit analysis. Siddhartha Chib and Edward Greenberg extended this idea and also proposed simulation-based inference methods for the multivariate probit model which simplified and generalized parameter estimation. Example: bivariate probit In the ordinary probit model, there is only one binary dependent variable Y and so only one latent variable Y^* is used. In contrast, in the bivariate probit model there are two binary dependent ...
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Newton's Method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions an ...
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Truncated Normal
In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics. Definitions Suppose X has a normal distribution with mean \mu and variance \sigma^2 and lies within the interval (a,b), \text \; -\infty \leq a < b \leq \infty . Then $X$ conditional on has a truncated normal distribution. Its , $f$, for $a\; \backslash leq\; x\; \backslash leq\; b$, is given by :$f(x;\backslash mu,\backslash sigma,a,b)\; =\; \backslash frac\backslash ,\backslash frac$ and by $f=0$ otherwise. Here, :$\backslash phi(\backslash xi)=\backslash frac\backslash exp\backslash left(-\backslash frac\backslash xi^2\backslash right)$ is ...
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