|
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 [pp. 8–22]. Reprinted in J. Eatwell ''et al.'', eds. (1990). ''Econometrics: The New Palgrave''p. 1[pp. 1–34].Abstract (The New Palgrave Dictionary of Economics, 2008 revision by J. Geweke, J. Horowitz, and H. P. Pesaran). More precisely, it is "the quantitative analysis of actual economic Phenomenon, phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference". An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships". Jan Tinbergen is one of the two founding fathers of econometrics. The other, Ragnar Frisch, also coined the term in the sense in which it is used toda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probit Model
In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from ''probability'' + ''unit''. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model. A probit model is a popular specification for a binary response model. As such it treats the same set of problems as does logistic regression using similar techniques. When viewed in the generalized linear model framework, the probit model employs a probit link function. It is most often estimated using the maximum likelihood procedure, such an estimation being called a probit regression. Conceptual framework Suppose a response variable ''Y'' is ''binary'', that is it can have only two possible outcomes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siddhartha Chib
Siddhartha Chib is an econometrician and statistician, the Harry C. Hartkopf Professor of Econometrics and Statistics at Washington University in St. Louis. His work is primarily in Bayesian statistics, econometrics, and Markov chain Monte Carlo methods. Key papers include Albert and Chib (1993) which introduced an approach for binary and categorical response models based on latent variables that simplifies the Bayesian analysis of categorical response models; Chib and Greenberg (1995) which provided a derivation of the Metropolis-Hastings algorithm from first principles, guidance on implementation and extensions to multiple-block versions; Chib (1995) where a new method for calculating the marginal likelihood from the Gibbs output is developed; Chib and Jeliazkov (2001) where the method of Chib (1995) is extended to output of Metropolis-Hastings chains; Basu and Chib (2003) for a method for finding marginal likelihoods in Dirichlet process mixture models; Carlin and Chib (1995) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latent Variable
In statistics, latent variables (from Latin: present participle of ''lateo'', “lie hidden”) are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or measured. Such ''latent variable models'' are used in many disciplines, including political science, demography, engineering, medicine, ecology, physics, machine learning/artificial intelligence, bioinformatics, chemometrics, natural language processing, management and the social sciences. Latent variables may correspond to aspects of physical reality. These could in principle be measured, but may not be for practical reasons. In this situation, the term ''hidden variables'' is commonly used (reflecting the fact that the variables are meaningful, but not observable). Other latent variables correspond to abstract concepts, like categories, behavioral or mental states, or data structures. The terms ''hypothetical variables'' or ''hypothetical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, maximizing a likelihood function so that, under the assumed statistical model, the Realization (probability), observed data is most probable. The point estimate, point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference. If the likelihood function is Differentiable function, differentiable, the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes the likelihood when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivariate Normal Distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be ''k''-variate normally distributed if every linear combination of its ''k'' components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Definitions Notation and parameterization The multivariate normal distribution of a ''k''-dimensional random vector \mathbf = (X_1,\ldots,X_k)^ can be written in the following notation: : \mathbf\ \sim\ \mathcal(\boldsymbol\mu,\, \boldsymbol\Sigma), or to make it explicitly known that ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
