CrimeStatAppendix.C_Anselin

CrimeStatAppendix.C_Anselin - Appendix C Ordinary Least...

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C-1 Appendix C Ordinary Least Squares and Poisson Regression Models by Luc Anselin University of Illinois Champaign-Urbana, IL This note provides a brief description of the statistical background, estimators and model characteristics for a regression specification, estimated by means of both Ordinary Least Squares (OLS) and Poisson regression. Ordinary Least Squares Regression With an assumption of normality for the regression error term, OLS also corresponds to Maximum Likelihood (ML) estimation. The note contains the statistical model and all expressions that are needed to carry out estimation and essential model diagnostics. Both concise matrix notation as well as more extensive full summation notation are employed, to provide a direct link to “loop” structures in the software code, except when full summation is too unwieldy (e.g., for matrix inverse). Some references are provided for general methodological descriptions. Statistical Issues The classical multivariate linear regression model stipulates a linear relationship between a dependent variable (also called a response variable) and a set of explanatory variables (also called independent variables, or covariates). The relationship is stochastic, in the sense that the model is not exact, but subject to random variation, as expressed in an error term (also called disturbance term). Formally, for each observation i , the value of the dependent variable, y i is related to a sum of K explanatory variables, x ih , with h =1,..., K , each multiplied with a regression coefficient , β h , and the random error term, ε i : K y i = Σ x ih β h + ε i (C-1) h=1 Typically, the first explanatory variable is set equal to one, and referred to as the constant term . Its coefficient is referred to as the intercept , the other coefficients are slopes . Using a constant term amounts to extracting a mean effect and is equivalent to using all variables as deviations from their mean. In practice, it is highly recommended to always include a constant term. In matrix notation, which summarizes all observations, i=1,...,N , into a single compact expression, an N by 1 vector of values for the dependent variable, y is related to an N by K matrix of values for the explanatory variables, X , a K by 1 vector of regression coefficients, β , and an N by 1 vector of random error terms, ε : y= X β + ε (C-2)
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C-2 This model stipulates that on average, when values are observed for the explanatory variables, X , the value for the dependent variable equals X β , or: E(y | X) = X β (C-3) where E[ | ] is the conditional expectation operator. This is referred to as a specification for the conditional mean, conditional because X must be observed. It is a theoretical model, built on many assumptions. In practice, one does not know the coefficient vector, β , nor is the error term observed.
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