Poisson Regression
The Poisson regression model is the benchmark model for count data in much the same way as the normal linear model is the benchmark for real-valued continuous data. Early references in econometrics include Gilbert (1982), Hausman, Hall and Griliches (1984
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3.1 Specification 3.1.1 Introduction The Poisson regression model is the benchmark model for count data in much the same way as the normal linear model is the benchmark for real-valued continuous data. Early references in econometrics include Gilbert (1982), Hausman, Hall and Griliches (1984), and Cameron and Trivedi (1986). The Poisson model is simple, and it is robust. If the only interest of the analysis lies in estimating the parameters of a log-linear mean function, there is hardly any reason (except for efficiency) to ever contemplate anything other than the Poisson regression model. In fact, its applicability extends well beyond the traditional domain of count data. The Poisson regression model can be used for any constant elasticity mean function, whether the dependent variable is a count or continuous, and there are good reasons why it should be preferred over the more common log transformation of the dependent variable. And yet, there are instances where the Poisson regression model is unsuited. Essentially, the Poisson model is always overly restrictive when it comes to estimating features of the population other than the mean, such as the variance or the probability of single outcomes. The simplicity of the Poisson regression model, an asset when modeling the mean, turns then into a liability, and more elaborate models are needed. 3.1.2 Assumptions of the Poisson Regression Model The basic Poisson regression model relates the probability function of a dependent variable yi (also referred to as regressand, endogenous, or dependent variable) to a vector of independent variables xi (also referred to as regressors, exogenous, or independent variable). Let k be the number of regressors (including, usually, a constant). xi is then a column vector of dimension (k × 1). Finally, n is the number of observations in the sample.
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3 Poisson Regression
The standard univariate Poisson regression model makes the following three assumptions: Assumption . 1 f (y|λ) =
e−λ λy y!
y = 0, 1, 2, . . .
where f (y|λ) is the conditional probability function of y given λ, and it must hold that λ > 0. Assumption . 2 λ = exp(x β) where β is a (k × 1) vector of parameters, and x is a (k × 1) vector of regressors, including a constant. Assumption . 3 observation pairs (yi , xi ), i = 1, . . . , n are independently distributed. Discussion Assumptions 1 and 2 can be combined to obtain the following conditional probability function: f (y|x) =
exp(− exp(x β)) exp(yx β) y!
y = 0, 1, 2, . . .
(3.1)
The Poisson distribution has only one parameter that simultaneously determines conditional mean and variance. Therefore, the Poisson regression model as defined by the assumptions above implies an exponential (or loglinear) mean function, E(y|x) = λ = exp(x β)
(3.2)
and an exponential conditional variance function Var(y|x) = λ = exp(x β)
(3.3)
The fact that conditional mean and conditional variance are equal in the Poisson regression model is a particular feature – equidispersion – that will be subject to further discussion. The
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