# **Annex D: Count Outcome Models**

This Annex describes how count outcome models can serve the purpose of testing the determinants of the PSP projects identified in our database. It borrows heavily from Long and Freese (2001).

# _The Poisson Distribution_

The univariate Poisson distribution is the foundation of regression models for counts. Let _y_ be a random variable indicating the number of times an event has occurred. If _y_ has a Poisson distribution, then: Pr( _y | μ_ ) = ( _e_<sup>_−μ_</sup> _* μ_<sup>_y_</sup> _)/ y_ ! for _y_ = 0 _,_ 1 _,_ 2 _, . . ._ (1) where _μ >_ 0 is the sole parameter defining the distribution

_μ_ is the mean of the Poisson distribution. _μ_ is also the variance. Thus, Var( _y_ ) = _μ_ , which is known as _equidispersion_ . With most datasets, many count variables have a variance greater than their mean, which is called _overdispersion_ . As _μ_ increases, the probability of a zero count decreases. Thus, for many count variables, there will be observed zeros than predicted by the Poisson distribution.

# _The Poisson Regression Model_

The Poisson regression model (PRM) extends the Poisson distribution by allowing each observation to have a different value of _μ_ . More formally, the PRM assumes that the observed count for observation _i_ is drawn from a Poisson distribution with mean _μi_ , where _μi_ is estimated from observed characteristics. This is sometimes referred to as incorporating _observed heterogeneity_ , and leads to the structural equation: _μi_ = _E_ ( _yi |_ x _i_ ) = exp(x _iβ_ ) (2)

Using the exponential of x _β_ forces _μ_ to be positive; this is necessary since counts can only be zero or positive. For each value of _μ_ , the distribution around the mean represents the probability of each count. Interpretation of the model involves assessing how changes in the independent variables affect the conditional mean and the probabilities of various counts.

# _Exposure time_

In our sample, not all countries introduce PSP in the water sector at the same time. Statistically, different observations will have different _exposure_ times i.e. each observation is ‘at risk’ of having a positive count for a different amount of time. In our example, each country will have been ‘at risk’ of having a positive count of private project for as long as a country’s legal framework allows private participation in the water sector. Before that date the probability of counting zero projects is by definition equal to unity, and the probability to count more than zero projects is nil.

We create an exposure variable (FIRSTPSP) which measures the number of years since the first private infrastructure project (not just in the water sector) was signed in country _i_ . For most countries this signals the beginning of reform of public utilities and the point after which a private water project becomes possible. The data for FIRSTPSP is drawn from the World Bank’s PPI database.

Exposure times can be incorporated quite simply into count models. Let _ti_ be the amount of time that observation _i_ is at risk. If the rate (i.e., the expected number of observations for a single unit of time) for that case is _μi_ , then we would expect _ti_ * _μi_ to be the expected count over a period of length _ti_ . Then, assuming only two independent variables for simplicity, our count equation becomes: This shows that the effect of different exposure times can be included as the log of the exposure time with a regression coefficient constrained to equal unity.

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