Poisson regression model:19
E[yi - exp(χiβ) ∣χi] = E[exp(x0β)(ηi - 1)∣χi] = 0.
This condition holds as long as ηi is uncorrelated with the regressors x0i and
E(ηi∣x0i) = 1.20 Mullahy considers the case where E(ηi∣x0i) 6= 1 and a vector
of instruments zi0 of dimension 1 × P with P ≥ K is available such that
E(ηi∣zi0) = 1. Then, the moment conditions
E[exp(-x0iβ)yi - 1∣zi0] = 0
can be used to consistently estimate the parameter vector β . We use Mul-
lahy’s approach with cross-sectional data under the assumption of endoge-
nous tax rates.
An alternative approach of addressing the problem of endogeneity arises
with panel data. Windmeijer (2006) discusses moment conditions that can
be used to estimate the parameter vector consistently allowing for correlation
between the individual effects and the regressors under different assumptions
about the exogeneity of the explanatory variables.
Define
yit = exp(x0itβ)exp(ηi)wit
= μitViWit
where i is an index for municipalities, and t = 1, ...T is an index for time with
T with denoting the maximum number of years of a municipality-specific time
series in the data. ηi is a time-invariant, unobservable, municipality-specific
component and wit is a time-variant disturbance term. Notice that νi and
wit are scaling factors for the municipality-specific mean. With the panel
data models, we allow that that E (xitwit-τ) 6= 0 for τ ≥ 0, maintaining that
E(witνi) = 0, E(wit) = 1 (both of them for t = 1, ..., T) and E(wiswit) = 0
(for any s 6= t).
We apply Chamberlain’s (1992) quasi-differencing transformation to elim-
inate the multiplicative fixed effect νi . The quasi-differencing approach is
based on the following equation (see Windmeijer, 2006):
sit = yit
μit-ι
μit
- Vit-I = μit-1νi(wit - wit-1)
19 The Poisson maximum likelihood estimator solves the first-order conditions
Pin=1[yi - exp(x0iβ)]x0i = 0.
20Whenever a constant term is included in x0i , then E(ηi) = 1 can be assumed without
loss of generality (see Mullahy, 1997).
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