How Low Business Tax Rates Attract Multinational Headquarters: Municipality-Level Evidence from Germany

Poisson regression model:¹⁹

E[y_i - exp(χiβ) ∣χi] = E[exp(x⁰β)(η_i - 1)∣χi] = 0.

This condition holds as long as ηi is uncorrelated with the regressors x⁰_i and
E(η_i∣x⁰_i) = 1.²⁰ Mullahy considers the case where E(η_i∣x⁰_i) 6= 1 and a vector
of instruments z_i⁰ of dimension 1 × P with P ≥ K is available such that
E(η_i∣z_i⁰) = 1. Then, the moment conditions

E[exp(-x⁰_iβ)yi - 1∣z_i⁰] = 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(x⁰_itβ)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 w_it is a time-variant disturbance term. Notice that ν_i and
w_it are scaling factors for the municipality-specific mean. With the panel
data models, we allow that that E (x_itw_it-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 ν_i . The quasi-differencing approach is
based on the following equation (see Windmeijer, 2006):

¹⁹ The Poisson maximum likelihood estimator solves the first-order conditions
^P_iⁿ₌₁[yi - exp(x⁰_iβ)]x⁰_i = 0.

²⁰Whenever a constant term is included in x⁰_i , then E(ηi) = 1 can be assumed without
loss of generality (see Mullahy, 1997).

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