Fixed effects panel data models

Eventually, one might doubt the possibility to circumvent the endogeneity problem in a cross section at all. In particular, one could push things further and assume that any information contained in the stock of legally indepen- dent foreign-owned firms hosted in a municipality could lead to endogeneity. Focusing on new headquarters could help reducing the endogeneity bias in tax rates significantly. For this, we resort to fixed effects panel data analysis. The corresponding models for municipality𝑖in year𝑡about outcome𝑦𝑖𝑡 may 26_{Notice that this is more than to say that municipalities use business tax rates to}

attract firms in general. In our application, the average municipality is not able to attract any foreign MNEs. This may be seen as an indication that the attraction of such firms is not the most important (or even an impossible) policy objective of the average German municipality. Hence, we expect the endogeneity issue as subordinate, here. However, we mention and apply suitable methods for completeness and as a robustness check to the conventional count data models.

be characterized as

𝑦𝑖𝑡=𝑥𝑖𝑡𝛽+𝜇𝑖+𝑢𝑖𝑡 (1.1)

where 𝑖 is an index for municipalities, 𝑡 = 1, ...𝑇 is an index for time with 𝑇 = 5 denoting the number of years our panel covers (2001-2005), 𝑥𝑖𝑡 is

a 1×𝐾 vector of explanatory variables (one of them being 𝑇 𝐴𝑋𝑖𝑡 or, al-

ternatively, 𝐹 𝐴𝑇 𝐴𝑋𝑖𝑡), 𝛽 is a corresponding 𝐾 ×1 parameter vector, 𝜇𝑖

is a fixed municipality-specific effect, and 𝑢𝑖𝑡 is a time-variant idiosyncratic

disturbance term.27

Fixed effects estimation of (1.1) identifies the parameter vector𝛽 by exploit- ing the time variation in 𝑦𝑖𝑡. In general, the model in (1.1) will be based only

on municipalities for which 𝑦𝑖𝑡 ∕=𝑦𝑖𝑠 at least for one tuple𝑡, 𝑠. Furthermore,

fixed effects estimation of (1.1) is immune against correlation of the elements in 𝑥𝑖𝑡 with 𝜇𝑖 +𝑢𝑖𝑡 as long as this correlation involves only 𝜇𝑖 but not 𝑢𝑖𝑡.

Hence, that model reduces problems of endogeneity of 𝑇 𝐴𝑋𝑖𝑡 or 𝐹 𝐴𝑇 𝐴𝑋𝑖𝑡

by allowing time-invariant unobserved heterogeneity to be correlated with the tax variables.

While some of the panel data regressions involve the strictly nonnegative 𝑦𝑖𝑡

as dependent variable, others will be based upon ln𝑦𝑖𝑡.28 In the former case,

the number of covered observations will be much larger than in the latter

27_{In principal, fixed effects model estimation is possible with nonlinear models such as}

Poisson, NB, or ZI versions thereof. However, it turns out that pooling cross-section and time-series data in a short panel such as ours leads to convergence problems with maximum likelihood estimation. Since we are mostly interested in conditional means, we therefore employ linear models and truncated models with sample selection in this case.

28_{In our context the problem of a large mass of zeros in the data is greatly reduced for}

case. The parameters are not directly comparable between models using 𝑦𝑖𝑡 and those using ln𝑦𝑖𝑡. In the latter case, the parameters on 𝑇 𝐴𝑋𝑖𝑡 and

𝐹 𝐴𝑇 𝐴𝑋𝑖𝑡 are semi-elasticities,29 while in the former case they are not.30 In

any case, log-transformation of 𝑦𝑖𝑡 leads to a truncated sample about which

– given that the number of zeros in 𝑦𝑖𝑡 is relatively large – the assumption

of random selection of municipalities into positive numbers of foreign-owned firms may be called into question.

Although the bias associated with either sample selection or endogeneity of 𝑇 𝐴𝑋𝑖𝑡 or 𝐹 𝐴𝑇 𝐴𝑋𝑖𝑡 is mitigated to some extent by fixed effects estimation

of (1.1), it is unlikely fully removed. For this reason, for models with 𝑦𝑖𝑡 as

dependent variables, we estimate a fixed effects two-stage least squares model assuming that 𝑇 𝐴𝑋𝑖𝑡 or𝐹 𝐴𝑇 𝐴𝑋𝑖𝑡 is correlated with𝑢𝑖𝑡 (see Baltagi, 2008).

When using a truncated sample in models which involve ln𝑦𝑖𝑡 instead of𝑦𝑖𝑡,

we follow an established literature in econometrics that model participation (the process of 𝑦𝑖𝑡 = 0 versus 𝑦𝑖𝑡 > 0) and outcome (the process of 𝑦𝑖𝑡 > 0)

by a bivariate model (see Wooldridge, 2002, or Cameron and Trivedi, 1998, for an overview).31 _{Specifically, we follow the approach of Wooldridge (1995)}

as adapted for endogenous regressors by Semykina and Wooldridge (2005) to account for endogeneity of 𝑇 𝐴𝑋𝑖𝑡 or𝐹 𝐴𝑇 𝐴𝑋𝑖𝑡 in the fixed effects model

(1.1) based on ln𝑦𝑖𝑡.

29_{Then, a one-percentage-point change in the tax rate induces a one-hundredth change}

in ln𝑦𝑖𝑡.

30_{Then, a one-percentage-point change in the tax rate induces a one-hundredth change}

in 𝑦𝑖𝑡.

31_{An alternative to a selection model which rests on somewhat stronger assumptions}

would be a two-part approach which models zero-versus-positive 𝑦𝑖𝑡 and positive 𝑦𝑖𝑡 as