Estimating Gravity Models of Trade with Correlated Time-Fixed Regressors: To IV or not IV?

Abstract

Gravity-type models are widely used in international economics. In these models, the inclusion of time-fixed regressors like geographical or cultural distance, language and institutional (dummy) variables is often of vital importance, e.g., to analyze the impact of trade costs on internationalization activity. This paper assesses the problem of parameter inconsistency due to a correlation of the time-fixed regressors with the combined error term in panel data settings. A common solution is to use instrumental variable (IV) estimation in the spirit of Hausman and Taylor (1981) since standard fixed effect model (FEM) estimation is not applicable. However, some potential shortcomings of the latter approach recently gave rise to the use of non-IV two-step estimators. Given their growing number of empirical applications, we aim to compare the performance of IV and non-IV approaches in the presence of time-fixed variables and right-hand-side endogeneity using Monte Carlo simulations, where we explicitly control for the problem of IV selection in the Hausman–Taylor case. The simulation results show that the Hausman–Taylor model with perfect knowledge about the underlying data structure (instrument orthogonality) has on average the smallest bias. However, compared to the empirically relevant specification with imperfect knowledge and instruments chosen by statistical criteria, simple non-IV rival estimators perform equally well or even better. We illustrate these findings by estimating gravity-type models for German regional export activity within the EU. The results show that the HT specification is likely to bias the role of trade costs proxied by geographical distance upwards.

Appendices

Appendix A: Monte Carlo Simulation Design

The starting point for the Monte Carlo simulation experiment is (6.4). The time-varying regressors $x_{1_{1}}, x_{1_{2}},x_{2_{1}}, x_{2_{2}}$ are generated by the following autoregressive process:

$$\begin{array}{*{20}c} {x_{n_m ,i,t = 1} = 0} & {{\rm with}\,n,m = 1,2,} \\ \end{array}$$

(6.8)

$$\begin{array}{*{20}c} {x_{1_1 ,i,t} = \rho _1 x1_{i,t - 1} + \delta _i + \xi _{i,t} } & {{\rm for}\,t = 2,...,T,} \\ \end{array}$$

(6.9)

$$\begin{array}{*{20}c} {x_{1_2 ,i,t} = \rho _2 x2_{i,t - 1} + \psi _i + \omega _{i,t} } & {{\rm for}\,t = 2,...,T,} \\ \end{array}$$

(6.10)

$$\begin{array}{*{20}c} {x_{2_1 ,i,t} = \rho _3 x3_{i,t - 1} + \mu _i + \tau _{i,t} } & {{\rm for}\,t = 2,...,T,} \\ \end{array}$$

(6.11)

$$\begin{array}{*{20}c} {x_{2_2 ,i,t} = \rho _4 x4_{i,t - 1} + \mu _i + \lambda _{i,t} } & {{\rm for}\,t = 2,...,T.} \\ \end{array}$$

(6.12)

For the time-fixed regressors $z_{1_{1}}, z_{1_{2}}, z_{2_{1}}$ we analogously define

$$z_{1_{1,i}} = 1,$$

(6.13)

$$ z_{1_{2,i}} = g_1 \psi _i + g_2 \delta _i + \kappa _i ,$$

(6.14)

$$z_{2_1 ,i} = \mu _i + \delta _i + \psi _i + \in _i .$$

(6.15)

The variable $z_{1_{1},i}$ simplifies to a constant term, $z_{2_{1},i}$ is the endogenous time-fixed regressor since it contains μ _i as right-hand-side variable, the weights g ₁ and g ₁ in the specification of $z_{1_{2},i}$ control for the degree of correlation with the time-varying variables $x_{1_{1},i,t}$ and $x_{1_{2},i,t}$.¹³ The remainder innovations in the data generating process are defined as follows:

$$v_{i,t} \sim N(0,\sigma _v^2 ),$$

(6.16)

$$\mu _i \sim N(0,\sigma _\mu ^2 ),$$

(6.17)

$$\delta _i \sim U( - 2,2),$$

(6.18)

$$\xi _{i,t} \sim U( - 2,2),$$

(6.19)

$$\psi _i \sim U( - 2,2),$$

(6.20)

$$\omega _{i,t} \sim U( - 2,2),$$

(6.21)

$$\tau _{i,t} \sim U( - 2,2),$$

(6.22)

$$\lambda _{i,t} \sim U( - 2,2),$$

(6.23)

$$ \in _i \sim U( - 2,2),$$

(6.24)

$$\kappa _i \sim U( - 2,2).$$

(6.25)

Except μ _i and ν _i,t , which are drawn from a normal distribution with zero mean and variance $\sigma_{\mu}^{2}$ and $\sigma _{\nu}^{2}$, respectively, all innovations are uniform on [−2,2]. For μ _i ,δ _i ,ψ _i ,ϵ _i ,κ _i the first observation is fixed over T. With respect to the main parameter settings in the Monte Carlo simulation experiment we set:

• $\beta_{1_{1}}=\beta_{1_{2}}=\beta_{2_{1}}=\beta_{2_{2}}=1$

• $\gamma_{1_{2}}=\gamma_{2_{1}}=1$

• ρ ₁=ρ ₂=ρ ₃=ρ ₄=0.7

All variable coefficients are normalized to one, the specification of ρ<1 assures that the time-varying variables are stationary. We also normalize σ _ν equal to one and define a load factor ξ determining the ratio of the variance terms of the error components as ξ=σ _μ /σ _ν . ξ takes values of (2, 1 and 0.5). We run simulations with different combinations in the time and cross-section dimension of the panel as N=(100,500,1000) and T=(5,10). All Monte Carlo simulations are conducted with 500 replications for each permutation in y and u. As in Arellano and Bond (1991) we set T=T+10 and cut off first 10 cross-sections, which gives a total sample size of NT observations.

We apply the FEVD and Hausman–Taylor estimators.¹⁴ As outlined above, one drawback in earlier Monte Carlo based comparisons between the HT model and rival non-IV candidates was the strong assumption made for IV selection in the HT case, namely that true correlation between right-hand-side variables and the error term is known. However, this may not reflect the identification and estimation problem in applied econometric work and Alfaro (2006) identifies it as one of the open questions for future investigation in Monte Carlo simulations. We therefore account for the HT variable classification problem by implementing algorithms from ‘model selection criteria’-literature, which combine information from Hansen/Sargan overidentification test for moment condition selection as outlined above and time-series information-criteria. Following Andrews (1999), we define a general model selection criteria (MSC) based on IV estimation as

$$ \mathit{MSC}_{n}(m) = J(m)-h(c)k_{n},$$

(6.26)

where n is the sample size, c as number of moment conditions selected by model m based on the Hansen J-statistic J(m), h(.) is a general function, k _n is a constant term. As (6.26) shows, the model selection criteria centers around the J-statistic.¹⁵ The second part in (6.26) defines a ’bonus’ term rewarding models with more moment conditions, where the form of function h(.) and the constant terms k _n are specified by the researcher. For empirical application Andrews (1999) proposes three operationalizations in analogy to model selection criteria from time series analysis:

• MSC-BIC: J(m)−(k−g)lnn

• MSC-AIC: J(m)−2(k−g)

• MSC-HQIC: J(m)−Q(k−g)lnlnn with Q=2.01

where (k−g) is the number of overidentifying restrictions, and depending on the form of the bonus term, the MSC may take the BIC (Bayesian), AIC (Akaike) and HQIC (Hannan Quinn) form. We apply all three information criteria in the Monte Carlo simulations motivated by the results in Andrews and Lu (2001) and Hong et al. (2003) that the superiority of one of the criteria over the others in terms of finding consistent moment conditions may vary with the sample size.¹⁶ For each of these MSC criteria, we specify the following algorithms:

• 1. Unrestricted form: For all possible IV combinations out of the full IV-set S=(QX1,QX2,PX1,PX2,Z1,Z2), where Q denote deviations from group means and P are group means. The IV set satisfies the order condition k ₁>g ₂ (giving a total number of 42 combinations). We calculate the value of the MSC criterion (for the BIC, AIC and HQIC separately) and choose that model as final HT specification, which has minimum MSC value over all candidates.

• 2. Restricted form: This algorithm follows the basic logic from above, but additionally puts the further restriction that only those models serves as MSC candidates for which the p-value of the J-statistic is a above a critical value K _crit., which we set to 0.05 to maximize the likelihood that the selected moment conditions are valid in terms of statistical pre-testing. The restricted (see Andrews 1999, for this point).

We present flow charts of the restricted and unrestricted MSC based search algorithm in Fig. 6.8. As Andrews (1999) argues, the above specified model selection criteria is closely related to the C-statistic approach by Eichenbaum et al. (1988) to test whether a given subset of moment conditions is correct or not.¹⁷

Fig. 6.8

MSC based model selection algorithm for HT-approach

Thus, alternatively to the above described algorithms, we specify a downward-testing approach based on the C-statistic: Here, we start from the HT model with full IV set in terms of the REM moment conditions as S ₁=(QX1,QX2,PX1,PX2,Z1,Z2). We calculate the value of the J-statistic for the model with IV-set S ₁ and compare its p-value with a predefined critical value K _crit., which we set in line with the above algorithm as K _crit.=0.05. If $P_{S_{1}}>K_{\mathit{crit.}}$ we take this model as a valid representation in terms of the underlying moment conditions. If not, we calculate the value of the C-statistic for each single instrument in S ₁ and exclude that instrument from the IV-set that has the maximum value of the C-statistic.

We then re-estimate the model based on the IV-subset S ₂ net of the selected instrument with the highest C-statistic and again calculate the J-statistic and its respective p-value. If $P_{S_{2}}>K_{\mathit{crit.}}$ is true, we take the HT-model with S ₂ as final specification and otherwise again calculate the C-statistic for each instrument to exclude that one with the highest value. We run this downward-testing algorithm for moment conditions until we find a model that satisfies $P_{S_{.}}>C_{\mathit{crit.}}$ or, at the most, until we reach the IV-sets S _n to S _m , where the number of overidentifying restrictions (k−g)=1, since the J-statistic is not defined for just-identified models. Out of S _n to S _m , we then pick the model with the lowest J-statistic value. The C-statistic based model selection algorithms is graphically summarized in Fig. 6.9.

Fig. 6.9

C-statistic based model selection algorithm for HT-approach

Appendix B: Variable Description for the Gravity Model

Table 6.3 Data description and source for export model