A generalized spatial error components model for gravity equations

Abstract

Since the publication of the paper by Anderson and van Wincoop (J Econ Litt 42:691–751, 2004), the estimation of gravity models has turned increasingly structural. The benefit of this development is that empirical models which are based on few parameters explain data on bilateral trade flows relatively well, while being consistent with general equilibrium. The latter permits using the estimated parameters for comparative static analysis. In general, in the cross section such models involve country-pair-specific variables and exporter-specific as well as importer-specific variables. The latter are determined through structural model constraints. The disturbances on this model are typically assumed to be independently if not also identically distributed. This paper illustrates that the assumption of independently distributed disturbances is likely flawed in practice. Ignoring such independence leads to inconsistent test statistics and standard errors of the parameters. We present a structural gravity model which permits the disturbances to be cross-sectionally interdependent in up to three dimensions and illustrate the consequences of doing so for inference.

Appendices

Appendices

1.1 Appendix A: The variance–covariance matrix of the disturbances given in (9)

$$\begin{aligned} {\varvec{\Omega }}_{\upsilon }= & {} E[\upsilon \upsilon ^{\prime }]= {\varvec{\Delta }}_{\mu }E[\mathbf {mm}^{\prime }]{\varvec{\Delta }}_{\mu }^{\prime } +{\varvec{\Delta }}_{\lambda }E[\mathbf {ll}^{\prime }]{\varvec{\Delta }}_{\lambda }^{\prime }+E[\varepsilon \varepsilon ^{\prime }] \\= & {} \sigma _{\mu }^{2}{\varvec{\Delta }}_{\mu }\left( ({\mathbf {I}}_{C}-\rho _{M} {\mathbf {M}})^{-1}({\mathbf {I}}_{C}-\rho _{M}{\mathbf {M}}^{\prime })^{-1}\right) {\varvec{\Delta }}_{\mu }^{\prime } \\&+\sigma _{\lambda }^{2}{\varvec{\Delta }}_{\lambda }\left( ({\mathbf {I}}_{C}-\rho _{L}{\mathbf {L}})^{-1}({\mathbf {I}}_{C}-\rho _{L}{\mathbf {L}}^{\prime })^{-1}\right) {\varvec{\Delta }}_{\lambda }^{\prime } \\&+\sigma _{\varepsilon }^{2}({\mathbf {I}}_{C^{2}} -\rho _{W}{\mathbf {W}})^{-1}( {\mathbf {I}}_{C^{2}} -\rho _{W}{\mathbf {W}}^{\prime })^{-1} \\= & {} \sigma _{\mu }^{2}( {\mathbf {I}}_{C} \otimes \iota _{C})\left( ( {\mathbf {I}}_{C}-\rho _{M}{\mathbf {M}})^{-1}({\mathbf {I}}_{C}-\rho _{M}{\mathbf {M}} ^{\prime })^{-1}\right) ( {\mathbf {I}}_{C} \otimes \iota _{C}^{\prime } )\\&+\sigma _{\lambda }^{2}( \iota _{C} \otimes {\mathbf {I}}_{C})\left( ( {\mathbf {I}}_{C}-\rho _{L}{\mathbf {L}})^{-1}({\mathbf {I}}_{C}-\rho _{L}{\mathbf {L}} ^{\prime })^{-1}\right) ( \iota _{C}^{\prime } \otimes {\mathbf {I}}_{C})\\&+\sigma _{\varepsilon }^{2}({\mathbf {I}}_{C^{2}}-\rho _{W}{\mathbf {W}})^{-1}( {\mathbf {I}}_{C^{2}}-\rho _{W}{\mathbf {W}}^{\prime })^{-1} \\= & {} C\sigma _{\mu }^{2}\left( \left( {\mathbf {A}} ^{\prime }{\mathbf {A}}\right) ^{-1} \otimes \bar{{\mathbf {J}}}_{C} \right) +C\sigma _{\lambda }^{2}\left( \bar{{\mathbf {J}}}_{C}\otimes \left( {\mathbf {B}}^{\prime }{\mathbf {B}}\right) ^{-1}\right) +\sigma _{\varepsilon }^{2}\left( {\mathbf {C}}^{\prime }{\mathbf {C}} \right) ^{-1}, \end{aligned}$$

1.2 Appendix B: The spatial ML estimator in unbalanced spatial panels

For estimation, one can partition $\varOmega _\upsilon$ given in (9) and its inverse as

$$\begin{aligned} \frac{1}{\sigma _{\varepsilon }^{2}}{\varvec{\Sigma }}_{\upsilon }^{-1} =\frac{1}{\sigma _{\varepsilon }^{2}}\left[ \begin{array}{cc} {\mathbf {S}}_{o}{\varvec{\Sigma }}_{\upsilon }^{-1}{\mathbf {S}}_{o}^{\prime } &{} {\mathbf {S}}_{o}{\varvec{\Sigma }}_{\upsilon }^{-1}{\mathbf {S}}_{f}^{\prime }\\ {\mathbf {S}}_{f}{\varvec{\Sigma }}_{\upsilon }^{-1}{\mathbf {S}}_{o}^{\prime } &{} {\mathbf {S}}_{f}{\varvec{\Sigma }}_{\upsilon }^{-1}{\mathbf {S}}_{f}^{\prime } \end{array} \right] :=\frac{1}{\sigma _{\varepsilon }^{2}}\left[ \begin{array}{cc} {\varvec{\Sigma }}_{\upsilon ,oo} &{} {\varvec{\Sigma }}_{\upsilon ,of} \\ {\varvec{\Sigma }}_{\upsilon ,fo} &{} {\varvec{\Sigma }}_{\upsilon ,ff} \end{array} \right] , \end{aligned}$$

(25)

where ${\mathbf {S}}_{f}$ is obtained from skipping all rows of ${\mathbf {I}}_{C^2}$ that refer to observed values.

The likelihood will be maximized numerically using

$$\begin{aligned}&\displaystyle {\varvec{\Sigma }}_{\upsilon ,oo} =\frac{\sigma _{\mu }^{2}}{\sigma _{\varepsilon }^{2}}{\varvec{\Delta }}_{\mu o}\left( {\mathbf {A}}^{\prime }{\mathbf {A}} \right) ^{-1}{\varvec{\Delta }}_{\mu o}^{\prime }+\frac{\sigma _{\lambda o}^{2} }{\sigma _{\varepsilon }^{2}}{\varvec{\Delta }}_{\lambda o}\left( {\mathbf {B}}^{\prime }{\mathbf {B}}\right) ^{-1}{\varvec{\Delta }}_{\lambda o}^{\prime }+{\mathbf {S}} _{o}\left( {\mathbf {C}}^{\prime }{\mathbf {C}}\right) ^{-1}{\mathbf {S}}_{o}^{\prime } \\&\displaystyle \widehat{\beta } =\left( {\mathbf {Z}}^{\prime } _{o}\widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}^{-1} {\mathbf {Z}}_{o} \right) ^{-1}{\mathbf {Z}}^{\prime }_{o} \widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}^{-1} {\mathbf {x}}_{o} \nonumber \\&\displaystyle \widehat{{\mathbf {u}}}_{o} =\left( {\mathbf {I}}_{H}-{\mathbf {Z}}_{o}\left( {\mathbf {Z}}^{\prime }_{o}\widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}^{-1} {\mathbf {Z}}_{o} \right) ^{-1}{\mathbf {Z}}^{\prime }_{o} \widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}^{-1} \right) {\mathbf {x}}_{o} \nonumber \\&\displaystyle \widehat{\sigma }_{\varepsilon }^{2} ={\mathbf {x}}^{\prime }_{o} \widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}^{-1}(\widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}-{\mathbf {Z}}_{o}\left( {\mathbf {Z}}^{\prime } _{o}\widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}^{-1} {\mathbf {Z}}_{o} \right) ^{-1}{\mathbf {Z}}^{\prime }_{o})\widehat{{\varvec{\Sigma }}}_{\upsilon ,oo}^{-1} {\mathbf {x}}_{o}\nonumber \end{aligned}$$

(26)

to obtain the concentrated likelihood

$$\begin{aligned} \ln L_{o}(\theta )=-\frac{n}{2}\ln 2\pi -\frac{1}{2}\ln \widehat{{\mathbf {u}}}_{o}^{\prime }{\varvec{\Sigma }}_{\upsilon ,oo}^{-1}\widehat{{\mathbf {u}}}_{o}-\frac{1}{2} \ln \det ({\varvec{\Sigma }}_{\upsilon ,oo})-\frac{n}{2} . \end{aligned}$$

(27)

1.3 Appendix C: The estimation of the variance–covariance matrix of the estimated parameters

The upper left diagonal block of the information matrix refers to the estimated vector slope parameters, $\widehat{\beta }$, and, for balanced data, the estimate of the block is given by

$$\begin{aligned} \widehat{{\varvec{\Sigma }}}_{\beta }=\widehat{\sigma }_{\varepsilon }^{2}({\mathbf {Z}}^{\prime }{\varvec{\Sigma }}_{\upsilon }(\widehat{\phi }_{\mu },\widehat{\phi }_{\lambda },\widehat{\rho }_{M}, \widehat{\rho }_{L},\widehat{\rho }_{W})^{-1}{\mathbf {Z}})^{-1}, \end{aligned}$$

(28)

where $\phi _{\mu }=\frac{\sigma _{\mu }^{2}}{\sigma _{\varepsilon }^{2}}$ and $\phi _{\lambda }= \frac{\sigma _{_{\lambda }}^{2}}{\sigma _{\varepsilon }^{2}}$. The result for the unbalanced case is analogous. The lower right block of the variance–covariance matrix contains the estimates of the sub-information matrix for the remaining parameters $\widehat{\gamma }=(\widehat{\sigma }_{\varepsilon }^{2},\widehat{\phi }_{\mu },\widehat{\phi }_{\lambda },\widehat{ \rho }_{M},\widehat{\rho }_{L},\widehat{\rho }_{W})$. Harville (1977) provides a useful general differentiation result to derive this lower block information matrix referring to $\widehat{\gamma }$:

$$\begin{aligned} J_{rs}=E\left[ -\frac{\partial ^{2}L}{\partial \gamma _{r}\gamma _{s}}\right] =\frac{1}{2}tr\left[ {\varvec{\Omega }}_{\upsilon }^{-1}\frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \gamma _{r}}{\varvec{\Omega }}_{\upsilon }^{-1}\frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \gamma _{s}}\right] \qquad r,s=1,...,6. \end{aligned}$$

(29)

Using

$$\begin{aligned} {\varvec{\Omega }}_{\upsilon }=C\sigma _{\mu }^{2}\left( \left( {\mathbf {A}}^{\prime }{\mathbf {A}}\right) ^{-1} \otimes \bar{{\mathbf {J}}}_{C}\right) +C\sigma _{\lambda }^{2}\left( \bar{{\mathbf {J}}}_{C}\otimes \left( {\mathbf {B}}^{\prime }{\mathbf {B}}\right) ^{-1} \right) +\sigma _{\varepsilon }^{2}\left( {\mathbf {C}}^{\prime }{\mathbf {C}} \right) ^{-1} \end{aligned}$$

(30)

for the balanced panel model, one obtains

$$\begin{aligned} \begin{aligned} \frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \sigma _{\varepsilon }^{2} }&=C\sigma _{\mu }^{2}\left( \left( {\mathbf {A}}^{\prime }{\mathbf {A}}\right) ^{-1} \otimes \bar{{\mathbf {J}}}_{C}\right) +C\sigma _{\lambda }^{2}\left( \bar{{\mathbf {J}}}_{C}\otimes \left( {\mathbf {B}}^{\prime }{\mathbf {B}}\right) ^{-1} \right) +\sigma _{\varepsilon }^{2}\left( {\mathbf {C}}^{\prime }{\mathbf {C}} \right) ^{-1} \\ \frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \phi _{\mu }}&=\sigma _{\varepsilon }^{2}C\left( ({\mathbf {A}}^{\prime }{\mathbf {A}})^{-1} \otimes \overline{{\mathbf {J}}}_{C} \right) , \\ \frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \phi _{\lambda }}&=\sigma _{\varepsilon }^{2}C\left( \overline{{\mathbf {J}}}_{C} \otimes ({\mathbf {B}}^{\prime }{\mathbf {B}} )^{-1}\right) , \\ \frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \rho _{M}}&=\sigma _{\varepsilon }^{2}C\left( \phi _{\mu }( {\mathbf {A}}^{\prime }{\mathbf {A}})^{-1}({\mathbf {M}}+{\mathbf {M}}^{\prime }-2\rho _{M}{\mathbf {M}}^{\prime }{\mathbf {M}})({\mathbf {A}}^{\prime }{\mathbf {A}} )^{-1} \otimes \overline{{\mathbf {J}}}_{C}\right) , \\ \frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \rho _{L}}&=\sigma _{\varepsilon }^{2}C\left( \overline{{\mathbf {J}}}_{C} \otimes \phi _{\lambda } \left( {\mathbf {B}}^{\prime }{\mathbf {B}}\right) ^{-1}({\mathbf {L}}+{\mathbf {L}}^{\prime }-2\rho _{L}{\mathbf {L}}^{\prime } {\mathbf {L}})({\mathbf {B}}^{\prime }{\mathbf {B}})^{-1} \right) , \\ \frac{\partial {\varvec{\Omega }}_{\upsilon }}{\partial \rho _{W}}&=\sigma _{\varepsilon }^{2}\left( ({\mathbf {C}}^{\prime }{\mathbf {C}})^{-1}({\mathbf {W}}+ {\mathbf {W}}^{\prime }-2\rho _{W}{\mathbf {W}}^{\prime }{\mathbf {W}})({\mathbf {C}}^{\prime }{\mathbf {C}})^{-1}\right) . \end{aligned} \end{aligned}$$

(31)

In the presence of missing observations, one can use (see Haining et al. 1989)

$$\begin{aligned} \frac{\partial {\varvec{\Omega }}_{\upsilon ,oo}}{\partial \gamma _{r}}=\frac{ \partial {\mathbf {S}}_{o}{\varvec{\Omega }}_{\upsilon }{\mathbf {S}}_{o}^{\prime }}{ \partial \gamma _{r}}={\mathbf {S}}_{o}\frac{\partial {\varvec{\Omega }} _{\upsilon }}{\partial \gamma _{r}}{\mathbf {S}}_{o}^{\prime } \end{aligned}$$

(32)

to derive estimates of the elements of the information matrix $J_{rs,o}$ of the observed model. Plugging in the estimated parameters gives the required estimate of the second block of the information matrix. However, the numerical calculation will be involved given the complicated structure of ${\varvec{\Omega }}_{\upsilon ,oo}^{-1}$ and $\frac{\partial {\varvec{\Omega }}_{\upsilon ,oo}}{\partial \gamma _{r}}$.

1.4 Appendix D

Table 5 Estimation results, unbalanced DOT trade flows