An empirical approach to measure unobserved cultural relations using music trade data

Abstract

Cultural relations between countries influence the exchange of cultural goods. This study provides novel knowledge of unobserved cultural relations by mea- suring the effect of cultural relations on the trade in recorded music compact discs, using the gravity model of international trade. We consider such relations as unob- served heterogeneity and introduce into the standard model a factor structure (multi- ple interactive fixed effect terms) to extract the features of unobserved relations, in- cluding cultural relations, between trading countries. We also consider the existence of multiple zero-trade country pairs and introduce a selectivity structure to account for zero flows. After the estimation procedure, we derive the implications of cultural relations from the estimated values of interactive terms using multivariate analysis. From the results of post-estimation analysis, the estimated values of our interactive terms could be interpreted as the effect of cultural relations. In addition to the positive effect of cultural proximity on trade, which existing studies have revealed, our inter- active terms could capture (i) the negative effect of cultural proximity on music trade, such as home consumption bias, (ii) the positive effect of modern music consumption trend on music trade, which is unexplained by cultural proximity based on traditional cultural studies.

Appendix

1.1 Interpretation of additive fixed effect parameters

Base on their theoretical setting, Helpman et al. (2008) derive the empirical gravity model with exporter- and importer-specific additive fixed effects, and the economic status of each country in terms of GDP, population, price level and other (possibly unobserved) country-specific trade frictions are incorporated into this model. These factors are assumed to be country-specific and independent of trading partners. Figures 6, 7, 8 and 9 show the choropleth maps where countries are differentiated based on the additive fixed effect estimates from the outcome and the selection equation. $\alpha _{y,i}$ ($\alpha _{d,i}$) is the fixed effect parameter of the i-th as an importer in the outcome equation (the selection equation), and $\gamma _{y,j}$ ($\gamma _{d,j}$) is the fixed effect parameter of the j-th as an exporter in the outcome equation (the selection equation). Darker red colors indicate larger estimate values.

In these maps, the additive fixed estimates seem to be proportional to their economic scale. This tendency is shared by both estimates of exporter- and importer-specific fixed effect parameters in both equations. These findings are consistent with the theoretical implications concerning the additive fixed effect parameters obtained in Helpman et al. (2008). Some country-specific, partner-independent factors, such as economic scale, are important determinants but non-identifiable determinants of trade in gravity models when we only use the dyadic, cross-sectional dataset. It is true that the additive fixed effects can control for these factors, but other factors remain that are not accounted for by the explanatory variables and the additive fixed effects in the gravity equation. Then, those factors could be interpreted as unobserved, heterogeneous institutional relations or cultural relations between the importing and exporting country.

Fig. 6

Values of $\alpha _{y,i}$ on the World Map

Fig. 7

Values of $\gamma _{y,j}$ on the World Map

Fig. 8

Values of $\alpha _{d,i}$ on the World Map

Fig. 9

Values of $\gamma _{d,j}$ on the World Map

1.2 Country list by importing/exporting groups

The lists of importing/exporting groups are in Tables 12 and 13.

Table 12 Country List by Importing Groups (${\mathscr {A}}\;_{y,i}$)

Table 13 Country List by Exporting Groups (${\mathscr {G}}_{y,j}$)

1.3 Estimation procedure

Denote the structural parameters (the coefficient vectors in (1) and (2), the variance and the correlation coefficient parameter) as ${\theta }'=(\varvec{\beta }',\varvec{\delta }',\sigma ,\rho )$ and the incidental parameters (the additive and interactive fixed effect parameters) as

$$\begin{aligned} \underbrace{\;\;\varvec{\pi }'_{ij}\;\;}_{1\times 2(2+2r)}=(\underbrace{\;\;\varvec{a}'_{i}\;\;}_{1\times (2+2r)},\underbrace{\;\;\varvec{g}'_{j}\;\;}_{1\times (2+2r)}),\; i = 1, 2, \ldots , N,\; j = 1, 2, \ldots , N, j\ne i. \end{aligned}$$

(8)

where

$$\begin{aligned} \varvec{a}'_{i}= & {} (\varvec{a}'_{y,i},\varvec{a}'_{d,i}) =(\underbrace{\alpha _{y,i}, {\mathscr {A}}\;_{y,i} }_{1+r}, \underbrace{\alpha _{d,y}, {\mathscr {A}}\;_{d,i} }_{1+r})\\ \varvec{g}_{j}^{'}= & {} (\varvec{g}'_{y,j},\varvec{g}'_{d,t}) =(\underbrace{\gamma _{y,t}, {\mathscr {G}}_{y,t} }_{1+r}, \underbrace{\gamma _{d,y}, {\mathscr {G}}_{d,t} }_{1+r}). \end{aligned}$$

For identification, we employ the same restrictions as Bai (2009):

$$\begin{aligned} \sum _{i=1}^{N}\alpha _{y,i} = \sum _{i=1}^{N}\alpha _{d,i} = \sum _{j=1}^{N}\gamma _{y,j} = \sum _{j=1}^{N}\gamma _{d,j} = 0, ~~ \sum _{i=1}^{N} {\mathscr {A}}\;'_{y,i} = \sum _{i=1}^{N}{\mathscr {A}}\;'_{d,i} = \sum _{j=1}^{N}{\mathscr {G}}'_{y,j} = \sum _{j=1}^{N}{\mathscr {G}}'_{d,j} = \underbrace{\varvec{0}}_{r\times 1}, \end{aligned}$$

$$\begin{aligned} {\mathscr {A}}\;'_{y}{\mathscr {A}}\;_{y} = {\mathscr {A}}\;'_{d}{\mathscr {A}}\;_{d} = \varvec{I}_{r},\; \underbrace{{\mathscr {A}}\;'_{y}}_{r\times N} = ({\mathscr {A}}\;'_{y,1}, {\mathscr {A}}\;'_{y,2}, \ldots , {\mathscr {A}}\;'_{y,N}),\; \underbrace{{\mathscr {A}}\;'_{d}}_{r\times N} = ({\mathscr {A}}\;'_{d,1}, {\mathscr {A}}\;'_{d,2}, \ldots , {\mathscr {A}}\;'_{d,N}) \end{aligned}$$

$$\begin{aligned} {\mathscr {G}}'_{y}{\mathscr {G}}_{y} = \varvec{\varLambda }_{y},\; {\mathscr {G}}'_{d}{\mathscr {G}}_{d} = \varvec{\varLambda }_{d}, \; \underbrace{{\mathscr {G}}'_{y}}_{r\times N} = ({\mathscr {G}}'_{y,1}, {\mathscr {G}}'_{y,2}, \ldots , {\mathscr {G}}'_{y,N}),\; \underbrace{{\mathscr {G}}'_{d}}_{r\times N} = ({\mathscr {G}}'_{d,1}, {\mathscr {G}}'_{d,2}, \ldots , {\mathscr {G}}'_{d,N}),\; \end{aligned}$$

where $\varvec{I}_{r}$ is the $r\times r$ identity matrix, and $\varvec{\varLambda }_{y}$ and $\varvec{\varLambda }_{d}$ are $r\times r$ diagonal matrices.

We employ an expectation-maximization (EM) algorithm (e.g., Dempster et al. (1977)) to obtain the maximum likelihood estimate of the unknown parameters in (1), (2), (3), and (4). The procedure follows two steps: expectation (the conditional expectation of the complete likelihood function given the observations, E-step) and maximization (the maximization of the expected likelihood with respect to unknown parameters, M-step). The actual procedure to obtain the maximum likelihood estimator (MLE) by the EM algorithm is as follows¹²:

E-step::

Given the initial values of the parameters {$\varvec{\beta }^{(s)}$, $\varvec{a}_{y}^{(s)},\varvec{g}_{y}^{(s)}$}, {$\varvec{\delta }^{(s)}$, $\varvec{a}_{d}^{(s)},\varvec{g}_{d}^{(s)}$}, $\rho ^{(s)}$, $\sigma ^{(s)}$, define the conditional means and variances of the latent dependent variables, $y_{ij}^{*}$ and $d_{ij}^{*}$,

$$\begin{aligned} \mathbb {E}_{y_{ij}^{*} | d=0}[y_{ij}^{*}],\; \mathbb {E}_{d_{ij}^{*} | y_{ij}, d_{ij}=1}[d_{ij}^{*}],\;\mathbb {E}_{d_{ij}^{*} | d_{ij}=0}[d_{ij}^{*}],\; \mathbb {V}_{y_{ij}^{*} | d_{ij}=0}[y_{ij}^{*}],\;\mathbb {V}_{d_{ij}^{*} | y_{ij}, d_{ij}=1}[d_{ij}^{*}],\; \text{ and }\;\mathbb {V}_{d_{ij}^{*} | d_{ij}=0}[ d_{ij}^{*} ] \end{aligned}$$

and define

$$\begin{aligned} \tilde{y}_{ij}= & {} d_{ij}\cdot y_{ij}+(1-d_{ij})\cdot \mathbb {E}_{y_{ij}^{*} | d_{ij}=0}[y_{ij}^{*}]\\ \tilde{d}_{ij}= & {} d_{ij}\cdot \mathbb {E}_{d_{ij}^{*} | y_{ij}, d_{ij}=1}[d_{ij}^{*}] +(1-d_{ij})\cdot \mathbb {E}_{d_{ij}^{*} | d_{ij}=0}[d_{ij}^{*}] \end{aligned}$$

M-step::

This step uses three conditional maximization sub-steps (e.g., Meng and Rubin (1993); McLachlan and Krishnan (2008)):

1.:

Given {$\varvec{\beta }^{(s)}$, $\varvec{a}_{y}^{(s)}, \varvec{g}_{y}^{(s)}$},{$\varvec{\delta }^{(s)}$, $\varvec{a}_{d}^{(s)}, \varvec{g}_{d}^{(s)}$}, $\rho ^{(s)}$, $\sigma ^{(s)}$, define

$$\begin{aligned} \hat{\tilde{y}}_{ij}^{(s)}\equiv \tilde{y}_{ij}^{(s)} - \rho ^{(s)}\sigma ^{(s)}(\tilde{d}_{ij}^{(s)}-\mathbf {z}'_{ij}\varvec{\delta }^{(s)}-\alpha _{d,i}^{(s)}-\gamma _{d,j}^{(s)}-{\mathscr {A}}\;_{d,i}^{(s)}({\mathscr {G}}_{d,j}^{(s)})').\end{aligned}$$

Then, update {$\varvec{\beta }$,$\varvec{a}_{y},\varvec{g}_{y}$} by the minimizer of the following criterion function:

$$\begin{aligned} (\varvec{\beta }^{(s+1)}, \varvec{a}_{y}^{(s+1)}, \varvec{g}_{y}^{(s+1)}) =\arg \min _{\varvec{\beta }, \varvec{a}_{y}, \varvec{g}_{y}} \sum _{i=1}^{N}\sum _{j=1}^{N} \left( \hat{\tilde{y}}_{ij}^{(s)}-\mathbf {x}'_{ij}\varvec{\beta }-\alpha _{y,i}-\gamma _{y,j}-{\mathscr {A}}\;_{y,i}{\mathscr {G}}'_{y,j} \right) ^{2}. \end{aligned}$$

2.:

Given {$\varvec{\beta }^{(s+1)}$,$\varvec{a}_{y}^{(s+1)},\varvec{g}_{y}^{(s+1)}$}, {$\varvec{\delta }^{(s)}$,$\varvec{a}_{d}^{(s)},\varvec{g}_{d}^{(s)}$}, $\rho ^{(s)}$,$\sigma ^{(s)}$, define

$$\begin{aligned}\hat{\tilde{d}}_{ij}^{(s)}\equiv \tilde{d}_{ij}^{(s)} - \frac{\rho ^{(s)}}{\sigma ^{(s)}}(\tilde{y}_{ij}^{(s)}-\mathbf {x}'_{ij}\varvec{\beta }^{(s+1)}-\alpha _{y,i}^{(s)}-\gamma _{y,j}^{(s)}-{\mathscr {A}}\;_{y,i}^{(s)}({\mathscr {G}}_{y,j}^{(s)})'). \end{aligned}$$

Then, update {$\varvec{\delta }^{(s)}$,$\varvec{a}_{d}^{(s)},\varvec{g}_{d}^{(s)}$} by the minimizer of the following criterion function:

$$\begin{aligned} (\varvec{\delta }^{(s+1)},\varvec{a}_{d}^{(s+1)},\varvec{g}_{d}^{(s+1)}) =\arg \min _{\varvec{\delta },\varvec{a}_{d},\varvec{g}_{d}} \sum _{i=1}^{N}\sum _{j=1}^{N} \left( \hat{\tilde{d}}_{ij}^{(s)}-\mathbf {z}'_{ij}\varvec{\delta }-\alpha _{d,i}-\gamma _{d,j}-{\mathscr {A}}\;_{d,i}{\mathscr {G}}'_{d,j} \right) ^{2}. \end{aligned}$$

3.:

Given {$\varvec{\beta }^{(s+1)}$,$\varvec{a}_{y}^{(s+1)},\varvec{g}_{y}^{(s+1)}$},{$\varvec{\delta }^{(s+1)}$,$\varvec{a}_{d}^{(s+1)},\varvec{g}_{d}^{(s+1)}$},$\rho ^{(s)}$,$\sigma ^{(s)}$, update $\rho$ and $\sigma$ by

$$\begin{aligned} \sigma ^{(s+1)}= & {} \left( T_{yy}^{(s)}+\frac{(T_{yd}^{(s)})^{2}}{T_{dd}^{(s)}}\cdot (1-T_{dd}^{(s)})\right) ^{1/2},\;\; \rho ^{(s+1)}=\frac{1}{\sigma ^{(s+1)}}\cdot \frac{T_{yd}^{(s)}}{T_{dd}^{(s)}} \end{aligned}$$

where ($\pi _{y,ij}\equiv \alpha _{y,i}-\gamma _{y,j}-{\mathscr {A}}\;_{y,i}{\mathscr {G}}'_{y,j}$, $\pi _{d,ij}\equiv \alpha _{d,i}-\gamma _{d,j}-{\mathscr {A}}\;_{d,i}{\mathscr {G}}'_{d,j}$), and

$$\begin{aligned} T_{yy}^{(s)}= & {} \frac{1}{N(N-1)}\sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N} \left\{ (\tilde{y}_{ij}^{(s)}-\mathbf {x}'_{ij}\varvec{\beta }^{(s+1)}-\pi _{y,ij}^{(s+1)})^{2} + (1-d_{ij})\cdot \mathbb {V}_{y^{*}|d=0}^{(s)}[y_{ij}^{*}]\right\} \\ T_{yd}^{(s)}= & {} \frac{1}{N(N-1)}\sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N} \left\{ (\tilde{y}_{ij}^{(s)}-\mathbf {x}'_{ij}\varvec{\beta }^{(s+1)}-\pi _{y,ij}^{(s+1)}) (\tilde{d}_{ij}^{(s)}-\mathbf {z}'_{ij}\varvec{\delta }^{(s+1)}-\pi _{d,ij}^{(s+1)})\right. \\&+\left. (1-d_{ij})\cdot \rho ^{(s)}\sigma ^{(s)}\cdot \mathbb {V}_{d^{*}|d=0}^{(s)}[d_{ij}^{*}]\right\} \\ T_{dd}^{(s)}= & {} \frac{1}{N(N-1)}\sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N} \left\{ (\tilde{d}_{ij}^{(s)}-\mathbf {z}'_{ij}\varvec{\delta }^{(s+1)}-\pi _{d,ij}^{(s+1)})^{2} \right. \\& \left. +d_{ij}\cdot \mathbb {V}_{d^{*}|y,d=1}^{(s)}[d_{ij}^{*}]+(1-d_{ij}) \cdot \mathbb {V}_{d^{*}|d=0}^{(s)}[d_{ij}^{*}]\right\} \end{aligned}$$

The procedure proposed here is easy to implement: the minimization problems in the first two conditional maximization sub-steps can be solved in the same way as Bai (2009), and the third sub-step has closed-form solutions.

The asymptotic distribution of the MLE has a non-zero mean vector since we are facing the situation where the sample sizes of exporting countries and importing countries simultaneously go to infinity at the same rate. The asymptotic properties of the bias-corrected estimator can be derived along the same lines as Hahn and Kuersteiner (2002), Arellano and Hahn (2016), and Fernández-Val and Weidner (2018): the bias-corrected estimator is defined as:

$$\begin{aligned} \tilde{\varvec{\theta }}=\hat{\varvec{\theta }}- \mathbf {b}_{N}-\mathbf {d}_{N} \end{aligned}$$

where $\hat{\varvec{\theta }}$ is the MLE, and $\mathbf {b}_{N}$ and $\mathbf {d}_{N}$ are bias correction terms defined in (10) and (11) in the Appendix. The asymptotic distribution of the estimator is given as follows:

$$\begin{aligned} \sqrt{N(N-1)}\left( \tilde{\varvec{\theta }}-\varvec{\theta }_{0}\right) {\mathop {\longrightarrow }\limits ^{d}} N(\mathbf {0},\;\mathscr {W}_{NN}^{-1}\varPsi _{NN}\mathscr {W}_{NN}^{-1})\end{aligned}$$

1.4 Conditional expectation of the latent variables

The conditional means and variances of the latent variables, $y_{ij}^*$ and $d_{ij}^*$, given observations, $y_{ij}$ and $d_{ij}=1$, or $d_{ij}=0$ are,

$$\begin{aligned} \mathbb {E}_{y^{*}_{ij}|d_{ij}=0}[y_{ij}^{*}]= &\, {} \mathbf {x}'_{ij}\varvec{\beta }+\pi _{y,ij}-\rho \sigma _{u}\cdot \frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }\\ \mathbb {E}_{d_{ij}^{*}|y_{ij},d_{ij}=1}[d_{ij}^{*}]= & \,{} (1-\rho ^{2})^{1/2}\cdot \left\{ \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}}+\frac{\phi \left( \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}}\right) }{\varPhi \left( \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}}\right) }\right\} \\ \mathbb {E}_{d^{*}_{ij}|d_{ij}=0}[d_{ij}^{*}]= & \,{} \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij} -\frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }\\ \end{aligned}$$

where $\pi _{y,ij}\equiv \alpha _{y,i}-\gamma _{y,j}-{\mathscr {A}}\;_{y,i}{\mathscr {G}}'_{y,j}$, $\pi _{d,ij}\equiv \alpha _{d,i}-\gamma _{d,j}-{\mathscr {A}}\;_{d,i}{\mathscr {G}}'_{d,j}$, and

$$\begin{aligned} \mathbb {V}_{y^{*}_{ij}|d_{ij}=0}[y_{ij}^{*}]= & {} \sigma _{u}^{2}\left\{ 1+\rho ^{2}\frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }\left\{ \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}-\frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }\right\} \right\} \\ \mathbb {V}_{d^{*}_{ij}|y_{ij},d_{ij}=1}[d_{ij}^{*}]= & {} (1-\rho ^{2}) \cdot \left\{ 1-\frac{\phi \left( \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}}\right) }{\varPhi \left( \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}}\right) }\left( \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}} +\frac{\phi \left( \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}}\right) }{\varPhi \left( \frac{\eta _{ij}}{(1-\rho ^{2})^{1/2}}\right) }\right) \right\} \\ \mathbb {V}_{d^{*}_{ij}|d_{ij}=0}[d_{i}^{*}]= & {} 1+\frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }\left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij} -\frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }\right) \\ \text{ cov }(y^{*}_{ij},d^{*}_{ij})= & {} \rho \sigma _{u}\left\{ 1+\frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) } \cdot \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij} -\frac{\phi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }{1-\varPhi \left( \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}\right) }\right) \right\} \\= & {} \rho \sigma _{u}\cdot \mathbb {V}_{d^{*}_{ij}|d=0}[d_{ij}^{*}] \end{aligned}$$

where

$$\begin{aligned}\eta _{ij} = \mathbf {z}'_{ij}\varvec{\gamma }+\pi _{d,ij}+\rho \cdot \frac{y_{ij}^{*}-\mathbf {x}'_{ij}\varvec{\beta }-\pi _{y,ij}}{\sigma _{u}}.\end{aligned}$$

1.5 Asymptotic properties of the bias-corrected estimator

Denote

$$\begin{aligned} \mu _{y,ij} \equiv \mathbf {x}'_{ij}\varvec{\beta } +\alpha _{y,i}+\gamma _{y,j}+{\mathscr {A}}\;'_{y,i}{\mathscr {G}}_{y,j} \equiv \mathbf {x}'_{ij}\varvec{\beta } + \pi _{y,ij}, \end{aligned}$$

and

$$\begin{aligned} \mu _{d,ij} \equiv \mathbf {z}'_{ij}\varvec{\delta } +\alpha _{d,i}+\gamma _{d,j}+{\mathscr {A}}\;'_{d,i}{\mathscr {G}}_{d,j} \equiv \mathbf {z}'_{ij}\varvec{\delta } + \pi _{d,ij}. \end{aligned}$$

The likelihood function of the sample $\{\;\{\; y_{ij},d_{ij}\; \}_{j=1, j\ne i}^{N}\; \}_{i=1}^{N}$ is

$$\begin{aligned} \log L(\varvec{\theta },\varvec{a}_{y},\varvec{g}_{y},\varvec{a}_{d},\varvec{g}_{d}) = \sum _{i=1}^{N}\sum _{j=1, j\ne i}^{N} \ell _{ij}(\varvec{\theta },\varvec{a}_{y,i},\varvec{g}_{y,j},\varvec{a}_{d,i},\varvec{g}_{d,j}) \end{aligned}$$

where $\ell _{ij}$ is the log-likelihood contribution of the (i, j)-th observation:

$$\begin{aligned} \ell _{ij} = d_{ij}\cdot \log \frac{1}{\sigma } \phi \left( \frac{y_{ij}-\mu _{y,ij}}{\sigma }\right) + d_{ij}\cdot \log \varPhi \left( \frac{ \mu _{z,ij} + \rho \cdot (y_{ij}-\mu _{y,ij})/\sigma }{ (1-\rho ^2)^{1/2} } \right) + (1-d_{ij})\cdot \log \varPhi \left( \mu _{z,ij} \right) , \end{aligned}$$

The parameter vectors are summarized into the following vectors,

$$\begin{aligned}\varvec{a}_{d,i}\equiv ( \alpha _{d,i}, {\mathscr {A}}\;_{d,i})',\; \varvec{a}_{y,i} \equiv ( \alpha _{y,i}, {\mathscr {A}}\;_{y,i} )',\; \varvec{g}_{d,i} \equiv (\gamma _{d,j}, {\mathscr {G}}_{d,j} )',\; \varvec{g}_{y,i} \equiv (\gamma _{y,j}, {\mathscr {G}}_{y,j} )',\end{aligned}$$

and

$$\begin{aligned} \varvec{a}_{i} \equiv (\varvec{a}'_{d,i},\varvec{a}'_{y,i})',\; \varvec{g}_{j} \equiv (\varvec{g}'_{d,i},\varvec{g}'_{y,i})', \varvec{\theta }\equiv (\varvec{\beta }', \varvec{\delta }', \sigma _{u}, \rho )'. \end{aligned}$$

The bias-corrected estimator of $\varvec{\theta }$ is asymptotically regarded as the solution of the following corrected score function (Fernández-Val and Weidner (2018) and Arellano and Hahn (2016)),

$$\begin{aligned} \frac{1}{N(N-1)}\sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N} \varvec{\psi }_{ij} (\varvec{\theta }) \equiv \frac{1}{N(N-1)}\sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N} \left( \frac{\partial \ell _{ij}}{\partial \varvec{\theta }}+\varXi '_{i\bullet }\frac{\partial \ell _{ij} }{ \partial \varvec{a}_{i}}+\varXi '_{\bullet j}\frac{\partial \ell _{ij}}{\partial \varvec{g}_{j}} \right) = \mathbf {0}, \end{aligned}$$

(9)

where the incidental parameters, $\varvec{a}_{i}$ and $\varvec{g}_{j}$, are evaluated at a given value of $\varvec{\theta }$ in the maximization process, and

$$\begin{aligned} \underbrace{\;\;\varXi _{i\bullet }\;\;}_{(2+2r)\times K}= & {} \mathop {\mathrm {plim}}_{N\rightarrow \infty }\left( - \sum _{j=1,j\ne i}^{N} \frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{a}_{i}\partial \varvec{a}'_{i}} \right) ^{-1}\left( \sum _{j=1,j\ne i}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{a}_{i}\partial \varvec{\theta }'} \right) \\ \underbrace{\;\;\varXi _{\bullet j}\;\;}_{(2+2r)\times K}= & {} \mathop {\mathrm {plim}}_{N\rightarrow \infty } \left( - \sum _{i=1, i \ne j}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{g}_{j}\partial \varvec{g}'_{j}} \right) ^{-1}\left( \sum _{i=1, i \ne j}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{g}_{j}\partial \varvec{\theta }'} \right) \end{aligned}$$

are terms which make the log-likelihood function informationally orthogonal between $\varvec{\theta }$ and the incidental parameters, $\varvec{a}_{i}$ and $\varvec{g}_{j}$. See Section 4.2 of Fernández-Val and Weidner (2018).

The asymptotic variance and the analytical expression of the bias correction term are analogously derived as Example 10 in Fernández-Val and Weidner (2018). The asymptotic variance of the bias-corrected estimator is given as $\mathscr {W}_{NN}^{-1}\varPsi _{NN}\mathscr {W}_{NN}^{-1}$ where

$$\begin{aligned} \varPsi _{NN}= & {} \frac{1}{N(N-1)}\sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N} \left( \frac{\partial \ell _{ij}}{\partial \varvec{\theta }}+\varXi '_{N,i\bullet }\frac{\partial \ell _{ij} }{ \partial \varvec{a}_{i}}+\varXi '_{N,\bullet j}\frac{\partial \ell _{ij}}{\partial \varvec{g}_{j}} \right) \cdot \left( \frac{\partial \ell _{ij}}{\partial \varvec{\theta }}+\varXi '_{N,i\bullet }\frac{\partial \ell _{ij} }{ \partial \varvec{a}_{i}}+\varXi '_{N,\bullet j}\frac{\partial \ell _{ij}}{\partial \varvec{g}_{j}} \right) ' \\ \mathscr {W}_{NN}= & {} \frac{1}{N(N-1)}\sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N} \left\{ \frac{ \partial ^{2}\ell _{ij} }{ \partial \varvec{\theta } \partial \varvec{\theta }' } +\varXi '_{N, i \bullet } \frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{a}_{i}\partial \varvec{a}'_{i}}\varXi _{N,i \bullet } +\varXi '_{N,\bullet j}\frac{\partial ^{2}\ell _{ij}}{ \partial \varvec{g}_{j}\partial \varvec{g}'_{j}}\varXi _{N,\bullet j}\right\} . \end{aligned}$$

where $\varXi _{N,i\bullet }$ and $\varXi _{N,\bullet j}$ are sample analogs of $\varXi _{i\bullet }$ and $\varXi _{\bullet j}$, respectively. According to the formula in page 129 in Fernández-Val and Weidner (2018), the k-th element of the bias correction terms are derived as follows (note that we use the following abbreviations, $\sum _{j\ne i}^{N}$ and $\sum _{i\ne j}^{N}$ for $\sum _{j=1, j \ne i}^{N}$ and $\sum _{i=1, i \ne j}^{N}$, respectively),

$$\begin{aligned}b_{k,N} &= -\frac{1}{N}\sum _{i=1}^{N}\text{ trace }\left[ \left( \sum _{j \ne i}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{a}_{i}\partial \varvec{a}'_{i}} \right) ^{-1}\left( \sum _{j \ne i}^{N}\frac{\partial \ell _{ij}}{\partial \varvec{a}_{i}}\frac{\partial \varvec{\psi }_{k,ij} }{ \partial \varvec{a}'_{i}} \right) \right]\\&\quad +\frac{1}{2N}\sum _{i=1}^{N}\text{ trace }\Biggl [ \left( \sum _{j \ne i}^{N}\frac{\partial ^{2}\varvec{\psi }_{k,ij} }{ \partial \varvec{a}_{i}\partial \varvec{a}'_{i}} \right) \left( \sum _{j \ne i}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{a}_{i}\partial \varvec{a}'_{i}} \right) ^{-1}\left( \sum _{j \ne i}^{N}\frac{\partial \ell _{ij} }{ \partial \varvec{a}_{i} } \frac{\partial \ell _{ij}}{\partial \varvec{a}'_{i}} \right) \left( \sum _{j \ne i}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{a}_{i}\partial \varvec{a}'_{i}} \right) ^{-1}\Biggl ] \end{aligned}$$

(10)

$$\begin{aligned}d_{k,N} &= -\frac{1}{N}\sum _{j=1}^{N}\text{ trace }\left[ \left( \sum _{i \ne j}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{g}_{j}\partial \varvec{g}'_{j}} \right) ^{-1}\left( \sum _{i \ne j}^{N}\frac{\partial \ell _{ij}}{\partial \varvec{g}_{j}} \frac{\partial \varvec{\psi }_{k,ij}}{\partial \varvec{g}'_{j}} \right) \right] \\&\quad+\frac{1}{2N}\sum _{j=1}^{N}\text{ trace }\Biggl [\left( \sum _{i \ne j}^{N}\frac{\partial \varvec{\psi }_{k,ij} }{ \partial \varvec{g}_{j}\partial \varvec{g}'_{j}} \right) \left( \sum _{i \ne j}^{N}\frac{\partial ^{2}\ell _{ij} }{ \partial \varvec{g}_{j}\partial \varvec{g}'_{j} } \right) ^{-1}\left( \sum _{i \ne j}^{N}\frac{\partial \ell _{ij}}{\partial \varvec{g}_{j}} \frac{\partial \ell _{ij}}{\partial \varvec{g}'_{j} } \right) \left( \sum _{i \ne j}^{N} \frac{\partial ^{2}\ell _{ij} }{\partial \varvec{g}_{j}\partial \varvec{g}'_{j}} \right) ^{-1} \Biggl ] \nonumber \\ & \quad k= {} 1,2,\ldots ,K. \end{aligned}$$

(11)

where $\varvec{\psi }_{k,ij}$ is the k-th element of $\varvec{\psi }_{ij}$ defined in (9). The partial derivatives in these formula are given as follows ($\eta _{ij}\equiv \mu _{d,ij}-\rho (y_{ij}-\mu _{yij})/\sigma$, $\zeta _{ij}\equiv \mu _{d,ij}$),

$$\begin{aligned} \underbrace{\;\; \frac{\partial \ell _{ij}}{\partial \varvec{a}_{i}}\;\;}_{(2+2r)\times 1}= & {} \left( \begin{array}{c} d_{ij} \cdot \frac{1}{\sigma }\cdot \left\{ \frac{y_{ij}-\mu _{y,ij}}{\sigma }-\lambda \left( \eta _{ij}\right) \cdot \frac{\rho }{(1-\rho ^{2})^{1/2}}\right\} \cdot \left( \begin{array}{c} 1\\ {\mathscr {G}}_{y,j}\\ \end{array}\right) \\ \left\{ d_{ij}\cdot \lambda \left( \eta _{ij}\right) \cdot \frac{1}{(1-\rho ^{2})^{1/2}}-(1-d_{ij})\cdot \lambda \left( -\zeta _{ij}\right) \right\} \cdot \left( \begin{array}{c} 1\\ {\mathscr {G}}_{d,j}\\ \end{array}\right) \end{array}\right) \\ \underbrace{\;\;\frac{\partial ^{2}\ell _{ij}}{ \partial \varvec{a}_{i}\partial \varvec{a}'_{i}}\;\;}_{(2+2r)\times (2+2r)}= & {} \left( \begin{array}{cc} \left\{ -1-\cdot \frac{\rho ^{2}\cdot \xi _{+}\left( \eta _{ij}\right) }{1-\rho ^{2}}\right\} \cdot \frac{d_{ij}}{\sigma ^{2}} \cdot \varDelta _{yy,j}^{{\mathscr {G}}} &{} \frac{\rho \cdot \xi _{+}\left( \eta _{ij}\right) }{1-\rho ^{2}}\cdot \frac{d_{ij}}{\sigma }\cdot \varDelta _{yd,j}^{{\mathscr {G}}} \\ \frac{\rho \cdot \xi _{+}\left( \eta _{ij}\right) }{1-\rho ^{2}}\cdot \frac{d_{ij}}{\sigma }\cdot \varDelta _{dy,j}^{{\mathscr {G}}} &{} -\left\{ \frac{d_{it}\cdot \xi _{+}\left( \eta _{ij}\right) }{1-\rho ^{2}}-(1-d_{ij})\cdot \xi _{-}\left( \zeta _{ij}\right) \right\} \varDelta _{dd,j}^{{\mathscr {G}}} \end{array}\right) , \\ \frac{\partial \ell _{ij}}{\partial \varvec{g}_{j}}= & {} \left( \begin{array}{c} d_{ij}\cdot \frac{1}{\sigma }\cdot \left\{ \frac{y_{ij}-\mu _{y,ij}}{\sigma }-\lambda \left( \eta _{ij}\right) \cdot \frac{\rho }{(1-\rho ^{2})^{1/2}}\right\} \cdot \left( \begin{array}{c} 1\\ {\mathscr {A}}\;_{y,i}\\ \end{array}\right) \\ \left\{ d_{ij}\cdot \lambda \left( \eta _{ij}\right) \cdot \frac{1}{(1-\rho ^{2})^{1/2}} -(1-d_{ij})\cdot \lambda \left( -\zeta _{ij}\right) \right\} \cdot \left( \begin{array}{c} 1\\ {\mathscr {A}}\;_{d,i}\\ \end{array}\right) \end{array}\right) \\ \frac{\partial ^{2}\ell _{ij}}{\partial \varvec{g}_{j}\partial \varvec{g}'_{j}}= & {} \left( \begin{array}{cc} \left\{ -1-\frac{\rho ^{2}\cdot \xi _{+}\left( \eta _{ij}\right) }{1-\rho ^{2}}\right\} \cdot \frac{d_{ij}}{\sigma ^{2}}\cdot \varDelta _{yy,i}^{{\mathscr {A}}\;} &{} \xi _{+}\left( \eta _{ij}\right) \cdot \frac{\rho }{1-\rho ^{2}}\cdot \frac{d_{ij}}{\sigma }\cdot \varDelta _{yy,i}^{{\mathscr {A}}\;} \\ \frac{\rho \cdot \xi _{+}\left( \eta _{ij}\right) }{1-\rho ^{2}}\cdot \frac{d_{ij}}{\sigma }\cdot \varDelta _{dy,i}^{{\mathscr {A}}\;} &{} -\left\{ d_{ij}\cdot \frac{\xi _{+}\left( \eta _{ij}\right) }{1-\rho ^{2}}-(1-d_{ij})\cdot \xi _{-}\left( \zeta _{ij}\right) \right\} \varDelta _{dd,i}^{{\mathscr {A}}\;} \end{array}\right) \end{aligned}$$

where $\varDelta ^{{\mathscr {G}}}_{uv,j}=\left( \begin{array}{c} 1\\ {\mathscr {G}}_{u,j} \end{array}\right) \left( \begin{array}{c} 1\\ {\mathscr {G}}_{v,j} \end{array}\right)^\prime$ and $\varDelta ^{\mathscr {A}}\;_{uv,i}=\left( \begin{array}{c} 1\\ {\mathscr {A}}\;_{u,i} \end{array}\right) \left( \begin{array}{c} 1\\ {\mathscr {A}}\;_{v,i} \end{array}\right)^\prime$, $u,v = y,d$, and

$$\begin{aligned} \lambda \left( \eta _{ij}\right) =\frac{\phi (\eta _{ij})}{\varPhi (\eta _{ij})},\, \xi _{+}\left( \eta _{ij}\right) =\lambda \left( \eta _{ij}\right) \left\{ \eta _{ij}+\lambda \left( \eta _{ij}\right) \right\} ,\, \xi _{-}\left( \zeta _{ij}\right) =\lambda \left( -\zeta _{ij}\right) \left\{ \zeta _{ij}-\lambda \left( -\zeta _{ij}\right) \right\} . \end{aligned}$$

Other partial derivatives, $\frac{\partial ^{3}{\ell}_{ij}}{\partial \theta _{k}\partial \varvec{a}_{i}\partial \varvec{a}^\prime_{i}}$, $\frac{\partial ^{2}{\ell}_{ij}}{\partial \theta _{k}\partial \varvec{a}^\prime_{i}}$, $\frac{\partial ^{2}{\ell}_{ij}}{\partial \theta _{k}\partial \varvec{g}^\prime_{j}}$, and$\frac{\partial^{3}{\ell}_{ij}}{\partial \theta _{k}\partial \varvec{g}_{j}\partial \varvec{g}^\prime_{j}}$ are also derived in a similar way.