Productivity Convergence in Vietnamese Manufacturing Industry: Evidence Using a Spatial Durbin Model

Abstract

This paper applies the $\beta$-convergence regression model in order to assess convergence of total factor productivity among Vietnamese provinces for manufacturing industries. Specifically, we express this model in the form of a Spatial Durbin Model (SDM), which allows us to take into account the presence of omitted variables that can be spatially correlated and correlated with the initial level of productivity. We calculate the annual total factor productivity (TFP) of 63 Vietnamese provinces and 6 manufacturing industries, using the results of the structural estimation of a value-added production function from firm data over the period from 2000 to 2012. The regression of growth rates of TFP over this period on the initial levels of productivity using SDM shows that there is convergence in most industries, i.e. the gap between lower-productivity and higher-productivity provinces decreases. These results also show the importance of modeling the indirect effect of the initial level of productivity of a province on its TFP growth rate, through its effect on neighboring provinces. The inclusion of these indirect effects is made possible by SDM and increases the speed of convergence for most considered manufacturing industries, except for metal and machinery, and transportation and telecommunication.

Appendices

Appendix 1: Derivation of the Spatial Durbin Model

Consider a set of n spatial units whose neighborhood can be described by a proximity matrix W. The matrix W is an $n \times n$ deterministic, non-negative spatial weight matrix. The elements of W are used to specify the spatial dependence structure among the observations. If observation unit i is related to observation j, then $w_{ij} >0$. Otherwise, $w_{ij} = 0$, and the diagonal elements $w_{ii}$ are set to zero as a normalization of the model. The matrix is also standardized to have row-sums of unity. We are interested in estimating the relationship between two variables and we assume that their data generating process can be written as

$$\begin{aligned} y= & {} \beta x + \varepsilon \\ \varepsilon= & {} \rho W \varepsilon + v\\ v= & {} \delta x + u \end{aligned}$$

where y is an $n \times 1$ vector of observations on the continuous dependent variable, x is an $n \times k$ matrix of observations on the explanatory variable, $\varepsilon$ is an $n \times 1$ vector of a spatially correlated omitted variable following a spatial autoregressive process with autoregressive coefficient $\rho$, and v and u are $n \times 1$ vectors of i.i.d. random error terms. The last equation shows that the omitted variable which is correlated with v, is also correlated with the explanatory variables when $\delta \ne 0$. Using this set of equations, [16] show that the resulting model is a spatial Durbin model (see also [10]). Indeed, the first equation can be written as

$$\begin{aligned} y = \beta x + (I - \rho W)^{-1} v \end{aligned}$$

by replacing $\varepsilon$ with $(I - \rho W)^{-1} v$ from the second equation. After rearranging terms, we get

$$\begin{aligned} y = \rho W y + \beta x - \rho W \beta x + v \end{aligned}$$

which can be expressed as

$$\begin{aligned} y = \rho W y + \gamma _1 x + \gamma _2 W x + u \end{aligned}$$

using the third equation and defining $\gamma _1 = \beta + \delta$ and $\gamma _2 = \rho \beta$. This last expression is known as the spatial Durbin model as Anselin [4] proposed the adaptation of the Durbin model in time series to the spatial context.

Appendix 2: Estimation Procedure of the Value Added Production Function

This appendix draws from lectures given by Jeffrey Wooldridge at the University of Wisconsin, Madison, in 2008.

Consider a value added production function for firm i in period t, i.e.

$$\begin{aligned} y_{it} = \alpha + \beta l_{it} + \gamma k_{it} + \omega _{it} + \varepsilon _{it} \end{aligned}$$

where $y_{it}$ denotes the firm output, here measured as value added, $l_{it}$ is the labor input, and $k_{it}$ the observed state variable, i.e. capital input, all in logarithmic form. The sequence $\{ \omega _{it}, t = 1,\ldots ,T \}$ is unobserved productivity, and $\{ \varepsilon _{it}, t = 1,\ldots ,T \}$ is a sequence of shocks. The key difference between $\omega _{it}$ and $\varepsilon _{it}$ is that the former is a state variable and, hence, impacts on the firm’s decisions. It is not observed by the econometrician, and it can impact on the choice of inputs, leading to the well-known simultaneity problem in production function estimation (see [2]).

The demand of intermediate input as a function of the state variable and unobserved level of productivity of a firm,⁷ or

$$\begin{aligned} m_{it} = m(k_{it},\omega _{it}) \end{aligned}$$

where $m_{it}$ denotes the intermediate input. For simplicity, [17] assume that the function m(., .) is time invariant and show that making mild assumptions on the firm’s production function, this demand function is monotonically increasing in $\omega _{it}$. This allows inversion of the intermediate input demand function and allows to write unobserved level of productivity of a firm as a function of the capital and intermediate inputs, i.e.

$$\begin{aligned} \omega _{it} = g(k_{it},m_{it}) \end{aligned}$$

Under the assumption

$$\begin{aligned} \mathop {\text{ E }}\left( \varepsilon _{it} | l_{it}, k_{it}, m_{it} \right) = 0 \end{aligned}$$

we get the following regression function

$$\begin{aligned} \mathop {\text{ E }}\left( y_{it} | l_{it}, k_{it}, m_{it} \right)= & {} \alpha + \beta l_{it} + \gamma k_{it} + g(k_{it},m_{it}) \\= & {} \beta l_{it} + h(k_{it},m_{it}) \end{aligned}$$

where $h(k_{it},m_{it}) = \alpha + \gamma k_{it} + g(k_{it},m_{it})$. Since g(., .) is a general function, $\gamma$ and $\alpha$ are not identified from the previous regression model.

As we are interested not only by the estimation of $\beta$ but also of $\gamma$, we can follow the direct estimation procedure proposed by Wooldridge [24].⁸ In order to identify $\gamma$ along with $\beta$, [24] follows [17, 22] by assuming that

$$\begin{aligned} \mathop {\text{ E }}\left( \varepsilon _{it} | l_{it}, k_{it}, m_{it}, l_{it-1}, k_{it-1}, m_{it-1}, \ldots , l_{i1}, k_{i1}, m_{i1} \right) = 0 \end{aligned}$$

The dynamic in the productivity process is also restricted such that

$$\begin{aligned} \mathop {\text{ E }}\left( \omega _{it} | k_{it}, l_{it-1}, k_{it-1}, m_{it-1}, \ldots , l_{i1}, k_{i1}, m_{i1} \right)= & {} \mathop {\text{ E }}\left( \omega _{it} | \omega _{it-1} \right) \\= & {} f(\omega _{it-1}) = f\left( g(k_{it-1},m_{it-1})\right) \end{aligned}$$

where the latter equivalence holds for some function f(.) because $\omega _{it-1} = g(k_{it-1},$ $m_{it-1})$. A consequence of this restriction is that the variable labor input $l_{it}$ is allowed to be correlated with the innovation $a_{it}$ in $\omega _{it} = f(\omega _{it-1}) + a_{it}$ while $k_{it}$, past values of $l(_{it}, k_{it}, m_{it})$, and functions of these are uncorrelated with $a_{it}$.

Plugging $\omega _{it} = f\left( g(k_{it-1},m_{it-1})\right) + a_{it}$ into the expression of the value added production function gives

$$\begin{aligned} y_{it} = \alpha + \beta l_{it} + \gamma k_{it} + f\left( g(k_{it-1},m_{it-1})\right) + u_{it} \end{aligned}$$

where $u_{it} = a_{it} + \varepsilon _{it}$. This equation allows to identify $\beta$ and$\gamma$ provided we have the orthogonality condition

$$\begin{aligned} \mathop {\text{ E }}\left( u_{it} | k_{it}, l_{it-1}, k_{it-1}, m_{it-1}, \ldots , l_{i1}, k_{i1}, m_{i1} \right) = 0 \end{aligned}$$

Effectively, $k_{it}$, $k_{it-1}$ and $m_{it-1}$ act as their own instruments and $l_{it-1}$ as an instrument for $l_{it}$. The functions g(., .) and f(.) can be approximated by low-order polynomials and then parameters involved in the previous equation can be estimated using instrumental variable technique.