Resolución 046/06 CGE
DE LOS ASCENSOS DE JERAQUIA EN EL NIVEL INICIAL
The analysis is centered on the economic performance of firms located in the 78 municipalities. Measures of production and productivity of firms are needed but there no survey data for productivity exists. In order to obtain such a variable, the most widely-used approach is the Cobb Douglas function (Romp et. al, 2007), which assumes constant returns to scale and perfect competition. The Cobb-Douglas function is expressed using labor and physical capital as inputs, and total factor productivity as the technological catalyst that drives production Y in the firm i at time t, as expressed in equation (51):
𝑌𝑖,𝑡,𝑟 = 𝑇𝐹𝑃𝑖,𝑡,𝑟𝐿𝛼𝑖,𝑡,𝑟𝐾 𝑖,𝑡,𝑟𝛽 (51) 0 2 4 6 8 10 12 14 2001 2002 2003 2004 2005 2006 2007 2008 Lo g o f P ro d u ct io n , C ap it al a n d L ab o r Years
Graph 1. Production, Capital and Labor
Production Private Capital Labor
Source: Banco de Guatemala
-10.05 -10 -9.95 -9.9 -9.85 -9.8 -9.75 0 1 2 3 4 5 6 7 8 2001 2002 2003 2004 2005 2006 2007 2008 Log of P roduc ti vi ty , P ubl ic C api ta l and R & D Years
Graph 2. Total Factor Productivity, Public Capital and R&D
TFP Public Capital R&D
Source: Own Calculation, Banco de Guatemala, FUNCEDE
2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 2001 2002 2003 2004 2005 2006 2007 2008 Log of N um be r of pe opl e Years
Graph 3. Number of People by Degree of Education
Elementary Middle S. High S.
The subscripts i = 1, 2, 3, …. N is the firm index; t = 2001, 2002, …, 2008 is the yearly index; and r = 1, …, 78 is the municipality index where firms belong. Y is the output of firm i at time t belonging to region r; L is the labor input; K is the private capital input; 𝛼 is the labor income share; and β is the private capital income share.
The mechanism for computing the TPF is to encompass output, employment and private capital as shown in equation (52):
𝑌𝑖,𝑡,𝑟 𝐿𝛼𝑖,𝑡,𝑟𝐾
𝑖,𝑡,𝑟𝛽
= 𝑇𝐹𝑃𝑖,𝑡,𝑟 (52)
From these formulations, it is possible to estimate equation (53), which describes the endogenous growth model, where "A" is the unexplained technical progress; 𝑇𝐹𝑃𝑖,𝑡,𝑟 is the total
factor productivity that represents the technical change driven by human capital (HC) available in the municipality where the firm locates its plant, research and development (R&D) available in the municipality that lead to innovations, and public investment (PK) executed in the municipality r at the time t, as shown below:
𝑇𝐹𝑃𝑖,𝑡,𝑟 = 𝐴𝑖,𝑡,𝑟𝐻𝐶𝑖,𝑡,𝑟𝑅𝐷𝑖,𝑡,𝑟𝑃𝐾𝑖,𝑡,𝑟 (53)
Transformed into log terms, (53) may be re-written as:
ln(𝑇𝐹𝑃𝑖,𝑡,𝑟) = ln(𝐴𝑖,𝑡,𝑟) + 𝛽1𝑙𝑛(𝐻𝐶𝑖,𝑡,𝑟) + 𝛽2ln(𝑅𝐷𝑖,𝑡,𝑟) + 𝛽3𝑙𝑛(𝑃𝐾𝑖,𝑡,𝑟) + 𝑒𝑖,𝑡,𝑟 (54)
Considering that the unexplained part of the TFP can be contained in two (time and region) fixed effects and derived from the form ln(𝐴𝑖,𝑡) = 𝜃𝑖+ 𝜃 𝑡, equation (55) below presents
the first baseline model to analyze the contribution of public investment in productivity. In addition to the elements expressed in this equation and the fixed effects, the model will include some control variables for agglomeration, the economic sector, and the distinction between core and periphery regions. For simplifying the notation, the log terms in the equation are presented in lowercase:
𝑡𝑓𝑝𝑖,𝑡,𝑟 = 𝜃𝑟+ 𝜃 𝑡+ 𝛽1ℎ𝑐𝑖,𝑡,𝑟 + 𝛽2𝑟𝑑𝑖,𝑡,𝑟+ 𝛽3𝑝𝑘𝑖,𝑡,𝑟+ ∑ 𝛾𝑛𝑧𝑖,𝑡,𝑟
𝑛
One advantage of considering the fixed effects of time and region is that they can provide estimates per region and per year, avoiding the unnecessary imposition of homogeneous coefficients across regions. At the same time, the fixed effects method offer a technique that can help to reduce endogeneity due to omitted variable bias problem. In other words, fixed effects on region help to capture time-invariant unobservable factors that do not change in each region, including the geography of the place. Fixed effects on time help to capture unobservable factors in the regions in average within each period of time, for instance an electoral year.
5.3.2.2 The extended model:
One purpose of this paper is to measure the spillover effects across municipalities; this will be accomplished using a spatial weight matrix (W). It is important to recognize that the inclusion of this variable is not without criticism in terms of the role it plays and the form in which it is included in the analysis. On the positive side, the inclusion of the estimation of economic linkages across the regions via distance based measures offers an opportunity to capture the effects of neighbors. On the other hand, there is concern about the degree to which bias might be introduced into the model, for instance a misspecification of the model (Anselin, 1988). Crespo and Feldkircher (2012) offer significant and critical empirical evidence about the importance of choosing a weight matrix. They handle spatial dependence among observations while taking care of uncertainty for choosing different spatial structures through a Bayesian model averaging. One of their goals is to evaluate how values of parameters changes with different W in a simulation study, and show that choosing the W matrix is critical for the estimation of the parameters of the covariates because W may lead to wrong conclusions. In spite of such efforts, there is still no consensus of how to determine the W, for instance, LeSage and Fischer (2008) argue that one way to choose the W is by using the “goodness of fit statistics”.
For dealing with this potential problem, the present paper presents different weight matrices to apply in the extended model (equation 58) for comparing the results and obtaining robust conclusions. The present paper uses the time-distance inverse matrix, the Queen matrix,
the Euclidean matrix and the 6th-nearest neighbor matrix; all of which are row-standardized such that:
𝑤𝑟,𝑠∗= 𝑤𝑟,𝑠
∑ 𝑤𝑠 𝑟,𝑠 (56)
Therefore, equation (57) below expresses the weighted physical public investment of the region r according to distance with respect to region s (where r ≠ s). Thus:
𝑃𝐾𝑊𝑟,𝑠 = ∑ 𝑤𝑟𝑠∗𝑃𝐾 𝑠
𝑠 (57)
In a similar manner to the basic model, the baseline extended model is expressed in log terms,
𝑡𝑓𝑝𝑖,𝑡,𝑟= 𝜃𝑟+ 𝜃 𝑡+ 𝛽1ℎ𝑐𝑖,𝑡,𝑟+ 𝛽2𝑟𝑑𝑖,𝑡,𝑟+ 𝛽3𝑝𝑘𝑖,𝑡,𝑟+ 𝛽4𝑝𝑘𝑤𝑖,𝑟,𝑠+ ∑ 𝛾𝑛𝑧𝑖,𝑡,𝑟 𝑛
+ 𝜀𝑖,𝑡,𝑟 (58)
5.3.2.3 The Multilevel Analysis:
The third model studies the impact of public investment in infrastructure on firm’s production by using multilevel analysis. The multilevel analysis is a hierarchical linear model that takes into account the spatial disaggregation of the economic and social units, i.e. in regional economics is common to observe nested regional information containing different level of aggregation: for instance, countries containing information of regions, regions containing information of cities, and cities containing information of individuals. It allows breaking the assumption that all observations have identical distributions, while still considering independence. Usually, the estimation of the multilevel analysis entails the estimation of the effect of the mean of a particular group of individuals, while estimating the effect of the mean of the region where the group of individuals belongs. Indeed, Corrado and Fingleton (2011) state that many economic researches containing spatial econometric models don’t consider the hierarchical information of the space, which misleads the analysis of the spatial lags variables when failing to provide inference of the effects coming from different hierarchical levels. This is because requiring the regression coefficients in all regions to be the same is generally too restrictive as each region has different economic and social characteristics. Chung and Hewings
(2013) demonstrate that when considering hierarchical models, is possible to achieve an accurate estimation of the influence of shocks with different levels of information. For example, when considering a hierarchical model, they found that the dependency on national production has a weaker influence on regional economies than when considering a non-hierarchical model.
The following model is intended to provide estimates of the influence of public capital on production of the firms. The model captures the effect of information coming from the firm and municipality level, and the goal is to examine the robustness of the influence of public investment on firms’ economic performance. Thus, the first step is to identify an equation that represents the random behavior on every level:
𝑌𝑖𝑟𝑡= 𝛽0𝑟𝑡+ 𝑣𝑖𝑟𝑡 (59)
The production 𝑌 of the firm i nested in the region r during year t is equal to the average of the production in region r for every year t plus an individual random error 𝑣𝑖𝑟𝑡. In addition,
there exists the possibility that all of the firms in a specific region can be affected by a common influence within the region 𝑢0𝑟𝑡. Hence:
𝛽0𝑟𝑡= 𝛾00𝑡+ 𝑢0𝑟𝑡 (60)
Therefore, taking into consideration the structural equation that comes from combining equations (59) and (60), the errors 𝑣𝑖𝑟𝑡 and 𝑢0𝑟𝑡 are assumed to have a zero mean and variances
of 𝜎𝑣2 and 𝜎𝑢2 , respectively. Therefore, 𝜎𝜀2= 𝜎𝑣2+ 𝜎𝑢2 represents the sum of the within-group
and between-group variances of production. Taking these into consideration, it is possible to calculate the intra-class correlation (𝜌), which is a measure of the proportion of the variance explained at the group level:
𝜌 = 𝜎𝑢
2
𝜎𝑣2+ 𝜎
𝑢2 (61)
Equation (61) makes it possible to detect whether the local conditions of the region during a specific year influence the production of the firm located in that region. When the rho value (𝜌) approaches zero, it is possible to assert that the production of the firm i is no longer attributable to regional influences r.
The next step consists of adding the fixed effects as represented in the equation (12). 𝑦𝑖,𝑟,𝑡= 𝛾0,0,𝑡+ 𝛽1ℎ𝑐𝑖,𝑟,𝑡+ 𝛽2𝑟𝑑𝑖,𝑟,𝑡+ 𝛽3𝑝𝑘𝑖,𝑟,𝑡+ ∑ 𝛾𝑛𝑧𝑖,𝑡,𝑟 𝑛 + 𝑢0,𝑟,𝑡+ 𝛿1𝑙𝑖,𝑟,𝑡 + 𝛿2𝑘𝑖,𝑟,𝑡+ 𝑣𝑖,𝑟,𝑡 (62)
In addition, an extended model (equation 63) will be used that includes the spatial effects from the other regions.
𝑦𝑖,𝑟,𝑡 = 𝛾0,0,𝑡+ 𝛽1ℎ𝑐𝑖,𝑟,𝑡+ 𝛽2𝑟𝑑𝑖,𝑟,𝑡+ 𝛽3𝑝𝑘𝑖,𝑟,𝑡+ 𝛽4𝑊𝑝𝑘𝑖,𝑟,𝑠+ 𝛽5𝑊ℎ𝑐𝑖,𝑟,𝑠
+ 𝛽6𝑊𝑟𝑑𝑖,𝑟,𝑠+ ∑ 𝛾𝑛𝑧𝑖,𝑡,𝑟
𝑛
+ 𝑢0,𝑟,𝑠+ 𝛿1𝑙𝑖,𝑟,𝑡+ 𝛿2𝑘𝑖,𝑟,𝑡+ 𝑣𝑖,𝑟,𝑡 (63)
The following specification (64) has two purposes. First, it provides an elasticity of the private inputs for a proxy of productivity. Second, it allows us to control for a potential multicollinearity problem involving private capital and labor variables. Indeed, the association between labor and private capital can be highly correlated in the sense that firms prepare their plan of production by considering the combination of both variables and not separately. One way to solve this problem is by normalizing the Cobb-Douglas production function with respect to labor inputs as follows:
(𝑦𝑖𝑟𝑡− 𝑙𝑖𝑟𝑡) = 𝛾00𝑡+ 𝛽1ℎ𝑐𝑖𝑟𝑡+ 𝛽2𝑟𝑑𝑖𝑟𝑡+ 𝛽3𝑝𝑘𝑖𝑟𝑡+ 𝛽4𝑊𝑝𝑘𝑖𝑟𝑡+ 𝛽5𝑊ℎ𝑐𝑖𝑟𝑡 + 𝛽6𝑊𝑟𝑑𝑖𝑟𝑡+ ∑ 𝛾𝑛𝑧𝑖,𝑡,𝑟
𝑛
+ 𝑢0𝑟𝑡+ 𝛿2(𝑘𝑖𝑟𝑡− 𝑙𝑖𝑟𝑡) + 𝑣𝑖𝑟𝑡 (64)