Constructing spatial weight matrix

An important component of spatial econometric models is the spatial weights matrix. It is a non-random matrix that specifies the spatial relationship between observations exogenously.

Hence, the spatial weights matrix W specifies what constitutes a neighbourhood and how and whether potential neighbours interact. There are many possible ways to define the spatial weight matrix. There are (queen and rook) contiguity matrices, inverted distance, nearest neighbours, k nearest neighbours, economically interactive, and on. While there is an ongoing discussion as to how best to determine empirically what is the correct matrix (which has led to some scepticism regarding the use of spatial econometrics in general) we side with a theory-driven approach advanced by the likes of Corrado and Fingleton (2012) and Cook, An, and Favero (2019).

23This is likely due in part to the persistent, sequential economic crises the country has endured amid economic restructuring characterized by rapid de-industrialization and a return to a focus on production and exports of commodities following the China Shock in the early 2000s (Jenkins and De Freitas Barbosa2012; Millar2014; Jenkins 2015; Da Costa Oreiro, Agostini, and Gala2020).

4.4. MODEL SPECIFICATION CHAPTER 4

Figure 4.6: Child labour in Regions of Brazil 2001-2015

We use two main spatial weights matrices. The first examines spatial dynamics looking at patterns of export competition. In this vein, international political economists have long noted that spatial dynamics often transcend simple geography (Beck, Gleditsch, and Beardsley 2006).

The fundamental idea is that, in the global market, companies and geographic units such as states compete with other producers of similar goods around the world, regardless of where they are located. In building our competition weight (W ) we follow Guler et al. (2002) and much of the succeeding literature, measuring similarity in export portfolios. This measure captures the simili-tude of states’ sector-level export profiles – i.e. looking at product similarities in exports portfolios with no discrimination on export destinations (Chatagnier and Kavaklı 2017; Wang 2017; Baccini and Koenig-Archibugi2014; Cao 2010; Simmons and Elkins 2004; Elkins, Guzman, and Simmons 2006; Polillo and Guillén2005; Guler et al. 2002).²⁴ We used exported good product classification based on the International Standard Industrial Classification of All Economic Activities (ISIC) rev. 3. In the baseline specification, we look at similarity in exported goods at two digits of the ISIC rev. 3 (United Nations1990).²⁵ The result is a W_ISIC,iz,t matrix that measures the Pearson correlation (similarity) of exported goods between state i and competitor z at time t.²⁶ There are practical reasons to prefer ISIC to the HS product classification. Most importantly, we can match child labour data to the specific ISIC sector in which children are working. At the same time, it is not possible to achieve similar concordance using the HS classification. Second, we believe that looking at similarity in exported goods by the economic activity that produced them allows to

24It must be said that the recent scholarship has developed an alternative approach that measures competition taking into account the similarity in export destination (Kim, Liao, and Imai2020). While this is a welcomed advancement, these measures are inadequate to address the questions of this paper. Including export destination as a measure of similarity - beyond being the main dependent variable - would be theoretically inconsistent. For instance, two states with completely different export destinations would not be competing even if they export the same goods. Hence, changes in child labour in one state would not affect the other. This is hardly justifiable theoretically. For instance, if a state that widely adopts child labour and trades exclusively with China starts implementing policies that reduce child labour, it may favour its competitors’ (at the sector-level) access to the Chinese market, even if initially they had no access to it. Hence, competition should focus on similarity of products or the productive industries rather than at the current export destination.

25The complete list of sectors at the two and the three-digit level of the ISIC rev.3 is presented in Table4.6in the Annexes. While it is possible to measure competition at the fourth digit and the first digit of the ISIC classification, these alternatives have problems. The first digit appears to be very broad, grouping together all manufacturing and all agriculture production, hence it is arguably a less precise measure of competition. Conversely, the fourth digit is very detailed. However, in developing countries, refined granular data for exported products suffer from severe problems of missing observations; this may be particularly true for the poorer states of Brazil. Hence, using fine-grained product-level measures risks creating a systematic bias as poorer states would regularly have a less precise measure of competition. Using a more accurate, yet, less detailed measure of the product specification is a compromise between the precision of the measure and the risk of bias.

26For computational reasons, in estimating the Spatial Durbin Model we average this matrix across all of the years creating a Wiz matrix. We also follow Cao and Prakash (2011) in replacing Wiz,t with 0 if Wiz,t < 0, therefore assuming that countries with very dissimilar export profiles are not competing with each other - i.e. are not neighbours.

4.4. MODEL SPECIFICATION CHAPTER 4

account for some degree of elasticity of the productive sector.²⁷

The second is the traditional W_GEO,iz matrix based on geography that accounts for the pos-sibility that geographical distance has frictional effects on market activity.²⁸ Workers, including child workers, prefer to find jobs in their local environment because commuting and moving entails monetary and psychological costs. Moreover, practices arguably diffuse between locations that are geographically proximate through processes of learning and emulation. For these reasons, we also employ a row-normalized queen contiguity matrix.²⁹ This means that neighbours are defined solely by whether states share a border. We then take the initial contiguity matrices and construct sparse block-diagonal matrices in which the diagonals are the original neighbour matrix surrounded by zero matrices on the off-diagonal blocks. This allows for relatively easier creation of spatially lagged variables and neighbour lists, given the multidimensional (state-sector-year) data being used. These block diagonal neighbour matrices are then used to construct neighbour lists in R to test for spatial dependence with the Moran’s I Test using first the ISIC 2-digit competition W matrix and then the geographic W matrix.³⁰ As this test is not specified for a particular spatial process, it can be applied directly to the data. The results are reported in 4.1and 4.2.

Table 4.1: Moran’s I Tests with ISIC 2dgt Competition W

Moran I statistic SD Moran I statistic Variance P-Value

Export Share to China 10.271 0.197 0.000 0.000

Child labour Rate 14.433 0.278 0.000 0.000

Table 4.2: Moran’s I Tests with Geo W

Moran I statistic SD Moran I statistic Variance P-Value

Export Share to China 10.82 0.421 0.002 0.000

Child labour Rate 13.545 0.528 0.002 0.000

A positive value for Moran’s I indicate positive spatial autocorrelation, that is, units near one another or with non-geographic ‘spatial’ dependence are similar with regards to either export share

27For instance, facilities producing textiles may be able to adapt to market demand in producing different goods.

28Do note, hence, that we use the word “competitor” to identify both neighbours and competitors in export profiles.

29Instead the ISIC W matrix is not row standardized. The reason for this difference is that the overall export competition that a state faces is not always the same. Some states may export a lot of goods that other states export as well, while other states may be alone in exporting their goods. Arguably the former states will face overall more pressure than their competitors when it comes to trade competition. On the other hand, looking at geography, we do not believe that we can make any assumption about the total export competition they face based on the number of borders they share with others. Hence, we row-normalize the data.

30For more information on this test, see, for example, Cook, An, and Favero (2019)

or child labour rates. A negative Moran’s I indicates the opposite, that geographically (or otherwise) related units are highly dissimilar. In the case of both export share to China and child labour rate, the test results reported in Tables4.1and 4.2indicate that there is significant (p-values well below 0.001) and strong (Moran’s I standard deviate of 10 and greater) positive spatial dependence.