metadata: authors: 0: author: Thomas,Timothy S.
1: author: You,Liangzhi
2: author: Wood-Sichra,Ulrike
3: author: Ru,Yating
4: author: Blankespoor,Brian
5: author: Kalvelagen,Erwin
count: Brazil
docna: 0: docna: Generating Gridded Agricultural Gross
            Domestic Product for Brazil : A Comparison of Methodologies
docty: Policy Research Working Paper
subtopic: Livestock and Animal Husbandry,Crops and Crop Management Systems,Climate Change and Agriculture,Energy and Natural Resources,Forests and Forestry,Forestry,Coastal and Marine Resources,Food Security
teratopic: Energy,Environment,Water Resources,Agriculture,Rural Development
abstracts: cdata!: This paper examines two new methods to
            generate gridded agricultural Gross Domestic Product (GDP)
            and compares the results with a traditional method. In the
            case of Brazil, these two new methods of spatial
            disaggregation and cross-entropy outperform the prediction
            of agricultural GDP from the traditional method that
            distributes agricultural GDP using rural population. The
            paper finds that the best prediction method is spatial
            disaggregation using a regression approach for all the key
            crops and contributors to agricultural GDP. However, the
            issue of degrees of freedom is an important limiting factor,
            as the approach requires sufficient subnational data. The
            cross-entropy method with readily available spatially
            distributed crop, livestock, forest, and fish allocation far
            outperforms the traditional method, at least in the case of
            Brazil, and can operate with national- and/or
            subnational-level data.
geo_reg_key: 119228,517191
repnme: repnme: Generating Gridded Agricultural Gross
            Domestic Product for Brazil : A Comparison of Methodologies
display_title: Generating Gridded Agricultural
            Gross Domestic Product for Brazil : A Comparison of Methodologies
listing_relative_url: /research/2019/08/31344296/generating-gridded-agricultural-gross-domestic-product-brazil-comparison-methodologies
topicv3: Agriculture and Food Security m1326205 435,Crop Production m1326627 174,Forestry m1326646 436,Population, Aging and Health m1327736 640,Livestock m1504490 4024
publishtoextweb_dt: 2019-08-19T00:00:00Z
historic_topic: Energy,Environment,Water Resources,Agriculture,Rural Development
no_of_pages: 20
author_keys: 0: author_key: 166335
1: author_key: 1072043
2: author_key: NA
3: author_key: NA
4: author_key: 957135
5: author_key: 1075297
model_extractions: Policy Research Working Paper 8985

# Generating Gridded Agricultural Gross Domestic Product for Brazil

A Comparison of Methodologies _Timothy S. Thomas Liangzhi You Ulrike Wood-Sichra Yating Ru Brian Blankespoor Erwin Kalvelagen_ Development Economics Development Data Group August 2019
Policy Research Working Paper 8985

# **Abstract**

This paper examines two new methods to generate gridded agricultural Gross Domestic Product (GDP) and compares the results with a traditional method. In the case of Brazil, these two new methods of spatial disaggregation and cross-entropy outperform the prediction of agricultural GDP from the traditional method that distributes agricultural GDP using rural population. The paper finds that the best prediction method is spatial disaggregation using a regression approach for all the key crops and contributors to agricultural GDP. However, the issue of degrees of freedom is an important limiting factor, as the approach requires sufficient subnational data. The cross-entropy method with readily available spatially distributed crop, livestock, forest, and fish allocation far outperforms the traditional method, at least in the case of Brazil, and can operate with nationaland/or subnational-level data.

This paper is a product of the Development Data Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at tim.thomas@cgiar.org (corresponding author) and bblankespoor@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Generating Gridded Agricultural Gross Domestic Product for Brazil: A Comparison of Methodologies<sup>1</sup>

Timothy S. Thomas<sup>2</sup> Liangzhi You<sup>3</sup> Ulrike Wood‐Sichra<sup>3</sup> Yating Ru<sup>3</sup> Brian Blankespoor<sup>4</sup> Erwin Kalvelagen<sup>3</sup> Keywords: agriculture, regional economic activity, gridded GDP, cross‐entropy, spatial disaggregation JEL: Q10, R12

> 1 The authors thank Juha Siikämaki (IUCN), Tim Robinson (FAO), Jean‐François Arvis (World Bank), Uwe Deichmann (World Bank) and Siobhan Murray (World Bank) for earlier discussions. We also thank the participants of the Workshop on Data Integration: Realising the Potential of Statistical and Geospatial Data that took place in Belgrade, Serbia and was organised in the context of the partnership agreement between UNECE and UN‐GGIM: Europe, with the support of Eurostat and EFTA. We appreciate Juha Siikämaki sharing the non‐timber data with us. We appreciate the support of our institutions and financing from DFID under the World Bank’s Strategic Research Program ‘Big Data’. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.

> 2 _Corresponding author_ . International Food Policy Research Institute (IFPRI), Email: tim.thomas@cgiar.org 3 International Food Policy Research Institute (IFPRI)

4 Development Data Group, World Bank, Email: bblankespoor@worldbank.org
# Introduction

Doll, Muller, and Elvidge (2000) were the first to use satellite data to spatially distribute GDP on a geographic grid. Their methodology relied on night‐time lights, which generally gave an indication of where cities were, while the intensity of the light gave an indication of both population density and intensity of production, with the more intense lights being associated with higher GDP per unit area. The methodology they used was especially helpful in countries for which finely distributed subnational data on economic activities were not available. It allowed researchers and policy makers to have a better idea as to which cities and locales were of high importance to the national economy, and which were under‐ developed.

However, this methodology performed poorly for countries with relatively large agricultural GDPs, since the night‐time lights were more of an indication of manufacturing and urban population rather than farming and rural population. In 2006, Nordhaus (2006) proposed using population to distribute GDP. If GDP per capita were relatively evenly distributed within administrative areas in which the values were aggregated, this could provide a reasonably accurate estimate of the geographic distribution of GDP. Several authors since then have used this idea and similar methodology to distribute agriculture GDP using rural population<sup>5</sup> or land cover (e.g. Ghosh, Powell, and Elvidge, et al., 2010; Gunasekera et al. 2015; Rafa, Moyer, and Wang et al., 2017).

In this paper, we propose two new methods for distributing agricultural GDP across a geographic grid. The first is a spatial allocation methodology which relies on several gridded data sets showing location of agricultural production. It starts with several components of agricultural GDP such as crop, livestock, fishery and timber production to provide a prior estimate of gridded agricultural GDP. The estimate is then reconciled with national and sub‐national agricultural GDP values using an entropy‐based data fusion method. The second is a “spatial disaggregation” method that relies on imposing a data‐ generating process at the gridcell level with the variable of interest (in our present case, agricultural GDP, though the technique can be applied to many types of questions) – which is not measured at the gridcell level ‐‐ being defined by a combination of known gridded explanatory variables and a set of parameters to be estimated. The equations are aggregated up to the level at which the variable of interest is known, and a multiple regression is used to solve for the parameters, which in turn can be used to make predictions at the gridcell level.

In this paper, we use Brazilian agricultural GDP data to test these two new techniques against the traditional technique of using rural population to distribute the agricultural GDP. We have data available at the municipio level, with 5,564 units. However, we instead use the data at a more aggregated microregion level, with 558 units. Then we make predictions at the gridcell level and aggregate up to the municipio, at which level we compute two measures of predictive accuracy.

## Review of Previous Work

Previous analyses use two primary geographically‐referenced data types to locate GDP. The one that has been used the longest is nighttime light emissions captured by satellite imagery (Doll, Muller, and

> 5 It should be pointed out that gridded population data sets are technically not satellite data sets but GIS raster (gridded) data sets, but they may include satellite data sets as inputs to their construction.

2
Elvidge 2000; Sutton and Costanza 2002; Ebener, Murray, Tandon et al. 2005, Doll, Muller, and Morley 2006; Henderson, Storeygard, and Weil 2012; Sutton, Elvidge, and Ghosh 2007; Chen and Nordhaus, 2011; Ghosh, Anderson, and Powell et al. 2009). Night lights are seen as indicators of where people – especially urban people – are concentrated, and more importantly, where economic activity (particularly non‐agricultural) is concentrated.

The second is population. Nightlights seemed to capture industrial and commercial activity reasonably well, but it performed poorly in capturing significant portions of total GDP in countries that have relatively large agricultural sectors.

Both types of methods use national or subnational GDP measures, and distribute the measures to each pixel based on the intensity of the indicator: night lights or population density. In the latter case, the assumption is normally that GDP per capita is uniform across people in the national or subnational unit for which data are available.

Often the distribution happens using separate measures for rural and non‐rural. If the two are distinguished, often nightlights are used with non‐rural GDP distribution while population is used to distribute to rural areas the agricultural GDP. Because this is a mechanistic approach, in whatever way the population data set has an error, there will be a corresponding error in the gridded agricultural GDP. Furthermore, in whatever way the true distribution deviates from the assumption of perfectly even distribution of agricultural GDP per person this will also produce errors.

Nordhaus (2006) uses subnational GDP data to distribute the GDP based on population per pixel. He then does post‐distribution analysis at a pixel level with climate and geographic variables. He does not distinguish between urban and rural populations, even though data for most countries show a gap between rural and urban income. Kummu, Taka, and Guillaume (2018) use a similar methodology, and Yetman, Gaffin, and Xing (2004) use it to generate a future gridded GDP based on a future gridded population. The method to grid GDP by UNEP and the World Bank distinguishes between rural and urban population assuming the latter to have a higher GDP per capita and allocates GDP using an estimated ratio of urban to rural GDP per capita by country (UNISDR 2011).<sup>6</sup> Ghosh, Powell, Elvidge, et al. (2010) use LandScan population grids (e.g. Dobson et al. 2000) to distribute national level agricultural GDP into the rural area, then use night lights to distribute the remaining GDP to urban areas. Rafa, Moyer, and Wang et al. (2017) also use a hybrid method for distributing rural and urban GDP in Uganda. Likewise, Gunasekera et al. (2015) use separate models for the non‐agricultural and agricultural sectors with refinements of the non‐agricultural model in Blanchard et al. (mimeo).

<mark>Turning to the first of two new methodologies for generating gridded agricultural GDP, we note that previous work of spatial allocation models includes GDP models (Ghosh et al. 2010; Nordhaus 2006; UNISDR 2011; and Gunasekera et al. 2015), population models (Tobler et al. 1997) and agricultural models such as MapSPAM (You et al. 2006) and Global Agro‐Ecological Zones (GAEZ) (FAO/IIASA 2012). GAEZ has a total crop production data set for the year 2000 that combines actual yield and production and FAO crop statistics (international) prices for 23 major commodities.</mark>

> 6  Available at Global Risk Data Platform: http://preview.grid.unep.ch.

3
# Methodology

## Distribution Based on Rural Population

The methodology for distributing rural population used in this analysis is relatively straightforward. First, as with all the methodologies compared here, the agricultural GDP is aggregated from the municipio level up to the microregion level. Then, the rural population grid was produced using the Gridded Population of the World (GPW) version 4 (CIESIN 2017) for 2010 adjusted to the United Nation’s World Population Prospects (UN WPP), and masking out urban area using the Global Human Settlement grid for 2015 (Pesaresi and Freire 2016).<sup>7</sup> Once these steps are done, it is straightforward to aggregate population to the 5 arc minute pixel used by all three methodologies, then for each microregion, distribute the agricultural GDP per rural person to each pixel based on the rural population in each pixel.

## Spatial Allocation Using Cross‐Entropy

<mark>The methodological framework is a spatial allocation model that estimates the subnational agricultural GDP to 5 arc minute pixels (or any other spatial resolution desired). The probabilistic model includes the control (constraint) of national or subnational agricultural GDP available from various sources of national ministries by country or reports. The five components of agricultural GDP (crops, livestock, fisheries, timber, and non‐timber forest products) are used to provide a prior estimate of agricultural</mark> GDP at the pixel level.

<mark>We define our spatial AgGDP allocation problem in a cross‐entropy framework following You et al. (2014). The first step is to transform all real‐value parameters into a corresponding probability form. Let S</mark> _i_ <mark>be the percentage of the total agricultural GDP allocated to pixel</mark> _<mark>i</mark>_ <mark>within a certain country (e.g. Country X) except the crop value for that pixel.</mark> _<mark>AgGDP</mark>_ i <mark>is the agricultural GDP allocated to pixel</mark> _<mark>i</mark>_ <mark>in Country X while</mark> _<mark>TotalAgGDP</mark>_ <mark>is the total agricultural GDP for that country. Therefore:</mark> <mark>Let</mark> _<mark>PreAlloc</mark> i_ <mark>be the prior allocation of AgGDP share from our best guess. One possibility for the first approximation is from pixel level components of AgGDP where we have gridded data sets:</mark> PreAgGDP� �Crop� �Livestock� �Forestry� �Hunting� �Fishing� <mark>As we have the major components of AgGDP, and the sum of these component should be close to the official value. At least, we make sure that the official AgGDP value (</mark> _<mark>TotalAgGDP)</mark>_ <mark>should be no less than the sum of all five components of agricultural GDP. Therefore, we first sum them up</mark> 7 We select GHS Settlement grid as an example of a geospatial definition of urban population to exclude from the model. The definition of urban is an active topic of research (e.g. Dijkstra and Poelmann 2014; Roberts et al. 2017; Angel et al. 2018; Bosker et al. 2018).

4
; = PreAgGDP; x ~~————————~~

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ij = XijB + eij

>.

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y=AWy+XB+e=AWy+XB+(1-pw)t y = (I—AW)71XB + (I — AW)“ — pw)7?

U — pW)U -—AW)y = UT — pW)xp +u V=(-AW)*U —pw)* 7I[d — pw)*)"[U —aw)*]" V =07°(l—AW)"* — pw) [CU — pw) *]" [Gd — aw)* J"
(i-pW)t=I1+pW+p? *+p? 24+p* 44-5 Var(A + B) = Var(A) + Var(B) + 2Cov(A,B) Cov(A + B,C + D) = Cov(A,C) + Cov(A,D) + Cov(B,C) + Cov(B,D) (i —aw)-1 (i —aw)-1 “= (I1-Aw) y=XBré 3 InL=—v ~~in~~ 2 2xn—-w ~~x~~ 2 2 ~~*~~ 2 in(|V1) _ ~~G-XB)V-*(y-XB)~~ 202
Because we are taking the log of the determinant, the log is equal to the sum of the log of each of the eigenvalues. If we let the eigenvalues be given by I, the log likelihood function can then be written as lnL�� �� ln2π� �� lnσ<sup>�</sup> � �� ∑���� ln�γ�� � ���<sup>�</sup> �y�X<sup>�</sup> β��V����y�X�β� (5) With the aggregated variance matrix for Brazil having 5,434 rows and columns, the calculation can take several minutes. One of the ways to speed this up is to compute a grid of eigenvalues, say for equal to 0 to 0.9 with a step of 0.1, and the same for . Then we could use bilinear interpolation on this grid of eigenvalues to approximate the log of the Jacobian for any pair of  and . In our program written in R, we create this as an option, but generally computed the exact Jacobian instead of the approximate Jacobian.

Once the parameters are estimated, it is then possible to compute the predicted value at the aggregated level of the dependent variable, and subtracting that from the actual value of the dependent variable leaves the residual. We can use this residual to make the prediction at the disaggregated level perfectly sum to the actual value of the dependent variable at the aggregated level. The residual value can be treated as a “fixed effect” for the aggregated unit: a characteristic of the unit that was not measured by variables of the regression but is by definition a characteristic of the aggregated unit.

The fixed effect at the aggregated level needs to be distributed to the disaggregated units. The easiest way to do that is simply divide the fixed effect by the number of disaggregated units in that aggregated unit, and this now becomes the fixed effect at the disaggregated level. However, this was to some extent an ad hoc method, and other methods of rescaling could be developed.

To solve the likelihood function relatively quickly, we would maximize the concentrated log likelihood function, concentrated for  and . Noting that the Jacobian and the aggregated variance matrix V (along with V<sup>‐1</sup> ) are functions of  and , but not for  or <sup>2</sup> . We can solve for  and <sup>2</sup> as functions of J and V<sup>‐1</sup> and therefore ultimately of  and . They are Once the optimal values of  and  are found, in principle the values of all the parameter estimates can be inserted into the full likelihood to find the proper standard errors of not only of  and , but also  and <sup>2</sup> . In practice, this requires inverting a large numerically computed hessian, which may fail. An alternative, used in this paper, is to loop through estimates of the log likelihood function computed for a regression which excludes only one variable. This allows for use of the likelihood ratio test against the full regression, which gives a probability for the parameter being non‐zero. This probability can be used to create a pseudo‐z‐statistic, which can be used with the parameter estimate to give a pseudo‐standard error. In practice, the standard errors computed this way were similar to the standard errors found through the concentrated likelihood function that assumed that the spatial parameters were known rather than estimated.

9
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in the models producing the data will contaminate our analysis which is based on the data. This might lead to large errors from the regression if, for example, a crop in SPAM is in large quantity in a given pixel, but in reality it is not in the pixel at all, or only in small quantity.

# Results and Discussion

## Spatial disaggregation

The regression results from using crop production from SPAM ‐‐ along with livestock, forest, and fisheries data – are found in Table 1.

For crops, the units are in metric tons, and the dependent variable is in thousands of reais. So, if we were to try to interpret the parameters as prices, the value for the parameter estimate for wheat is 0.456, or R$456 per metric ton (MT), which at the end of 2010 would be equal to USD274 per MT. This is within the price range it could have sold for. Maize price from the regression, on the other hand, is low at R$101 per metric ton. Given the explanatory variables are all modeled values, it is not entirely surprising that the parameters do not reflect prices as well as we might hope. They do, however, allow us to use these parameters to project agricultural GDP to each pixel, which was the goal.

Table 1 shows that most of the agricultural outputs that are important for the Brazilian agricultural sector have statistically significant parameters. In addition to key cereals, cassava, beans, soybeans, and a number of cash crops can be found, along with chickens, cattle, and pigs, and non‐timber forest products.

_Table 1. Regression using pixel‐level production data_

|**Variable**|**Param**|**Std**<br>**error**|**t‐stat**|**Prob**|
|---|---|---|---|---|
|Wheat|0.456|0.083|5.473|0.000|
|Rice|0.061|0.028|2.167|0.030|
|Maize|0.101|0.025|3.982|0.000|
|Barley|‐0.296|0.612|0.483|0.629|
|Sorghum|0.321|0.311|1.035|0.301|
|Other cereals|0.572|0.552|1.035|0.301|
|Potatoes|0.167|0.074|2.263|0.024|
|Sweet potatoes|2.786|0.526|5.298|0.000|
|Yams|‐1.037|1.619|0.640|0.522|
|Cassava|0.081|0.029|2.761|0.006|
|Beans|0.504|0.212|2.381|0.017|
|Other pulses|11.812|9.957|1.186|0.236|
|Soybeans|0.043|0.019|2.204|0.028|
|Groundnuts|1.128|0.664|1.699|0.089|
|Coconuts|0.446|0.077|5.808|0.000|
|Oil palm|0.079|0.098|0.802|0.422|
|Sunflower|‐1.708|3.836|0.445|0.656|

11
|**Variable**|**Param**|**Std**<br>**error**|**t‐stat**|**Prob**|
|---|---|---|---|---|
|Rapeseed|71.528|197|0.363|0.717|
|Sesame|‐726.78|1,899|0.383|0.702|
|Other oils|‐0.214|1.348|0.159|0.874|
|Sugar cane|0.009|< 0.001|> 10|0.000|
|Cotton|0.384|0.133|2.882|0.004|
|Other fibers|0.120|0.391|0.308|0.758|
|Coffee, arabica|0.367|1.111|0.330|0.741|
|Coffee, robusta|1.547|1.703|0.908|0.364|
|Cocoa|1.072|0.768|1.395|0.163|
|Tea|32.869|20.193|1.628|0.104|
|Tobacco|1.369|0.233|5.883|0.000|
|Bananas|0.379|0.052|7.244|0.000|
|Fruit, tropical|0.050|0.011|4.681|0.000|
|Fruit, temperate|0.146|0.047|3.089|0.002|
|Vegetables|0.172|0.048|3.623|0.000|
|All others|0.990|0.539|1.838|0.066|
|Chickens|0.003|0.000|7.299|0.000|
|Cattle|0.045|0.008|5.591|0.000|
|Goats|‐0.007|0.064|0.103|0.918|
|Pigs|0.069|0.015|4.537|0.000|
|Sheep|0.087|0.050|1.735|0.083|
|Inland water|0.107|0.092|1.159|0.246|
|Forest loss|0.211|0.117|1.801|0.072|
|Non‐timber products|0.000|0.000|3.181|0.001|
|Constant|‐7.538|27.877|0.270|0.787|
|Log likelihood|‐7199.47||||
|Lam|0.6223|0.169|3.677|0.000|
|Rho|‐0.5343|0.419|1.274|0.203|

_Source: Authors._ _Notes: We were unable to compute the inverse of the Hessian matrix. So we iterated through the parameters, assigning the value of 0 to one at a time, and recomputing the value of the maximized likelihood function. We then applied the likelihood ratio test to get a probability. We treated the probability as if it were from a normal distribution to get the t‐statistic (z‐statistic). Then we used the t‐statistic with the parameter estimate to derive a pseudo‐standard error. The pseudo‐standard error was very close to the uncorrected standard errors that were computed as part of the concentrated likelihood function._

# Comparing to cross‐entropy estimate

We compare the predictive accuracy of the three methods by making predictions at each pixel and then aggregating up to the municipio level. We have the true agricultural GDP measures at the municipio 12
level, even though we only used in the regressions the values at the microregion level. This allows us to test how well each method works, and it gives us a way to compare the methods.

We see from Table 2 that the spatial disaggregation method performed best in terms of both mean absolute deviation and root means square error. The cross‐entropy method performed second best, and not that much less successively, with a mean absolute deviation that is only 15 percent higher than that for the spatial disaggregation method, and a root mean square error that is only 10 percent higher. Both methods performed better than that of the traditional method of using rural population to map agricultural GDP. In fact, the traditional method had a mean absolute deviation that was 4 times higher than that of the spatial disaggregation method. The root mean square error was only 50 percent higher, meaning that it was much less likely to generate large outliers.

Figure 2 shows the complete distribution of actual and predicted values for municipios for the three methods. The spatial disaggregation method had several municipios with predicted values of 0 but actual values greater than 0. In fact, there are several large outliers using this method in which the predicted value was much higher than the actual. This may suggest that the somewhat ad hoc method for adjusting the fixed effects was less than optimal and an alternative method should be found. Generally, though, the spatial disaggregation method cluster near the diagonal line is narrower than that of the cross‐entropy spatial allocation method, which in turn is narrower than the one for the rural population density method.

_Table 2. Deviations of pixel agricultural GDP summed to municipio from actual value_

||**Mean absolute**<br>**deviation**|**Root mean**<br>**square error**|
|---|---|---|
|Rural population density|28,744|25,397|
|Cross‐entropy spatial allocation|8,249|18,347|
|Spatial disaggregation from regression on agricultural<br>production|7,214|16,673|

_Source: Authors._ Figure 3 shows the pixel‐level results produced by the three methods. The difference between the white and the orange should mostly be overlooked. It shows different assumptions that fixed certain municipios and their underlying pixels to be valued at 0 (the white areas). Ignoring a somewhat artificial contrast between the white and orange, the cross‐entropy method and the spatial disaggregation method appear to be very similar in most locations. The results using the population density are less smooth, looking more speckled, based simply on the nature of the underlying rural population data set.

# Conclusions

When there are enough data to allow high enough degrees of freedom to use a regression approach for all the key crops and contributors to agricultural GDP, the spatial disaggregation approach is superior. In principle, we could have used many more parameters – though the hedonic price interpretation would 13
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