## The Effect of Foreign Aid on Sub-national Development

|# 4. Estimation Strategy

## Texte intégral

1Following the econometric specification predominantly used in the aid effectiveness literature, the baseline reduced-form model is given by the following equation,

2where ln(*Light*_{ic,t}) represents the logarithm of night-time luminosity in grid cell *i *= 1*,...,N *of country *c *= 1*,...,C *in year *t *= 1*,...,T *(to which a constant of 0.01 was added). *Aid*_{ic,t}_{−}_{k }refers to the four definitions of grid-cell-level foreign aid, as described in the previous section. The model includes a distributed lag specification to account for differences in the timing of the aid effect across projects. **X**_{ic,t }is a vector of geospatial explanatory variables to control for local conditions. *δ*_{i }and *λ*_{t }denote grid and year fixed effects, respectively, to control for any unobserved heterogeneity. *ε*_{ic,t }represents the grid-specific error term.

3As noted earlier, it is possible for the above model to suffer from spatial dependence, and that a given grid cell may be susceptible to spillover effects from neighbouring grid cells. I account for this issue by estimating a spatial autoregressive model as defined by Anselin *et al. *(2008), which includes spatially lagged dependent and treatment variables. The specification can be expressed as follows,

4where most variables and coefficients maintain the same interpretation as in Equation (4.1). ln(*Ligĥt*_{ic,t}) and_{ }*Aîd*_{ic,t}_{−}_{k }correspond to the spatial lag operators, and *ρ** *and *φ*_{k }denote the spatial autoregressive parameters.

5For each grid cell *i*, the operators are defined as weighted averages of the values of the contiguous neighbouring cells, such that ẑ_{ic,t} = ∑_{j} ŵ_{ij }z_{jc,t}. The spatial weights ŵ_{ij} are constructed based on a row-standardized spatial contiguity matrix **W**, an *N*×*N *positive matrix describing the spatial relationship between the cross-sectional grid cells. Each element corresponds to *w*_{ij}/∑_{j} *w*_{ij}, where w_{ij} = 1 if *i *and *j *are neighbours, and w_{ij} = 0 otherwise; diagonal elements are equal to 0 by construction.

6One problem when using this model is that the spatial lag term is endogenous, as a result of the bidirectional nature of the spatial relation (Anselin *et al.*, 2008). This implies that the spatial distribution of *y*_{ic,t }for each cross-section is determined by both the explanatory variables at each location *i *and those at neighbouring locations. The simultaneity can however be accounted for through instrumentation or by fully specifying a distributional model.

# 4.1 Quantile regression with fixed effects

7The main empirical strategy relies on a quantile regression approach suitable for panel data, developed by Koenker (2004). The benefit of this method is that it allows one to determine the effects of foreign aid at different levels of development within countries, providing a better indication of donors’ performance. In addition, the fixed effects at the grid cell level, rather than the country or administrative region level, can more accurately control for any time-invariant unobserved heterogeneity.

8To outline the mechanics behind this approach, consider the following model for the conditional quantile functions for a classical linear random effects model:

9In this specification, the effects of *x*_{ij }are allowed to vary dependent on the quantile *τ*, whereas the fixed effects *α*_{i }are not, as they only have an *i*-specific location shift effect on the conditional quantiles.

10Assuming that the *x*_{ij }component contains an intercept and that *n *is large relative to *m*_{i}, a penalized approach is most convenient to estimate the conditional quantiles simultaneously. The optimal FE estimator is therefore based on minimizing a weighted sum of *q *ordinary quantile regression objective functions, such that

11where *ρ*_{τ}(*u*) = *u*(*τ** *− I(*u < *0)) denotes the piecewise linear quantile loss function. The weights *w*_{k }control the relative influence of the *q *quantiles for the estimation of the *α*_{i}* *parameters. These intercepts are shrunk toward a common value using an *l*_{1}_{ }penalty term with associated penalty parameter *λ*, as defined by the second term in the equation. As *λ** *→ ∞, → 0 for all *i*, and the model is purged of the fixed effects. Koenker (2004) demonstrates that the penalized fixed-effect estimator is asymptotically unbiased and Gaussian.

- 1 Bose, A., & Chatterjee, S. (2003). Generalized Bootstrap for Estimators of Minimizers of Convex Fun (...)

12For the main estimations, I analyse the effects of foreign aid on night-time luminosity at four quantiles - 25^{th}, 50^{th}, 75^{th }and 95^{th }- and assign equal weights *w*_{k }of 0.25 to each. The penalty parameter *λ** *is set equal to 1, and standard errors are computed using a weighted generalised bootstrap.1

13Unfortunately, the large cross-sectional sample of over 10,600 grid cells poses computational constraints for the estimation of a fixed effects spatial lag model in quantile regressions. To address the potential simultaneity bias, I follow the specification in Bitzer & Gören (2018) and instead include the first time lag of the spatially lagged dependent variable (i.e. ln(*Ligĥt*_{ic,t})).

# 4.2 Instrumental variable approach

14Despite the use of both a highly disaggregated dataset and the quantile fixed effects approach, there may still be endogeneity in the models that remains unaccounted for. I therefore complement the above estimation strategy with one that uses an instrumental variable to identify the causal effect of foreign aid at the grid level. I adopt the methodology used in Galiani *et al. *(2017) and Dreher & Lohmann (2015), which exploits a plausible quasi-experiment created by the income threshold set by the International Development Association (IDA) for receiving concessional aid.

15Eligibility for IDA support is contingent on two factors: lack of creditworthiness, and a country’s relative poverty, which is determined by the GNI per capita being below a set “operational cut-off” measured in current US dollars. It was initially set at $580 in 1987, and has since been updated annually to account for inflation ($1,215 at the end of the sample period in 2014). Figure 1 below displays the historical evolution of the threshold.

16This arbitrary threshold – and consequently a country’s position in relation to it – should therefore be exogenous to the level of economic activity within a recipient grid cell, unlike actual graduation from the IDA which is dependent on a country’s economic performance (i.e. policy, creditworthiness, vulnerability to shocks). In the case this condition does not hold, the indicator for a country being below the income threshold is interacted with a grid cell’s probability of receiving aid.

17Once the levels of the two variables are controlled for, the resulting interaction term provides a plausibly exogenous instrument. Therefore, even if the time-varying component is endogenous to the outcome variable, the exclusion restriction should only be violated if the unobserved variables driving the endogeneity were also correlated with the grid-specific component (Lang, 2016).

18I thus estimate a two-stage least squares (2SLS) regression to identify the causal impact of foreign aid. For the sake of simplicity, I only use the first lag for each of the aid indicators as the treatment, so as to have only one endogenous variable.

19The estimation strategy is given by the following 2 equations,

- 2 The annual operational cut-offs for the years 1987-2010 are obtained from the replication datasets (...)

20where *IDA*_{c,t}_{−4 }is defined as a binary variable equal to 1 if the GNI per capita of country *c *is below the IDA threshold at least three periods prior to the aid disbursement, and 0 otherwise.2 The lagged variable is based on the intuition that countries do not become ineligible for concessional aid immediately after crossing the income threshold, and should in fact remain above it for at least three consecutive years in order to begin the graduation process.

21represents a grid cell’s probability of receiving aid, defined as the number of years out of the whole sample period in which at least one project was located in a given cell. The remaining variables maintain the same interpretation as in the baseline model described in Equation (4.1). The levels of the interacted variables are accounted for through the inclusion of the grid-specific and country-year fixed effects.

22I expect the coefficient associated with the interaction term to be positive, as countries that have crossed the operational cut-off would be on track towards graduation from the IDA, and are thus likely to see a fall in aid inflows. In addition, grid cells with a higher probability of receiving aid would be more likely to have an active project at any given time.

23On the other hand, there is a possibility that any changes in IDA lending would be counteracted by lending from the IBRD. Results from Galiani *et al. *(2017) would suggest that this may not be an issue, and that other donors adjust their contributions in line with IDA aid.

24A primary concern in using this methodology pertains to the strength of the instrument. In fact, in the case of this sample, only 18 countries out of the total 54 crossed the IDA threshold during the sample period. As a result, the instrument would remain constant across all years for the majority of observations, offering little variation to predict within-grid aid flows.

25To overcome this shortcoming, I use an alternative continuous indicator for the time-varying component, which measures a country’s distance from the IDA income threshold at any given point in time (i.e. *d*_{ct }= *GNI*_{ct }− *IDA*_{t}). In addition, I implement both instruments only on the sub-sample of countries that have crossed the threshold at one point, corresponding to approximately 3,000 grid cells.

## Notes

1 Bose, A., & Chatterjee, S. (2003). Generalized Bootstrap for Estimators of Minimizers of Convex Functions. *Journal of Statistical Planning and Inference, 117(2)*, 225-239.

2 The annual operational cut-offs for the years 1987-2010 are obtained from the replication datasets by Galiani *et al. *(2017); the cut-offs for the remaining years between 2011 and 2014 are obtained from the World Bank’s Operational Policies Manual 3.10 Annex D.

© Graduate Institute Publications, 2020