# 4. Main results

We estimate the benchmark specification using both Ordinary Least Squares (OLS) and Weighted Least Squares (WLS). For the latter, we apply the sample weights at the household level provided by the NSS 2011-12, which ensure that the full data are representative for India. There is considerable debate on whether using OLS or WLS is preferable. No doubt, weighted summary statistics present a representative picture for the underlying population when survey data is used. But when it comes to regression analysis, WLS does not necessarily generate more consistent or more efficient estimators than OLS. Fortunately, the two methods yield very similar coefficients (Table 2).

To check whether the difference between the estimators from the two approaches is statistically significant we first apply a test described by Deaton (1997). The test consists of running a weighted regression of the OLS residuals on all the explanatory variables, and evaluating whether the estimated coefficients are jointly equal to zero. The resulting F statistic is 1.36, which is significant at the 0.01 level. However, the R-square of the regression is only 0.041, suggesting a limited difference in explanatory power between the two approaches. We further check the correlation between the parameters estimated with OLS and WLS (Figure 1). For the parameters on household characteristics the correlation coefficient is 0.98; for location effects it is 0.95. Despite these similarities, we conduct the analysis using both OLS and WLS and systematically verify that the conclusions are not dependent on the estimation method. For brevity, in what follows we only present results based on OLS estimators. Results based on WLS estimators are available upon request.

In applying OLS and WLS we implicitly assume that the error terms in the benchmark specification are independently distributed across households and locations. However, the literature on spatial econometrics shows that observations from nearby locations often exhibit similar properties and tend to be spatially correlated. This spatial correlation raises problems similar to those created by the serial autocorrelation of residuals in time-series analysis (Anselin 2003, and Anselin and Rey 2010).

To assess whether there is spatial autocorrelation in our data we run several tests on the residuals of the benchmark specification. First, we average these residuals across the 1,406 places and confirm that the mean residuals by place are distributed closely around zero. This indicates that there is no clustering of residuals at the place level. We then compute the correlation coefficients between mean residuals by place, for all size groups. We do this within each district, and also across districts at distance intervals of 50 kms, up to a maximum distance of 400 km. The resulting correlation coefficients turned out to be small and mostly insignificant (Figure 2). In the case of small rural, large rural and small urban areas, 23 of the 24 correlation coefficients between mean residuals for the same population size groups are statistically insignificant. The only exception is for small urban areas that are distant from each other between 0 and 50 km where the coefficient is around 0.19 and statistically significant. The cross-correlations of group means of residuals that belong to different population size groups shows a similar pattern: 80 of the 81 coefficients are statistically insignificant. In the case of large urban areas, the standard deviations of the correlation coefficients are much larger, because there are much fewer observations. But a vast majority of the correlation coefficients are statistically insignificant.

The lack of spatial autocorrelation of residuals implies that the results of the benchmark regression are not biased, but it does not imply a lack of spatial correlation in household expenditures per capita. Similar to the procedure used for the mean residuals by place, we compute the correlation coefficients between location effects, both within the same district and across districts, at intervals of 50 km. In sharp contrast to 11