For what concerns the sign of the IR, the introduction of the square terms allows to capture different possible relationships, such as a U shape or inverted-U shape relationship between farm size and productivity. Based on the sign and significance of the coefficients �� and �� we can compute a threshold level (maximum for inverted U-shape, and minimum for U-Shape) for farm size that implies a sign-switch in the relationship. If a nonlinear relationship exists, the pattern of thresholds will be even more interesting to analyze in the context of a quantile regressions for panel data (Powell, 2016), as we can analyze nonlinearities and sign switches along the whole distribution of productivity.

## **_4.2.Robustness checks and testing for changes in agriculture productivity_**

We complement our analysis with some robustness checks to test the validity of the panel quantile on the one hand, and for analyzing how the sign of the ALP impacts changes with agriculture productivity.

We use first the panel structure of our data to corroborate our results by using a quantile regression on the second year of the survey with lagged independent variables to avoid endonogeneity. While the empirical application of cross-sectional conditional quantile regression models has been widely developed and tested, corresponding methods for panel data have been developed only recently. The large majority of the panel quantile studies has thus dealt with macroeconomic applications, longer time spans and fewer numbers of observations (Koenker, 2004; Lamarche, 2010; Kato and Galvao, 2010; Kato, Galvao and Montes-Rojas, 2012). The main problems associated with fixed effect quantile regression models is that, as it is the case with nonlinear panel data models, the method of differencing out the fixed effects used for the conditional linear-mean model does not carry over to the conditional quantiles. Panel quantile methods also suffer from the standard incidental parameter problem (Neyman and Scott, 1948). This results in an inconsistent estimate of the coefficient of interest (e.g., the land area) in an individual fixed effects quantile regression model when the number of individuals grows large (tends to infinity) and the number of time periods is fixed (small) (Kato and Galvao, 2011, Flores et al., 2014).

Given this background, we use the same functional form and estimate the relationship of interest with a quantile regression model for 2009, the second year of the survey (Table 4). Agriculture ALP is thus regressed on its lagged value, on farm size and its square, on lagged non-farm income and on a set of exogenous controls. As farm size (and not land operated) and past off-farm incomes are the results of past decisions of farmers, no endogeneity arises. As an additional test, we also run the cross-section quantile regression on ALP changes between 2004 and 2009 (Table 5).

# **5. Estimation Results**

Table 2 presents results for a simple OLS pooled regression with year dummy (Column 1), and for fixed-effect regressions (column 2) with gross agriculture income per hectare (our ALP) as the dependent variable, using the two rounds of data, 2004 and 2009.

By measuring ALP as gross revenue per ha, we interpret the functions that we estimate as a normalized revenue function, in the spirit of McFadden, i.e. as a gauge function that characterizes farm production along a, possibly bounded, optimization path. This implies that the revenue per ha observed already incorporates the same period farm choices relative to input levels and is a function only of exogenous variables, including prices and farm size.

We find a nonlinear significant U-shape relationship between farm size and productivity. This suggests the presence non-constant returns to scale in agriculture, first decreasing up to the threshold (3.44 ha) and then increasing.

However, the analysis of panel quantile regression suggests a different picture. Even though not significant, the relationship between ALP and farm size for low performers (low agriculture productivity) is an inverted U-shaped form, and after median productivity is reached, switches to a U-shaped form. This in turn suggests that returns to scale switch across the distribution of both productivity and farm size.

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