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a b s t r a c tIn this paper, we consider a semiparametric single-index panel data model with cross-sectional dependence and stationarity. Meanwhile, we allow fixed effects to be correlated with the regressors to capture unobservable heterogeneity. Under a general spatial error dependence structure, we then establish some consistent closed-form estimates for both the unknown parameters and the link function for the case where both cross-sectional dimension (N) and temporal dimension (T ) go to infinity. Rates of convergence and asymptotic normality are established for the proposed estimates. Our experience suggests that the proposed estimation method is simple and thus attractive for finite-sample studies and empirical implementations. Moreover, both the finite-sample performance and the empirical applications show that the proposed estimation method works well when the cross-sectional dependence exists in the data set.

Harvest maturity is closely related to peach fruit quality and has a very important effect on the fresh fruit market. Unfortunately, at present, it is difficult to determine the maturity level of peach fruits by artificial methods. The objectives of this study were to develop quadratic polynomial regression models using near-infrared spectroscopy that could determine the peel color difference, fruit firmness, soluble solids content (SSC), soluble sugar, organic acid components, and their relationships with the absorbance of chlorophyll (index of absorbance difference, IAD) in late maturing ‘Xiahui 8’ peach and ‘Xiaguang’ nectarine fruits. The analysis was based on data for fruits at veraison, fruits at harvesting maturity, and all fruits. The results showed that firmness has the highest correlation coefficient with IAD. Prediction models for fruit maturity were established between firmness and the IAD of the two cultivars using the quadratic polynomial regression method. Further variance analysis on the one degree term and quadratic term of each equation showed that every partial regression coefficient reached a significant or extremely significant level. No significant difference was observed between estimated and observed values after regression prediction. The regression equations seem to fit well. Other peach and nectarine varieties were used to test the feasibility of maturity prediction by this method, and it was found that maturity was successfully predicted in all the samples. The result indicated that the IAD can be used as an index to predict peach fruit maturity.