We study estimation and hypothesis testing in single-index panel data models with individual effects. Through regressing the individual effects on the covariates linearly, we convert the estimation problem in single-index panel data models to that in partially linear single-index models. The conversion is valid regardless of the individual effects being random or fixed. We propose an estimating equation approach, which has a desirable double robustness property. We show that our method is applicable in single-index panel data models with heterogeneous link functions. We further design a chi-squared test to evaluate whether the individual effects are random or fixed. We conduct simulations to demonstrate the finite sample performance of the method and conduct a data analysis to illustrate its usefulness.