Unbiasedness property under trimmed data Suppose we want to estimate a simple regression model by OLS and the selected model is the true model. Randomly trimming data does not change the unbiasedness property of the coefficient. However, when we trim the data in a certain way, would the unbiasedness property of the coefficient change?
To give an example, say we are trying to estimate the effect of income on spending by simply regressing spending on income. Suppose that we are actually estimating the true model and this means that higher powers of income is not in the true model. Then we trim the top %10 earners from the data and perform the regression again. Would the unbiasedness of the coefficient of income variable change?
 A: I think you are confusing sampling bias with bias of estimator. Your sample can be biased if your sampling does not produces samples that reflect the properties of the population of interest. For example, if you want to find out the average age of some population but deliberately sample only individuals that are between 18 and 65 years old, then obviously you won't get sample that is representative for the population and you could not expect the average age in your sample to truthfully reflect the average age in the population. Also your estimator can be biased, so no matter of your data, it inflicts bias in the estimates. Those are two different kinds of bias.
Trimming your data is basically the same as if you deliberately decided not to sample those cases, so you are inflicting bias in your sample. Obviously, by trial and error you could trim some ranges of your data to influence your estimates, and so e.g. if your estimator systematically produces overestimates, then some kind of trimming of the values that influence the estimates may help in correcting the bias. However, this would not change anything about estimator bias, it will influence your data and your estimates.
