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?