Suppose b represents the OLS estimator, and B the true coefficient in the regression model without intercept
y = Bx + u.
Under certain assumptions b is unbiased so that
E[b | X] = B.
Suppose that I want to have an estimator of
E[b | X].
The estimator of B is b. Hence it should be the estimator of this conditional mean. Then
Est. E[b | X] = b
where Est. stands for estimator. What does this tell me? The OLS estimate I obtain with the sample data at hand gives me the estimate of the mean of the sampling distribution of b. So we assume that the sample data at hand is so typical that it produces a b that is an estimate of the mean of the sampling distribution of the estimator. Is this correct?