Econometrics Assignment help Watch

Because you the dependent variable is binary, the expected value of the dependent variable will be the probability of success. The variance of the dependent variable will be the variance of a binary random variable. The only difference is the probability of success depends on x. This means the variance, since it depends on the probability of success i.e P(crime=1) will also depend on x. Look up linear probability models on youtube.

An unbiased estimator has a sampling distribution with expectation equal to the parameter being estimated. So if you can calculate the expectation of the estimator you can confirm whether or not it is indeed the parameter being estimated. An estimator is an estimation procedure which can be applied to a sample. An estimate is just one possible realization. The sampling distribution of the estimator is what you would get if you took many samples and applied the procedure to each one to get many estimates. In practice you would only calculate one sample. But with the sampling procedure you may be able to calculate the expected value of the estimator, it's variance, or even see how it will be distributed.

In OLS, if the x's are not random, since the expected value of the errors is assumed to be zero. The estimators of the population coefficients are unbiased. If there is a misspecification error in your model like the omission of an important variable then there could be bias.