Question

Problem 1: Consistency and unbiasedness Let's first review the may ways we evaluate an estimator's performance. 1a. What is the formal (mathematical) definition of unbiasedness? (Your answer should include expected values.) 1b. Give a more intuitive definition of unbiasedness (no expected values). 10. How does consistency differ from unbiasedness? 1d. Why do we care if an estimator is unbiased/consistent? le. Assume we do not have autocorrelated disturbances. Assume contemporaneous exogeneity is satisfied. For which of the following time-series models is OLS unbiased? Briefly explain your answer. • static models • models with lagged explanatory variables • models with lagged outcome variables 1f. Assume we do not have autocorrelated disturbances. Assume contemporaneous exogeneity is satisfied. For which of the following time-series models is OLS consistent? Briefly explain your answer. • static models • models with lagged explanatory variables • models with lagged outcome variables 1g. Now assume we have autocorrelated disturbances. For which of the following time-series models is OLS consistent? Briefly explain your answer. • static models • models with lagged explanatory variables • models with lagged outcome variables

Answer 1

Problem I consistency and unbiased ness a) What is the formal defination of unbiasedness? tems) (in mathematics) unbiasedness? (in mo memands teams, an estimator is unliased it, El= 0, where ô » Horoskopat sample parameter o> lopulation parameter Aix le An estimator of a given barameler is said to be unblases its expeched value is equal to the huo walue of the parameter. For exampbtor (6) Crive a more intuitive detination of unbiased ness ß.. An unbiased estimabor is an accurate statistic that is wed to approximahe a population parameter. "Accurate in this sense, that it is neither an overestimate nor an ondo estimate of an overestimate and underestimare dores happen, the mean of the difference is called a bias, c) How does consistency differ from unbiasedness? consistency means : lim Elô) = 0; when sample size ne becomes al no large, medte vores the sample parameter a aúll be equal to loopulation barometes ! je consistency of an estimator means that as sample size gets large, the estimare gets closer and closer to the foue value of the barameter, whereas unbiased ness is a finihe sample brobestu that is not athecting by sample size. LIEP BYP

d) Why do we cane if an estimator is unbiased ) consistent ? Generally an unbiased stecktie statistics is breferred over a biased statistics. This is because there is a long oun tendency of the blased statistics to under lover estimate of the true value of the paramener. A based parameter gives us biased l unne aliable result, and sample results will not be healiable