ECN 425: Introduction to Econometrics Alvin Murphy Arizona State University: Fall 2018 Assignment #1 Due at the beginning of class on Thursday, September 6th PART I: DERIVING OLS ESTIMATORS (You must show all work to receive full credit) 1) 1) Suppose the population regression function can be written as: uxy 10 , where 0uE and 0| xuE . The sample equivalents to these two restrictions imply: 0ˆ 1 :1 n i i u n and 0ˆ 1 :1 n i ii ux n . Parts (a)-(c) of this problem ask you to derive the OLS estimators for 0 and 1 . Please show all of your work. (20 points: 5/5/10) (a) Use 0ˆ 1 :1 n i i u n to demonstrate that the OLS estimator for 0 can be written as: xy 10 ˆˆ , where n i i y n y :1 1 and n i i x n x :1 1 . (b) Use 0ˆ 1 :1 n i ii ux n together with the result from (a) to demonstrate that the OLS estimator for 1 can be written as: n i ii n i ii xxx yyx 1 :1 1 ̂ . (c) Use your result from (b) together with the definition of the variance and covariance to demonstrate that i ii x yx var ,covˆ 1 . 2 2) Suppose the population regression function is uzy i 10 , and you estimate the following sample regression function: iii uxy ˆˆˆ 10 , where zx . (20 points: 10/10) (a) Express your estimator, 1 ̂ , in terms of the data and parameters of the population regression function, ii zx ,, 1 , and i u . (b) Use your result from (a) to demonstrate that 1 ̂ is generally a biased estimator for 1 . PART II: USING A FAKE DATA EXPERIMENT TO INVESTIGATE OLS ESTIMATORS A fake data experiment can be a useful way to investigate the properties of an estimator. This process begins by specifying the “true” economic model (i.e. the population regression function). The next step is to use this model to generate some data that represent a population. Finally, by taking repeated samples from the population and using these samples to estimate the sample regression function several times, you can evaluate how well your estimator performs (e.g. bias and variance) under specific conditions. 3) In this problem, you will use a fake data experiment to demonstrate the importance of correctly specifying the form of the sample regression function. More precisely, you will compare the bias of the OLS estimator when the model is correctly specified, to the bias when the model is incorrectly specified to use the wrong explanatory variable. In the file “fake1.dta”, I have generated a population of 500 observations from the (true) regression equation: uzy 10 , such that 0uE , 0| zuE , and 2|var zu . (25 points: 5/5/5/5/5) a) Use these data to calculate the population paramete.