5 Topic: Measurement Error This question illustrates conceptual material, and thus it has extra exposition. In class so far (and most of the rest of class too!), we have taken for granted that whatever that the X and Y variables in our data set were measured perfectly. That is, they really contained the information they were supposed to. But this is not always the case! Think of our favorite example: house prices. It is reasonable to assume that the price variable really was the sale price, but do you think square feet is perfectly measured for any house? Of course not- In this problem, we will mrplom this topic of measureth error (sometimes called enors-invariables). We will only look at the simplest, easiest type of measurement error, so the problem is usually worse than this in real life! We have an outcome Y and a predictor X, related through the equation (nothing different yet): Y=u+1X+. ~N(0,02)- (1) The parameters and predictions that we actually care about for decisiOn-making are always those from this model. But what if X and/or Y are not perfectly measured? Instead of the true X and Y, we will' imagine we have data on one or both of X and Y, which are just X and Y corrupted by Normally distributed errors: 2:\"... my\"... where ewes) and ammo- (2) [n words, instead of the true variable(s) we instead have the true variable(s) plus an idiosyncratic error tem that corrupts the truth up or down. As usual, we will assume that the errors are independent, and in this case, we assume they are independent of all other variables. (Is this realistic? For example, do you think the error in measuring square footage is the same for all sizes of house?) This is called classical measurement error, and it's far from the only type, but it's all we'll deal with in this class- (3) Suppose that there is no measurement error at all, so we have n i.i.d. observations (H, Xi): like usual. Use the code le homevork2HeasurementErrorHonteCarlc.R, set 63 = 6: = 0, and verify that 51 estimated with this data has all the nice properties found in class- (b) Suppose that only Y is measured with error, so we have n i.i.d. observations (171',X{)- We will based cur estimation on the model: ' = so+slx+e shamans. (3) instead of (1). Using this model and the data on (17,-, X,-) we form the estimator 51 of 31- Remember, ,61 from (1) is what we care about at the end of the day, not 61- That is, we assume throughout that (1) is true. (i) Recall from slide 47 of week 1 that ll: cov(X, Y) / var(X). Following the same steps, show that .81: cov(X, Y) / var(X)- Then compute cov(X, Y) by plugging' m Y: Y + e\" to nd a relationship between ,61 and (31. What does this tell you about the value of b1 as an estimator of 1? (ii) Use_ (1) to compute the variance of Y given X. Use (1) and (2) to compute the variance of Y given X (equivalently, compute 53 in terms of other parameters)- Compare the two- What does this tell you about the standard error of In as an estimator of ,61? (iii) Use the code homeworkZ-HeasurementrrorHon'teCarlo . R with 6: = D and describe what happens to the sampling distribution of 51 as 5; increases, and discuss how this relates to what you found in (b) (i) and (b)(ii)