An important property of a conditional expectation function is its relationship to other estimators. Suppose we...

 

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An important property of a conditional expectation function is its relationship to other estimators. Suppose we have two variables, Y, and X. Fix a specific value of X = x. Then, let's define: R(fa): = nx n ; - f.)2 iE Sx Where fe is a value (which we can think of as a function as x changes: an estimator of Y; given X), S is the set of i with X = x, and nx = |S| (i.e. the number of observations in Sx). R is the RMSE of fe on this value. A good thing to have would be to find the value of fa which minimizes the RMSE. What is this? To find out, answer the following questions. (a) (5 points) Find the value of f which minimizes R(f)2 by taking the derivative of R(f)2. What is this value? How does it relate to the CEF? Why does this value also minimize R(fx)? (b) (5 points) What does this say about the CEF in terms of all possible estimators of Y given X? An important property of a conditional expectation function is its relationship to other estimators. Suppose we have two variables, Y, and X. Fix a specific value of X = x. Then, let's define: R(fa): = nx n ; - f.)2 iE Sx Where fe is a value (which we can think of as a function as x changes: an estimator of Y; given X), S is the set of i with X = x, and nx = |S| (i.e. the number of observations in Sx). R is the RMSE of fe on this value. A good thing to have would be to find the value of fa which minimizes the RMSE. What is this? To find out, answer the following questions. (a) (5 points) Find the value of f which minimizes R(f)2 by taking the derivative of R(f)2. What is this value? How does it relate to the CEF? Why does this value also minimize R(fx)? (b) (5 points) What does this say about the CEF in terms of all possible estimators of Y given X?

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Solution Step1 STEP2 The value of fx that minimizes Rf... View the full answer