Suppose that we use the model Y = Xi + i, where E[i] = 0 and...

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Suppose that we use the model Y = Xi + i, where E[i] = 0 and Var(e;) = . In other words, there is no intercept in this model. This is called regression through the origin. (a) Find the least squares estimator for this model. (b) Show that is unbiased; that is, E[1] = 1. Note: To simplify the notation, we have written E[1] instead of E[ 1 | X,... Xn]. Because we've conditioned on Xi, you can treat X; as fixed (not random) in your calculations. (c) Suppose you use your estimator from (a), but the true model is actually Yi = Bo + BiXi ti, where o 0. Show that the estimator you derived in (a) is biased, and calculate the bias. (That is, calculate E[1] - B. Again, treat X; as fixed.) Suppose that we use the model Y = Xi + i, where E[i] = 0 and Var(e;) = . In other words, there is no intercept in this model. This is called regression through the origin. (a) Find the least squares estimator for this model. (b) Show that is unbiased; that is, E[1] = 1. Note: To simplify the notation, we have written E[1] instead of E[ 1 | X,... Xn]. Because we've conditioned on Xi, you can treat X; as fixed (not random) in your calculations. (c) Suppose you use your estimator from (a), but the true model is actually Yi = Bo + BiXi ti, where o 0. Show that the estimator you derived in (a) is biased, and calculate the bias. (That is, calculate E[1] - B. Again, treat X; as fixed.)