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Question 3 Suppose we want to predict a continuous response Y. The true conditional expectation of Y is given by, ?(Y | X, Z) = ?? + ??X + ??Z here Z is a continuous but an unobserved predictor and X is an observed continuous predictor. From the real life data we can only observe y?, y?, ..., y? and x?, x?, ..., x? and fit the following model, ?? = ??? + ???x? Then answer the following problems, (a) [4 Marks] The estimator ??? is a biased estimator. Calculate the bias of the estimator. That is calculate ?? ? ?(???). Show the detailed derivations. (b) [2 Marks] Calculate the Var(??? | X?, X?) using the fitted model under the constant variance assumption. Show the detailed derivations. (c) [4 Marks] Now imagine that there Y is a n ! 1 vector, X? is a n ! (k + 1) matrix (where the first column is the column of 1s). ?? = (??, ??, ..., ??)? is a (k + 1) ! 1 vector, X? is a n ! (p ? k ? 1) matrix and ?? = (????, ????, ..., ??)? is a (p ? k ? 1)) ! 1 vector. The true model is provided by, ?(Y | X?, X?) = X??? + X??? and the fitted model is given by ? = X???? again the ??? is a biased estimator. Calculate the bias that is, ?? ? ?(???). Show the detailed derivations.

          Question 3
Suppose we want to predict a continuous response Y. The true conditional expectation of Y is given by,
?(Y | X, Z) = ?? + ??X + ??Z
here Z is a continuous but an unobserved predictor and X is an observed continuous predictor. From the real life data we can only observe y?, y?, ..., y? and x?, x?, ..., x? and fit the following model,
?? = ??? + ???x?
Then answer the following problems,
(a) [4 Marks] The estimator ??? is a biased estimator. Calculate the bias of the estimator. That is calculate ?? ? ?(???). Show the detailed derivations.
(b) [2 Marks] Calculate the Var(??? | X?, X?) using the fitted model under the constant variance assumption. Show the detailed derivations.
(c) [4 Marks] Now imagine that there Y is a n ! 1 vector, X? is a n ! (k + 1) matrix (where the first column is the column of 1s). ?? = (??, ??, ..., ??)? is a (k + 1) ! 1 vector, X? is a n ! (p ? k ? 1) matrix and ?? = (????, ????, ..., ??)? is a (p ? k ? 1)) ! 1 vector. The true model is provided by,
?(Y | X?, X?) = X??? + X???
and the fitted model is given by
? = X????
again the ??? is a biased estimator. Calculate the bias that is, ?? ? ?(???). Show the detailed derivations.

Question 3
Suppose we want to predict a continuous response Y. The true conditional expectation of Y is given by,
?(Y | X, Z) = ?? + ??X + ??Z
here Z is a continuous but an unobserved predictor and X is an observed continuous predictor. From the real life data we can only observe y?, y?, ..., y? and x?, x?, ..., x? and fit the following model,
?? = ??? + ???x?
Then answer the following problems,
(a) [4 Marks] The estimator ??? is a biased estimator. Calculate the bias of the estimator. That is calculate ?? ? ?(???). Show the detailed derivations.
(b) [2 Marks] Calculate the Var(??? | X?, X?) using the fitted model under the constant variance assumption. Show the detailed derivations.
(c) [4 Marks] Now imagine that there Y is a n ! 1 vector, X? is a n ! (k + 1) matrix (where the first column is the column of 1s). ?? = (??, ??, ..., ??)? is a (k + 1) ! 1 vector, X? is a n ! (p ? k ? 1) matrix and ?? = (????, ????, ..., ??)? is a (p ? k ? 1)) ! 1 vector. The true model is provided by,
?(Y | X?, X?) = X??? + X???
and the fitted model is given by
? = X????
again the ??? is a biased estimator. Calculate the bias that is, ?? ? ?(???). Show the detailed derivations.

Added by Tina P.

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