Question

You have data on Xi, Yi and Ci drawn i.i.d. from their joint distribution. You also know that each of Yi and Xi has finite nonzero fourth moments and Ci has finite nonzero eighth moment.

Hint: conditioning on Ci is the same as conditioning on Ci and Ci2. This is because Ci2 contains no extra information beyond that contained in Ci. Therefore, E[ui|Ci] = E[ui|Ci, Ci2] and similarly for other conditional expectations.

Answer 1

Now E[(Y − Yb)
2
] = E[(Y − bX)
2
] = E(Y
2 − 2bXY + b
2X2
) = E(Y
2
) −
2bE(XY ) + b
2E(X2
). Since d
2
db2 E[(Y − Yb)
2
] = 2E(X2
) > 0, E[(Y − Yb)
2
]
is maximized when 0 = d
dbE[(Y − Yb)
2
] = 2bE(X2
) − 2E(XY ). Thus, the
unique best predictor of the indicated form is given by Yb = βX, where
β =
E(XY )
E(X2)
.
The mean squared error of the best predictor of the indicated form is
given by E[(Y − Yb)
2
] = E(Y
2
) − 2βE(XY ) + β
2E(X2
) = E(Y
2
) −
2[E(XY )]2
E(X2) +
[E(XY )]2
E(X2) = E(Y
2
) −
[E(XY )]2
E(X2)
.