Suppose that an experimenter postulates a model of the

Chapter 11, Problem 11.65

(choose chapter or problem)

Suppose that an experimenter postulates a model of the type

\(Y_{i}=\beta_{0}+\beta_{1} x_{1 i}+\epsilon_{i}, \quad i=1,2, \ldots, n.\)

when in fact an additional variable, say \(x_2\), also contributes linearly to the response. The true model is then given by

\(Y_{i}=\beta_{0}+\beta_{1} x_{1 i}+\beta_{2} x_{2 i}+\epsilon_{i}, \quad i=1,2, \ldots, n.\)

Compute the expected value of the estimator

\(B_{1}=\frac{\sum_{i=1}^{n}\left(x_{1 i}-\bar{x}_{1}\right) Y_{i}}{\sum_{i=1}^{n}\left(x_{1 i}-\bar{x}_{1}\right)^{2}}\).