Linear Regression Model: Interpreting Error Term Expectations

Let (Y1;X1;1;X1;2); :::; (Yn;Xn;1;Xn;2) be an i.i.d sample from (Y;X1;X2) satisfying Y = 0 + 1X1 + 2X2 + U where (1;X1;X2) is not perfectly colinear, E(Y 4);E(X4 j ) < 1 for j = 1; 2. You the researcher wish to interpret this regression as the best linear predictor of Y given X1 and X2.

b. Is it necessarily true that E(U) = 0? What about E(X1U) = 0 or E(X2U) = 0?

No, it is not necessarily true that E(U) = 0. The error term U represents the part of the dependent variable Y that cannot be explained by the independent variables X1 and X2. Therefore, its expected value may not necessarily be zero.

Similarly, it is not necessarily true that E(X1U) = 0 or E(X2U) = 0. The error term U represents the unobserved factors that affect Y, and X1 and X2 may be correlated with these unobserved factors. In such cases, there may be a non-zero covariance between X1 and U, or between X2 and U, resulting in non-zero expectations.