Linear Regression Analysis_Estimate Parameter

Question

Suppose that the true model $y_i=\beta_0+\beta_1 x_i+\epsilon_i$ and we fit the model $y_i=\beta^{\prime}_1x_i+\epsilon^{\prime}_i$ and we obtain $\hat y=\hat{\beta^{\prime}_1}x_i$. Find the expression of $\hat{\beta_1}$ and investigate its biasedness.
I have to tried MSE to find $\hat{\beta_1}$ but I got an expression which still includes $\hat{\beta_0}$. Please help or give some hints. Thanks

Answer 1

I guess it is a problem of model misspecification.

You can show that the OLS estimator of $\beta '$ is $\frac{\sum_{i=1}^n X_i Y_i}{\sum X^2_i}$. Hence, $$ var(\hat{\beta}')=\frac{\sum X_i^2var(Y_i)}{(\sum X_i^2 )^2}=\frac{\sum X_i^2var(\beta_0 +\beta_1X_i +\epsilon_i)}{(\sum X_i^2 )^2} = \frac{\sigma^2}{\sum X_i^2}. $$ And the expected value is given by, $$ E (\hat{\beta}')=\frac{\beta_0\sum X_i+\beta_1 \sum X_i^2}{\sum X_i^2} = \beta_0 \frac{\sum X_i}{\sum X_i^2} + \beta_1, $$ thus $b = E(\hat{\beta}'- \beta_1)=\beta_0 \frac{\sum X_i}{\sum X_i^2} $, so th e MSE is given by $$ MSE(\hat{\beta}')=\frac{\sigma^2}{\sum X_i} + \beta_0^2 \frac{(\sum X_i)^2}{(\sum X_i^2)^2} \, . $$ You have a biased estimator of $\beta_1$ such that the bias does not seems to vanish asymptotically, so the estimator is not consistent with respect to the "real" model.