Due: Sept. 28th. Worth 3% of your mark. For each answer include the R code that you use, as well as a brief explanation. Upload your answers to the assignment 1 dropbox on UM Learn.

  1. Download any data set that contains at least 3 variables.

  2. In R, use LS to estimate a population model of the form: \(\boldsymbol{y} = X\boldsymbol{\beta} + \boldsymbol{\epsilon} \nonumber\)

    On the right hand side of your model, only include 2 regressors and the intercept (like in the Cobb-Douglas example).

  3. Verify that the $x$ variables are orthogonal to the LS residuals.

  4. Verify that the LS residuals from your estimated model sum to zero.

  5. Verify that the regression line (it is actually a 2-dimensional “plane”) passes through the sample mean of the data.

  6. Verify that the fitted values and residuals are invariant to a non-singular linear transformation.

  7. Use the Frisch-Waugh-Lovell theorem and partial regression to get the LS estimate for just one of the $\beta$.

    Recall that the FWL theorem suggests that, for the model:

    \[y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon \nonumber\]

    the LS estimator for $\beta_2$ (for example) can be obtained by:

    1. Regressing $x_2$ on $x_1$ and the constant, saving the residuals.

    2. Regressing $y$ on $x_1$ and the constant, saving the residuals.

    3. Regressing the residuals from (ii) onto (i), without a constant.

For the R code required for this assignment, click on this tutorial.