Raising the bar

Chapter 13. Bridge to econometrics

Unit 13.6. Questions for repetition

1. Using Example 13.2 (UoL exam 2012, zone B) as an illustration, explain why an LSE exam problem may be formally simple (only properties of variance and covariance plus simple regression are needed) and yet research-level in difficulty.

2. State the omitted-variable scenario: the true model is $latex y_i = \beta_0 + \beta_1 x_i + \beta_2 z_i + u_i$, but $latex z_i$ is unobserved and you fit $latex y_i = \beta_0 + \beta_1 x_i + v_i$. Show that the new error is $latex v_i = \beta_2 z_i + u_i$ and explain why $latex E v_i$ may be non-zero so that the slope estimator for the wrong model is biased.

3. Aggregation in economics means summation. Starting from the individual consumption function $latex c_{i, t} = \beta_0^i + \beta_1^i y_{i, t} + u_{i, t}$, derive the aggregate equation $latex C_t = \beta_0 + \beta_1 Y_t + u_t$ and identify $latex \beta_0$ and $latex u_t$. Explain why $latex \beta_1$ has to be "forced into" the equation.

4. Distinguish model misspecification from functional misspecification and give an example of each (use Table 13.1).

5. State the elimination method for solving the 2-by-2 linear system (Eq. 13.2). Define the determinant of a 2-by-2 matrix and derive the solution formulas $latex x_1 = \det B_1 / \det A$, $latex x_2 = \det B_2 / \det A$ (Eq. 13.6).

6. In the two-regressor model $latex y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$, apply linearity of covariance with respect to the first argument (and the formal condition $latex \textrm{cov}(u, x_j) = 0$) to derive the system of two equations in $latex \beta_1, \beta_2$, and write out the resulting expression for $latex \hat \beta_1$.

7. State both definitions of linearly dependent vectors and prove that they are equivalent. Why is the requirement "not all $latex c_i$ are zero" essential in the first definition?

8. Define linearly independent vectors. Prove that on the plane the vectors $latex X = \bigl( \begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \bigr)$, $latex e_1 = \bigl( \begin{smallmatrix} 1 \\ 0 \end{smallmatrix} \bigr)$, $latex e_2 = \bigl( \begin{smallmatrix} 0 \\ 1 \end{smallmatrix} \bigr)$ are linearly dependent, but $latex e_1, e_2$ alone are linearly independent. Define the dimension of a space.

9. State Theorem 13.1 (Criterion of linear dependence) and explain why it is computationally useful. Write down the matrix formula for the OLS estimator $latex \hat \beta = (X^T X)^{-1} X^T y$ and the variance formula $latex \textrm{Var}(\hat \beta) = \sigma^2 (X^T X)^{-1}$, and use them to explain the consequences of (a) exact multicollinearity and (b) approximate multicollinearity for the OLS estimator.

10. Define the dummy variable trap. Illustrate it with the four-seasons example, and state the two ways to avoid it. What is the interpretation of the coefficients in each case?

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