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Econ 410 Practice Problems

Day 9

We began today (and we will continue on Day 11) updating our formal regression assumptions for multiple regression. Together, the following assumptions are known as the Classical Linear Regression Model:

• MLR.1 (Linear in Parameters) The population model (in other words, the true model) can be written as:

y = β0 + β1x1 + · · · + βkxk + u

where β0, . . . , βk are the population parameters and u is the unobservable random error.

• MLR.2 (Random Sampling) We have a simple random sample of size n, {(yi , x1i , . . . , xki) : i = 1, 2, . . . , n}, following the population model defined in MLR.1.

• MLR.3 (No Perfect Collinearity) In the sample, there are no exact linear relation-ships among the independent variables (including the constant term).

• MLR.4 (Zero Conditional Mean) The error term (u) has an expected value of zero given any value of the explanatory variables. In other words, E(u|x1, . . . , xk) = 0.

• MLR.5 (Homoskedasticity) The error term (u) has the same variance given any value of the explanatory variables. In other words, Var(u|x1, . . . , xk) = σ 2 .

These assumptions are useful because of the following key results:

1. MLR.1 - MLR.4 ⇒ OLS estimators of β0 through βk are unbiased

2. MLR.1 - MLR.5 ⇒ OLS estimators of β0 through βk are BLUE (Gauss-Markov The-orem)

Exercise 1 True or False: The following model violates MLR.1:

yi = β0 + β1xi + β2x 2i + ui

Exercise 2: True or False: The Gauss-Markov theorem states that the ordinary least squares estimators always have the smallest variance among all unbiased estimators.