Week 3: Chapter 5&9 Flashcards
GM assumptions 1-5, what’s the sample property
finite-sample property
- Holds for any sample size n, so long as n > k+1
CLM assumptions 1-6
Exact sampling property
- Violation of MLR 6 invalidates our inference
Two implications of a large sample (as n approaches infinity)
- Bj has an approximately normal distribution as n n approaches infinity)
- t and F statistics have approximately t and F distributions when as n approaches infinity
Unbiased VS Consistency
Unbiasedness:
- On average B^ equals B
- The midpoint of the distribution of B^ is B
- Nothing to do with spread or distribution of B^
Consistency:
- As we add more observations, the distribution of B^ gets more tightly distributed around B
- Tells you something about the speed and distribution of B^
Consistency definition
B^j is consistent when Var(B^j) and E(B^j) become more tightly distributed around Bj, until they collapse into a single value, as n tends to infinity
Consistency formula P(Wn…)
notes
Show how consistency follows from unbiasedness
Notes
What happens when Cov(u,x) / Var ( x) = 0?
Plim B^1 = B1
MLR4’
E(u) = 0 and Cov(xi,u) = 0.
- Zero Mean and Zero Correlation assumption
Requires that each xj is uncorrelated with u and E(u)
- Still consistent
What does MLR1-4 Imply
OLS estimators are unbiased and consistent
What does MLR1-4’ Imply
OLS estimators are consistent
What does RESET test stand for and what does it test for?
Regression Estimation Specification Error Test
- Tests for non-linearities within the model
- RESET tests for functional form misspecification brought about by the exclusion of higher order polynomials of our x’s
RESET equation and why there is no y^4
check notes for the equation.
- No y^4 as it uses up too many dofs
What is the null of RESET test?
What does it mean when we fail to reject it?
H0: δ1 = δ2 = 0
If we fail to reject, means the original model was correct
What are the drawbacks of the RESET test?
A drawback with RESET is that it provides no real direction on how to proceed if the model is rejected. THUS IT IS JUST A FUNCTIONAL FORM TEST
Mizon-Richard Test
- focuses on log transformations
- y = g0 + g1x1 + g2x2 + g3log(x1) + g4log(x2) + u (g = a number symbol)
Null and alternative hypothesis for MR test
H0: g3 = 0, g4 = 0 (original equation correctly specified)
H1: g1 = 0, g2 = 0
David-MacKinnon Test
States that the fitted values from one model should be insignificant when added to another model, yˇ and y^ (fitted values)
1. y = b0 + b1x1 + b2x2 + d1yˇ + error
2. y = b0 + b1log(x1) + b2log(x2) + q1yˆ + error
H0 : d1 = 0 and H0 : q1 = 0
How do you know if its a good proxy variable?
The closer Cor(proxy, variable) is to 1, the better the proxy
The proxy variable final model
y = alpha0 + b1x1 + b2x2 + alpha3x3 + e
What can we use instead of a proxy variable?
Lagged dependent variable
- accounts for historical factors that cause current differences
Examples of measurement errors
Self-reported income, weight etc.
Always have errors, people misinterpret the information
Proxy VS Measurement Error
- In the proxy case, the omitted variable is important to the extent to which it affects our other independent variables
- In the measurement error case, the mismeasurement in the independent variable is the issue
Measurement error in the dependent variable:
- Model
- Implications
y* = 𝛽0 + 𝛽1x1 + . . . + 𝛽kxk + u
y = b0 + b1x1 + . . . + bkxk + v
V: e0 = y - y* (measurement error)
GM assumptions still hold, still consistent
