Chapter 5-6: hypothesis testing

Null hypothesis

default assumption that there is no significant effect, difference, or relationship between variables in a population, suggesting any observed results are due to chance

Types of tests

Rejection region C^α =

values of the test statistic that result in rejecting the null at the significance level α.

Rejection rule and Critical value(s)

Types of errors

Level and Power of a Test

Assumption A.5

then …

We found that …

Testing single restrictions

Step 1

Testing single restrictions

Step 2

Testing single restrictions

Step 3

Testing multiple restrictions

Step 1

Testing multiple restrictions

Step 2.1

Testing multiple restrictions

Step 2

Testing multiple restrictions

Step 3

p is the number of restrictions being tested

Goal: We want to test whether the parameters on the regressors are jointly equal to zero.

+ F test using R²

Restricted least squares (RLS) =

Example

RLS estimator

We can analyse the properties of the RLS estimator - similar to our analysis of the OLS estimator. We find:

E(B_RLS) =

Var(B_RLS) =

s_RLS² =

t statistic

F-statistic using SSR

Used in Chow test

F-statistic using R²

Unrestricted vs Restricted

Chow test

Idea

The Chow test statistic is a version of the following test statistic:

The unrestricted model:

The restricted model (B1 = B2):

Chow test statistic

Confidence intervals

Due to the symmetry of the t-distribution we have that:

Maximum likelihood estimation - recap

Likelihood function

Log-likelihood function

Log-likelihood

First and second derivative

Maximum likelihood estimation

Model

Likelihood function

Model

Log-likelihood function

Model

FOCs

Solving FOCs

Biased or Unbiased?

Consistency

Expectation

Variance

Cramer-Rao lower bound

Information matrix

Derivation information matrix

The Cramer-Rao lower bound is

Frisch-Waugh-Lovell Theorem

Motivation and idea

Frisch-Waugh-Lovell Theorem

Frisch-Waugh-Lovell Theorem Simplified

Frisch-Waugh-Lovell Theorem

Interpretation

In empirical research, the true specification of the regressor matrix is unknown:

The empirical researcher can make two mistakes:

1. Include irrelevant regressors

2. Omit relevant regressors

The empirical researcher can make two mistakes:

1. Include irrelevant regressors

The empirical researcher can make two mistakes:

2. Omit relevant regressors

The empirical researcher can make two mistakes

Summary table

Violation of Assumption A.2

Violation of Assumption A.4

How does this affect the properties of the OLS estimator?

95% CI for β₂−β₃

F-test formula for single restriction: