Topic 7: Classic Hypothesis Testing

Question 1

What is the distribution of this symbol h _~ N(μ, Ω), and what are it’s proporties?

Answer 1

A normal distribution with mean μ and variance Ω.

Where: h_{(p x 1)}, Ω_{(p x p)}

1. h = μ + Ω^1/2z, where z _~ N(0,1)
2. Rh _~ N( Rμ, RΩR’ )
   Where R is q x p, rank q matrix.
3. R(h-μ) _~ N(0,RΩR)
4. (RΩR)^1/2 R(h-μ) _~ N(0,I_q)

Question 2

For z _~ N(0, I_m), how is w = z‘z distributed?

Answer 2

On a Chi-Squared Distibution,

χ² _~ (m)

Where m is the degrees of freedom, or the ‘number of normal distributions used to generate the vector’.

Question 3

What are the properties of the Chi-Squared Distribution?

Answer 3

1. If w _~ χ²(m) then E(w) = m
2. For Chi-Sqr Independantly Distributed w₁ & w_2,
   w₁ = w₁ + w_{<span>2</span> ~} χ²(m)
   Where m = m₁ + m₂
3. Where h_{p x 1} ~ N(0,Ω)
   h‘Ω^-1h _~ χ²(p)
4. For a projection matrix P_{n x n} of rank k, z_{n x 1 ~} N(0,I_n) and n > k,
   z‘Pz _~ χ²(k)

Question 4

How would the variable below be distributed, where:

z _~ N(0,1)
w ~ χ(0,1)

Answer 4

This is a student t distribution!

t _~ t(m)

Question 5

What is the distribution of this variable?

What are the properties of this distribution?

Answer 5

The F distribution!

The expectation of the denominator and nominator are one.

A student t dist squared, t² _~ F(0,m).

Question 6

What is the distribution of β^ - β

Answer 6

Question 7

What are the steps for hypothesis testing?

Answer 7

1. Fix null and the alternative hypothesis.
2. Choose a statistic for testing H₀.
3. Set the critical region.
4. Reject or do not reject.

Question 8

What is type one error?

Answer 8

Probability of rejecting the null hypothesis given it is true.

Question 9

What is type 2 error?

Answer 9

When the null hypothesis is not rejected, given it is false.

Question 10

What is the distribution of a single parameter estimate?

e_i‘(β^-β)

Answer 10

Question 11

What is the distribution of the following?

Answer 11

As the left and right parts of the second equality are distributed on the normal distribution, this can be modelled by a chi-squared distribution with n-k degrees of freedom.

Question 12

What is the t-test formula for a coefficient?

Answer 12

Question 13

Say some theory predicts:

β₂+β₃=0

β₁=0

What restriction matrix should be constructed?

Answer 13

Question 14

What are the steps for a restriction test?

Answer 14

1. Set up restriction matrix.
2. Get distribution of R(β^-β), called n_β
3. Make a Chi-Square of n_β
4. Make an F Dist
5. Convert β to R’r
6. Test

Question 15

Get the distribution of R(β^-β).

Construct a chi-distribution from this statistic.

Answer 15