12. Multivariate Inference

Question 2

What is the standard error formula when there are multiplex (regressors)

Answer 2

𝑦𝑖= 𝛽0+ ∑(𝛽𝑗𝑥𝑗𝑖) 𝑘 𝑗=1 +𝑢𝑖

𝑠𝑒(𝛽ℓ ̂)=√1𝑛 𝑉𝑎𝑟(𝑢𝑖̂) 𝑉𝑎𝑟(𝑥ℓ𝑖̃ ̂)

Above, 𝑥ℓ𝑖̃ ̂ are the residuals from a regression of 𝑥ℓ𝑖 on all other 𝑥.

Question 3

What is the t stat for the

H0: B1 - B2 = 0
H1: B1 - B2 ≠ 0

Answer 3

Need SE(𝛽1 ̂−𝛽2 ̂)

𝑉𝑎𝑟(𝛽1 ̂−𝛽2 ̂) = 𝑉𝑎𝑟(𝛽1 ̂)+𝑉𝑎𝑟(𝛽2 ̂)−2𝐶𝑜𝑣(𝛽1 ̂,𝛽2 ̂).

Thus:

𝑡-𝑠𝑡𝑎𝑡 = (𝛽1 ̂−𝛽2 ̂−0)/ (√𝑉𝑎𝑟 ̂ (𝛽1̂)+𝑉𝑎𝑟 ̂ (𝛽2̂)−2𝐶𝑜𝑣 . ̂ (𝛽1 ̂,𝛽2 ̂))

Question 4

What is a joint hypothesis?

Answer 4

Hypothesis that requires 2 or more equal signs

H0: B1=0 & B2=0

Question 5

Could you do separate T-tests for each

Answer 5

No as it would be imprecise as each test ignored half of the hypothesis

Question 6

What test do you do for joint hypothesis then?

Answer 6

We need to do an F-test

Question 7

Why is that the formula for s.e. w multiple regressors

Answer 7

Question 8

Answer 8

Question 9

how to evluate t stat w multiple coeff

Answer 9

take se(x1 + x2 … etc)

Calculate the Var(“ “)
Sqaure root it for standard deviation
Replace with sample estiamtes (putting a hat on it)

WE have formula for Var(b1har) and Var(b2) from earlier

Question 10

What is Var(b1 hat) and Var(B2 hat)?

Answer 10

from s.d. of multiple coeef

Question 11

Answer 11

Question 12

When is the F test valid

Answer 12

If there is homoskedasticity

Question 13

For the test of a single restriction, how are the f and t test equivalent

Answer 13