HANDOUT 2

Question 1

How many OLS conditions for multivariate analysis?

Answer 1

1 for intercept

K for K slope coefficients

Question 2

DOF for multivariate analysis

Answer 2

DOF = n - (k + 1)

Question 3

3 stages of partitioned regression to find coefficient on X1 only

Answer 3

1. regress Y on X2–>Xk and save residuals
2. regress X1 on X2–>Xk and save residuals
3. regress Y tilda on X1 tilda = we’ve extracted the effects of X2–>Xk on both variables

Question 4

What does partitioned regression do?

Answer 4

Partial derivatives

Question 5

b1 formula

Answer 5

sum(yi tilda - y tilda bar)(x1i tilda - x1 tilda bar) / sum(x1i tila - x1 tilda bar)^2

Question 6

v(b1) formula

Answer 6

sigma^2 / sum (x1i tilda - x1 tilda bar)^2

Question 7

How does v(b1) compare in bivariate and partitioned regression case?

Answer 7

bivariate: denom of v(b1) = TSS
partitioned: denom = RSS
RSS <=TSS
So V(b1) smaller in bivariate case

Question 8

Interpret B1 if linear in X and Y

Answer 8

dYi/dX1i = B1 = change in Y for unit increase in X1, ceteris paribus

Question 9

Interpret B1 if linear in Y, non-linear in X

Answer 9

B1/100 = change in Y for 1% increase X1
or B1 x ln(1+g) if g>0.1
Ceteris paribus

Question 10

Interpret B1 if non-linear in Y, linear in X

Answer 10

100B1 = %change in Y for 1 unit increase X1
or 100(e^B1 - 1) = "
Ceteris paribus

Question 11

Interpret B1 if Y = alpha + B1X1i + B2X1i^2 + …

Answer 11

dY/dX1 = B1 + 2B2X1
B1 = change in Y for 1 unit rise X1 when X1=0

Question 12

Why is R^2 bad?

Answer 12

Can increase it by adding irrelevant variables
Because the coeff on that variable will always be ≠ 0 due to statistical variation.
So it increases ESS = higher R^2.

Question 13

R^2 bar formula

Answer 13

1 - [(RSS/DOF)/(TSS/n-1)]
More variables = RSS falls = R^2 bar rises
But DOF also falls = R^2 bar falls
Punishment factor

Question 14

Akaike info criterion

Answer 14

AIC = 2k/n + ln(RSS/n)

Weakest punishment of 2/n for additional variables.

Question 15

Schwarz Bayesian

Answer 15

BIC = kln(n)/n + ln(RSS/n)

Strongest punishment as long as n>=8

Question 16

Hannan-Quinn

Answer 16

HQC = 2k ln(ln(n))/n + ln(RSS/n)

Middle punishment

Question 17

What test do we do for a SINGLE restriction? Why?

Answer 17

T-TEST

b is normally distributed, but since sigma^2 unknown we use s^2 = t-test instead

Question 18

DOF for t-test

Answer 18

DOF = n - (k + 1)

Question 19

V(b1 + b2 - 2b3)

Answer 19

= V(b1) + V(b2) + 4V(b3) + 2COV(b1, b2)

-4COV(b1, b3) - 4COV(b2, b3)

Question 20

What test do we do for a MULTIPLE restrictions?

Answer 20

F-TEST

Question 21

F test standard formula

Answer 21

F = [(RSS^R - RSS^U) / d] / [RSS^U / DOF]

Question 22

DOF in the formula refers to

Answer 22

DOF of the unrestricted model

Question 23

d in the formula refers to

Answer 23

Number of equality signs in H0

= difference in number of parameters between restricted & unrestricted models

Question 24

IF H0 is true, what should our test statistic be?

Answer 24

RSSR - RSSU = 0

As ESS equal for both.

Question 25

IF H0 is FALSE, what should our test statistic be?

Answer 25

RSSR - RSSU > > 0

As ESS higher for U, so RSS lower for U.

Question 26

Critical values for F test

Answer 26

F^c d, dof

Only one +VE CV

Question 27

F test formula in terms of R^2

Answer 27

F = [(R^2 U - R^2 R) / d] / [(1 - R^2 U) / dof]

Question 28

When can we NOT use the F formula in terms of R^2?

Answer 28

When the dependent variables are different
When we test NON-ZERO restrictions
Because known coefficients can only be on LHS = become part of dependent variable

Question 29

F test of model significance H0

Answer 29

H0: all slope coefficients = 0

Question 30

Restricted model in F test of significance

Answer 30

Y = alpha + Ei
Model explains no variation in Y
ESS = R^2 = 0
RSS^R = TSS

Question 31

R^2 F stat specifically for test of overall significance

Answer 31

As R^2 R = 0

F = [R^2 U / d] / [(1 - R^2 U)/dof]

Question 32

If we do NOT reject H0 what does this imply?

Answer 32

If test stat < CV –> do NOT reject
Do NOT reject means all coefficients = 0
Model is rubbish - explains no variation in Y.