HANDOUT 2 Flashcards
How many OLS conditions for multivariate analysis?
1 for intercept
K for K slope coefficients
DOF for multivariate analysis
DOF = n - (k + 1)
3 stages of partitioned regression to find coefficient on X1 only
- regress Y on X2–>Xk and save residuals
- regress X1 on X2–>Xk and save residuals
- regress Y tilda on X1 tilda = we’ve extracted the effects of X2–>Xk on both variables
What does partitioned regression do?
Partial derivatives
b1 formula
sum(yi tilda - y tilda bar)(x1i tilda - x1 tilda bar) / sum(x1i tila - x1 tilda bar)^2
v(b1) formula
sigma^2 / sum (x1i tilda - x1 tilda bar)^2
How does v(b1) compare in bivariate and partitioned regression case?
bivariate: denom of v(b1) = TSS
partitioned: denom = RSS
RSS <=TSS
So V(b1) smaller in bivariate case
Interpret B1 if linear in X and Y
dYi/dX1i = B1 = change in Y for unit increase in X1, ceteris paribus
Interpret B1 if linear in Y, non-linear in X
B1/100 = change in Y for 1% increase X1
or B1 x ln(1+g) if g>0.1
Ceteris paribus
Interpret B1 if non-linear in Y, linear in X
100B1 = %change in Y for 1 unit increase X1 or 100(e^B1 - 1) = " Ceteris paribus
Interpret B1 if Y = alpha + B1X1i + B2X1i^2 + …
dY/dX1 = B1 + 2B2X1 B1 = change in Y for 1 unit rise X1 when X1=0
Why is R^2 bad?
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.
R^2 bar formula
1 - [(RSS/DOF)/(TSS/n-1)]
More variables = RSS falls = R^2 bar rises
But DOF also falls = R^2 bar falls
Punishment factor
Akaike info criterion
AIC = 2k/n + ln(RSS/n)
Weakest punishment of 2/n for additional variables.
Schwarz Bayesian
BIC = kln(n)/n + ln(RSS/n)
Strongest punishment as long as n>=8
Hannan-Quinn
HQC = 2k ln(ln(n))/n + ln(RSS/n)
Middle punishment
What test do we do for a SINGLE restriction? Why?
T-TEST
b is normally distributed, but since sigma^2 unknown we use s^2 = t-test instead
DOF for t-test
DOF = n - (k + 1)
V(b1 + b2 - 2b3)
= V(b1) + V(b2) + 4V(b3) + 2COV(b1, b2)
-4COV(b1, b3) - 4COV(b2, b3)
What test do we do for a MULTIPLE restrictions?
F-TEST
F test standard formula
F = [(RSS^R - RSS^U) / d] / [RSS^U / DOF]
DOF in the formula refers to
DOF of the unrestricted model
d in the formula refers to
Number of equality signs in H0
= difference in number of parameters between restricted & unrestricted models
IF H0 is true, what should our test statistic be?
RSSR - RSSU = 0
As ESS equal for both.
IF H0 is FALSE, what should our test statistic be?
RSSR - RSSU > > 0
As ESS higher for U, so RSS lower for U.
Critical values for F test
F^c d, dof
Only one +VE CV
F test formula in terms of R^2
F = [(R^2 U - R^2 R) / d] / [(1 - R^2 U) / dof]
When can we NOT use the F formula in terms of R^2?
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
F test of model significance H0
H0: all slope coefficients = 0
Restricted model in F test of significance
Y = alpha + Ei
Model explains no variation in Y
ESS = R^2 = 0
RSS^R = TSS
R^2 F stat specifically for test of overall significance
As R^2 R = 0
F = [R^2 U / d] / [(1 - R^2 U)/dof]
If we do NOT reject H0 what does this imply?
If test stat < CV –> do NOT reject
Do NOT reject means all coefficients = 0
Model is rubbish - explains no variation in Y.
