Lecture notes 2 Multivariate regression? Flashcards
Why may multivariate data be better than bi-variate?
Bivariate can be biased as it does not take into account other relevant variables for explaining the relationship.
How do you interpret coefficients in multi-variate?
you partially differentiate them W.R.T variable you are interpreting.
-What is a difference when interpreting coefficients compared to the bivariate case?
- In multi-variate case it is the change in Y from a change in xi holding all else constant.
In a Y = B0 + X1B1 + XKBK
how do we interpret B2
What should you remember as it is multivariate?
A 1 unit change in X1 causes a B1 unit change in Y (HOLDING ALL ELSE CONSTANT)
In Y = B0 + B1log(X1) … Bk how
How do we interpret Beta 1 for small change and non small change!
What should you remember as it is multivariate?
for small change a 1% increase in X1 causes a B1/100 increase in Y
For a big change must take exact values of log(x) and then interpret beta
HOLDING ALL ELSE CONSTANT!
How do you interpret log of outcome
log(Y) = B0 + X1B1 + BkXk
What should you remember as it is multivariate?
for small change a 100 x beta
for big change you must take expectation of beta.
so it is 100 x exp(beta -1)
(HOLDING ALL CONSTANT)
How do you intepret a log log model?
log(Y) = B0 + B1log(X) + B2log(Xk)
For small changes a 1% increase in X is a Beta % increase in Y
For big changes will have to take exact.
How do you interpret a squared model like
Y = X1B1 + X^2B2 …. XkBk
a 1 unit change in X causes a B1 + 2XB2 change in Y
How do you find where the conditional expectation peaks?
You = the derivative to 0
What are the assumptions made for Multivariate regression models?
Same CLRM as ever:
- E(Xi | ei) = 0
- V(xi | ei) = sigma squared
-Cov(ei , ej) = 0
-Ei varies N(0, sigma squared)
1.What is an alternative method to get the b1 estimate in a model like
Y = B0 + B1X1….bkxik
2.What is this method essentially doing?
3.. What does this theoretically explain?
- Partition regression
You first do a regression of Y on everything but X1 and save the residual. Call this U
(This is portion of Y not explained by any other variables in the model).
Then regress X1 on all explanatory variables and save residual call this V
This explains the portion of Xi that is not explained by the other variables.
Then regress you on U on VB1 and the b1 is the coefficient of X1
- You are removing all effects from everything other than X1.
- This theoreticlly epxlains why we can interpret multivariate coefficients as holding constant.
what are the steps for a hypothesis test for a multivariate regression on one variable
State hypothesis:
B0 =
B1 not equaled to
Find CV, this will be from T table with
S. L and DOF.
Then compute T stat which is estimate - hypothesis / Se(b)
Compare to CV from t tables
Then reject or accept H0.
How do you deal with the scenario when you are hypothesis testing more than 1 coefficient for calculating t stat?
- What happens if there is a se(100b2)
There are two se(b1) + se (b2)
This is the same as square root of variance.
So it is V(b1) + V(b2) + 2Cov(b1,b2)
- You square the 100 and put it outside the V then put 100 out the cov
How to test multiple restrictions in hypothesis testing?
What are the steps and what is a trick?
Use an F test:
F test = (RSS^r - RSS^u /d ) / RSS^u / DOF
Then find CV. C = 0.05 or something
CV F d, DOF numerator and denominator degrees of freedom.
Then compare this to F test to accept or reject H0.
