Section 4

Question 1

In a multiple regression model, how many explanatory variables are there and how many parameters to estimate are there?

Answer 1

K-1 explanatory variables

K parameters to estimate

Question 2

What does βj where j=2…k represent?

Answer 2

Each β (other than β1) represents PARTIAL slope coefficients

Tf β2 measures change in mean Y per unit change in X2 (Ceteris paribus)

Question 3

What is the technique for minimising sum of squares in multiple regression analysis?

Answer 3

Find derivatives dS/dβi, set all equal to 0 and solve

Question 4

What is assumption must be modified to ensure that multiple regression OLS estimators are still BLUE?

Answer 4

Assumption 4 must be modified: it now must be extended across all explanatory variables in the model, so EACH regressor is uncorrelated with the error

Question 5

What is assumption must be added to ensure that multiple regression OLS estimators are still BLUE?

Answer 5

No exact collinearity can exist between any of the variables (tf no exact linear relationship between any of the regressors)

Question 6

Another name for exact collinearity?

Answer 6

Pure collinearity

Note: exact collinearity is very rare

Question 7

How are the sampling distributions for the OLS estimators distributed?

Answer 7

Normally:

β(hat)j ~ N(βj,σ^2)

Question 8

How do we know the sampling distributions of the OLS estimators are unbiased?

Answer 8

The means equal the true (but Unknown) values

Question 9

What is the use of the R(bar)^2 statistic?

Answer 9

R^2 statistic is often used to compare models with the same dependent variable to see which model is better at explaining it. However, as more explanatory variables are added to a model, R^2 will naturally increase therefore can lead to incorrect conclusions about models. To penalise the use of the extra explanatory variables we use the R(bar)^2 statistic

The R(bar)^2 statistic will only increase if new variables ADD to the analysis

Question 10

What test statistic, with how many degrees of freedom, would be used for a test involving one parameter? (Eg. βj=βj*) how would you do a test of significance?

Answer 10

T statistic, n-k DofF

Sub in βj*=0

See 4.4.1 to learn how to do it

Question 11

Give two examples of testing a linear restriction?

Answer 11

Testing if Σparameters=1

Or

Testing if β2=β3

Question 12

How would you go about testing if β2=β3 (LINEAR RESTRICTION) via a hypothesis test? (H0? Distribution of it? Test statistic?)

Answer 12

H0: β2-β3=0
Distribution: β2-β3 ~N(β2-β3, σ^2β2 +σ^2β3 -2cov(β2β3))

SEE 4.4.2
(T statistic)

Question 13

What test statistic, with how many degrees of freedom, would be used for the testing of joint restrictions? (Eg. β2=β4=0)

Answer 13

F statistic with q DofF for numerator and n-k for denominator, where q=number of restrictions(number of β involved) and n=sample size and k=number of β in total

See 4.4.3 and learn it!

Question 14

See more complex example for tests of joint restrictions in notes

Answer 14

Now

Question 15

Thing to remember when doing an F-test on joint restrictions?

Answer 15

Don’t worry about two/one tailed test, if it says 5% level of significance then use 5% graph

Question 16

What is the test of overall significance(test of significance of regression) and what is the H0 and H1 for it?

Answer 16

The test that all k-1 slope parameters are jointly equal to 0

H0: β2=β3=…=βk=0
H1: at least one is not equal to 0

Question 17

What form has the restricted model for the test of overall significance?

Answer 17

Yi=β1+εi

Question 18

How do you then do the test of overall significance?

Answer 18

Same as the joint restrictions test with q=k-1 (doesn’t test significance of beta intercept)

Question 19

What is the alternative test to the test of overall significance?

Answer 19

Test:
H0: R^2=0
H1: R^2 not equal to 0

Question 20

How can you test a single restriction hypothesis using F-statistic? (And note)

Answer 20

Use same method and equation as normal F statistic test, with q=1

Note: will find F-value=(t-value)^2

Question 21

Define multicollinearity?

Answer 21

The situation where explanatory variables are highly correlated BUT not exactly

Question 22

What is a consequence of multicollinearity?

Answer 22

Assumption for BLUE is there should be no exact MC.
We are considering ‘close’ but not exact MC, tf doesn’t strictly violate assumptions BUT there are still consequences for OLS estimators since although they still may be the ‘best’ doesn’t necessarily mean they are good (see page 14)

Question 23

5 signs of multicollinearity?

Answer 23

1) parameter estimates have large σ^2
2) may find very small t statistics
3) despite insignificant variables in model, R^2 may still be high (contradictions t vs R^2)
4) coefficient estimates may have signs contrary to what theory predicts
5) estimators may be highly sensitive (small Δdata values-> largeΔOLS estimates)

Question 24

Best 3 ways to detect multicollinearity?

Answer 24

1) compare R^2 with t stats
2) compare correlation coefficients between explanatory variables
3) run additional regressions between explanatory variables then look at R^2 (see page 15)

Question 25

4 ways to deal with multicollinearity?

Answer 25

1) remove problem variable(s)
2) changing/extending sample data
3) changing functional form of model
4) use previous studies to get some β values then use model to calculate other β values

Question 26

Problem with removing variable(s) to deal with MC?

Answer 26

Could lead to misunderstanding of model

Question 27

Problem with using other study’s β values to deal with MC?

Answer 27

Their data may be inaccurate/different to yours

Question 28

SEE alternative function forms in notes

Answer 28

Now

Question 29

What is a log linear and log semi-linear model? What is a special feature of a log linear model?

Answer 29

Log linear model - sides in βln(X) or ln(Y) form
Parameters can be interpreted as elasticities!

Semi log linear - only left side (ln(Y)) is in log form

Question 30

Explain what the difference is between doing a joint test of restrictions and doing several individual ones?

Answer 30

Joint test: tests whether both beta 1 and beta 2 are 0

Individual tests: tests whether beta 2 is 0 when beta 4 is free to be whatever it is estimated to be and vice versa

Question 31

Explain how doing a joint test might lead to a different conclusion to doing 2 individual ones?

Answer 31

If two regressors beta 1 and beta 2 are highly correlated with each other (multicollinearity), then when doing a t test on X1 when X2 is already in the model, X1 is likely to appear insignificant since its ADDITIONAL explanatory power is negligible, and vice versa. Therefore for a t test they may both come out as insignificant and hence be excluded from the model

Question 32

For a complex example of joint restriction testing, what do you have to remember?

Answer 32

When rearranging the model, the final form of the rearrangement should only have terms with betas or errors, if any variables with neither in front they must be moved to the left to create a new variable Z (see notes example)
Also some X variables may be attached to the same beta, these must be replaced with a new variable W