term 1 - multiple regressors - revise this for week 2

Question 1

what is ommitted variable bias?

Answer 1

when an ommited varibale Z is a determinant of Y and is correlated with the regressor X, then the OLS estimator will be biased

Question 2

are all ommitted variables equal?

Answer 2

no we cannot include all ommitted variables and we dont need to. we only need to include those which are a determinant of Y and correlated with X as ommitted variable bias will occur

Question 3

what are the two uses of regression?

Answer 3

for prediction and to estimate causal effects

Question 4

what does randomisation imply?

Answer 4

that any differences between the treatment and control groups are random and not systematically related to the treatment

Question 5

what are the three ways to overcome OVB

Answer 5

run a randomised controlled experiment in which treatment is randomly assigned.
2) adopt the cross tabulation approach with finer gradations of Y and X
3) use a regression in which the ommitted variable is no longer ommitted

Question 6

wbat does the OLS estimator solve?

Answer 6

the OLS estimator minimises the average squared differences between the actual values of Y_i and the prediction based on the estimate line. this yields OLS estimators of B_0 and B_1

Question 7

what is R^2?

Answer 7

the fraction of the varience which is expained by the regression line

Question 8

what is the adjusted R^2?

Answer 8

the adjusted r squared corrects the problem that r squared always increases when you add another regressor by penalising you for including another regressor.

Question 9

what is the equation of the adjusted R^2?

Answer 9

1- [(n-1)/(n-k-1)]* SSR/TSS Where k is the number of regressors

Question 10

what occurs to the difference between the adjusted R^2 and R^2 when n is large?

Answer 10

when n is large the difference between the adjusted R^2 and R^2 decreases as k becomes relatively smaller compared to n. the adjusted R^2 is still smaller however

Question 11

what is least squares assumptions for casual inference in multiple regression?

Answer 11

1) the conditional distribution of u given the X’s has mean zero that is, E(u| X_1i =x1,…,x_ki=xk)=0
2) (X_1i,…..,X_ki, Y_i), i =1,…,n are i.i.d
3) large outliers are unlikely : X1,…,Xk and Y have finite fourth moments
4) there is perfect multicollinearity

Question 12

what least square assumption failure leads to OVB?

Answer 12

the condition distribution of u given X’s has mean 0, if this fails OVB occurs

Question 13

what type of sampling leads to iid of the regressors and variables?

Answer 13

simple random sampling creates independent and identical distributions of the variabels

Question 14

why do outliers need to be rare?

Answer 14

the OLS can be sensitive to large outliers, so you need to check your data to make sure there are no crazy values

Question 15

what is perfect multicollinearity?

Answer 15

when one of the regressors is an exact linear function of the other regressors

Question 16

what is the dummy variable trap?

Answer 16

when there is a set of multiple binary variables which are mutally exclusive and exhaustive. if you include all these variables and a constant then there is perfect multicollinearity

Question 17

what are the solutions to the dummy variable trap?

Answer 17

omit one of the groups or omit the intercept

Question 18

what is the solution to perfect multicollinearity

Answer 18

The solution to perfect multicollinearity is to modify your list of regressors so that you no longer have perfect multicollinearity.

Question 19

what is imperfect multicollinearity?

Answer 19

Imperfect multicollinearity occurs when two or more regressors are very highly correlated.
Imperfect multicollinearity implies that one or more of the regression coefficients willbe imprecisely estimated (estimators will have higher standard error

Question 20

what is a control variable?

Answer 20

A control variable W is a regressor included to hold constant factors that, if neglected, could lead the estimated causal effect of interest to suffer from omitted variable bias.

Question 21

what are three interchangable statements about what makes an effective contorl variable?

Answer 21

i. An effective control variable is one which, when included in the regression, makes the error term uncorrelated with the variable of interest
.ii. Holding constant the control variable(s), the variable of interest is “as if ” randomly assigned.
iii. Among individuals (entities) with the same value of the control variable(s), the variable of interest is uncorrelated with the omitted determinants of Y

Question 22

do control variables need to be causal?

Answer 22

no the control variables need not be causal and their coefficients generally do not have a causal interpretation

Question 23

how do they test a single coefficenient in multiple regression?

Answer 23

hypothesis tests and confidence intervals for a single coefficient in multiple regression follow the same logic and recipe as the slope in a single regressor model

Question 24

what is a joint hypothesis?

Answer 24

a joint hypothesis specifies a value for two or more coefficients, that is, it imposes a restriction on two or more coefficients

Question 25

why cant we test the coefficients one at a time ?

Answer 25

because the rejection rate under the null hypothesis isnt 5%. one at a time method rejects the null hypothesis too often

Question 26

what are the two solutions for joint hypothesis testing?

Answer 26

use a different critical value in this procedure- not 1.96. this is the bonferroni method but this method is rarely used in practise because of the low power
use a different test statistic designed to test both B1 and B2 at once ie the F statistic

Question 27

what is the F statistic

Answer 27

the F statistic tests all parts of a joint hypothesis at once.

Question 28

what is the formula for the F statistic in the special case of the joint hypothesis B1=B1,0 and B2=B2,0 in a regression with two regressors

Answer 28

F= 1/2 * [(t_1^2 +t_2^2 - 2p_t1,t2 t_1t_2)/(1-p_t1,t2 ^2)] where p estimates the correlation between t1 and t2 and t is the test statistic for the independent tests of the coefficents

Question 29

what is the chi squared distribution?

Answer 29

the chi squared distribution with q degrees of freddom is defined to be the distribution of the sum of q independent squared standard normal random variables

Question 30

in large samples how is F distributed?

Answer 30

in large samples, F is distributed as χ^2/q. where X is the chi squared distribution

Question 31

what is the 5% critical value for the F test?

Answer 31

chi squared value with q degrees of freedom divided by q

Question 32

what is the two methods for testing single restrictions on multiple coefficients?

Answer 32

1) rearrange the regression - rearrange the regressors so that the restriction becomes a restriction on a single coefficient in an equivalent regression
2) perform the test directly - use stata

Question 33

what is the method to rearrange the regression?

Answer 33

Y=B0+B1X1+B2X2 +U
H0: B1=B2 vs H1:B1 not=B2
add and subtract B2X1 and replace B1-B2 with y and X1+X2 as w:
Y= B0 +yX1 +B2W +u
hypothesis test :
H0: y=0 or y not=0

Question 34

does the rearranged transformation have the same R^2, predicted values and same residuals?

Answer 34

yes

Question 35

what is the 95% joint confidence set?

Answer 35

a set valued function of the data that contains the true coefficents in 95% of hypothetical repeated samples

Question 36

what does a high R^2 value mean?

Answer 36

it means the high R^2 value means that the regressors explain the variation in Y

Question 37

what does a high R^2 value not mean?

Answer 37

it does not mean that you have eliminated ommitted variable bias,
it does not mean that you have an unbiased estimator of a causal effect
it does not mean that the included variables are statistically significant - this must be done through hypothesis tests