Descriptive Analysis and Linear Regression

Question 1

Linear Regression Model

Answer 1

Yi = B1 + B2X2i + BkXki + ui
Yi = dependent variable
Xi = explanatory/independent/regressor
B1 = intercept/constant (average value of Y when X=0
B2 = slope coefficient

Question 2

ui

Answer 2

stochastic error term

average effect of all unobserved variables

Question 3

objective of regression analysis

Answer 3

estimate values of Bs based on sample data

Question 4

OLS

Answer 4

Ordinary Least Squares - used to estimate regression coefficients
finds the pair of B1 and B2 (b1 and b2) that minimise RSS

Question 5

OLS assumptions

Answer 5

• LRM is linear in its parameters
• regressors = fixed/non-stochastic
• exogeneity - expected value of error term = 0 given values of X
• homoscedasticity - constant variance of each u given values of X
• no multicollinearity - no linear relationship between regressions
• u follows normal distribution

Question 6

OLS estimators are BLUE

Answer 6

best linear unbiased estimators

• estimators are linear functions of Y
• on average they are = to the true parameter values
• they have minimum variance i.e. efficient

Question 7

standard deviation of error term =

Answer 7

standard error

= RSS/df

Question 8

n-k

Answer 8

degrees of freedom
n = sample size
k = no. of regressors

Question 9

hypothesis testing

Answer 9

construct Ho and Ha e.g B2 = 0 and B2 x 0

t = b2/se(b2)

Question 10

if t > cv from table

Answer 10

reject null

Question 11

type 1 error

Answer 11

incorrect rejection of true null

detecting an affect that is not present

Question 12

type 2 error

Answer 12

failure to reject false null

failing to detect present effect

Question 13

low p-value

Answer 13

suggests that estimated coefficient if statistically significance

Question 14

p-value < 0.01, 0.05, 0.1

Answer 14

statistically significant at 1%, 5%, 10% levels

Question 15

dummy variables

Answer 15

0 = absence
1 = presence

Question 16

e.g 1 if female, 0 if male

Answer 16

B2 would measure changes when you go from male to female
b1 = estimated wage for men
b2 = estimated diff btw men and women
b1+b2 = estimated wage for women

Question 17

if exogeneity assumption doesn’t hold

Answer 17

leads to bias estimates and therefore we need to adjust for omitted variables

Question 18

quadratic terms

Answer 18

capture increasing/decreasing marginal effects

have to generate a new variable and add it to regression

Question 19

marginal effect

Answer 19

first derivative of regression functioned wrt variable of interest

Question 20

interaction variable

Answer 20

constructed by multiplying two regressors

allows the magnitude of the effect X has on Y to vary depending on the level of another X

Question 21

interpreting

Answer 21

how does the regression function respond to a change in a variable

Question 22

if it is not linear (log-log)

Answer 22

log-log model so that it is linear in parameters

take logs and add error term

Question 23

log-lin model

Answer 23

dependent variable in logs – %
explanatory variables in levels – units
B2 measures relative change in output Q for an absolute change in input

Question 24

lin-log model

Answer 24

estimates % growth in dependent variable for an absolute change in explanatory variable

Question 25

lin-lin model

Answer 25

using a linear production function

Question 26

testing for linear combinations

Answer 26

se – t-stat – compare to critical value – create p-value – reject/don’t reject null

Question 27

TSS

Answer 27

total sum of squares = ESS + RSS sum of squared deviations from the sample mean = how well we could predict outcome w/o any regressors

Question 28

ESS

Answer 28

explained sum of squares = how much of that variation do our regressors predict

Question 29

RSS

Answer 29

residual sum of squares = outcome variation that regressors don’t explain

Question 30

R^2

Answer 30

ESS/TSS
overall measure of goodness-of-fit of the estimated regression line
how much of variation is explained by regressors
increases when u add more regressors

Question 31

F-stat

Answer 31

tests significance of all coeffs
(ESS/k-1) / (RSS/n-k)
>critical value =reject null

Question 32

dummy variable trap

Answer 32

situation of multicollinearity

to distinguish btw m categories we can only have m-1 dummies

Question 33

perfect collinearity

Answer 33

perfect linear relationship between two or more regressors

one predictor variable can be used to predict another

Question 34

imperfect collinearity

Answer 34

one dependent variable always equals to a linear combination of the other dependent variables plus a small error term

Question 35

consequences of multicollinearity in the data

Answer 35

larger standard errors – smaller t-ratio – wider CI – less likely to reject null

Question 36

homoscedasticty

Answer 36

assumption that error term has has the same variance for all observations (doesn’t always hold)

Question 37

heteroscedasticity

Answer 37

error terms have unequal variances for different observations

Question 38

consequences of heteroscedasticity

Answer 38

• OLS still consistent and unbiased
• se either too large or too small so t-stats, F-stats, p-values etc will be wrong
• OLS no longer efficient

Question 39

dealing with heteroscedasticity

Answer 39

• use log transformation
• keep using OLS and compute heteroscedasticty
• weighted least squares

Question 40

using a logarithmic transformation of the outcome variable

Answer 40

e.g. ln(wage) - these variables tend to have more variance at higher values

Question 41

continuing to use OLS and computing heteroscedasticity - robust standard errors

Answer 41

regress y on x

corrects se to allow for heteroscedastcity

Question 42

weighted least squares

Answer 42

more efficient than OLS in presences of heteroscedastcity

Question 43

omission of relevant variables

Answer 43

they’ll be captured by the error term

if they are correlated to the ones included then parameters are biased and exogeneity assumption doesn’t hold