EXAM 2

Question 1

How do we draw a line through data to estimate the population slope?

Answer 1

Ordinary Least Squares

Question 2

What is the OLS Estimator

Answer 2

The OLS estimator minimizes the average squared difference b/w the actueal values of Y, and the prediction based on the estimated line

Question 3

Regression R^2
1. R^2=0 means ESS=0
2. R^2=1 means ESS=TSS
3. 0<=R^2<=1

• for regression w/ a single X, R^2 is equal to the square of the correlation coefficient b/w X&Y

Answer 3

Measures the fraction of the variance of Y that is explained by X
1. is unitless
2. between 0 (no fit) and 1 (perfect fit)

Question 4

Standard Error of the Regression

Answer 4

Measures the magnitude of a typical regression residual in the units of Y
- measures the spread of the distribution of u
- units of u, which are the units of Y and measures the average size of the OLS residual

Question 5

TSS=ESS+SSR(sum of squared residuals)

R^2=TSS/ESS (total sum of squares/explained sum of squares)

Answer 5

R^2=1-SSR/TSS

Question 6

Root mean squared error (RMSE ) is closely related to the SER

Question 7

The Least Squares Assumptions
1. The conditional distribution of ui, given Xi, has the mean = zero
2. (Xi,Yi), i=1,2,3,…n are iid
3. Large outliers in X and/or Y are rare

Question 8

1. The conditional distribution of ui, given Xi, has the mean = zero

Failure of this leads to omitted variable bias
- means if there is a omitted variable that correlations with the equation, the condition fails and there is OV bias

Answer 8

Is equivalent to assuming that the population regression line is the conditional mean of Yi given Xi

1. Because X is assigned randomly, all other individual characteristics–the things that make up u– are distributed independently of X , so u and X are independent.
2. Thus, in an ideal randomized controlled experiment, E [ui |Xi ] = 0

In actual experiments, or with observational data, we will need to think hard about whether E [ui |Xi] = 0 holds.

Question 9

2. (Xi,Yi), i=1,2,3,…n are iid

Answer 9

Assumption automatically applies if sampled by random sampling because chosen from same location
- they’re selected at random so the values are independently distributed

We can expect to encounter non-i.i.d. data when information is
recorded over time for the same entity (panel data and time series
data)

Question 10

3. Large Outliers are rare

Answer 10

assuming that Xi and Yi have nonzero
finite fourth moments, i.e., 0 < E [X 4
i ] < ∞ and 0 < E [Y 4
i ] < ∞, or
in other words, that the distributions of Xi and Yi have finite kurtosis - outliers are often data glitches (coding or recording
problems). Sometimes they are observations that really shouldn’t be
in your data set

Question 11

R^2

Answer 11

This measures the variance of Y that is explained by X

Question 12

SER- standard error of the regression

Answer 12

Measures the magnitude of a typical regression residual in the units of Y.

Question 13

t= (estimator- hypothesized value)/
(SE of estimator)

Answer 13

t= (average of Y- mean y) / (Sy/sqrt(n))

Question 14

95% Confidence Interval

Answer 14

• the set of points that cannot be rejected at the 5% significance level;
• a set-valued function of the data (an interval that is a function of the
  data) that contains the true parameter value 95% of the time in
  repeated samples.

Question 15

Homoskedastic

Answer 15

If variance of the conditional distribution of u given X doesn’t depend on X

E[u|x]=0

Question 16

Heteroskedastic

Answer 16

If variance of the conditional distribution of u given X does depend on X
V [ui |Xi = x] changes with x

Question 17

Homoskedastic- only SE are valid if errors are homoskedastic

-

Answer 17

Both homoskedasticity and heteroskedasticity formula differs, and will get different SE

Question 18

Usual SE are heteroskedasticity- robust SE because they’re valid whether or not errors are heteroskedastic

Question 19

Advantages of homoskedasticity

Answer 19

Equation is simpler

Disadvantage:
- formula is only correct if errors are homoskedastic

Question 20

Homoskedasticity

Answer 20

• ## default setting in regression software

Question 21

Population vs Sample Parameter

Answer 21

A parameter is a measure that describes the whole population. A statistic is a measure that describes the sample.

Question 22

Slope in population regression line

Answer 22

Expected effect of on Y of a unit change in X

Question 23

Regression Error

Answer 23

consists of omitted factors that affect measurement of Y, and error in measuring Y

Question 24

Omitted Variable Bias

Answer 24

Variables that are omitted but causes bias in the OLS estimator

Question 25

Causality

Answer 25

effect measured in ideal randomized controlled experiment

Question 26

IDEAL
RANDOMIZED
CONTROLLED
EXPERIMENT

Answer 26

Ideal: subjects all follow the treatment protocol –perfect compliance,
no errors in reporting, etc.

Randomized: subjects from the population of interest are randomly
assigned to a treatment or control group (so there are no confounding
factors).

Controlled: permits measuring the differential
effect of the treatment.

Experiment: the treatment is assigned as part of the experiment: the
subjects have no choice, so there is no “reverse causality” in which
subjects choose the treatment they think will work best.

Question 27

Three ways to overcome omitted variable bias

Answer 27

1. Run a randomized control experiment in which STR is randomly assigned
2. Adopt cross tabulation approach
   - data tables that present the results of the entire group of respondents, as well as results from subgroups of survey respondents.
3. Use regression in which omitted variable is no longer omitted

Question 28

Adjusted R^2

Answer 28

Penalized you for including another regressor
- this is because R^2 always increases when adding another regressor
———————-
adjusted R^2< R^2
values are close when n is larger

Question 29

4. There is no perfect multicollinearity
   - Perfect collinearity is when one of the regressors is an exact linear function of other regressors
   - accidentally include same variable twice

Answer 29

Imperfect multicollinearity occurs when two or more regressors are very
highly correlated.

• If two regressors are very highly
  correlated, then their scatterplot will pretty much look like a straight
  line –they are “colinear” –but unless the correlation is exactly 1 or
  −1, that collinearity is imperfect.
  Imperfect multicollinearity implies that one or more of the regression
  coefficients will be imprecisely estimated

Question 30

imperfect multicollinearity
- results in large standard errors
for one or more of the OLS coefficients

Question 31

p value > alpha
pvalue<alpha

Answer 31

p value > alpha
ACCEPT NULL HYPOTHESIS

pvalue<alpha
REJECT NULL HYPOTHESIS

Question 32

Joint Hypothesis

Answer 32

specifies a value for 2 or more coefficients; imposes a restriction on 2+ coefficients

Question 33

How to test join hypothesis

Answer 33

F STATISTICS

Question 34

F-statistics

Answer 34

1. Large when t1 and t2 is large

when n is large
F=.5(t1+t2)

Question 35

Chi Squared Distribution

Answer 35

q degrees of freedom

Question 36

Control Variable W

Answer 36

Correlated with + controls for; an omitted causal factor in regression of Y on X, but doesn’t have causal effect on Y

Question 37

Three interchangeable statements about effective control variable

Answer 37

1. when included in the
   regression, makes the error term uncorrelated with the variable of
   interest.
   2 Holding constant the control variable(s), the variable of interest is “as
   if” randomly assigned.

3 the variable of interest is uncorrelated with the omitted
determinants of Y

Question 38

When control variables are included, the LSA (1) E [ui |X1,i , …, XK ,i ] = 0
must not hold

Question 39

Conditional mean independence

Answer 39

Given the control variable, the mean of ui doesn’t depend on variable of interest
E [ui |Xi , Wi ] = E [ui |Wi ]