Final Exam

Question 1

The classical assumptions must be

Answer 1

met in order for OLS estimators to be the best available

Question 2

Classical Assumption #1

Answer 2

The regression model is linear, is correctly specified, and has an additive error term

Question 3

Classical Assumption #2

Answer 3

The error term has a zero population mean

Question 4

Classical Assumption #3

Answer 4

All explanatory variables are uncorrelated with the error term

Question 5

Classical Assumption #4

Answer 5

Observations of the error term are uncorrelated with each other (no serial correlation)

Question 6

Classical Assumption #5

Answer 6

The error term has a constant variance (no heteroskedasticity)

Question 7

Classical Assumption #6

Answer 7

No explanatory variable is a perfect linear function of any other explanatory variable(s) (no perfect multicollinearity)

Question 8

Classical Assumption #7

Answer 8

The error term is normally distributed

Question 9

Omitted Variable Bias (Conditions)

Answer 9

relevant (β2 ≠ 0) – X1 and X2 are correlated

Question 10

Expected bias

Answer 10

Expected bias in መ 𝛽1 has two components: the sign of β2 and the sign of Corr(X1, X2)

Question 11

Limited Dependent Variables

Answer 11

We have discussed dummy variables (indicator variables, binary variables) as a tool for measuring qualitative/categorical independent variables (gender, race, etc.)

Question 12

linear probability model

Answer 12

simply running OLS for a regression, where the dependent variable is a dummy (i.e. binary) variable:
where Di is a dummy variable, and the Xs, βs, and ε are typical independent variables, regression coefficients, and an error term, respectively

Question 13

the term, linear probability model

Answer 13

comes from the fact that the right side of the equation is linear while the expected value of the left side measures the probability that Di = 1

Question 14

Some issues with LPM

Answer 14

෠ 𝑌 𝑖 ≤ 0 or ෠ 𝑌 𝑖 ≥ 1 a more fundamental problem with the linear probability model: nothing in the model requires ෠ 𝑌 to be between 0 and 1!  If ෠ 𝑌 is not between 0 and 1, how do we interpret it as a probability?  A related limitation is that the marginal effect of a 1-unit increase in any X is forced to be constant, which cannot possibly be true for all values of X  E.g., if increasing X by 1 always increases Y by a particular amount, ෠ 𝑌 must exceed 1 when X is sufficiently large

Question 15

The Binomial Logit Model

Answer 15

The binomial logit is an estimation technique for equations with dummy dependent variables that avoids the unboundedness problem of the linear probability model

Question 16

Logits cannot be estimated using OLS

Answer 16

but are instead estimated by maximum likelihood (ML), an iterative estimation technique that is especially useful for equations that are nonlinear in the coefficients  Again, for the logit model is bounded by 1 and 0

Question 17

Basic Procedure for Random Assignment Experiments

Answer 17

 Recruit sample of subjects  Randomly assign some to treatment group and some to control group ◼ Random assignment makes treatment uncorrelated with individual characteristics  Measure average difference in outcomes between treatment and control groups

Question 18

natural experiments (or quasi-experiments

Answer 18

attempt to utilize the “treatment-control” framework in the absence of actual random assignment to treatment and control groups

Question 19

Difference-in-difference estimator:

Answer 19

Policy impact = (Tpost – Tpre) – (Cpost– Cpre)  T: treatment group outcome, C: control group outcome  The DD estimate is the amount by which the change for the treatment group exceeded the change for the control grou

Question 20

Panel data:

Answer 20

repeated observations of multiple units over time (combination of cross-sectional and time-series)

Question 21

Main advantages of panel data

Answer 21

 Increased sample size  Ability to answer types of questions that cross-sectional and time-series data cannot accommodate  Enables use of additional methods to eliminate omitted variables bias

Question 22

Panel Data Notation

Answer 22

 𝑖 subscript indexes the cross-sectional unit (individual, county, state, etc.)  t subscript indexes the time period in which the unit is observe

Question 23

First-differenced estimator

Answer 23

∆𝑌 𝑖 = 𝛼0 + 𝛽1∆𝑋𝑖 + ∆𝜀𝑖

Question 24

Advantages of random effects estimator (if assumption about 𝑎𝑖 is correct):

Answer 24

 Allows time-invariant regressors to be included  More degrees of freedom (only estimates parameters of distribution from which 𝑎𝑖 is assumed to be drawn; fixed effects estimator uses one degree of freedom per fixed effect)

Question 25

Disadvantages of random effects estimator:

Answer 25

 Biased if assumption that 𝑎𝑖 is uncorrelated with regressors is incorrect (while FE estimator allows arbitrary correlation between 𝑎𝑖 and regressors)

Question 26

❑ Fixed effects estimator widely preferred when regressors of interest are time-varying

Answer 26

 It rarely seems likely that 𝑎𝑖 is uncorrelated with any regressors; fixed effects model is generally far more convincing

Question 27

Hausman test

Answer 27

❑ Fixed and random effect estimators can be compared with a Hausman test (previously seen in instrumental variables context as test for endogeneity)

Question 28

Fixed vs. random effects - General Advice

Answer 28

 Under random effects hypothesis, both RE and FE estimators are consistent (should give similar results); under alternative hypothesis, FE consistent but RE is not  Therefore, if estimates are significantly different, can reject null hypothesis of random effects

Question 29

Fixed vs. random effects - Concept

Answer 29

❑ General advice: use fixed effects estimator if it’s feasible

Question 30

T-test

Answer 30

Divide the coefficient by the standard error to get the t-value

Question 31

Omitted Variables – Bias Assessment

Answer 31

Sign (β2) * Sign (Corr (X1, X2)) = Sign of Bias (β1)

Question 32

Irrelevant Variables - Inclusion Criteria

Answer 32

Theory: is there sound justification for including the variable? Bias: do the coefficients for other variables change noticeably when the variable is included? T-Test: is the variable’s estimated coefficient statistically significant? R-square: has the R-square (adjusted R-square) improved?

Question 33

Serial Correlation

Answer 33

First-order serial correlation occurs when the value of the error term in one period is a function of its value in the previous period; the current error term is correlated with the previous error term.

Question 34

DW Test

Answer 34

compare DW(d) to the critical values (𝐝_𝐋, 𝐝_𝐔)

Question 35

Pure Heteroskedasticity

Answer 35

occurs in correctly specified equations

Question 36

Impure Heteroskedasticity

Answer 36

arises due to model misspecification

Question 37

Multicollinearity

Answer 37

Multicollinearity exists in every equation & the severity can change from sample to sample.  There are no generally accepted true statistical tests for multicollinearity. VIF > 5 as a rule of thumb

Question 38

Binary Dependent Variable Models

Answer 38

Linear Probability Model (LPM) & Logit / Probit Model

Question 39

Linear Probability Model (LPM)

Answer 39

- Similar to OLS regression - R-squared is no longer an accurate goodness-of-fit measure - Interpretation: probability that Y=1 on a percentage point scale

Question 40

Logit / Probit Model

Answer 40

- Restricted between 0 and 1 - Automatically corrects for heteroskedasticity - Marginal effect of X is not constant - Not linear in the coefficients

Question 41

LPM Interpretation (example)

Answer 41

On average, a 1-unit increase in DISTANCE is associated with a negative 7.2 percentage point change in the probability of choosing Cedars Sinai, holding all else constant

Question 42

LPM Limitation #1

Answer 42

Unboundedness | The linear probability model produces nonsensical forecasts (>1 and <0)

Question 43

LPM Limitation #2

Answer 43

Adj-R^2 is no longer accurate measure of overall fit

Question 44

LPM Limitation #3

Answer 44

Marginal Effect (slope) of a 1-unit increase in X is forced to be constant

Question 45

LPM Limitation #4

Answer 45

Error term is neither homoskedastic nor normally distributed

Question 46

Logit Model Interpretation - | Coefficient Interpretation:

Answer 46

The sign and significance can be interpreted just as in linear models. B1 is the effect of a 1-unit increase in X1 on the log-odds ratio

Question 47

Calculating Marginal Effects

Answer 47

margin, dydx (X) at mean

Question 48

LPM Interpretations

Answer 48

B*100 percentage point change in the probability that Y=1

Question 49

Logit Interpretations

Answer 49

B change in the log-odds ratio of Y=1

Question 50

Marginal Effects Interpretations

Answer 50

(dy/dx)*100 percentage point change in the probability that Y=1

Question 51

Experimental Methods - | Selection Problems

Answer 51

Treatment is not necessarily randomly assigned because of other systematic differences in the error term (endogeneity) which would cause bias in the treatment’s effect. We need a valid counterfactual to truly understand the effect of the intervention / treatment.

Question 52

Valid Counterfactual

Answer 52

a control group that is exactly the same as the treatment group except it does not receive the treatment

Question 53

Solution to Selection Problems

Answer 53

Randomization | -Researcher randomly assigns subjects to either a treatment or control group to estimate the treatment effect

Question 54

Natural / Quasi-Experiments

Answer 54

Randomized experiments are hard to do in the social sciences, so researchers often rely upon natural experiments where an exogenous event mimics the treatment and control group framework in the absence of actual random assignment

Question 55

Counterfactual Challenge

Answer 55

Counterfactual Challenge | Hard to find an untreated group that really is otherwise identical to the treated group

Question 56

Panel Data - definitions expanded

Answer 56

Formed when cross-sectional and time-series data sets are combined to create a single data set. Main reason for working with panel data (beyond increasing sample size) is to provide insight into analytical questions that can’t be answered by using time-series or cross-sectional data alone

Question 57

Panel Data Advantages

Answer 57

Increased sample size so more degrees of freedom & sample variability Able to answer new research questions Can eliminate omitted variable bias with fixed effects (controlling for unobserved heterogeneity)

Question 58

Panel Data Concerns

Answer 58

Heteroskedasticity & Serial Correlation

Question 59

Panel Data - Fixed Effects Model

Answer 59

Does a good job of estimating panel data equations, and it also helps avoid omitted variable bias due to unobserved heterogeneity.

Question 60

Fixed Effects Model Assumptions

Answer 60

Each cross sectional unit has its own intercept. A fixed effects analysis will allow arbitrary correlation between all time-varying explanatory variables and 𝑎_𝑖

Question 61

Fixed Effects Model Drawback

Answer 61

measurement error, autocorrelation, heteroskedasticity

Question 62

Fixed Effects Model

Answer 62

The omitted variable bias arising from unobserved heterogeneity can be mitigated with panel data and the fixed effects model.

Question 63

How Fixed Effect Model address Omitted Variable Bias

Answer 63

How? Estimates panel data by including enough dummy variables to allow each cross-sectional of individual i and time period t to have a different intercept. These dummy variables absorb the time-invariant, individual-specific omitted factors in the error term

Question 64

Panel Data - Random Effects Model | When to use

Answer 64

When the explanatory variable of interest is time-invariant

Question 65

Panel Data - Random Effects Model | Assumption

Answer 65

Ai and regressors (Xit) are uncorrelated

Question 66

Panel Data - Random Effects Model | Advantages

Answer 66

- Can handle time-invariant variables | - Uses fewer degrees of freedom than FE because of the lack of subject dummies

Question 67

Hausman Test

Answer 67

Compares fixed and random effect estimators to see if their difference is statistically significant. If different  fixed effects model preferred (reject the null hypothesis of random effects) If not different  random effects model to conserve degrees of freedom (or provide estimates of both the fixed effects and random effects models) If both models predict VERY DIFFERENT results, it suggests RE model has omitted variable bias and endogeneity present, making FE more statistically accurate

Question 68

What is the purpose of determining the Cook’s D, what is it used to detect?

Answer 68

Cook’s D is used to detect influential outliers.

Question 69

Linear Probability Model Example

Answer 69

𝐻𝑆𝑖 = 0.42 + 0.028𝑚𝑒𝑑𝑢𝑐𝑖 + 0.002𝑚𝑒𝑑𝑢𝑐𝑖 | 2 + 0.06𝑤𝑜𝑟𝑘�

Question 70

Problems with Linear Probability Model

Answer 70

1) HS෢i could be ≤ 0 or ≥ 1. Linear probability model is difficult to interpret as a probability because HS෢i is not bounded by 0 and 1. The linear probability model produces nonsensical forecasts (greater than 1 and less than 0). 2) The marginal effect of a 1-unit increase in any X is forced to be constant, which cannot possibly be true for all values of X. 3) 𝑅ത2 is no longer an accurate goodness-of-fit measure. The predicted values of Y are forced to change linearly with X, so you could obtain a low 𝑅ത2 for an accurate model.

Question 71

What is the difference between a random and natural experiment?

Answer 71

Random experiments involve researcher’s randomly assigned subjects to either a treatment or control group to estimate the treatment effect. Natural experiments or quasi-experiments, attempt to utilize the “treatment-control” framework in the absence of actual random assignment to treatment and control groups. Instead of researcher randomly assigning treatment, rely on some exogenous event to create treatment and control groups. When the event or the policy is truly exogenous, treatment is as good as randomly assigned.

Question 72

Problems in random experiments.

Answer 72

1) Random experiments are often very costly or cannot be carried out due to being unethical. 2) Non-random samples. They often lack generalizability since the sample may not be randomly drawn from the entire population of interest. 3) Attrition bias because treatment or control units non-randomly drop out of the experiment. 4) Hawthorne effects (people behave differently when observed, may respond to treatment/control status). 5) Randomization failure (can only control for observed treatment-control differences; bias may result if unobservable characteristics not perfectly balanced).

Question 73

Explain briefly the difference-in-differences estimator

Answer 73

This method estimates the impact of a treatment by comparing the outcomes of a treatment group and a control group before and after the treatment is received.

Question 74

Main underlying assumption of difference-in-differences estimator

Answer 74

The main underlying assumption: in the absence of the treatment, the difference between the outcomes of the two groups would not have changed (i.e., they would have followed a common trend). The change in outcomes of the control group is viewed as the counterfactual for the change in outcomes of the treatment group.

Question 75

What is panel data?

Answer 75

Panel data are repeated observations of multiple units over time. It is a combination of cross-sectional and time-series.

Question 76

Advantages of Panel Data

Answer 76

1) More degrees of freedom and more sample variability than cross sectional data alone or time series data alone allow for more accurate inference of the model parameters and hence increase the efficiency of estimates. 2) Eliminate omitted variables bias. It is often argued that the real reason someone finds an effect is because of ignoring specific variables that are correlated with the explanatory variables when specifying the model. Panel data allows us to control for missing or unobserved variables. 3) Ability to answer types of questions that cross-sectional and time-series data cannot accommodate. For example transitions from employment to unemployment, from employment to retirement, changes on health status or any other variables that can change through time.

Question 77

differences between the fixed effects and random effects panel data models.

Answer 77

The fixed effects model allows for 𝑎𝑖 to be correlated with the regressors and the random effects estimator assumes 𝑎𝑖 is not correlated with the regressors.

Question 78

Advantages of the random effects model

Answer 78

Advantages of the random effects model are that allows time-invariant regressors to be included and it includes more degrees of freedom.

Question 79

Disadvantages of the random effects model

Answer 79

The main disadvantage of random effects estimator is that is biased if the assumption that 𝑎𝑖 is uncorrelated with regressors is incorrect.

Question 80

Advantage of the fixed effects model

Answer 80

An advantage of the fixed effects model is that allows arbitrary correlation between 𝑎𝑖 and any regressors

Question 81

Disadvantage of the fixed effects model

Answer 81

One of the main disadvantages is that drops out time-invariant regressors.

Question 82

Preference between fixed effects model and random effects model

Answer 82

Unless we wish to estimate the effect of a time invariant variable, fixed effects are generally preferred over random effects due to having less restrictive assumptions.