Cross Sectional

Question 1

What are two requirements for linear regression

Answer 1

1) Additive errors 2) One multiplicative parameter per term

Question 2

What are some important requirements for dummy variables?

Answer 2

They cannot depend on Y You cannot include intercept and all dummies. Would give perfect collinearity (Xs are not linearly independent)

Question 3

Derive the OLS beta estimator

Answer 3

Question 4

What is the formula for R^2 centered?

Answer 4

RSS/ TSS = 1 - SSE/TSS

Question 5

What is TSS, RSS and SSE

Answer 5

Question 6

When to use centered and uncentered R^2?

Answer 6

When model does not contains constant, use uncentered

Question 7

What are the large-sample asymptotic assumptions of regression?

Answer 7

1) Stationary ergodicity 2) Full rank (E[Xi’Xi]) 3) Martingale difference sequence (error times regressor) 4) Moment existence. Regressors have finite 4th moments and errors have finite variance

Question 8

What is estimator bias?

Answer 8

Difference between expected value of estimator and true value of parameter

Question 9

What is estimator consistency?

Answer 9

And estimator is consistent if its probability limit is the true parameter

Question 10

What is the problem with ommited variables?

Answer 10

If the estimated excludes relevant variables that are correlated with the included variables, the parameter estimates will be biased. Also, parameter variance will not be estimated consistently. It is safe to exclude dummy variables since they are orthogonal to regressors

Question 11

What happens if you include extraneous (non-relevant variables)

Answer 11

You parameter estimates will still be unbiased and variance will still be consistent. But you will have more variance (potentially too low t-stat)

Question 12

What are the problems of working with heteroscedastic data?

Answer 12

Parameters will be unbiased, but variance estimator will be inconsistent. One solution is to use White’s robust variance estimator. Using White’s estimator on homoscedastic data will however give worse finite sample properties and increases likelihood of size distortions. Another solution to heteroscedastcity is to use GLS

Question 13

How can use test for heteroscedasticity?

Answer 13

Use White’s test. Estimate your model. Then regress squared errors on squares and cross products of all regressors (including constants). Null hypothesis: all parameters (except for intercept) are zero. Test statistic n*R^2. It is chi^2 squared distributed with df = number of regressors in auxiliary regression (excluding intercept)

Question 14

What happens when errors are correlated with regressors?

Answer 14

If measuring regressors with noise. Regressors and errors may be correlated. This gives downwards bias. Endogeneity will also give bias.

Question 15

What is a type 1 error

Answer 15

Rejecting a true null

Question 16

What is a type 2 error?

Answer 16

Failure to reject the null when the alternative is true

Question 17

What is the size of a test?

Answer 17

Pr(Type 1 error) = alpha

Question 18

What is the power of a test?

Answer 18

Pr(1- type 2 error). Probability of rejecting a null when the opposite is true

Question 19

What is a linear equiality hypothesis?

Answer 19

R*beta - r = 0 Where R is an mxk matrix r is mx1 vector m i number of restrictions k is number of regressors

Question 20

What are the 3 types of hypothesis tests?

Answer 20

Wald test Lagrange Multiplier test Likelihood Ratio test

Question 21

What are the differences in power between the 3 types of hypothesis tests?

Answer 21

W is about the same as LR which is larger than LM in terms of test statistics. Since they follow the same distribution, larger test statistics gives more power

Question 22

How to implement a Wald test

Answer 22

Run the unrestricted regression and estimate parameters and covariances. Compute test statistics using null hypothesis. Compare against chi-squared distribution

Question 23

How to implement a Lagrange Multiplier test

Answer 23

That is, the Lagrange Multiplier Test examines the size of the “shadow price” of the constraint regarding we are trying to test in the OLS optimization framework. Estimate the restricted model and compute its errors. Calculate the score using errors from the restricted model and the regressors from the unrestricted model Calculate the average score and the varianc of the score Compute the test statistic

Question 24

How to implement a Likelihood ratio test

Answer 24

The Likelihood Ratio test is based on testing whether the difference of the scores, evaluated at the restricted and unrestricted parameters, is large Estimate the unrestricted model and its parameter covariance Compute the test-statistic using the restrictions Compare against CVs from Chi^2_m distribution

Question 25

What are the main model selection techniques?

Answer 25

General to specific (GTS) Specific to general (STG) Information Criteria Cross-valiadation

Question 26

Explain general to specific

Answer 26

Start with the largest possible number of possible models If at least one regressor is insignificant, remove the regressors with the lowest t-stat. Restimate the model. Continue until all regressors are significant

Question 27

Explain specific to general

Answer 27

Start with the variable with the lowest p-value. At new variables sequentially as long as most recently added variable is significant

Question 28

Discuss pros and cons of GTS and STG

Answer 28

GTS has a positive probability of including irrelevant variables, it will however never exclude relevant variables. In STG, the variance is NOT consistently estimated in the beginning, which can lead to wrong inference. For both, we have the problem that t-stats do not follow standard distributions when used sequentially.

Question 29

Explain Information Criteria

Answer 29

If possible, search through all models and pick the one where the information criteria is lowest. Typical information criteria are AIC and BIC. If not feasible, run GTS or STG type search through models. Information criteria are of the form: - log-likelihood + penalty term for including regressors. In OLS use ln(error variance) instead of ll.

Question 30

Discuss pros and cons of different information criteria

Answer 30

The AIC asymptotically may select a model with irrelevant regressors

Question 31

Explain k-fold cross validation

Answer 31

Divide the data into k group. For each group, estimate beta excluding the data from the group. Calculate SSE error using the beta estimate on the data in the group. Sum SSE for all group. Select the model with lowest total SSE.

Question 32

What are some important specification tests?

Answer 32

Chow test. For stability / structural breaks

RESET test: for misspecified model

Rollig/recursive parameters

Residual plots

Question 33

What is a chow test?

Answer 33

it is a test for parameter stability. Regress y on xs and x*indicator(s) Null is that indicator-parameter is zero. Test using Wald, LR or LM

Question 34

What is a RESET test?

Answer 34

Test for a misspecified model. Estimate the model. Calculate y_hat Run regression of y on original regressors, plut squared y_hat, cubed y_hat etc. Null: parameters for y_hat are zero. If rejected, some non-linear model may be better

Question 35

How should outliers in regression by defined?

Answer 35

In terms of errors. Could upper an lower quantiles (e.g. 0.025). Defining errors in terms y would lead to downwards bias

Question 36

Explain two methods of deadling with outliers?

Answer 36

Trimming. Removing outliers observations Winzorisations. Setting observations above threshold to threshold (in terms of errors) After this, rerun the regression

Question 37

Explain residual bootstrap and when to use it

Answer 37

Estimate beta_hat and errors in full sample Resample errors and xs independetly. Compute y_tilde from resample x and errors and beta_hat Estiamte beta_tilde Repeat k times (e.g. 500) Take variance of beta-tildes Can only be used if data is homoscedastic

Question 38

Explain non-parametric bootstrap

Answer 38

Resample x and y pairwise. Estimate beta_tilde. Repeat k times (e.g. 500). Take variance of beta_tildes

Question 39

What is the test statistic in a Wald test?

Answer 39

Distributed Chi^2 m, where m is number of restrictions

Question 40

What are some key factor that affect the power of a test?

Answer 40

Sample size

Distance between null and true value

Estimator precision