Tutorial 5 - Panel data, Pooled OLS, Random-Effects-Estimator, Fixed-Effects-Estimator

Question 1

What is panel data?

Answer 1

panel data or longitudinal data are multi-dimensional data involving measurements over time. Panel data contain observations of multiple phenomena obtained over multiple time periods for the same firms or individuals.

Question 2

How does a Linear Panel Data Model look like?

Answer 2

• i = 1, .., N cross-sectional dimension (e.g. persons, firms, countries)
• t = 1, …, T time-series dimension (e.g. years)

Question 3

How does a stacked model for all NT observations look like?

Answer 3

• i = 1, .., N cross-sectional dimension (e.g. persons, firms, countries)
• t = 1, …, T time-series dimension (e.g. years)

Question 4

What does Pooled OLS estimator mean and how is it defined?

Answer 4

pooling all NT observations and applying OLS

Question 5

Which condition needs to be met for the POLS to be consistent?

Answer 5

POLS is only consistent under the assumption of weak exogeneity of the regressors

Question 6

What is the variance of the POLS error term?

Answer 6

NT × NT matrix:

Question 7

What may the error variance matrix Ω of POLS contain and what not?

Answer 7

it may contain:

• heteroscedasticity, i.e. different variances of error terms across individuals
• autocorrelation of error terms within individuals over time

BUT:

• Assumption: no correlation of error terms across individuals

Question 8

What does the Breusch-Godfrey-Test test?

Answer 8

Test for serial correlation

Question 9

How does a model with serial correlation look like?

Answer 9

Question 10

What are the steps for applying the Breusch-Godfrey test?

Answer 10

1. Estimate an (P)OLS regression to estimate the model
2. Using the estimated residuals ûᵢ as an estimate for û, estimate following model (below)
3. Test (n − p)R² against critical value from X²ₚ (p needs to be justified, e.g. how many observations per person (cluster))

Question 11

What is μᵢ in a model with random effects?

Answer 11

individual-specific, time-constant unobserved effect

Question 12

What are the assumptions for Random Effects?

Answer 12

see (1), (2), (3) below

with (3): the regressors are strictly exogenous, i.e. mean independence of composite error term and regressors of all (!) time periods. Assumption is violated if the individual-specific effect μᵢ is correlated with xᵢₜ!

Question 13

What effect does μᵢ have on the composite error (in random effects models)?

Answer 13

Cov (ϵᵢₛ, ϵᵢₜ) ≠ 0

Note: RE explicitly models the serial correlation in a GLS framework

Question 14

How does the variance between two individual observations look like in a model with random effects?

Answer 14

Question 15

Show that there is serial correlation of error terms within individuals over time in a model with random effects.

Answer 15

Question 16

Is there autocorrelation across individuals in a model with random effects?

Answer 16

No autocorrelation across individuals

Question 17

How does the Var(ϵ) look like in a model with random effects?

Answer 17

• Var(ϵ) = Ω, a NT × NT matrix.
• However, under the RE assumptions, it only includes two distinct parameters: σ_μ² and σₑ².
• In practice, Ω has to be consistently estimated -> ^Ω

Question 18

What is the random effects estimator?

Answer 18

Question 19

What are problems both with POLS and RE that Fixed Effects and First Differences can fix?

Answer 19

POLS and RE are inconsistent if the individual-specific effect μᵢ is correlated with xᵢₜ (example: individual-specific ability could be correlated with education, or with joining a union)

Question 20

How do Fixed Effects and First Differences handle the case that the individual-specific effect μ_i is correlated with x_it ?

Answer 20

Fixed Effects and First Differences allow for this correlation by using only variation in xᵢₜ within individuals over time -> μᵢ cancels out -> controls for time-constant selection bias

Question 21

How can you find an expression for fixed effects?

Answer 21

Fixed Effects = POLS on time-demeaned data:

Question 22

How can you find an expression for first differences?

Answer 22

First Differences = POLS on first-differenced data:

Question 23

How would a fixed effects model be different if estimated on the data of the model shown below?

Answer 23

• variables educ and black drop out because they don’t change over time for a given individual
• FE is the same as a “Least Squares Dummy Variable Estimator” - here: POLS with a dummy for each individual
• Time-constant variables like educ drop out, but we can estimate the interaction of educ with time dummies -> i.e., we can’t estimate the level of the return to education, but we can estimate whether the return changed over time

Question 24

What do the different terms in the model mean?

Answer 24

index ᵢₜ: not only identify the person, but also the time of the observation
λₜ stands for control variables (one for each year, except the base year) = like a dummy for each year
μᵢ = 545 values, one for each observation

Question 25

Which four characteristics/assumptions describe the multiple linear regression mode?

Answer 25

• Linearity:
• Independence:
• Strict Exogeneity:
• Error Variance:

Question 26

What is linearity in this model?

Answer 26

Linearity: The model is linear in parameters α,β,γ, effect c_i and error u_it

Question 27

What is independence in this model?

Answer 27

Independence: X_i, z_i, y_i are i.i.d. = independent and identically distributed. The observations are independent across individuals but not necessarily across time.

Question 28

What is exogeneity in this model?

Answer 28

Strict Exogeneity: The idiosyncratic error term u_it is assumed uncorrelated with the explanatory variables of all past, current and future time periods of thesame individual. This is a strong assumption which e.g. rules out lagged dependent variables.

Question 29

What is assumed about error variance in this model?

Answer 29

Question 30

What is c_i in the random effects model?

Answer 30

In the random effects model, the individual-specific effect is a random variable that is uncorrelated with the explanatory variables

Question 31

Which three additional assumptions make this a random effects model?

Answer 31

• Unrelated effects
• Effect variance
• Identifiability

Question 32

What is the assumption of related effects in the fixed effects model?

Answer 32

That the assumption from the random effects does not hold:

E [c_i | X_i, z_i] ≠ 0

assumes that the individual-specific effect is a random variable that is correlated with the explanatory variables of all past, current and future time periods of the same individual

Question 33

What is the assumption of unrelated effects in the random effects model?

Answer 33

E [c_i | X_i, z_i] = 0

assumes that the individual-specific effect is a random variable that is uncorrelated with the explanatory variables of all past, current and future time periods of the same individual

Question 34

What is the assumption of effect variance in the fixed effects model?

Answer 34

absence of assumption of constant variance of the individual specific effect

Question 35

What is the assumption of effect variance in the random effects model?

Answer 35

assumes constant variance of the individual specific effect

Question 36

What is c_i in the fixed effects model?

Answer 36

In the fixed effects model, the individual-specific effect is a random variable that is allowed to be correlated with the explanatory variables

Question 37

Which three additional assumptions make this a fixed effects model?

Answer 37

• Related effects
• Effect variance
• Identifiability

Question 38

What happens if this random effects model is estimated by Pooled OLS?

Answer 38

• The pooled OLS estimator of α, β and γ is unbiased
• It is consistent and approximately normally distributed
• However, the pooled OLS estimator is not efficient
• The usual standard errors of the pooled OLS estimator are incorrect -> Correct standard errors can be estimated with the cluster-robust covariance estimator treating each individual as a cluster

Question 39

What happens if this fixed effects model is estimated by Pooled OLS?

Answer 39

The pooled OLS estimators of α, β and γ are biased and inconsistent, because the variable c_i is omitted and potentially correlated with the other regressors

Question 40

What is the first difference etimator?

Answer 40

• The first-difference (FD) estimator is an approach used to address the problem of omitted variables with panel data.
• The estimator is obtained by running a pooled OLS estimation for a regression of Δ y_it on Δx_it