Panel data-FE

Question 1

The model for fixed effects is
yi = β0 + β1xi + ci + ui
What are some examples of what ci could be?

Answer 1

ci could represent the effects of ability, health, motivation, intelligence, parental resources, managerial quality, organizational culture, state/local policies or regulations, etc.

Question 2

What does the regression of the de-meaned y on de-meaned x look like, mechanically?

Answer 2

yit −y ̄i =β1(xit −x ̄i)+(uit −u ̄i)

Question 3

What does it mean to de-mean in the context of fixed effects?

Answer 3

within each panel unit i, take the average over t on both sides and subtract the average from each it observation:

Question 4

What are some examples of time-invariant explanatory variables that fall out of the fe model?

Answer 4

gender, race

Question 5

Why do time-invariant explanatory variables fall out of the fe model?

Answer 5

They all equal their within-group mean, so the within-transformation equals zero

Question 6

write out first difference model

Answer 6

∆yi =β1∆xi +∆ui

Question 7

How do OLS assumptions apply to first difference model?

Answer 7

The new error term ∆ui is uncorrelated with the new explanatory variable, ∆xi .

This requires that we have no cross-period correlations between u and x: called strict exogeneity

The xi must vary over time for at least some i, else they difference out (same as the within transformation)

Question 8

What does strict exogeneity require?

Answer 8

no cross-period correlations between u and x

Question 9

In theory, what happens to constant when you estimate first difference model?

Answer 9

Differences out–if you want, you can include it to allow for year-to-year trend

Question 10

What happens when you apply first difference model to multiple years?

Answer 10

each year of data is differenced with previous year, so you lose the first year in your dataset

Question 11

True or false: In the one-way fixed effects model, we treat ci as a parameter to be estimated

Answer 11

True

Question 12

Mechanically, what are we doing when we estimate ci?

Answer 12

Effectively we are allowing for a unique intercept for every cross-sectional
unit i. This is feasible to estimate since each i is observed multiple times.

Question 13

model for fixed effects?

Answer 13

yit = β0 + β1xit + ci + uit

Question 14

What paramaters are we estimating when using fe?

Answer 14

intercept (B0), slope(B1), and fixed effects (which are n-1 intercepts)

Question 15

what does the LSDV (least squares dummy variable) model do?

Answer 15

includes (n-1) dummy variables in the regression

Question 16

drawbacks of LSDV approach?

Answer 16

• time-consuming
• soaks up degrees of freedom
• often not interested in the fixed effects themselves–(exception is the teacher effects work)

Question 17

When is FE more efficient than first difference?

Answer 17

FE is more efficient (smaller standard errors) than first differencing if the error terms are serially uncorrelated and T > 2

Question 18

True or false: FE Assumes no correlation in u across units of panel i

Answer 18

True

Question 19

Consistency and unbiasedness of fixed effects themselves in large samples?

Answer 19

The estimates of the fixed effects themselves (ci ) are unbiased but inconsistent in large samples. (Why? As the number of panel units grows (N → ∞) the number of parameters to estimate grows).

Question 20

What model does stata fit when you run xtreg?

Answer 20

(yit −y ̄i +y ̄)=β0 +β1(xit −x ̄i +x ̄)+(uit −u ̄i +u ̄)

Question 21

Things to ask yourself when you run fe

Answer 21

Where is the identification coming from?

How much variation is there within panel units?

Question 22

What happens when there is little variation within panel units?

Answer 22

You risk imprecise estimates

Question 23

xtreg ouput: What does the f test tell you? What is the null?

Answer 23

F-test for joint significance of fixed effects (null hypothesis H0 is that all fixed effects are zero). If rejected, fixed effects model is a reasonable assumption and regular OLS would provide inconsistent estimates. In practice, rarely rejected.

Question 24

xtreg output: what does R-squared within tell you?

Answer 24

variance “explained” by within-group deviations from mean

Question 25

xtreg output; what does R-squared between tell you?

Answer 25

variance in group means y ̄i “explained” by the group mean x’s: x ̄i

Question 26

xtreg output; what does sigma_u tell you?

Answer 26

estimate of the standard deviation in fixed effects (ci )

Question 27

Assumptions for FE?

Answer 27

FE.1: linear model yit = β1xit1 + … + βkxitk + ci + uit
FE.2: cross-sectional units are a random sample
FE.3: xit varies over time for some i, no perfect collinearity
FE.4: ∀t, E(uit|Xi,ci) = 0 or the expected value of u given x in all time periods is zero (strict exogeneity)
FE.5: Var (uit |Xi , ci ) = Var (uit ) = σu2 - homoskedasticity
FE.6: for t ̸= s errors are uncorrelated: Cov (uit , uis |xi , ci ) = 0. No serial correlation.

Question 28

What assumptions do you need for unbiasedness for FE and first difference?

Answer 28

FE.1: linear model yit = β1xit1 + … + βkxitk + ci + uit
FE.2: cross-sectional units are a random sample
FE.3: xit varies over time for some i, no perfect collinearity
FE.4: ∀t, E(uit|Xi,ci) = 0 or the expected value of u given x in all time periods is zero (strict exogeneity)

Question 29

What assumptions do you need for FE model to be BLUE?

Answer 29

FE.1: linear model yit = β1xit1 + … + βkxitk + ci + uit
FE.2: cross-sectional units are a random sample
FE.3: xit varies over time for some i, no perfect collinearity
FE.4: ∀t, E(uit|Xi,ci) = 0 or the expected value of u given x in all time periods is zero (strict exogeneity)
FE.5: Var (uit |Xi , ci ) = Var (uit ) = σu2 - homoskedasticity
FE.6: for t ̸= s errors are uncorrelated: Cov (uit , uis |xi , ci ) = 0. No serial correlation.

Question 30

When is fixed effects more efficient than the first difference model?

Answer 30

FE.6: for t ̸= s errors are uncorrelated: Cov (uit , uis |xi , ci ) = 0. No serial correlation.

Question 31

Where is variation in FE (within) model coming from?

Answer 31

uses deviations from unit means, e.g., mean “pre” vs. mean “post”

Question 32

Where is variation in first difference model coming from?

Answer 32

uses variation in successive time periods, e.g., just prior to and just after a “treatment” (a change in x)

Question 33

Is the assumpion that errors ui are iid typically satisfied in panels?

Answer 33

No–With repeat observations on the same cross-sectional unit, it is likely that errors are correlated across observations for the same i.

Question 34

How do you cluster standard errors in fe?

Answer 34

The “cluster” is typically the cross-sectional unit, although when the regressor of interest is aggregated at a higher level (e.g., state), can cluster at that level. Theory requires large N and that higher levels nest the cross-sectional units.

Question 35

Two advantages of fixed effects models?

Answer 35

• Unobserved ui can be correlated with the explanatory variables
• β1 is estimated using within-group (i) variation in x,y

Question 36

5 disadvantages of fixed effects models?

Answer 36

Cannot estimate slope coefficients for time-invariant x
Fixed effects “remove” a lot of the variation in y
The “within” model is less efficient (higher standard errors)
There may be more measurement error (and attenuation bias) when relying on within-group changes vs. levels
Group intercepts use up a lot of degrees of freedom