26. Panel Data, First Differences and Fixed Effects

Question 1

What is cross-sectional data?

Answer 1

Data on several individuals at a single point in time

Question 2

What notation did we use for cross sectional data?

Answer 2

𝑦𝑖 = 𝛽0 + 𝛽1𝑥1𝑖 + 𝛽2𝑥2𝑖 + 𝑣i

for which the subscript 𝑖 denotes individual 𝑖.

Question 3

What is Panel Data

Answer 3

Observe data for several individuals, and observe each individual at several points in time e.g. N individuals for T time periods

Question 4

What notation do we use for panel data

Answer 4

𝑦𝑖𝑡 = 𝛽0 + 𝛽1𝑥1𝑖𝑡 + 𝛽2𝑥2𝑖𝑡 + 𝑣𝑖𝑡.

Subscript 𝑖𝑡 denotes individual 𝑖 at time 𝑡.

Question 5

Hoiw do we control for time-invariant effects with the error term in panel data?

Answer 5

Decompose error term 𝑣𝑖𝑡 into** 𝑎𝑖, representing a time-invariant piece, and 𝑢𝑖𝑡, representing a
time-varying piece.**

ai is essentially the effect of being an individual

Question 6

What is the concern with ai (time-invariant effects)

Answer 6

Unobserved and possibly correlated with our regressors –> confounder

Concern always there but discuss in panel data because data is strong enough to allow us to overcome the concern

Question 7

How do we overcome the confounder of time invariant effects?

Answer 7

1. First Differences
2. Fixed Effects

Question 8

How do we do First Differences to overcome time-invariant effects?

Answer 8

For any variable, 𝑤, define the notation Δ𝑤𝑖𝑡 ≔ 𝑤𝑖𝑡 − 𝑤𝑖,𝑡−1.

1. Perform Δ𝑦𝑖𝑡 = 𝑦𝑖𝑡 − 𝑦𝑖,𝑡−1

Δ𝑦𝑖𝑡 = 𝛽0 + 𝛽1𝑥1𝑖𝑡 + 𝛽2𝑥2𝑖𝑡 + 𝑎𝑖 + 𝑢𝑖𝑡 − (𝛽0 + 𝛽1𝑥1𝑖,𝑡−1 + 𝛽2𝑥2𝑖,𝑡−1 + 𝑎𝑖 + 𝑢𝑖,𝑡−1)

2. Thus

Δ𝑦𝑖𝑡 = 𝛽1Δ𝑥1𝑖𝑡 + 𝛽2Δ𝑥2𝑖𝑡 + Δ𝑢𝑖𝑡

ai is differenced away thus ai is no longer a confounder because it does not affect the difference across time periods in either treatment or outcome

Question 9

Does only ai get removed by differencing?

Answer 9

No, any time invariant effects (constant, other variables) are removed

Cost of removing bias is that we lose all these time invariant effects

Question 10

How do we normally overcome bias due to confounders

Answer 10

we take it out of the error term and include it directly in the model

we can do an analgous operation for time-invariant effects tthrough dummy variables (fixed effects)

Question 11

How do we perform fixed effects to overcome time-invariant effects

Answer 11

we include the dummy variable, 𝛿𝑖 for all individuals except one (we must exclude one
dummy variable to avoid perfect collinearity due to the dummy variable trap).

The individual without
a dummy variable in the model is often called the “omitted group” or “comparison group.”

Question 12

Interpret coefficients for the fixed effects formula

Answer 12

𝛽1 and 𝛽2 are “the average change in the outcome associated with 𝑥1 (or 𝑥2) increasing by 1, holding fixed all other 𝑥 and holding fixed who the individual is.”

𝛽0 is “the expected outcome for the omitted group when all 𝑥 are 0.”

𝑎𝑖 is “the average change in the outcome associated with being individual 𝑖 compared to the omitted
group, holding fixed all 𝑥.

Question 13

Can we include time-invariant regressors in a fixed effects regression

Answer 13

No, including the variable violates no perfect collinearity

The heuristic explanation is that the dummy variable, 𝛿𝑖, and effect 𝑎𝑖, capture the effect of all time-invariant characteristics of person 𝑖.

If a variable, 𝑥2𝑖, does not change over time, we could lump its effect in with 𝑎𝑖, and do not need to separately estimate the effect.

Question 14

What is the cost of removing time-invariant bias with fixed effects

Answer 14

Again can’t estimate the effect of any time-invariant variables

Any time invariant variables must be excluded

Question 15

How do econometricians generally use the term fixed effects?

Answer 15

Use it to refer to any situation in which dummy variables are included all possible values of a variable

commonly applied to time periods

Question 16

What are two way fixed effects?

Answer 16

When we control for both individual and time fixed effects

Question 17

Is first differences or fixed effects more common

Answer 17

For the purpose of the exam, just know that first differences and fixed effects are two methods of overcoming the bias caused by 𝑎𝑖. Know the mechanics as described above.

In practice, fixed effects is more common because of the “simplicity” of implementation and because of the desirability of directly estimating the 𝑎𝑖.

Question 18

what si the equation of the first difference estimator D

Answer 18

time invariant effects exlcuded

Question 19

interpret fixed effects

Answer 19

dummy variables add the time-invarient effect - average change in y associated difference in y associated with being individual i compared to the excluded individual

Question 20

How do fixed effects estimators work graphically

Answer 20

allows intercept to differ for each indv. and quantity the diff. in intercept for an indv. compared to control indv.

Question 21

can we incl. regressors that do not change over tiem for an indv. when using the fixed effects estimator with dummy vairables

Answer 21

Incl. any time-invariant variable will fail no perfect collinearity when using dummy

Question 22

what is more commonly used? Fixed effects or first differences?

Answer 22

Fixed effects more commonly used to measure the ‘fixed effects’ of ai. First differences, differences ai away

Simplicity of implementation too

Question 23

What are time fixed effects

Answer 23

Including a time dummy that represents the effect of beign in time period t, since this will also induce bias

Question 24

Write notation for a dummy variable fixed effects estimator with time and individual fixed effects

Answer 24

Question 25

If we have panel data, what is the default strategy?

Answer 25

Estimate a fixed effects model with time fixed effects First difference estimator superior only if time periods is small

Question 26

Answer 26

Question 27

what else do u lose w first diff apart from ai

Answer 27

1st observation

Question 28

demeaning method explain

Answer 28