term 2- regression with panel data- revise

Question 1

what is a panel dataset?

Answer 1

a panel dataset contains observations on multiple entities where each entity is observed at two or more points

Question 2

what is a balanced panel?

Answer 2

no missing observations; that is, all observations are observed for all entities and all time periods

Question 3

what is an unbalanced panel?

Answer 3

some observations are missing; that is some observations are not observed for some entities and time periods

Question 4

what can we control for with panel data?

Answer 4

we can control for factors that:
vary across entities but do not vary over time
could cause committed variable bias if they are committed
are unobserved or unmeasured - and therefore cannot be included in the regression using multiple regression

Question 5

how does panel data control for ommitted variable bias?

Answer 5

if an ommitted varibale does not change over time, then any changes in Y over time cannot be caused by the ommitted variable

Question 6

what if there is more than two time periods?

Answer 6

if there are more than two time periods you can rewrite the regression in two ways:
1) “n-1 regressor” regression model
2) “ fixed effects” regression model

Question 7

express the fixed effects model in n-1 binary regressor form?

Answer 7

Y_it = B0 +B1X_it + s_2 D_2i +… + s_n D_ni + u_it
where D_2i = 1 for i=2 or 0 otherwise

Question 8

express the fixed effects model in the fixed effects form?

Answer 8

Y_it =a_i +B1 X_it +u_it where a_i is called a state fixed effect and is the constant effect of being in state i

Question 9

when is the n -1 binary regressors OLS regression pratical?

Answer 9

it is only pratical when n isnt too big

Question 10

how do you do n-1 binary regressors OLS regresiion?

Answer 10

first create the binary vairable D2i,…, Dni
then estimate 1 by OLS
inference (hypothesis tests, confidence intervals) is as usual (using heteroskedasticity robust standard errors)
this is impractical when n is large ie n=1000

Question 11

how do you complete entity demeaned OLS regression?

Answer 11

first construct the entity demeaned variables Y~_it and X~_it
then estimate 2 by regressing Y~_it on X~_it using OLS
(it is like the changes approach but instead Y_it deviated from the state average instead of Y_i1. )

Question 12

give examples of an ommitted variable which might vary over time but not across states?

Answer 12

safer cars such as airbags; changes in national law
these produce intercepts that change over time

Question 13

what are the two formulations of regression with time fixed effects?

Answer 13

T-1 binary regressor formulation
time effects formulation

Question 14

what is the T-1 binary regressor formulation?

Answer 14

Y_it =B0 +B1X_it+ 𝛿2B_2t+….+𝛿_TB_2t+u_it
where B2t= 1 when t=2 ,0 otherwise, etc for B3 up to BT

Question 15

what is the time effects formulation

Answer 15

Y_it=B0 +B1X_it + 𝜆_t+u_it

Question 16

under the anel data version of the least squares assumtion, is the ordinary least squares fixed effects estimator B1 normally distributed?

Answer 16

yes

Question 17

what are the fixed effects regression assumptions?

Answer 17

1) E(u_it|X_i1,….,X_iT,a_i) =0
2) (X_i1,…,X_iT,u_i1,…,u_iT), i =1,..,n are iid draws from their joint distribution
3) large outliers are unlikely (X_iT,u_iT) have finite fourth moments
4) there is no perfect multicollinearity (multiple X’s)

Question 18

when is the assumption 2 of the fixed effects regression assumptions satisfied?

Answer 18

(Xi1,…,XiT,ui1,…,uiT), i =1,…,n, are i.i.d.draws from their joint distribution.
it is satisfied when entities are randomly sampled from their populations from simple random sampling

Question 19

what does the assumption 2 of the fixed effect regression assumption not need to satisfy?

Answer 19

the observations do not require to be iid over time for the same entity

Question 20

what is autocorrelation?

Answer 20

autocorrelation means correlation with itself. suppose a variable Z is observed at different dates so observations are on Z_t t=1,..,T, then Z is autocorrelated if Corr(Z_t,Z_t+j) not equal to zero for some dates j when j is not equal to zero

Question 21

if ommitted factors are serially correlated, what is the error term?

Answer 21

the error term is also serially correlated

Question 22

why are OLS standard errors in general are wrong for panel data?

Answer 22

they assume that the error term is serially uncorrelated. in reality the OLS standard errors often underestimate the true sampling uncertainty

Question 23

what are clustered standard errors?

Answer 23

clustered standard errors estimate the variance of B1 when the variables are iid across entities but are potentially autocorrelated within an entity

Question 24

what is the equation of the clustered SE of Y^_?

Answer 24

square root[{s^2(mean of Y)}/n] where s^2(mean of Y)} = 1/(n-1) * Σ( sample mean of Y for entity i - mean Y)

Question 25

what is the one key features of the clustered SE’s?

Answer 25

in the cluster SE derivation we never assumed that observations are iid within an entity thus we have implicitly allowed for serial correlation within an entity

Question 26

why might panel data help?

Answer 26

potential OV bias from variables that vary across states but are constant over time
potential OV bias from variables that vary over time but are constant across states

Question 27

what are the advantages of fixed effects regression?

Answer 27

you can control for unobserved variables that vary across states but not over time and/or vary over time but not across states
more observations give you more information
estimations involves relatively straightforward extensions of multiple regressions

Question 28

what are the limitations of regression with fixed effects

Answer 28

need variation over time with entities
time lag effects can be important