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METRICS TERM 2 > HANDOUT 5 > Flashcards

HANDOUT 5 Flashcards

1
Q

Panel data allows us to…

A

observe multiple individuals over multiple points in time.

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2
Q

restriction on individuals for panel data

A

MUST be the SAME individuals that we follow over time.

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3
Q

Strongly balanced data =

A

same number of observations for every individual

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4
Q

Why may the data be unbalanced? Is this an issue?

A

Some individuals drop out the study = attrition rate. Problem if attrition rate = f(X variables) –> self-selection bias = sample no longer representative.

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5
Q

2 types of unobserved heterogeneity

A
  1. unobserved individual heterogeneity = ai

2. unobserved time heterogeneity = dt

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6
Q

Omitted relevant variable formula for Eit Ai

A

E(b1) = B1 + B2 COV(Ai, Eit) / Var(Eit)

E(b1)≠B1

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7
Q

Pooled OLS =

A

same intercept, same slope
Use all NT observations
No dummies = no heterogeneity
Treat same individual at 2 points in time as 2 separate individuals.

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8
Q

Pooled OLS equation

A

Yit = alpha + BXit+ €it

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9
Q

4 assumptions for pooled OLS

A
  1. Each individual randomly selected
  2. No individual heterogeneity
  3. No time-specific heterogeneity = parameters constant across time (no structural change)
  4. E(€it I Xit) = 0 - strict exogeneity
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10
Q

Are estimates for pooled OLS good?

A

They’re consistent for large N / large T / large NT.

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11
Q

beta interpretation for pooled ols

A

average increase in Y for unit increase in X, averaged across all individuals and all times.

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12
Q

formula for pooled OLS estimate of b

A

sum i=1,..,N sum t=1,…,T (Xit - X bar)(Yit - Y bar)

/ sum i=1,..,N sum t=1,…,T (Xit - X bar)^2

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13
Q

How can we have different intercepts and different slopes for all individuals?

A

Yit = alpha i + Bi Xit + €it

Run N separate regressions - get alpha i and bi for every individual. get sigma^2 for every individual.

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14
Q

We can only do OLS with different intercepts and different slopes if…

A

T is large enough for every i = consistency.

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15
Q

bi formula for different slopes, different intercepts

A

bi = sum t=1,..,T (Xit - Xi bar)(Yit - Yi bar) /
sum t=1,..,T (Xit - Xi bar)^2
De-mean using individual’s own time average.

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16
Q

How else can we run a regression that allows intercepts and slopes to differ for all individuals?

A

Run one large regression with additive and multiplicative dummies for N-1 individuals - this way we only get one sigma^2 estimate.

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17
Q

Fixed effects / within-groups model allows…

A

Different intercepts, same slopes

= allows for individual heterogeneity ai

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18
Q

Equation for FE model

A

Yit = alpha + BXit + ai + €it

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19
Q

b formula for FE

A

b = sum i=1,..,N sum t=1,…,T (Xit - Xi bar)(Yit - Yi bar)

/ sum i=1,..,N sum t=1,…,T (Xit - Xi bar)^2

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20
Q

Why is FE model also know as “within-groups” model?

A

Because we de-mean by an individuals own time average (then average across all individuals)

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21
Q

How many parameter estimates do we get for FE?

A

1 x slope coefficient Beta

N x intercept estimates

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22
Q

Slope coefficients in FE are consistent if…

A

Large N / large T / large NT

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23
Q

Intercepts in FE are consistent if…

A

large T

24
Q

De-meaning method for FE

A

Yit = alpha + B1Xit + B2Ai + €it
Yi bar = alpha + B1 Xi bar + B2 Ai + €i bar
(Yit - Yi bar) = B1(Xit - Xi bar) + (€it - €i bar)

25
Q

Does it matter if we omit Ai for FE model?

A

NO - because de-meaning eliminates Ai anyway = we can control for unobserved individual heterogeneity. So don’t worry about bias.

26
Q

3 stages of partitioned regression for FE

A
  1. Regress Yit on Ai & save resid (Yit - Yibar)
  2. regress Xit on Ai & save resid
  3. Regress Yi tilda on Xi tilda
    = we’ve stripped out the individual heterogeneity.
27
Q

In a FE model, our X variables must…

A

MUST vary with time. So we cannot estimate coefficients on race, gender etc. we can only control for them as part of FE.

28
Q

Ai captures…

A

all the time invariant variables.

29
Q

Random effects model equation

A

Yit = alpha + BXit + ai + €it

30
Q

How does RE differ to FE? What 3 assumptions do we make?

A

WE make some assumptions about ai, which are random drawings from a distribution:
E(ai) = 0
V(ai) = sigma^a - time invariant
COV(ai, aj) = 0

31
Q

In a RE model, what happens to ai?

A

It becomes part of the error term

Ui = ai + €i

32
Q

Does Ui in RE model satisfy CLRM assumptions?

A
  1. E(Ui) = 0 yes
  2. V(Ui) = sigma^2 a + sigma^2 yes time invariant
  3. COV(Uit, Uis) = sigma^2 a ≠ 0 violates CLRM.
33
Q

Why is there a non-zero covariance between Uit and Uis for RE?

A

Looks like serial correlation

Because ai is time invariant and is part of error term.

34
Q

Solution to serial correlation in RE model

A

Transform the equation so that the error term is serially uncorrelated within i.

35
Q

lamba to transform RE model =

A

Lambda = 1 - sqrt[sigma^2 / (sigma^2 + Tsigma^2 a)]

36
Q

Transformed RE model equation

A

(Yit - lamba Yi bar) = (1-lambda) alpha +

B(Xit - lamba Xibar) + (Rit - lamba Ri bar)

37
Q

If lamba=0, what does RE model become?

A

lamba = 0 –> POOLED OLS
If lambda = 0, sigma^2 a = 0
So NO individual heterogeneity
Yit = alpha + BXit + Rit

38
Q

If lamba=1, what does RE model become?

A

lamba = 1 –> FE Model
If lambda = 1, sigma^2 a –> infinity
HUGE individual heterogeneity
Yit - Yi bar = B(Xit - Xi bar) + (Rit - Ri bar)

39
Q

If 0 < lambda < 1, which model is most efficient?

A

RE model

40
Q

FE vs RE when COV(ai, Xit) = 0

A 
RE = unbiased, efficient
FE = unbiased, but inefficient
41
Q

FE vs RE when COV(ai, Xit) ≠ 0

A 
RE = efficient still, but BIASED
FE = unbiased still, but inefficient
42
Q

Why is FE model inefficient? But then why is it useful?

A

inefficient as estimate a lot of parameters - different intercept for every individual.
But good if huge individual heterogeneity.

43
Q

Test for FE vs RE

A

Hausman Test
H0: COV(ai, Xit) = 0 - RE correct and efficient
H1: COV(ai, Xit)≠0 - RE incorrect
FE correct under either

44
Q

Test stat for Hausman test

A

H = (b FE - b RE)^2 / V(b FE) - V(b RE)

45
Q

Hausman test: CVs from…

A

Chi-squared distribution

dof = number of slope coefficients we estimate

46
Q

Hausman test stat under H0 and H1

A

H0: b FE = b RE so H –> 0
H1: b FE ≠ b RE so H –> infinity as we square
Under both V(b FE) > V(b RE) so denom > 0

47
Q

We can also do the Hausman test for…

A

OLS vs IV
H0: COV(Xi, €i) = 0
OLS is efficient, but biased under H1
IV is always ok, but inefficient

48
Q

First difference model equation

A

change Yt = B change Xt + change €it

Ai eliminated.

49
Q

In first difference model, t goes from…

A

t = 2,…,T

2 because we have a first difference so decrease no observations by 1.

50
Q

OLS estimate of b for first difference model

A

b = sum i =1,…,N t=2,…,T (changeXit - changeX bar)(changeYit - change Y bar) / sums (change Xit - change X bar)^2

51
Q

In first difference model, change X bar =

A

change X bar = sums change Xit / n(T - 1)

52
Q

When is first difference model same as FE?

A

when T=2

53
Q

When is first difference model NOT same as FE?

A

When T>2

54
Q

When T>2, worry with first difference model =

A

The error term is MA(1) process

change €it = €it - €it-1

55
Q

Complications with dynamic model for panel data

A

OLS -> biased estimators
As t–>infinity, and N fixed, estimates consistent
But usually in panel data T small & large N

56
Q

What is the bias caused by in dynamic model?

A

BY having to eliminate ai from each observation –> correlation of order (1/T) between lagged dependent variable and residuals.

57
Q

Solution to OLS bias in dynamic model

A

Take first differences & then use instruments to do IV estimation. Since error term = MA1 = 1 period memory, use lags yt-2 or anything before as replacement for yt-1.

METRICS TERM 2 (6 decks)

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