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METRICS TERM 1 > HANDOUT 9 > Flashcards

HANDOUT 9 Flashcards

1
Q

If Y2i is NOT non-stochastic do we have an issue?

A

YES - this means Y2i is endogenous

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2
Q

Variables that determine attendance

A

Attend = f(quality, time, location, past-performance) + €2i

Where €2i = motivation, interest, ability = unobservables

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3
Q

What variables determine €1i

A

€1i = random luck + same unobservables

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4
Q

COV(€1i, €2i)

A

> 0
higher ability = higher performance
higher ability = higher attendance

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5
Q

COV(attend, €2i)

A

> 0 by definition

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6
Q

Therefore, COV(attend, €1i)

A

≠ 0 –> VIOLATES CLRM

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7
Q

Why can’t we use OLS when we have an endogenous variable?

A 
As E(€1i I endogenous variable) ≠ 0
Therefore OLS is BIASED
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8
Q

Implication for OLS estimate of coefficient on attend

A

As COV(€1i, €2i) > 0 –> UPWARD BIAS

  • OLS will overestimate the coefficient on attend
  • We get a very positive significant coefficient on attend
  • attend could actually have no effect on performance, but appears to have on driven by unobservables
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9
Q

IF COV(€1i, €2i) < 0, OLS –>

A

DOWNWARD BIAS

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10
Q

Solution to endogeneity

A

= IV estimation

Two stage least squares

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11
Q

What does IV estimation try to do?

A

We want a variable that looks like attendance, but it unrelated to €1i.

Replace attend by variables that determine attend but are unrelated to €1i.

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12
Q

2 components of attendance

A
  1. systematic - could be instruments

2. random = €2i = get rid of

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13
Q

two stages of IV

A
  1. regress attend on instruments that are relevant and exogenous & save fitted values
  2. Replace attend by attend^ in original regression equation
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14
Q

Is IV unbiased?

A

NO - but it is CONSISTENT

As n–>infinity, E(b3)–>B3

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15
Q

How can TIME of seminars be a valid instrument for attendance?

A

If time of seminars is randomly allocated by tabula = unrelated to motivation, interest and ability.
So include 1. Mon/Fri dummy and 2. 9am dummy

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16
Q

Why do we include the original exogenous variables female and a-level when we regress attend on its instruments?

A

We want the coefficient on attend hat to explain variation in perf over and above females and a-levels “holding all else constant”. If we didn’t include them, maybe the instruments are related to females and a-levels.

17
Q

2 conditions for IV instruments

A
  1. Instrument relevance

2. Instrument exogeneity

18
Q

What’s the issue if our instruments are only weakly correlated with the endogenous regressor?

A

Weak correlation –> IV coefficients inconsistent–> t-stats unreliable

19
Q

Test statistic for weak instruments

A

B^IV = B + (B^OLS - B)/F

20
Q

What is F in the test statistic for weak instruments?

A

F = f-stat from test of joint significance of the coefficients on the instruments when we regress the endogenous regressor on those instruments (+ exogenous variables)
H0: d3 = d4 =0
H1: d3/3 ≠ 0

21
Q

Rule of thumb for F in the test statistic for weak instruments

A

F < 10 –> weak instruments

22
Q

If F=1 in the test statistic for weak instruments

A

F=1 –> bias of order 100%
We are back to the biased OLS estimate
So IV estimation does NOT help - coefficients are inconsistent.

23
Q

As F–>infinity in the test statistic for weak instruments

A 
F--> infinity
Instruments very significant 
Bias --> 0
B^IV --> B true coefficient 
So IV is CONSISTENT.
24
Q

Instrument exogeneity means

A 
E(€1i I instruments) = 0
Therefore E(€1i I attend^) = 0
25
Q

Test for instrument exogeneity

A 
J = mF
m= number of instruments
F = f-test of joint significance of instruments in equation of IV residuals on instruments and exogenous variables
26
Q

F for instrument exogeneity

A

IV residuals = perf - (b0 + b1attend + b2female + b3 alevels)

Iv resid = d0 + d1 female + d2 a-level + gamma1 mon/fri + gamma 2 9am + Vi

H0: gammas = 0
- We want the instruments to be unrelated to the error term = exogenous

27
Q

What distribution do we get CVs from for test of instrument exogeneity?

A

J follows Chi-squared with dof=m-k

m = no instruments; k = no endogenous regressors.

28
Q

We can only carry out the test for instrument exogeneity IF…

A

m > k
Cannot have less than a chi-squared 1
So we need no instruments > no endogenous regressors.

29
Q

H0 in J test

A

H0: instrument exogeneity
H1: instruments are INVALID

30
Q

2 Problems with instrument exogeneity test

A
  1. Need m>k but hard to find 1 never-mind 2 instruments = most of the time CANNOT test, can only use reasoning.
  2. Test is LOW-POWERED = often tells us instruments are exogenous when they’re actually INVALID.
31
Q

V(b1) OLS

A

sigma^2 / sum Y2i tilda^2

Y2i tilda = residuals from regression of Y2i on other exogenous variables X1i…

32
Q

V(b1) IV

A

sigma^2 / sum Y2i^ tilda^2

Y2i^ tilda = residuals from regression of Y2i on other exogenous variables AND instruments

33
Q

How do V(b1 OLS) and V(b1 IV) compare?

A

Variation you predict < the variation that actually happens.
So RSS IV < RSS OLS
So V(b1 IV) > V(b1 OLS)
SE IV > SE OLS
IV produces bigger se and so smaller t-ratios

34
Q

Test for detecting endogeneity

A

Original OLS regression with Y1i on exogenous variables, the endogenous variable and e2i residuals from endogenous regressor (attend) equation on exogenous variables + instrument.

35
Q

equation for e2i for our performance example. What is e2i a measure of?

A

e2i = attend - (d0 + d1 a-levels + d2female + d3mon/fri + d4 9am)
We strip out the systematic variation in attend = left with variation due to unobservables so e2i = best guess of motivation, interest, ability.

36
Q

H0 in detecting endogeneity test

A

H0: delta = 0

  • Coefficient on e2i = 0
  • So perf is unrelated to unobservables = no endogeneity problem.

METRICS TERM 1 (11 decks)

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