Flashcards in L5: IV regression Deck (35)

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1

## When might we use instrumental variables? (3 and what these issues have in common)

###
1) OVB from a variable that is correlated with X but is unobserved (tf cannot be incl. in regression eqn.)

2) Simultaneous causality bias (ie. X causes Y AND Y causes X)

3) Errors-in-variables bias (X is measured with error)

All 3 problems -> E(u|X) not equal to zero

2

## What does IV regression do?

### Eliminates bias when E(u|X) not equal to zero, using an instrumental variable, Z

3

## What are endogenous and exogenous variables?

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Endogenous - a variable correlated with u

Exogenous - a variable not correlated with u

4

## What are the two conditions for a VALID INSTRUMENT?

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1) Instrument relevance: corr(Zi, Xi) /=0

2) Instrument exogeneity: corr(Zi, ui) = 0

5

## Explain carefully how to estimate when using an IV?

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2 stage least squares:

1) ISOLATE part of X that is uncorrelated with u by regressing X on Z using OLS:

EQN: Xi= π0+ π1Zi+ vi

Because Zi is uncorrelated with ui, π0+ π1Zi is also tf so is Xi! From here, we then compute predicted values of Xi, where: Xi(hat)=π0(hat)+ π1(hat)Zi

2) Replace Xi by Xi(hat) in the regression of interest, and regress Y on Xi(hat) using OLS:

ie. Yi=B0+B1Xi(hat)+ui

Since Xi(hat) is uncorrelated with ui, E(u|X(hat))=0 tf it works! (Then can estimate B1(hat)(TSLS))

6

## What does 2SLS require?

### n to be large so π0 and π1 are estimated precisely

7

## Show that the 2SLS estimator is equal to the ratio of the covariances: S(YZ)/S(XZ)

### see notes bottom page 1 side 1

8

## Is the 2SLS estimator consistent?

### YES see notes for why (ie. both the sample covariances are consistent tf the estimator tends with probability to true value of B1)

9

## What is inference like using TSLS?

### Same as usual

10

## Why are OLS standard errors from the 2nd stage regression wrong?

### They do not take into account the estimation of the first stage where Xi(hat) is estimated (stata can solve this with a command that computes the TSLS with corrects SEs) (HTSK-robust SEs)

11

## Why would a regression that relates quantity (Y) to price (X) likely suffer from bias? What type of bias would this be?

### This regression only gives equilibrium point at the crosssover of S and D, but when collecting data in a market only get price and quantity at equilibrium tf no D and S function and tf this gives rise to simultaneity bias (ie. change in D causes change in Quantity supplied and vice versa?)

12

## See

### cigarette demand example in notes

13

## See

### General IV regression model notes

14

## What is the problem in the generalised IV regression model with adding more IVs?

### see notes

15

## Explain the three cases of identification relevant to 2SLS? When can 2SLS be done?

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Exact identification if m=k

Underidentified if m less than k

Overidentified if m>k

Can only be done with exact/overidentification - where m is number of IVs and k is number of ENDOgenous regressors

16

## See notes

### Bottom of side 2 check I understand how to do TSLS with a single endogenous regressor (X) and multiple exogenous regressors (W1...Wi) (go over cig example too!)

17

## If you have 2 suitable IVs, Z1 and Z1, that are both correlated with the endogenous variable and uncorrelated withe error, which should you use and why?

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BOTH!

regress the endogenous variable on both Z1 and Z2 - this is a case of overidentification and therefore will reduce the SE of the results (so long as additional IVs are appropriate): more information -> BETTER ESTIMATES!

18

## Explain under what assumptions does TSLS hold and its t-statistic is normally distributed?

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1. E(ui|W1i,...,Wri) = 0 the exogenous regressors are exogenous.

2. (Yi,X1i,...,Xki,W1i,...,Wri,Z1i,...,Zmi) are i.i.d

3. The X’s, W’s, Z’s, and Yhave nonzero, finite 4th moments

4. The instruments (Z1i,...,Zmi) are valid (ie. Corr(Zmi,ui)=0 and Corr(Zmi,Xi)=/0 for m=1 to M)

19

## In MRM generalised IVs, when are instruments said to be relevant? And when are they said to be weak?

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In the first stage, if at least one π is not equal to zero then the instruments are relevant

If they are all equal to zero (or v. close to zero) the instruments are weak

20

## What do weak instruments do?

### They explain very little of the variation in X BEYOND what is explained by the W's

21

## What is a consequence of IVs being weak?

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TSLS sampling distribution and t-stat are not at all normal, even when n is large!

(Why? Because makes S(XZ) v small tf beta1(hat)TSLS becomes very large!) (ie. no correlation between X and Z and tf Z does not explain X tf Z does not explain Y either!) (see notes bottom of S2P2 and top of S1P3)

22

## How do you test instrument strength?

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F-test that tests that all the coefficients on Z1,...,Zm DO NOT ENTER first stage regression (ie. are all equal to zero)

Rule of thumb: if F-stat is less than 10 then the set of instruments is weak! (tf -> biased 2SLS)

23

## What does comparing to F=10 actually allow us to do?

### Compare if the bias (relative to OLS) is greater or less than 10% (IF F is less than 10, bias is more than 10% and vice versa!!!)

24

## 2 solutions to weak instruments?

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1) Find better instruments/drop ones you think may be weak

2) Use other estimators (can be very complicated though)

25

## What criteria must be fulfilled to test for instrument exogeneity? What is the consequence for TSLS if this assumption does not hold?

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Criteria: the model must be overidentified to do this test!

If the assumption of instrument exogeneity fails, then TSLS is INCONSISTENT!

26

## When to use J-test of overidentifying restrictions?

### If given say 2 IVs, Z1 and Z2, and computer TSLS for both and the estimates for beta are very different, then know that one of Z1 or Z2 must be invalid

27

## See

### bottom of p2s2 on how to conduct a J-test

28

## What are the hypotheses for a J-test?

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H0: All instruments are exogenous

H1: At least one instrument is not exogenous

29

## J-statistic distribution? How many DofF in a J-test?

### Chi-squared, with m-k DofF

30