instruments hanes

8/13/2019 Instruments Hanes

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How to use Instrumental Variables

Questions that can be answered with experiments:

1) If I give people aspirin, what happens to their body temperature?

2) If I use fertilizer on tomato plants, do I get more tomatoes?

Other conditions could affect outcome (hot room for body temperature, rainfall for plants)

If I have the power to hold other conditions fixed, I do so.

If I cant control other conditions,

I randomizetreatment so that treatment is uncorrelated with other changes in conditions.

Questions that cant be answered with experiments:

1) What would happen to quantity demanded if I raise price,

leave other determinants of demand the same? (Price elasticity of demand)Fool with price, see effect on quantity demanded. P Q

2) What would happen to real GDP if I raise real interest rate,

leave other determinants of real GDP the same? (Slope of IS curve)

Fool with real interest rate, see effect on real GDP. r Y


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How youcantanswer nonexperimental questions:

collect data where variables of interest wriggle

across time (time-series)units (cross-section)

time & units (panel)

and observe co-movements between variables.

because other determinants/relevant conditions will be correlated with treatment

1) Anything that changes one economic variable changes others too -general equilibrium!

2) Reverse causality.

In quantity demanded, treatment is

P

1) other variables that affect , perhaps correlated with P:I, other PsQ D

2) supply (P)Q S

In IS curve, treatment is r

1) other variables that affect Y, perhaps correlated with r: G, Y* (exports), Ye

2) Taylor rule r ' r % (Y&Y) % (&()


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How to approach nonexperimental questions

1) Ignore reverse causality and correlation with other determinants.

- old papers

- econometrics field, where goal is to demonstrate technique not answer questions

2) Answer question for a model, not real world

In a model, you can exogenously wriggle variable of interest.

To quantify parameters, assume model is true and fit reduced forms to data

3) Natural experiments

Find some wriggles in treatment variable that are uncorrelated with other determinants

Often involves instrumental variable:

- something measureable that causes or indicates wriggles in treatment variable (relevant)

- uncorrelated with other determinants (valid)

e.g.in quantity demanded, variable that affects supply not demand

in IS curve, exogenous variations in central bank setting of r


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In terms of regressions

Y ' 1X

1 %

2X

2 % ( '

3X

3%.....)

Troublesome variable Control variable ... Unmeasurable determinantsX1

X2

X3

General equilibrium problems: X3

'31X

1%.... X

1'

13X

3%.... X

1 '

14X

4 % ....

X3

' 34X

4 % ....

Reverse causality:

X1 ' Y1Y % 15X5%... X1 ' Y11X1 % 2X2 % % 15X5%...

correlation with

Instrumental variables Z:

1) Relevant:Z is correlated with X1

X1

' Z1

Z1

% Z2

Z2

% ........

2) Valid: Z isuncorrelated with (not cause or effect or common cause or...)

Exclusion restrictionX3

' 0Z1

% 0Z2

% ........


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Methods

1) 2SLS Good for small samplesand/or probably-homoskedastic s

1)First stage Regress onZ.X1

2)Use first-stage coefficients to get predicted values X1

Note theres lots of variation in not captured byX1

X1

3) Second stage Regress Yon X1

Note that it wouldnt be right to use SEs from second stage to judge significance -

this wouldnt account for uncertainty about coefficients of first-stage regression.

2SLS SEs account for this.

2) GMM Good for large samples, heteroskedastic s

Can allow for correlations across observations residuals.

E.g. clustering (common in panels):

s correlated across some observations, uncorrelated across others

Problem!if instruments are weak(relevant but only weakly correlated with )X1

SEs too smalland estimated coefficients biassed toward OLS values

(though consistentas )n Y 4

even in very largesamples


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How do you find instruments?

How do you know if possible instruments arerelevantand valid?

1) Think about why varies. List all possible reasons.X1

2) Think about other determinants of Y. List all possible s.X3

3) Which reasons for -variation are unrelated to and measurable?X1

X3

Of course, you must think about what causesZ to vary, and on and on...

This is not a statistical issue. Its knowledge about the world (including economics).

but there are..


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Statistical methods to test whether candidateZs are relevant, strong & valid

1) Look at first-stage regression.

Check sign, significance, magnitude (plausible?) of coefficient onZ.

Because of weak instrument problem, t-stats (or F-stats for multipleZs)

need to be lots bigger than 2. Rule of thumb: at least 10.

2) Look at reduced form regression of YonZ

Y ' 1 Z1Z1 % Z2Z2 % ........ % 2X2 %

Sign, magnitudes of coefficients should be consistent with story.

3) If you have more than oneZ( is overidentified) you can test oneZagainst others,X1

see if results are consistent acrossZs.

Validity/exclusion restriction means allZs uncorrelatedwith (Y & 1X

1&

2X

2)

a) Get estimates ,1

2

b) Calculate (Y& 1X

1&

2X

2)

c) AreZs is uncorrelated with ?(Y& 1X1& 2X2)

Sargan test (for 2SLS), Hansen J-test (for GMM)

Zs are OK if you fail to reject hypothesis.


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Are lagged Ys and/orXs valid instruments?

One of the most mechanical and naive, yet common, approaches to the choice of instruments uses

atheoretical and hard-to-assess assumptions about dynamic relationships to construct instrumentsfrom lagged variables in time series or panel data. The use of lagged endogenous variables..is

problematic if the equation error or omitted variables are serially correlated

(Angrist & Krueger, 2001)

E.g. in the Arellano-Bond panel estimator, identification results from lack of serial correlation in

the errors (p. 293)


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Are lagged Ys and/orXs valid instruments?(cont.)

Example: supply and demand

You want to estimate demand elasticity.

Q D ' &bPt

%t

where t

' D

t&1%

t

D$0

Q S' cPt

% et

where et

' Se

t&1% e

t

S$0

( , )1

'&b Y1

' 1/c

so Pt

'1

b%c(

t& e

t) Q

t'

b

b%c(

t% e

t)

1) For or to be relevantinstruments for ,Pt&1 Qt&1 Pt and/or must be greater than zero.D

S

2) but if , they are not validD

0

because they are correlated with t


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Bottom line for you

In every empirical anything,

1) Try to tell me why varies across observations within your sampleX1

2) Choose one from below

a) do not talk about effects (as in demand elasticity, IS curve slope, ...)

b) argue that most variation in is uncorrelated with unmeasurable determinants of YX1

variable

c) use BS (fitted model)

d) use instruments

3) If you choose d),

i) Tell me what causesZs to vary, why they are correlated with , why they are validX1

ii) Show first-stage and reduced-form regressions

iii) ProveZs are not weakwith rule-of-thumb F-tests (or s) or other testsR 2

iv) Unless you can make very strong a prioricase for validity, show Sargan or J-test

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