Endogeneity and Instrumental variable estimation method - PowerPoint Presentation

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Endogeneity and Instrumental variable estimation method

Obid

A.Khakimov

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Revew

Four complications that induce correlation between

X

and

e

Omitted Variables Bias

Measurement Error

Simultaneous Causality

Using Lagged Values of the Dependent Variable as

Explanators

, in the presence of serial correlation

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Endogeneity

1. Omission of relevant variables

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Measurement error

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Measurement error: independent variable

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Simultaneity

Consider the simultaneous system

Reduced forms

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7

Responses to Endogeneity

Remedial-1: IV estimation method

Remedial-2: Differencing methods:

Limitations:

- will not eliminate selection bias.

- only eliminate

fixed

variables; sometimes endogenous variables change values over time

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8

Responses to Endogeneity

Approach #3: Difference it out -- continued

Limitations:

- DD models will not eliminate selection bias.

- DD models only eliminate

fixed

variables; sometimes endogenous variables change values over time

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9

Omitted Variables

1. Find additional data so that every relevant variable is included.

Ignore it: if omitted variable is uncorrelated with all included variables

Find proxy variable.

Proxy z must be redundant (= ignorable)

E (y | x, q, z) = E (y | x, q

)

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Measurement Error

1. Improve measurement

2. Argue that the degree of error is small

- Use outside data for validation

3. Argue that error is uncorrelated with included variables

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Instrumental Variables

An

Instrumental Variable

is a variable that is correlated with

X

but uncorrelated with

e.If Z

i

is an instrumental variable:

E(

Z

i

X

i

)

≠ 0

E(

Z

i

ei

) = 0

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Instrumental Variables

The econometrician can use an instrumental variable

Z

to estimate the effect on

Y

of only that part of

X that is correlated with Z.

Because

Z

is uncorrelated with

e

, any part of

X

that is correlated with

Z

must also be uncorrelated with

e

.

An instrumental variable lets the econometrician find a part of X that behaves as though it had been randomly assigned

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Instrumental Variables

One way to see this is in terms of two regression equations

Y

i

=

β

0 + β

1

X

i

+

ε

i

X

i

=

γ

0

+

γ1Zi

+ ηi

Note that, in this model

X

is endogenous (may be correlated with

ε

)

The instrumental variables model requires that:

1.

γ

1

≠

0 so that

Z

predicts

X

, and

2. Z

uncorrelated with

ε

(

Z

is exogenous) [

Cov

{

ε

, Z

} = 0]

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Implication

Estimate model by OLS and by IV, and compare estimates

If

If

But test INDIRECTLY using Wu-

Hausman test.

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THE INSTRUMENTAL VARIABLES (IV) ESTIMATOR

Suppose that one or more of the regressors in X is not independent of the equation error term, even in the limit as the sample size goes to infinity. That is, X is correlated with u, the equation disturbance.

However, suppose we have another variable, Z, (an instrument for X) that has the properties:

Z and X are correlated

(2

) Z and u are uncorrelated

Now define the IV estimator as:

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Generalised IV estimator (GIVE)

A more general form of the IV estimator where we have more instrumental variables than “endogenous” X variables

GIVE is potentially more efficient than simple IV, if instruments are well-chosen

Test whether instruments are ‘valid’ using Sargan’s test

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How do we find an instrumental variable?

There are two methods:

Arbitrary search and test.

Two stage least squares

.

Two Stage Least Squares (2SLS) offers an excellent direct estimation method in the case of exactly or over-identified equations.

18

Finding a suitable Instrumental Variable

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Strong IVs

A strong instrument has a high correlation with the endogenous variable.

How strong a correlation? Staiger & Stock (1997) recommend a partial F statistic of 5 or greater.

- Run 1

st stage with and without the IV.

- Compare the overall F statistics: a difference of 5 or

more is sufficient evidence of strength.

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20

Weak IVs

If the IVs are weak,

2SLS and 2SRI are consistent, but there can be considerable bias even in large samples

standard errors are too small

2SLS and 2SRI perform poorly

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Weak IVs

What to do if IVs are weak?

If there is a single endogenous variable, use a

conditional likelihood ratio

(CLR) test:

* perform a regular likelihood ratio test

* adjust the critical values

* available in Stata; see Stata Journal, 3, 57-70

and

http://elsa.berkeley.edu/wp/marcelo.pdf

by Moreira

and Poi

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The

first stage

involves the creation of an instrument. Use the reduced from equation for P to get its fitted value, Phat.

The

second stage

involves a variant of instrumental variables estimation. Replace P by Phat in the supply equation and use OLS in this second stage of the estimation process

So it is in fact a special way and perhaps less arbitrary way of doing instrumental variables estimation.

22

Two stage least squares as IV estimation

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Although one could undertake 2SLS estimation manually, running the reduced form regression, saving the fitted values and then running the second stage (structural form) regression, modern software allows you to get the results automatically with one set of instructions.

You need to tell the software which RHS variable is endogenous and which other variables should be used as regressors in the reduced form (first stage) of the regression.

Using the automatic IV procedure will also guarantee appropriate estimates of the second stage standard errors.

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Two stage least squares estimation with modern econometric software

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IV over identification test

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Hausman specification test

Test compares two coefficients and follows

chi-square

distribution

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Paper replication (IV method)

The Colonial Origins of Comparative Development: An Empirical Investigation Author(s): Daron

Acemoglu

, Simon Johnson, James A. Robinson Source: The American Economic Review, Vol. 91, No. 5 (Dec., 2001), pp. 1369-1401

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Introduction

Research question:

How accurately to measure the effect of institutions on

economic development process?

“

Countries with better "institutions," more secure property rights, and less distortionary policies will invest more in physical and human capital, and will use these factors more efficiently to achieve a greater level of income (e.g., Douglass C. North and Robert P. Thomas, 1973; Eric L. Jones, 1981; North, 1981).”

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Introduction

Theory:

1. There were different types of colonization policies:

A)

European powers set up "extractives states"

B) Setting new colonies with European institutions with strong emphasis on private property and checks against government power

2.

States where disease environment was not favorable lead to less migration of

colonizators

and more likely development "extractives states“

3.

Colonial institutions persisted even after independence

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Literature and theory

William H. McNeill (1976), Crosby (1986), and Jared M. Diamond (1997) have discussed the

influence of diseases on human history.

Diamond (1997), in particular, emphasizes

comparative development, but his theory is based on the geographical determinants of the incidence of the

neolithic

revolution.Ronald E. Robinson and John Gallagher (1961), Lewis H. Gann and Peter Duignan (1962), Donald Denoon (1983), and Philip J. Cain and Anthony G. Hopkins (1993) emphasizes that settler colonies such as the United States and New Zealand are different from other colonies, and point out that these differences were important for their economic success.

Frederich

A. von Hayek (1960) argued that the

British common law tradition was superior to the French civil law, which was developed during the Napoleonic

era to restrain judges' interference with state policies

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Literature and theory

Curtin (1964) documents

how early British expectations for settlement in West Africa were dashed by very high mortality among early settler.

Pilgrim decided to migrate to the United States rather than Guyana because of the high mortality rates in Guyana (see Crosby, 1986 pp. 143-44).

Robinson and Gallagher (1961), Gann and

Duignan

(1962), Denoon (1983), and Cain and Hopkins (1993), have documented the development of "settler colonies," where Europeans settled in large numbers, and life was modeled after the home country.When the establishment of European like institutions did not arise naturally, the settlers were ready to fight for them against the wishes of the home country.

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Literature and theory

There are a number of economic mechanisms that will lead to institutional persistence:

1.

Setting up institutions that place restrictions on government power and enforce property rights is costly. It may not pay the elites at independence to switch to extractive institution.

2. The gains to an extractive strategy may depend on the size of the ruling elite.

3. If agents make irreversible investments that are complementary to a particular set of institutions, they will be more willing to support them, making these institutions per-

sist

(see, e.g.,

Acemoglu

, 1995)

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Nigeria, which has approximately the 25th percentile of the institutional measure in this

sample, 5.6,

and Chile, which has approximately the 75th percentile of the institutions

index, 7.8

. The estimate in column (1), 0.52, indicates that there should be on average a 1.14- log-point difference between the log GDPs of the corresponding countries (or approximately a 2-fold difference-e1

. 14-

1 2.1). In practice, this GDP gap is 253 log points (approximately 1-fold)

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R - Current institutions (protection against expropriation between 1985 and 1995),

C - Early (circa 1900) institutions,

S - European settlements in the colony (fraction of the population with European descent in 1900),

M - mortality rates faced by settlers.

X - vector of covariates that affect all variables.

where Y is income per capita in country

i

, R is the protection against expropriation measure, X

i

is a vector of other covariates

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Robustness check

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