Panel Data I

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PANEL DATA WORKSHOP

BRUNEL UNIVERSITY

February 29, 2008.

PART I:

THE ABC OF INSTRUMENTAL VARIABLES

AND GMM ESTIMATION


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The ABC of instrumental variables and

GMM estimation

Presentation outline

1. Introduction

2. Instrumental variables estimation

3. Empirical example

4. Three important tests

5. Empirical example continued

6. Summary


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1. Introduction

Econometrics is concerned with the analysis of financial

business and /economic data– time series, cross sectional

or panel data.

The analysis can be at individual, firm, industry or country

level.

Often the aim is to establish a causal relationship between

various variables.

1. Do bank loans help firms export more?

2. Are women discriminated against in the labour market?

3. Does financial development foster aggregate growth?


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How about the endogeneity

problem???


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1. Introduction

Suppose OLS regression of exports on bank loans shows a

positive correlation between the two variables.

Does this correlation imply that bank loans are the cause of

increased export? -- Not necessarily so!!!

May be exporters are more successful in obtaining bank

loans or banks favour exporters more than non-exporters! Thus the causality might be from exports (dependent

variable) to bank loans (independent variable).

This is known as the problem of reverse causality or more

generally endogeneity.

Bank loans are potentially endogenous in the model.


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1. Introduction

As another example, suppose that OLS regression of per

capita GDP growth on financial development shows a

positive relationship between the two variables.

Since it is possible that financial markets develop in

anticipation of future GDP growth, financial development

could be a lead indicator of growth rather than an

exogenous cause of growth.

This creates the problem of endogeneity because the

finance-growth relationship is simultaneously determined.

It is important to remember that OLS would be biased when

one or more regressors are endogenous.


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1. Introduction

The examples illustrate that (i) correlation does not

necessarily imply causality, and (ii) OLS may not always be

an adequate empirical tool in finance.

The objective of this lecture is to first introduce an estimation

technique which is effective at dealing with the problem of

endogeneity.

This technique is known as instrumental variables (IV)

estimation, or more generally as generalised method of

moments (GMM).

Unlike OLS, IV/GMM offer the chance of testing for causal

relationships between economic variables.


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2. IV estimation

Consider the following regression model

For OLS to be unbiased, the matrix of regressors X and the

error term ε should be uncorrelated. That is

In this case we say that the regressors are exogenous.

When at least one of the regressors are correlated with ε,

Regressors that are correlated with the error term are called

endogenous regressors.


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2. IV estimation

Endogeneity could result from a variety of reasons including:

1. Reverse causality.

2. Simultaneity bias

3. Omitted variables bias.

4. Measurement errors.

Whatever the reason behind endogeneity, the OLS estimator

of β

will be biased and inconsistent.

If there are several regressors and just one of them is

endogenous, the OLS estimator would still be biased.


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2. IV estimation

In order to obtain a valid estimator of β and make correct

inference about the relationship between y and X, we need

some additional variables.

The variables which help obtain a consistent estimator of β

are known as instrumental variables (say Z).

Instrumental variables should satisfy two properties:

1. They have to be correlated with the endogenous

regressors X:

Instrument relevance.

2. They have to be uncorrelated with the error term ε:

Instrument exogeneity.


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2. IV estimation

The instrumental variables should only affect the dependent

variable (y) indirectly through their relationship with the

endogenous regressors (X).

In other words Z should not be part of the model.

It is not always easy to come up with valid instruments that

are exogenous to the model AND correlated with X.

Suppose y = per capita GDP growth and X includes an

indicator of financial development.

The concern of endogeneity arises because faster per capita

GDP growth is conducive to financial development.


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2. IV estimation

If valid instruments that satisfy the properties of relevance

and exogeneity are available, a consistent estimator of β

can be obtained.

This consistent estimator is called the instrumental variables

(IV) estimator, and is denoted as .

Consistency means that as the sample size gets large, the

estimator converges to the true value β. The formula for the basic IV estimator is

The IV estimator has approximate normal distribution in

large samples. So statistical inference such as t-tests can be

conducted in a standard fashion.


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2. IV estimation

The basic IV estimator can be obtained as a two-stage least

squares estimation process:

1. Regress each endogenous regressor on all instruments

and exogenous regressors, and generate predicted values.

2. Estimate the model by OLS, replacing the endogenous

regressors with their predicted values.

If the error term is heteroskedastic or serially correlated,there are two options:

a. can be used with robust standard errors.

This option corresponds to the use of robust standard errors

in OLS regressions

It is the "safest” option, though not the most efficient one.


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2. IV estimation

b. An efficient version of the IV estimator called the

generalised method of moments (GMM) can also be used.

This option corresponds to the use of Generalised Least

Squares (GLS) in the standard regression analysis.

In small samples, the GMM estimator tends to be inaccurate.

IV-GMM estimation requires at least as many instrumental

variables as endogenous regressors.

When there are more instruments than endogenous

regressors we say that the model is overidentified.


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IV, GMM, relevance,

exogeneity, overidentification.


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The aim is to test whether access to finance (bank loans)

causes an increase in private firms exports in China.

The following model is specified (i indexes firm)

EXPORT is log of export, BANK is log of bank loans, DIST is

log of distance from the city the firm is located in to the

nearest port, LAB is a dummy variable showing whether thefirm is in a labour intensive industry or not.

The model has three regressors, two of which, DIST and

LAB are arguably exogenous (why?)

BANK is potentially endogenous, however.

One the one hand, bank loans might help firms export by

providing them with the necessary financial resources.


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On the other hand, banks might prefer to lend to exporting

firms. So exporting could help secure more bank loans.

Because of this potential problem of simultaneity bias, we

employ IV/GMM.

To start with, explore the following three variables as

potential instruments:

1. POL: A dummy variable indicating whether the firm has

political connections or not.

2. STATE: The share of state-owned enterprises (SOEs) in

the region the firm is located in.

3. EQUITY: The amount of equity/collateral the firm has.


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Arguably, these instrumental variable candidates are

correlated with the endogenous regressor (BANK). Thus

they are likely to be relevant instruments.

Political connection and high level of collateral help obtain

more bank loans; while high presence of SOEs is likely to

reduce private firms’ access to finance.

On the other hand, the property of exogeneity requires that

the instruments affect exporting through bank loans alone,

rather than being fundamental drivers of export.

First estimate the model by OLS with robust standard errors,

and then by IV/GMM.


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A peek at the cross sectional data ( N=5167)


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OLS with robust standard error:

Bank loans are positively correlated with export, but with

marginal statistical significance.

Should we trust these results? Probably not, because of

simultaneity bias.


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Two-stage least squares (IV) with robust standard errors: First stage regression

1.These are the instrumental variables.

2. Usually first-stage regressions are not reported in applied work.

But it is important to routinely inspect them.


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Two-stage least squares (IV) with robust standard errors: Second stage regression

1. All interpretation of the model should be based on the second stage

regression.

2. Compared to OLS, 2SLS results appear to be counterintuitive: bank

loans hurt exports; and distance to port does not seem to matter.

3. We should test for the validity of the instruments before taking these

results seriously!!


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When working with instrumental variables three importanttests should be performed as a matter of routine. These are:

1. Testing or checking for the relevance of the instrumental

variable candidates: If the instruments have no or little

correlation with the endogenous regressors, they are called

weak instruments and would bias the IV estimator.

2. Testing for the exogeneity of the instruments: If theinstruments are correlated with the error term, IV would be

invalid.

3. Testing whether the endogenous regressors are really

endogenous: If the regressors are not endogenous after all,

OLS would be the most efficient estimation method.


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1. TESTING FOR INSTRUMENT RELEVANCE:

The idea is to check whether the instruments are sufficiently

correlated with the endogenous regressors.

The simplest way is to test for the joint significance of the

instruments in the first stage regression.

As a rule of thumb, if the calculated F statistic is more than10 and the p-value is 0, the instruments are likely to be

relevant.

If the instruments are weak, it is advisable to look for

other/additional instruments.


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2. TESTING FOR INSTRUMENT EXOGENEITY

The instruments should have no correlation with the error.

In order to test for instruments exogeneity, we need to have

more instruments than endogenous regressors. The number

of excess instruments is called the number of overidentifying

restrictions (in our example this number equals 2).

The Sargan/Hansen can be used to test for IV exogeneity. The null hypothesis of the test is “ All instruments are

valid”.

If the null hypothesis is rejected, it means that at least one of

the instruments is not valid.

The test does not pinpoint which instruments are invalid.


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3. TESTING FOR ENDOGENEITY OF REGRESSORS.

Even if the instruments are found to be valid, it is a good

idea to test whether it is really necessary to use IV/GMM.

This can be achieved through the Hausman test for the

endogeneity of regressors

Where VIV and VOLS are the variance of the IV and OLS

estimators respectively.

The null hypothesis is : “All regressors are exogenous”.

Under the null, H is distributed as Chi-squared random variable

with degrees of freedom equal to the number of regressors.

If the null hypothesis is not rejected, stick with OLS!!


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Relevance

Exogeneity

Endogeneity

TEST TEST TEST!


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5. Empirical example cont.

We can test for the relevance and exogeneity of the

instruments in our export model as follows:

The command “estat first” tests for instrument relevance.

Since the F statistic is greater than 10 and the p-value = 0,

the problem of weak instruments is probably not too serious.


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The command “ estat overid ” test for the exogeneity of

instruments.

The null hypothesis of the test is “ POL, EQUITY and STATE

are all exogenous instruments”.

Under the null, the test statistics is distributed as a Chi-

squared random variable with 2 degree of freedom ( 3

instrumental variables – 1 endogenous regressor).

The p-value of the test = 0, so we reject the null hypothesis

and conclude that at least one of the instruments is not valid.

Thus the IV results reported earlier should be discarded.

Let’s try re-estimating the model by dropping one of the

instruments (EQUITY).


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POL and STATE

are valid instruments


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With valid instruments, the results suggest that bank loans

play a positive and highly significant role in boosting exports.

To check the robustness of this finding, re-estimate the last

model by GMM.

The two sets of results are practically the same, which is

reassuring.


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5. Empirical example cont

.

Finally test for the endogeneity of BANK

The Hausman test suggests that BANK is indeed

endogenous. So using OLS would have been problematic.

REJECT THE NULL


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6. Summary

1. The problem of endogeneity is common in applied

econometrics.

2. IV/GMM offer a way of tackling the problem of

endogeneity.

3. It is important to test for the validity of the

instruments before taking the results from IV/GMM

estimation seriously.

THANK YOU!

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