rossi-instruments and fixed effects

8/12/2019 Rossi-Instruments and Fixed Effects

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Omitted Variables, Instruments and

Fixed Effects

Structural Econometrics Conference

July 2013

Peter Rossi

UCLA | Anderson


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VariationImagine that our goal is to determine the pure orcausal effect of changing the variablex1on y.

What is the ideal source of variation? Exogenous

variation by which we mean experimental variation.As though we conducted an experiment where we

randomly changedx1. This means that all variation

is independent of any other variables influencing y.

Then we could simply regress yonx1

y =!1x1+ "


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Omitted variablesHowever, we live in an non-experimental world.

or

Our experiments are imperfect and dont have full

randomization.

There exist other variables which belong in the

equation in the sense that they are correlated with x.

y =!1x1+ "

y=!

1x1+ !

2x2+#( )


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Omitted variablesIf we regress y on x1, what does least squaresestimate?

E y| x1

!

"

#

$=E %

1

x1

+ %2

x2

& +'| x1

!

"

#

$=%

1x

1+ %

2E x

2| x

1!"

#$

=%1x

1+ %

2(x

1

)

)x1

E y| x1!" #$ =%1 + %2( = &

Direct Effect ofx1 Indirect Effect ofx1


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Omitted variablesThis means that there can be an asymptotic bias orinconsistency from omission of variables.

Implies a bias term which does not go away even ininfinite samples.

plim !OLS(yonx1) = !

!" #1=$#

2


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Is Omitted Variable Bias A Problem?Not necessarily.

Supposex1is under firms control butx2is not.

If we use our data to estimate the relationship

between x1and x2then this is the same using OLS

from y on x1.

Suppose both variables are under firms control.

It is still not clear. If x1is price, x2is promotion (like adisplay). If my display policy doesnt change (the

frequency with which I support price changes bydisplay) then I want delta not beta1!


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What Can Be Done?Bringx2into the model and use multiple regression.

What does multiple regression do?

Instead of using all of the variability ofx1to estimate

the coefficient, multiple regression uses only that

portion uncorrelated tox2(regressx1onx2and usethe residuals). Throwing out data!

y =

b1x1+

b2x2+

ey = b

1e1.2

+v

x1= c

0+ c

1x2+e

1.2


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Omitted Variables and EndogeneitySuppose the omitted variable is not observed by theresearcher.

The omitted variable bias is now

y =!x+ "w+#y( )

x=w+ $x

E y| x!" #$ =%x+E &w+'y | x!"

#$

=%x+&E w| x!" #$

=%x+&(

w

2

(w

2+(

)x

2

*

+

,,

-

.

//x


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Omitted Variables and EndogeneityThe omitted variable bias is

Another derivation:

!

"w

2

"w

2+"

#x

2

$

%

&&

'

(

))

y =!x+"

! = X'X( )#1X'y

plim !( ) =!+plim X'XN

$

%&'

()

#1X'"

N

$

%&&

'

())


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Omitted Variables and EndogeneityIf there is an asymptotic bias.

errors are correlated with the rhs variable, x

plim X'X

N

!"#

$%& =Var(x) = '

w

2+'

(x

2

plim X')

N

!

"#$

%& = cov x,*w+ (

y( ) = * 'w2( )

+ plim, = *'w

2

'w

2+'

(x

2

!

"##

$

%&&

plimX'!

N

"

#$%

&'( 0


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Endogeneity ProblemIn this example, the omitted variable induces acorrelation between the rhs variable and the error

term. This creates the asymptotic bias. We cant use

estimators that are based on a regression estimator.

This means that this is not a valid regression function

and you cant use the wrong likelihood (the onebased on the assumption that it is a regression)!

y =!x+ "y

E "y| x#

$ %

&= f x( )' 0


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IV SolutionWhat is the problem? The variation in x is notindependent of y as would be the case if we had

experimentally or randomly induced variation.

Solution: assume that a part of the variability of x isclean or independent of any other relevant

variables. Partition the variability by the use of an

instrument.

I.E. throw out some portion of the variability of xand hope (assume) the rest is clean.


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A Simple Linear IV Modely =!x+ "

y

x= z'# + "x

"y

"x

$

%&&

'

())~N 0,*( )

Here only that portion of the variation of x which canbe explained by the instruments (z) can be used to

infer about beta.


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Identification Intuition

p = 4z+ !1

q ="1p + !2

A simple example

corr !1,!

2( ) = .8

Colors

holderror

variation

fixed


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Intuition on IV Estimators

y =!"z+ !#x+ #

y( ) =$1z+v1x= "z+ #

x =$

2z+v

2

v1

v2

%

&''

(

)**~N 0,+( )

Consider the case of only one instrument. Computethe conditional distribution of y,x | z called the

reduced form:


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!ILS

=

"1

"2

=

z'z( )#1

z'y

z'z( )#1

z'x

= z'x( )#1

z'y

The reduced form is simply a Multivariate RegressionModel for which MLE is least squares. We can then

form the ratio of least squares estimates.

The intuition here is that if we move z, then both thex, y outcomes are altered. The ratio is the effect of x

on y.


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Asymptotic Distribution

plim !ILS( ) =plim

z'x

N

"

#$%

&'

(1z'y

N=plim !+

z'x

N

"

#$%

&'

(1z')

y

N

"

#$$

%

&''

=!+ covz,x( )p limz')

y

N

"

#$$

%

&'' =!

N !" !( ) = z'xN

#

$%&

'(

"1

Nz')

y

N

Nz')

y

N

d* +* N 0,,2( )


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Asymptotics is normal since the denominator isassumed to converge to a non-zero constant.

But, suppose zx has a distribution concentrated onsmall values. Then we have problems. The

sampling distribution behaves like a ratio of normals.

18


!ILS

=

z'y

z'x


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Another way of motivating the IV estimator is twostage least squares: 1. regress x on z; 2. put the

fitted value into the structural equation.

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2SLS

x= x+ex = !z+e

x

fit

y = "2SLS

x+ey

ory = "

2SLSx+ ce

x

If you put x and the residualinto a multivariate regression,

only that portion of x whichvaries independently of the

residual is used.

Sounds like Petrin and Train!


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1. Reduces variation in rhs variables2. Requires a valid instrument. There is no test for

validity of instruments. Validity must stem from

economic reasoning. Validity means that the IVmust not enter the outcome or structural equation.

(exclusion or (y, z) indep given x).

3. There is no valid test for endogeneity- Hausmantest (compare IV and non-IV procedures) requiresa valid instrument.

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Costs|Dangers of the IV approach

y,


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4. Obviously, we want a strong instrument so wecan have maximum precision in estimation. Can

there ever be a strong, yet, valid instrument? Or

would we ever care about endogeneity in thecase of a strong instrument?

5. IV estimators are biased (finite sample) withsampling distributions that are hard to

approximate. Standard Asymptotics fails in the

case of many and weak instruments. Asymptoticstandard errors are much too small and nominal

95 CI can have coverage less than .75.

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Costs|Dangers of the IV approach


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In econometrics, there has been a lot of emphasis onimproved inference getting CI with the correct size.

Are we testing or are we making predictions?

Perhaps, we might prefer an estimator with bias but

smaller variance.

For marketing mix problems, what we really want is a

good estimate of the profit function and how this

varies with the marketing mix variables. For somesituations, this can be obtained without the use of

instruments!

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Estimation Versus Inference


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Consider a situation with a good deal of endogeneity(1/4 variation in the structural error co-moves with x).

First stage R-squared of 1 percent. N=200

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Weak Instruments


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Weak Ins Ex

Posterior distribution showing pronounced non-normality and

huge long tails.

You can see why asymptotics fail. Asymptotics are based on

local curvature of the likelihood (or moment) criterion

function. This likelihood is very curved locally!

beta

-6 -4 -2 0 2 4 6

0

2000

4000

6000


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An instrument is a variable that moves around x buthas no direct effect on y. Suppose we use an invalid

instrument.

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Invalid Instruments


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One definition of endogeneity problem:

Any situation in which there is an unobservable

that influences an outcome (e.g. sales) that is

observed by the firm and incorporated in thesetting the rhs marketing mix variables (enter into

the FOC for firm optimization).

In this sense, endogeneity is ubiquitous. It also

applies at any level data aggregation (from market tothe individual level).

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When Should We Worry about Endogeneity?


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Endogeneity of Price (it this a fad?)How did we get so fixated on price endogeneity?

Fitted sales models have too small elasticities.

too small: implied optimal prices higher than

actual prices.

Explanation:

1. Failure to include competing stores or chains.

2. Mis-specified functional forms andheterogeneity bias.

3. Endogeneity bias


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Price EndogeneityConsider a generic example:

i is the index of cross-sectional unit (consumer,

store, market)

j is the brand index

t is time

yijt

| pijt


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Some ExamplesTypes of endogeneity concerns wrt topijt.

Over j: unobserved product characteristics

solution: brand intercepts or brand specific

parameters

Over i: unobserved market characteristics.

solution: market-specific effects.

Over t: what is this unobserved demand shock thatvaries over time? If t is weekly or higher frequency,

this is difficult to believe.

solution: dont do anything about this or iv?


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Price Endogeneity

2008 2009 2010 2011 2012

2.5

3.0

3.5

4.0

4.5

5.0

Price

Prices of Kelloggs Frosted Mini-Wheats over time atone store. What is the source of this variation?


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Price EndogeneityVariation in Regular or Base Prices

Temporary price promotions or sales

Those express concerns regarding endogeneity

would have you believe that retailers change prices

on a week to week basis because they anticipatemarket wide demand shocks.

What are these demand shocks? (at the product

level, from week to week) Can they possibly beimportant? If they are advertising, isnt this just a

standard omitted var problem?


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Price EndogeneityIt looks like all or at least the vast majority of pricevariation is related to long and short run price

changes. If there is an endogeneity problem, it cant

be of economic consequence.

If I instrument with wholesale prices, then I eliminate

variation in prices due to temporary price cuts or

sales. That will reduce NOT increase price

elasticity.

Isnt the more important question to understand how

to model long run vs short run price changes? -- seethe bald man and Erdem, Imai and Keane.


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Some SuggestionsUse Fixed Effects and Controls instead ofinstruments where possible.

Dont use weak instruments.

Obviously, well executed randomized experiments

are the ultimate valid instruments. Ironically, we seelittle of this in marketing relative to economics.


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Fixed EffectsConsider the general sales response model:

The term fixed effects refers to the possibility that

each unit of observation (market, product, or time

period) could have unique parameters. In principle,

all model parameters could be unit-specific. In this

case, we simply estimate the model independentlyon all units.

yijt = !

ijt + "

ijtx

ijt + #

ijt


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Fixed EffectsClearly this is inefficient and possibly infeasible(singular X matrix for some units). For this reason,

methods which combine data from multiple units

(pooling) are used. One possibility is to simply allow

the intercepts to vary, only.

In this model, we say there are market, product, andtime specific fixed effects. This is clearly a

saturated model. Lets remove the time component.

yijt = !

ijt + "x

ijt + #

ijt


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Fixed Effects

Because this is a linear model, we can eliminate the

intercept fixed effects parameters by differencing orcentering:

yijt = !

ij + "x

ijt + #

ijt

yijt!y

ij,t!1=" x

ijt!x

ij,t!1( )+ #ijt! #ij,t!1( )y

ijt!y

ij.=" x

ijt!x

ij.( )+ #ijt! #ij.( )


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Fixed EffectsWhat are we doing?

Centering: we now use ONLY variation for unit ij over

time. All cross-sectional variation is thrown out.

Differencing: only changes over time within unit

matter.

Simple to implement. However, errors are no longer

iid and have autocorrelation within unit. A cluster

covariance estimate should be used.


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Fixed Effects RationaleThere is some unobservable that varies across unitsand effects both y and x. By including the fixed

effect, we are actually estimating the level of the

unobservable in a so-called non-parametric way

(we make no restrictions on the distribution of the

unobservable).

The presence of the unobservable means that units

with similarxs are not necessarily similar. However,

a unit is always similar to itself. So the validexperiment is time series or within-in variation in

x.


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Errors in the VariablesLets review. If a regressor is measured with error,there is a downward bias in the OLS estimates.

y =!x* + "

x= x*+#

or

y =! x$#

( )+ " =!x+ "$ !#

( )

plim !OLS( )" ! =!

#$

2

#$

2+#

x*2

%

&''

(

)**


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Fixed Effects and Measurement ErrorThe Error-In-the-Variables asymptotic bias isminimized by maximizing total variation in the true

construct. If you reduce variation in the true value of

x, then you are increasing the error-in-the-variables

bias.

Fixed effects procedures radically reduce total

variation (particularly if you put in unit fixed effects)

and, therefore, maximize errors-in-the-variables bias.

Many report that estimates of beta decline as you put

in fixed effects. Does this mean you haveunobservables or errors-in-the-variables?


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Fixed Effects and Measurement ErrorWe may be in much better shape than manyeconomists in that we might argue that price is

measured with little error. Although this is not true for

many datasets where prices are merged in from

other sources.

Are promotion and advertising variables measured

with little error?

All variables collected via a survey style method mayhave a serious EIV problem.


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An Example Involving Fixed EffectsLarry (JP), Moe (Peter), and Curly (Guenter) areinterested in the determinants of demand for Private

Labels. In particular, we want to estimate the income

expenditure elasticity.

Sis PL share of total expenditure for household i in

month t. Panel of about 35,000 householdsobserved for at least one year. Xare covariates such

as education etc.

Sit = ! ln Income

it( )+ xit'" + #

it


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An Example Involving Fixed Effects

There are two sources of variation: 1. acrosspanelists and 2. across time for the same panelist.

There is a great deal of income variation across

panelists and there is a very large sample of

panelists.

JP and Guenter have received conventional applied

econometrics/IO. Their training has taught them to be

suspicious of effects estimated by cross-sectional

regressions.


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Why?

Standard argument: there is some unobservable that

is correlated with both income and demand for

private labels. Perhaps, subscription to CR. Or lowerincome folks shop at stores where the relative price

of PL is lower/different than for higher income folks.

Larry and Curly want to throw out ALL of the cross-

sectional variation in income by putting in household

fixed effects.

Sit! S

i.= " lnInc

it! Inc

i.( )+ x

it!x

i.( )

'

# + $it


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To be fair, Larry and Curly dont necessarily reject thecross-sectional results but they wouldnt accept them

as valid unless the cross-section results agree with

the fixed effects results. They regard this as proof

that there arent any unobservables.

Problems with this argument:

1. Assumes that income effect measured acrosshouseholds has the same economic meaning as

the effect measured over time in the same

household.

2. Assumes that measurement error in income issmall.


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Moe thinks there is a great deal of measurementerror in income (panelists dont arent forced to

update income each year and it is not clear what

year the income applies to).

His preferred specification is a cross-sectional

regression with average income to reduce

measurement error. Average income probably

should be interpreted as a proxy for permanent

income.

What about responses to year to year changes in

income? This is transitory income.


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In this example, we expect the cross-sectionalregression to yield higher income elasticities than the

within or fixed effects regression from an

elementary application of the theory of permanent

income.

It doesnt mean there is a correlated unobservable

lurking out there!

Before you throw in fixed effects, I think you have to

have a good reason. A better set of controls is

always more valuable than throwing in the toweland putting in fixed effects.


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The Dangers of Fixed Effects

Fixed effects estimation methods throw out a greatdeal of the variation in the data.

Measurement error problems are frequently ignored.

Fixed effects approach does not extend in any

meaningful way to nonlinear models like choicemodels.

If results change when you dump in fixed effects, this

does not mean there is an unobservable!

You need a strong argument that your controls are

not adequate.


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Conclusions

We are very fortunate in marketing to have suchexcellent data.

The key characteristic of our data is the large

variation in marketing mix variables.We should be very cautious about adopting methods

which are profligate with data.

There has to be a strong argument that we might

expect not only endogeneity but a high degree ofendogeneity bias before resorting to IV methods.


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Conclusions

Ultimately, the only way to properly evaluateinstruments is in the context of a joint model of

supply and demand.

There may be few valid and/or useful instruments formany Marketing Mix vars like advertising!

There is a stronger argument that fixed effects

approaches are more useful that IV methods in

marketing applications, particularly ones where the

rhs variable of interest is measured well.


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Conclusions

Theoretically, arguments for endogeneity do notdepend on the level of data aggregation. However, it

is very hard to make the argument that endogeneity

bias is large with individual consumer data as the

relative variance of the common component to thetotal error variation will be small.

rossi-instruments and fixed effects

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