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A Report on Work in Progress
INFERENCE WITH IMPERFECT INSTRUMENTAL
VARIABLES
Aviv Nevo
Northwestern University and NBER
Adam Rosen
Northwestern University
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Motivation
• Finding valid IV, that are uncorrelated with the error-term, is a concern
facing much of applied micro research.
• In practice, in many cases the validity of the IV is somewhat questionable;
• Leading example: estimation of demand for differentiated products.
Depending on what controls are included the error will typically include:
unobserved quality and/or unobserved promotional activities;
Common IV include: observed characteristics or prices in other markets;
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Our Focus
• In this paper we acknowledge the imperfection of our IV and ask:
Can we get partial identification even with imperfect IV?
Can we use the imperfect IV to learn the direction of the OLS bias?
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The Model
Consider the simple linear (population) model
We focus on estimating the slope parameter, .
Assumptions:
A1: Random sampling – we can use a random sample of size n from the population
model.
A2: No perfect collinearity.
A3: and .
If instead we assumed ( ) then OLS would yield consistent and
unbiased estimates.
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Denote , and .
Let Z be an imperfect instrumental variable such that
A4: and .
A5: The covariance of X and ,, and Z and , have the same sign: .
A6: Z is less correlated with , than X: .
If instead (of A5 and A6) we made the stronger assumption then the
standard IV estimator would yield consistent estimates.
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• Define:
and
so that and are the probability limits of the OLS and traditional IV (using Z
as an instrument) estimators for .
Note: can be greater or smaller than $, depending of the sign of
• Let,
• Finally, define
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Identification
Result 1: Assume A2-A5. If then $ is between and , i.e., we have
an informative bound.
From the definitions: and .
Note: If instead of A5, , then we require .
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Result 2: Assume A2-A5. If or then
and .
Y Since is unknown, unless , then does not inform of the direction of bias.
Note: if is small, i.e., the correlation between Z and , is small, then the second
condition is likely to be satisfied.
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What can we hope to get from A6?
< By the definition of , , which implies .
< Therefore, .
< is unknown so this is not a feasible estimator.
< By A6, , so if is a continuous and monotonic function (of ) on
[0,1], then can be bounded by and .
Result 3: Assume A2-A6. If then $ is between and , i.e., we have
an informative bound.
; if and if (if );
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$T(8)IV as a function of 8: DXZ=.2, 8*=0.25
$T(8)IV as a function of 8: DXZ=.3, 8*=0.25
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$T(8)IV as a function of 8: DXZ=-.1, 8*=0.25
$T(8)IV as a function of 8: DXZ=-.5, 8*=0.25
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Result 4: Assume A2-A6. If then
As the correlation between X and Z decreases, helps improve on the bound
provided in Result 1.
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Result 5: Assume A2-A6 and that . If then
If we are willing to sign the correlation between X and ,, then for we can
bound . But all we can say about $ is that
Up to now: Assuming A5, if we got informative bounds, but the results are less
promising for the case.
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Alternative Approaches
• Replace A6 with an assumption of the sort:
It is unclear where comes from. Indeed the previous approach can be seen as an
attempt to replace with .
• Suppose we have two 2 (or more) IV
Let Z1 and Z2 denote these IV. Suppose each of them satisfies A4-A6.
If and , then the additional IV can tighten bounds;
If and , then using Z2 we can achieve an informative bound;
If and ,
Difference Z1 and Z2 to create a variable that let’s us exploit previous results;
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and , wlog assume ;
For simplicity assume Y ;
Define .
if and ;
There exists a ( that satisfies both these conditions iff ;
Assuming such a ( exists it should be in [0, ), the closer it is to
the more likely it is that , but the lower .
Result 6: Assume A2-A6, and , then
Furthermore, we can combine with Result 3 to improve bound.
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Estimation
• The above bounds can be estimated by replacing population moments with their
sample counterparts. In most cases these are easy to compute (OLS or IV)
estimators;
• Standard errors can be computed using Imbens and Manski (2004);
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Extension
• The above analysis extends directly to multiple regression with a single endogenous
variables, by netting out the other variables and defining thing accordingly (Note:
the interpretation of the error and the conditions are different after netting out the
effects.)
• In principle, the idea of replacing equalities with inequalities is not limited to our
model (just find the set of parameters that satisfy the inequalities). However, we still
have not provided conditions that characterize these sets and will help in search for
IVs.
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Application: Estimating Demand for Differentiated Products
Logit Model
• The indirect utility for individual i from UPC j in market t
where: xjt observable characteristics; pjt is price, >jt unobserved product
characteristic, and gijt is a mean zero stochastic term.
• Individuals can also choose the “outside option”:
• Each individual chooses exactly one good;
• Assuming is distributed iid extreme value, then
and
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The error term and instruments
• The error-term is the unobserved characteristic and depending on the application it
will include either unobserved quality or promotional activities;
• In most cases it is not hard to come up with models of supplier behavior that lead to
correlation between prices and these error terms;
• Below we think of the error as unobserved promotional activities and variation in
valuation of unobserved quality that are likely to be correlated with price reductions.
• In this setting common IV are prices (of the product) in other markets
correlated with prices due to common costs shocks;
uncorrelated with the error term if demand errors are independent across mkts;
• The independence could be violated, for example, if
promotional activity is correlated across markets;
national advertising;
• Common response: argue theoretically or compare to other evidence;
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Data
C Scanner data for cereal at the brand-quarter-MSA level;
20 quarters: 1988-1992;
focus on 25 top cereal brands;
47-65 markets, focus on SF and Boston;
C Key variables:
market shares (quantity) – volume converted to servings; one serving per day;
prices – revenue/quantity : pre-coupon real transaction per serving price;
quantity sold on promotion – estimated fraction sold on weeks with promotion;
advertising – national quarterly brand level from LNA;
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PRICES AND MARKET SHARES OF BRANDS IN SAMPLE
Description Mean Median Std Min Max BrandVariation
CityVariation
QuarterVariation
Prices (¢ per serving)
19.4 18.9 4.8 7.6 40.9 88.4% 5.3% 1.6%
Advertising(M$ perquarter)
3.56 3.04 2.03 0 9.95 66.2% -- 1.8%
Share withinCereal Market(%)
2.2 1.6 1.6 0.1 11.6 82.3% 0.5% 0%
Source: IRI Infoscan Data Base, University of Connecticut, Food Marketing Center.
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Results
assume: (not needed for all the results);
characteristics include: advertising, brand and city dummies;
error: unobserved promotions and differences (over time and cities) in perceived quality;
Case 1: single IV
• consider Z = quantity sold on promotion – clearly not a valid IV: ;
• we find: , (0.71),
Y from Result 1: ;
• Y from Result 3: ;
Note: (1) For most of what we care about the effect is multiplicative, thus the gain
from the second bound is meaningful;
(2) It is not clear that A6 should hold;
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Case 2: multiple IV – prices in other markets
• prices in other markets have been used in these setting (e.g, HLZ, 1994; Hausman,
1996, Nevo, 2001) but have been criticized (e.g., Bresnahan, 1996);
• Let Z1 = average price of brand in other markets in the region (NE for Boston and
Northern California for SF) ;
• Let Z2 = average price of brand in markets in the other region (NE for SF and
Northern California for Boston) ;
• In the data: , , and ;
so there exists a ( s.t and ;
• Assume that these inequalities hold for ( );
Y we’ll use , , and to compute our bounds
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We find:
point estimates
- 2.21 (0.71) ;
- 4.08 (0.87) ;
Bounds:
using quantity sold on promotion: [-10.60, -3.94];
using prices in other markets:
-11.47 Y [-10.97, -4.08]
-4.85 Y [-10.97, -4.85]
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Conclusions and Extensions
• With a single IV we derived informative bounds for .
• Suggests a strategy for searching for imperfect IV;
• With several IV we were able to derive informative bounds also for the case where
the IV are positively correlated with the endogenous variable;
• In the application we studied the proposed bounds were reasonably tight;
• Where are we going next?
multiple endogenous variables;
improve bounds;
more detailed application(s) – maybe better fit for cases with ;