Identification
Lukˊaˇs Laffˊers
Matej Bel University, Dept. of Mathematics
MUNI Brno
1.10.2021
What can be learnt from the data?
DATA + MODEL → CONCLUSIONS
Why should we study identification?
Conceptual framework of what could we potentially learn from the data and
model.
What are the crucial components of this.
What if some of the model assumptions are incorrect.
Identification
Is a different and separate topic from the statistical inference.
Primarily based on:
Lewbel, Arthur. ”The identification zoo: Meanings of identification in
econometrics.” Journal of Economic Literature 57.4 (2019): 835-903.
History of Identification
Working (1925, 1927): ”By intelligently applying proper refinements, and
making corrections to eliminate separately those factors which cause
demand curves to shift and those factors which cause supply curves to shift,
it may be possible even to obtain both a demand curve and a supply curve
for the same product and from the same original data.”
History of Identification 2
Frisch (1934, 1938) - confluency in linear regression
Hurwicz (1950) - introduced the term ”structure”
Koopmans and Reiersol (1950): “Scientific honesty demands that the
specification of a model be based on prior knowledge of the phenomenon
studied and possibly on criteria of simplicity, but not on the desire for
identifiability of characteristics that the researcher happens to be interested
in”
Phillips (1989): ”it seems important that we should understand the
implications of identification failure for statistical inference. Yet, this is a
subject that seems to be virtually untouched in the literature”
Reviews: Dufour and Hsiao (2008), Tamer (2010)
Notation
m - a model
φ - what can be known from data
θ - a parameter
s - a structure
Model m
set of functions or constants (regression function, utility functions,
coefficient vectors)
that satisfy given restrictions (linear/monotone regression function,
normal errrors, parameters bounded)
a particular model value m
a set M of model values
any m implies a particular DGP (data generating process)
Data φ
Set of constants and/or functions about the DGP that are assumed to
be known or knowable from data
Examples: data distribution functions, conditional mean functions,
linear regression coefficients, or time series autocovariances
Parameter θ
Set of constants and/or functions that summarize relevant features of a
model.
The thing we wish to estimate.
May include nuisance parameters that are not of direct interest, but
may be necessary for identification/estimation of other objects
Structure s
m implies a particular value of φ and of θ
BUT, there may be multiple ms that imply the same φ and θ
Let Structure s(φ, θ) be the collection of all m that imply φ and θ
Two parameter values θ and ˜θ are said to
be observationally equivalent if there exists
φ such that s(φ,θ) and s(φ, ˜θ) are both not
empty.
(in other words: both θ and ˜θ could be true,
based on observed φ)
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Types of identification
Point identification of θ
Global identification
Point identification of m
Local identification
Partial identification
Parametric/Semi-/Non- identification
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Point identification (of a parameter θ)
There do not exist any pairs θ and ˜θ that
are different and observationally equivalent.
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Global identification (of a parameter θ)
Let θ ∈ Θ and let θ0 be the true value
θ0 is point identified if there isn’t any θ
∈ Θ0 that is observationally equivalent
to θ0
But we don’t know what θ0 is.
So, if we require that no two elements
of Θ are obs. equivalent.
Then θ0 is identified no matter what it
happens to be. (hence the word global)
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Point identification (of a model m)
There do not exist any pairs m and ˜m that
are different and observationally equivalent
(now treating the whole models m and ˜m as
parameters).
Stronger than a point identification of θ.
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Local identification (of a parameter θ)
There exists a neigborhood of θ0 so that for
all values of θ = θ0 in this neighborhood, θ
is not observationally equivalent to θ0
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Partial (set) identification (of a parameter θ)
There exist some parameter values θ that
are not observationally equivalent to θ0 (so
that not all θ ∈ Θ are obs. equivalent).
The collection of all θ that are obs.
equivalent to θ0 is called an identified set.
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Partial Identification
Manski (2003): “...it has been commonplace to think of identification as a binary
event – a parameter is either identified or not – and to view point identification
as a pre-condition for inference. Yet there is enormous scope for fruitful inference
using data and assumptions that partially identify population parameters.“
Reviews: Manski (2003), Tamer(2010)
Semi-/Non- parametric identification (of a parameter θ)
Parametric - φ and θ are finite
Non-parametric - θ includes functions
or infinite sets
Semi-parametric - θ includes both
vector of constants and functions
(Not always easy to distinguish
between them)
Notation
m - a model
φ - what can be known
from data
θ - parameter
s - structure
Nonparametric Models
Pros
more credible assumptions
more flexible
economic restrictions
Cons
curse of dimensionality
more difficult to implement
sometimes harder to interpret
Why Non-parametric?
(DiNardo and Tobias 2001)
Parametric model:
Why Non-parametric?
(DiNardo and Tobias 2001)
Non-parametric model:
Example 1 - Median
M - set of all possible distributions of rv W with
strictly increasing distribution function.
φ is the distribution function of W, F(w)
θ is the median of W
Structure s(φ,θ) contains a single element if
F(θ) = 1/2 where φ = F or is empty if
F(θ) = 1/2.
F(θ) = 1/2 and F(˜θ) = 1/2 implies θ = ˜θ and
hence θ is point identified
Notation
m - a model
φ - what can be
known from
data
θ - parameter
s - structure
Example 2a - Linear regression
M - Set of joint distributions of (ε,X) that satisfy
y = Xθ +ε
E(XT
ε) = 0
E(XT
X) is non singular
both ε and X have finite first and second moments
φ is the set of first and second moments of X and y
θ is the vector of parameters
s(φ,θ) is non-empty when E[XT
(y −Xθ)] = 0 is satisfied.
θ is uniquely determined by θ = E(XT
X)−1
E(XT
y) and hence it is point
identified
parametric or semi-parametric?
Example 2b - Linear regression
M - Set of joint distributions of (ε,X) that satisfy
y = Xθ +ε
E(XT
ε) = 0
E(XT
X) is non singular
both ε and X have finite first and second moments
φ is the joint distribution of (X,y)
θ is the vector of parameters and the distribution function of ε
s(φ,θ) is non-empty when E[XT
(y −Xθ)] = 0 is satisfied.
θ is uniquely determined by θ = E(XT
X)−1
E(XT
y) and hence it is point
identified.
parametric or semi-parametric?
Example 3a- Treatment effects
M - all possible joint distributions of (Y(1),Y(0),T)
(Y(1),Y(0)) ⊥ T
if T = t then Y = Y(t)
φ is the joint distribution of (Y,T) (alternatively E[Y|T = 1] and
E[Y|T = 0])
θ is the average treatment effect θ = E[Y(1)−Y(0)]
s(φ,θ) is non-empty whenever θ = E[Y|T = 1]−E[Y|T = 0] is satisfied.
Under (Y(1),Y(0)) ⊥ T we have that θ = E[Y|T = 1]−E[Y|T = 0] and
hence it is point identified.
Does there exist an unique value of θ for every possible φ?
Example 3b- Treatment effects
M - all possible joint distributions of (Y(1),Y(0),T)
(Y(1),Y(0)) ⊥ T
if T = t then Y = Y(t)
φ consists of E[Y|T = 1] and E[Y|T = 0]
θ is the average treatment effect θ = E[Y(1)−Y(0)]
s(φ,θ) is non-empty whenever θ = E[Y|T = 1]−E[Y|T = 0] is satisfied.
Under (Y(1),Y(0)) ⊥ T we have that θ = E[Y|T = 1]−E[Y|T = 0] and
hence it is point identified.
Example 3c- Treatment effects
M - all possible joint distributions of (Y(1),Y(0),T)
E[Y(t)|T] = E[Y(t)] (mean unconfoundedness)
if T = t then Y = Y(t)
φ consists of E[Y|T = 1] and E[Y|T = 0]
θ is the average treatment effect θ = E[Y(1)−Y(0)]
s(φ,θ) is non-empty whenever θ = E[Y|T = 1]−E[Y|T = 0] is satisfied.
Under (Y(1),Y(0)) ⊥ T we have that θ = E[Y|T = 1]−E[Y|T = 0] and
hence it is point identified.
Example 3d- Treatment effects
M - all possible joint distributions of (Y(1),Y(0),T)
ymin ≤ Y(t) ≤ ymax (Y(1),Y(0)) ⊥ T no randomization here!
if T = t then Y = Y(t)
φ consists of E[Y|T = 1], E[Y|T = 0] and Pr(T = 1)
θ is the average treatment effect θ = E[Y(1)−Y(0)]
E[Y(1)] = E[Y(1)|T = 1]Pr(T = 1)+E[Y(1)|T = 0]
unobserved
Pr(T = 0)
ymin ≤ E[Y(1)|T = 0] ≤ ymax
θ ∈ [θL,θH] and θ is partially identified
Bounds on Average Treatment Effect
E[Y(t)] = E[Y|T = t]·P(T = t)
Observed
+E[Y(t)|T = t]
Unobserved
·P(Z = t)
Observed
Observed quantities
Unobserved quantities
Assumption of Bounded support
Suppose that ymin ≤ Yi(t) ≤ ymax
LBE[Y(t)] = E[Y|T = t]·P(T = t)+ymin ·P(T = t)
≤
E[Y(t)] = E[Y|T = t]·P(Z = t)+E[Y(t)|T = t]·P(T = t)
≤
UBE[Y(t)] = E[Y|z = t]·P(T = t)+ymax ·P(T = t)
E[Y(t)] (and hence also ATE) is partially identified and the interval
(LBE[Y(t)],UBE[Y(t)]) is called an identified set.
Example 4 - Supply and Demand
Demand: Q = b ·P +c ·Z +U,
Supply: Q = a ·P +ε
M - all possible joint distributions of (I,U,ε) and coeffs (a,b,c)
E(U) = E(ε) = 0 and (U,ε) ⊥ Z
φ is coeffs (φ1,φ2) from Q = φ1Z +V1 and P = φ1Z +V2, where
E(V1) = E(V2) and (V1,V2) ⊥ Z
θ = a - coeff of price in supply eqn.
For any m in s(φ,θ), we need to have θ = a, φ1 = ac
a−b
, φ2 = c
a−b
if c = 0 we get a = φ1
φ2
and s(φ,θ) contains many elements
if c = 0 then any θ and ˜θ will be obs. equivalent with φ = (0,0)
In other words: we need the instrument Z to appear in the demand eqn.
Point identification?
So far ”by construction”:
Ex 1: θ = F−1
(0.5)
Ex 2: θ = E(XT
X)−1
E(XT
y)
Ex 3: θ = E[Y|T = 1]−E[Y|T = 0]
Ex 4: θ = φ1/φ2
Other strategies?
True θ0 is an unique maximizer of a optimization problem defined by the
model. (e.g. Likelihood function is globally concave)
Identification logically precedes estimation.
What is ”knowable” φ?
Distribution based on IID data: Glivenko-Cantelli theorem
Expected values: Law of Large Numbers
In many cases, it is assumed that the parameter is identified (GMM).
Example:
Preferences θ may be identified from demand functions φ.
But how do we identify these demand functions?
φ is the starting point. We assume this is knowable from the data.
Reasons for not (point) identification
model is incomplete
perfect collinearity
non-linearity
simultaneity
endogeneity
unobservability
Some remarks
We keep asking this: ”Does there exist an unique value of θ for every
possible φ ?”
There are different ways how to achieve identification.
Stronger assumptions are more difficult to defend but easier to work
with.
Weaker assumptions may not be sufficient to guarantee identification.
Some assumptions are difficult to interpret (mean unconfoundedness is
sensitive to transformation of Y)
What if the identification fails?
If we treat unidentified model as if it was identified:
Parameters, Tests and Confidence sets have no clear interpretation
Consistent estimation is not possible
Statistical inference methods are not valid
Numerical problems (inverting singular matrices)
”Harmful econometrics” vs. ”Cuteonomics”
Structural
model of economic behaviour is
built up based on economic
theory
focus on deep parameters
allows to answer rich set of
questions
Reduced form
as few assumptions as possible
focus on reduced form
parameters (e.g. ATE, ATT,
MTE, LATE, QTE)
attempts to do or mimic RCT
prefers simplicity and
transparency
Lewbel’s JEL Zoo paper (section 5.1) suggests to use both and gives many
examples.
Example: Y = a +bT +e
Structural model
variables U1,U0,V1,V0
individual effect U1
y = U0 +U1T and T = V0 +V1T
E(V1) = 0
(U1,U0,V1,V0) ⊥ Z
cov(e,Z) = 0 (this implies
cov(U1,V1) = 0)
=⇒
E[Y(1)−Y(0)] = E(U1) = b
Reduced form model
variables Y(1),Y(0),T(1),T(0)
individual effect Y(1)−Y(0)
Y(t,z) satisfies Y(t,0) = Y(t,1)
E(T(1)−T(0)) = 0
(Y(1),Y(0),T(1),T(0)) ⊥ Z
P(T(1) = 0,T(0) = 1) = 0
=⇒
E[Y(1)−Y(0)|T(1) = 1,T(0) = 0] = cov(Z,Y)
cov(Z,T)
Structural model
identifies ATE
cov(U1,V1) = 0 is a restriction
on the heterogeneity of the
treatment effect U1
stronger assumptions about the
outcome Y
can we justify cov(e,Z) = 0?
Reduced form model
identifies LATE
No defiers condition is a
restriction on the heterogeneity
of types, not about outcomes
stronger assumptions about the
treatment T
who are the compliers?
what do we know about the
rest?
how about non-binary
treatments?
Examples of restrictions from economic theory
shape restrictions: concavity, continuity or monotonicity of functions
(utility function, demand function, production function)
implications of optimization (first order conditions)
equilibrium conditions
exclusion restrictions (an instrument does not appear in the equation of
interest)
long-run restrictions on covariance matrix of errors in VAR models
(money-supply shock has no long-run effect on output)
Example 5 - Cost function and Revenue distribution
Matzkin (1994)
A firm operating in a perfectly competitive market decides whether to
invest in a development of a new product.
We wish to know
cost function of a typical firm
distribution of the revenues
We observe input prices (x1
,x2
,...,xN
) for the N firms and whether they
invested (yi
= 1) or not (yi
= 0).
We take revenue (ε ≥ 0) as a random variable.
Example 5 - Model Restrictions
Properties of the production function:
monotonous
convex
homogeneous of degree one in prices
Further assumptions
revenue is independent of input prices
the distribution of revenue ε, F is strictly increasing.
the value of the cost function h is known for a particular vector of input
prices. h(x∗) = α
M - Set of joint distributions of (x,y,ε), cost function h and production
function that they jointly satisfy all the assumptions.
φ is the probability of not investing given prices x: P(y = 0|x)
θ is the cost function h and the distribution of revenues F
Question of Identification
Will the assumptions enable us to recover the cost function (h) and the
distribution of revenues (F)?
It turns out that yes
g(x) ≡ P(y = 0|x) = Pr(ε ≤ h(x)) = F(h(x))
F(t)
norm
= F((t/α)h(x∗
))
h.o.d.1
= F(h((t/α)x∗
)) = g((t/α)x∗
)
h(x)
mono
= F−1
g(x)
=⇒ (h,F) is identified.
parametric model: h(x) = x β, F ∼ lnN(µ,σ2
), θ = (β,µ,σ2
)
semi-parametric model: h(x) = x β, θ = (β,F)
nonparametric model: no parametric restrictions on both (h,F),
θ = (h,F)
Application: Gandhi, Navarro and Rivers (2013)
Example 6: Demand Function under Slutsky Condition
Blundell, Horowitz and Parey (2012)
Heterogenous demand function for gasoline in the U.S.
Additive separability only under very restrictive assumptions about
preferences
Nonparametric estimate is noisy (DWL < 0)
Identification:
Q - Demand, P - Price, Y - Income, U - Unob. Heterogeneity
Q = g(P,Y,U) increasing in U
U is independent of (P,Y)
Slutsky restriction: ∂g(P,Y,α)
∂P
+g(P,Y,α)g(P,Y,α)
∂Y
≤ 0
Results:
Middle income group shows
strongest price responsiveness
highest DWL
Slutsky restriction
∂g(P,Y,α)
∂P
total effect
+g(P,Y,α)
g(P,Y,α)
∂Y
−income effect
substitution effect
≤ 0
Slutsky matrix is negative semi-definite.
Simpler: In one dimension: own price elasticity is negative.
Even simpler: cost minimizing consumer will buy less of a certain good if it
gets more expensive.
Thank you for your attention!
References
Lewbel, Arthur. ”The identification zoo: Meanings of identification in econometrics.” Journal of Economic Literature 57.4 (2019): 835-903.
Working, Holbrook. ”The statistical determination of demand curves.” The Quarterly Journal of Economics 39.4 (1925): 503-543.
Working, Elmer J. ”What do statistical “demand curves” show?.” The Quarterly Journal of Economics 41.2 (1927): 212-235.
Frisch, Ragnar. ”Circulation planning: proposal for a national organization of a commodity and service exchange.” Econometrica, Journal of the Econometric Society
(1934): 258-336.
Frisch, Ragnar. ”The double-expenditure method.” Econometrica: Journal of the Econometric Society (1938): 85-90.
Koopmans, Tjalling C., and Olav Reiersol. ”The identification of structural characteristics.” The Annals of Mathematical Statistics 21.2 (1950): 165-181.
Phillips, Peter CB. ”Partially identified econometric models.” Econometric Theory 5.2 (1989): 181-240.
Hsiao, Cheng and Jean-Marie Dufour. ”The New Palgrave Dictionary of Economics Online.” (2008).
Tamer, Elie. ”Partial identification in econometrics.” Annu. Rev. Econ. 2.1 (2010): 167-195.
Manski, Charles F. Partial identification of probability distributions. Springer Science & Business Media, 2003.
DiNardo, John, and Justin L. Tobias. ”Nonparametric density and regression estimation.” Journal of Economic Perspectives 15.4 (2001): 11-28.
Matzkin, Rosa L. ”Restrictions of economic theory in nonparametric methods.” Handbook of econometrics 4 (1994): 2523-2558.
Gandhi, Amit, Salvador Navarro, and David A. Rivers. ”On the identification of gross output production functions.” Journal of Political Economy 128.8 (2020):
2973-3016.
Blundell, Richard, Joel L. Horowitz, and Matthias Parey. ”Measuring the price responsiveness of gasoline demand: Economic shape restrictions and nonparametric
demand estimation.” Quantitative Economics 3.1 (2012): 29-51.