Expert Answer:ECON803 Econometrics, The Counterfactual Model

  

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Policy assignment
A review of “the review of the sentence of home detention
2007-2011”
Due date: Friday, March 22th, 12’noon
Hand-in: WY215 or email: pskov@aut.ac.nz
The purpose of this assessment is to write a guided review of the Ministry of Justice report
“Home Detention – A review of the sentence of home detention 2007-2011”. The report
has a total word limit of 1500 words and needs to be completed individually.
In 2007 the New Zealand Government introduced home detention as a stand-alone
sentence. The policy change meant that eligible offenders sentenced to short-term
imprisonment could serve their sentence at an approved residence under electronic
monitoring. As part of the 2008 New Zealand general election debate the National Party
committed to an evaluation of the “appropriateness” of home detention sentence. In
October 2011 the Ministry of Justice (MoJ) released the report “Home Detention – A review
of the sentence of home detention 2007-2011”.
The report can be found here:
https://thehub.sia.govt.nz/assets/documents/41113_A_review_of_the_sentence_of_home
_detention_2007-2011_0.pdf
1. (~350 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤). Write a short summary of the MoJ report. Please make sure you
include information on:
a. Research question(s)
b. Data
c. Method
d. Conclusion
2. (~150 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤). Re-state the (main) MoJ research question (from Q1) using the
notation from the lecture on the counterfactual framework.
3. (~500 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤). Discuss the empirical strategy in the MoJ report. Make sure to use
the notation from the lecture on the counterfactual framework. You are encouraged
to derive and sign any possible bias in the MoJ estimate(s) and discuss what
assumptions could justify the MoJ approach.
4. (~500 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤). In an imaginary world without any restrictions what would, in your
view, be the ideal experiment to identify the causal effect of home detention?
Note that (~ 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤) is a suggestion, feel free to distribute the 1500 words differently if
that works better for your answers.
ECONOMETRICS
Econ803
The Counterfactual Model
Peer Ebbesen Skov
Motivation
Consider the following question:
• What is the effect of class size on the educational
achievements?
• What is the effect of an increase in the minimum wage
on employment?
• What is the effect of terrorism on economic growth?
• What is the effect of increased cigarette excise tax on
smoking?
• What is the effect of marriage on crime?
The answers to these questions (and many others which
affect our daily life) involve the identification and
measurement of causal links: an old problem in
philosophy and statistics.
We need a framework to study causality
2
A formal framework to think about causality
We have a population of units; for each unit we observe a
variable 𝐷 and a variable 𝑌.
We observe that 𝐷 and 𝑌 are correlated. Does correlation
imply causation?
Reference: David Card. Chapter 30 in Handbook of Labor Economics,
1999, vol. 3, Part A, pp 1801-1863. Elsevier.
3
A formal framework to think about causality
Does correlation imply causation?
In general no, because of:


Confounding factors;
Reverse causality.
We would like to understand in which sense and under
which hypothesis one can conclude from the evidence
that 𝐷 causes 𝑌.
4
Terminology
𝑖 index for the units in the population under study
𝐷𝑖 is the (binary) treatment status where
𝐷𝑖 = ቊ
1
𝑢𝑛𝑖𝑡 𝑖 ℎ𝑎𝑠 𝑏𝑒𝑒𝑛 𝑒𝑥𝑝𝑜𝑠𝑒𝑑 𝑡𝑜 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
0 𝑢𝑛𝑖𝑡 𝑖 ℎ𝑎𝑠 𝑛𝑜𝑡 𝑏𝑒𝑒𝑛 𝑒𝑥𝑝𝑜𝑠𝑒𝑑 𝑡𝑜 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
𝑌𝑖 𝐷𝑖 indicates the potential outcome according to
treatment where
𝑌𝑖 𝐷𝑖 = ቊ
𝑌𝑖 1
𝑌𝑖 0
𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑖𝑛 𝑐𝑎𝑠𝑒 𝑜𝑓 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑖𝑛 𝑐𝑎𝑠𝑒 𝑜𝑓 𝑛𝑜 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
The observed outcome for each unit can be written as:
𝑌𝑖 = 𝐷𝑖 𝑌𝑖 1 + 1 − 𝐷𝑖 𝑌𝑖 0
This forces us to think in terms of counterfactuals or socalled potential outcomes.
Note that the terminology is borrowed from experimental
analysis.
5
The fundamental problem of causal inference
DEFINITION: Causal effect
For a unit 𝑖 the treatment 𝐷𝑖 has a causal effect on the
outcome 𝑌𝑖 if the event 𝐷𝑖 = 1 instead of 𝐷𝑖 = 0 implies
that 𝑌𝑖 = 𝑌𝑖 1 instead of 𝑌𝑖 = 𝑌𝑖 0 . In this case the
causal effect of 𝐷𝑖 on 𝑌𝑖 is
∆𝑖 = 𝑌𝑖 1 − 𝑌𝑖 0
The identification and the measurement of this effect is
logically impossible.
PROPOSITION 1: The Fundamental Problem of Causal Inference
It is impossible to observe for the same unit 𝑖 the values
𝐷𝑖 = 1 and 𝐷𝑖 = 0 as well as the values 𝑌𝑖 1 and 𝑌𝑖 0
and therefore it is impossible to observe the effect of 𝐷 on
𝑌 for unit 𝑖 (Holland, 1986)
Another way to phrase this problem is to say that we
cannot infer the effect of a treatment because we do not
have the counterfactual evidence, i.e. what would have
happened in the absence of treatment.
6
The fundamental problem of causal inference
𝑌 1
𝑌 0
Treatment status (𝐷 = 1)
Treatment status (𝐷 = 0)
It is not possible to observe the potential outcome under
the treatment state for those observed in the control
state. Just like you cannot observe the potential outcome
under the control state for those observed in the
treatment state. (Morgan and Winship, 2007).
Yet another way of thinking about it: even if you have
access to all individuals level values of 𝑦𝑖 in the
population you only observe half of the information you
need: individuals contribute information only from the
treatment state in which they are observed. (Morgan and
Winship, 2007).
7
Unit homogeneity solution
One “solution” to the missing counterfactual problem is
to assume unit homogeneity:
∆𝑖 = ∆ for all 𝑖
If 𝑌𝑖 1 and 𝑌𝑖 0 are constant across individual
units, then cross-sectional comparisons will
recover ∆ = ∆𝑖
If Yi 1 and Yi 0 are constant across time, then
before-and-after comparisons will recover
∆ = ∆i
While this might work in physical sciences the assumption
seems highly unlikely to be realistic in social sciences
Before we get to the statistical ‘solution’ let’s digress for a
moment and consider another challenge…
Note the following notation:
𝑌𝐷𝑖 ≡ Yi 𝐷
8
Stable Unit Treatment Value Assumption (SUTVA)
Recall the observed outcome for each unit can be written
as:
𝑌𝑖 = 𝐷𝑖 𝑌𝑖 1 + 1 − 𝐷𝑖 𝑌𝑖 0
This notation implicitly makes the following assumption:
SUTVA:
𝑌 𝐷1, 𝐷2 , … , 𝐷𝑁 𝑖 = 𝑌

𝐷1′ ,𝐷2′ , …,𝐷𝑁
𝑖
𝑖𝑓 𝐷𝑖 = 𝐷𝑖′
In other words:
▪ There is no interference between units:


Potential outcomes for a unit most not be affected
by treatment for any other units.
Spill-over effects, contagion, dilution
No different versions of treatment

Nominally identical treatments are in fact identical
Variable levels of treatment, technical errors
9
Causal Inference without SUTVA
Let 𝐷 = 𝐷1 , 𝐷2 be a vector of binary treatments for N =
2
How many different values can D possible take?
How many potential outcomes for unit 1?
How many causal effects for unit 1?
How many observed outcomes for unit 1?
𝑌1 = 𝑌 𝐷1 ,𝐷2 1
Without SUTVA, causal inference becomes exponentially
more difficult as N increases
10
The statistical solution
Statistics proposes to approach the problem by focusing
on the average causal effect for the entire population or
for some interesting sub-groups.
The effect of treatment on a random unit (ATE):
𝐸 ∆𝑖 = 𝐸 𝑌𝑖 1 − 𝑌𝑖 0
= 𝐸 𝑌𝑖 1
− 𝐸 𝑌𝑖 0
Or equivalently
𝑁
1
𝐸 ∆𝑖 = ෍ 𝑌1𝑖 − 𝑌𝑜𝑖
𝑁
𝑖=1
Note that ATE 𝐸 ∆𝑖 is still unidentified. The majority of
this paper is devoted to various assumptions under which
we can identify ATE from observed information.
11
The statistical solution
Statistics proposes to approach the problem by focusing
on the average causal effect for the entire population or
for some interesting sub-groups.
The effect of the treatment on the treated (ATT):
𝐸 ∆𝑖 |𝐷𝑖 = 1 = 𝐸 𝑌𝑖 1 − 𝑌𝑖 0 |𝐷𝑖 = 1
= 𝐸 𝑌𝑖 1 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 0 |𝐷𝑖 = 1
Or equivalently
𝑁1
𝑁
𝑖=1
1
1
𝐸 ∆𝑖 =
෍ 𝑌1𝑖 − 𝑌𝑜𝑖 where 𝑁1 = ෍ 𝐷𝑖
𝑁1
When would 𝐴𝑇𝐸 ≠ 𝐴𝑇𝑇? When 𝐷𝑖 and 𝑌𝑑𝑖 are
associated.
Exercise: define the treatment on control units (ATC).
12
The statistical solution
Statistics proposes to approach the problem by focusing
on the average causal effect for the entire population or
for some interesting sub-groups.
The conditional average treatment effect (CATE)
𝐸 ∆𝑖 |𝑋𝑖 = 𝑥 = 𝐸 𝑌𝑖 1 − 𝑌𝑖 0 |𝑋𝑖 = 𝑥
13
Illustration: Average Treatment Effect
Suppose we observe a population of 4 units:
i
𝑫𝒊
𝒀𝒊
1
1
3
2
1
1
3
0
0
4
0
1
What is ATE: 𝐸 ∆𝑖 = 𝐸 𝑌𝑖 1
𝒀𝟏𝒊
𝒀𝟎𝒊
− 𝐸 𝑌𝑖 0
∆𝑖
?
Naïve estimator:
ATE: 𝐸 ∆𝑖 = 𝐸 𝑌𝑖 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 |𝐷𝑖 = 0
(Note this is the observed difference in means)
𝐸 ∆𝑖 =
Is this (likely) an unbiased estimate of ATE?
Let’s expand the table
14
Illustration: Average Treatment Effect
Suppose we observe a population of 4 units:
i
𝑫𝒊
𝒀𝒊
1
1
3
2
1
1
3
0
0
4
0
1
What is ATE: 𝐸 ∆𝑖 = 𝐸 𝑌𝑖 1
𝒀𝟏𝒊
𝒀𝟎𝒊
− 𝐸 𝑌𝑖 0
∆𝑖
?
Naïve estimator is likely biased i.e. we over/under
estimate the average treatment effect.
To obtain an unbiased estimate of the ATE we need
potential outcomes that we do no observe:
But suppose we did – let’s complete the expanded table
15
Illustration: Average Treatment Effect
Suppose we observe a population of 4 units:
i
𝑫𝒊
𝒀𝒊
1
1
3
2
1
1
3
0
0
4
0
1
What is ATE: 𝐸 ∆𝑖 = 𝐸 𝑌𝑖 1
𝒀𝟏𝒊
− 𝐸 𝑌𝑖 0
𝐸 𝑌𝑖 1
=
𝐸 𝑌𝑖 0
=
− 𝐸 𝑌𝑖 0
=
𝐸 ∆𝑖 = 𝐸 𝑌𝑖 1 − 𝑌𝑖 0
=
𝐸 ∆𝑖 = 𝐸 𝑌𝑖 1
𝒀𝟎𝒊
∆𝑖
?
or
16
Illustration: Average Treatment Effect
Suppose we observe a population of 4 units:
i
𝑫𝒊
𝒀𝒊
1
1
3
2
1
1
3
0
0
4
0
1
𝒀𝟏𝒊
𝒀𝟎𝒊
∆𝑖
What is ATT: 𝐸 ∆𝑖 |𝐷𝑖 = 1 = 𝐸 𝑌𝑖 1 − 𝑌𝑖 0 |𝐷𝑖 = 1
𝐸 ∆𝑖 |𝐷𝑖 = 1 =
Why is the 𝑛𝑎𝑖𝑣𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑟 = 𝐴𝑇𝑇 ≠ 𝐴𝑇𝐸. Will this
always be the case?
17
Is comparison by treatment status informative?
A comparison of outcome between treatment status (the
naïve estimator) often gives a biased estimate of the ATT:
𝐴𝑇𝐸 = 𝐸 𝑌𝑖 1 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 0 |𝐷𝑖 = 1
𝑁𝑎𝑖𝑣𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟 = 𝐸 𝑌𝑖 1 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 0 |𝐷𝑖 = 0
= 𝐸 𝑌𝑖 1 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 0 |𝐷𝑖 = 1
+ 𝐸 𝑌𝑖 0 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 0 |𝐷𝑖 = 0
= 𝐴𝑇𝑇 + 𝐸 𝑌𝑖 0 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 0 |𝐷𝑖 = 0
Note 𝐴𝑇𝑇 = 𝐸 ∆𝑖 |𝐷𝑖 = 1 and the second term
𝐸 𝑌𝑖 0 |𝐷𝑖 = 1 − 𝐸 𝑌𝑖 0 |𝐷𝑖 = 0 is often referred to
as sample selection bias.
The difference between the left hand side (which we can
estimate) and 𝐴𝑇𝑇 is the sample selection bias equal to
the difference between the outcomes treated and control
subjects in the counterfactual situation of no treatment
(i.e. at the baseline).
The problem is that the outcome of the treated and the
outcome of the control are not identical in the notreatment situation.
18
Is comparison by treatment status informative?


1.
Causal inference requires a good identification
strategy
The treatment assignment mechanism determines
whether average causal effects are identifiable
Treatment is randomized by the researcher:
1.
2.
3.
2.
Natural experiments
1.
2.
3.
4.
3.
Birthday cut-offs
Weather
Close elections
Arbitrary administrative rules/policy
Treatment is “as-if” random after statistical control
1.
2.
4.
Laboratory experiments
Survey experiments
Field experiments
Marriage (controlling for age ,education and
income)
Earnings (controlling for age, education and
experience)
Treatment is self-selected and no plausible control is
available.
19
ECONOMETRICS
Econ803
The Naïve Estimator
Peer Ebbesen Skov
Livvy Mitchell
Naïve Estimator
Naïve estimator for ATT
Compares outcomes of participants (D=1) and nonparticipants (D=0) as follows:
𝑁𝑎𝑖𝑣𝑒 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟 = 𝐸 𝑌 𝐷 = 1 − 𝐸 𝑌 𝐷 = 0
It is unbiased under the assumption of no selection bias
(on observed and/or unobserved characteristics)
whereby:
𝐸 𝑌0 𝐷 = 1 = 𝐸 𝑌0 𝐷 = 0
Generally we don’t believe that to be the case.
2
Naïve Estimator
Naïve estimator for ATT
The 𝑁𝑎𝑖𝑣𝑒 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑜𝑟 is unbiased under the assumption
of no selection bias on observed and/or unobserved
characteristics:
We generally distinguish between two source of bias

Differences in observed characteristics

Non-overlap (B1)

Different distribution of observables (wrong
weighting scheme) (B2)

Selection on unobserved characteristics

Omitted variable bias, e.g., ‘ability bias’ (B3)
3
Naïve Estimator
Assessing comparability of groups in terms of
observables
𝐹 𝑋 𝑡𝑟𝑒𝑎𝑡𝑒𝑑 =? = 𝐹 𝑋 𝑛𝑜𝑛𝑡𝑟𝑒𝑎𝑡𝑒𝑑
But, how can you compare the joint empirical distribution
of all the X’s between two samples?
Instead..
(a) Variable-by-variable measures


Moments: means, variances
Empirical distributions: densities, CDF, boxplots
(b) Overall measures across all X’s


Or, across some X’s have interaction terms
pstest allows factor variables (e.g., foreign##c.age)
4
Naïve Estimator
Propensity-Score matching (“pstest”)
Use pstest to easily compare the characteristics in two
groups:
pstest var_list , raw treated (treated) scatter
Use it also to quickly graph the non-parametric density or
boxplot of a continuous variable (“var”) for two groups:
pstest var [if] , raw treated (treated) density|box
5
Naïve Estimator
Variable-by-variable: pstest output
6
Naïve Estimator
Summaries: pstest output
7
Naïve Estimator
Continuous variables: pstest output
8
Naïve Estimator
Overall indicators: pstest output
9
Naïve Estimator
How do we get a credible counterfactual?

If no convincing comparison group exists, fancy
statistical work can’t recover the true impact.

Robustness checks
Different methods differ in:


How they construct the counterfactual

Assumptions they make

Data they require
At times: what parameter (ATE, ATT,…) they recover.
Naïve comparisons of e.g., participants and
nonparticipants or simple before-after differences will not
provide the correct counterfactual.
10
Naïve Estimator
Validity concepts
Internal validity is concerned with the validity of the estimates.
• Does the study successfully uncover causal effects for the
sample studied?
• Are the estimates unbiased?
External validity is concerned with the generalisability of the
estimates.
• Do the study’s findings inform us about different
populations?
11
Naïve Estimator
Take-away points
❑ How did the non-treated ‘escape’ treatment?
❑ People deciding to participate and those decided not to are
general fundamentally different.
❑ Assess comparability in terms of observables
❑ Make your life easy by choosing your comparison group
wisely:
How to choose a comparison group

Randomly deny program access to a sub-group of
participants (i.e. randomised experiment)

Arbitrary rules that locally ‘randomise’ people (i.e.
regression discontinuity design)

Sources of natural variation in treatment assignment: two
very similar groups on average, but one ‘randomly’ has more
exposure to treatment (i.e. instrumental variables)

Non-participants who look similar to participants (i.e.
regression, matching)

Remove selection bias under pre-program data on the two
12
groups (i.e. difference-in-differences)

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