Bayes Theorem & Share Trading on the ASXIain McLean
The A-Line Market Index is a game-changer in market trend analysis. We've spent decades innovating and improving low risk methods of share market trading.
We believe you deserve the chance to secure your financial freedom. The A-Line Newsletter is Australia's best value newsletter for share trading on the ASX.
In 2014 we were invited to present to the Melbourne Chapter of the Australian Technical Analysts Association (ATAA) in relation to our indicator The A-Line and how to use probabilities to forward test trading strategies.
Bayes Theorem & Share Trading on the ASXIain McLean
The A-Line Market Index is a game-changer in market trend analysis. We've spent decades innovating and improving low risk methods of share market trading.
We believe you deserve the chance to secure your financial freedom. The A-Line Newsletter is Australia's best value newsletter for share trading on the ASX.
In 2014 we were invited to present to the Melbourne Chapter of the Australian Technical Analysts Association (ATAA) in relation to our indicator The A-Line and how to use probabilities to forward test trading strategies.
Talk at the modcov19 CNRS workshop, en France, to present our article COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
Message: @Pi_vendor_247 on telegram if u want to sell PI COINS.
1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
Key Features of USDA Loans:
Zero Down Payment: USDA loans require no down payment, making homeownership more accessible.
Competitive Interest Rates: These loans often come with lower interest rates compared to conventional loans.
Flexible Credit Requirements: USDA loans have more lenient credit score requirements, helping those with less-than-perfect credit.
Guaranteed Loan Program: The USDA guarantees a portion of the loan, reducing risk for lenders and expanding borrowing options.
Eligibility Criteria:
Location: The property must be located in a USDA-designated rural or suburban area. Many areas in California qualify.
Income Limits: Applicants must meet income guidelines, which vary by region and household size.
Primary Residence: The home must be used as the borrower's primary residence.
Application Process:
Find a USDA-Approved Lender: Not all lenders offer USDA loans, so it's essential to choose one approved by the USDA.
Pre-Qualification: Determine your eligibility and the amount you can borrow.
Property Search: Look for properties in eligible rural or suburban areas.
Loan Application: Submit your application, including financial and personal information.
Processing and Approval: The lender and USDA will review your application. If approved, you can proceed to closing.
USDA loans are an excellent option for those looking to buy a home in California's rural and suburban areas. With no down payment and flexible requirements, these loans make homeownership more attainable for many families. Explore your eligibility today and take the first step toward owning your dream home.
what is the best method to sell pi coins in 2024DOT TECH
The best way to sell your pi coins safely is trading with an exchange..but since pi is not launched in any exchange, and second option is through a VERIFIED pi merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and pioneers and resell them to Investors looking forward to hold massive amounts before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade pi coins with.
@Pi_vendor_247
how to sell pi coins on Bitmart crypto exchangeDOT TECH
Yes. Pi network coins can be exchanged but not on bitmart exchange. Because pi network is still in the enclosed mainnet. The only way pioneers are able to trade pi coins is by reselling the pi coins to pi verified merchants.
A verified merchant is someone who buys pi network coins and resell it to exchanges looking forward to hold till mainnet launch.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
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how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
The secret way to sell pi coins effortlessly.DOT TECH
Well as we all know pi isn't launched yet. But you can still sell your pi coins effortlessly because some whales in China are interested in holding massive pi coins. And they are willing to pay good money for it. If you are interested in selling I will leave a contact for you. Just telegram this number below. I sold about 3000 pi coins to him and he paid me immediately.
Telegram: @Pi_vendor_247
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Even tho Pi network is not listed on any exchange yet.
Buying/Selling or investing in pi network coins is highly possible through the help of vendors. You can buy from vendors[ buy directly from the pi network miners and resell it]. I will leave the telegram contact of my personal vendor.
@Pi_vendor_247
Turin Startup Ecosystem 2024 - Ricerca sulle Startup e il Sistema dell'Innov...Quotidiano Piemontese
Turin Startup Ecosystem 2024
Una ricerca de il Club degli Investitori, in collaborazione con ToTeM Torino Tech Map e con il supporto della ESCP Business School e di Growth Capital
how can I sell my pi coins for cash in a pi APPDOT TECH
You can't sell your pi coins in the pi network app. because it is not listed yet on any exchange.
The only way you can sell is by trading your pi coins with an investor (a person looking forward to hold massive amounts of pi coins before mainnet launch) .
You don't need to meet the investor directly all the trades are done with a pi vendor/merchant (a person that buys the pi coins from miners and resell it to investors)
I Will leave The telegram contact of my personal pi vendor, if you are finding a legitimate one.
@Pi_vendor_247
#pi network
#pi coins
#money
how to sell pi coins in South Korea profitably.DOT TECH
Yes. You can sell your pi network coins in South Korea or any other country, by finding a verified pi merchant
What is a verified pi merchant?
Since pi network is not launched yet on any exchange, the only way you can sell pi coins is by selling to a verified pi merchant, and this is because pi network is not launched yet on any exchange and no pre-sale or ico offerings Is done on pi.
Since there is no pre-sale, the only way exchanges can get pi is by buying from miners. So a pi merchant facilitates these transactions by acting as a bridge for both transactions.
How can i find a pi vendor/merchant?
Well for those who haven't traded with a pi merchant or who don't already have one. I will leave the telegram id of my personal pi merchant who i trade pi with.
Tele gram: @Pi_vendor_247
#pi #sell #nigeria #pinetwork #picoins #sellpi #Nigerian #tradepi #pinetworkcoins #sellmypi
If you are looking for a pi coin investor. Then look no further because I have the right one he is a pi vendor (he buy and resell to whales in China). I met him on a crypto conference and ever since I and my friends have sold more than 10k pi coins to him And he bought all and still want more. I will drop his telegram handle below just send him a message.
@Pi_vendor_247
What website can I sell pi coins securely.DOT TECH
Currently there are no website or exchange that allow buying or selling of pi coins..
But you can still easily sell pi coins, by reselling it to exchanges/crypto whales interested in holding thousands of pi coins before the mainnet launch.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and resell to these crypto whales and holders of pi..
This is because pi network is not doing any pre-sale. The only way exchanges can get pi is by buying from miners and pi merchants stands in between the miners and the exchanges.
How can I sell my pi coins?
Selling pi coins is really easy, but first you need to migrate to mainnet wallet before you can do that. I will leave the telegram contact of my personal pi merchant to trade with.
Tele-gram.
@Pi_vendor_247
1. Arthur Charpentier, SIDE Summer School, July 2019
# 11 Observations, Experiments & Causality
Arthur Charpentier (Universit´e du Qu´ebec `a Montr´eal)
Machine Learning & Econometrics
SIDE Summer School - July 2019
@freakonometrics freakonometrics freakonometrics.hypotheses.org 1
2. Arthur Charpentier, SIDE Summer School, July 2019
Causal Inference
“Social scientists know that large amounts
of data will not overcome the selection
problems that make causal inference so dif-
ficult”, Grimmer (2015, We Are All Social
Scientists Now: How Big Data, Machine
Learning, and Causal Inference Work To-
gether)
(source Pearl & Mackenzie (2018, The
Book of Why))
@freakonometrics freakonometrics freakonometrics.hypotheses.org 2
3. Arthur Charpentier, SIDE Summer School, July 2019
Causality and Counterfactuals
“we may define a cause to be an object, followed by another, and where all the
objects similar to the first are followed by objects similar to the second. Or in
other words where, if the first object had not been, the second never had existed”,
Hume (1748, An Enquiry Concerning Human Understanding)
Classical conditional : if X occurred, then Y occurred
Counterfactual : if X had not occurred, then Y would not have occurred
“no causation without manipulation”, Holland (1986, Statis-
tics and Causal Inference)
Importance of Directed Acyclic Graphs (DAG) to describe
causal effects between variables
@freakonometrics freakonometrics freakonometrics.hypotheses.org 3
4. Arthur Charpentier, SIDE Summer School, July 2019
Berkeley Gender Bias
Graduate admissions data from Berkeley, 1973
• men : 8442 applications, 44% admission rate
• women : 4321 applications, 35% admission rate
Discrimination towards women ?
A B C D E F
applied 825 560 325 417 191 373
M
admitted 62% 63% 37% 33% 28% 6%
applied 108 25 593 375 393 341
F
admitted 82% 68% 34% 35% 24% 7%
see Bickel et al. (1975, Sex bias in graduate admissions)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 4
5. Arthur Charpentier, SIDE Summer School, July 2019
Simpson’s Paradox & Ecological Fallacy
hosp. hosp.
A B
total 1000 1000
survivors 800 900
deads 200 100
rate (%) 80% 90%
hosp. hosp.
A B
total 600 900
survivors 590 870
deads 10 30
rate (%) 98% 97%
hosp. hosp.
A B
total 400 100
survivors 210 30
deads 190 70
rate (%) 53% 30%
@freakonometrics freakonometrics freakonometrics.hypotheses.org 5
9. Arthur Charpentier, SIDE Summer School, July 2019
Simpson’s Paradox & Ecological Fallacy
Very important concept in political science
see Gelman (1986, Red State, Blue State, Rich State,
Poor State: Why Americans Vote the Way They Do)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 9
10. Arthur Charpentier, SIDE Summer School, July 2019
Conditional Independence or could a machine create a DAG ?
Conditional Independence
X ⊥⊥ Y |Z if and only if
• f(x, y|z) = f(x|z) · f(y|z)
• f(x, y, z) · f(z) = f(x, z) · f(y, z)
• f(y|x, z) = f(y|z)
i.e. once we know Z, learning the value of x does not provide additional
information about X
Partial Correlation
ρxy|z =
ρxy − ρxz · ρyz
(1 − ρ2
xz) · (1 − ρ2
yz)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 10
11. Arthur Charpentier, SIDE Summer School, July 2019
Conditional Independence or could a machine create a DAG ?
as X ⊥⊥ Y =⇒ ρxy = 0, X ⊥⊥ Y |Z =⇒ ρxy|z = 0
and if (X, Y, Z) is Gaussian, the converse is true
as ρxy = 0 =⇒ X ⊥⊥ Y, ρxy|z = 0 =⇒ X ⊥⊥ Y |Z
@freakonometrics freakonometrics freakonometrics.hypotheses.org 11
12. Arthur Charpentier, SIDE Summer School, July 2019
Conditional Independence or could a machine create a DAG ?
Consider variables X1, X2, X3 and Y
We have
• X1 ⊥⊥ X2
• X1 ⊥⊥ X3
• X2 ⊥⊥ Y |X3
Using conditional independence tests,
remove edges, identify colliders and chains
(might not be a unique solution...)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 12
13. Arthur Charpentier, SIDE Summer School, July 2019
Conditional Independence or could a machine create a DAG ?
Assuming a Gaussian setting, Spirtes et al. (2000, Causation, Prediction, and
Search) suggested to test ρxy|z = 0 using
zn =
1
2
n − dim[z] − 3 log
|1 + ρxy|z|
|1 − ρxy|z|
One can also consider a non-parametric test, based on the distance
d(x, y, z) = f(x, y, z) · f(z) − f(x, z) · f(y, z)
(main issue here: curse of dimensionality)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 13
14. Arthur Charpentier, SIDE Summer School, July 2019
From DAG to Structural Econometric Equations
There is a one-to-one mapping between a system of structural
equations and a graphical model
• X2 = β23X3 + β21X1 + η2
• X3 = β31X1 + η3
• Y = β3X3 + ε
@freakonometrics freakonometrics freakonometrics.hypotheses.org 14
15. Arthur Charpentier, SIDE Summer School, July 2019
Observation and Experiment
Can we use observational data ?
Consider the following dataset (from Imai (2222) - resume.csv)
1 firstname
2 sex
3 race
4 callback
first name of the fictitious job applicant
sex of applicant, sex ∈ {female,male}
race of applicant, race ∈ {black,white}
whether a call back was made, call ∈ {0, 1}
See
1 firstname sex race callback
2 1 Allison female white 0
3 2 Kristen female white 0
4 3 Lakisha female black 0
5 4 Latonya female black 0
6 5 Carrie female white 0
7 6 Jay male white 0
@freakonometrics freakonometrics freakonometrics.hypotheses.org 15
16. Arthur Charpentier, SIDE Summer School, July 2019
Observation and Experiment
It is experimental data, collected from an experimental research design, in which
a treatment variable is manipulated in order to examine its causal effects on an
outcome variable.
The treatment refers to the race of a fictitious applicant, implied by the name
given on the r´esum´e.
The outcome variable is whether the applicant receives a callback.
We are interested in examining whether or not the r´esum´es with different names
yield varying callback rates.
no call call total
black 2278 157 2435
white 2200 235 2435
total 4478 392 4870
@freakonometrics freakonometrics freakonometrics.hypotheses.org 16
17. Arthur Charpentier, SIDE Summer School, July 2019
Observation and Experiment
Conditional probabilities
no call call total
black 50.87% 40.05% 50.00%
white 49.13% 59.95% 50.00%
total 100.00% 100.00% 100.00%
no call call total
black 93.55% 6.45% 100.00%
white 90.35% 9.65% 100.00%
total 91.95% 8.05% 100.00%
3.2% points difference...
@freakonometrics freakonometrics freakonometrics.hypotheses.org 17
18. Arthur Charpentier, SIDE Summer School, July 2019
Observation and ExperimentAisha
Rasheed
Keisha
Tremayne
Kareem
Darnell
Tyrone
Hakim
Tamika
Lakisha
Tanisha
Todd
Jamal
Neil
Brett
Geoffrey
Brendan
Greg
Emily
Anne
Jill
Latoya
Kenya
Matthew
Latonya
Leroy
Allison
Ebony
Jermaine
Laurie
Sarah
Meredith
Carrie
Kristen
Jay
Brad
0.00
0.05
0.10
0.15
Aisha
Rasheed
Keisha
Tremayne
Kareem
Darnell
Tyrone
Hakim
Tamika
Lakisha
Tanisha
Todd
Jamal
Neil
Brett
Geoffrey
Brendan
Greg
Emily
Anne
Jill
Latoya
Kenya
Matthew
Latonya
Leroy
Allison
Ebony
Jermaine
Laurie
Sarah
Meredith
Carrie
Kristen
Jay
Brad
0.00
0.05
0.10
0.15
@freakonometrics freakonometrics freakonometrics.hypotheses.org 18
19. Arthur Charpentier, SIDE Summer School, July 2019
Looking for a Conterfactual
1 firstname sex race callback
2 1 Allison female white 0
Would the same employer would have called back if the applicant’s name were
instead a stereotypically African-American.
Unfortunately, we would never observe this counterfactual outcome, because the
researchers who conducted this experiment did not send out the same r´esum´e to
the same employer using Lakisha as the first name.
Let t denote the treatment (the first name0) which sound African-American
(t = 1) or not (t = 0). Let y(t) denote the response (call back) as a function of
the treatment t. We do observe various covariate x = (x1, x2, · · · , xk) such as the
age, the education, etc.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 19
20. Arthur Charpentier, SIDE Summer School, July 2019
Looking for a Conterfactual
The dataset is here
black-sounding callback age education
i name t y(t = 1) y(t = 0) x1 x2
1 1 1 ? 20 college
2 0 ? 0 55 high school
3 0 ? 1 40 graduate school
We would like to quantify ceteris paribus y(1) − y(0) (called causal effect)
Fundamental problem of causal inference : we cannot observe the counterfactual
outcomes
@freakonometrics freakonometrics freakonometrics.hypotheses.org 20
21. Arthur Charpentier, SIDE Summer School, July 2019
Looking for a Conterfactual
In observational studies, researchers do not conduct an intervention.
But they still want to quantify the impact of a policy. For instance the impact of
an increase of minimum wage on unemployment (see Card & Krueger (1994),
minwage.csv) )
In 1992, New Jersey (NJ) raised the minimum wage from 4.25to5.05 (per hour).
1 chain
2 location
3 wage_before
4 wage_after
5 full_before
6 full_after
7 part_before
8 part_after
name of fast-food restaurant chain
location of restaurants (centralNJ, northNJ, PA, shoreNJ, southNJ)
wage before minimum-wage increase
wage after minimum-wage increase
number of full-time employees
(before and after)
number of part-time employees
(before and after)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 21
22. Arthur Charpentier, SIDE Summer School, July 2019
Looking for a Conterfactual
One can look at a counterfactual, with a neighboring state - Pennsylvania (PA) -
that will be our control group
This would be a cross-section comparison design
NJ PA
below below
mean $5.5 mean $5.5
before $4.61 91.06% $4.65 94.03%
after $5.08 0.34% $4.61 95.52%
@freakonometrics freakonometrics freakonometrics.hypotheses.org 22
23. Arthur Charpentier, SIDE Summer School, July 2019
Looking for a Conterfactual
NJ PA
partial full proportion partial full proportion
before 5423 2319 29.9% 1291 714 35.6%
after 5351 2446 31.4% 1339 547 29.0%
Difference-in-differences estimate,
DID = yafter
t=1 − ybefore
t=1
difference in the treatment group
− yafter
t=0 − ybefore
t=0
difference in the control group
@freakonometrics freakonometrics freakonometrics.hypotheses.org 23
24. Arthur Charpentier, SIDE Summer School, July 2019
Observation and Experiment
q
q
q
q
Averageproportionofsalariesbelow$5.5(%)
q
q
q q
q q
020406080100
averagecausaleffect
NJ
PA
BEFORE AFTER
q
q
q
q
Averageproportionoffull...timeemployees(%)
q
q
q
q
q
q
20253035
averagecausaleffect
NJ
PA
BEFORE AFTER
(the counterfactual outcome for the treatment group has a time trend parallel to
that of the control group)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 24
25. Arthur Charpentier, SIDE Summer School, July 2019
Randomized Experiments
n individuals, that are either treated (ti = 1) or not (ti = 0). We observe
outcome yi for covariates xi.
We want to study potential outcomes yi(1) and yi(0)
turnout
yi(1) yi(0) ti x1,i x2,i
1 y1 ? 1 x1,1 x2,1
2 ? y2 0 x1,2 x2,2
...
...
...
...
...
...
n yn ? 1 x1,n x2,n
The causal effect is yi(1) − yi(0)
Assume no simultaneity, no interference between individuals and same treatment
@freakonometrics freakonometrics freakonometrics.hypotheses.org 25
26. Arthur Charpentier, SIDE Summer School, July 2019
Randomized Experiments
ATE - Average Treatment Effect
Average Treatment Effect E Y (1) − Y (0)
Sample Average Treatment Effect
1
n
n
i=1
yi(1) − yi(0)
Assume randomization of the treatment assignment, n1 treated, n0 non treated
Assumption : (Y (1), Y (0)) ⊥⊥ T
Crude estimator (difference in means), τ =
1
n1 i:ti=1
yi −
1
n0 i:ti=0
yi, or
τ =
n
i=1
tiyi
n1
−
(1 − ti)yi
n0
Then E τ = E Y (1) − Y (0)
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27. Arthur Charpentier, SIDE Summer School, July 2019
Randomized Experiments
Intuition with a simple regression, yi(ti) = α + βti + εi, where E ε = 0.
Causal effect is measured by yi(1) − yi(0) = β
More generally, assume that ε is ε(t) - with E ε(T) = 0, then
yi(1) − yi(0) = β + εi(1) − εi(0). Thus β = E[Yi(1) − Yi(0)] (and β = E[Yi(1))
and α = E[Yi(0)]
One can consider a more general model, with yi(ti) = α + βti + ε(ti) where
E[ε(t)] = 0.
Then β = E[Yi(1) − Yi(0)] and α = E[Yi(0)] (as previously)
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28. Arthur Charpentier, SIDE Summer School, July 2019
Randomized Experiments
In this model - introduced in Neyman (1923, Sur les applications de la th´eorie des
probabilit´es aux exp´eriences agricoles) - (called Rubin-Neyman) let σ2
t = Var[ε(t)]
Since τ =
1
n1 i:ti=1
yi −
1
n0 i:ti=0
yi =
1
n1
n
i=1
ti · yi −
1
n0
n
i=1
(1 − ti) · yi
The variance estimate of τ is usually biased
E
σ2
(ti − t)2
−
σ2
0
n0
+
σ2
1
n1
=
(n1 − n0)(n − 1)
n1n0(n − 2)
σ2
1 − σ2
0
so bias is null when either n1 = n0 or σ2
1 = σ2
0.
But generally biased (even asymptotically).
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29. Arthur Charpentier, SIDE Summer School, July 2019
(Cluster) Randomized Experiments
One can also consider some cluster randomized experiments : treatment is at
cluster level, see Bland (2004, Cluster randomised trials in the medical literature) or
Boruch et al. (2004, Estimating the effects of interventions that are deployed in
many places: place-randomized trials)
Consider clusters j = 1, · · · , m, outcome is yi,j, with treatment tj ∈ {0, 1}.
Assume random assignment, i.e. (Yi,j(1), Yi,j(0)) ⊥⊥ Tj.
And for convenience, assume that nj’s are equal...
τ =
1
m1 j:tj =1
yj −
1
m0 j:tj =0
yj where yj =
1
nj
nj
i=1
yi,j
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30. Arthur Charpentier, SIDE Summer School, July 2019
(Cluster) Randomized Experiments
As before, E τ = E Y (1) − Y (0)
Exact variance is Var[τ] =
Var[Y (1)]
m1
+
Var[Y (0)]
m0
,
where Var[Y (t)] =
σ2
t
n
[1 + (n − 1)ρt] ≤ σ2
t
(due to (possible) intracluster correlation ρt)
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31. Arthur Charpentier, SIDE Summer School, July 2019
Observational Studies & Propensity Score
In the regression discontinuity design, we want to find an (arbitrary) cutoff point
c that determines the treatment assignment, i.e. ti = 1xi≥c.
We want to estimate E[Yi(1) − Yi(0)|Xi = c]
Identification of the Average Treatment Effect :
Assumption 1: Overlap, ∀x, P[T = 1|X = x] ∈ (0, 1)
Assumption 2: Ignorability (Y (1), Y (0)) ⊥⊥ T|X = x, ∀x
Set µ(t, x) = E[Yi(t)|T = t, X = x]
Crude estimator (difference in means),
τ =
1
n
n
i=1
µ(1, x) − µ(0, x)
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32. Arthur Charpentier, SIDE Summer School, July 2019
Observational Studies & Regression Discontinuity
See Imbens & Lemieux (2008, Regression Discontinuity Designs).
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