Quantitative Analysis For Management 11th Edition Render Test BankRichmondere
Full download : http://alibabadownload.com/product/quantitative-analysis-for-management-11th-edition-render-test-bank/ Quantitative Analysis For Management 11th Edition Render Test Bank
A Fuzzy Mean-Variance-Skewness Portfolioselection Problem.inventionjournals
A fuzzy number is a normal and convex fuzzy subsetof the real line. In this paper, based on membership function, we redefine the concepts of mean and variance for fuzzy numbers. Furthermore, we propose the concept of skewness and prove some desirable properties. A fuzzy mean-variance-skewness portfolio se-lection model is formulated and two variations are given, which are transformed to nonlinear optimization models with polynomial ob-jective and constraint functions such that they can be solved analytically. Finally, we present some numerical examples to demonstrate the effectiveness of the proposed models
Math 221 Massive Success / snaptutorial.comStephenson164
1. (TCO 1) An Input Area (as it applies to Excel 2010) is defined as______.
2. (TCO 1) In Excel 2010, a sheet tab ________.
3. (TCO 1) Which of the following best describes the AutoComplete function?
4. (TCO 1) Which of the following best describes the order of precedence as it applies to math operations in Excel?
Data Mining: Concepts and Techniques — Chapter 2 —Salah Amean
the presentation contains the following :
-Data Objects and Attribute Types.
-Basic Statistical Descriptions of Data.
-Data Visualization.
-Measuring Data Similarity and Dissimilarity.
-Summary.
Quantitative Analysis For Management 11th Edition Render Test BankRichmondere
Full download : http://alibabadownload.com/product/quantitative-analysis-for-management-11th-edition-render-test-bank/ Quantitative Analysis For Management 11th Edition Render Test Bank
A Fuzzy Mean-Variance-Skewness Portfolioselection Problem.inventionjournals
A fuzzy number is a normal and convex fuzzy subsetof the real line. In this paper, based on membership function, we redefine the concepts of mean and variance for fuzzy numbers. Furthermore, we propose the concept of skewness and prove some desirable properties. A fuzzy mean-variance-skewness portfolio se-lection model is formulated and two variations are given, which are transformed to nonlinear optimization models with polynomial ob-jective and constraint functions such that they can be solved analytically. Finally, we present some numerical examples to demonstrate the effectiveness of the proposed models
Math 221 Massive Success / snaptutorial.comStephenson164
1. (TCO 1) An Input Area (as it applies to Excel 2010) is defined as______.
2. (TCO 1) In Excel 2010, a sheet tab ________.
3. (TCO 1) Which of the following best describes the AutoComplete function?
4. (TCO 1) Which of the following best describes the order of precedence as it applies to math operations in Excel?
Data Mining: Concepts and Techniques — Chapter 2 —Salah Amean
the presentation contains the following :
-Data Objects and Attribute Types.
-Basic Statistical Descriptions of Data.
-Data Visualization.
-Measuring Data Similarity and Dissimilarity.
-Summary.
Researchers use several tools and procedures for analyzing quantitative data obtained from different types of experimental designs. Different designs call for different methods of analysis. This presentation focuses on:
T-test
Analysis of variance (F-test), and
Chi-square test
Invited talk at the Focus Fortnight 8: ""The analysis of discrete choice experiments", organized by the Centre for Bayesian Statistics in Health Economics, University of Sheffield (UK), September, 2007.
Bayesian, frequentist and fiducial (BFF) inferences are much more congruous than they have been perceived historically in the scientific community. Most practitioners are probably more familiar with the competing narratives of the two dominant statistical inferential paradigms, Bayesian inference and frequentist inference. The third, lesser known fiducial inference paradigm was pioneered by R.A. Fisher in an attempt to define an inversion procedure for inference as an alternative to Bayes' theorem. Although each paradigm has its own strengths and limitations subject to their different philosophical underpinnings, this talk intends to bridge these three different inferential methodologies through the lenses of confidence distribution theory and artificial sampling procedures. The talk attempts to understand how uncertainty quantifications in these three distinct paradigms, Bayesian, frequentist, and fiducial inference, can be unified and compared on a foundational level, thereby increasing the range of possible techniques available to both statistical theorists and practitioners across all fields.
InstructionDue Date 6 pm on October 28 (Wed)Part IProbability a.docxdirkrplav
InstructionDue Date: 6 pm on October 28 (Wed)
Part IProbability and Sampling Distributions1.Thinking about probability statements. Probability is measure of how likely an event is to occur. Match one of probabilities that follow with each statement of likelihood given (The probability is usually a more exact measure of likelihood than is the verbal statement.)Answer0 0.01 0.3 0.6 0.99 1(a) This event is impossible. It can never occur.(b) This event is certain. It will occur on every trial.(c) This event is very unlikely, but it will occur once in a while in a long sequence of trials.(d) This event will occur more often that not.2. Spill or Spell? Spell-checking software catches "nonword errors" that result in a string of letters that is not a word, as when "the" is typed as "the." When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the following distribution:Value of X01234Probability0.10.20.30.30.1(a) Check that this distribution satisfies the two requirements for a legitimate assignment of probabilities to individual outcomes.(b) Write the event "at least one nonword error" in term of X (for example, P(X >3)). What is the probability of this event?(c) Describe the event X ≤ 2 in words. What is its probability? 3. Discrete or continuous? For each exercise listed below, decide whether the random variable described is discrete or continuous and explains the sample space.(a) Choose a student in your class at random. Ask how much time that student spent studying during the past 24 hours.(b) In a test of a new package design, you drop a carton of a dozen eggs from a height of 1 foot and count the number of broken eggs.(c) A nutrition researcher feeds a new diet to a young male white rat. The response variable is the weight (in grams) that the rat gains in 8 weeks.4. Tossing Coins(a) The distribution of the count X of heads in a single coin toss will be as follows. Find the mean number of heads and the variance for a single coin toss.Number of Heads (Xi)01mean:Probability (Pi)0.50.5variance:(b) The distribution of the count X of heads in four tosses of a balanced coin was as follows but some missing probabilities. Fill in the blanks and then find the mean number of heads and the variance for the distribution with assumption that the tosses are independent of each other.Number of Heads (Xi)01234mean:Probability (Pi)0.06250.0625variance:(c) Show that the two results of the means (i.e. single toss and four tosses) are related by the addition rule for means. (d) Show that the two results of the variances (i.e. single toss and four tosses) are related by the addition rule for variances (note: It was assumed that the tosses are independent of each other). 5. Generating a sampling distribution. Let's illustrate the idea of a sampling distribution in the case of a very small sample from a very small .
PSY520 – Module 7Answer SheetSubmit your answers in the .docxwoodruffeloisa
PSY520 – Module 7
Answer Sheet
Submit your answers in the boxes provided. No credit will be given for responses not found in the correct answer area.
Chapter 19:
19.9Randomly selected records of 140 convicted criminals reveal that their crimes were committed on the following days of the week:
DAYS WHEN CRIMES WERE COMMITTED
FREQUENCY
MON.
TUE.
WED.
THU.
FRI.
SAT.
SUN.
TOTAL
Observed (ƒₒ)
17
21
22
18
23
24
15
140
Question:
Calculations or Logic:
Answer:
Using the .01 level of significance, test the null hypothesis that in the underlying population, crimes are equally likely to be committed on any day of the week.
Step 1
What is the research problem?
Step 2
What is the null hypothesis?
What is the alternative hypothesis?
Step 3
What are the degrees of freedom?
What is the decision rule?
Step 4
What is the critical X2?
What is the value of X2? (you will need to calculate this)
Step 5
What is the decision? (retain or reject the null hypothesis at the specified level of significance; note the relationship between the observed and critical X2 scores)
Step 6
What is your interpretation of the decision in relation to the original research problem?
Specify the p -value for this test result.
How might this result be reported in the literature?
19.10While playing a coin-tossing game in which you are to guess whether heads or tails will appear, you observe 30 heads in a string of 50 coin tosses.
Question:
Calculations or Logic:
Answer:
Test the null hypothesis that this coin is unbiased, that is, that heads and tails are equally likely to appear in the long run.
Step 1
What is the research problem?
Step 2
What is the null hypothesis?
What is the alternative hypothesis?
Step 3
What are the degrees of freedom?
What is the decision rule?
Step 4
What is the critical X2?
What is the value of X2? (you will need to calculate this)
Step 5
What is the decision? (retain or reject the null hypothesis at the specified level of significance; note the relationship between the observed and critical X2 scores)
Step 6
What is your interpretation of the decision in relation to the original research problem?
Specify the p -value for this test result.
19.13In 1912, over 800 passengers perished after the ocean liner Titanic collided with an iceberg and sank. The table below compares the survival frequencies of cabin and steerage passengers.
ACCOMMODATIONS ON THE TITANIC
SURVIVED
CABIN
STEERAGE
TOTAL
YES
299
186
485
NO
280
526
806
TOTAL
579
712
1291
Source: MacG. Dawson, R .J. (1995). The “unusual” episode data revisited. Journal of Statistical Education, 3, no. 3.
Question:
Calculations or Logic:
Answer:
Using the .05 level of significance, test the null hypothesis that survival rates are independent of the passengers’ accommodations (cabin or steerage).
Step 1
What is the research problem?
Step 2
What is the null hypothesis?
What is the alternative hypothesis?
Step 3
What are the degrees of freedom?
...
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
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In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Generating a custom Ruby SDK for your web service or Rails API using Smithyg2nightmarescribd
Have you ever wanted a Ruby client API to communicate with your web service? Smithy is a protocol-agnostic language for defining services and SDKs. Smithy Ruby is an implementation of Smithy that generates a Ruby SDK using a Smithy model. In this talk, we will explore Smithy and Smithy Ruby to learn how to generate custom feature-rich SDKs that can communicate with any web service, such as a Rails JSON API.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
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This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...
High-Dimensional Methods: Examples for Inference on Structural Effects
1. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
High-Dimensional Methods: Examples for
Inference on Structural Effects
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M.
Taddy
July 16, 2013
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
2. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Example 1: 401(k)’s and Assets
Estimate the effect of 401(k) eligibility on measure of accumulated
assets, (e.g. PVW 1994, 1995, 1996)
yi = di α0 + xi β + ζi
yi = net financial assets or total wealth,
di = eligible for 401(k),
xi = controls for individual characteristics. PVW argue
important to control for income
income (<10k, 10k-20k, 20k-30k, 30k-40k, 40k-50k, 50k-75k,
75k+), age, age2
, family size, education (high school, some
college, college), married, two-earner, defined benefit, ira,
home-owner
n = 9915
Baseline estimates: Net-TFA: 9216.5 (1340.6); TW: 6612.0
(2110.1)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
3. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Did we control sufficiently for income?
Previous model provides nice baseline, but we might wonder
Are 7 dummies for income categories sufficient to control for
income?
More complex nonlinearity?
Interactions?
Could we improve efficiency?
Even if eligibilty were randomly assigned, might want to
introduce controls to improve efficiency
“Over-controlling”?
Other important variables?
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
4. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Regularization
All treatment effects estimation problems involve
regularization/variable selection.
The only valid inference without dimension reduction is to
conclude that one cannot learn from the data.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
5. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Regularization
Dimension reduction strategies:
Randomized treatment assignment: Reduces dimension of
control vector to 0.
May still want to do variable selection to improve efficiency
(“single-selection” procedures)
Stratified sampling
Intuition
Formal Model Selection
Will fail without good intuition (Needle in a haystack: Want a
big needle or a small haystack)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
6. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Intuitive Dimension Reduction
401(k) Example Baseline:
Data Selection
Wave 4 of 1990 SIPP
Ignore panel aspect
Impose criteria (e.g. age selection) to limit sample to
“interesting” population (control)
Variable Selection
655 raw variables (obviously many administrative/technical,
highly unlikely to be related to problem of interest)
Select 9 as controls: income, age, family size, education,
married, two-earner, defined benefit, ira, home-owner
Functional form
no interactions, dummies for income categories, quadratic in
age, dummies for schooling levels
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
7. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
“Formal” Dimension Reduction: Model
Model:
yi = di α0 + g(xi ) + ζi ≈ di α0 + zi βg + ζi
di = m(xi ) + ui ≈ zi βm + ui
E[ζi |xi , di ] = E[ui |xi ] = 0
where zi is a function of xi .
“Reduced forms”:
yi = zi β + ζi
di = zi βm + ui
β = βg + α0βm
ζi = ζi + α0ui
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
8. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
“Formal” Dimension Reduction: Intuition
Intuition for what we get from each reduced form:
Reduced form for treatment:
Find variables that are strongly related to treatment
Ignoring these potentially leads to omitted-variables-bias
(OVB)
Reduced form for outcome:
Find variables that are strongly related outcome
Improve efficiency
Reduce OVB which would result if associated coefficient in
treatment equation is small but non-zero
Reduced forms are parts of problem data are informative about
(predictive relationships)
Good methods for estimating these parts of the model
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
9. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
“Formal” Dimension Reduction: Choices
Choice 1 (Data Set): Same as above
Choice 2 (Baseline Variables): Same as above
Could consider adding additional variables
Each variable makes the haystack bigger
Think carefully. Do we want more variables or more flexibility
in other dimensions?
Choice 3 (Functional Form): Want to try to be very flexible in
income (Details to follow)
Choice 4 (Selection Method/Auxiliary Parameters): LASSO
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
10. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Review: Feasible LASSO Allowing Heteroskedasticity
LASSO (heteroskedastic):
βL = arg min
n
i=1
(yi − xi β)2
+ λ
p
j=1
|γj βj |
for a generic outcome (y) and set of regressors (x)
Need to fill in values for λ and γj for j = 1, ..., p.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
11. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Review: Feasible LASSO Allowing Heteroskedasticity
γj :
Want γj ≈ 1
n
n
i=1 x2
ij ε2
i
Estimate iteratively for a given value of λ.
Start with initial guess for β (β∗
) → ei = yi − xi β∗
(initial
guess for {εi }n
i=1)
Estimate new value of β by solving above problem for given
value of λ with γj = 1
n
n
i=1 x2
ij e2
i
Take new value of β to form new residuals and new set of γj
estimates
Iterate to convergence (or max number of iterations)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
12. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Review: Feasible LASSO Allowing Heteroskedasticity
λ:
1. Cross-validation
10-fold CV in all examples
Use 1 s.e. rule as discussed by M. Taddy previously
Iterative estimation of penalty loadings inside of CV loop
Less stable than vanilla LASSO CV
1 s.e. rule seems to add some stability
2. Theoretical value:
Simple bound 1: 2.2Φ−1
(1 − q
2p )
Simple bound 2: 2.2 2n log(2p/q)
Theoretically need q → 0, q size of test of hypothesis that
“biggest” coefficient equals 0 when all coeffients equal 0. We
use q = .05 or q = .1/ max{p, n} in our examples. (Bounds
above are bounds on this critical value.)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
13. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
“Formal” Dimension Reduction: Variables
Dummies: married, two-earner, defined benefit, ira,
home-owner
Schooling: 5th degree orthogonal polynomial (generated by
Stata orthpoly)
Family Size: 3rd degree orthogonal polynomial (generated by
Stata orthpoly)
Age: Cubic spline with 10 equally spaced knots (30 terms)
Income: Cubic spline with 15 equally spaced knots (45 terms)
Interactions:
1. Dummies with Schooling, Family Size, and Age terms (190
interactions)
2. Income terms interacted with A. Dummies, Schooling,
Family Size, and Age terms and B. interactions in 1. (10,485
interactions)
10,763 total variables
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
14. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Dimension Reduction
So, n = 9,915; p = 10,763
Cannot identify effect of 401(k) if we think we actually need all of
these terms to adequately control for income.
Have already intuitively reduced dimensionality by focusing on 9
controls that seem plausibly related to income and assets
Baseline results resolve this identification problem by assuming the
functional form is known.
Rather than assume functional form, try to learn it from data using
variable selection (from a flexible but still very parsimonious
specification).
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
15. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Results:
Note: Before constructing splines, income and age were put on
[0,1] interval.
401(k) Eligibility:
Selected variables (CV and Plug-in): income,
max(0, income − .33), two–earner, defined–benefit,
home–owner, education3, max(0, age3 − .4),
max(0, age3 − .5), home–owner ∗ income
Estimate 401(k) effects obtained by combining these variable
with those selected for relevant outcome and running OLS
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
16. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Results:
Net Financial Assets:
Selected variables (CV and Plug-in): income, two–earner,
age, two–earner ∗ family–size, ira ∗ age, home–owner ∗ age2,
ira ∗ income, home–owner ∗ income, home–owner ∗ income2
Estimated Effect: 8687.43 (1274.76) [Recall baseline
estimates: 9216.5 (1340.6)]
Total Wealth:
Selected variables (CV and Plug-in): income, two–earner,
ira ∗ age, home–owner ∗ age, home–owner ∗ age2,
ira ∗ income, home–owner ∗ income, age ∗ income
Estimated Effect: 5374.79 (1990.94) [Recall baseline
estimates: 6612.0 (2110.1)]
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
17. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
ATE with Heterogeneous Treatment Effects:
We could also estimate the ATE allowing for full heterogeneity of
treatment effects.
Model:
yi = di g1(xi ) + (1 − di )g0(xi ) + ζi
di = m(xi ) + ui (as before)
Following Hahn (1998), can estimate ATE (α) as
α =
1
N
N
i=1
di (yi − g1(xi ))
m(xi )
−
(1 − di )(yi − g0(xi ))
1 − m(xi )
+ g1(xi ) − g0(xi )
where we obtain estimates of the functions g0(·), g1(·), and m(·)
as above using variable selection methods.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
18. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
ATE with Heterogeneous Treatment Effects:
Estimates of m(·) were obtained previously.
Net Financial Assets and Total Wealth:
Use the same set of covariates broken out by values of 401(k)
eligibilty dummy
Use the CV penalty parameter from before scaled for the
number of observations in each category (3682 are eligible,
6233 not-eligible)
I.e. multiply CV penalty parameter by 3682/9915 in eligible
models and by 6233/9915 in not-eligible models
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
19. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
ATE Results:
Financial Assets:
Selected Variables (eligible): income, IRA, age,
two–earner ∗ family–size, IRA ∗ age, home–owner ∗ age,
IRA ∗ income, home–owner ∗ income
Selected Variables (non-eligible): IRA, home–owner, age,
IRA ∗ age, home–owner ∗ age
Estimates: 8032.54 (1136.6)
Total Wealth:
Selected Variables (eligible): income2
, two–earner, age,
IRA ∗ age, home–owner ∗ age, IRA ∗ income,
home–owner ∗ income, age ∗ income
Selected Variables (non-eligible): income, IRA, IRA ∗ age,
home–owner ∗ age, home–owner ∗ age2
, IRA ∗ income,
home–owner ∗ income
Estimates: 6180.29 (1828.5)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
20. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Example 2: Effect of Abortion on Crime
Goal: Understand causal effect of dit (abortion) on yit (crime).
(Donohue and Levitt 2001)
Problem: Abortion rates are not randomly assigned
Key concern:
states are different for lots of reasons
crime rates in states evolve differently for lots of reasons
factors that are associated to differences in states, state
evolutions, etc. may also be related to differences in abortion
rates, abortion rate evolution, etc.
Most of the favorite stories for confounding obviously fit here.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
21. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Example 2: Baseline Model
Donohue and Levitt (2001) baseline model
yit = ditα0 + xitβg + γt + δi + εit
yit = crime-rate (violent, property, or murder per 1000)
dit = “effective” abortion rate
xit = eight controls: log of lagged prisoners per capita, the log
of lagged police per capita, the unemployment rate, per-capita
income, the poverty rate, AFDC generosity at time t − 15, a
dummy for concealed weapons law, and beer consumption per
capita
γt time effects
δi state effects
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
22. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Baseline Results:
Violent Property Murder
Estimator Effect Std. Err. Effect Std. Err. Effect Std. Err.
DL Table 4 -.129 .024 -.091 .018 -.121 .047
First-Diff -.152 .034 -.108 .022 -.204 .068
Use first-differences from now on.
Assumes confounds are time invariant, state invariant, or captured
by small set of variables in xit
State-specific characteristics related to features of abortion only
allowed to be related to level of crime rate
I.e. evolution of abortion rates and crime rates unrelated after
subtracting the mean
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
23. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Flexible Trends
“Model” with flexible trends:
yit = ditα0 + g(xi1, xi2, . . . , xiT , t, i) + ζit
dit = m(xi1, xi2, . . . , xiT , t, i) + uit
Want to allow flexible state-specific trends BUT clearly can’t learn
about about effect if trends are allowed to do anything.
I.e. if m(·) and g(·) can vary arbitrarily across i and t clearly can’t
identify α0
Need to reduce the dimension.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
24. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Intuitive Dimension Reduction
Abortion Baseline:
Functional form
g(xi1, xi2, . . . , xiT , t, i) = xitβg + γt + δi
additively separable
(Correlated) Evolution of state crime and abortion rates
captured by macro-economy (γt), constant state-specific level
shifts from aggregate (δi ), and small number of time varying
variables
Variable Selection
Select 8 time varying state-level control variables (of the many
state-level macro series available)
What if there are (correlated) differences in abortion and crime
evolution not captured by aggregate evolution?
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
25. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Formal Dimension Reduction
Model (First-Differences):
∆yit = ∆ditα0 + γt + witπ1 + εit
∆dit = κt + witπ2 + uit
Use first-difference to remove state effects
time effects (not-selected over), included in both reduced form
models
With wit = ∆xit first-difference version of Donohue and Levitt
(2001) model. (Results presented above.)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
26. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Formal Dimension Reduction: Variables
wit:
1. differences in eight original controls
2. initial conditions of controls, abortion rate, crime rate
3. within state averages of controls, abortion rate
4. t, t2
5. interactions of 1-3 with 4
corresponds to a model for crime and abortion rates with a
cubic trend that may depend on baseline state characteristics
p = 284
n = 576
Variables in 2-5 motivated by a desire to have a flexible, sensible
model of evolution
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
27. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Kitchen Sink:
Why not use “everything”?
Violent Property Murder
Estimator Effect Std. Err. Effect Std. Err. Effect Std. Err.
All Controls .006 .755 -.154 .224 2.240 2.804
A flexible cubic trend arguably isn’t going crazy, but everything is
rendered very imprecise.
Probably a lot of things added aren’t really important
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
28. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Variable Selection:
Variables in each equation selected using LASSO.
10-Fold CV
Plug-in penalty parameter: 2.2 2n(log(2p/.05))
Penalty loadings estimated using iterative Lasso with 100 max
iterations
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
29. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Variable Selection:
Violent crime selection:
Abortion equation (CV and Plug-in): (11) lagged prisoners,
lagged police, lagged unemployment, initial income, initial
income difference × t, initial beer consumption difference × t,
initial income × t, initial prisoners squared × t2, average
income, average income × t, initial abortion rate
Crime equation (CV): No variables
Crime equation (Plug-in): Initial difference in abortion rate ×
t, Initial abortion rate × t
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
30. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Variable Selection:
Property crime selection:
Abortion equation (CV): (7) lagged prisoners, lagged income,
initial income, initial income difference, initial income
difference × t, average income, initial abortion rate
Abortion equation (Plug-in): (12) lagged prisoners, lagged
police, lagged income, Initial income difference, initial income,
initial income difference × t, initial beer difference × t, initial
prisoners squared × t, initial prisoners squared × t2, initial
beer squared × t2, average income, initial abortion rate
Crime equation (CV): No variables
Crime equation (Plug-in): (3) Initial income squared × t,
Initial income squared × t2, average AFDC squared
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
31. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Variable Selection:
Murder rate selection:
Abortion equation (CV): (5) lagged prisoners, lagged
unemployment, initial income × t, average income × t, initial
abortion rate
Abortion equation (Plug-in): (9) lagged prisoners, lagged
unemployment, initial unemployment difference squared,
initial prisoners × t, initial income times t, initial beer
difference × t2, average income × t, initial abortion rate,
initial abortion rate × t
Crime equation (CV and Plug-in): No variables selected
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
32. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Estimated Effects of Abortion on Crime
Violent Property Murder
Estimator Effect Std. Err. Effect Std. Err. Effect Std. Err.
DL Table 4 -.129 .024 -.091 .018 -.121 .047
First-Diff -.152 .034 -.108 .022 -.204 .068
All Controls .006 .755 -.154 .224 2.240 2.804
Post-DS (CV) -.119 .120 -.042 .059 -.122 .131
Post-DS (Plug-in) -.174 .120 -.052 .070 -.123 .148
“Post-DS” Results in-line with critique raised by Foote and Goetz
(2008).
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
33. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Example 3: Institutions and Growth (AJR 2001)
Equation of interest:
log(GDP per capitai ) = α(Protection from Expropriationi ) + xi β + εi
Endogeneity/Simultaneity:
better institutions may lead to higher incomes
higher incomes may lead to the development of better
institutions
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
34. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Instrument
Instrument = European Settler Mortality:
First Stage: settlers set up better institutions in places they
might stick around in (i.e. where they were less likely to die)
and institutions are highly persistent
Exclusion: GDP, while persistent, is unlikely to be strongly
influenced by the factors that determined the exact
development of institutions 100+ years ago except through
the institutions established
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
35. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Controls
There may be other factors that are highly persistent and
related to development of institutions and GDP
leading candidate - geography (Geographic Determinism)
Want to control for geography and use variation in mortality
not captured by geography
Baseline AJR results control linearly for latitude
AJR consider continent dummies, split by continent, first-stage
gets weak with some of these
Baseline estimates find strong positive effect of institutions:
First-stage: -0.5372 (0.1545)
α: 0.9692 (0.2128)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
36. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Setting
Want to control flexibly for geography but may lose power to
identify effect of institutions.
IV with one instrument and unknown controls:
yi = αdi + xi β + εi
di = π1zi + xi Π2 + vi
zi = xi γ + ui
Believe zi is a valid instrument after controlling for xi .
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
37. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Setting
Same as estimating regression coefficients after conditioning but
now have three reduced form/prediction equations:
yi = xi β + εi
di = xi Π2 + vi
zi = xi γ + ui
Do variable selection on the three equations and use union of
selected variables as controls.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
38. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Variable Selection
Select over a flexible function of geography:
Africa, Asia, North America, South America (dummies)
latitude, latitude2, latitude3, (latitude-.08)+, (latitude-.16)+,
(latitude-.24)+, ((latitude-.08)+)2, ((latitude-.16)+)2,
((latitude-.24)+)2, ((latitude-.08)+)3, ((latitude-.16)+)3,
((latitude-.24)+)3
Using all these variables results in a very weak first-stage:
First-stage: -0.2164 (0.2191)
α: 0.9480 (0.7384) (and unreliable)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
39. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Variable Selection:
Variables for each equation selected with Lasso:
Penalty loading calculated with iterative method
10-Fold CV and Plug-in [2.2 2n(log(2p/γ)) with
γ = .1/ log(n)] give same results
GDP - Africa
Expropriation - Africa
Mortality - Africa
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
40. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Results:
Latitude All Controls Selection
First Stage -0.5372 -0.2164 -0.5429
(0.1545) (0.2191) (0.1719)
Second Stage 0.9692 0.9480 0.7710
(0.2128) (0.7384) (0.1971)
First Stage - Coefficient on Settler Mortality
Second Stage - Coefficient on Protection from Expropriation
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
41. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Example 4: Eminent Domain
Estimate economic consequences of the law of takings or eminent
domain (following Chen and Yeh (2010))
Consider effect on Case-Shiller Price Index.
Eminent domain (or law of takings): when a government actor
physically acquires the property rights of one or more individuals
Laws/judicial decisions may not be exogenous.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
42. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Instruments
Random assignment of judges to three judge federal appellate
panels
⇒
Panel demographics randomly assigned conditional on the
distribution of characteristics of federal circuit court judges in a
given circuit-year
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
43. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Why variable selection?
Many characteristics of three judge panels
Any set of characteristics of the three judge panel unrelated to
structural unobservable
Some instruments may be more valuable than others
Could attempt to solve through intuition. Number of judges who
are democrats:
Judges’ political affiliation known to predict decisions for
many outcomes
First Stage: 0.0664 (0.0713)
Second Stage: -0.2583 (0.5251) (Unreliable)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
44. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Model
Econometric model:
yct = ac + bt + gct + θ Takings Lawct + Wctd + ct
Takings Lawct = αc + βt + γct + Wctδ + zctΠ + vct
c circuit; t time
yct: log(house price index) or log(GDP)
Takings Lawct: number of pro-plaintiff (overturn gov’t taking)
apellate takings decisions
(ac, αc), (bt, βt), and (gct, γct): circuit-specific effects,
time-specific effects, and circuit-specific time trends.
(d, δ) coefficients on exogenous variables
zct instruments with coefficients Π
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
45. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Model
Controls (Wct):
≈ 30 probability controls for panel demographics
a dummy for no cases in that circuit-year
number of takings appellate decisions
θ: effect of an additional decision upholding individual property
rights on an economic outcome
Any set of characteristics of three judge panels is potentially an
instrument.
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
46. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Instruments
Do ex ante dimension reduction by intuively selecting
characteristics thought to have strong signal about judge
preferences over government vs. individual property rights:
42 baseline variables
gender, race, religion (jewish, catholic, protestant, evangelical,
none) , political affiliation, bachelor obtained in-state, bachelor
from public university, JD from a public university, has an LLM
or SJD, elevated from a district court
number of panels with 1, 2, or 3 members with each
characteristic
+
cubic in number panels democrat, number with JD from public
university, number elevated from district
first order interactions between all variables
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
47. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Additional pre-processing
Remove instruments likely to be irrelevant based on features of
instrument vector alone:
remove any instrument with mean < .05, standard deviation
after partialling out controls < .000001
remove one from each pair of any pair with bivariate
correlation > .99 in absolute value
Note: Selection based on characteristics of z cannot introduce bias
under exclusion restriction.
Leaves 147 instruments (n = 183)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
48. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Obtaining the Estimates
Method: Post-LASSO
Penalty parameters and loadings:
Penalty loading calculated through iterative scheme
10-fold Cross-validation to obtain penalty
Plug-in penalty: 2.2
√
nΦ−1(1 − γ/(2p)), γ = .1/ log(n)
(Gives same results as 10-fold CV)
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects
49. Example 1: 401(k) Example 2: Crime Example 3: Institutions Example 4: Eminent Domain
Results: Case-Shiller Prices
LASSO Selection Results:
1+ JD public squared
First Stage: 0.4495 (0.0511) [Using 1+ democrat: 0.0664
(0.0713)]
Second Stage: 0.0648 (0.0240) [Using 1+ democrat: -0.2583
(0.5251) (Unreliable)]
V. Chernozhukov, M. Gentzkow, C. Hansen, J. Shapiro, M. Taddy
High-Dimensional Methods: Examples for Inference on Structural Effects