Chapter 7. Basic Methods for Establishing Causal Inference Chapter 8. Advances Methods for Establishing Causal Inference Initial Postings: Read and reflect on the assigned readi
Chapter 7. Basic Methods for Establishing Causal Inference
Chapter 8. Advances Methods for Establishing Causal Inference
Initial Postings: Read and reflect on the assigned readings for the week. Then post what you thought was the most important concept(s), method(s), term(s), and/or any other thing that you felt was worthy of your understanding in each assigned textbook chapter.Your initial post should be based upon the assigned reading for the week, so the textbook should be a source listed in your reference section and cited within the body of the text. Other sources are not required but feel free to use them if they aid in your discussion.
Also, provide a graduate-level response to each of the following questions:
- Causal inference is used as a secondary or tertiary tool in root cause analysis. Please explain how causal inference and root cause analysis are used in problem detection. Respond to this discussion board (DB) in the context of your field of employment. For example, if you are in I.T., respond to this DB by explaining the cause of a network failure; or if you are the in the food industry, use this DB to explain the cause of a recent decline in customer satisfaction. Please address each component of the discussion board. Also, cite examples according to APA standards.
[Your post must be substantive and demonstrate insight gained from the course material. Postings must be in the student's own words – do not provide quotes!]
[Your initial post should be at least 450+ words and in APA format (including Times New Roman with font size 12 and double spaced). Post the actual body of your paper in the discussion thread then attach a Word version of the paper for APA review]
Basic Methods for Establishing Causal Inference
Chapter 7
© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or distribution without the prior written consent of McGraw-Hill Education
Learning Objectives
Explain the consequences of key assumptions falling within a causal model
Explain how control variables can improve causal inference from regression analysis
Use control variables in estimating a regression equation
Explain how proxy variables can improve causal inference from regression analysis
Use proxy variables in estimating a regression equation
Explain how functional form choice can affect causal inference from regression analysis
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The assumptions to estimate the parameters of a regression equation are:
The data-generating process for an outcome, Y, can be expressed as: Yi = α + β1X1i + … βKXKi + Ui
{Yi, Xi, …, XKi) is a random sample
E[U] = E[U × X1] = … = E[U × XK] = 0
If these assumptions hold, we can use our regression equation estimates as “good guesses” for the parameters.
Assessing Key Assumptions within a Causal Model
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Assumption 1 states that: the determining function is linear in the parameters, and that other factors—in the form of the error term—are additive (they simply add on at the end)
For example:
Total Costs = Fixed Costs + f1Factor1 + … + fKFactorJ
FactorJ represents a factor of production and f1 its price
If we have data on Factor1 through FactorK , where K < J
Total Costs = α + β1Factor1i + … βKFactorKi + Ui
Assessing Key Assumptions within a Causal Model
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Assumption 2 states that our sample is random
There are many ways to collect a random sample, but all start with first defining the population
For example, we may define the population as all individuals in the United States, and then randomly draw Social Security numbers to build the sample.
When dealing with populations that span multiple periods of time, we treat what was observed for a given period of time as realization from a broader set of possibilities
Random Sample
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Key merit of drawing a random sample is that, on average, it should look like a smaller version of the population from which we are drawing
The information in a random sample should “represent” the population
For any given sample of data, randomness does not guarantee that it represents the population well
Random vs Representative Sample
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Random vs Representative Sample
If we have a random sample of 20 people asking them about their age and rating of the product from all the customers
But problem with this sample is it is not representative of a population of age over 40
To avoid situations like this, it is common practice to take measures to collect a representative sample
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Age and Rating Data for a New Product
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Representative sample: a sample whose distribution approximately matches that of the population for a subset of observed, independent variables
Constructing a representative sample:
Step 1: Choose the independent variables whose distribution you want to be representative
Step 2: Use information about the population to stratify (categorize) each of the choses variables
Step 3: Use information about the population to pre-set the proportion if the sample that will be selected from each stratum
Step 4: Collect the sample by randomly sampling from each stratum, where the number of random draws from each stratum is set according to the proportions determined in Step 3
Random vs. Representative Sample
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We are interested in how rating depend on age, so we have age in the role of independent variable:
Step 1: With just one independent variable, this step is trivial—we want a representative sample according to age
Step 2: We need to utilize information we have about the population. We know that 30% of the population is over the age of 40. We can stratify the data into two groups: over 40 and 40 and under.
Random vs Representative Sample
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Random vs Representative Sample
Step 3: We use our knowledge of the population to determine the proportion of our same coming from these two strata: 30% should be over 40 and 70% 40 and under. If our sample size is N = 1,000, we will have 300 who are over 40 and 700 who are 40 and under
Step 4: We may collect a random sample larger than 1,000 to ensure there are at least 300 who are over 40 and at least 700 who are 40 and under. Then, randomly select 300 from the subgroup who are over 40, and randomly select 700 from the group who are 40 and under
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The concepts of random and representative are not mutually exclusive when it comes to data samples. A sample can be both
If we construct a representative sample, then by construction it is not truly a random sample
Constructing a representative sample ensures that we observe the pertinent range of our independent variables
Construction of a representative sample often ensures that we have substantial variation in the independent variables
Random vs. Representative Sample
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Consequences of Nonrandom Samples
The construction of a representative sample generally results in nonrandom sample
A sample that is nonrandom is also known as selected sample
Two fundamental ways in which a sample can be nonrandom or selected. It can be selected according to:
The independent variables (Xs)
The dependent variable (Y)
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Selection by independent variable
THE REGRESSION LINE FOR THE DATA SET IS:
RATING = 40 + 0.5AGE.
USING JUST DATA FOR AGE < 30 WILL SIMPLY LIMIT WHERE, ALONG THE LINE, WE ARE OBSERVING DATA.
USING JUST THESE DATA POINTS WILL SKEW OUR ESTIMATES FOR THE REGRESSION LINE.
Assessing Key Assumptions within a Causal Model
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Selection by dependent variable
SAMPLE IS SELECTED SUCH THAT THE ONLY OBSERVATIONS WHERE THE RATING IS ABOVE 60 (ABOVE THE GREEN LINE).
SELECTION OF SAMPLE DEPENDING ON RATING (DEPENDENT VARIABLE) MAY CAUSE PROBLEMS WHEN ESTIMATING REGRESSION EQUATION.
SELECTION OF SAMPLE DEPENDING ON DEPENDENT VARIABLE MAY CREATE A SITUATION WHERE E[Ui] = E[Xi[Ui] = 0 MAY HOLD TRUE FOR THE FULL POPULATION, BUT E[Ui] 0 and E[Xi[Ui] 0 FOR THE SELECTED SUBSET OF THE POPULATION.
Assessing Key Assumptions within a Causal Model
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Selection by depended variable
SELECTING DATA POINTS WHERE RATING IS ABOVE 60, HAS TWO IMPORTANT CONSEQUENCES:
THE MEAN VALUE OF THE ERRORS IS POSITIVE FOR THE SELECTED SUBSET AND,
THE ERRORS AND AGE ARE NEGATIVELY CORRELATED.
Assessing Key Assumptions within a Causal Model
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Assumption 3 states that E[U] = E[U × X1] = … = E[U × XK] = 0. This means we assume the errors have a mean of zero and are not correlated with the treatments in the population
Violation of this assumption, meaning there exists correlation between the errors and at least one treatment, is known as an endogeneity problem
The component(s) of the error, Ui, that are correlated with a treatment(s), X, as confounding factors
No Correlation Between Errors and Treatment
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Three main forms in which endogeneity problems generally materialize:
Omitted variable: Any variable contained in the error term of a data generating process, due to lack of data or simply a decision not to include it
Measurement error: When one or more of the variables in the determining function (typically at least one of the treatments) is measured with error.
Simultaneity: This can arise when one or more of the treatments is determined at the same time as the outcome; often occurs when some amount of reverse causality occurs
No Correlation Between Errors and Treatment
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Control variable: any variable included in a regression equation whose purpose is to alleviate an endogeneity problem
Confounding factor that is added to a determining function
Control Variables
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Yi = α + β1X1i + … βKXKi + Ui
If the variable C is a confounding factor within the data-generating process, if…
C affects the outcome, Y
C is correlated with at least one treatment (Xj)
Then…
C is a good control, and its inclusion as part of the determining function can help mitigate an endogeneity problem
Criterion for a Good Control
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Dummy variable is a dichotomous variable (one that takes on values 0 or 1)that is used to indicate the presence or absence of a given characteristic
Typically utilized in regression equations in lieu of categorical, ordinal, or interval variables
Dummy Variables
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Categorical variable
Indicates membership to one of a set of two or more mutually exclusive categories that do not have an obvious ordering
Ordinal variable
Indicates membership to one of a set of two or more mutually exclusive categories that do not have an obvious ordering, but the difference in values is not meaningful
Interval variable
Indicates membership to one of a set of two or more mutually exclusive categories that have an obvious ordering, and the difference in values is meaningful
Types of Variables
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Suppose we have a data-generating process as:
Salesi = α + β1Commisioni + β2Locationi + Ui
We cannot regress “Sales” on “Commission” and “Location” since Location does not take on numerical values
Instead include the dummy variables created for Location as part of the determining function, rather than the Location variable itself:
Salesi = α + β1Commisioni + β2LosAngelesi + β2Chicagoi + Ui
Base group is the excluded dummy variable among a set of dummy variables representing a categorical, ordinal, or interval variable
Dummy Variables
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Selecting Controls
The variables that theory says should affect the outcome should all be included in the regression
All these variables belong as part of the data-generating process
These variables can serve as valuable data sanity checks
A data sanity check for a regression is a comparison between the estimated coefficient for an independent variable in a regression and the value for that coefficient as predicted by theory
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When Selecting Controls:
Identify variables that theoretically should or might affect the outcome
Include variables that theoretically should affect the outcome
For variables that theoretically might affect the outcome, include those that prove to affect the outcome empirically through a hypothesis test
For variables that theoretically might affect the outcome, discard those that prove irrelevant through a hypothesis test
Selecting Controls
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Proxy variable is a variable used in a regression equation in order to proxy for a confounding factor, in an attempt to alleviate the endogeneity problem caused by that confounding factor
Proxy Variables
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Functional form choice can affect causal inference from regression analysis
Assuming the following data-generation function:
Salesi = α + βHoursi + Ui
Implies that value of sales change with hours at a constant rate of β (e.g. if β is 12 then each increase in hours will increase sales by 12)
Form of the Determining Function
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Functional form choice can affect causal inference from regression analysis
Hours may affect Sales in a non-linear way, such that they have a large effect for the first few hours, but the effect diminishes as hours become large
A quadratic determining function might be better than the linear determining function
The causal relationship between Sales and Hours:
Salesi = α + βHoursi + β2Hours2i + Ui
Form of the Determining Function
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Salesi = α + βHoursi + β2Hours2i + Ui
Where we set Hours = X1 and Hours2 = X2 and it looks like a generic multiple regression equation
Form of the Determining Function
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Consequences of using the wrong function form:
Constrains the shape of the relationship between sales and hours
If we assume it is linear, the effect is constant β.
If we assume it is quadratic, the effect is not constant – simple calculus will show it is + hours.
Use Weierstrass approximation theorem: if a function is continuous, it can be approximated as closely as desired with polynomial function
Form of the Determining Function
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Quadratic Relationship Between Y and X
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THIS FUNCTION CLEARLY CANNOT BE APPROXIMATED BY LINEAR OR QUADRATIC FUNCTION. HOWEVER THERE IS A POLYNOMIAL THAT CAN GET EXTREMELY CLOSE TO THIS HIGHLY IRREGULAR FUNCTION.
Example of a Continuous but Highly Irregular Function
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Laffer Curve
THE LAFFER CURVE IS BASED ON THE IDEA THAT TAX REVENUE WILL BE ZERO BOTH WITH A ZERO TAX RATE AND A 100% TAX RATE BUT IS POSITIVE FOR TAX RATES IN BETWEEN
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Interpretations of β for Different Log Functional Forms
Log-log measures elasticity, the percentage change in one variable with a percentage change in another
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Advanced Methods for Establishing Causal Inference
Chapter 8
© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or distribution without the prior written consent of McGraw-Hill Education
Learning Objectives
Explain how instrumental variables can improve causal inference in regression analysis
Execute two-state least square regression
Judge which type of variables may be used as instrumental variables
Identify a difference-in-difference regression
Execute regression incorporating fixed effects
Distinguish the dummy variable approach from a within estimator for a fixed effect regression model
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Instrumental variables
In the context of regression analysis, a variable that allows us to isolate the causal effect of a treatment on an outcome due to its correlation with the treatment and the lack of correlation with the outcome
Can improve causal inference in regression analysis
Instrumental Variables
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A firm attempting to determine how its sales depend on price it charges for its product
Beginning with a simple data-generating process:
Salesi = α + β1Pricei + Ui
If local demand factor depends on local income, then local income is a confounding factor:
Salesi = α + β1Pricei + β2Incomei + Ui
Instrumental Variables: An Example
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Including income in the model removes local income as confounding factor
Does its inclusion ensure that no other confounding factors still exist?
Many possibilities may come to mind, including local competition, market size, and market growth rate
Instrumental Variables: An Example
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We may be unable to collect data on all confounding factors or find suitable proxies
Then we are unable to remove the endogeneity problem by including controls and/or proxy variables
A widely used method for measuring causality that can circumvent this problem involves instrumental variables
Instrumental Variables
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Suppose we know price differences across some of the stores were solely due to differences in fuel costs
When two locations have different prices, we generally cannot attribute differences in sales to price differences, since these two locations likely differ in local competition
Rather than use all of the variation in price across the stores to measure the effect of price on sales, we focus on the subset of price movements due to variation in fuel costs
Instrumental Variables
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WHEN TWO LOCATIONS HAVE DIFFERENT PRICES ONLY BECAUSE THEIR FUEL COSTS DIFFER, ANY DIFFERENCE IN SALES CAN BE ATTRIBUTED TO PRICE, SINCE FUEL COSTS DON’T IMPACT SALES PER SE
Instrumental Variables: An Example
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Suppose we have the following data-generating function:
Yi = α + β1X1i + β2X2i + … + βKXKi + Ui
Variable Z is a valid instrument for Xi if Z is both exogenous and relevant, if:
Exogenous: It has no effect on the outcome variable beyond the combined effects of all variables in the determining function (X1…XK)
Relevant: For the assumed data-generating process, Z is relevant as an instrumental variable if it is correlated with X1 after controlling for X2….XK
Instrumental Variables
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Two-stage least squares regression (2SLS) is the process of using two regressions to measure the causal effect of a variable while utilizing an instrumental variable
The first stage of 2SLS determines the subset of variation in Price that can attributed to changes in fuel costs; we can call the variable that tracks this variation
The second stage determines how Sales change with the movements of
This means that if we see Sales correlate with , there is reason to interpret this co-movement as the causal effect of Price
Two-Stage Least Square Regression
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For an assumed data-generating process:
Yi = α + β1X1i + β2X2i + … + βKXKi + Ui
Suppose X1 is endogenous and Z is a valid instrument for X1. We execute 2SLS, in the first stage we assume:
X1i = γ + δ1Zi + δ2X2i + … + δKXKi + Vi
Then regress X1 on Z, X2…,XK and calculate predicted values for X1, defined as:
= + 1Z + 2X2 + … + XK
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