Basic Methods for Establishing Causal Inference

Chapter 7

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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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