What is an instrumental variable in linear regression? How do instrumental variables improve causal inference?
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Also, provide a graduate-level response to each of the following questions:
- What is an instrumental variable in linear regression? How do instrumental variables improve causal inference? Please answer both of these questions and give an example of an instrumental variable.
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Advanced Methods for Establishing Causal Inference
Chapter 8
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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
Two-Stage Least Square Regression
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In the second stage, regress Y on , X2, …, XK
From the second stage regression, the estimated coefficient for is a consistent estimate for β1 (the causal effect of X1 on Y) and the estimated coefficient on X2 is a consistent estimate for β2
Run two consecutive regressions using the predictions from the first as an independent variable in the second
Statistical software combines this process into a single command
Two-Stage Least Square Regression
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2SLS Estimates for Y Regressed on X1, X2, and X3
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Summary of 2SLS where we have J endogenous variables and L J instrumental variables
Yi = α + β1X1i + β2X2i + … + βKXKi + Ui
Suppose X1, …, XJ are endogenous and Z1, …, ZL are valid instruments for X1, …, XJ
Execution of 2SLS proceeds as follows:
Two-Stage Least Square Regression
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Two-Stage Least Square Regression
Regress X1, …, XJ on Z1, …, ZK , XJ+1 , … XK in J separate regressions
Obtain predicted values , …, using the corresponding estimated regression equations in Step 1. This concludes “Stage 1”
Regress Y on , …, , XJ+1 , … XK , which yields consistent estimates for α, β1, …, βK. This is “Stage 2”
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An instrumental variable must be exogenous and relevant, and if so, we can use 2SLS to get consistent estimates for the parameters of the determining function
Can we assess whether the instrumental variable possesses these two characteristics?
Evaluating Instruments
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An instrumental variable is exogenous if it is uncorrelated with unobservables affecting the dependent variable
For a data-generating process Yi = α + β1X1i + … + βKXKi + Ui , an instrumental variable Z must have Corr(Z, U) = 0
To prove this, regress Y on X1,…..XK, and calculate the residuals as: ei = Yi – ‒ X1i ‒ … ‒ XKi
We could then calculate the sample correlation between Z and the residuals, believing this to be an estimate for the correlation between Z and U
Exogeneity
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The problem is that the residuals were calculated using a regression with an endogenous variable
Our parameter estimates are not consistent, meaning the sample correlation between Z and the residuals generally is not an estimator for the correlation between Z and U
If the number of instrumental variables is equal to the number of endogenous variables, there is no way to test for exogeneity
If the number of instrumental variables is greater than the number of endogenous variables, there are tests that can be performed to find evidence that at least some instrumental variables are not exogenous, but there is no way to test that all are exogenous
Exogeneity
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Testing for relevance is simple and can be added when conducting 2SLS
For a data-generating process: Yi = α + β1X1i + … + βKXKi + Ui where X1 is endogenous, Z is relevant if it is correlated with X1 after controlling for X1, …, XK
We can assess whether this is true by regressing X1 on Z, X2…,XK
Relevance
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Regression Output for Price Regressed on Income and Fuel Costs
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It is important to establish convincing evidence that an instrumental variable(s) is relevant
Doing so avoids common criticism of instrumental variables centered on the usage of weak instruments
A weak instrument is an instrumental variable that has little partial correlation with the endogenous variable whose causal effect on an outcome it is meant to measure
Relevance
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Regression Results for X1 Regressed on X2, X3,Z1, and Z2
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Regression Results for Y Regressed on , X2, and X3
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Classical Applications of Instrumental Variables for Business
Cost variables are popular choices as instrumental variables, particularly in demand estimations
Any variable that affects the costs of producing the good or service (input prices, cost per unit, etc.) can be to be a valid instrument for Price
Prices charged typically depend on costs
Cost variables are often both relevant and exogenous when used to instrument for Price in a demand equation
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Classical Applications of Instrumental Variables for Business
Policy change is another popular choice as an instrumental variable
Local sales tax and/or price regulations can serve as instrumental variables for Price in a demand equation
Labor laws can serve as instrumental variables for wages when seeking to measure the effect of wages on productivity
Policy changes often affect business decisions (making them relevant) but often occur for reasons not related to business outcomes (exogenous)
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With panel data we are able to observe the same cross-sectional unit multiple times at different points in time
Difference-in- difference regression
Fixed-effects model
Dummy variable estimation
Within estimation
Panel Data Method
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Consider an individual who owns a large number of liquor stores in the states of Indiana and Michigan
Suppose Indiana state government decides to increase the sales tax on liquor sales by 3%
The owner may want to know the effect of this tax increase on her profit
Difference-in-Differences
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To learn the effect of tax increase on the profit, the store owner collects data for two years as shown below:
Difference-in-Differences
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To assess the effect of a tax hike on profit, the store owner may assume the following data-generating process:
Profitsit = α + βTaxHikeit + Uit
Profitsit is the profit of store i during Year t, and TaxHikeit equals 1 if the 3% tax hike was in place for store i during Year t and 0 otherwise
We could regress Profits on TaxHike, but difficult to argue that TaxHike is not endogenous
TaxHike equals 1 for a specific group of stores at a specific time; this method of administering the treatment may be correlated with unobserved factors affecting Profits
Difference-in-Differences
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Control for a cross-sectional group (g = Indiana, Michigan) and for time (t = 2016, 2017)
Assume the following model:
Profitsigt = α + β1Indianag + β2Yeart + β3TaxHikegt Uigt
The data-generating process can also be written as:
Profitsigt = α + β1Indianag + β2Yeart + β3Indianag × Yeart + Uigt
Difference-in-Differences
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β3 is the diff-in-diff for profits in this example
Difference in profits between 2017 and 2016 for Indiana:
α + β1 + β2 + β3 + Uigt ‒ (α + β1 + Uigt)= β2 + β3
Difference in profits between 2017 and 2016 for Michigan:
α + β2 + Uigt ‒ (α + Uigt)= β2
Take the difference between the change in profits in Indiana and Michigan to get the diff-in-diff:
β2 + β3 ‒ β2 = β3
Difference-in-Differences
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Difference-in-Differences for Liquor Profits in Indiana and Michigan
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Difference-in-Differences
Difference-indifferences (diff-in-diff) is the difference in the temporal change for the outcome between the treated and untreated group
Diff-in-diff highly effective and applies for dichotomous treatments spanning two periods
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Fixed effects model is a data-generating process for panel data that includes controls for cross-sectional groups
The controls for cross-sectional groups are call fixed effects
For a data-generating process to be characterized as a fixed effects model, it need have only controls for the cross-sectional groups
Can control for time periods by including time trends
Outcomeigt = α+ δ2Group2g + … + δGGroupGg + γTimet + βTreatmentgt+ Uigt
The Fixed-Effects Model
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The Fixed-Effects Model
By controlling for the groups and periods, many possible confounding factors in the data-generating process are eliminated
Can add controls (Xigt’s) beyond the fixed effects and time dummies to help eliminate some of the remaining confounding factors
Two ways of estimating the fixed-effects model include: dummy variable estimation and within estimation
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Dummy variable estimation uses regression analysis to estimate all of the parameters in the fixed effects data-generating process
Regress the Outcome on dummy variables for each cross-sectional group (except the base unit), dummy variables for each period (except the base period), and the treatment
The Fixed-Effects Model: Dummy Variable Estimation
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Subset of Dummy Variable Estimation Results for Sales Regressed on Tax Rate
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The Fixed-Effects Model: Dummy Variable Estimation
Interpreting the table from the previous slide:
Each state coefficient measures the effect on a store’s profits of moving the store from the base state (State 1) to that alternative state, for a given year and tax rate
Each year coefficient measures the effect on a store’s profits of moving the store from the base year (Year 1) to that alternative year, for a given state and tax rate
The coefficient on Tax Rate measures the effect on a store’s profits of changing the Tax Rate, for a given state and year
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The Fixed-Effects Model: Within Estimation
Within estimation uses regression analysis of within-group differences in variables to estimate the parameters in the fixed effects data-generating process, except for those corresponding to the fixed effects (and the constant)
Eliminates the need to estimate the coefficient for each fixed effect
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The Fixed-Effects Model: Within Estimation
Outcomeigt = α+ δ2Group2g +…+ δGGroupGg + γTimet + Treatmentgt+ Uigt
We estimate the parameters γ2, …, γT, β via within estimation:
Determine the cross-sectional groups and calculate group-level means: = and =
Create new variables: Outcome*igt = Outcomeigt ‒ , Treatment*igt = Treatmentgt ‒
Regress Outcome* on Treatment* and the Period dummy variables
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Comparing Estimation Methods
Dummy variable estimation provides estimates for the fixed effects (the effects of switching groups on the outcome), whereas within estimation does not
For dummy variable estimation R-squared is often misleadingly high, suggesting a very strong fit
For within estimation, R-squared is more indicative that the variation in Treatment is explaining variation in the Outcome
Both estimation models eliminate confounding factors that are fixed across periods for the groups or are fixed across groups over time
Both estimation models could yield inaccurate estimates if there are unobserved factors that vary within a group over time
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