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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.

Chapter 8

Advanced Methods for Establishing Causal Inference

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Learning Objectives

1. Explain how instrumental variables can improve causal

inference in regression analysis

2. Execute two-state least square regression

3. Judge which type of variables may be used as instrumental

variables

4. Identify a difference-in-difference regression

5. Execute regression incorporating fixed effects

6. Distinguish the dummy variable approach from a within

estimator for a fixed effect regression model

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Instrumental Variables

Instrumental variables

â€¢ 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

â€¢ Can improve causal inference in regression analysis

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Instrumental Variables: An Example

â€¢ A firm attempting to determine how its sales depend on price

it charges for its product

â€¢ Beginning with a simple data-generating process:

Salesi = Î± + Î²1Pricei + Ui

â€¢ If local demand factor depends on local income, then local

income is a confounding factor:

Salesi = Î± + Î²1Pricei + Î²2Incomei + Ui

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Instrumental Variables: An Example

â€¢ Including income in the model removes local income as

confounding factor

â€¢ Does its inclusion ensure that no other confounding factors still

exist?

â€¢ Many possibilities may come to mind, including local

competition, market size, and market growth rate

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Instrumental Variables

â€¢ We may be unable to collect data on all confounding factors or

find suitable proxies

â€¢ Then we are unable to remove the endogeneity problem by

including controls and/or proxy variables

â€¢ A widely used method for measuring causality that can

circumvent this problem involves instrumental variables

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Instrumental Variables

â€¢ Suppose we know price differences across some of the stores

were solely due to differences in fuel costs

â€¢ 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

â€¢ 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

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Instrumental Variables: An Example

WHEN TWO LOCATIONS HAVE

DIFFERENT PRICES ONLY BECAUSE

THEIR FUEL COSTS DIFFER, ANY

DIFFERENCE IN SALES CAN BE

ATTRIBUTED TO PRICE, SINCE FUEL

COSTS DONâ€™T IMPACT SALES PER SE

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Instrumental Variables

â€¢ Suppose we have the following data-generating function:

Yi = Î± + Î²1X1i + Î²2X2i + â€¦ + Î²KXKi + Ui

â€¢ Variable Z is a valid instrument for Xi if Z is both exogenous and

relevant, if:

1. Exogenous: It has no effect on the outcome variable beyond the

combined effects of all variables in the determining function

(X1â€¦XK)

2. Relevant: For the assumed data-generating process, Z is relevant

as an instrumental variable if it is correlated with X1 after

controlling for X2â€¦.XK

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Two-Stage Least Square Regression

â€¢ 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

â€¢ The first stage of 2SLS determines the subset of variation

in Price that can attributed to changes in fuel costs; we

à·£

can call the variable that tracks this variation ð‘ƒð‘Ÿð‘–ð‘ð‘’

â€¢ The second stage determines how Sales change with the

à·£

movements of ð‘ƒð‘Ÿð‘–ð‘ð‘’

à·£ ,

â€¢ This means that if we see Sales correlate with ð‘ƒð‘Ÿð‘–ð‘ð‘’

there is reason to interpret this co-movement as the

causal effect of Price

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Two-Stage Least Square Regression

â€¢ For an assumed data-generating process:

Yi = Î± + Î²1X1i + Î²2X2i + â€¦ + Î²KXKi + Ui

â€¢ Suppose X1 is endogenous and Z is a valid instrument for X1.

We execute 2SLS, in the first stage we assume:

X1i = Î³ + Î´1Zi + Î´2X2i + â€¦ + Î´KXKi + Vi

â€¢ Then regress X1 on Z, X2â€¦,XK and calculate predicted values for

X1, defined as:

à· ð›¾+

ð‘‹=

à·œ ð›¿áˆ˜ 1Z + ð›¿áˆ˜ 2X2 + â€¦ + ð›¿áˆ˜ KXK

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Two-Stage Least Square Regression

à·¢1, X2, â€¦, XK

â€¢ In the second stage, regress Y on ð‘‹

â€¢ From the second stage regression, the estimated coefficient for

à·¢1 is a consistent estimate for Î²1 (the causal effect of X1 on Y)

ð‘‹

and the estimated coefficient on X2 is a consistent estimate for

Î²2

â€¢ Run two consecutive regressions using the predictions from

the first as an independent variable in the second

â€¢ Statistical software combines this process into a single command

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2SLS Estimates for Y Regressed on

X1, X2, and X3

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Two-Stage Least Square Regression

â€¢ Summary of 2SLS where we have J endogenous variables and L

â‰¥ J instrumental variables

Yi = Î± + Î²1X1i + Î²2X2i + â€¦ + Î²KXKi + Ui

Suppose X1, â€¦, XJ are endogenous and Z1, â€¦, ZL are valid

instruments for X1, â€¦, XJ

â€¢ Execution of 2SLS proceeds as follows:

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Two-Stage Least Square Regression

1. Regress X1, â€¦, XJ on Z1, â€¦, ZK , XJ+1 , â€¦ XK in J separate

regressions

à·¢1, â€¦, ð‘‹à·¡ð½ using the corresponding

2. Obtain predicted values ð‘‹

estimated regression equations in Step 1. This concludes

â€œStage 1â€

à·¢1, â€¦, ð‘‹à·¡ð½, XJ+1 , â€¦ XK , which yields consistent

3. Regress Y on ð‘‹

estimates for Î±, Î²1, â€¦, Î²K. This is â€œStage 2â€

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Evaluating Instruments

â€¢ 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

â€¢ Can we assess whether the instrumental variable possesses

these two characteristics?

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Exogeneity

â€¢ An instrumental variable is exogenous if it is uncorrelated with

unobservables affecting the dependent variable

â€¢ For a data-generating process Yi = Î± + Î²1X1i + â€¦ + Î²KXKi + Ui , an

instrumental variable Z must have Corr(Z, U) = 0

â€¢ To prove this, regress Y on X1,â€¦..XK, and calculate the residuals

à·¢1X1i â€’ â€¦ â€’ ð›½

à·¢ð¾XKi

as: ei = Yi â€“ à·ð›¼ â€’ ð›½

â€¢ 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

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Exogeneity

â€¢ The problem is that the residuals were calculated using a regression

with an endogenous variable

â€¢ 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

â€¢ If the number of instrumental variables is equal to the number of

endogenous variables, there is no way to test for exogeneity

â€¢ 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

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Relevance

â€¢ Testing for relevance is simple and can be added when

conducting 2SLS

â€¢ For a data-generating process: Yi = Î± + Î²1X1i + â€¦ + Î²KXKi + Ui

where X1 is endogenous, Z is relevant if it is correlated with X1

after controlling for X1, â€¦, XK

â€¢ We can assess whether this is true by regressing X1 on Z,

X2â€¦,XK

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Regression Output for Price Regressed on

Income and Fuel Costs

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Relevance

â€¢ It is important to establish convincing evidence that an

instrumental variable(s) is relevant

â€¢ Doing so avoids common criticism of instrumental variables

centered on the usage of weak instruments

â€¢ 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

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Regression Results for X1 Regressed on X2,

X3,Z1, and Z2

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à·¢ðŸ, X2,

Regression Results for Y Regressed on ð‘¿

and X3

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Classical Applications of Instrumental

Variables for Business

â€¢ Cost variables are popular choices as instrumental variables,

particularly in demand estimations

â€¢ 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

â€¢ Prices charged typically depend on costs

â€¢ Cost variables are often both relevant and exogenous when

used to instrument for Price in a demand equation

Â© 2019 McGraw-Hill Education.

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Classical Applications of Instrumental

Variables for Business

â€¢ Policy change is another popular choice as an instrumental

variable

â€¢ Local sales tax and/or price regulations can serve as

instrumental variables for Price in a demand equation

â€¢ Labor laws can serve as instrumental variables for wages when

seeking to measure the effect of wages on productivity

â€¢ Policy changes often affect business decisions (making them

relevant) but often occur for reasons not related to business

outcomes (exogenous)

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Panel Data Method

â€¢ With panel data we are able to observe the same crosssectional unit multiple times at different points in time

â€¢ Difference-in- difference regression

â€¢ Fixed-effects model

â€¢ Dummy variable estimation

â€¢ Within estimation

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Difference-in-Differences

â€¢ Consider an individual who owns a large number of liquor

stores in the states of Indiana and Michigan

â€¢ Suppose Indiana state government decides to increase the

sales tax on liquor sales by 3%

â€¢ The owner may want to know the effect of this tax increase on

her profit

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Difference-in-Differences

â€¢ To learn the effect of tax increase on the profit, the store

owner collects data for two years as shown below:

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Difference-in-Differences

â€¢ To assess the effect of a tax hike on profit, the store owner may

assume the following data-generating process:

Profitsit = Î± + Î²TaxHikeit + Uit

â€¢ 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

â€¢ We could regress Profits on TaxHike, but difficult to argue that

TaxHike is not endogenous

â€¢ 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

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Difference-in-Differences

â€¢ Control for a cross-sectional group (g = Indiana, Michigan) and

for time (t = 2016, 2017)

â€¢ Assume the following model:

Profitsigt = Î± + Î²1Indianag + Î²2Yeart + Î²3TaxHikegt Uigt

â€¢ The data-generating process can also be written as:

Profitsigt = Î± + Î²1Indianag + Î²2Yeart + Î²3Indianag Ã— Yeart + Uigt

Â© 2019 McGraw-Hill Education.

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Difference-in-Differences

â€¢ Î²3 is the diff-in-diff for profits in this example

â€¢ Difference in profits between 2017 and 2016 for Indiana:

Î± + Î²1 + Î²2 + Î²3 + Uigt â€’ (Î± + Î²1 + Uigt)= Î²2 + Î²3

â€¢ Difference in profits between 2017 and 2016 for Michigan:

Î± + Î²2 + Uigt â€’ (Î± + Uigt)= Î²2

â€¢ Take the difference between the change in profits in Indiana

and Michigan to get the diff-in-diff:

Î²2 + Î²3 â€’ Î²2 = Î²3

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Difference-in-Differences for Liquor Profits in

Indiana and Michigan

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Difference-in-Differences

â€¢ Difference-indifferences (diff-in-diff) is the difference in the

temporal change for the outcome between the treated and

untreated group

â€¢ Diff-in-diff highly effective and applies for dichotomous

treatments spanning two periods

Â© 2019 McGraw-Hill Education.

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The Fixed-Effects Model

â€¢ Fixed effects model is a data-generating process for panel data

that includes controls for cross-sectional groups

â€¢ The controls for cross-sectional groups are call fixed effects

â€¢ For a data-generating process to be characterized as a fixed

effects model, it need have only controls for the cross-sectional

groups

â€¢ Can control for time periods by including time trends

â€¢ Outcomeigt = Î±+ Î´2Group2g + â€¦ + Î´GGroupGg + Î³Timet +

Î²Treatmentgt+ Uigt

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The Fixed-Effects Model

â€¢ By controlling for the groups and periods, many possible

confounding factors in the data-generating process are

eliminated

â€¢ Can add controls (Xigtâ€™s) beyond the fixed effects and time

dummies to help eliminate some of the remaining confounding

factors

â€¢ Two ways of estimating the fixed-effects model include:

dummy variable estimation and within estimation

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The Fixed-Effects Model: Dummy Variable

Estimation

â€¢ Dummy variable estimation uses regression analysis to

estimate all of the parameters in the fixed effects datagenerating process

â€¢ Regress the Outcome on dummy variables for each crosssectional group (except the base unit), dummy variables for

each period (except the base period), and the treatment

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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

â€¢ Interpreting the table from the previous slide:

â€¢ Each state coefficient measures the effect on a storeâ€™s profits of

moving the store from the base state (State 1) to that

alternative state, for a given year and tax rate

â€¢ Each year coefficient measures the effect on a storeâ€™s profits of

moving the store from the base year (Year 1) to that alternative

year, for a given state and tax rate

â€¢ The coefficient on Tax Rate measures the effect on a storeâ€™s

profits of changing the Tax Rate, for a given state and year

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The Fixed-Effects Model: Within Estimation

â€¢ 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)

â€¢ Eliminates the need to estimate the coefficient for each fixed

effect

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The Fixed-Effects Model: Within Estimation

Outcomeigt = Î±+ Î´2Group2g +â€¦+ Î´GGroupGg + Î³Timet + Treatmentgt+ Uigt

â€¢ We estimate the parameters Î³2, â€¦, Î³T, Î² via within estimation:

1. Determine the cross-sectional groups and calculate group-level

1

ð‘ð‘” ð‘‡

Ïƒð‘–=ð‘– Ïƒð‘¡=ð‘– ð‘‚ð‘¢ð‘¡ð‘ð‘œð‘šð‘’ð‘–ð‘”ð‘¡ and ð‘‡ð‘Ÿð‘’ð‘Žð‘¡ð‘šð‘’ð‘›ð‘¡ =

means:

=

ð‘ ð‘‡

ð‘”

1

ð‘” Ïƒð‘‡

Ïƒð‘

ð‘‡ð‘Ÿð‘’ð‘Žð‘¡ð‘šð‘’ð‘›ð‘¡ð‘–ð‘”ð‘¡

ð‘ð‘” ð‘‡ ð‘–=ð‘– ð‘¡=ð‘–

2. Create new variables: Outcome*igt = Outcomeigt â€’ ð‘‚ð‘¢ð‘¡ð‘ð‘œð‘šð‘’ð‘” ,

Treatment*igt = Treatmentgt â€’ ð‘‡ð‘Ÿð‘’ð‘Žð‘¡ð‘šð‘’ð‘›ð‘¡ð‘”

3. Regress Outcome* on Treatment* and the Period dummy variables

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Comparing Estimation Methods

â€¢ Dummy variable estimation provides estimates for the fixed effects

(the effects of switching groups on the outcome), whereas within

estimation does not

â€¢ For dummy variable estimation R-squared is often misleadingly

high, suggesting a very strong fit

â€¢ For within estimation, R-squared is more indicative that the

variation in Treatment is explaining variation in the Outcome

â€¢ Both estimation models eliminate confounding factors that are

fixed across periods for the groups or are fixed across groups over

time

â€¢ Both estimation models could yield inaccurate estimates if there

are unobserved factors that vary within a group over time

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