Instrumental Variables Liang Dai

Regression For illustration purpose, I have been using binary treatment variables In reality, Di could be continuous. So we regress Yi on Di To control for factors affecting Yi other than Di, we add in control variables, i.e. regress Yi on Di and these variables What if we can’t observe all these factors?

Three commonly used methods Instrumental variables (IV) Difference-in-Difference (DID) Regression Discontinuity (RD) Today: IV

Alarm clock example again Alarm clock users Non-users Treatment (alarm clock used): Di 1 Average outcome: Yi 8 7 We already know that E[Y1i|Di=1]-E[Y0i|Di=0] is a biased estimator

The lottery Assume we know that, in addition, a promotion of alarm clocks had been made to this class before: The class tossed a fair coin one by one, whoever got a head won an alarm clock for free (Zi=1, and 0 if lost). Call Zi instrument Assume the lottery has a causal effect on the usage of alarm clocks: E[Di|Zi=1]=0.9, E[Di|Zi=0]=0.1 Also, Zi is independent of the omitted variables (independence assumption) Zi affects Yi only through Di, not directly (exclusion assumption) A valid instrument has to meet all these three assumptions. Zi captures the exogenous part of the endogenous variable Di. Independence assumption guarantees Zi is an “experiment” on Yi, and thus rules out endogeneity.

The graph Why causality is established? The magnitude of treatment effect Why reverse causality and selection bias are resolved?

Treatment effect Alarm clock winners losers Treatment (alarm clock won): Zi 1 Average outcome: Yi 6 8 Causal effect of winning on wake-up time (ρ)=Causal effect of winning on alarm clock usage (φ) * causal effect of alarm clock usage on wake-up time (λ) We use a two-step estimation: The first stage: Causal effect of winning on alarm-clock usage: φ=E[Di|Zi=1]-E[Di|Zi=0]=0.9-0.1=0.8 Causal effect of winning on wake-up time: ρ=E[Y1i|Zi=1]-E[Y0i|Zi=0]=6-8=-2. This is called reduced form. Then, causal effect of alarm clock usage on wake-up time λ=ρ/φ=(-2)/0.8=-2.5. This is called local average treatment effect (LATE)

LATE != ToT IV works only if it affects treatment, thus it estimates the treatment effect on the compliers, i.e., those who use the alarm clocks only if assigned one; It says nothing about those who always use clocks or those who never use clocks, whether or not assigned one. Treatment effect on the Treated (ToT) is the treatment effect on both the treated compliers and always-takers.

Two-Step Least Square (2SLS) 2SLS is a generalization of the previous 2-step estimation to the case where Treatment (Di) takes multiple values, there are control variables, and more than one instrumental variables First stage: regress Di on IV’s, and obtain predicted value of Di Second stage: regress Yi on predicted value of Di and controls

Example 1 Angrist & Krueger, QJE 91 Does compulsory school attendance affects schooling and earnings? Treatment: years of compulsory schooling Outcome: years of schooling, earnings Challenge: ?

Example 1 Angrist & Krueger, QJE 91 Does compulsory school attendance affects schooling and earnings? Treatment: years of compulsory schooling Outcome: years of schooling, earnings Challenge: Hard to generate variation in compulsory school attendance without selection bias

The IV birthday 1. Causal impact on treatment: Students required to attend school if turned 6 y.o. by Jan 1 of the year; Born 2000/12/31-> start schooling in 2007 Born 2001/1/2->starting schooling in 2008 Born in early month=>start schooling at older age

The IV (cont.) 1. causal impact on treatment Students can’t drop out until reaching 16 y.o. Born in early month=>can drop out with fewer years of schooling Birthday->years of compulsory education

The IV (cont.) 2. independence: birthday unlikely correlated with personal attributes other than age at school entry 3. exclusion: birthday does not affect years of schooling and earnings directly Data: three decennial censuses

Example 2 Bennedsen et al, QJE 07 Does the decision to appoint a family or external CEO affect performance of family firms? Theoretically ambiguous: Family CEO could be beneficial, due to nonmonetary rewards, firm-specific knowledge, trust from key shareholders, longer term horizon, etc Family CEO could hurt firm, due to tension between family and business objectives, smaller pool for selection Empirical challenge: ?

Example 2 Bennedsen et al, QJE 07 Does the decision to appoint a family or external CEO affect performance of family firms? Theoretically ambiguous: Family CEO could be beneficial, due to nonmonetary rewards, firm-specific knowledge, trust from key shareholders, longer term horizon, etc Family CEO could hurt firm, due to tension between family and business objectives, smaller pool for selection Selection bias: appointment correlates with unobservable firm characteristics, e.g. family member saver of the firm, endogenously weak board, anti-takeover provisions.

IV Gender of departing CEO’s first child 1. Causal impact on Treatment Male first-child firms are more likely to appoint family CEO 2. Independence assumption >80% of first-child births happened before 1980, when technology to detect gender of children was not available Literature shows no evidence of “missing women” in Denmark 3. Exclusion assumption Gender of first-child determined several decades ago does not directly affect firm performance today Is family size a valid IV?

First stage

Reduced form

OLS vs IV

Example 3 Duranton & Turner, AER 11 “Fundamental Law of Road Congestion” Traffic, as measured by vehicle-kilometers travelled (VKT), increases one for one with provision of roads. Thus, increase of provision of roads will NOT ease congestion. Empirical evidence?

This paper: Fundamental Law of HIGHWAY Congestion Subjects: metropolitan statistical areas (MSA) Treatment: highway provision Outcome: VKT Challenge: ?

IVs Planned interstate highway kilometers from the 1947 highway plan; 1898 railroad route plan; Incidence of major expeditions of exploration between 1835 and 1850. Why valid?