The Plan for Day Two

1
The Plan for Day Two

• Practice and pitfalls
• (1) Natural experiments as interesting sources of
  instrumental variables
• (2) The consequences of weak instruments for
  causal inference
• (3) Some useful IV diagnostics
• (4) Walk through an empirical application
• Goal provide concrete examples of instrumental
  variables methods

2
Instrumental Variables and Natural Experiments

• What is a natural experiment?
• situations where the forces of nature or
  government policy have conspired to produce an
  environment somewhat akin to a randomized
  experiment
• Angrist and Krueger (2001, p. 73)
• Natural experiments can provide a useful source
  of exogenous variation in problematic regressors
• But they require detailed institutional knowledge

3
Instrumental Variables and Natural Experiments

• Some natural experiments in economics
• Existing policy differences, or changes that
  affect some jurisdictions (or groups) but not
  others
• Minimum wage rate
• Excise taxes on consumer goods
• Unemployment insurance, workers compensation
• Unexpected shocks to the local economy
• Coal prices and the Middle East oil embargo
  (1973)
• Agricultural production and adverse weather events

4
Instrumental Variables and Natural Experiments

• Some potential pitfalls
• Not all policy differences/changes are exogenous
• Political factors and past realizations of the
  response variable can affect existing policies or
  policy changes
• Generalizability of causal effect estimates
• Results may not generalize beyond the units under
  study
• Heterogeneity in causal effects
• Results may be sensitive to the natural
  experiment chosen in a specific study (L.A.T.E.)

5
Instrumental Variables and Natural Experiments

• Some natural experiments of criminological
  interest
• Levitt (1996) prison population ? crime rate
• Levitt (1997) police hiring ? crime rate
• Apel et al. (2008) youth employment ?
  delinquency
• Some natural experiments not of criminological
  interest, but interesting nonetheless
• Angrist and Evans (1998) fertility ? labor
  supply

6
Levitt (1996), Q.J.E.

• Large decline in crime did not accompany the
  large increase in prison population (1971-1993)
• Prima fascia evidence of prison ineffectiveness
• But...increased prison use could mask what would
  have been a greater increase in crime
• Underlying determinants of crime probably
  worsened
• And...prison population probably responded to
  crime increase

7
Levitt (1996), Q.J.E.

• Prison overcrowding legislation
• Population caps, prohibition of double celling
• In 12 states, the entire prison system came under
  court control
• AL, AK, AR, DE, FL, MS, NM, OK, RI, SC, TN, TX
• Relationship between legislation and prisons
• Prior to filing, prison growth outpaced national
  average by 2.3 percent
• After filing, prison growth was 5.1 percent slower

8
Levitt (1996), Q.J.E.

• Logic of the instrumental variable in this study
• Court rulings concerning prison capacity cannot
  be correlated with the unobserved determinants of
  crime rate changes
• Or...the only reason court rulings are related to
  crime is because they limit prison population
  growth

9
Levitt (1996), Q.J.E.

• 2SLS model yields a prison effect on crime at
  least four times as high as the LS model
• Violent crime rate
• bLS .099 (s.e. .033)
• bIV .424 (s.e. .201)
• Property crime rate
• bLS .071 (s.e. .019)
• bIV .321 (s.e. .138)
• A 10 increase in prison size produces a 4.2
  decrease in violent crime and a 3.2 decrease in
  property crime

10
Levitt (1996), Q.J.E.

• L.A.T.E. effect of prison growth on crime among
  states under court order to slow growth
• Some relevant observations
• Generalizability predominately Southern states
• Large prison populations, unusually fast prison
  growth
• T.E. heterogeneity (slowed) prison growth due
  to court-ordered prison reductions may be
  differentially related to crime rates
• Other IVs could lead to different causal effect
  estimates

11
Levitt (1997), A.E.R.

• Breaking the simultaneity in the police-crime
  connection
• When more police are hired, crime should decline
• But...more police may be hired during crime waves
• Election cycles and police hiring
• Increases in size of police force
  disproportionately concentrated in election years
• Growth is 2.1 in mayoral election years, 2.0 in
  gubernatorial election years, and 0.0 in
  non-election years

12
Levitt (1997), A.E.R.

• However...can election cycles affect crime rates
  through other spending channels?
• Ex., education, welfare, unemployment benefits
• If so, all of these other indirect channels must
  be netted out

13
Levitt (1997), A.E.R.
Reduced-form coefficients
First-stage coefficients
14
Levitt (1997), A.E.R.

• Comparative estimates of the effect of police
  manpower on city crime rates
• Violent crime rate
• Levels bLS .28 (s.e. .05)
• Changes bLS .27 (s.e. .06)
• Changes bIV 1.39 (s.e. .55)
• Property crime rate
• Levels bLS .21 (s.e. .05)
• Changes bLS .23 (s.e. .09)
• Changes bIV .38 (s.e. .83)

15
Levitt (1997), A.E.R.

• Follow-up instrumental variables studies of the
  police-crime relationship in the U.S.
• Levitt (2002) Number of firefighters
• Klick and Tabarrok (2005) Washington, DC,
  terrorism alert levels post-9/11
• Evans and Owens (2007) Grants from the federal
  Office of C.O.P.S.
• These findings basically replicated those from
  Levitts (1997) original study

16
Apel et al. (2008), J.Q.C.

• What effect does working have on adolescent
  behavior?
• Prior research suggests the consequences of work
  are uniformly negative
• Focus on work intensity rather than work per se
• Youth Worker Protection Act
• Problem of non-random selection into youth labor
  market
• Especially pronounced for high-intensity workers

17
Apel et al. (2008), J.Q.C.

• Something interesting happens at age 16
• Youth work is no longer governed by the federal
  Fair Labor Standards Act (F.L.S.A.)

18
Apel et al. (2008), J.Q.C.

• F.L.S.A. governs employment of all 15 year olds
  during the school year
• No work past 700 pm
• Maximum 3 hours/day and 18 hours/week
• But, F.L.S.A. expires for 16 year olds
• And...every state has its own law governing
  16-year-old employment
• Thus, youth age into less restrictive regimes
  that vary across jurisdictions

19
Apel et al. (2008), J.Q.C.

• Change in work intensity at 15-16 transition
  among 15-year-old non-workers

Magnitude of change is an increasing function of
the number of hours allowed at age 16
20
Apel et al. (2008), J.Q.C.
State Child Labor Law Model 1 Model 2 Model 3
Hours per Week 0.32 (.05)
No Limit 11.43 (1.6)
Hours per Weekday 1.19 (.19)
No Limit 9.37 (1.3)
Work Curfew 2.19 (.27)
No Limit 23.83 (2.7)

R-square .400 .401 .409
?R-square with IVs .014 .015 .023
Partial R-square for IVs .023 .025 .037
F-test for IVs 26.2 28.3 41.9
Approx. relative bias .000 .000 .000
21
Apel et al. (2008), J.Q.C.

• A 20-hour increase in the number of hours worked
  per week reduces the variety of delinquent
  behavior by 0.47 (.0233?20)

22
Angrist and Krueger (1991), J.L.E.

• Returns to education (Y wages)
• Problem of omitted ability bias
• Years of schooling vary by quarter of birth
• Compulsory schooling laws, age-at-entry rules
• Someone born in Q1 is a little older and will be
  able to drop out sooner than someone born in Q4
• Q.O.B. can be treated as a useful source of
  exogeneity in schooling

23
Angrist and Krueger (1991), J.L.E.

• People born in Q1 do obtain less schooling
• But pay close attention to the scale of the
  y-axis
• Mean difference between Q1 and Q4 is only 0.124,
  or 1.5 months
• So...need large N since R2X,Z will be very small
• AK had over 300k for the 1930-39 cohort

24
Angrist and Krueger (1991), J.L.E.

• Final 2SLS model interacted QOB with year of
  birth (30), state of birth (150)
• OLS b .0628 (s.e. .0003)
• 2SLS b .0811 (s.e. .0109)
• Least squares estimate does not appear to be
  badly biased by omitted variables
• But...replication effort identified some pitfalls
  in this analysis that are instructive

25
Bound, Jaeger, and Baker (1995), J.A.S.A.

• Potential problems with QOB as an IV
• Correlation between QOB and schooling is weak
• Small Cov(X,Z) introduces finite-sample bias,
  which will be exacerbated with the inclusion of
  many IVs
• QOB may not be completely exogenous
• Even small Cov(Z,e) will cause inconsistency, and
  this will be exacerbated when Cov(X,Z) is small
• QOB qualifies as a weak instrument that may be
  correlated with unobserved determinants of wages
  (e.g., family income)

26
Bound, Jaeger, and Baker (1995), J.A.S.A.

• Even if the instrument is good, matters can be
  made far worse with IV as opposed to LS
• Weak correlation between IV and endogenous
  regressor can pose severe finite-sample bias
• Andreally large samples wont help, especially
  if there is even weak endogeneity between IV and
  error
• First-stage diagnostics provide a sense of how
  good an IV is in a given setting
• F-test and partial-R2 on IVs

27
Useful Diagnostic Tools for IV Models

• Tests of instrument relevance
• Weak IVs ? Large variance of bIV as well as
  potentially severe finite-sample bias
• Tests of instrument exogeneity
• Endogenous IVs ? Inconsistency of bIV that makes
  it no better (and probably worse) than bLS
• Durbin-Wu-Hausman test
• Endogeneity of the problem regressor(s)

28
Tests of Instrument Relevance

• Diagnostics based on the F-test for the joint
  significance of the IVs
• Nelson and Startz (1990) Staiger and Stock
  (1997)
• Bound, Jaeger, and Baker (1995)
• Partial R-square for the IVs
• Shea (1997)
• There is a growing econometric literature on the
  weak instrument problem

29
Tests of Instrument Exogeneity

• Model must be overidentified, i.e., more IVs
  than endogenous Xs
• H0 All IVs uncorrelated with structural error
• Overidentification test
• 1. Estimate structural model
• 2. Regress IV residuals on all exogenous
  variables
• 3. Compute N?R2 and compare to chi-square
• df IVs endogenous Xs

30
Durbin-Wu-Hausman (DWH) Test

• Balances the consistency of IV against the
  efficiency of LS
• H0 IV and LS both consistent, but LS is
  efficient
• H1 Only IV is consistent
• DWH test for a single endogenous regressor
• DWH (bIV bLS) / v(s2bIV s2bLS) N(0,1)
• If DWH gt 1.96, then X is endogenous and IV is
  the preferred estimator despite its inefficiency

31
Durbin-Wu-Hausman (DWH) Test

• A roughly equivalent procedure for DWH
• 1. Estimate the first-stage model
• 2. Include the first-stage residual in the
  structural model along with the endogenous X
• 3. Test for significance of the coefficient on
  residual
• Note Coefficient on endogenous X in this model
  is bIV (standard error is smaller, though)
• First-stage residual is a generated regressor

32
Software Considerations

• I have a strong preference for Stata
• Classic routine (-ivreg-) as well as a
  user-written one with a lot more diagnostic
  capability (-ivreg2-)
• Non-linear models -ivprobit- and -ivtobit-
• Panel models -xtivreg- and -xtivreg2-
• Useful post-estimation routines
• Overidentification -overid-
• Endogeneity of X in LS model -ivendog-
• Heteroscedasticity -ivhettest-

33
Software Considerations

• Basic model specification in Stata
• ivreg y (x z) w weight wtvar, options
• y dependent variable
• x endogenous variable
• z instrumental variable
• w control variable(s)
• Useful options first, ffirst, robust,
  cluster(varname)

34
Software Considerations

• For SAS users Proc Syslin (SAS/ETS)
• Basic command
• proc syslin datadataset 2sls options1
• endogenous x
• instruments z w
• model y x w / options2
• weight wtvar
• run
• Useful options1 first
• Useful options2 overid

35
Software Considerations

• For SPSS users 2SLS
• Basic command
• 2sls y with x w
• / instruments z w
• / constant.
• For point-and-click aficionados
• Analyze ? Regression ? Two-Stage Least Squares
• DEPENDENT, EXPLANATORY, and INSTRUMENTAL

36
Software Considerations

• For Limdep users 2SLS
• Basic command
• 2SLS Lhs y
• Rhs one, x, w
• Inst one, z, w
• Wts wtvar
• Dfc

37
Application Adolescent Work and Delinquent
Behavior

• Prior research shows a positive correlation
  between teenage work and delinquency
• Reasons to suspect serious endogeneity bias
• 2nd wave of the NLSY97 (N 8,368)
• Y 1 if committed delinquent act (31.9)
• X 1 if worked in a formal job (52.6)
• Z1 1 if child labor law allows 40 hours
  (14.2)
• Z2 1 if no child labor restriction in place
  (39.6)

38
Regression Model Ignoring Endogeneity

• . reg pcrime work if nomiss1 wave2
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 1, 8366) 6.33
• Model 1.37395379 1 1.37395379
  Prob gt F 0.0119
• Residual 1815.97786 8366 .217066443
  R-squared 0.0008
• -------------------------------------------
  Adj R-squared 0.0006
• Total 1817.35182 8367 .217204711
  Root MSE .4659
• --------------------------------------------------
  ----------------------------
• pcrime Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• work .0256633 .0102005 2.52
  0.012 .0056677 .0456588
• _cons .3053242 .0074009 41.26
  0.000 .2908167 .3198318
• --------------------------------------------------
  ----------------------------
• Teenage workers significantly more delinquent
• Modest effect but consistent with prior research

39
First-Stage Model

• . reg work law40 nolaw if nomiss1 wave2
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 2, 8365) 626.64
• Model 271.829722 2 135.914861
  Prob gt F 0.0000
• Residual 1814.33364 8365 .216895832
  R-squared 0.1303
• -------------------------------------------
  Adj R-squared 0.1301
• Total 2086.16336 8367 .249332301
  Root MSE .46572
• --------------------------------------------------
  ----------------------------
• work Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• law40 .0688902 .0154383 4.46
  0.000 .0386274 .099153
• nolaw .3818684 .0110273 34.63
  0.000 .3602521 .4034847
• _cons .3655636 .0074883 48.82
  0.000 .3508847 .3802425
• --------------------------------------------------
  ----------------------------
• State child labor laws affect probability of work
• This is a really strong first stage (F, R2)

40
Two-Stage Least Squares Model

• . ivreg pcrime (work law40 nolaw) if nomiss1
  wave2
• Instrumental variables (2SLS) regression
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 1, 8366) 6.86
• Model -19.5287923 1 -19.5287923
  Prob gt F 0.0088
• Residual 1836.88061 8366 .219564978
  R-squared .
• -------------------------------------------
  Adj R-squared .
• Total 1817.35182 8367 .217204711
  Root MSE .46858
• --------------------------------------------------
  ----------------------------
• pcrime Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• work -.0744352 .0284206 -2.62
  0.009 -.1301466 -.0187238
• _cons .3580171 .0158135 22.64
  0.000 .3270187 .3890155
• --------------------------------------------------
  ----------------------------
• Instrumented work
• Instruments law40 nolaw

41
What Do the Models Suggest Thus Far?

• Completely different conclusions!
• OLS Teenage work is criminogenic (b .026)
• Delinquency risk increases by 8.5 percent (base
  .305)
• 2SLS Teenage work is prophylactic (b .074)
• Delinquency risk decreases by 20.7 percent (base
  .358)
• Which model should we believe?
• We still have some additional diagnostic work to
  do to evaluate the 2SLS model
• Overidentification test, Hausman test

42
Regression-Based Overidentification Test

• . reg IVresid law40 nolaw if nomiss1 wave2
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 2, 8365) 0.25
• Model .111648085 2 .055824043
  Prob gt F 0.7755
• Residual 1836.76895 8365 .219577878
  R-squared 0.0001
• -------------------------------------------
  Adj R-squared -0.0002
• Total 1836.8806 8367 .219538735
  Root MSE .46859
• --------------------------------------------------
  ----------------------------
• IVresid Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• law40 .010988 .0155334 0.71
  0.479 -.0194613 .0414374
• nolaw .0016436 .0110953 0.15
  0.882 -.020106 .0233931
• _cons -.0022127 .0075344 -0.29
  0.769 -.0169821 .0125567
• --------------------------------------------------
  ----------------------------
• Overidentification test 8,368 .0001 .8368
  ?2(1)

43
Overidentification Test from the Software

• . overid
• Tests of overidentifying restrictions
• Sargan NR-sq test 0.509 Chi-sq(1)
  P-value 0.4757
• Basmann test 0.508 Chi-sq(1)
  P-value 0.4758
• IVs jointly pass the exogeneity requirement
• Notice that -overid- provides a global test,
  whereas the regression-based approach allows you
  to test the IVs jointly as well as individually

44
Durbin-Wu-Hausman (DWH) Test Estimated by Hand

• Summary coefficients
• OLS model b .026, s.e. .010
• 2SLS model b .074, s.e. .028
• Notice the size of the 2SLS standard error
• DWH (.074 .026) / v(.0282 .0102) 3.82
• CONCLUSION Least squares estimate of the work
  effect is biased and inconsistent
• The 2SLS estimate is preferred

45
Regression-Based DWH Test

• . reg pcrime work FSresid if nomiss1 wave2
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 2, 8365) 10.40
• Model 4.50567523 2 2.25283761
  Prob gt F 0.0000
• Residual 1812.84614 8365 .216718009
  R-squared 0.0025
• -------------------------------------------
  Adj R-squared 0.0022
• Total 1817.35182 8367 .217204711
  Root MSE .46553
• --------------------------------------------------
  ----------------------------
• pcrime Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• work -.0744352 .0282357 -2.64
  0.008 -.1297842 -.0190862
• FSresid .1150956 .0302771 3.80
  0.000 .0557449 .1744462
• _cons .3580171 .0157106 22.79
  0.000 .3272204 .3888139
• --------------------------------------------------
  ----------------------------
• Coeff. on work is bIV, while t-test on FSresid is
  DWH
• Standard error for work is underestimated, though

46
Or Just Let the Software Give You the DWH Test

• . ivendog
• Tests of endogeneity of work
• H0 Regressor is exogenous
• Wu-Hausman F test 14.45067
  F(1,8365) P-value 0.00014
• Durbin-Wu-Hausman chi-sq test 14.43093
  Chi-sq(1) P-value 0.00015
• Notice that -ivendog- provides a chi-square test
  for DWH, but the z-test that we computed by hand
  is easily recovered
• v(?2) z ? v(14.43) 3.80

47
Alternative Specifications for the
Work-Delinquency Association

• IV probit model
• Without IVs b .072 (s.e. .029)
• With IVs b .207 (s.e. .078)
• Continuous work hours
• Without IVs b .0015 (s.e. .0003)
• With IVs b .0024 (s.e. .0009)
• Indicator for intensive work (gt20 hours)
• Without IVs b .043 (s.e. .012)
• With IVs b .095 (s.e. .036)

48
Alternative Specifications for the
Work-Delinquency Association

• Control variables gender, race, child, dropout,
  family structure, family size, urbanicity,
  dwelling, school suspension, unemployment rate,
  mobility
• Binary work status
• Without IVs b .013 (s.e. .010)
• With IVs b .061 (s.e. .029)
• Continuous work hours
• Without IVs b .0007 (s.e. .0003)
• With IVs b .0023 (s.e. .0010)
• Intensive work indicator
• Without IVs b .020 (s.e. .012)
• With IVs b .085 (s.e. .040)

49
So Where Do We Stand with the Work-Delinquency
Question?

• Are child labor laws correlated with work?
• YES first-stage F is large
• Are child labor laws good IVs?
• YES overidentification test is not rejected
• Is teenage work endogenous?
• YES Hausman test is rejected
• Prior research findings that teenage work is
  criminogenic are selection artifacts

50
Stata Commands for the Foregoing Example

• Regression model ignoring endogeneity
• reg y x w
• First-stage regression model
• reg x z1 z2 w
• With controls and multiple IVs, test relevance
• test z1 z2
• 2SLS regression model
• ivreg y (x z1 z2) w

51
Stata Commands for the Foregoing Example

• Manual post hoc commands
• Get residuals for regression-based overid. test
• After 2SLS model predict IVresid if e(sample),
  resid
• Then reg IVresid z1 z2
• Get residuals for regression-based DWH test
• After first-stage model predict FSresid if
  e(sample), resid
• Then reg y x w FSresid
• Canned post hoc commands
• After 2SLS model overid and ivendog

52
NowWhat Happens if I Throw in a Potentially
Bogus Instrument?

• Now there are three instrumental variables
• Z1 1 if child labor law allows 40 hours
  (14.2)
• Z2 1 if no child labor restriction in place
  (39.6)
• Z3 1 if high unemployment rate in county
  (20.1)
• A little more difficult to tell a convincing
  story that the unemployment rate is only related
  to delinquency through work experience
• But lets see what happens

53
First-Stage Model

• . reg work law40 nolaw highun if nomiss1
  wave2
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 3, 8364) 427.28
• Model 277.229696 3 92.4098987
  Prob gt F 0.0000
• Residual 1808.93366 8364 .216276144
  R-squared 0.1329
• -------------------------------------------
  Adj R-squared 0.1326
• Total 2086.16336 8367 .249332301
  Root MSE .46505
• --------------------------------------------------
  ----------------------------
• work Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• law40 .0636421 .0154519 4.12
  0.000 .0333525 .0939317
• nolaw .3775975 .0110447 34.19
  0.000 .3559472 .3992479
• highun -.0636009 .0127283 -5.00
  0.000 -.0885517 -.0386502
• _cons .3808061 .0080759 47.15
  0.000 .3649754 .3966368
• --------------------------------------------------
  ----------------------------
• So far so good and consistent with expectation

54
Two-Stage Least Squares Model

• . ivreg pcrime (work law40 nolaw highun) if
  nomiss1 wave2
• Instrumental variables (2SLS) regression
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 1, 8366) 5.47
• Model -16.0635514 1 -16.0635514
  Prob gt F 0.0194
• Residual 1833.41537 8366 .219150773
  R-squared .
• -------------------------------------------
  Adj R-squared .
• Total 1817.35182 8367 .217204711
  Root MSE .46814
• --------------------------------------------------
  ----------------------------
• pcrime Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• work -.0657624 .0281159 -2.34
  0.019 -.1208765 -.0106483
• _cons .3534516 .0156602 22.57
  0.000 .3227537 .3841496
• --------------------------------------------------
  ----------------------------
• Instrumented work
• Instruments law40 nolaw highun

55
Post-Hoc Diagnostics

• . overid
• Tests of overidentifying restrictions
• Sargan NR-sq test 5.301 Chi-sq(2)
  P-value 0.0706
• Basmann test 5.301 Chi-sq(2)
  P-value 0.0706
• . ivendog
• Tests of endogeneity of work
• H0 Regressor is exogenous
• Wu-Hausman F test 12.32811
  F(1,8365) P-value 0.00045
• Durbin-Wu-Hausman chi-sq test 12.31438
  Chi-sq(1) P-value 0.00045
• Overidentification gives cause for concern
• The p-value shouldnt be anywhere near 0.05

56
Follow-Up Overidentification Test

• . reg IVresid law40 nolaw highun
• Source SS df MS
  Number of obs 8368
• -------------------------------------------
  F( 3, 8364) 1.77
• Model 1.1613555 3 .387118499
  Prob gt F 0.1511
• Residual 1832.25406 8364 .21906433
  R-squared 0.0006
• -------------------------------------------
  Adj R-squared 0.0003
• Total 1833.41541 8367 .219124586
  Root MSE .46804
• --------------------------------------------------
  ----------------------------
• IVresid Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• law40 .0080993 .0155512 0.52
  0.603 -.0223849 .0385836
• nolaw -.0035329 .0111156 -0.32
  0.751 -.0253223 .0182565
• highun -.0277671 .0128101 -2.17
  0.030 -.0528781 -.0026561
• _cons .0058369 .0081277 0.72
  0.473 -.0100955 .0217693
• --------------------------------------------------
  ----------------------------
• Okayunemployment rate is problematic as IV

57
Conclusion from Diagnostic Tests

• 2SLS work effect is similar
• Without unemployment, b .074 (s.e. .028)
• With unemployment, b .066 (s.e. .028)
• Butthe second model is invalidated because the
  unemployment rate is not exogenous
• If affects criminality through other channels
• We need to control for all other indirect
  pathways, or
• It should not be used as an IV at all

58
Closing Comments about Instrumental Variables
Studies

• In general, a lagged value of the endogenous
  regressor is not a good instrument
• Traditional structural equation model uses lagged
  values of X and Y as instruments to break the
  simultaneity between the current values of X and Y

These models impose the awfully strong
assumption that lagged values of X and Y only
affect the outcomes through current values
59
Closing Comments about Instrumental Variables
Studies

• Good IV models are generally interesting in their
  own right, and should not be treated as tack on
  analyses
• Practice varies widely across disciplines
• Some researchers write papers about their
  discovery and application of a clever IV for
  some problem
• Other researchers tack on IV models at the end
  of their analysis, often poorly, as a way to
  convince readers that their results are robust

60
Rules for Good Practice with Instrumental
Variables Models

• IV models can be very informative, but its your
  job to convince your audience
• Show the first-stage model diagnostics
• Even the most clever IV might not be sufficiently
  strongly related to X to be a useful source of
  identification
• Report test(s) of overidentifying restrictions
• An invalid IV is often worse than no IV at all
• Report LS endogeneity (DWH) test

61
Rules for Good Practice with Instrumental
Variables Models

• Most importantly, TELL A STORY about why a
  particular IV is a good instrument
• Something to consider when thinking about whether
  a particular IV is good
• Does the IV, for all intents and purposes,
  randomize the endogenous regressor?

62
Other Interesting IV Topics I Just Dont Have
Time to Cover

• 2SLS with a continuous treatment
• Instrumental variables for sample selectivity
• Generalized method of moments (IV-GMM)
• Non-linear two-stage least squares (N2SLS)
• Two-sample instrumental variables (TSIV)
• Fixed-effects instrumental variables (FEIV)
• Dynamic panel data estimators

63
References

• Angrist. (2006). Instrumental variables methods
  in experimental criminology research What, why
  and how. Journal of Experimental Criminology, 2,
  23-44.
• Angrist Evans. (1998). Children and their
  parents labor supply Evidence from exogenous
  variation in family size. American Economic
  Review, 88, 450-477.
• Angrist Krueger. (1991). Does compulsory school
  attendance affect schooling and earnings.
  Quarterly Journal of Economics, 106, 979-1014.
• Angrist Krueger. (2001). Instrumental variables
  and the search for identification From supply
  and demand to natural experiments. Journal of
  Economic Perspectives, 15, 69-85.
• Apel, Bushway, Paternoster, Brame Sweeten.
  (2008). Using state child labor laws to identify
  the causal effect of youth employment on deviant
  behavior and academic achievement. Journal of
  Quantitative Criminology, 24, 337-362.

64
References

• Bound, Jaeger Baker. (1995). Problems with
  instrumental variables estimation when the
  correlation between the instruments and the
  endogenous explanatory variables is weak. Journal
  of the American Statistical Association, 90,
  443-450.
• Evans Owens. (2007). COPS and crime. Journal of
  Public Economics, 91, 181-201.
• Imbens Angrist. (1994). Identification and
  estimation of local average treatment effects.
  Econometrica, 62, 467-475.
• Kelejian. (1971). Two-stage least squares and
  econometric systems linear in parameters but
  nonlinear in the endogenous variable. Journal of
  the American Statistical Association, 66,
  373-374.
• Klick Tabarrok. (2005). Using terror alert
  levels to estimate the effect of police on crime.
  Journal of Law Economics, 48, 267-279.
• Levitt. (1996). The effect of prison population
  size on crime rates Evidence from prison
  overcrowding litigation. Quarterly Journal of
  Economics, 111, 319-351.
• Levitt. (1997). Using electoral cycles in police
  hiring to estimate the effect of police on crime.
  American Economic Review, 87, 270-290.

65
References

• Levitt. (2002). Using electoral cycles in police
  hiring to estimate the effect of police on crime
  Reply. American Economic Review, 92, 1244-1250.
• Nelson and Startz. (1990). The distribution of
  the instrumental variables estimator and its
  t-ratio when the instrument is a poor one.
  Journal of Business, 63, S125-S140.
• Permutt Hebel. (1989). Simultaneous-equation
  estimation in a clinical trial of the effect of
  smoking on birth weight. Biometrics, 45, 619-622.
• Sexton Hebel. (1984). A clinical trial of
  change in maternal smoking and its effect on
  birth weight. Journal of the American Medical
  Association, 251, 911-915.
• Shea. (1997). Instrument relevance in
  multivariate linear models A simple measure.
  Review of Economics and Statistics, 79, 348-352.
• Sherman Berk. (1984). The specific deterrent
  effect of arrest for domestic assault. American
  Sociological Review, 49, 261-272.
• Staiger and Stock. (1997). Instrumental variables
  regression with weak instruments. Econometrica,
  65, 557-586.