The Wald Estimator

The simplest IV estimator uses a single binary (0-1) instrument to estimate a model with one endogenous regressor and no covariates. Without covariates, the causal regression model is

Y i = a + pSi + (4.1.11)

where Pi and Si may be correlated. Given the further simplification that Zi is a dummy variable that equals 1 with probability p, we can easily show that

Cov(Yi, Zi) = {E[Yi|Zi = 1] – E[Yi|Zi = 0]gp(1 – p),

with an analogous formula for Cov(Si, Zi). It therefore follows that

Подпись: (4.1.12)= E[Yi|Zi = 1] – E[Yi|Zi = 0]

P = E[Si|Zi = 1] – E[Si|Zi = 0] ‘

A direct route to this result uses (4.1.11) and the fact that E[^|Zi] = 0, so we have

Подпись: (4.1.13)E[Yi|Zi] = a + pE[Si[Zi].

Solving this equation for p produces (4.1.12).

Equation (4.1.12) is the population analog of the landmark Wald (1940) estimator for a bivariate regres­sion with mismeasured regressors.[43] The Wald estimator is the sample analog of this expression. In our context, the Wald formula provides an appealingly transparent implementation of the IV strategy for the elimination of omitted variables bias. The principal claim that motivates IV estimation of causal effects is that the only reason for any relation between the dependent variable and the instrument is the effect of the instrument on the causal variable of interest. In the context of a binary instrument, it therefore seems nat­ural to divide—or rescale—the reduced-form difference in means by the corresponding first-stage difference in means.

The Angrist and Krueger (1991) study using quarter of birth to estimate the economic returns to schooling shows the Wald estimator in action. Table 4.1.2 displays the ingredients behind a Wald estimate constructed using the 1980 census. The difference in earnings between men born in the first and second halves of the year is -.01349 (s. e.=.00337), while the corresponding difference in schooling is -.1514. The ratio of these two differences is a Wald estimate of the economic value of schooling in per-year terms. This comes out to be.0891 (s. e.=.021). Not surprisingly, this estimate is not too different from the 2SLS estimates in Table 4.1.1. The reason we should expect the Wald and 2SLS estimates to be similar is that they are both constructed from the same information: differences in earnings by season of birth.

The Angrist (1990) study of the effects of Vietnam-era military service on the earnings of veterans also shows the Wald estimator in action. In the 1960s and early 1970s, young men were at risk of being drafted for military service. Concerns about the fairness of US conscription policy led to the institution of a draft lottery in 1970 that was used to determine priority for conscription. A promising instrumental variable for Vietnam veteran status is therefore draft-eligibility, since this was determined by a lottery over birthdays. Specifically, in each year from 1970 to 1972, random sequence numbers (RSNs) were randomly assigned to each birth date in cohorts of 19-year-olds. Men with lottery numbers below an eligibility ceiling were eligible for the draft, while men with numbers above the ceiling could not be drafted. In practice, many draft-eligible men were still exempted from service for health or other reasons, while many men who were draft-exempt nevertheless volunteered for service. So veteran status was not completely determined by randomized draft-eligibility,

Table 4.1.2: Wald estimates of the returns to schooling using quarter of birth instruments


Born in the 1st or 2nd quarter of year


Born in the 3rd or 4th quarter of year


Difference (std. error) (1)-(2)

ln (weekly wage)





Years of education





Wald estimate of return to education



OLS estimate of return to education



Notes: Adapted from a re-analysis of Angrist and Krueger (1991) by Angrist and Imbens (1995). The sample includes native-born men with positive earnings from the 1930-39 birth cohorts in the 1980 Census 5 percent file. The sample size is 329,509.

but draft-eligibility provides a binary instrument highly correlated with Vietnam-era veteran status.

For white men who were at risk of being drafted in the 1970 draft lottery, draft-eligibility is clearly associated with lower earnings in years after the lottery. This is documented in Table 4.1.3, which reports the effect of randomized draft-eligibility status on average Social Security-taxable earnings in column 2. column 1 shows average annual earnings for purposes of comparison. For men born in 1950, there are significant negative effects of eligibility status on earnings in 1971, when these men were mostly just beginning their military service, and, perhaps more surprisingly, in 1981, ten years later. In contrast, there is no evidence of an association between draft-eligibility status and earnings in 1969, the year the lottery drawing for men born in 1950 was held but before anyone born in 1950 was actually drafted.

Because eligibility status was randomly assigned, the claim that the estimates in column 2 represent the effect of draft-eligibility on earnings seems uncontroversial. The information required to go from draft- eligibility effects to veteran-status effects is the denominator of the Wald estimator, which is the effect of draft-eligibility on the probability of serving in the military. This information is reported in column 3 of Table 4.1.3, which shows that draft-eligible men were almost 16 percentage points more likely to have served in the Vietnam era. The Wald estimate of the effect of military service on 1981 earnings, reported in column 4, amounts to about 15 percent of the mean. Effects were even larger in 1971 (in percentage terms), when affected soldiers were still in the army.

An important feature of the Wald/IV estimator is that the identifying assumptions are easy to assess and

Table 4.1.3: Wald estimates of the effects of military service on the earnings of white men born in 1950

Earnings year


Veteran Status












Estimate of























Notes: Adapted from Angrist (1990), Tables 2 and 3. Standard errors are shown in parentheses. Earnings data are from Social Security administrative records. Fig­ures are in nominal dollars. Veteran status data are from the Survey of Program Participation. There are about 13,500 individuals in the sample.

interpret. Suppose Dj denotes Vietnam-era veteran status and zj indicates draft-eligibility. The fundamental claim justifying our interpretation of the Wald estimator as capturing the causal effect of Dj is that the only reason why E[Yj|Zj] changes as Zj changes is the variation in E[Dj|Zj]. A simple check on this is to look for an association between Zj and personal characteristics that should not be affected by Dj, for example, age, race, sex, or any other characteristic that was determined before Dj was determined. Another useful check is to look for an association between the instrument and outcomes in samples where there is no relationship between Dj and Zj. If the only reason for draft-eligibility affects on earnings is veteran status, then draft – eligibility effects on earnings should be zero in samples where draft-eligibility status is unrelated to veteran status.

This idea is illustrated in the Angrist (1990) study of the draft lottery by looking at 1969 earnings, an estimate repeated in the last row of Table 4.1.3. It’s comforting that the draft-eligibility treatment effect on 1969 earnings is zero since 1969 earnings predate the 1970 draft lottery. A second variation on this idea looks at the cohort of men born in 1953. Although there was a lottery drawing which assigned RSNs to the 1953 birth cohort in February of 1972, no one born in 1953 was actually drafted (the draft officially ended in July of 1973). The first-stage relationship between draft-eligibility and veteran status for men born in 1953 (defined using the 1952 lottery cutoff of 95) therefore shows only a small difference in the probability of serving by eligibility status. Importantly, there is also no significant relationship between earnings and draft-eligibility status for men born in 1953, a result that supports the claim that the only reason for draft-eligibility effects is military service.

We conclude the discussion of Wald estimators with a set of IV estimates of the effect of family size on mothers’ employment and work. Like the schooling and military service studies, these estimates are used for illustration elsewhere in the book. The relationship between fertility and labor supply has long been of interest to labor economists, while the case for omitted variables bias in this context is clear: mothers with weak labor force attachment or low earnings potential may be more likely to have children than mothers with strong labor force attachment or high earnings potential. This makes the observed association between family size and employment hard to interpret since mothers who have big families may have worked less anyway. Angrist and Evans (1998) solve this omitted-variables problem using two instrumental variables, both of which lend themselves to Wald-type estimation strategies.

The first Wald estimator uses multiple births, an identification strategy for the effects of family size pioneered by Rosenzweig and Wolpin (1980). The twins instrument in Angrist and Evans (1998) is a dummy for a multiple third birth in a sample of mothers with at least two children. The twins first-stage is.625, an estimate reported in column 3 of Table 4.1.4. This means that 37.5 percent of mothers with two or more children would have had a third birth anyway; a multiple third birth increases this proportion to 1. The twins instrument rests on the idea that the occurrence of a multiple birth is essentially random, unrelated to potential outcomes or demographic characteristics.

The second Wald estimator in Table 4.1.4 uses sibling sex composition, an instrument motivated by the fact that American parents with two children are much more likely to have a third child if the first two are same-sex than if the sex-composition is mixed. This is illustrated in column 5 of Table 4.1.4, which shows that parents of same-sex sibling birth are 6.7 percentage points more likely to have a third birth (the probability of a third birth among parents with a mixed-sex sibship is.38). The same-sex instrument is based on the claim that sibling sex composition is essentially random and affects family labor supply solely by increasing fertility.

Twins and sex-composition instruments both suggest that the birth of a third child has a large effect on employment rates and on weeks and hours worked. Wald estimates using twins instruments show a precisely-estimate employment reduction of about.08, while weeks worked fall by 3.8 and hours per week fall by 3.4. These results, which appear in column 4 of Table 4.1.4, are smaller in absolute value than the corresponding OLS estimates reported in column 2. This suggests the latter are exaggerated by selection bias. Interestingly, the Wald estimates constructed using a same-sex dummy, reported in column 6, are larger than the twins estimates. The juxtaposition of twins and sex-composition instruments in Table 4.1.4 suggests that different instruments need not generate similar estimates of causal effects even if both are valid. We expand on this important point in Section 4.4. For now, however, we stick with a constant-effects framework.

Table 4.1.4: Wald estimates of labor supply effects







IV Estimates using:



First stage


Wald estimates


First stage


Wald estimates














Weeks worked
























Note: The table reports OLS and Wald estimates of the effects of a third birth on labor supply


using twins and sex-composition instruments. Data are from the Angrist and Evans (1998) extract including married women aged 21-35 with at least two children in the 1980 Census. OLS models include controls for mother’s age, age at first birth, dummies for the sex of first and second births, and dummies for race.




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