Econometricians Are Known by Their… Instruments

It’s the Law

Economists think people make important choices such as those related to schooling by comparing anticipated costs with expected benefits. The cost of staying in secondary school is determined partly by compulsory schooling laws, which punish those who leave school too soon. Since you avoid punishment by staying in school, compulsory schooling laws make extra schooling seem cheaper relative to the alternative, dropping out. This generates a causal chain reaction leading from compulsory schooling laws to schooling choices to earnings that might reveal the economic returns to schooling. The ’metrics methods behind this idea are those of Chapters 3 and 5: instrumental variables and dif f erences-indif f erences.

As always, IV begins with the first stage. One hundred years ago, there were few compulsory attendance laws, while today most American states keep students in school until at least age 16. Many states also forbid school-aged children from working, or require school authorities to give permission for a child to work. Assuming that some students would otherwise drop out if not for such laws, stricter compulsory school requirements should increase average schooling. Provided changes in state compulsory attendance laws are also unrelated to the potential earnings of residents in each state (as determined by things like family background, the states’ industrial structure, or other policy changes), these laws create valid instruments for schooling in equations like Г6.1У

But compulsory attendance laws probably are related to potential earnings. In the early twentieth century, for example, agricultural Southern states had few compulsory attendance requirements, while compulsory schooling laws were stricter in the more industrial North. Simple comparisons of earnings across U. S. regions typically reveal vast differences in earnings, but these are mostly unrelated to the North’s more rigorous schooling requirements. Compulsory schooling requirements also grew stricter over time, but here, too, simple comparisons are misleading. Many features of the American economy changed as the twentieth century progressed; compulsory schooling laws are but a small part of this ever-evolving economic story.

A creative combination of DD and IV offers a possible way around OVB roadblocks in this context. Compulsory schooling requirements expanded and tightened most dramatically in the first half of the twentieth century. Masters Joshway and Daron Acemoglu collected state-by-year information on the compulsory schooling laws applicable to those who might have been in school at this time.- These laws include child labor provisions as well as compulsory attendance requirements. Child labor laws that require a certain amount of schooling be completed before children are allowed to work seem to have increased schooling more than attendance requirements. A useful simplification in this context uses the laws in effect in census respondents’ states of birth at the time they were 14 years old to identify states and years in which 7, 8, and 9 or more years of schooling were required before work was allowed. The resulting set of instrumental variables consists of dummies for each of these three categories; the omitted category consists of states and years in which 6 or fewer years of schooling were required before work was allowed.

Because child labor instruments vary with both state and year of birth, they can be used to estimate a first-stage equation that controls for possible time effects through the inclusion of year-of-birth dummies, while controlling for state characteristics through the inclusion of state-of-birth dummies. Control for state effects should mitigate bias from regional differences that are correlated with compulsory schooling provisions, while the inclusion of year-of-birth effects should mitigate bias from the fact that earnings differ across birth cohorts for many reasons besides compulsory schooling laws. The resulting first-stage equation looks like the Chapter 5 regression DD model (described by equation (5J5)) used to estimate the effect of state and year changes in the MLDA on death rates. Here, however, year-of-birth dummies replace dummies for calendar time.

The Acemoglu and Angrist compulsory schooling first-stage equation was estimated with an extract of men in their forties, drawn from each of the U. S. census samples available every decade from 1950 to 1990. Stacking these five censuses produces a single large data set in which different censuses contribute different cohorts. For example, men in their forties observed in the 1950 Census were born from 1900 to 1909 and subject to laws in effect in the 1910s and 1920s, while men in their forties observed in the 1960 Census were born from 1910 to 1919 and subject to laws in effect in the 1920s and 1930s.

The first-stage estimates reported in column (1) of Table 6.3 suggest that child labor laws requiring 7 or 8 years of schooling before work was allowed increased schooling (measured as highest grade completed) by about two-tenths of a year. Laws requiring 9 or more years of schooling before work was allowed had an effect twice as large. A parallel set of reduced-form estimates appear in column (3) of the table. These come from regression models similar to those used to construct the first-stage estimates reported in column (1), with the log weekly wage replacing years of schooling as the dependent variable. Laws requiring 7 or 8 years of schooling before work was allowed appear to have raised wages by about 1%, while laws requiring 9 or more years of schooling before work increased earnings by almost 5%, though only the latter estimate is significant. The 2SLS estimate generated by these estimates is.124 (with an estimated standard error of.036).

A 12% wage gain for each additional year of schooling is impressive, all the more so since the schooling increase in question is involuntary. Stronger compulsory schooling laws appear to raise schooling, and this in turn produces higher wages for the men constrained by these laws (compulsory schooling compliers, in this case). Especially interesting is the fact that the 2SLS estimate of the returns to schooling generated by compulsory schooling instruments exceeds the corresponding OLS estimate of .075. This finding weighs against the notion of upward ability bias in the OLS estimate.

TABLE 6.3

Returns to schooling using child labor law instruments

Dependent variable

Ye ars о f sc ho о Li ng Log weekly wages

_____________________________ (1) (2) (3) 14)

A. First-stage and reduced-torm estimates

Child labor law ret]. 7 years

l=*

G

0

-.024

.010

-.013

(.067)

(.048)

(.011)

(.011)

Child labor Jaw ret]. S years

.191

.024

.013

.005

(.062)

(.051,.

(.010)

(.010)

Child labor law req. 9 years

.400

.016

.046

.008

or more

(.093)

(.053)

(.017)

(.014)

Б. Second-stage estimates

Years of education

.124

.399

(.036)

(.360)

State of birth dummies x linear

No

Yes

No

Yes

year of birth trends

Notes: This table shows 2SLS estimates of the returns to schooling using as instruments three dummies indicating the years of schooling required by child labor laws as a condition for employment. Panel A reports first-stage and reduced – form estimates controlling for year and state of birth effects and for census year dummies. Columns (2) and (4) show the results of adding state-specific linear trends to the list of controls. Panel В shows the 2SLS estimates of the returns to schooling generated by the first-stage and reduced-form estimates in panel A. Sample size is 722,343. Standard errors are reported in parentheses.

Before declaring mission accomplished, a master looks for threats to validity. The variation in schooling generated by compulsory schooling laws produces a DD-style first stage and reduced form. As discussed in Chapter 5. the principal threat to validity in this context is omitted state-specific trends. Specifically, we must worry that states in which compulsory schooling laws grew stricter simultaneously experienced unusually large wage growth across cohorts for reasons unrelated to schooling. Perhaps wage growth and changes in schooling laws are both driven by some third variable, say, changes in industrial structure.

The case for omitted variables bias in this context grows even stronger once we recognize that most of the action in the compulsory schooling research design comes from comparisons of Northern and Southern states. Southern states saw enormous economic growth in the twentieth century, while at the same time, social legislation in these states proliferated. The relative growth in earnings in Southern states might have been caused in part by more restrictive compulsory attendance provisions. But it might not.

Chapter 5 explains that a simple check for state-specific trends adds a linear time trend for each state to the model of interest. In this case, the relevant time dimension is year of birth, so the model with state-specific trends includes a separate linear year-of-birth variable for each state of birth in the sample (the regression model with year-of-birth trends looks like equation Г5.6А.

Columns (2) and (4) in Table 6.3 report the results of this addition. The estimates in these columns offer little evidence that compulsory schooling laws matter for either schooling or wages. First-stage and reduced-form estimates both fall precipitously in the model with trends, and none are significantly different from zero. Importantly, the first – stage estimates in column (2) are more precise (that is, have smaller standard errors) than those estimated without state-specific trends. Lack of statistical significance therefore comes from the fact that the estimates with trends are much smaller and not from reduced precision. The reduced-form estimates in column (4) similarly offer little evidence of a link between compulsory school laws and earnings. The 2SLS estimate generated by columns (2) and (4) comes out at an implausibly large.399, but with a standard error almost as large. Sad to say for Master Joshway, Table 6.3 reveals a failed research design.

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