Permanent Income and the Black-White Test Score Gap

May combine two claims.

The black-white test score gap shrunk rapidly during the 1970s and 1980s, but progress largely stopped among cohorts born since the early 1970s.

“while the gap shrunk rapidly during the 1970s and 1980s, progress has largely stopped among cohorts born since the early 1970s (Neal 2006; Chay, Guryan, and Mazumder 2009).”

• cites Neal 2006 sourceNot checked — no full text of the source
• cites Chay, Guryan, and Mazumder 2009 sourcePartly supported

  The Introduction and Section III (page numbers unavailable) report substantial narrowing in the 1980s and no further convergence for cohorts born from 1973 to 1989. The paper does not support rapid narrowing throughout the 1970s and 1980s: it identifies the 1980s as the period when the overall gap fell significantly, although some age groups began converging earlier. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

May combine two claims.

Annual income is a poor proxy for a family's consumption and investment possibilities and may be measured with substantial error in population surveys.

“As has long been recognized (Modigliani and Brumberg 1954; Friedman 1957), annual income is a poor proxy for a family's consumption and investment possibilities, and in any case it may be measured with substantial error in population surveys.”

• cites Modigliani and Brumberg 1954Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Friedman 1957 sourcePartly supported

  Chapter VI distinguishes measured income from permanent income and describes consumption as adjusted to permanent income, supporting the point that annual income can be a poor proxy for consumption possibilities. In “The Relative versus the Absolute Income Hypotheses” (page unavailable), Friedman questions the accuracy of income and consumption data for a sample of farm families. That passage does not establish substantial income-measurement error in population surveys generally, and the passages do not directly address investment possibilities. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

Classified as probably not a substantial claim.

Fryer and Levitt (2004) find that a short list of covariates can fully explain the black-white gap in reading and (to a lesser extent) math scores among entering kindergarteners in the ECLS sample.

“They find that these variables can fully explain the black-white gap in reading and (to a lesser extent) math scores among entering kindergarteners in the ECLS sample.”

• cites Fryer and Levitt (2004) sourceSupported

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Fryer and Levitt showed that differences in covariates explained roughly the same absolute black-white gap across specifications for Kindergarten, first, and third grade scores, but that the unexplained gap grew monotonically across these grades.

“FL (see also Fryer and Levitt 2004), showed that differences in covariates explained roughly the same absolute black-white gap across specifications for Kindergarten, 1 st , and 3 rd grade scores, but that the unexplained gap grew monotonically across these grades.”

• cites Fryer and Levitt 2006Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Fryer and Levitt 2004 sourcePartly supported

  Section V reports that the covariate-adjusted Black–White gap grew from fall kindergarten to spring first grade, supporting that part of the claim. But the paper compares only those two time points; it does not report third-grade scores or establish that covariates explained roughly the same absolute gap across all three grades and specifications. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

May combine two claims.

In longitudinally consistent data, the black-white test score gap is smaller for children born since 1970 than for those born before 1960, though there was little additional progress for children born between the early 1970s and late 1980s.

“In longitudinally consistent data, it is smaller for children born since 1970 than for those born before 1960, though there was little additional progress for children born between the early 1970s and late 1980s (Chay, Guryan, and Mazumder 2009).”

• cites Chay, Guryan, and Mazumder 2009 sourceSupported

Classified as probably not a substantial claim.

If families face credit constraints or are uncertain about future incomes, current and past income may have larger effects on investment decisions and test scores than does future income, producing direct effects of {Y1,...,YT} on e.

“If families face credit constraints or are uncertain about future incomes, current and past income may have larger effects on investment decisions and test scores than does future income, producing direct effects of {Y 1 , …, Y T } on e”

• cites Carneiro and Heckman, 2002 sourcePartly supported

  The Introduction and Section 2 say that family resources during childhood can shape skills and college readiness, and discuss credit constraints as a possible influence on education decisions. Our copy does not make the claimed comparison with future income, address uncertainty about future income in this account, or state the proposed direct effects of {Y1,…,YT} on e. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

May combine two claims.

In a specification where measured permanent income is instrumented with current income, the income coefficient is larger and the conditional black-white gap is smaller.

“we find that the income coefficient is larger and the conditional black-white gap smaller in a specification where (measured) permanent income is instrumented with current income.”

May combine two claims.

Adding permanent income to the controls used by Fryer and Levitt (2006) shrinks the unexplained black-white gap in 3rd grade below 0.15 standard deviations, less than half of what is found with their controls.

“When we add permanent income to the controls used by Fryer and Levitt (2006), the unexplained gap in 3rd grade shrinks below 0.15 standard deviations, less than half of what is found with their controls.”

• cites Fryer and Levitt (2006)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Adding permanent income to Fryer and Levitt's short vector of controls (birth weight, number of children's books in the home, mother's age at first birth, and a WIC recipient indicator) reduces the remaining black-white gap in 3rd grade to 0.15 standard deviations, less than half of the 0.34 obtained with the Fryer and Levitt controls.

“When we add permanent income to the short vector of controls used by Fryer and Levitt -birth weight, number of children's books in the home, mother's age at first birth, and a WIC recipient indicator -we find that the remaining black-white gap in 3 rd grade falls to 0.15 standard deviations, less than half of the 0.34 that we obtain with the Fryer and Levitt controls.”

May combine two claims.

In that IV specification, the conditional black-white gap is only 0.32 standard deviations, down from 0.56 without income controls, 0.43 when annual income is controlled, and 0.36 when permanent income is controlled in an OLS specification.

“In this specification, the conditional black-white gap is only 0.32 standard deviations, down from 0.56 without income controls, 0.43 when annual income is controlled, and 0.36 when permanent income is controlled in an OLS specification.”

May combine two claims.

In ECLS, the gap without income controls is 0.62 standard deviations and the traditionally-estimated gap conditional on annual income is 0.38.

“By contrast, the gap without income controls is 0.62 standard deviations and the traditionally-estimated gap conditional on annual income is 0.38.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Log current incomes average 10.72 (SD 0.97), while log permanent incomes average 10.77 (SD 0.70).

“Log current incomes average 10.72 (standard deviation 0.97), while the log permanent incomes average 10.77 (SD 0.70).”

May combine two claims.

Using family income six years before the test date as an instrument yields an income coefficient of 0.326 and a black coefficient of -0.365.

“We find exactly this when we repeat the IV specification using family income six years before the test date (when the child was aged 4 or 5) as an instrument: The income coefficient drops to 0.326, and the black coefficient increases in magnitude to -0.365.”

May combine two claims.

Instrumenting current income with annual income four years prior yields an income coefficient of 0.283, larger than OLS but significantly lower than the 15-year income average, and a gap of 0.376.

“In fact, the income coefficient is 0.283, larger than in the OLS specification but significantly lower than in the specification that uses the 15-year income average.The gap is 0.376, similarly between these two.”

May combine two claims.

Adding log current family income yields a small but significant income coefficient, while the black coefficient shrinks slightly but is generally similar to that in Column 6.

“The income coefficient is small but significant, while the black coefficient shrinks slightly but is generally similar to that seen in Column 6.”

May combine two claims.

The raw gap in permanent incomes is somewhat smaller than the current income gap, but most of the difference disappears when the maternal age control is added.

“The raw gap in permanent incomes (Column 2) is somewhat smaller, but most of the difference disappears when the maternal age control is added.”

May combine two claims.

In Column 10, the long-run average income coefficient is more than double the current income coefficient in Column 9, while the black coefficient is only -0.15.

“Applying our estimator, in Column 10, we see that the long-run average income coefficient is more than double the current income coefficient in Column 9, while the black coefficient is only -0.15.”

May combine two claims.

The raw black-white gap is much larger on the PPVT than on the PIAT math, and is somewhat smaller on the PIAT reading.

“The raw black-white gap is much larger on the PPVT than on the PIAT math, and is somewhat smaller on the PIAT reading.”

May combine two claims.

The Column 10 black coefficient is just over half of the estimate from a specification with a current income control and much less than half of what is estimated without income controls at all.

“This is just over half of the estimate from a specification with a current income control and much less than half of what is estimated without income controls at all (with or without the SES control).”

May combine two claims.

Replacing current income with the average of family income across all four ECLS survey waves yields an income coefficient about one-third larger and shrinks the black-white gap to 0.34.

“The income coefficient is about one-third larger here, and the black-white gap shrinks to 0.34.”

May combine two claims.

The unexplained gap is smaller in the TS2SLS specification than in Fryer and Levitt's specification in each grade; in Kindergarten black students earn higher math scores than white students with similar observables, though the difference is insignificant, and the gap grows as students progress from Kindergarten through fifth grade.

“The unexplained gap is smaller in the TS2SLS specification than in Fryer and Levitt's specification in each grade -indeed, in Kindergarten we find that black students earn higher math scores than white students with similar observables, though the difference is insignificant -but as in their results it grows as students progress from Kindergarten through 5 th grade.”

• cites Fryer and Levitt 2006Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The income patterns are quite robust to inclusion or exclusion of demographic controls, which have little explanatory power for either income measure once the other is controlled.

“Similarly, a comparison between Panels A and B demonstrates that the income patterns are quite robust to the inclusion or exclusion of our demographic controls, which have little explanatory power for either income measure once the other is controlled.”

May combine two claims.

In the ECLS TS2SLS specification, the income coefficient is over 50% larger than in Column 6 and more than double that in Column 5.

“The income coefficient is over 50% larger than in Column 6 (and more than double that in Column 5).”

May combine two claims.

When more information is used to construct income measures, the income coefficient gets larger and the black coefficient shrinks toward zero.

“As expected, when we use more information to construct our income measures, the income coefficient gets larger and the black coefficient shrinks toward zero.”

May combine two claims.

In the two-stage-least-squares specification using log current income as an instrument for log long-run average income, the income coefficient is larger and the black coefficient is smaller in magnitude than in the corresponding OLS specification in Column 7.

“The income coefficient is larger and the black coefficient smaller (in magnitude) than in the corresponding OLS specification in Column 7.”

May combine two claims.

In the subsample with nonmissing, non-imputed family income, the black-white gap both unconditional and conditional on the Fryer-Levitt covariates is notably smaller, but the conditional gap remains large and significant.

“The black-white gap, both unconditional and conditional on the FL covariates, is notably smaller in this subsample, but the conditional gap remains large and significant.”

May combine two claims.

The imputation algorithm allowed formation of a usable current income for 99% of CNLSY children (unweighted) and a permanent income measure for 94%.

“Our full imputation algorithm -described in the data appendix -allowed us to form a usable current income for 99% of CNLSY children (unweighted) and a permanent income measure for 94%, with missing values arising primarily when mothers permanently attrited from the NLSY sample before age 39.”

May combine two claims.

The conditional black-white gap in the ECLS TS2SLS specification is 0.18, less than a third of the raw gap and just over half of the gap controlling for average income over the ECLS panel.

“The conditional black-white gap is 0.18, less than a third of the raw gap and just over half of the gap controlling for the average income over the ECLS panel.”

May combine two claims.

Without the SES index, the specification without family income control yields a slightly larger black-white gap, but the specification with current income control yields a notably smaller gap and much larger income coefficient than when the SES index is included.

“The specification without our family income control yields a slightly larger black-white gap, but that with a control for current income yields a notably smaller gap (and much larger income coefficient) than when the SES index is included.”

May combine two claims.

In the regression of current income on race and long-run average income with demographic controls, the average income coefficient is statistically indistinguishable from one and the R2 is just over 0.6.

“When we include our simple demographic controls, the average income coefficient is statistically indistinguishable from one and the R 2 just over 0.6.”

May combine two claims.

In the reverse regression, the black coefficient is -0.27 and highly significant.

“The black coefficient is negative, -0.27, and highly significant.”

May combine two claims.

Empirically, both current income and human capital measures are very poor proxies for long-run measures of families' financial circumstances.

“There is little theoretical justification for believing that current income, rather than permanent income, is an important determinant of student achievement, and empirically both current income and human capital measures turn out to be very poor proxies for long-run measures of families' financial circumstances.”

May combine two claims.

In the preferred 2SLS and TS2SLS estimates, a 10% increase in family permanent income is associated with an increase in child math scores of 0.04 standard deviations in the CNLSY and 0.07 standard deviations in the ECLS.

“In our preferred 2SLS and TS2SLS estimates, a 10% increase in family permanent income is associated with an increase in child math scores of 0.04 (CNLSY) to 0.07 (ECLS) standard deviations.”

May combine two claims.

α rises monotonically, while the transitory component is less variable for 9-year-old children than for older or younger children.

“α rises monotonically, while the transitory component is less variable for 9-year-old children than for older or younger children.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The average mother in the NLSY sample gives birth in her mid 20s but there is substantial variation around this average.

“the average mother in our NLSY sample gives birth in her mid 20s but there is substantial variation around this average.”

May combine two claims.

When the ECLS sample is reweighted to match the maternal age distribution in the CNLSY sample, the raw black-white gap is somewhat smaller (0.62 vs. 0.78 standard deviations) but the pattern of coefficients across specifications is similar.

“When we reweight the ECLS sample to match the maternal age distribution in the CNLSY sample, the raw black-white gap is somewhat smaller (0.62 vs. 0.78 standard deviations) but the pattern of coefficients across specifications is similar.”

May combine two claims.

α_t begins low but rises to about 1.1 by the early 30s and stays relatively constant through the end of the 30s.

“α t begins low but rises to about 1.1 by the early 30s and stays relatively constant through the end of the 30s.”

May combine two claims.

As mean income is lower for blacks than for whites, attenuation from using annual income leads to overstatement of the black-white test score gap conditional on income.

“As mean income is lower for blacks than for whites, this attenuation leads to overstatement of the blackwhite test score gap conditional on income.”

Classified as probably not a substantial claim.

Nearly all of the black-white wage gap can be attributed to differences in premarket factors such as test scores.

“Neal and Johnson (1996) show that nearly all of the black-white wage gap can be attributed to differences in “premarket” factors such as test scores”

• cites Neal and Johnson (1996) sourceNot checked — no full text of the source

May combine two claims.

Fryer and Levitt (2004, 2006) find that the black-white gap remaining after controlling for demographic and behavioral variables is nearly 0.4 standard deviations by the end of 3rd grade, down from a raw gap of 0.88 SDs in their sample.

“Fryer and Levitt (2004, 2006) are the most successful at explaining the gap via differences in observable characteristics, but even they find that the gap that remains after controlling for a vector of demographic and behavioral variables is nearly 0.4 standard deviations by the end of the 3 rd grade (down from a raw gap of 0.88 SDs in their sample).”

• cites Fryer and Levitt (2004, 2006)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Important shortcomings in the way income is measured have led the existing literature to dramatically understate the role of family income differences in the black-white test score gap and overstate the gap among children with the same family incomes.

“In this paper we argue that important shortcomings in the way that income is measured have led the existing literature to dramatically understate the role of family income differences in accounts of the black-white test score gap and, therefore, to dramatically overstate the gap among children with the same family incomes.”

Classified as probably not a substantial claim. May combine two claims.

Income inequality between blacks and whites appears to play some role in the test score gap, but it is quite small.

“Income inequality between blacks and whites appears to play some role in the test score gap, but it is quite small”

• cites Jencks and Phillips (1998)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

The key identifying assumption that transitory variations in annual income are unrelated with outcomes conditional on permanent income frequently follows immediately from the permanent income hypothesis and may be a tolerably close approximation in many contexts.

“This assumption frequently follows immediately from the permanent income hypothesis (PIH) and may be a tolerably close approximation in many contexts.”

• AssumptionThe authors state this as an assumption: The permanent income hypothesis holds: families can freely borrow or save across periods.
• AssumptionThe authors state this as an assumption: Transitory variations in annual income are unrelated with outcomes conditional on permanent income.

May combine two claims.

The black-white gap in mean scores is large and persistent, nearly always above 0.5 standard deviations and more commonly in the 0.75 to 1 range.

“The gap in mean scores between black and white children is large and persistent, nearly always above 0.5 standard deviations and more commonly in the 0.75 to 1 range.”

May combine two claims.

The black-white test score gap is robust across many different samples and tests, and is apparent on both aptitude and achievement tests.

“It is robust across many different samples and different tests, and in particular is apparent on both aptitude and achievement tests.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Dahl and Lochner (2010) find that the causal effect of income is quite large.

“The Dahl and Lochner study in particular finds that the causal effect of income is quite large.”

• cites Dahl and Lochner (2010)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

There would be no correlation between current income and children's books if the household behaved according to the permanent income hypothesis and faced no credit constraints.

“(Note, however, that there would be no correlation if the household behaved according to the permanent income hypothesis and faced no credit constraints.)”

Classified as probably not a substantial claim. May combine two claims.

Even with the SES index excluded, the exclusion restriction may be incorrect; it would be violated if current income were correlated with the number of children's books in the home conditional on permanent income.

“Even with the mechanically related SES index excluded, the exclusion restriction may be incorrect.It would be violated, for example, if current income were correlated with the number of children's books in the home conditional on the family's permanent income.”

• AssumptionThe authors state this as an assumption: The transitory component of income is uncorrelated with achievement, conditional on race and the other controls.

May combine two claims.

The positive coefficient for early-career income is consistent with the view that early-career income is more important, but it is only marginally significant at the 10% level.

“The positive coefficient for the former is consistent with the view that early-career income is indeed more important, but it is only marginally significant (at the 10% level).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The result in Column 4 could derive from measurement error in permanent income.

“However, the result could also derive from measurement error in permanent income.”

May combine two claims.

The first-stage regression of permanent income on current income and race has plim λ_y = R_b/α_t and plim λ_b = (1-R_b)δ(p).

“In this regression, it is straightforward to show that plim λ y = R b / α t and plim λ b = (1-R b )δ(p).”

May combine two claims.

Replacing permanent income with annual income rescales income by α_t and, if e_it is nondegenerate, produces attenuation in the income effect and biases the black coefficient downward.

“First, it rescales income by the multiplicative factor α t .Thus, if var(e it ) = 0, θ y ' = θ p / α t .Second, if e it is nondegenerate, it produces attenuation in the income effect and biases the black coefficient downward:”

May combine two claims.

The Column 7 results support the exclusion restriction and indicate that the Column 4 result is unlikely to derive from a permanent income hypothesis violation.

“As credit constraints are unlikely to produce a causal role for future income conditional on current income, we view this result as support for our exclusion restriction and as an indication that the result in Column 4 is unlikely to derive from a PIH violation.”

May combine two claims.

Simple regressions are unlikely to identify the causal effect of income, and therefore the conditional black-white gap cannot be interpreted as the gap that would remain after an intervention to equalize incomes.

“The interpretation of conditional test score gaps is not entirely straightforward, as simple regressions are unlikely to identify the causal effect of income and therefore the conditional black-white gap cannot be interpreted as the gap that would remain after an intervention to equalize incomes.”

May combine two claims.

Both an OLS regression of s on p and b and an IV regression that instruments for p with y will overstate the magnitude of the black-white gap conditional on p.

“The same errors-in-variables results used above imply that both an OLS regression of s on and b and an IV regression that instruments for with y will overstate the magnitude of the black-white gap conditional on p.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The income measures in the ECLS are of much lower quality than those in the NLSY.

“The income measures in the ECLS are of much lower quality than those in the NLSY.”

May combine two claims.

By the time the same children finished third grade, the raw black-white gap had grown and the share explained by the Fryer and Levitt covariates had fallen to two thirds.

“By the time the same children finished third grade, the raw black-white gap had grown and the share explained by the FL covariates had fallen to two thirds (Fryer and Levitt 2006).”

• cites Fryer and Levitt 2006Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

The TS2SLS estimator identifies β if the four second moment matrices consistently estimate the corresponding population moments.

“If the four second moment matrices in (13), when scaled by the appropriate sample sizes n 1 and n 2 , consistently estimate the corresponding population moments, the TS2SLS estimator identifies β.”

Probably not a substantial claim: may not pass S4 (based).

With the permanent income hypothesis, educational investments and consumption decisions depend only on the discounted value of lifetime income, not on income in any particular year conditional on this.

“With this assumption, educational investments and consumption decisions depend only on the discounted value of lifetime income, Σ t Y t (1+r) -t , and not on the family's income in any particular year conditional on this.”

• AssumptionThe authors state this as an assumption: The permanent income hypothesis holds: families can freely borrow or save across periods.

May combine two claims.

If the ECLS current income measure is less reliable, the income coefficient in Column 7 remains somewhat attenuated and the black coefficient is somewhat negatively biased.

“If so, the income coefficient in Column 7 remains somewhat attenuated, and the black coefficient somewhat negatively biased.”

• AssumptionThe authors state this as an assumption: The current income measure in the ECLS is equivalent to that in the NLSY.

Probably not a substantial claim: may not pass S3 (load-bearing).

Under more general e_it processes, the asymptotic bias in the IV estimator may be larger or smaller than that in OLS.

“Under more general e it processes, however, the asymptotic bias in the IV estimator may be larger or smaller than that in OLS.”

May combine two claims.

Fryer and Levitt's raw third-grade math gap is 0.88 standard deviations, and inclusion of their controls reduces it to 0.38.

“The raw gap is 0.88 standard deviations, and the inclusion of the FL controls reduces this to 0.38.”

• cites Fryer and Levitt 2006Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

R_b is less than or equal to 1, with R_b = 1 only if var(e_it) = 0.

“Thus, R b ≤ 1 with R b =1 only if var(e it ) = 0.”

May combine two claims.

The noise in annual family income diminishes as mothers reach middle age, so an IV estimator constructed using income from that period is less biased by measurement error in permanent income than is OLS.

“The noise in annual family income diminishes as mothers reach middle age, so an IV estimator constructed using income from that period is less biased by measurement error in permanent income than is OLS.”

Probably not a substantial claim: may not pass S4 (based).

With covariates V available only in sample 1, the hybrid estimator is consistent for β under the additional assumption E[V'(y - pλ_p - Wλ_W)]=0.

“We use these to form a hybrid two-sample 2SLS estimator: . ( 16) This is consistent for β.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In most analyses of the CNLSY data, they control for a quadratic in the mother's age at the child's birth.

“Accordingly, in most of our analyses of the CNLSY data we control for a quadratic in the mother's age at the child's birth.”

Classified as probably not a substantial claim.

The IV estimator's assumption might be violated if families are credit constrained and their educational investments are sensitive to cash on hand.

“This assumption might be violated if, for example, families are credit constrained and their educational investments are thus sensitive to the amount of cash that they have on hand.”

• AssumptionThe authors state this as an assumption: Current income has no direct effect on student achievement once long-run income is controlled; current income is uncorrelated with the residual in equation (6).

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The sample standard deviation of PPVT scores is much larger than those for PIAT scores, perhaps indicating a problem with the NLSY score norms.

“Recall from Table 1A that the sample standard deviation of PPVT scores is much larger than those for PIAT scores, perhaps indicating a problem with the NLSY score norms.”

Probably not a substantial claim: may not pass S4 (based).

Previous research has found that family income and other variables measuring a family's external circumstances do a relatively poor job of explaining the black-white test score gap.

“Previous research has found that family income and other variables measuring a family's external circumstances do a relatively poor job of explaining the black-white test score gap.”

May combine two claims.

R_b will be low if the transitory component of income is large or if current income is measured with a great deal of error.

“R b will be low if the transitory component of income is large or if current income is measured with a great deal of error.”

Probably not a substantial claim: may not pass S3 (load-bearing).

In some cases, the testing protocol was not followed perfectly.

“In some cases, the testing protocol was not followed perfectly.”

Classified as probably not a substantial claim.

Dahl and Lochner estimate that a permanent $1,000 increase in family income causes test scores to rise by about 0.06 standard deviations.

“They estimate that a permanent $1,000 increase in family income causes test scores to rise by about 0.06 SDs.”

• cites Dahl and Lochner (2010)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

The sample is based on the child's age rather than the mother's.

“our sample is based on the child's age rather than the mother's.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Using the reliability ratio R_b to correct the coefficient from (7) using the errors-in-variables formula (9) is exactly equivalent to the re-arrangement of equation (10).

“This is exactly equivalent to using the reliability ratio R b to correct the coefficient from (7) using the errors-in-variables formula (9).”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Nothing that follows would be changed if the preference parameter γ entered directly into the educational production function f().

“Nothing that follows would be changed if the preference parameter γ entered directly into the educational production function f()”