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Have We Put an End to Social Promotion?

Changes in Grade Retention Rates among Children Aged 6 to 17 from 1972 to 2003

Carl B. Frederick

Robert M. Hauser

Center for Demography of Health and Aging

University of Wisconsin-Madison

August 1, 2005

Prepared for presentation at the 2005 meetings of the American Sociological Association, Philadelphia, Pennsylvania. This research has been supported in part by the Russell Sage Foundation, by the Vilas Estate Trust at the University of Wisconsin-Madison, and by a center grant for population research from the National Institute of Child Health and Human Development to the Center for Demography of Ecology at the UW-Madison. We thank Jeremy Freese and Megan Andrew for methodological advice. The opinions expressed herein are those of the authors. Address correspondence to Carl B. Frederick (cfrederi@ssc.wisc.edu) and Robert M. Hauser (hauser@ssc.wisc.edu) at 1180 Observatory Drive, Madison, Wisconsin 53706.

Have We Put an End to Social Promotion?

Changes in Grade Retention Rates among Children Aged 6 to 17 from 1972 to 2003

Carl B. Frederick

Robert M. Hauser

Center for Demography of Health and Aging

University of Wisconsin-Madison

Abstract

We examine trends over time in age-grade retardation in schooling at ages 6 to 17 and in the effects of its demographic and socioeconomic correlates. We estimate a stereotype logistic regression model of age-grade retardation in almost 900,000 observations of school-age children and their families in the annual October school enrollment supplements of the Current Population Survey. This model identifies systematic variation in the effects of social background across age and time from 1972 to 2003. While the effects of social and economic background variables on progress through school become increasingly powerful as children grow older, that typical pattern has been attenuated across the past three decades by a steady, secular decline in the influence of those variables across all ages. A great deal of concern has been expressed about rising levels of economic and social inequality in the United States since the middle 1970s, and about the potential intergenerational effects of such inequality. However, there has been an opposite trend in the effects of social origins on age-grade retardation, which is one of the more important indicators of progress through schooling. A trend is not a law, and there is reason to be concerned about the recent deceleration of the secular decline in effects of social background.

Have We Put an End to Social Promotion?

Changes in Grade Retention Rates among Children Aged 6 to 17 from 1972 to 2003

Grade retention is one of the methods often proposed and used to help poor performing students catch-up to their peers. At best, most research on the effects of grade retention portrays it as a practice that provides no benefit to the students; at worst it is considered a damaging practice. For example, Jimerson’s (2001: 434) meta-analysis concludes that,

“studies examining the efficacy of grade retention on academic achievement and socioemotional adjustment that have been published during the past decade report results that are consistent with the converging evidence and conclusions of research from earlier in the century that fail to demonstrate that grade retention provides greater benefits to students with academic or adjustment difficulties than does promotion to the next grade.”

Jimerson found that retained students performed worse than their peers both academically and socially. Nagaoka and Roderick (2004) compare students close, both above and below, to the Chicago School System’s test based retention cutoff score. They find that third graders who were retained showed no difference in test scores to those who were promoted and sixth graders who were retained scored worse than those who were promoted.

Previous studies have shown that retained students are more likely to drop out of high school (Rumberger and Larson, 1998; Alexander, Entwisle, and Dauber, 2003; Shepard, 2004; Jimerson, 2004). Allensworth (2004) finds that the Chicago Public Schools’ efforts to end social promotion did not change the overall dropout rate but that failing to pass the eighth grade promotional gate increased the likelihood that low achieving students would drop out. The most disturbing result was that the promotional gates policy exacerbated dropout differentials by race and gender; there was little change in the dropout rates of males and African Americans. Similarly, using national data from October Current Population Surveys, Hauser, Simmons, and Pager (2004) find that being over-age for grade is a powerful, proximate antecedent of high school dropout. Jimerson (1999) also reports that results from a 20 year prospective longitudinal study show that children who have been retained are less likely to earn a diploma by age 20, less likely to attend post-secondary education, were paid less and have lower employment ratings than a group of promoted low achieving students.

Beebe-Frankenburger et al (2004) call the entire practice of retention into question. They find evidence that retained students are the same age and have lower IQ scores than their at-risk and promoted peers. Two of the assumptions underpinning the rationale for retention are that it benefits children who are immature and have the capability to meet standards during the extra year. If one takes age as a rough proxy for maturity, as the authors do, then the children who are retained in practice will not benefit from it. Shepard (2004) echoes this sentiment, arguing that based on the research literature to date, retention does not meet the Food and Drug Administration’s guidelines of a safe and effective treatment as used in the drug approval process.

Some recent studies, not included in Jimerson’s meta-analysis, portray a more favorable view of the effects of retention. Alexander, Entwisle, and Dauber (2003) find positive results of retention on academic achievement in a longitudinal study of Baltimore school children. Their favorable finding is that retention halts the downward slide that retainees were experiencing previously in their academic careers. These positive results only appear for children retained only once in elementary school and after the first grade. However, Alexander, et al. acknowledges that, consistent with other studies, these effects diminish over time. In their critique of the first edition, Shepard, Smith and Marion (1998) note that in the multivariate analyses, the positive effects of retention wash out.

Shepard (2004) raises the following critiques of the analysis. The authors misinterpret the greater test score gains of retained students relative to their peers and fail to fully consider regression to the mean of the test scores or effects of repeated testing. They also overemphasize the beneficial effects experienced by second and third grade retainees while first grade retainees show detrimental effects of retention. Hauser (2005) notes that the analyses performed by Alexander, et al. are circular in nature; they control for events that are subsequent to the retention decision. He adds that effects of race and gender are not reported in the book, and the descriptive statistics reported in the book are wrong because they are unweighted in spite of the oversampling of white, higher status students in the Baltimore study.

Eide and Showalter (2001) use an instrumental variable approach¹ with data from the High School and Beyond (HSB) and the National Education Longitudinal Study (NELS:88). Their study attempts to isolate the effects of retention on high school completion and labor force earnings while avoiding possible simultaneity between retention and academic outcomes. They give a cautiously optimistic reading of findings that indicate beneficial effects of retention that are not statistically significant. Likewise, Jacob and Lefgren (2004) use an instrumental variable approach with administrative data from the Chicago Public School system and find that for those students within a narrow range of the cutoff test score determining retention and summer school, there are modest positive effects on achievement among white third graders and no effects on 6^th graders who have been retained. Consistent with other studies, Jacob and Lefgren’s findings indicate that whatever advantage exists for retained students is temporary; the effects on test scores two years after the repeated grade are smaller than those one year after retention.

Despite these negative or weak findings, the popular sentiment in America is that schools and teachers need to be more accountable to ensure that children progress at appropriate rates (National Education Goals Panel, Goal #8, The National Commission on Excellence in Education, 1983). Many politicians, including Presidents Clinton and Bush, have made direct pleas to end social promotion (Clinton, 1999; Bush, 2004). Hauser (2005, 2004) warns that the annual testing in third through eighth grades mandated by the No Child Left Behind Act (US Congress, 2002) may increase the incidence of test-based retention. That is, whatever the purposes, strengths, or weaknesses of tests, once given they tend to be used to make decisions about test-takers.

There is some agreement that the rate of children who have ever been retained is growing (Hauser, Pager, and Simmons, 2004; Jimerson and Kaufman, 2003; McCoy and Reynolds, 1999; Allington and McGill-Franzen 1992 for New York State; Shepard and Smith² 1989 for 14 states and Washington, D.C.) but it may not affect all subpopulations equally. Indeed retained students tend to be male (Hauser, Pager, Simmons, 2004; Corman, 2003; Zill, Loomis, and West, 1997) , black or Hispanic (Hauser, Pager, Simmons, 2004; Alexander, Entwisle, and Dauber, 2003; Jimerson, Carlson, Rotert, Egeland, and Sroufe, 1997; Zill et al, 1997), younger relative to their peers (Corman, 2003; Shepard and Smith, 1986) and from disadvantaged backgrounds, e.g., from broken families or with lower parental educational attainment and lower family income (Hauser, Pager, Simmons, 2004; Corman, 2003; Hauser, 2001).

Has the pressure to end social promotion translated into more children being retained? A definitive answer to this question is not easy to come up with, as the intense debate implies – and the full testing regime mandated by the No Child Left Behind Act is not yet in place. Shepard (2004) notes that retention and social promotion are not mutually exclusive. The prevalence of policies limiting double retentions means that a student may be retained in one year and socially promoted in following years. Corman (2003) estimates retention rates using the National Household Education Survey (NHES). He finds that, between 1991 and 1996, 10 to 15% of six year olds and 28 to 30% of fifteen year olds have been retained. Wheelock (2005) reports estimates from NELS:88 that 20% of eighth graders have been retained, and 21% of adolescents in the ADD Health Study have been retained.

The first place one would think to look for information about grade retention is the National Center for Education Statistics (NCES). While the Digest of Education Statistics 2003 does not mention social promotion or grade retention, the NCES data have improved since 2002 regarding grade retention. The Condition of Education in 2005 discusses delayed entry to and retention in kindergarten. It compares differentials between on-time kindergarteners, delayed-entry kindergarteners and kindergarten repeaters from the Early Childhood Longitudinal Study – Kindergarten Class of 1998 (ECLS-K). However, there is no report of the overall prevalence of even this single measure of retention and no mention of retention in elementary or secondary school at all.

Hauser (2004, 2001) mentions three possible sources of national-level data where one might find national trends in grade retention over time. The first source of retention information is exemplified by the state data collected by {Shepard & Smith 1989 #530} and, more recently, by the National Research Council (Heubert and Hauser, 1999).³ These data have the advantage of being direct reports from state education agencies of the incidence of grade retention. The disadvantage is exemplified by the NRC report: the average number of years in which the 26 states plus Washington DC provide estimates of retention is 4.22 years, and most of these years occur in the mid-1990s. Thus, the data are no longer timely, and geographic coverage is limited. Shepard and Smith (1989) note that not much about the comparability of state retention data is known, specifically whether the population used in the denominator is from beginning or end of the year enrollment data.

Retention estimates have also been made using the Census (Hauser, 2004) and the October Supplement to the Current Population Survey (CPS) (Hauser, 2004; Hauser, Pager, and Simmons, 2004; Eide and Showalter, 2001; Hauser, 2001; Bianchi, 1984). These estimates yield the longest time span but the drawback is that retention has to be inferred from age and current grade enrollment in the CPS or educational attainment by age in historical census data.⁴ Although it lacks a repeated measure of grade retention or measures of academic achievement, the CPS is the best data with which to construct comparable yearly estimates of the incidence of grade retention on a national level (Hauser 2001). In order to estimate the retention rates in the CPS data, we constructed a measure from the age and currently enrolled grade of each student between the ages of 6 and 17. This measure identifies children who are below the modal grade for their age and has been used to estimate retention rates in prior studies (see Hauser, Page, and Simmons, 2004; Corman, 2003 uses a similar measure with month and year and state cutoff dates to corroborate retrospective reports).

INCIDENCE OF RETENTION

Figure 1 reports the proportion of children at selected ages who are enrolled below the modal grade for their age (hereafter, BMG). In order to smooth the lines, the trend lines are three year moving averages and the horizontal axis indicates the year in which the cohort was born. Read vertically, it shows the within-cohort change in the incidence of BMG. As expected, the proportion of BMG children increases at each age except for a crossover between ages 15 and 17 for the 1983 through 1985 cohorts. Through the 1986 cohort, the last year for which we have complete cohort data, the later cohorts have higher overall incidence levels than the earlier ones. The increase in the incidence of BMG occurred within a few years. On each side of this increase the rates were relatively stable. Cohorts born after 1986 seem to show a decrease in the proportion of children who are behind their peers at ages six and nine.

Does this trend reflect actual changes in the retention rates or is it merely an artifact of increasing age at entry into school or other non-retention phenomena? The lines are very nearly parallel to the trend line for six year olds suggesting that the overall BMG rates are driven by age at school entry and/or by retention at early ages. There is a direct retrospective question about grade retention in the school enrollment section of the CPS but unfortunately it is only asked in 1992, 1995, and 1999. There is yet a third way to construct estimates of retention trends with CPS data. After 1994, respondents were asked not only about the current grade but also the grade in which students were enrolled in the previous year (Eide and Showalter 2001). This measure is different from the other two because it only picks up retentions in the last year, not ever having been retained as the other two measures do. Table 1 compares the constructed BMG measure with the retrospective report of whether the student has ever been retained. The correlation is adequate at 0.45. The correlation coefficient is driven down because the BMG measure identifies more not retained students as being below their modal grade who report that they have never been retained.

Why would the BMG measure err in the direction of false positives? Among other things,⁵ it cannot distinguish between children who had been retained in grade and those whose parents decided to delay school entry. Hauser and colleagues (Hauser 2004; Hauser, Pager, and Simmons 2004, Heubert and Hauser 1999) have used the school enrollment supplements to the CPS to document the rates of children who have been retained between ages 6 and 17. They used 6 year-olds who were below the modal grade for their age as a baseline measure of BMG for that cohort in order to isolate retentions that happened after first grade. By subtracting this cohort baseline for each of the successive age groups, they are able to identify retention that happens after entry into elementary school. Figure 2 updates their series. The horizontal axis is the birth year of the cohort. The trend lines are 3 year moving averages in order to smooth fluctuations.

There is complete cohort data through the 1986 cohort. The BMG rates after age 6 were fairly steady with 8 to 10% of children BMG by age 8, 13 to 17% by age 11, 17 to 21% by age 14, and 22 to 24% by age 17 through the cohort of 1978. Then there was a sharp decline in BMG at all ages to 7%, 11%, 14% and 17% for the four age groups respectively. The rates have seemed to stabilize around these values since then. The most recent data from the two youngest age groups indicate that there might be another upswing in retention during the elementary school grades. Figure 2 corroborates the evidence from figure 1 that the rise in retention may be driven by pre-first grade retentions or delays in school entry. It appears that retentions at older ages are being replaced by retentions at younger ages.

The rise in 6 year olds who are BMG from 1978 to 1992 (the cohorts of 1972 to 1986) coincides with Shepard’s 1989 observation that, “Holding children back in kindergarten in large numbers is a phenomenon of the 1980s.”⁶ She concludes that retention in kindergarten is still retention; some of the increase in BMG 6 year olds is attributable to retention per se. Another reason for the increase is “academic red-shirting.” Marshall (2003) reviews research on academic red-shirting. Whether the decision to red-shirt children is made by their parents alone or with teacher input, the reasons given are similar to those given for retaining children. Either the child needs more time to mature, or the extra year would give a lower performing student a chance to catch up to meet the expectations of kindergarten. The effects of red-shirting reported by Marshall are similar to effects of retention in that there is a temporary advantage to the red-shirted child, but it disappears by third grade. Red-shirted children are also more likely to be placed in special education classes than comparable peers. Because of the similarity of red-shirting and retention, the number of false positives identified among BMG children may not be as great as it appears.

DATA

The CPS data on school enrollment come from a nationally representative probability sample of the civilian, non-institutionalized population each October. While they do not have detailed educational measures, such as the previous year’s enrollment (prior to 1994) or information about transitional or special education classrooms, the data do provide repeated cross sections of the population over a long period of time. The fact that the data are repeated cross-sections, rather than true longitudinal observations does not pose a problem because we are interested in aggregate retention rates, rather than in the consequences of retention for individuals. We are able to construct a uniform CPS file from 1968 onward because a common set of social background questions have been asked every year, along with information about age and grade enrollment of school-aged persons (Hauser & Hauser, 1993; Hauser, Jordan, & Dixon, 1993); however, prior to 1972, the data do not capture Hispanic ethnicity.

Table 2 lists the two sets of covariates used in the analysis: demographic and socioeconomic variables. The demographic covariates include gender, race, age, year of participation, region of the country, metropolitan status and number of siblings. The socioeconomic variables include the education and occupational status of the household head and his or her spouse, family income in the twelve months immediately prior to the survey,⁷ whether or not the child’s family owns their home, and whether the child comes from a broken family.⁸ The education measures varied over time. Through 1991, the variable was measured as years of education from no years of education to 6 years of college or more. From 1992 forward, education was measured as highest category of school or degree completed. Hauser (1997) discusses the incompatibilities and inconsistencies of these two measures of education. For the analysis presented here, we converted the educational credentials to the metric of putative years of school completed. This introduces a degree of measurement error by assuming, for example, that 16 years of education is identical to a bachelor’s degree.

There are two education variables for each ‘parent’ in the final model. The first captures the number of years completed through high school graduation, and the second captures college education. Thus, a person who completed the 10^th grade has a value of 10 for the first variable and 0 on the second. The corresponding scores for high school graduates with no college and people with a bachelor’s degree are 12, 0 and 12, 4, respectively. This scheme was used to allow for piecewise linear effects of education before and after the high school to college transition, especially in light of the differences in college attendance over time among the parents in this sample.

The CPS data include 864,878 children aged 6 to 17 clustered within 256,608 households over 32 years. All of the models reported here were estimated using robust standard errors that have been adjusted for household clustering. There were missing data in some observations on seven continuous variables. The fourth and fifth columns of Table 2 show the differences between the data without missing data and the data with imputations. The final column of Table 2 shows the total number of non-missing observations for each variable. Appendix A shows the percent missing on each variable by year.

There are two types of missing data on the variables used in this analysis. Data on household income and head’s education are sometimes missing but there are real values associated with these characteristics out in the world.⁹ We used multiple imputation to replace this type of missing data. When data are not missing completely at random (MCAR), listwise deletion can yield biased results (Allison, 2001:6). Multiple imputation yields consistent, asymptotically efficient, and asymptotically normal estimates under the weaker assumptions that the data are missing at random (MAR) and the model is correctly specified (Allison, 2001). Of the two variables with this type of missing data, income is the only one for which there is evidence that it is not MAR. However, only seven percent of cases lack income data, so this violation should not strongly affect the estimates.

The data on spouse’s education and occupational status as well as head’s occupational status are missing for a different reason; values do not exist because of unemployment or because there is no spouse in the household. In order to account for this type of missing data we used a dummy variable adjustment technique. We substitute the missing data on these variables with mean values of non-missing cases, conditional on the age of their child and the period of the survey. Then we included a dummy variable in each model that we estimated, indicating whether the observation had missing values. Where missing observations truly do not exist, this method of accounting for missing data is statistically sound (Allison, 2001; King, Honaker, Joseph, and Scheve, 2001).

MODEL

There are two prototypical ways to estimate trends over time and age using conventional logistic regression (logit) models. Equation 1 is a logit model with BMG as the response variable, where i indexes individuals, j indexes age, and k indexes time periods. The social background variables are indexed by l. The 32 years are divided into eight groups of four years that coincide with presidential terms. In order to accomplish this division, the first period covers five years, and the last period covers three years. Thus, the alphas represent separate intercepts for each age*period combination, the variables of interest. This model is unsatisfactory because the only evidence of change is in the different levels of the intercepts; the effect sizes do not change.

In a more nuanced treatment, Hauser, Pager, and Simmons used separate logistic regressions by age to examine the effects of demographic and social background variables as well as time on being BMG. This allowed them to look at how the effects differed among children aged 6, 9, 12, 15, and 17. A similar set of logistic regressions is required to assess differences in how the effects of social background variables change with age and time. Equation 2 is the full interaction model in which the social background effects can vary independently in each age*period category. Again, the alpha term represents differential intercepts. Now the beta term represents the effects of social background variables at the baseline category, age = 6 and period = 1972 – 1978. The gamma term contains the differences in effects over age and period. In each of these models, the effects vary independently over each joint category of age and period. This model allows all the effect sizes to vary unconstrained but uses many more of degrees of freedom, especially as the number of categories of j and k gets bigger.

An equivalent method is to estimate the logit model in equation 1, leaving out the intercepts in the alpha term separately for each age*period combination. Either way, this model yields unwieldy results because there are more than 2000 estimated parameters.

Equation 3 is a stereotype logit model (SLM) the main model of interest in this paper.¹⁰ The alpha term represents an intercept for each level of age, indexed by j, and time period, indexed by k. The next term consists of a vector of parameters, β, for the vector of explanatory variables x_ijkl. The final two terms, containing λ_j and λ_k, capture the proportional change in the linear predictor separately for each level of j and k. The proportional change, λ_j, is the factor by which the linear predictor changes as age increases, while λ_k is the factor by which the linear predictor changes for each time period.

The advantage of this model is that it is more detailed than the model that constrains change in the effects to be identical across ages and time, but more parsimonious than the full interaction model in which effects may vary independently by age and by time.

If λ is positive, it means that the effects of the explanatory variables increase in magnitude. Similarly if λ is negative, the effects decrease in magnitude. This can be shown by factoring the linear predictor out of the final three terms:

The λ variables can either be continuous, ordinal, or nominal. In order to test for model fit, we ran four different models in which we treated both age and period as continuous variables, both as nominal variables, and one as nominal while the other was continuous. When the lambda variable is continuous, λ₅ is half the size of λ₁₀. When the lambda is nominal, the interpretation of λ_k is relative to an omitted baseline category.

FINDINGS

Morris (1993) found that retention rates across grade levels were not linear but were better described by a negative growth exponential function. To check for such non-linearities, we treated both age and period as nominal variables. The parameter estimates from the simple logistic model and the SLM model are shown in Table 3. Each of the models tells us two things about the trends in grade retention as measured by the proportion of children who are BMG. The intercepts indicate the level of retention for each age group and each period, and the coefficients describe how the effects of demographic and socioeconomic characteristics on retained students have, or have not changed (in the case of the simple logit model).

Table 4 lists the F-statistics of the model comparison test for multiply imputed data proposed by Allison (2001:68). Both the SLM and full interaction models are preferred over the logit model, and the full interaction model is preferred over the SLM model. The statistical evidence implies that the demographic and socioeconomic effects have changed and that this change was not captured very well by the SLM model.

However, given the statistical power that a sample size of nearly 900,000 cases imparts, it is not surprising that the model with the most parameters is the preferred model. Rather than presenting a table with the 2,112 coefficients in the full interaction model, we first review estimates from the simple logistic model and then focus on estimates from the SLM model. A later section explores similarities and differences in the change in coefficients between the SLM and full interaction models.

MAIN FINDINGS

While the simple logistic model is patently crude, the estimates are still of substantive interest because they provide a general, baseline picture of the sources of variation in age-grade retardation. Across all ages combined, the demographic and socioeconomic effects are similar to those in earlier analyses of data from the October CPS {Hauser, Pager, et al. 2004 #420}. With all other variables controlled, age-grade retardation is negligibly different between African Americans and non-Hispanic whites (the reference group). Hispanics and persons of other races are actually less likely than non-Hispanic whites to be age-grade retarded. Males are far more likely to fall below the modal grade than females. Central cities of major metropolitan areas (the reference group) have slightly higher levels of age-grade retardation than their suburbs, but lower levels than smaller central cities, their suburbs, or other (non-metropolitan) areas. Age-grade retardation is at similar levels in the North (the reference category) and West regions, but it is substantially higher in the Midwest and South. Children and youth whose parents own their homes and who have higher incomes are substantially less likely to fall behind in school, and the effects of parent’s educational attainments and occupational statuses are equally salutary. Age-grade retardation increases modestly if a parent is missing from the child’s household, and it increases with each additional child in the household.

Figures 3 and 4 display the age and period intercepts of the simple logistic model. As one should expect, age-grade retardation increases regularly with age, but a decreasing rate from ages six through fourteen (ignoring an anomalous, small decline between ages 13 and 14), and it increases sharply at the beginning of the high school years. The most striking finding from this simple model (Figure 4) is that, net of other demographic and socioeconomic characteristics, age-grade retardation increased sharply between the early 1970s and 1990. That is, other things being equal, the odds of being behind the modal grade for age almost doubled between 1972-1976 and 1989-1992. Briefly, because social background characteristics have become more favorable – largely because of increases in parents’ educational attainments and a smaller number of children in the home – the observable decline in age-grade retardation (in figures 1 and 2) did not merely disappear, but was reversed. After the early 1990s, there has been a very modest decline in age-grade retardation, and no “end to social promotion.” Even in 2001-2003, the overall odds of age-grade retardation were almost 80 percent higher than in 1972-1976.

Within the SLM model, it is far more difficult to describe overall trends in levels of age-grade retardation because the effects of social background variables vary with age and period. Rather, we focus on the ways in which the effects of demographic and socioeconomic variables vary across ages and periods. The main feature of the SLM model is that there are two common factors by which the effects of each of the 21 demographic and socioeconomic variables vary across ages and periods. One describes gradients by age, and the other describes gradients by time period.

Figure 5 shows variation with age in the effects of the socioeconomic variables on age-grade retardation. On a logarithmic scale, there is a sharp increase in the average effects between ages 6 and 7 followed by a lesser, but nearly linear increase in the logistic regression coefficients from age 7 to age 17. That is, there is exponential growth in the effects of demographic and socioeconomic variables on age-grade retardation. The effects of the background variables are almost 8 times larger at age 17 than at age 6.

Figure 6 shows that there is an opposite secular trend in effects of the demographic and socioeconomic variables on age-grade retardation. There is a modest decline in the effects from 1972-1976 to 1985-1988, followed by a sharp decline until 1997-2000, and a leveling of the decline to 2001-2003. Over the entire period, there has been more than a 60 percent decline in typical effects of social and economic variables on age-grade retardation.

The combined influence of age and period changes in the effects of social background variables is illustrated in table 5. The estimates are based on the SLM models. Because of the constraints of the SLM model, all of the variables that follow the dominant pattern have similar trends in their effect sizes. (Indeed, all of the variables in the beta term of the model are constrained this way.) Because the model was set up with additive lambda terms it is easy to examine the trends in age and period separately. Recall from the rearrangement of the SLM model in equation 4 that the coefficient size changes by a factor of (1+λ_j+λ_k). The lambda coefficients for both age and period are less than one so the direction of the effect is determined by the sign of the beta coefficient.

As table 5 shows, the effects are in the expected directions: boys are more likely to be retained than girls; each additional child in the household increases the likelihood of being BMG and higher levels of education and income decrease the likelihood of being BMG.

Because the lambda at each age is positive and increasing, the effect of each background variable increases with age. Net of period effects, as children age the effects of gender, income, home ownership, number of children in the household, and head’s education and occupational status increase. Boys are more likely to be retained than girls, children with less financial resources more likely than those with more, and so on. The never retained group becomes more homogeneous as children age because of the differential retention rates at every age.

Recall that the lambda values for period are all negative and decline at three different rates. Unlike age, the differences between those who are BMG and not are decreasing as time passes and this has especially been the case after 1989. The slowdown after 2000 might signal stabilization in the effects of social background variables, but only time will tell.

The group of ever retained children more closely resembles the group of never retained children in terms of social background characteristics now than it did from 1972 through 1976. One of two things has taken place: either fewer traditionally retained students are being held back, or more traditionally promoted students are being held back. Coupled with the rising overall incidence of being BMG over the same time period, it would appear that the latter is the case. There would be tenuous evidence that efforts to end social promotion since 1972 have been effective, if more of the traditionally promoted children are being retained.

In summary, while the effects of social and economic background variables on progress through school become increasingly powerful as children grow older, that typical pattern has been attenuated across the past three decades by a steady, secular decline in the influence of those variables across all ages. A great deal of concern has been expressed about rising levels of economic and social inequality in the United States since the middle 1970s, and about the potential intergenerational effects of such inequality. However, there has been an opposite trend in the effects of social origins on age-grade retardation, which is one of the more important indicators of progress through schooling. To be sure, a trend is not a law, and there is reason to be concerned about the recent deceleration of the secular decline in effects of social background.

MODEL FIT

The underlying assumption of the SLM model as specified in equation 3 is that there are two common factors by which the effects of each of the 21 parameters in the beta term change in the dimensions of age and time. Given that the statistical tests have rejected the SLM model in favor of the full interaction model, why doesn’t the SLM model fit? The possibilities include the following: each of the 21 parameters changes unlike any of the others, there are two groups of variables that each follow distinct trends, or the case that the changes in age are not uniform in each period and vice versa, to name a few. We examined differences between coefficients of the SLM model and of the unrestricted (full interaction) model at each age*period level in order to understand why the SLM model is rejected.

In order to make these comparisons, we needed criteria with which to judge whether or not the SLM model adequately approximated the trends in the full interaction model. We developed these criteria by looking at surface plots of the implied coefficients of each model, by examining differences between the two coefficients (the residuals),¹¹ and by comparing the tables of the coefficients themselves.

For example, figure 7 shows surface plots of the relationships among age, period, and the effect of household head’s primary and secondary schooling on a child’s age-grade retardation. In this case, the shape of the full interaction surface (top panel) appears to approximate that of the SLM estimates (middle panel), and we see no particular pattern in the residual coefficients by age or period.

Our first criterion of fit has to do with the shape of the SLM surface relative to the shape of the full interaction surface. (These are relative criteria because each coefficient is in its own metric.) The first question is whether there is any trend in the SLM model. If the SLM surface is relatively horizontal in both dimensions, or ‘flat’ for short, while the full interaction model moves around systematically, then the SLM model is not picking up any trend. A second question regarding the SLM surface is, if it is not flat, does it follow the general trend that the full interaction model follows? More succinctly, does it tilt the same direction along both dimensions? If the SLM and full interaction surfaces do not have a similar tilt, then the SLM model does not approximate the full interaction model.

The second criterion utilizes the information in the surface plots of the coefficient residuals, compared to the full interaction surfaces. Is the residual surface flatter than the full interaction model? If this is the case, the trend in identified by the SLM model accounts for the overall trend in the full interaction model. Is the residual surface closer to zero? If this is the case, the SLM model not only captures the trend but also estimates the magnitude of the effects as well.

Table 6 summarizes our judgments using the criteria listed above for each demographic and socioeconomic background variable in the beta term of equation 3. If the comparison met all the criteria, then the SLM model does adequately approximate the full interaction model for that variable. The last column in table four lists the divergences of the two models if the SLM performed poorly on that variable or whether the SLM trend over- or underestimated the observed trend. Overestimation in this case means that the residual surface tilted down in the opposite direction, back and to the right instead of up and to the left of the full interaction model. Underestimation occurs if the surfaces tilt in the same direction but the residual surface is more horizontal.

We will not discuss our findings about each demographic and socioeconomic variable in detail here. Rather, we simply note that the SLM model works rather well at capturing age and period variations in the effects of a core set of demographic, socioeconomic, and family variables: gender, income, home ownership, head’s K-12 education, head’s occupational status, spouse’s occupational status, and number of children in the household. In other cases, the trends are unclear, e.g., the other measures of parents’ educational attainments, or the SLM model plainly fails to capture trend, e.g., in the case of the geographic variables (metropolitan location and region), race-ethnicity, and non-intact family.

In our judgment, we will get sharper and more valid estimates of trend by re-estimating the model, excluding the proportional restrictions on effects of variables for which the current SLM specification does not fit. We plan to explore two ways of doing this. One is to let the effects of geography, race-ethnicity, and non-intact family vary freely by age and period. That is, we will introduce all interactions of the effects of those variables with age and with period while maintaining the specification of proportional change in other effects. A second alternative specification will introduce proportional effects that are specific to one or more of the variables that do not follow the overall trend. A third possibility, possibly combined with the other two, will be to impose and test the restriction of no trend by age or period, e.g., in the case of race-ethnicity.

CONCLUSION

After controlling demographic and socioeconomic background variables, the incidence of children being below the modal grade for their age has increased overall from 1972 to 2003. This is a serious matter because age-grade retardation is a major, proximate cause of high school dropout. The composition of the BMG population as measured by the effects of demographic and socioeconomic variables does not change in a single uniform manner but three trends emerge from the data. The dominant pattern of change, as measured by the SLM model, is driven by gender and socioeconomic factors. At least two other trends appear to be behind trends in region of residence and city status. A possible next step in this investigation¹² is to examine results from a model similar to the SLM model in equation 3 but with three distinct beta terms each with its own set of lambda estimates to allow the city and region variables to follow their own trends.

The dominant trend identified by the SLM is a useful tool in describing the general pattern of change in effects of social background variables. This model yields diminishing effect sizes over time. What does this tell us about social promotion? The retained and never retained groups are increasingly more similar in social background, especially after 1988. The trend estimated by our relatively simple SLM model is rejected in favor of the full interaction model. While this does not change the trends in effects of the variables that follow the dominant trend, several effects do have different trends under the full interaction model, most notably race, region, city status, and the spouse’s K-12 education. We think that it will be informative to improve the specification of our model and to continue to follow trends and differentials in age-grade retardation as the effects of No Child Left Behind come to dominate educational outcomes in primary and secondary schools.

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Table 1: Comparison of Retention Measures in the CPS
BMG	Direct Measure of Repetition
	Yes	No
Yes	5,768	12,418
No	822	49,792
Correlation	0.4509

Table 2 - Covariate Descriptives
Variable	Nonmissing		Imputed		Nonmissing obs
Variable	Proportion	SD			Nonmissing obs
White	.7397	0.44			864878
Black	.1340	0.34			864878
Hispanic	.0855	0.28			864878
Other Race	.0409	0.20			864878
Male	.5103	0.50			864878
Major Central City	.0900	0.29			864878
Major Suburb	.1338	0.34			864878
Other Central City	.1312	0.34			864878
Other Suburb	.2172	0.41			864878
Rural	.4278	0.49			864878
East	.2135	0.41			864878
Midwest	.2588	0.44			864878
South	.2992	0.46			864878
West	.2285	0.42			864878
Home Ownership	.7180	0.45			864878
Broken Family	.2172	0.41			864878
	Mean	SD	Mean	SD
Total Children in the Household	2.73	1.45			864878
Logged Income	9.91	0.85	0.30	0.46	803533
Head's K-12 Education	11.10	2.07	11.10	2.07	864054
Heads Postsecondary Education	1.27	1.96	1.27	1.96	864054
Spouse's K-12 Education	11.29	1.82	11.30	1.58	646356
Spouse's Postsecondary Education	1.04	1.72	0.94	1.62	646356
Head's Occupational Status	37.89	20.33	37.89	19.18	769035
Spouse's Occupational Status	39.54	18.81	39.39	13.22	420439

Table 3: Effects of Age, Period, Demographic and Socioeconomic Variables on Age-Grade Retardation, 1972 to 2003:

Logistic Regression (Equation 1) and Stereotype Logit Model (Equation 3)

Variable		Logistic Regression		Stereotype Logit
Variable		Coefficient	Std. Error	Coefficient	Std. Error
Alpha j	Age 7	0.4795	0.01481	1.5277	0.07791
	Age 8	0.6484	0.01582	2.0567	0.08379
	Age 9	0.7812	0.01563	2.3550	0.08504
	Age 10	0.8508	0.01558	2.6306	0.08557
	Age 11	0.9345	0.01547	2.9045	0.08702
	Age 12	0.9744	0.01546	3.0756	0.08844
	Age 13	1.0206	0.01542	3.2102	0.08942
	Age 14	0.9903	0.01548	3.4806	0.09250
	Age 15	1.1223	0.01535	3.8274	0.09492
	Age 16	1.2552	0.01529	4.1555	0.09829
	Age 17	1.2867	0.01553	4.3638	0.10193
Alpha k	1977-1980	0.1097	0.01293	0.0435	0.07002
	1981-1984	0.2923	0.01314	0.1455	0.07199
	1985-1988	0.4889	0.01324	0.2239	0.07343
	1989-1992	0.6441	0.01284	0.0053	0.07197
	1993-1996	0.6262	0.01296	-0.3811	0.07414
	1997-2000	0.6279	0.01412	-0.7859	0.08304
	2001-2003	0.5864	0.01618	-0.8654	0.09486
Beta	African American	0.0103	0.01128	0.0079	0.00543
	Hispanic	-0.0686	0.01428	-0.0047	0.00708
	Other Race	-0.1147	0.01913	-0.0324	0.00983
	Male	0.4855	0.00632	0.2305	0.01202
	Major Suburb	-0.0360	0.01628	-0.0402	0.00803
	Other Central City	0.1957	0.01522	0.0651	0.00801
	Other Suburb	0.1294	0.01483	0.0340	0.00735
	Non-Metropolitan	0.2475	0.01391	0.0737	0.00784
	Midwest	0.2366	0.01062	0.0900	0.00715
	South	0.2112	0.01054	0.0939	0.00684
	West	-0.0350	0.01148	-0.0175	0.00553
	Logged Income	-0.1966	0.00533	-0.0995	0.00567
	Home Ownership	-0.2332	0.00849	-0.1275	0.00757
	Broken Family	0.0539	0.01066	0.0379	0.00480
	Head's K-12 Education	-0.0604	0.00197	-0.0280	0.00164
	Head's Postsecondary Education	-0.0208	0.00276	-0.0121	0.00145
	Spouse's K-12 Education	-0.0601	0.00257	-0.0296	0.00190
	Spouse's Postsecondary Education	-0.0136	0.00331	-0.0128	0.00172
	Head's Occupational Status	-0.0040	0.00024	-0.0018	0.00015
	Spouses Occupational Status	-0.0032	0.00031	-0.0014	0.00016
	Number of Children in Household	0.0800	0.00260	0.0375	0.00221

Lambda j	Age 7			0.7128	0.07576
	Age 8			0.9605	0.09067
	Age 9			1.0740	0.09617
	Age 10			1.2141	0.10255
	Age 11			1.3443	0.10856
	Age 12			1.4327	0.11336
	Age 13			1.4923	0.11628
	Age 14			1.6955	0.12610
	Age 15			1.8362	0.13295
	Age 16			1.9587	0.13930
	Age 17			2.0709	0.14504
Lambda k	1977-1980			-0.0546	0.04639
	1981-1984			-0.1105	0.04632
	1985-1988			-0.1948	0.04558
	1989-1992			-0.4414	0.04224
	1993-1996			-0.6813	0.04505
	1997-2000			-0.9372	0.05453
	2001-2003			-0.9590	0.06208
Missing Head Occupation Indicator		0.1064	0.01106
Missing Spouse Occupation Indicator		0.0290	0.00878
Constant		0.4866	0.05958	-0.9041	0.08062

Table 4: F-statistics for Nested Model Tests
Test	Stat	df1	df2	95% CV
Simple nested in SLM	225.77	16	2466366	1.57*
SLM nested in full interaction	3.51	2246	1144970	1
Simple nested in full interaction	5.08	2264	1826113	1
*This is the critical value for df1 = 20

Table 5: Odds ratios for the Dominant Pattern variables for a representative group of ages and periods
Dominant Pattern	Male
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	1.259	1.612	1.752	1.922	2.029
		1989-1992	1.137	1.456	1.583	1.737	1.833
		2001-2003	1.010	1.293	1.405	1.541	1.627
	Logged Income
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.905	0.814	0.785	0.754	0.737
		1989-1992	0.946	0.850	0.820	0.788	0.770
		2001-2003	0.996	0.895	0.864	0.830	0.811
	Home Ownership
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.880	0.768	0.733	0.696	0.676
		1989-1992	0.931	0.812	0.776	0.737	0.715
		2001-2003	0.995	0.867	0.828	0.787	0.764
	Head's K-12 Education
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.972	0.944	0.934	0.924	0.917
		1989-1992	0.984	0.955	0.946	0.935	0.929
		2001-2003	0.999	0.969	0.959	0.949	0.942
	Head's SEI (100 points)
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.832	0.683	0.640	0.594	0.569
		1989-1992	0.903	0.741	0.694	0.644	0.617
		2001-2003	0.992	0.815	0.763	0.708	0.679
	Spouse's SEI (100 points)
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.868	0.745	0.708	0.668	0.647
		1989-1992	0.924	0.793	0.754	0.712	0.688
		2001-2003	0.994	0.854	0.811	0.766	0.741
	Number of Children in Household
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	1.038	1.081	1.096	1.112	1.122
		1989-1992	1.021	1.063	1.078	1.094	1.104
		2001-2003	1.002	1.043	1.057	1.073	1.083

Table 5: Odds Ratios for City Status, Region, Spouse K-12 Education and African American variables for a representative group of ages
City Status	Major Suburb
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.932	0.919	0.958	0.778	0.743
		1977-1980	1.178	0.914	0.792	0.678	0.708
		1981-1984	1.450	1.063	0.704	0.631	0.686
		1985-1988	1.541	1.207	1.224	0.866	0.774
		1989-1992	1.194	1.514	1.173	0.920	0.722
		1993-1996	1.590	1.113	1.232	1.001	0.826
		1997-2000	1.414	0.947	1.075	0.760	0.786
		2001-2003	0.933	1.062	1.395	1.112	1.115
	Other Central City
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	1.177	1.061	1.185	0.938	0.910
		1977-1980	1.396	1.284	0.792	0.955	0.938
		1981-1984	2.410	1.386	0.704	0.729	0.738
		1985-1988	1.762	1.615	1.224	0.999	0.925
		1989-1992	1.190	1.929	1.173	1.469	1.098
		1993-1996	1.806	1.364	1.232	1.690	1.347
		1997-2000	1.599	1.061	1.075	1.410	1.343
		2001-2003	1.115	1.445	1.395	1.686	1.237
	Other Suburb
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	1.106	0.943	1.169	0.916	0.900
		1977-1980	1.369	1.100	1.078	0.877	0.848
		1981-1984	2.024	1.484	1.110	0.783	0.757
		1985-1988	1.904	1.379	1.160	1.001	0.976
		1989-1992	1.245	1.924	1.330	1.108	0.865
		1993-1996	1.791	1.251	1.181	1.333	0.990
		1997-2000	1.636	1.013	1.405	1.083	0.984
		2001-2003	1.084	1.197	1.744	1.445	1.141
	Rural
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	1.441	1.074	1.112	0.777	0.784
		1977-1980	1.788	1.369	1.203	0.974	0.808
		1981-1984	2.477	1.449	1.069	0.694	0.741
		1985-1988	2.337	1.713	1.272	0.971	0.863
		1989-1992	1.579	1.950	1.613	1.370	0.858
		1993-1996	2.173	1.477	1.499	1.602	1.090
		1997-2000	2.198	1.258	1.473	1.319	1.124
		2001-2003	1.347	1.502	1.853	1.538	1.365
Region	Midwest
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	2.297	1.267	1.237	1.100	1.037
		1977-1980	2.456	1.442	1.249	0.995	1.275
		1981-1984	1.991	1.339	1.280	1.051	1.095
		1985-1988	1.770	1.238	0.973	1.126	1.008
		1989-1992	1.789	1.298	1.366	1.281	1.030
		1993-1996	1.958	1.560	1.236	1.002	1.177
		1997-2000	2.062	1.646	1.441	1.384	1.315
		2001-2003	2.086	1.224	1.496	1.429	1.061
	South
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.954	1.086	1.267	1.322	1.160
		1977-1980	1.195	1.183	1.166	1.155	1.411
		1981-1984	1.287	1.263	1.286	1.405	1.198
		1985-1988	1.215	1.119	1.188	1.425	1.358
		1989-1992	1.174	1.111	1.305	1.382	1.209
		1993-1996	1.152	1.370	1.257	1.161	1.130
		1997-2000	1.482	1.443	1.261	1.262	1.256
		2001-2003	1.561	1.234	1.271	1.402	0.972
	West
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	1.314	1.106	0.913	0.937	0.919
		1977-1980	1.617	1.001	1.067	0.916	1.234
		1981-1984	1.401	1.057	1.125	0.875	1.178
		1985-1988	1.319	0.886	0.805	0.897	0.916
		1989-1992	1.326	0.971	0.914	0.874	0.886
		1993-1996	0.987	1.063	0.842	0.749	0.798
		1997-2000	1.197	0.987	1.001	0.854	0.999
		2001-2003	1.556	0.949	0.961	0.866	0.787
	Spouse's K-12 Education
			Age 6	Age 9	Age 12	Age 15	Age 17
		1972-1976	0.965	0.919	0.930	0.901	0.894
		1977-1980	0.948	0.917	0.894	0.901	0.898
		1981-1984	0.979	0.926	0.940	0.931	0.919
		1985-1988	1.010	0.950	0.948	0.928	0.947
		1989-1992	1.010	0.991	0.946	0.925	0.944
		1993-1996	1.038	1.019	0.964	0.978	0.942
		1997-2000	0.942	0.973	0.997	0.958	0.959
		2001-2003	0.955	1.011	1.043	0.991	0.968

Figure 7: Surface Plots for the effects of Head’s K-12 Education

Appendix A: The percentage of missing values on each variable by year.
Year	Income	Head's Education^a	Head's Occupational Status	Spouse's Education^a	Spouse's Occupational Status
1972	6.00	0	10.15	18.26	60.52
1973	6.95	0	9.87	18.41	58.94
1974	6.72	0.03	10.92	19.70	58.54
1975	7.50	0	11.23	20.60	58.08
1976	7.86	0	10.74	20.32	56.77
1977	7.78	0	10.46	20.87	55.09
1978	7.95	0	10.55	22.16	53.57
1979	7.46	0	9.72	22.25	52.68
1980	5.91	0	10.74	23.17	53.06
1981	4.78	0	10.82	23.96	52.47
1982	4.97	0	10.88	24.49	51.87
1983	4.30	0	10.95	24.92	52.45
1984	4.38	0	11.10	25.18	51.06
1985	3.56	0	11.09	25.36	49.61
1986	2.36	0	10.41	26.00	49.15
1987	3.43	0	10.79	26.44	48.94
1988	4.88	0	10.99	26.84	48.04
1989	6.40	0	10.98	26.89	47.27
1990	6.60	0	11.01	26.93	47.61
1991	5.22	0.13	12.49	27.78	48.26
1992	5.33	0.14	12.44	28.31	48.05
1993	5.68	0.18	11.81	28.17	47.44
1994	6.93	0.08	13.20	28.51	47.50
1995	8.64	0.26	13.16	28.38	47.18
1996	8.57	0.17	11.91	28.52	46.52
1997	8.85	0.17	11.49	28.72	47.47
1998	9.57	0.33	10.83	29.27	47.82
1999	10.82	0.40	11.18	29.68	48.47
2000	12.66	0.33	11.09	30.14	49.02
2001	12.21	0.35	10.89	30.22	48.43
2002	12.31	0.39	10.97	30.75	49.68
2003	14.70	0.48	11.48	30.27	49.79
^a The two variables for both head and spouse's education are constructed from one variable so they are missing on the same cases

Appendix B: Sample Stata code to estimate the stereotype logit model

Each period represents a new line in the program

.ml model lf slm_ml18 (alpha: bmg = age7 age8 age9 age10 age11 age12 age13 age14 age15 age16 age17 yr7780 yr8184 yr8588 yr8992 yr9396 yr9700 yr0103) (beta: black hispanic othrace male majsub othercc othersub other midwest south west lninc ownhome brkfam headed headcolge spsed spscolge hsei ssei totkids, nocons) (lambda1: age7, nocons) (lambda2: age8, nocons) (lambda3: age9, nocons) (lambda4: age10, nocons) (lambda5: age11, nocons) (lambda6: age12, nocons) (lambda7: age13, nocons) (lambda8: age14, nocons) (lambda9: age15, nocons) (lambda10: age16, nocons) (lambda11: age17, nocons) (lambda12: yr7780, nocons) (lambda13: yr8184, nocons) (lambda14: yr8588, nocons) (lambda15: yr8992, nocons) (lambda16: yr9396, nocons) (lambda17: yr9700, nocons) (lambda18: yr0103, nocons), maximize difficult robust cluster(hhid)

.ml display

The first command calls the following program, slm_ml18.ado, to do the estimation

.program define slm_ml18

.args lnf alpha beta lambda1 lambda2 lambda3 lambda4 lambda5 lambda6 lambda7 lambda8 lambda9 lambda10 lambda11 lambda12 lambda13 lambda14 lambda15 lambda16 lambda17 lambda18

.tempname theta

.gen double `theta' = `alpha' + `beta' + (`lambda1'*`beta') + (`lambda2'*`beta') + (`lambda3'*`beta') + (`lambda4'*`beta') + (`lambda5'*`beta') + (`lambda6'*`beta') + (`lambda7'*`beta') + (`lambda8'*`beta') + (`lambda9'*`beta') + (`lambda10'*`beta') + (`lambda11'*`beta') + (`lambda12'*`beta') + (`lambda13'*`beta') + (`lambda14'*`beta') + (`lambda15'*`beta') + (`lambda16'*`beta') + (`lambda17'*`beta') + (`lambda18'*`beta')

.quietly replace `lnf' = ln(exp(`theta')/(1+exp(`theta'))) if $ML_y1==1

.quietly replace `lnf' = ln(1/(1+exp(`theta'))) if $ML_y1==0

.end

¹ They use the difference in days between the cutoff date of kindergarten entry and the child’s birthday as their instrumental variable.

² table 1 p 6

³ High Stakes Testing table 6-1.

⁴ Eide and Showalter (2001)compare the grade enrolled in last year with the grade enrolled in this year. These data however are only available in the CPS in and after 1994. Below is a comparison of the two measures.

⁵ Other reasons could be children whose birthday occurs between state mandated cutoff dates and the administration of the survey or children who have missed a year of school due to health reasons.

⁶Shepard, 1989:65

⁷ Family income is collected in ranges in the October CPS that have changed over the years. The variable is the natural log of the midpoints of these ranges adjusted to constant dollars with the CPI-U series published by the BLS.

⁸ This is defined as not living with a mother and father (female head with male spouse or male head with female spouse).

⁹ A third type of missing data is present but is imputed using a hot deck method by the Census Bureau. For the sake of this analysis, we treat these imputations as unproblematic.

¹⁰ The model in equation 3 is easily estimated with standard statistical software. Appendix B contains an example of the Stata code used in the estimation. This model is similar to the stereotype models discussed by Anderson (1984) and Lunt (2001). The similarities between our model and these models came to our attention after we developed the code in Appendix B.

¹¹ See Hauser and Andrews 2005 for a similar comparison using change along one dimension.

¹² Currently underway

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