Is History Destiny? Resources, Transitions and Child Education Attainments in Canada - December 2002

7. The Impact of Earlier Attainments on Subsequent Attainments

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In chapters 5 and 6, we considered the impacts of 'history' — as represented by characteristics at the time of cycle 1 — and transitions — as represented by shocks or changes between cycles 1 and 2, 2 and 3 and 1 and 3, on child attainments. In this chapter, we complement that analysis by considering another dimension of 'history', namely the impact of a child's earlier attainments on what she subsequently achieves. We have already had a glimpse of this in the analysis presented in chapter 4 where we saw that there were correlations between attainments measured in cycle 1 and attainments in cycle 3. However, while suggestive, such analysis is hardly conclusive because such correlations may be driven by some third factor that is not taken into account in that bivariate analysis.

In this chapter, we use a multivariate analysis of the relationship between earlier attainments, as measured in cycles 1 and 2, and attainments as measured in cycle 3. That is, we estimate relationships of the following form:

Where Y_ij is the attainment of child i, living in household j, , and y are parameters to be estimated, X_ij is a vector of child, pmk and household characteristics, Z_ij is a measure of earlier attainments and u_ij is a disturbance term. Before continuing, however, we must note the following concern. Z_ij is not an exogenous variable; in fact it is an outcome determined by child, pmk and household characteristics in earlier periods as well as unobserved characteristics that are absorbed into u_ij. As a result, it is reasonable to expect that E(Z_iju_ij) 0. For example, children with a greater interest or aptitude for reading (an unobservable characteristics that is absorbed into u_ij) will, holding all other factors constant, have better attainments as measured both by Y_ij and Z_ij. Consequently, all parameter estimates are vulnerable to bias.

Our results are reported in Tables 5 and 6a to 6f For the purposes of comparison, the first column replicates the core findings of Table 2, which we call 'specification' (1). Next, we report the results obtained by treating the earlier attainments as exogenous. In specification (2), we include the attainment measured in cycle 1. For children aged 8 and 9 in 1998 (i.e. children 4 or 5 at the time of cycle 1), this is the PPVT score. For all older children, the first stage attainment is the math score. In specification (3), the earlier attainment is the test score from cycle 2. In the case of reading, this is the reading score obtained during cycle 2 and in the case of mathematics, it is the math score obtained during cycle 2. An attraction of this approach is that we can compare the impact of other characteristics, such as pmk education and household income, once these earlier attainments are taken into account. We then introduce two changes to this specification. First, we treat earlier attainments as endogenous, using two different sets of instruments. The first set of instruments are the 'shocks' and 'transition' events that occur between 1994 and 1996. These are the development, or loss of a child's activity limitation, the acquisition or loss of full or step siblings, changing school or care giver, pmks marrying, divorcing, moving or gaining an educational diploma and the household moving into or out of poverty. Second, we include as an additional set of regressors the same set of shock and transition variables, but for the period 1996 to 1998 (i.e. changes between cycles 2 and 3). Lastly, note that because we do not have an 'attainment' for children aged 0-2 at the time of the start of the NLSCY, we restrict our attention to the three older cohorts. Even with this, there are four new specifications for three cohorts, each with two attainments. Consequently, we describe our results thematically.

Table 5
PPVT Score for Children 4-6

	(1)	(2)
PMK characteristics
Age	0.40 (4.07)**	0.41 (3.97)**
Did not complete high school	-3.86 (3.07)**	-3.75 (3.02)**
Obtained post-high school diploma	1.40 (1.59)	1.28 (1.44)
Obtained university degree	3.66 (2.95)**	3.26 (2.60)**
Lone Parent	1.13 (0.91)	1.24 (1.01)
Household characteristics
Income (x 1000)	0.23 (3.87)**	0.23 (3.71)**
Income squared (x 100000)	-0.00009 (1.81)*	-0.00009 (1.58)
Earlier attainments
Low birth weight		-4.57 (2.38)**
Mean, dependent variable	98.6
Notes: (1) Specification reported in Table 2. (2) Specification reported in Table 2 plus PPVT score (treated as being exogenous) from round 1.

 

Table 6a
Math Score for Children 8, 9

	(1)	(2)	(3)	(4)	(5)
PMK characteristics
Lone Parent	-18.14 (0.84)	-5.56 (0.64)	-22.89 (2.27)**	-16.19 (1.02)	-
Age	0.27 (0.51)	0.37 (0.64)	0.69 (0.82)	0.77 (0.63)	-
Did not complete high school	-16.00 (1.89)*	-13.37 (1.50)	-17.99 (1.45)	-21.95 (1.27)	-
Obtained post-high school diploma	-1.52 (0.25)	-4.57 (0.73)	-11.42 (1.23)	-2.48 (0.20)	-
Obtained university degree	30.59 (3.57)**	21.94 (2.43)**	34.51 (2.79)**	19.79 (1.17)	-
Household characteristics
Income (x 1000)	-0.0003 (0.01)	-0.0007 (0.02)	-0.205 (0.45)	1.25 (1.02)	-
Income squared (x 100000)	-0.0002 (0.56)	-0.00001 (0.35)	-0.0002 (0.60)	-0.0003 (1.67)*	-
Earlier attainments
PPVT from cycle 1	-	0.47 (2.20)**	-	-	-
Math score from cycle 2	-	-	0.38 (5.07)**	0.58 (2.92)**	0.72 (6.37)**
F test on instruments for earlier attainments	-	-	-	2.91**	3.75**
Mean, dependent variable	401.3
Notes: (1) Specification reported in Table 2; (2) Specification reported in Table 2 plus PPVT score (treated as being exogenous) from round 1. (2) Specification reported in Table 2 plus math score (treated as being exogenous) from round 2. (3) Specification reported in Table 2 plus math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child. PMK and household 'shocks' experienced between 1994 and 1996. (4) Math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996 as well as child. PMK and household characteristics observed as of 1994.

 

Table 6b
Reading Score for Children 8, 9

	(1)	(2)	(3)	(4)	(5)
PMK characteristics
Lone parent	2.87 (0.63)	8.35 (1.92)*	-0.05 (0.01)	11.05 (0.96)	-
Age	0.09 (0.25)	0.21 (0.75)	0.18 (0.32)	0.34 (0.44)	-
Did not complete high school	-24.18 (4.08)**	-19.69 (3.63)**	-18.74 (1.54)	-5.51 (0.39)	-
Obtained post-high school diploma	2.18 (0.58)	-0.22 (0.06)	-4.36 (0.84)	0.55 (0.07)	-
Obtained university degree	11.32 (1.60)	-1.14 (0.21)	9.65 (1.24)	-0.006 (0.00)	-
Household characteristics
Income (x 1000)	0.52 (1.96)**	0.44 (2.00)**	-0.205 (0.63)	1.31 (1.34)	-
Income squared (x 100000)	-0.00025 (1.07)	-0.0008 (0.45)	-0.0002 (0.68)	-0.0002 (1.14)	-
Earlier attainments
PPVT from cycle 1	-	0.84 (6.04)**	-	-	-
Reading score from cycle 2	-	-	0.62 (6.79)**	1.02 (3.79)**	0.92 (5.43)**
F test on instruments for earlier attainments	-	-	-	4.63**	5.75**
Mean, dependent variable	223.9
Notes: (1) Specification reported in Table 2. (2) Specification reported in Table 2 plus PPVT score (treated as being exogenous) from round 1. (3) Specification reported in Table 2 plus math score (treated as being exogenous) from round 2. (4) Specification reported in Table 2 plus math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996. (5) Math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996 as well as child, PMK and household characteristics observed as of 1994.

 

Table 6c
Math Score for Children 11, 13

	(1)	(2)	(3)	(4)	(5)
PMK characteristics
Lone parent	-6.52 (0.56)	21.31 (1.29)	12.39 (1.15)	5.12 (0.39)	-
Age	1.41 (1.67)*	2.64 (2.94)**	0.47 (0.62)	0.30 (0.33)	-
Did not complete high school	-14.12 (1.42)	0.84 (0.06)	-2.39 (0.21)	-0.88 (0.07)	-
Obtained post-high school diploma	-4.22 (0.53)	-1.82 (0.16)	8.27 (1.20)	4.20 (0.46)	-
Obtained university degree	23.48 (2.04)**	27.89 (1.75)*	43.08 (3.50)**	41.74 (3.20)**	-
Household characteristics
Income (x 1000)	-1.08 (1.06)	0.78 (0.65)	1.61 (1.81)*	2.02 (1.28)	-
Income squared (x 100000)	-0.0007 (0.44)	-0.0005 (0.22)	-0.002 (1.47)	-0.002 (1.37)	-
Earlier attainments
Math score from cycle 1	-	0.43 (5.69)**	-	-	-
Math score from cycle 2	-	-	0.54 (8.28)**	0.30 (0.82)	0.82 (7.00)**
F test on instruments for earlier attainments	-	-	-	1.59*	8.73**
Mean, dependent variable	518.5
Notes: (1) Specification reported in Table 2. (2) Specification reported in Table 2 plus math score (treated as being exogenous) from round 1. (3) Specification reported in Table 2 plus math score (treated as being exogenous) from round 2. (4) Specification reported in Table 2 plus math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996. (5) Math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996 as well as child, PMK and household characteristics observed as of 1994.

 

Table 6d
Reading Score for Children 11, 13

PMK characteristics
Lone parent	0.71 (0.18)	-0.20 (0.03)	2.86 (0.85)	5.05 (1.03)	-
Age	0.58 (1.85)*	1.04 (2.49)**	0.03 (0.13)	0.18 (0.50)	-
Did not complete high school	-7.95 (1.67)*	-7.12 (1.04)	3.04 (0.59)	-1.84 (0.28)	-
Obtained post-high school diploma	-0.77 (0.24)	1.52 (0.29)	1.77 (0.56)	3.71 (0.96)	-
Obtained university degree	8.40 (2.02)**	4.76 (0.84)	9.57 (2.32)**	9.61 (2.32)**	-
Household characteristics
Income (x 1000)	0.37 (1.00)	-0.44 (0.76)	0.42 (1.10)	0.64 (1.28)	-
Income squared (x 100000)	-0.00012 (0.22)	0.001 (1.28)	-0.0005 (0.99)	-0.0006 (1.02)	-
Earlier attainments
Math score from cycle 1	-	0.14 (3.77)**	-	-	-
Reading score from cycle 2	-	-	0.51 (10.85)**	0.20 (1.39)	0.43 (5.52)**
F test on instruments for earlier attainments	-	-	-	2.48**	10.99**
Mean, dependent variable	268.6
Notes: (1) Specification reported in Table 2. (2) Specification reported in Table 2 plus math score (treated as being exogenous) from round 1. (3) Specification reported in Table 2 plus reading score (treated as being exogenous) from round 2. (4) Specification reported in Table 2 plus reading score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996. (5) Reading score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996 as well as child, PMK and household characteristics observed as of 1994.

 

Table 6e
Math Score for Children 15

	(1)	(2)	(3)	(4)	(5)
PMK characteristics
Lone parent	-12.99 (0.67)	-6.95 (0.43)	2.26 (0.15)	-11.81 (0.52)	-
Age	2.09 (1.71)*	0.49 (0.35)	0.38 (0.35)	1.72 (1.13)	-
Did not complete high school	6.04 (0.37)	-18.16 (1.07)	19.71 (1.42)	23.31 (1.22)	-
Obtained post-high school diploma	14.77 (1.01)	2.04 (0.15)	16.58 (1.01)	20.47 (1.13)	-
Obtained university degree	63.64 (3.52)**	30.82 (1.60)	34.53 (1.76)*	59.88 (2.61)**	-
Household characteristics
Income (x 1000)	2.95 (1.99)**	2.17 (1.57)	0.29 (0.22)	1.28 (0.74)	-
Income squared (x 100000)	-0.0039 (2.10)**	-0.003 (1.76)*	-0.0004 (0.24)	-0.0013 (0.57)	-
Earlier attainments
Math score from cycle 1	-	0.62 (6.94)**	-	-	-
Math score from cycle 2	-	-	0.74 (12.77)**	0.13 (0.48)	0.86 (5.67)**
F test on instruments for earlier attainments	-	-	-	2.48**	4.17**
Mean, dependent variable	631.7
Notes: (1) Specification reported in Table 2. (2) Specification reported in Table 2 plus math score (treated as being exogenous) from round 1. (3) Specification reported in Table 2 plus math score (treated as being exogenous) from round 2. (4) Specification reported in Table 2 plus math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996. (5) Math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996 as well as child, PMK and household characteristics observed as of 1994.

 

Table 6f
Reading Score for Children 15

	(1)	(2)	(3)	(4)	(5)
PMK characteristics
Lone parent	-3.69 (0.56)	1.11 (0.14)	-0.00 (0.00)	-2.68 (0.36)	-
Age	1.14 (2.39)**	1.18 (2.44)**	0.35 (0.68)	0.61 (0.96)	-
Did not complete high school	0.99 (0.12)	-8.26 (0.63)	2.76 (0.28)	2.39 (0.21)	-
Obtained post-high school diploma	1.73 (0.26)	0.65 (0.11)	-2.63 (0.45)	-3.51 (0.52)	-
Obtained university degree	8.08 (1.44)	-2.79 (0.49)	-6.94 (1.25)	-0.16 (0.02)	-
Household characteristics
Income (x 1000)	1.02 (1.70)*	0.96 (1.69)*	0.46 (0.70)	-0.29 (0.49)	-
Income squared (x 100000)	-0.0016 (2.02)**	-0.002 (2.14)**	-0.0006 (0.60)	0.0002 (0.18)	-
Earlier attainments
Math score from cycle 1	-	0.13 (3.33)**	-	-	-
Reading score from cycle 2	-	-	0.57 (7.88)**	0.22 (0.81)	0.53 (2.83)**
F test on instruments for earlier attainments	-	-	-	1.12	4.56**
Mean, dependent variable	289.2
Notes: (1) Specification reported in Table 2. (2) Specification reported in Table 2 plus math score (treated as being exogenous) from round 1. (3) Specification reported in Table 2 plus math score (treated as being exogenous) from round 2. (4) Specification reported in Table 2 plus math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996. (5) Math score (treated as being endogenous) from round 2 plus child, PMK and household 'shocks' experienced between 1996 and 1998. Instruments are child, PMK and household 'shocks' experienced between 1994 and 1996 as well as child, PMK and household characteristics observed as of 1994.

One area of interest is whether pmk and household characteristics have any additional impact over and above any relationship that they have to an attainment at an earlier period. To see whether this is the case, we compare the results of specification (1) — where earlier attainments are excluded — with the results of specifications (2) through (4). The general pattern across all six tables is the following. A number of pmk and household characteristics, as measured during cycle 1, have an impact on attainments as measured in cycle 3 when an earlier measure of attainment is excluded. However, once we include the measure of earlier attainment, the impact of these characteristics begins to diminish. Most strikingly, when we compare specification (4) — where we treat earlier attainments as endogenous - with that of (1), the impact of earlier pmk and household characteristics all but disappears. The only meaningful impact that persists is the cases where the pmk has obtained a university degree, which continues to raise math scores where children are 11, 13 or 15. Although the quadratic on income in Table 6a and possession of a university degree in table 6d are both statistically significant, the magnitude of both effects is, at less than 3 per cent, trivial.

Next, we consider the impact of attainments measured four years previously on subsequent attainments, controlling for the same set of child, pmk and household characteristics described in chapter 5. However, in the absence of any viable instruments, we treat these earlier attainments as exogenous. In all six specifications, we find a statistically significant relationship. Further, the magnitudes of some of these associations are large. As an example, consider Table 6e which reports the impact of the math score obtained when the child was 11 on her math score at age 15. To see this, first note that the coefficient on the earlier math score is 0.62 meaning that for every additional point obtained on the math attainment in 1994 was associated with a 0.62 point increase on the score in 1998. Next, recall that in Table 3, we simulated the impact of various changes in pmk and household characteristics on attainments. Consider the case of two children who are otherwise identical in terms of observable characteristics save the education of their pmk. Table 3 tells us that the child whose pmk did not complete high school would have a math score at age 11 that was 2.8% or 14 points below the base. The child whose pmk has a university degree would have a math score at age 11 that was 4.7% or 24 points above the base. Thus, these children are predicted to have, at age 11, a difference in math scores of 38 points. At age 15, the difference in their attainments — assuming that they are exposed to exactly the same shocks and transition events — is due to both the direct effect of pmk education (the coefficients —18.16 and 30.82 respectively) plus the impact of this pmk characteristic on the earlier math score. The difference in their predicted math scores is the sum of the absolute values of the pmk education characteristics (18.16 + 30.82) plus the difference in predicted math score in cycle 1 — 38 times 0.62. Collectively, this produces a difference in predicted attainments at age 15 of 72.5 points, a 13 per cent difference relative to the mean.

Also instructive is the comparison between the results for prior attainments, as measured in cycle 1, with those obtained in cycle 2. Because of problems associated with the reading test in cycle 1, we can only do a 'like' for like comparison for the math scores for children aged 11 and13 and 15 in 1998. In both cases, as is shown by comparing specifications (2) and (3) in Tables 6c and 6e, the coefficients on the math score are higher when we use the more recent prior attainment.

Our next step is to consider the potential impact of the endogeneity of prior attainments. Our first attempt at addressing this is found in specification (4). Recall we use the representations of shocks and transitions between 1994 and 1996 described above as instruments. We want to compare these results with those from specification 3. Doing so across all six tables produces an ambiguous set of results. In two cases - math score for children 8, 9; reading score for children 8, 9 — the effect on instrumenting is to increase the size of the parameter estimate on the earlier attainment. But in the other four cases, instrumenting causes the parameter estimate to fall by considerable magnitudes and to become statistically insignificant. It would seem, therefore, that for older children there is no impact of earlier attainments on subsequent attainments once the endogeneity of the former is taken into account.

Such a conclusion, however, is premature. Bound, Jaeger and Baker (1995) show that when instruments have poor statistical power, that the use of two stage least squares is unlikely to correct for endogeneity bias and, even more grieviously, lead to parameter estimates that are downwardly biased. One way of examining the explanatory power of these instruments is to note the results of a F test on their joint significance in the first stage regression that predicts the endogenous variable. These are reported in the last row of Tables 6a through 6f. Echoing the findings reported earlier about the relative unimportance of transition variables, these F-statistics are all remarkable small, well below the target value of 8 to 10 suggested by Bound, Jaeger and Baker. Rectifying this problem with the instruments requires that we find additional variables with explanatory power in the first stage regression (i.e. variables that determine the earlier attainment) that can be plausibly excluded from our second stage regression (i.e. do not affect the later attainment). Recall that we have already noted that pmk and household characteristics, as measured in 1994, have little impact on attainments in 1998, once a measure of earlier attainments is taken into account. This suggests that these characteristics might also serve as plausible instruments.

Accordingly, Tables 6a to 6f, contain the results of a second attempt at estimating a two stage least squares regression (what we will call specification (5)). The attainment measured in cycle 2 is treated as endogenous with child, pmk and household shocks experienced between 1994 and 1996 as well as child, pmk and household characteristics observed as of 1994 used as instruments. Additional determinants of attainments in cycle 3 are the child, pmk and household shocks and transitions that occur between 1996 and 1998. There are four findings across these six tables that should be noted. First, the F test on instruments rises in value, suggesting that we are doing a better job of predicting the endogenous variable (although these values are not always as high as we might like). Second, with this improved set of instruments, we once again find a statistically significant relationship between earlier and later attainments. Third, in four cases — all math attainments as well as the reading score for children aged 8, 9 in 1998 — the parameter estimate rises in value compared to specification (3) where the same measure of earlier attainment is treated as being exogenous. In the other two cases, the parameter estimates are basically the same. This indicates that failing to account for the endogeneity of earlier attainments leads us to underestimate their correlation across time. Fourth, the magnitudes of these associations generally tend to be larger for math scores than for reading scores.

Another way to understand the importance of past attainments (i.e., 'history') for current attainments is to use our estimated regression models to calculate the change in current attainments which would be predicted for some given change in past attainments. Results of such an exercise are reported in Table 7, which illustrates for each specification reported in Table 5 and Tables 6a-6f (and for each cohort of children), the predicted consequences of increasing earlier test scores by 15 percent. (The exception is for the 'baby cohort' for whom we can only predict the implications of having been a low-birth-weight baby versus not.)

Table 7
Simulating the Impact of Changes in Prior Attainments (percent change from the base)

	Age 4-6	Age 8-9		Age 11, 13		Age 15
	PPVT	Math	Reading	Math	Reading	Math	Reading
Child was low birth weight	-4,7 %	-	-	-	-	-	-
Increase cycle 1 attainment	-	1.8 %	5.6 %	4.9 %	2.9 %	8.3 %	3.3 %
Increase cycle 2 attainment	-	4.1 %	8.3 %	7.3 %	7.1 %	10.4 %	8.5 %
Increase cycle 1 attainment (instrumental variable spec)	-	7.8 %	11.8 %	10.7 %	6.1 %	11.3 %	6.6 %
Notes: This simulation uses specifications (2) in table 5 and (2), (3) and (5) in tables 6a-6f to predict attainments. In each case the simulation is to increase the prior attainment by 15%. For the age 4-6 group, the simulation was to make the base case child a low birth weight baby.

While results for all estimated models are reported in Tables 6a-6f, we focus the discussion on simulation results corresponding with Specification 5 since we have argued in the previous section that this is our preferred model. Consider, first, the cohort of children who were aged 4 or 5 in 1994 and hence 8 or 9 in 1998. Figure 13 illustrates that for a 'base' child (i.e. all change variables set to 0 and the continuous variable, the previous attainment, set at the mean), a 15 percent increase in 1996 math score causes a 7.8 percent increase in 1998 math score; a 15 percent increase in 1996 reading score causes an 11.8 percent increase in 1998 reading score. For children aged 9 and 11 in 1998, a 15 percent increase in 1996 math score is associated with a 10.7 percent increase in 1998 math score; a 15 percent increase in reading score is associated with a 6.1 percent increase in 1998 reading score. Finally, for our oldest cohort of children aged 15 in 1998, a 15 percent increase in 1996 math score is associated with an 11.3 percent increase in 1998 math score; a 15 percent increase in 1996 reading score is associated with a 6.6 percent increase in 1998 reading score.

However, even these simulations do not entirely depict the full extent to which 'history' can matter for child attainments. To further illustrate the possible 'snow-balling' of past events into current outcomes, consider the following thought experiment which compares the development of two otherwise identical boys who were both 7 years old in 1994. Suppose the first boy is our average 'base case' (i.e., he is a Caucasian boy, born in the first quarter of the year, living in urban Ontario with a pmk of average age who has completed high school ; family income is the average for this sample). The other is 'disadvantaged' insofar as he comes from a family with income at the 25^th percentile, his mother is 10 years younger than the average and has less than a high-school education. Using our base model for the 'middle cohort' (i.e., that reported in Table 2), we would predict that by the time both children are 11 years old, the disadvantaged child will have a math score which is 6.7 percent lower (469 versus 502.6). (We have already illustrated this point in Figure 11). Now, if assume that the disadvantaged child remains disadvantaged relative to the base, and as well, has by age 11 less developed math skills, then both factors will contribute to limiting further development to age 15. Suppose we go on to predict for the now 11-year old children, the difference in their expected math scores by age 15 (using Specification 2 from Table 5¹⁹). Our calculations suggest that the difference in math scores would increase to 9.3 percent, of which 5.4 percent could be attributed to the direct effect of the continued disadvantage (ie., lower income and younger, less well-educated pmk) and an additional 3.9 percent could be attributed to the indirect consequences of that disadvantage through the persistence of problems accumulated from the past — the 'snowball' effect.

We have reviewed many results in this chapter, so it is helpful to conclude by summarizing our principal findings. History is destiny. Earlier attainments are causally associated— both in terms of statistical significance as well as magnitude — with subsequent attainments. This causal relationship tends to be larger when we account —albeit imperfectly — for the endogeneity of these earlier attainments and also tend to be larger for the math scores. pmk and household characteristics have little impact on subsequent attainments over and above their impact on earlier attainments.

• ¹⁹We now plug in the lower math score predicted at age 11 and continue to assume the same elements of disadvantage. Notice that this procedure assumes that the regression estimates would be stable across cohorts and over time.