Predictive Model & Verdict

Statistical Formulations, Parameter Estimation, 95% Confidence Intervals, and Empirical Performance

1. Data Generating Mechanism (DGM) Formulations

To predict individual educational attainment (\(Y_{\text{educ}}\)) conditioned on childhood background traits, we evaluate two statistical models: a primary Multinomial Logistic Model for milestone degree tiers and a Linear Regression (OLS) Model for continuous schooling years.

1.1 Multinomial Logistic DGM

We model the log-odds of achieving educational tier \(k \in \{\text{Some College}, \text{Bachelor Degree}, \text{Graduate Degree}\}\) relative to baseline \(K_0 = \text{High School or Less}\):

\[\ln\left(\frac{P(Y_{\text{educ}} = k \mid \mathbf{X})}{P(Y_{\text{educ}} = K_0 \mid \mathbf{X})}\right) = \beta_{k0} + \boldsymbol{\beta}_k^T \mathbf{X}\]

where \(\mathbf{X}\) is the vector of childhood socio-demographic covariates:

\[\mathbf{X} = \Big[ \text{parent\_educ}, \text{incom16}, \text{sibs}, \text{childs}, \text{age}, \text{year}, \text{sex}, \text{race}, \text{res16}, \text{family16}, \text{born}, \text{parborn} \Big]^T\]

The predicted probability \(P(Y_{\text{educ}} = k \mid \mathbf{X})\) across all 4 tiers is obtained via the soft-max transformation:

\[P(Y_{\text{educ}} = k \mid \mathbf{X}) = \frac{\exp\left(\beta_{k0} + \boldsymbol{\beta}_k^T \mathbf{X}\right)}{1 + \sum_{j=1}^{3} \exp\left(\beta_{j0} + \boldsymbol{\beta}_j^T \mathbf{X}\right)}\]

1.2 Linear Regression DGM

As a baseline comparison, completed schooling years (\(Y_{\text{years}} \in [0, 20]\)) are modeled as a linear conditional expectation:

\[Y_{\text{years}} = \beta_0 + \boldsymbol{\beta}^T \mathbf{X} + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2)\]

2. Model Parameters & 95% Confidence Intervals

Note: Tables 2.1 & 2.2 highlight key predictors. Complete parameter tables across all variables are available in the Sources tab.

2.1 Multinomial Logistic Model Parameters

Parameters are exponentiated into Odds Ratios (\(\text{OR} = \exp(\beta)\)) with 95% Confidence Intervals.

Table 2.1: Multinomial Odds Ratios & 95% Confidence Intervals for Bachelor’s Degree Attainment (Ref: High School or Less)
Outcome Tier Predictor Variable Odds Ratio (OR) p-value 95% CI Lower 95% CI Upper
Bachelor Degree Parental Education (Per Year) 1.333 < 0.001 1.318 1.347
Bachelor Degree Childhood Relative Income (Tier 1-5) 1.172 < 0.001 1.132 1.214
Bachelor Degree Sibling Count (Per Sibling) 0.854 < 0.001 0.840 0.868
Bachelor Degree Residence: Big-City Suburb (vs Farm) 1.420 < 0.001 1.362 1.481
Bachelor Degree Single Parent Family (vs Both Parents) 0.562 < 0.001 0.530 0.596
Bachelor Degree Parents Nativity: Neither U.S. Born (vs Both) 1.770 < 0.001 1.713 1.828

Interpretation: An Odds Ratio (OR) > 1 indicates increased likelihood of degree completion, while OR < 1 indicates decreased likelihood. For instance, each additional year of parental education increases Bachelor’s degree completion odds by 33.3% (\(\text{OR} = 1.333\)).

2.2 OLS Model Parameters

Table 2.2: OLS Linear Regression Parameters & 95% Confidence Intervals for Completed Schooling Years
Predictor Estimate (β) p-value 95% CI Lower 95% CI Upper
Parental Education (Per Year) 0.326 < 0.001 0.318 0.333
Childhood Relative Income (Tier 1-5) 0.129 < 0.001 0.101 0.157
Sibling Count (Per Sibling) -0.182 < 0.001 -0.195 -0.170
Residence Origin: Big-City Suburb 0.246 < 0.001 0.163 0.329
Family Structure: Single Parent -0.607 < 0.001 -0.672 -0.541
Parents Nativity: Neither U.S. Born 0.508 < 0.001 0.401 0.615

Interpretation: Estimate (\(\beta\)) measures change in total completed schooling years per 1-unit increase in the predictor. For example, each year of parental education adds +0.326 years of completed schooling.

3. Visualizing Model Uncertainty

3.1 Multinomial Odds Ratio Uncertainty

Figure 3.1 displays Odds Ratios (\(\text{OR} = \exp(\beta)\)) and 95% Confidence Intervals for predicting Bachelor Degree and Graduate Degree attainment relative to High School baseline.

Figure 3.1: Multinomial Logistic Model Odds Ratios & 95% Confidence Intervals for Bachelor’s and Graduate Degrees.

3.2 OLS Regression Uncertainty

Figure 3.2 displays continuous linear parameter estimates (\(\beta\)) and 95% Confidence Intervals for total completed schooling years (\(Y_{\text{years}}\)). Dashed line (\(\beta = 0\)) represents no effect.

Figure 3.2: OLS Parameter Estimates (β) & 95% Confidence Intervals for Completed Schooling Years.

4. Practical Profile Predictions

To illustrate model behavior, we evaluate two individual profiles with contrasting childhood backgrounds.

Table 2.3: Model-Predicted Educational Attainment Probabilities & Expected Schooling Years
Model Metric / Outcome Individual A (Advantaged) Individual B (Disadvantaged)
Childhood Background College-educated parents, High income, Two parents, Suburban, 1 Sibling Middle-school parents, Low income, Single parent, Farm origin, 4 Siblings
High School or Less Probability 6.7% 70.4%
Some College Probability 25.4% 21.3%
Bachelor Degree Probability 39.7% 5.1%
Graduate Degree Probability 28.1% 3.2%
Expected Total Schooling Years 16.7 Years 11.9 Years
  • Individual A: High parental education and economic stability yield a 39.7% chance of a Bachelor’s degree (67.9% chance of any college degree) and 16.7 expected schooling years.
  • Individual B: Cumulative background challenges reduce Bachelor’s degree probability to 5.1% (70.4% chance of stopping at High School) and 11.9 expected schooling years.

5. Summary & Empirical Predictor Importance

To rigorously evaluate feature importance across our statistical DGMs, predictors are ranked below by empirical test statistics (\(t\)-statistics for continuous schooling and \(z\)-scores for degree tier multinomial logistic regression) alongside 95% confidence intervals and effect sizes.

5.1 Predictor Importance Ranking (Most to Least Important)

  1. Parental Educational Background (\(t = 84.48, z = 52.54, p < 0.001\)):
    • OLS Linear Effect (\(\beta\)): \(+0.326\) years of completed schooling per parent year (\(95\%\text{ CI: } [0.318, 0.333]\)).
    • Multinomial Odds Ratio: \(\text{OR} = 1.333\) (\(95\%\text{ CI: } [1.318, 1.347]\)) for a Bachelor’s degree; \(\text{OR} = 1.405\) (\(95\%\text{ CI: } [1.388, 1.423]\)) for a Graduate degree.
    • Statistical Verdict: The single most powerful background predictor across both linear and multinomial specifications.
  2. Family Structure / Single-Parent Household (\(t = -18.18, z = -19.27, p < 0.001\)):
    • OLS Linear Effect (\(\beta\)): \(-0.607\) schooling years (\(95\%\text{ CI: } [-0.672, -0.541]\)) compared to two-parent households.
    • Multinomial Odds Ratio: \(\text{OR} = 0.562\) (\(95\%\text{ CI: } [0.530, 0.596]\)) for a Bachelor’s degree.
    • Statistical Verdict: Substantial negative penalty associated with single-parent childhood environments.
  3. Family Sibling Size (\(t = -27.77, z = -18.84, p < 0.001\)):
    • OLS Linear Effect (\(\beta\)): \(-0.182\) schooling years per additional sibling (\(95\%\text{ CI: } [-0.195, -0.170]\)).
    • Multinomial Odds Ratio: \(\text{OR} = 0.854\) (\(95\%\text{ CI: } [0.840, 0.868]\)) per sibling for a Bachelor’s degree.
    • Statistical Verdict: Significant dilution of per-child educational resources in larger families.
  4. Parental Immigrant Status (\(t = 9.30, z = 34.18, p < 0.001\)):
    • OLS Linear Effect (\(\beta\)): \(+0.508\) schooling years (\(95\%\text{ CI: } [0.401, 0.615]\)) for non-U.S. born parents vs. U.S. born parents.
    • Multinomial Odds Ratio: \(\text{OR} = 1.770\) (\(95\%\text{ CI: } [1.713, 1.828]\)) for a Bachelor’s degree.
    • Statistical Verdict: Strong generational upward mobility demonstrated by children of foreign-born parents.
  5. Childhood Relative Income (\(t = 9.17, z = 9.00, p < 0.001\)):
    • OLS Linear Effect (\(\beta\)): \(+0.129\) schooling years per income level (\(95\%\text{ CI: } [0.101, 0.157]\)).
    • Multinomial Odds Ratio: \(\text{OR} = 1.172\) (\(95\%\text{ CI: } [1.132, 1.214]\)) per tier for a Bachelor’s degree.
    • Statistical Verdict: Consistent, positive linear returns to childhood relative economic standing.
  6. Suburban Origin Geography (\(t = 5.82, z = 16.31, p < 0.001\)):
    • OLS Linear Effect (\(\beta\)): \(+0.246\) schooling years (\(95\%\text{ CI: } [0.163, 0.329]\)) for big-city suburbs vs. farm origin.
    • Multinomial Odds Ratio: \(\text{OR} = 1.420\) (\(95\%\text{ CI: } [1.358, 1.479]\)) for a Bachelor’s degree.
    • Statistical Verdict: Moderate positive boost associated with suburban growing environments.

5.2 Overall Model Performance

  • College Degree Accuracy (72.3%): The multinomial model correctly predicts college completion (Bachelor’s or higher vs. non-college) for 72.3% of individuals.
  • Continuous Schooling Accuracy (60.8%): The OLS model predicts total completed schooling within \(\pm 2\) years for 60.8% of individuals (Mean Absolute Error = 1.95 years).
  • Core Insight: Parental education, family structure, sibling count, parental nativity, childhood income, and suburban geography form a robust, highly predictive framework for educational attainment in the United States.