Mathematical Models

This page presents the statistical models used to estimate the causal effect of income on discretionary spending across life stages.

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Tables

The final model allows the income effect to differ across life stages — the core question of this project.

Model 1 — Income × Life Stage (No Children)

Model 1: Discretionary share ~ income × life stage (no children term)

Term	Estimate	Std Error	t stat	p value	CI Low	CI High
(Intercept)	0.0537	0.0087	6.1938	0.0000	0.0367	0.0707
income_scaled	-0.0026	0.0031	-0.8124	0.4166	-0.0087	0.0036
life_stageWorking Adult	0.4365	0.0102	42.9445	0.0000	0.4166	0.4564
life_stageElderly	-0.0116	0.0123	-0.9413	0.3466	-0.0357	0.0125
income_scaled:life_stageWorking Adult	-0.0141	0.0032	-4.4419	0.0000	-0.0203	-0.0079
income_scaled:life_stageElderly	0.0032	0.0033	0.9527	0.3408	-0.0033	0.0097

Model 2 — Income × Life Stage + Children

Model 2: Discretionary share ~ income × life stage + children

Term	Estimate	Std Error	t stat	p value	CI Low	CI High
(Intercept)	0.0556	0.0085	6.5204	0.0000	0.0389	0.0723
income_scaled	-0.0042	0.0031	-1.3586	0.1743	-0.0103	0.0019
life_stageWorking Adult	0.3815	0.0106	36.0641	0.0000	0.3608	0.4022
life_stageElderly	-0.0135	0.0121	-1.1200	0.2627	-0.0372	0.0102
num_children	0.0594	0.0038	15.8167	0.0000	0.0521	0.0668
income_scaled:life_stageWorking Adult	-0.0120	0.0031	-3.8347	0.0001	-0.0181	-0.0059
income_scaled:life_stageElderly	0.0046	0.0033	1.4201	0.1556	-0.0018	0.0110

The interaction terms (income_scaled:life_stageWorking Adult and income_scaled:life_stageElderly) measure how much the income slope for each group differs from the Student baseline. Non-zero interactions mean income affects spending differently depending on life stage — the central finding of this project.

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Models

Data Generating Model (DGM)

The data generating model describes the assumed process that produced our observations. We model discretionary spending share as a linear function of income, life stage, their interaction, and family size, with normally distributed error:

\[Y_i = \beta_0 + \beta_1 \cdot \text{Income}_i + \beta_2 \cdot \mathbf{1}[\text{WA}_i] + \beta_3 \cdot \mathbf{1}[\text{Elderly}_i]\] \[+ \beta_4 \cdot \text{Income}_i \cdot \mathbf{1}[\text{WA}_i] + \beta_5 \cdot \text{Income}_i \cdot \mathbf{1}[\text{Elderly}_i]\] \[+ \beta_6 \cdot \text{Children}_i + \varepsilon_i, \qquad \varepsilon_i \overset{\text{iid}}{\sim} \mathcal{N}(0,\, \sigma^2)\]

where \(Y_i\) is household \(i\)’s discretionary spending share, Income is measured in $10,000 increments, and Students are the reference (baseline) life stage.

What each term means:

• \(\beta_0\) — the intercept: predicted spending share for a childless Student with zero income (far outside the data range; not directly interpretable on its own)
• \(\beta_1\) — the income slope for Students: how much discretionary share changes per additional $10,000 for a Student household
• \(\beta_2,\, \beta_3\) — life stage intercept shifts: baseline differences between Working Adults / Elderly and Students at the same income
• \(\beta_4,\, \beta_5\) — interaction terms: how much the income slope for Working Adults and Elderly differs from the Student slope
• \(\beta_6\) — children effect: change in discretionary share per additional child, within the same life stage and income level
• \(\varepsilon_i\) — the residual: individual household deviation from the group average, assumed to follow a normal distribution with mean zero and constant variance \(\sigma^2\)

Fitted Equation

Substituting the estimated coefficients from Model 3:

\[\widehat{Y}_i = 0.0556 -0.0042 \cdot \text{Income}_i +0.3815 \cdot \mathbf{1}[\text{WA}_i] -0.0135 \cdot \mathbf{1}[\text{Elderly}_i]\] \[-0.0120 \cdot \text{Income}_i \cdot \mathbf{1}[\text{WA}_i] +0.0046 \cdot \text{Income}_i \cdot \mathbf{1}[\text{Elderly}_i] +0.0594 \cdot \text{Children}_i\]

Income is in $10,000 increments; all coefficients are in percentage-point units of discretionary spending share.