How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside D’Ambrosio Mobility Models in Microsoft Excel

In quantitative welfare economics and relative deprivation research, assessing social mobility requires measuring the social psychological distress caused by socio-economic decline. The D’Ambrosio mobility framework incorporates relative deprivation theory into directional transition dynamics, weighting downward shifts according to the cumulative mass of individuals surpassing the declining cohort. By implementing dynamic LAMBDA and VSTACK algorithms in Microsoft Excel, quantitative researchers can construct real-time D’Ambrosio downside mobility models without external statistical packages.

Mathematical Foundations of D’Ambrosio Downside Mobility

The D’Ambrosio downside framework incorporates a deprivation-sensitivity parameter $lambda ge 0$. For an $n imes n$ decile transition matrix $P$ where $P_{ij}$ represents the probability of moving from decile $i$ to decile $j$:

$$DA_{down} = sum_{i > j} P_{ij} cdot left(rac{i – j}{n} ight) cdot left(rac{n – j}{n} ight)^lambda$$

When $lambda > 0$, downward descents landing in lower deciles carry heightened deprivation weight due to the greater proportion of the population occupying superior economic ranks.

Authoring the Dynamic LAMBDA Formula

Create a named formula DAMBROSIO_DOWNSIDE inside Excel’s Name Manager:

=LAMBDA(transition_matrix, lambda_param,
    LET(
        n, ROWS(transition_matrix),
        row_seq, SEQUENCE(n, 1, 1, 1),
        col_seq, SEQUENCE(1, n, 1, 1),
        is_downside, row_seq > col_seq,
        rel_drop, IF(is_downside, (row_seq - col_seq) / n, 0),
        deprivation_weight, IF(is_downside, ((n - col_seq) / n) ^ lambda_param, 0),
        weighted_loss, IF(is_downside, transition_matrix * rel_drop * deprivation_weight, 0),
        da_score, SUM(weighted_loss),
        da_score
    )
)

Stacking Multi-Period Longitudinal Results with VSTACK

Consolidate multi-decade panel survey data into an automated summary dashboard using VSTACK:

=LET(
    cohort1990, Panel1990[#Data],
    cohort2000, Panel2000[#Data],
    cohort2010, Panel2010[#Data],
    cohort2020, Panel2020[#Data],
    lambda, 1.4,
    d1, DAMBROSIO_DOWNSIDE(cohort1990, lambda),
    d2, DAMBROSIO_DOWNSIDE(cohort2000, lambda),
    d3, DAMBROSIO_DOWNSIDE(cohort2010, lambda),
    d4, DAMBROSIO_DOWNSIDE(cohort2020, lambda),
    headers, HSTACK("Observation Window", "D'Ambrosio Downside Deprivation Index"),
    r1, HSTACK("1990-2000", d1),
    r2, HSTACK("2000-2010", d2),
    r3, HSTACK("2010-2020", d3),
    r4, HSTACK("2020-2030 (Est)", d4),
    VSTACK(headers, r1, r2, r3, r4)
)

Sensitivity Testing and Model Verification

Ensure that all transition matrix rows sum to 100%. Adjusting $lambda$ in linked cells allows researchers to stress-test differing degrees of social risk aversion across all stacked cohort comparisons simultaneously.