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

Evaluating multi-period welfare changes requires directional mobility indices grounded in information-theoretic entropy foundations. The Maasoumi mobility framework applies generalized entropy aggregation to transition probability distributions, quantifying the divergence between starting deciles and destination states. By authoring custom LAMBDA and VSTACK algorithms in Microsoft Excel, quantitative researchers can construct dynamic Maasoumi downside mobility models without external statistical scripting.

Mathematical Foundations of Maasoumi Downside Mobility

The Maasoumi framework measures distance from the identity matrix using a generalized entropy divergence parameter $gamma$. For an $n imes n$ decile transition matrix $P$ where $P_{ij}$ represents the probability of moving from decile $i$ to decile $j$:

$$M_{down} = rac{1}{gamma(gamma – 1)} sum_{i > j} P_{ij} cdot left[left(rac{i}{j} ight)^gamma – 1 ight]$$

When the destination decile $j$ represents a severe plunge below origin $i$, the ratio $i/j$ scales rapidly, penalising downward social descents in proportion to the entropy parameter $gamma$.

Authoring the Dynamic LAMBDA Formula

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

=LAMBDA(transition_matrix, gamma_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, 1),
        entropy_term, IF(is_downside, (rel_drop ^ gamma_param) - 1, 0),
        weighted_val, IF(is_downside, transition_matrix * entropy_term, 0),
        maasoumi_score, SUM(weighted_val) / (gamma_param * (gamma_param - 1)),
        maasoumi_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],
    gamma, 1.5,
    m1, MAASOUMI_DOWNSIDE(cohort1990, gamma),
    m2, MAASOUMI_DOWNSIDE(cohort2000, gamma),
    m3, MAASOUMI_DOWNSIDE(cohort2010, gamma),
    m4, MAASOUMI_DOWNSIDE(cohort2020, gamma),
    headers, HSTACK("Observation Window", "Maasoumi Downside Vulnerability Index"),
    r1, HSTACK("1990-2000", m1),
    r2, HSTACK("2000-2010", m2),
    r3, HSTACK("2010-2020", m3),
    r4, HSTACK("2020-2030 (Est)", m4),
    VSTACK(headers, r1, r2, r3, r4)
)

Sensitivity Testing and Model Verification

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