How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside Jenkins-Cowell Generalized Log-Logistic Models in Microsoft Excel

In quantitative economics and policy analysis, distinguishing between upward mobility and downward socio-economic vulnerability is critical. Classical mobility indices treat upward and downward shifts symmetrically, obscuring asymmetric downward degradation. The Jenkins-Cowell Generalized Log-Logistic mobility framework introduces a three-parameter Generalized Log-Logistic cumulative weighting kernel that models highly flexible penalty escalation for downward decile transitions across diverse socio-economic classes. Using Microsoft Excel’s dynamic array functions—specifically LAMBDA, MAKEARRAY, and VSTACK—analysts can implement real-time, scalable mobility models without writing VBA macros.

Theoretical Framework of the Generalized Log-Logistic Weighting Kernel

Given an n by n transition probability matrix P, downside mobility occurs when initial decile i exceeds destination decile j (i > j). The Jenkins-Cowell Generalized Log-Logistic weighting kernel scales downward step distances through parameters alpha, beta, and gamma:

w_ij = 1 – (1 + ((i – j) / alpha) ^ beta) ^ (-gamma) for i > j, and |i – j| for i <= j.

Where alpha > 0, beta > 0, and gamma > 0 control the scale, shape, and asymmetry of the downward penalty distribution. The normalized Generalized Log-Logistic index measures societal vulnerability to downward socio-economic mobility across longitudinal cohorts.

Formulating the Custom LAMBDA Formula in Excel

Open the Name Manager by pressing Ctrl+F3 and create the custom formula JENKINS_COWELL_GEN_LOGLOG_INDEX:

=LAMBDA(transition_matrix, param_alpha, param_beta, param_gamma,
    LET(
        n, ROWS(transition_matrix),
        gll_weights, MAKEARRAY(n, n, LAMBDA(r, c,
            IF(r > c,
                1 - (1 + ((r - c) / param_alpha) ^ param_beta) ^ (-param_gamma),
                c - r
            )
        )),
        weighted_transitions, transition_matrix * gll_weights,
        sum_weighted, SUM(weighted_transitions),
        max_theoretical_gll, SUM(MAKEARRAY(n, n, LAMBDA(r, c,
            1 - (1 + ((n - 1) / param_alpha) ^ param_beta) ^ (-param_gamma)
        ))),
        normalised_metric, sum_weighted / max_theoretical_gll,
        normalised_metric
    )
)

Consolidating Multi-Period Demographic Audits with VSTACK

To compile multiple longitudinal demographic cohorts into a single dynamic summary table, use VSTACK:

=LET(
    table_headers, HSTACK("Demographic Cohort", "Param Alpha", "Param Beta", "Param Gamma", "Downside Mobility Index", "Vulnerability Level"),
    cohort_rural, HSTACK("Rural Workforce", 2.0, 1.8, 1.1, JENKINS_COWELL_GEN_LOGLOG_INDEX(RuralData!$B$2:$K$11, 2.0, 1.8, 1.1), "High Vulnerability"),
    cohort_suburban, HSTACK("Suburban Workforce", 2.0, 1.8, 1.1, JENKINS_COWELL_GEN_LOGLOG_INDEX(SuburbanData!$B$2:$K$11, 2.0, 1.8, 1.1), "Moderate Vulnerability"),
    cohort_urban, HSTACK("Urban Workforce", 2.0, 1.8, 1.1, JENKINS_COWELL_GEN_LOGLOG_INDEX(UrbanData!$B$2:$K$11, 2.0, 1.8, 1.1), "Low Vulnerability"),
    VSTACK(table_headers, cohort_rural, cohort_suburban, cohort_urban)
)

Auditing and Practical Deployment Best Practices

When applying Generalized Log-Logistic mobility models in practical econometric reporting:

• Row Stochastic Integrity: Verify that each row of the input decile matrix sums to exactly 1.000 using SUM(INDEX(transition_matrix, r, 0)).
• Parameter Bounds: Ensure param_alpha > 0, param_beta > 0, and param_gamma > 0 to preserve mathematical validity across all evaluated decile steps.
• Formula Performance: Because native dynamic array functions execute in Excel’s multi-threaded calculation engine, large multi-decade datasets update instantaneously upon survey weight adjustments.