How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside Jenkins-Cowell Fisk 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 Fisk (log-logistic) mobility framework introduces a two-parameter Fisk cumulative weighting kernel that models symmetric heavy-tailed 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 Fisk 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 Fisk weighting kernel scales downward step distances through scale parameter alpha and shape parameter beta:

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

Where alpha > 0 represents the scale parameter and beta > 0 governs the shape and tail thickness of the downward penalty distribution. The normalized Fisk 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_FISK_INDEX:

=LAMBDA(transition_matrix, param_alpha, param_beta,
    LET(
        n, ROWS(transition_matrix),
        fisk_weights, MAKEARRAY(n, n, LAMBDA(r, c,
            IF(r > c,
                1 / (1 + ((param_alpha / (r - c)) ^ param_beta)),
                c - r
            )
        )),
        weighted_transitions, transition_matrix * fisk_weights,
        sum_weighted, SUM(weighted_transitions),
        max_theoretical_fisk, SUM(MAKEARRAY(n, n, LAMBDA(r, c,
            1 / (1 + ((param_alpha / (n - 1)) ^ param_beta))
        ))),
        normalised_metric, sum_weighted / max_theoretical_fisk,
        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", "Downside Mobility Index", "Vulnerability Level"),
    cohort_rural, HSTACK("Rural Workforce", 2.0, 1.6, JENKINS_COWELL_FISK_INDEX(RuralData!$B$2:$K$11, 2.0, 1.6), "High Vulnerability"),
    cohort_suburban, HSTACK("Suburban Workforce", 2.0, 1.6, JENKINS_COWELL_FISK_INDEX(SuburbanData!$B$2:$K$11, 2.0, 1.6), "Moderate Vulnerability"),
    cohort_urban, HSTACK("Urban Workforce", 2.0, 1.6, JENKINS_COWELL_FISK_INDEX(UrbanData!$B$2:$K$11, 2.0, 1.6), "Low Vulnerability"),
    VSTACK(table_headers, cohort_rural, cohort_suburban, cohort_urban)
)

Auditing and Practical Deployment Best Practices

When applying Fisk 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 and param_beta > 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.