How to Use VSTACK and LAMBDA to Author Dynamic Shorrocks Asymmetric Decile Downside Jenkins-Cowell Weibull 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 Weibull mobility framework introduces an asymmetric Weibull cumulative weighting kernel that models variable failure-rate acceleration for downward decile transitions. 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 Weibull Weighting Kernel

Given an ( n imes n ) transition probability matrix ( P ), downside mobility occurs when initial decile ( i ) exceeds destination decile ( j ) (( i > j )). The Jenkins-Cowell Weibull weighting kernel is expressed as:

[ w_{ij} = egin{cases} 1 – e^{-left(rac{i – j}{eta} ight)^eta} & ext{if } i > j \ |i – j| & ext{if } i le j end{cases} ]

Where ( eta > 0 ) is the scale parameter and ( eta > 0 ) represents the shape parameter. When ( eta > 1 ), the marginal penalty for downward transitions escalates super-linearly before asymptotic saturation. The normalized Weibull index measures societal vulnerability 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_WEIBULL_INDEX:

=LAMBDA(transition_matrix, param_eta, param_beta,
    LET(
        n, ROWS(transition_matrix),
        weibull_weights, MAKEARRAY(n, n, LAMBDA(r, c,
            IF(r > c,
                1 - EXP(-(((r - c) / param_eta) ^ param_beta)),
                c - r
            )
        )),
        weighted_transitions, transition_matrix * weibull_weights,
        sum_weighted, SUM(weighted_transitions),
        max_theoretical_weibull, SUM(MAKEARRAY(n, n, LAMBDA(r, c,
            1 - EXP(-(((n - 1) / param_eta) ^ param_beta))
        ))),
        normalised_metric, sum_weighted / max_theoretical_weibull,
        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 Eta", "Param Beta", "Downside Mobility Index", "Vulnerability Level"),
    cohort_rural, HSTACK("Rural Workforce", 3.5, 1.8, JENKINS_COWELL_WEIBULL_INDEX(RuralData!$B$2:$K$11, 3.5, 1.8), "High Vulnerability"),
    cohort_suburban, HSTACK("Suburban Workforce", 3.5, 1.8, JENKINS_COWELL_WEIBULL_INDEX(SuburbanData!$B$2:$K$11, 3.5, 1.8), "Moderate Vulnerability"),
    cohort_urban, HSTACK("Urban Workforce", 3.5, 1.8, JENKINS_COWELL_WEIBULL_INDEX(UrbanData!$B$2:$K$11, 3.5, 1.8), "Low Vulnerability"),
    VSTACK(table_headers, cohort_rural, cohort_suburban, cohort_urban)
)

Auditing and Practical Deployment Best Practices

When applying Weibull 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 ( eta > 0 ) and ( eta > 0 ). Setting ( eta approx 1.5 ext{ to } 2.5 ) models sharp sensitivity to severe socio-economic displacement.
• 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.