Labor Markov chain — weekly Bayesian flows

Upload or paste weekly gross flows in any of three shapes — the parser auto-detects. Long: week_start, from_state, to_state, count. Wide (matrix): week_start, EE, EU, EN, UE, UU, UN, NE, NU, NN. Row-oriented: week_start, from, E, U, N (one row per from-state). States are E / U / N. Dates accept YYYY-MM-DD, common locale formats, or Excel serials. .xlsx / .xls / .ods files are supported directly.

Model

Each row of the 3×3 transition matrix P has an independent Dirichlet prior. The number of people transitioning from state i to j in week k is multinomial.

Initial prior: uniform Dirichlet(1,1,1) for each origin row. Each weekly posterior becomes the next week's prior — fully recursive Bayesian filtering.

We model each origin row of P as a multinomial logit so that the nine transition probabilities sum to one within each row. Let $p_{ij,t}^c$ be the probability that a worker in country c moves from state i to state j in week t. With the diagonal element $p_{ii,t}^c$ as the reference category:

${\mu }_i^c$ are country fixed effects; ${\tau }_{i,t}$ are time (seasonal × business-cycle) effects; ${\mathbf{x}}_{ij,t}^c$ is the cell-specific covariate vector listed in the tables below. Each cell's equation is (4) evaluated with its own covariate set.

Cell-by-cell determinants & equations

For each of the nine transitions we list the determinants $x_k$, their data source, the sign predicted by labor-market theory, and the regression equation. Diagonal cells (EE, UU, NN) are the reference categories within their row — their probability is recovered from the constraint that row probabilities sum to one.

Estimation methods

1. Pooled multinomial logit (MLE). Stack weekly origin–destination counts $n_{ij,t}^c$ and maximize
   $$\ell (\mathbf{\beta })=\sum _{c,t,i}\sum _j{n}_{ij,t}^c\log p_{ij,t}^c(\mathbf{\beta })$$

   (15)
   by Newton–Raphson or BFGS. Standard errors are clustered at the country level.
2. Fractional / quasi-binomial GLM. Treat each observed row share ${\hat{p}}_{ij,t}^c=n_{ij,t}^c/N_{i,t}^c$ as a fraction and estimate by quasi-MLE with a logit link (Papke-Wooldridge). Robust to mis-specification of higher moments.
3. Bayesian Dirichlet-Multinomial regression. Place priors ${\mathbf{\beta }}_{ij}\sim \mathcal{N}(0,{\sigma }^{2}I)$ and sample via Hamiltonian Monte Carlo (Stan, PyMC, NumPyro). Posterior predictive draws integrate naturally with the recursive Dirichlet update used on this page.
4. Panel fixed effects. Include country fixed effects ${\mu }_i^c$ to absorb time-invariant institutional differences (EPL regime, demographic stock) and week fixed effects for global shocks (COVID, GFC).
5. Endogeneity & IV. Tightness $\theta =V/U$, wages, and UI generosity are jointly determined with flows. Use instruments: Bartik shift-share for local labor demand (Goldsmith-Pinkham-Sorkin-Swift); state-level UI rule changes (Hagedorn-Manovskii-Mitman); monetary-policy surprises (Romer-Romer, Gertler-Karadi). Implement as control function or 2SLS on the logit linear predictor.
6. Time-varying coefficients. Allow ${\mathbf{\beta }}_{ij,t}={\mathbf{\beta }}_{ij,t-1}+{\mathbf{\nu }}_t$ (random-walk state-space) and estimate via Kalman filter / particle filter; this captures structural breaks (Beveridge-curve shifts post-2009, post-2020).
7. Identification of partial effects. Reported coefficients are log-odds. Average marginal effects on probabilities use
   $$\frac{\partial p_{ij,t}^c}{\partial x_k}=p_{ij,t}^c({\beta }_{ij,k}-\sum _{m\ne i}{p}_{im,t}^c{\beta }_{im,k})$$

   (16)
   so cross-effects across destinations are non-trivial and must be evaluated at the sample (or counterfactual) means.

Counterfactual prediction workflow

To forecast what happens in country c when an independent variable $x_k$ changes by $\Delta x_k$:

1. Estimate ${\hat{\mathbf{\beta }}}_{ij}$ for all six off-diagonal cells using the equation system above and the full country-week panel.
2. Build the counterfactual covariate vector ${\tilde{\mathbf{x}}}_{ij,t}^c={\mathbf{x}}_{ij,t}^c+\Delta x_k{\mathbf{e}}_k$.
3. Compute counterfactual transition probabilities ${\tilde{p}}_{ij,t}^c$ via (5) and assemble the counterfactual matrix ${\tilde{\mathbf{P}}}_t^c$.
4. Iterate the chain on the current state vector ${\mathbf{\pi }}_t^c=(E_t,U_t,N_t)/{\text{Pop}}_t$:
   $${\mathbf{\pi }}_{t+h}^c={\mathbf{\pi }}_t^c({\tilde{\mathbf{P}}}_t^c{)}^h$$

   (17)
5. Read off the steady-state employment, unemployment, and participation rates from the left eigenvector of ${\tilde{\mathbf{P}}}^c$ with eigenvalue 1:
   $${\mathbf{\pi }}^{\ast c}{\tilde{\mathbf{P}}}^c={\mathbf{\pi }}^{\ast c},\sum _i{\pi }_i^{\ast c}=1.$$

   (18)
6. Compute the steady-state unemployment rate $u^{\ast }={\pi }_U^{\ast }/({\pi }_E^{\ast }+{\pi }_U^{\ast })$ and the participation rate ${\text{LFPR}}^{\ast }={\pi }_E^{\ast }+{\pi }_U^{\ast }$. The change relative to the baseline matrix is the policy / shock impact.
7. Bootstrap or posterior-sample $\hat{\mathbf{\beta }}$ to obtain credible bands around $u^{\ast }$ and LFPR.

Interpretation guide for labor economists

• Beveridge-curve diagnostics. If ${\beta }_{UE,\theta }$falls and ${\beta }_{UU,\text{Mism}}$ rises across the sample, the matching function has deteriorated — Beveridge-curve outward shift.
• UI design trade-off. Compare the marginal effect of UI generosity on UE (job-finding tax) and on UN (search-retention benefit). Optimal UI (Baily-Chetty) balances liquidity against moral hazard: $\frac{b}{w}\approx \frac{\sigma \Delta c/c}{1+{\epsilon }_{u,b}}$.
• Hysteresis test. A persistent positive coefficient on long-term-U share in UU implies true duration dependence rather than dynamic selection.
• Demographic vs. cyclical decomposition. Demean each covariate by country-time means: cyclical determinants (GDP_g, θ, VIX) drive short-run variation; structural ones (Age65, EPL, DI, Care) drive cross-country level differences.
• Aggregation back to the published matrix. The estimated logits substitute directly for the Dirichlet posterior means above when forecasting; the Bayesian update remains the natural smoother of the realized counts each week.

Transition matrices vary sharply by age and sex. The aggregate $\hat{\mathbf{P}}$ is a population-weighted mix of group matrices${\hat{\mathbf{P}}}^{(g)}$; ignoring composition causes Simpson-style bias when subgroup shares shift (ageing, female participation, youth schooling).

Labor-market flows differ sharply by occupation. Routine-manual and gig-heavy occupations churn faster; skilled trades and healthcare have thicker U→E channels; office and production carry secular automation risk. The whole-economy transition matrix is the employment-weighted mix of individual occupations:

Each lever below is tagged with the transition cell it moves, the expected sign, and a concrete implementation path. Under the multinomial-logit spec, $\Delta \log (p_{ij}/p_{ii})={\beta }_{ij,k}\Delta x_k$, so a policy that shifts covariate $x_k$ feeds directly into the counterfactual $\tilde{\mathbf{P}}$ and the steady-state $u^{\ast }$, ${\text{LFPR}}^{\ast }$.

Levers link to the estimated coefficient set ${\hat{\mathbf{\beta }}}_{ij}$ from the country's multinomial-logit fit. To simulate a specific policy quantitatively, translate the intervention into a $\Delta x_k$ for the mapped covariate (UI generosity, EPL index, childcare cost, ALMP spending, θ) and iterate the counterfactual chain in the simulator above.