Identifiability Constraints

GAPC models are not identifiable without constraints because the predictor \(\eta_{xt}\) is invariant under certain transformations of the parameters. pyStMoMo applies post-fit constraints that project parameters onto the identifiable subspace without changing fitted values.

Lee-Carter

\[\sum_x \beta_x = 1, \quad \frac{1}{T}\sum_t \kappa_t = 0 \text{ (mean absorbed into } \alpha_x)\]

APC

\[\frac{1}{T}\sum_t \kappa_t = 0, \quad \sum_c \gamma_c = 0, \quad \sum_c c \cdot \gamma_c = 0\]

The linear trend in \(\gamma_c\) is redistributed into \(\kappa_t\) and \(\alpha_x\), then zero-mean is re-enforced.

CBD Family (M6, M7, M8)

\[\sum_c \gamma_c = 0\]

M7 also removes the linear and quadratic trend from \(\gamma_c\).

Renshaw-Haberman

Combines the LC constraint (sum(\(\beta_x\))=1, mean(\(\kappa_t\))=0) with zero-mean cohort.