Identifiability issues of age-period and age-period-cohort models of the Lee-Carter type

The predominant way of modelling mortality rates is the Lee-Carter model and its many extensions. The Lee-Carter model and its many extensions use a latent process to forecast. These models are estimated using a two-step procedure that causes an inconsistent view on the latent variable. This paper considers identifiability issues of these models from a perspective that acknowledges the latent variable as a stochastic process from the beginning. We call this perspective the plug-in age-period or plug-in age-period-cohort model. Defining a parameter vector that includes the underlying parameters of this process rather than its realisations, we investigate whether the expected values and covariances of the plug-in Lee-Carter models are identifiable. It will be seen, for example, that even if in both steps of the estimation procedure we have identifiability in a certain sense it does not necessarily carry over to the plug-in models.