1
$\begingroup$

I'm running the Bayesian version of Lee-Carter model on jags, using rjags R package. Given a matrix of data $M$ such that $M_{x,t}=\log m_x(t)$ where $m_x(t)$ is defined as the central death rate at age $x$ in calendar year $t$, the Lee-Carter model states that: $$\log m_x(t) = \alpha_x + \beta_x\kappa_t+\epsilon_{x,t}$$

where $\epsilon_{x,t} \sim \mathcal{N}(0,\sigma^2_\epsilon)$. In a frequentist analysis, in order to estimate the parameters, two constraints are done:

$$ \sum_x \beta_x = 1\quad \text{and}\quad \sum_t \kappa_t = 0$$

and they allow a specific interpretation of parameters. My question is: when I run this model in JAGS, are these constraints necessary in order to obtain estimations? Or they just allow to interpretate in the same way the parameters?

$\endgroup$
  • $\begingroup$ I edited your title to something more informative, feel free to edit it again if you find it unsuitable. $\endgroup$ – Tim Jun 11 '18 at 12:02
  • $\begingroup$ Well maybe my question was not very direct to the problem. I meant if the algorithm performance could be affected by the constraints used in frequentist approach. $\endgroup$ – zick094 Jun 11 '18 at 12:08
1
$\begingroup$

I'm not an expert on Lee-Carter model, but from what I gasp, $\alpha_x$ is the mean per age group, $\beta_x$ is the effect per age group and $\kappa_t$ is the effect per time. As noted in the Understanding the Lee-Carter Mortality Forecasting Method paper by Girosi and King (I adapted the notation):

The constraint $\sum_t \kappa_t = 0$ immediately implies that the parameter $\alpha_x$ is simply the empirical average over time of the age profile in age group $x$ . . . We therefore rewrite the model in terms of the mean centered log-mortality rate . . .

What leads to the following model

$$ \log \tilde{m_{xt}} = \beta_x \kappa_t +\varepsilon_{xt} $$

where $\log \tilde{m_{xt}}$ is the centered $\log m_{xt}$, what looses the constraint on $\kappa_t$. As about $\beta_x$, you can use Dirichlet distribution for this parameter, as it already satisfies the constraint and is a common choice for a "distribution over probabilities".

$\endgroup$
  • $\begingroup$ Thank you for the answer. Well I used Dirichlet distribution but it is not very appropriate since it does not allows negative values, while in some applications it's possible to see that for some ages $\beta_x$ is negative. Anyway my question was if I don't use these constraints, could this affect the estmation in JAGS? I mean not in absolute values term, but in performance of algorithm. $\endgroup$ – zick094 Jun 11 '18 at 12:06
  • $\begingroup$ @zick094 as far as I understand, the constraints are only for interpretability. The model does not seem to differ from a regression model with interaction terms and the regression models do not have any such constraints, so no, it won't affect anything. $\endgroup$ – Tim Jun 11 '18 at 12:09
  • $\begingroup$ This answers to my question. The problem was that while in regression we have observable variables, here everything is latent. $\endgroup$ – zick094 Jun 11 '18 at 12:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.