In the classical Lee-Carter model, central death rates are modelled as follows:

$\log(m(x,t)) = a(x) + b(x)\kappa(t) +\varepsilon(x,t)$

for some $x=0,\ldots,\omega$ and $t=1,\ldots, T$ and where $a(x)$, $b(x)$ and $k(t)$ are real-valued parameters and $\varepsilon(x,t)\sim\mathcal{N}(0,\sigma^2)$ iid.

One usually imposes the following identifiability conditions

  1. $\sum_t k_t = 0$ and
  2. $\sum_x b_x^2 = 1$.

One then applies (ordinary) least squares:

$\sum_{t,x} \left( \log(m(x,t)) - a(x) -b(x)\kappa(t) \right)^2 \rightarrow \text{min}$.

Note that it is not quite ordinary regression, as all parameters are unknown and thus there are no regressors. From the conditions, it then follows immediately that

$\hat{a}_x = \frac{1}{T}\sum_{t} \log(m(x,t)).$

One defines the matrix $Z:=(Z_{x,t}:= log(m(x,t)) - \hat{a}_x)_{x,t}$

Then, either by solving the normal equations or using singular value decomposition, it follows that $b$ has to be the maximal (unit) eigenvector of $Z\cdot Z^t$ and that $k=Z^t b.$

It is then often claimed that this OLS approach breaks down, if the error terms $\varepsilon(x,t)$ are no longer iid normal or at least no longer homoscedastic.

Yet, in this context, I am not quite why this should be such a big deal. As in classical OLS, I would assume that the least quare estimator is still unbiased and that the condition of homoscedasticity ensures that the OLS estimator is equal to the MLE with all its nice properties. Moreover, the normality assumptions allows for inference.

Yet, from a standpoint of pure approximation, I don't quite see what the big deal would be here if the errors were not homoscedastic.

Any insights to this?



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