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When $X_1$ is unobserved, at iteration $k=1$ of EM, the posterior mean value when $X_2=3$ is $5.18$ by using an inference algorithm, i.e. Junction tree. Then the sufficient statistics for $X_1$ is: $s_1=\Sigma_{i=1}^nx_{i1}=9+4+5.18$ and $s_{11}=\Sigma_{i=1}^nx_{i1}^2=9^2+4^2+(5.18^2+\sigma_{11.2})$. where $\sigma_{11.2}$ is the posterior conditional ...

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An alternative approach would be to use a procedure similar to that described in Chow, G. C. and Lin, A.-l. (1971). Best linear unbiased interpolation, distribution, and extrapolation of time series by related series, The Review of Economics and Statistics 53(4): 372 – 75. This procedure is used to produce, for example, quarterly national accounts ...

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You need to think in terms of autocorrelations and volatility to make your choice: in your example you have the change in the number of workers $Y_{t,q}$ what is the meaning of the average change per quarter compared to the yearly production ? probably you should sum your quarterly changes to have a yearly one : I would recommend $\sum_q Y_{t,q}$. if you ...

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the geometric mean is appropriate. rule of thumb: geometric mean for percentage numbers arithmetic mean for absolute numbers and continuous rates

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Why don't you construct the annual value of $Y_t$ from the data, so in your example it would be $Y_{t,annual} = \sum_{i = 1}^{4}Y_{i, quart}$. This is of course only relevant if levels are important, and the time series is in absolute values. If it is a percentage, the geo-mean would be the correct (see https://en.wikipedia.org/wiki/Geometric_mean#...

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