How can I use a time series consisting of both Mean and SE to obtain predictions with Mean and SE This is a slightly modified version of a programming question that I asked in that forum and was redirected here to get a better sense of the statistics involved.
https://stackoverflow.com/questions/74839826/how-can-i-pass-a-time-series-consisting-of-both-mean-and-se-to-the-predict-funct/74840556?noredirect=1#comment132089220_74840556
I have a time series but each value in the time series is a mean and an SE. I do not have access to the raw data that calculated these Means and standard errors. I might get the sample size for each year, but I do not have that at the moment as well.
I would like to use this information to create an exponential model.
However, I am unsure how to use the SE values of my time series for this prediction.
My understanding is that whatever be the SE in the time series, the predicted values of the Mean remains the same. But the SE of the model itself should have a dependency on the SE of the time series. This SE is very important for me and I would later introduce that in the plot as well.
I got some good advice that I can calculate the CI for each year and predict Mean + CI and Mean - CI. However, that prediction itself will have an SE and how to add them up is not clear to us.
I also got an advice to throw multiple points for each year using the Mean/SE of that year. Then use those points to fit a model and predict. Intuitively, that seems fine to me but it would be great to obtain a confirmation.
If there are other tactics to solve this, I would be much grateful.
 A: My advice is to start by spelling out the data generating process, or a model of the situation at hand. For example, are the underlying data normally distributed? How (why) does the standard deviation change over time? Is the mean constant, but unobservable, or does it change over time? If so, what is the dynamic equation for the mean? This model should then guide your choices of statistical methods.
For example, if the mean is constant over time, but unknown and you want to gradually learn it from the data, you can easily calculate predictive distribution in closed form via Bayes rule. The best prediction (in terms lowest mean-squared error) is the conditional expected value of the mean, conditional the information so far.
If I understand your question correctly, this is kind of a state-space situation here. The true mean of the process is unobservable, but it changes over time (you want to forecast it), possibly an AR(1) process, and you have access to some measurements of this mean. The measurement is subject to error with a given SE. This seems like a filtering problem to me, take a look at my other answer regarding state-space models, Kalman filter and predictions here
If, on the other hand, the std. dev. itself exhibits time series patters and is of separate interest to you (you want to forecast it as well), take a look at stochastic volatility models. The Stan manual has a good TLDR here
Update
Based on your comment, it seems that the model you have in mind is as follows. You are after the "true" mean, $\mu_t$. But, you can only see an imperfect measurement of it, say $y_t$. The two are linked as follows,
$$
\begin{eqnarray*}
y_t &=& \mu_t + \varepsilon_t, \qquad& \varepsilon_t \sim N(0,\sigma_{\varepsilon}), \\ 
\mu_{t+1} &=& \rho \mu_t + \nu_t, \qquad& \nu_t \sim N(0,\sigma_{\nu})
\end{eqnarray*}
$$
This is a Kalman filtering problem. See my answer linked above to learn how you estimate $\rho$, and both "sigmas" by maximum likelihood estimation. Once you know those, you can show that the optimal (in the squared error sense) predictive density is
$$
\mu_{t+1} \sim N(a_{t+1},P_{t+1})
$$
where the mean and stdev can be calculated recursively as
$$
\begin{eqnarray*}
a_{t+1} &=& \rho a_t + (P_t/F_t) (y_t - a_t), \\ 
F_t &=& P_t + \sigma_{\varepsilon}^2,\\
P_{t+1} &=& P_t(1-P_t/F_t) + \sigma_{\nu}^2
\end{eqnarray*}
$$
For some initial $a_0, P_0$. Since you have the whole distribution of your predicted mean, $N(a_{t+1},P_{t+1})$, you can easily compute confidence intervals. See also Chapters 2 and 3 in Durbin and Koopman "Time Series Analysis by State Space Methods" (2nd ed)
