Forecasting time-series ahead by multiple time horizons

Question

Suppose that I have daily data on the population of a small village, given by $Y(t)$, as well as daily data on various factors that are relevant to the size of the population in the future, given by vector $X(t)$. These explanatory variables include untransformed variables as well as features engineered to be informative over long horizons (e.g. one of the variables captures the number of deaths over the last 30 days). I have collected this data for 8 years.

My objective is to forecast $Y(t)$ ahead by 1,2,3,...,365 days. I expect long-run forecasts to be different to short-run forecasts. If a holiday season is coming up I might expect a downwards spike in a few months time (people visiting the city), but if someone is on their deathbed then I will expect a downwards spike in a few days.

Since the population is sufficiently small that $\Delta Y(t+k)$ is typically in $\{-2,-1,0,1,2\}$ for the forecasting horizon under question, I will use a multiple categorical response variable classification model that will assign probabilities to the various class labels being observed.

My question centers on the specific considerations I need to make when constructing forecasts of the change from $Y(t)$ to $Y(t+k)$ where $k$ is large (e.g. 100 days).

Basically there will be the most hideous autocorrelation structure in $\Delta Y(t+k)$ over these time scales. If someone dies on day $2$, they are also dead on day $3, 4, ..., k$, meaning a string of $k$ or so $\Delta Y(t+k)$ will contain this same information.

These queries result:

• What are some ways of dealing with this immense autocorrelation structure in my response. Is it even a problem?
• Are there alternative methodologies to the ones I've proposed for forecasting these horizons (aside from typical machine learning methods such as random forests which I'm already working with).
• Any other handy advice.

Answer 1

You could consider working with ARIMAX or SARIMAX models.

1. These models will allow you to deal with any problem of autocorrelation. You can include autoregressive (AR) or moving average (MA) terms as needed, to eliminate autocorrelation.
2. You can easily include any independent variables along with the AR and MA terms.
3. You can model any seasonal behaviour as you need to account for fall or rise in population during holiday seasons or any other season.
4. They usually provide good forecasts. If you are using independent variables, you will need data on those variables to make forecasts for the future. Dynamic forecasts can only be made if the model is purely AR/MA process.