# Coefficient Averaging to Predict Value at end of Time-Series/Possible Random Effects Model?

I have some discrete time series data that consist of the following variables of interest:

• Project ID ($$project\_id$$)
• The total budget for a particular project ($$tot\_budget$$)
• The person who runs the project ($$person\_id$$) and spends all project monies up to the amount of the total budget specified above
• The number of days until the project ends ($$t$$) (the project ends at $$t=0$$, but projects have different starting values of $$t$$.
• The cumulative amount of money spent (in dollars) on the project at time $$t$$, $$(cum\_budget\_expenditures)$$
• The cumulative percent of the budget spent for the project at time $$t$$, $$(cum\_pct\_budget\_spent)$$. This is simply calculated as $$cum\_pct\_budget\_spent=cum\_budget\_expenditures/tot\_budget$$

I have data on a large number of different projects, each with different start dates, end dates and $$t$$ measured at possibly different points in time for the various projects—the dataset was updated every time an expenditure for a given project was made, and those are essentially "random" (in the laymen's sense of the word). Each person ($$person\_{id}$$) may have worked on multiple projects as well.

What I was hoping to do is build a general regression model that will allow me to predict, for the average project and at any point in time, the total (cumulative) percentage of the budget that will be spent when the project comes to a close. In other words, at some point $$t\ne0$$, I want to estimate what an average project's $$cum\_pct\_budget\_spent$$ at $$t=0$$ will be by using $$cum\_pct\_budget\_spent$$ at $$t\ne0$$ as the independent variable.

Initially, I thought about simply taking all the values of $$cum\_pct\_budget\_spent$$ across all projects at regressing these onto $$t$$, which would give me a regression equation. I could then plug $$t=0$$ into this equation to obtain my estimate of the total percentage of the budget spent at the end of the project. But then I realized that the parameter estimates of my regression equation might lead to pretty poor predictions since I'm not taking into account the individual projects here. For example one project might spend very little, but another quite a bit. So I was thinking what might be better is to fit a regression model to each project, and then average across all the parameter estimates to obtain a final regression model that could be used for any new project. Can I do this? Can I average model parameters like this? Is this similar to letting $$project\_{id}$$ be a random effect in my model? It seems to me if I let the $$project\_{id}$$ be a random effect in my model, this could account for the correlation in my data, which I know is problematic.

A second, less import question I have at this stage is if there other time-series type models that might be better suited for this. If so, what are they?

Thanks.