# Forecasting monthly demand given yearly shocks

I have a dataset of monthly demand in dollars. We'd like to forecast the next year by month using an ARIMA technique. We have a pool of 500 clients who submit invoices on behalf of hundreds of thousands of individual customers who are actually driving the "demand". Our clients WILL invoice, we are unsure of how much, though prior invoice levels tend to predict fairly well. Currently we're planning on modeling each of our clients individually and aggregating to the total-yearly demand estimate.

We'd like to control for exogenous factors though - specifically we have 2 "shocks" that occur once per year. These shocks are the different decision making bodies setting out the values for the next year.

One of them is that we have a rate we pay out for each invoicing body. That rate is set on the first day of the fiscal year and persists all year. We also have an inflow of customers that our clients have access to on the first as well.

We observe both of these factors for all years, but we are unsure how to incorporate them into the model since they are observed once per year while we are attempting to model monthly frequency. We don't want to disaggregate them because it's not as if customer_pool/12 is the pool of customers in a given month.

I've been looking into transfer functions, but I'm unsure how one would be specifically implemented in our case. I'm getting lost in the literature and was hoping someone could give some direction in how something like this would be modeled.

Below is some example data for a single client. Note that the Customer_pool is constant across all clients, while Single_rate is client-specific. Notice the change from 200712 to 200801

 +------------+-------------+---------------+-----------------+ | Year_Month | Single_Rate | Customer_pool | Invoiced Amount | +------------+-------------+---------------+-----------------+ | 200701 | 75 | 800 | 5,525.99 | | 200702 | 75 | 800 | 3,004.50 | | ... | | | | | 200712 | 75 | 800 | 4,961.10 | | 200801 | 85 | 865 | 5,900.55 | | ... | | | | | 201709 | 125 | 1020 | 15,029.00 | | 201710 | 125 | 1020 | 9,014.00 | +------------+-------------+---------------+-----------------+