# Can state-space models be used to solve this question?

Suppose there is a baseball stadium. The stadium has a food stand - let's assume that to make a purchase at the food stand, fans must purchase a ticket to watch the baseball game. This means that all fans who purchased food also purchased a ticket - but not all fans who purchased a ticket, also purchased food.

Let's assume that a private company rents the food stand and is not directly affiliated with the stadium. The food stand is interested in forecasting their future weekly sales (i.e. a univariate time series) - however, let's pretend that the food stand does not have access to the ticket sales information.

Of course, the food stand could use standard ARIMA models to forecast future sales - but could the food stand treat the "ticket sales information" as a "hidden state" and attempt to use this information to create a state-space time series model (e.g. https://sidravi1.github.io/blog/2020/06/20/linear-gaussian-state-space-models)? If the food stand has some very general ideas about the ticket sales (e.g. mean and standard deviation) - could they treat this information as the "initial conditions" for the state space model, and then attempt to better forecast their sales compared to a standard ARIMA model? Maybe the kalman filter can be used for filtration/smoothing, and the hidden markov model can indirectly be used as a proxy to determine how well the home team is playing (e.g. use the following assumption: more wins statistically results in higher attendance, which statistically results in higher sales )?

Can someone please confirm if this general idea describes the "hidden state" concept in state-space models?

Thanks

Note: Let's assume that the food stand has enough data (e.g. past 10 years)

Short answer: Yes this is possible (but it relies on a few assumptions being valid).

## Assumptions

• There is a hidden state (which we can't observe) that is influencing the Food Sales. By definition we can't know what this is but we can use our intuition and give it a name like "Ticket Sales Level" for example. Or like you said it could be "Home Team Performance". Or "Home Support Excitement".
• The hidden state is Markovian, i.e. its next value only depends on its current value. So we assume that just knowing how full the stadium is today will be useful for predicting how full the stadium will be next time round.

At any rate, it's completely possible that there could be a HMM with (for example) 3 hidden states (Good, Neutral, Bad sales) that can predict future sales.

Your HMM would tell you: "We're currently in a state of Good sales. When we're in this state we have average sales of 60,000±2,000. There's a 20% chance we'll switch to Neutral sales in the next game, 10% chance to switch to Bad Sales and 70% to stay in the Good Sales state."

Just to reiterate we need these assumptions to hold:

• There is a useful hidden state.
• The hidden state is Markovian.