# Forecasting time-series data with regression variables

## Goal

The goal is to forecast the prices all participating companies will set in period $t0$.

## Problem description

It is a real world application, in which multiple suppliers offer the same product for the purchaser for a certain period in a reverse blind auction. Only one price per supplier is submitted for each period. Lowest price wins, and the winner takes all the sales for the period it won.

The problem, as far as I understand it, is that the prices a supplier sets depend on both observable variables and latent time-series variables.

Firstly, as an example of the observable part in the attached data set, the price company A (denoted by a) will set on period $t0$ is very likely based on the prices all other companies set on period $t-1$, and possibly even on periods $(t-2 \dots t-k)$. Were the supply of the product for all companies endless, I presume the sensible thing to do would be to just build a regression tree (or some other model) predicting company price at $t0$ and using as predictive variables all other companies' prices at ($t-1\dots t-x$), where $x$ is "suitable" amount of lags. Additionally, the price changes for every company month-to-month are not linear.

Secondly, there is a latent part present also. The products have a relatively low shelf life, high stock value and long lead times in procurement. This combination leads quite often to a company running out of stock after one, and even more often after two consecutive wins. This is usually indicated either by a significant price hike or by withdrawing from the offer round. Additionally, there are some examples of a company dumping their price after a long series of non-wins in order to get rid of stock which would run out of shelf life. So there is a real possibility for autocorrelation in some of the time series. Again, if it was just a simple task of observing how a single company works in the market without any competition, I would probably use a some time series model.

The problem, as far as I understand is a combination of these two separate approaches:
1) the causality of which the other companies' pricing decisions have on the company's pricing decisions
2) the hidden model of the supply chain capabilities and pricing tactics of the company as observed in the time series

## Question:

1) are my assumptions correct on the dual nature of the problem?
1.1) if so, which approach or model would best take into account this duality? (Dynamic regression models?)
1.2) what requirements this poses to the data?
2) If my assumption is incorrect, which do you think would be best?
3) Which, if any, additional variables should be created (periods since last win, difference to winning price etc?)

## Data example:

+---------+-------+-----------+-----+-------+-------+-------+-------+ | Company | p_t-k | p_t-(k+1) | ... | p_t-3 | p_t-2 | p_t-1 | p_t-0 | +---------+-------+-----------+-----+-------+-------+-------+-------+ | A | 127.3 | 20.9 | | 16.1 | 19.3 | 170.2 | a | | B | 18.2 | 17.6 | | 14.4 | 15.4 | 13.3 | b | | C | 22.6 | 8.0 | | 31.5 | 24.0 | 31.0 | c | +---------+-------+-----------+-----+-------+-------+-------+-------+

I apologize for all potential misuse of terminology and notation.