# Time series data: How I measure influence of new product sales on existing product sales (statistically)?

Here my goal is…

• Find Product 5 (New Product) is really influencing other product sales (product 1 to 4) or not?
• If it is influencing other product sales, how much?

New to R and tried several related posts but didn’t find exact answer to my question. I love R and learning every day something new which helping us in taking data driven decisions.

My sample Dataset is like below (Week and Product 1 to Product 5 Sales per each week) Here my new product is Product 5 and launched on Week 5.

Week    Product-1   Product-2   Product-3   Product-4   Product-5
1   2   4   5   5
2   4   4   6   4
3   4   4   6   5
4   4   4   6   6   4
5   4   6   5   3   5
6   2   7   6   4   3
7   3   8   7   5   6
8   2   9   9   3   6


## Here my questions are

• What is the best process or model to show the influence of product 5 (statistically)?
• Do I need to run co-integration tests before I run correlation? Example some of these products are never be correlated with Product 5 (example: growth in cockle growth vs. growth in electricity demand)
• How I know correlation vs. causation in this mix?
• Since my new product launched on week 5, where I can start my correlations? Is it from week 5 or from earlier weeks?
• Do I need to test for stationarity first? and bring the data to stationary?

You could build an ARIMAX model. This would permit to include autoregressive (AR) terms as well as well as the sales in product 5 as an Exogenous Input (X). This would give you a potential model where the sales for a product $i$ at time $t$ is given by $s^i_t$ then,

$s^1_t=\alpha_1 s^1_{t-1} + \alpha_2 s^1_{t-2} + \ldots + \beta_0 s^5_t + \beta_1 s^5_{t-1} + \ldots$

Note that you may need to make the series stationary first, but see more on that below. You could estimate this model with the seasonal R package that relies on the X-13ARIMA-SEATS software developed by the US Census Bureau.

I would recommend to ensure that your time series are all stationary, see for example this post before you use X13. I would also run cointegration tests. See for more explanation this excellent post.

Since you only have data on week 5 I would start modeling in week 5 but you could include autoregressive (AR) terms related to the sales of product 1 prior to week 5.

• How I know correlation vs. causation in this mix?

Finding the causal affect of one variable on another is a difficult one because there are probably a few hidden variables that are the driving factors behind product 5 and all others. For example the true causal effect could be that the weather improved causing sales to increase for product 5 and others, making them correlated but not having any causal relationship.

One way of trying to remove bias in determining causality is through the use of the following https://en.wikipedia.org/wiki/Instrumental_variable.