Models to Analyze the Impact of Advertising on Sales

To model the relationship between advertising mediums (TV, newspapers, radio) and sales, you can consider the following two models:

1. **Multiple Linear Regression Model**: This model helps understand how each advertising medium affects sales:

   ```

   Y = β0 + β1 * TV + β2 * Newspapers + β3 * Radio + ε

   ```

   Where:

   - `Y` is the sales.

   - `β0` is the intercept.

   - `β1`, `β2`, and `β3` are the coefficients for TV, newspapers, and radio, respectively.

   - `ε` is the error term.

   This model estimates how much each medium contributes to sales, assuming linear relationships between the advertising expenditures and sales.

2. **Time Series Analysis (if data is temporal)**: If your data is collected over time, consider a time series model that accounts for trends and seasonality, along with the effects of advertising:

   - **Autoregressive Integrated Moving Average with Exogenous Variables (ARIMAX)**:

     ```

     Y_t = α + β1 * TV_t + β2 * Newspapers_t + β3 * Radio_t + ARIMA(p,d,q) + ε_t

     ```

     Where:

     - `Y_t` is the sales at time `t`.

     - `TV_t`, `Newspapers_t`, and `Radio_t` are the values for each advertising medium at time `t`.

     - `ARIMA(p,d,q)` represents the autoregressive, differencing, and moving average components.

     - `ε_t` is the error term at time `t`.

   This model accounts for both the impact of advertising and time-dependent patterns in the data, such as trends and seasonal effects.