An additive model is a statistical model that consists of several independent variables, which are weighted and summed together to obtain a predicted outcome. The concept of an additive model can be traced back to the early 1900s when it was used to model the relationship between temperature and mortality rates in different cities. 

In an additive model, each independent variable is assumed to have a linear relationship with the outcome, and the model is specified by estimating the coefficients that represent the relative importance of each variable. The process of estimating these coefficients is called regression analysis, which involves minimizing the differences between the predicted values and the actual values. 

Uses of Additive Model in Various Fields

The additive model has been widely used in various fields, including economics, finance, and social sciences. It provides a flexible framework for modeling complex relationships among different variables, and it can be extended to include interactions and nonlinear effects. One of the advantages of the additive model is its interpretability. Since each variable has a well-defined coefficient, it is easy to understand the contribution of each variable to the predicted outcome. This makes it an attractive choice for applications where explaining the results is crucial, such as in policy-making or decision analysis. 

Additive Model with Explanatory Variables

An additive model is a model in which the explanatory variables have an additive effect on the response variable. So, for example, if variable A has an effect of size a on some response measure and variable B one of size b on the same response, then in an assumed additive model for A and B their combined effect would be a + b. One of the best examples of explaining the additive model is the multiple regression model, wherein one dependent variable is influenced by different independent variables, such that the presence of one independent variable is not impacted by the other independent variables within the same model. The additive model can be further explained using the following linear equation-

                                          𝑦 = 𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2+ ∈

Such that, y is the independent variable, now when the value of 𝑥2

is altered by adding 1 to it, such that 𝑥2̃ = 𝑥2 + 1, the equation becomes

                                          𝑦̃ = 𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2̃+∈ 

                                          𝑦̃ = 𝛼 + 𝛽1𝑥1 + 𝛽2(𝑥2 + 1)+∈ 

                                          𝑦̃ = 𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝛽2+ ∈ 

                                          𝑦̃ = 𝑦 + 𝛽2

Now, in this case, it can be found that despite the value assumed by 𝑥2, the incremental effect of 𝑥1 on y remains the same, and it shows the additive effect, and the model is known as the additive model. It is also possible to represent an additive model in time series.

Problem: Businesses are often faced with the challenge of making decisions based on data that can be difficult to interpret. 

Agitate: Without a reliable way to analyze and interpret complex data sets, businesses risk making costly mistakes or missing out on important opportunities. 

Solution: An additive model is an effective tool for analyzing and interpreting large datasets. It provides a comprehensive view of the relationships between different variables in a dataset, allowing businesses to make informed decisions quickly and accurately. With an additive model, businesses can easily identify trends and patterns in their data that would otherwise remain hidden.

In summary, the additive model is a powerful tool for modeling the relationship between different variables. Its flexibility, interpretability, and ability to handle complex data make it a popular choice for many researchers and practitioners. When determined statistically, the additive model is laid down as the weighted sum of different independent variables, such that the impact of each independent variable on the dependent variable is not impacted by other independent variables.