Testing Additivity and Linearity in a Financial Income Model

# How to test the linearity assumption of a model?


# How to test the linearity assumption of a model?












Let's say I want to have a model that projects income over a stressed period. I have a marked-to-market component that shows the P&L of trading book positions during this stressed period. Along with that, I have Gross Revenue data, Expenses, Carry, Transfer Pricing, Treasury Costs, etc, such that: MTM P&L + Forecasted Revenue + Forecasted Carry + Forecasted Transfer Pricing + Forecasted Treasury Costs - Expenses = Income

I want to check whether this linear assumption holds; that Income is simply additive across the variables mentioned. How could I test this?

## Answer by python_enthusiast (score 1)

https://quant.stackexchange.com/a/44397

To test the linearity of a model, you look at the residuals obtained from the regression. For example, check out this code snippet from R:

```
# Import a library that contains data:
library(car)
head(mtcars)

# Fit a multiple regression on this data:
# (We are trying to predict miles per galon from some car variables)
fit <- lm(mpg~disp+hp+wt+drat, data=mtcars)

# Evaluate Nonlinearity
# component + residual plot 
crPlots(fit) 

# Notice that the relationship is nonlinear with respect to the variable 'disp', for example.
```

You could also have used, instead of `crPlots`, the function `ceresPlots`, which is slightly different, but serves the same purpose of checking for non-linearities:

```
# Ceres plots 
ceresPlots(fit)
```

## Answer by Richi Wa (score 0)

https://quant.stackexchange.com/a/43812

Do you have multiple observations of this relation? If so then you could perform linear regression and perform all the regression diagnostics.

## Answer by Dave Harris (score 0)

https://quant.stackexchange.com/a/43849

You would perform a multiple regression, except that instead of using the standard default of $\beta=0$, for all of your $\beta$s, instead, you would use $\beta=1$ for all $\beta$s. If the F test is statistically significant, then your null hypothesis is falsified. Depending upon the computer language, it will require a manual intervention into the code.

## Answer by Drew (score 0)

https://quant.stackexchange.com/a/43851

It sounds like you are asking about the adequacy of a linear model fit to data on these variables. In this context, you need to suggest the functional form of an alternative to test it against. A model including linear as well as (at least some) squares and cross-products of your variables seems a standard alternative, and can sometimes be rationalized as an approximation (Taylor) of a more generic differentiable functional form. In short, you're going to compare the fit of your linear model to one including these other terms in a regression context.