Linear Regression #8: Linearity

Very often in order to come closer to the “true” model, regression models contain non-linear terms. In 1969 James Ramsey developed the Regression Equation Specification Error Test (RESET) as a general specification test. It examines whether non-linear combinations of the fitted values explain the target variable.

Linear Model:

\[H_0:\ \mathbb{E}\left[y_i\middle|\ x_i\right]=x_i\beta\]

Non-Linear Model:
\[H_1:\ \mathbb{E}\left[y_i\middle|\ x_i\right]\neq x_i\beta\]

The procedure of testing is described hereafter:

  1. Build the standard OLS linear regression model.
  2. Use the fitted values of the linear regression and add them to a new model. Set \(\small{ K }\) (often \(\small{ K=1,2,3 }\) )
    \[{\hat{y}}_{RESET}=x_i\beta+\sum_{k=1}^{K}\omega_k\ {\hat{y}}^{k+1}\]
  3. Use the common F-test to decide if \(\small{ {H_0:\ \omega}_{1..K}=0 }\).
  4. If \(\small{ H_0 }\) is rejected, the linear model is misspecified and hence there are non-linear terms.
    In R, the test can be run by the command \(\small{ resettest(model) }\).