Variable and its dynamics in one multiple regression model

Question

I am trying to find the dependence between default rates in bank and macroeconomic variables with linear regression. To do so I created a code which estimates every possible model - every combination of variables is tested. As an output I obtain R-squared, statistics for Chow, Breusch-Pagan, Breusch-Godfrey, RESET and Shapiro-Wilk tests as well as VIF. The only model which pass all tests, has satisfying R-squared and low VIF is as follows: $ y_t = \beta_0 + \beta_1 x_t + \beta_2 \varDelta x_t$ where $ \varDelta x_t = x_t - x_{t-1} $. Altough using variable and its dynamics in one model seems a bit strange, I did not find any reason to reject the model. I would be grateful if someone could help me motivate accepting or rejecting such a model.

Answer 1

If you plugging-in $\Delta x_t$ into the model, you have $$ y_t = \beta_0 + \beta_1x_t + \beta_2x_t - \beta_2x_{t-1} + \epsilon_t, $$ denote $\theta_1 = \beta_1 + \beta_2$ and $-\beta_2 = \theta_2$ your model is $$ y_t = \beta_0 + \theta_1 x_t + \theta_2 x_{t-1} + \epsilon_t. $$ Which, from statistical point of view has nothing extraordinary. I.e., the expected value of $Y$ (rate?) at time $t$ given the history of the process $X_t$ up to time $t$ depends only on its current ($t$) and previous ($t-1$) values. Whether it make-sense from an economical point of view - is a matter of economic theory, not statistics.