Variable and its dynamics in one multiple regression modelI 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 = β 0 + β 1 x t + β 2 Δ x t where Δ 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.

Question

Variable and its dynamics in one multiple regression modelI 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    =      β    0    +      β    1        x    t    +      β    2    Δ      x    t   where   Δ      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.

spornya1 · Accepted Answer

If you plugging-in   Δ      x    t   into the model, you have      y    t    =      β    0    +      β    1        x    t    +      β    2        x    t    −      β    2        x          t      −      1        +      ϵ    t    ,denote       θ    1    =      β    1    +      β    2   and   −      β    2    =      θ    2   your model is      y    t    =      β    0    +      θ    1        x    t    +      θ    2        x          t      −      1        +      ϵ    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.

Variable and its dynamics in one multiple regression model

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