I am trying to understand the possible causes to this regression issue I am having.

I have a basic time-series regression:

(1) $y_t = \alpha_0 + \sum_{j=0}^{J=8}\beta_jx_{t-j} + \epsilon_t$

I am particularly interested (of course) in the $\beta_{j}$. $X_t$ is a continuous variable (e.g stock return).

I also have this model: (2) $y_t = \alpha_0 + \sum_{j=0}^{J=8}\beta_jx_{t-j} + \sum_{j=0}^{J=8}\gamma_jz_{t-j} +\epsilon_t$

where $z_t$ is also a continuous variable [0, +infinity] (e.g. trading activity).

In model (1), my $\beta_{t}$ are very similar to $\beta_{t}$ in (2).

I am particulary interested in the interaction effect between $x_t$ and $z_t$, so I estimate the following regression

(3) $y_t = \alpha_0 + \sum_{j=0}^{J=8}\beta_jx_{t-j} + \sum_{j=0}^{J=8}\gamma_jz_{t-j} + \sum_{j=0}^{J=8}\kappa_jx_{t-j}z_{t-j} + \epsilon_t$.

Now what troubles me is that my $\beta_j$ in (3) are quite significantly different from model (1) and (2). What can explain this? What is a proper diagnostic to investigate this issue?

  • $\begingroup$ I assume that your $\beta¨j$ have changed, but that at the same time their standard erros have changed ? I would have to see the results but with what you describe I think you may have multicollinearity among the independent variables by introducing $x_t,z_t$ ? (By the way, $t-j$ is not to be read as $t$ mnius $j$ but as $x_{tj}$ I assume ?) $\endgroup$ – user83346 Aug 23 '15 at 9:10
  • $\begingroup$ The standard errors have changed and it is $t-j$ since I control for autocorrelation in the lag obervations of x and z. I also believe it is a multicollinearity issue ... how problematic can this be? $\endgroup$ – CharlesM Aug 24 '15 at 4:20

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