How do I interpret interaction coefficients in a temporally lagged regression model? I am running a linear regression on daily price returns and I want to capture the interaction between subsequent returns:
$r_{t+1} = \alpha + \beta_1 r_t + \beta_2 r_{t-1} + \beta_3 r_t \cdot r_{t-1} + \epsilon$
Where $r$ is a geometric return ($p / p_{t-1}$).  
Each coefficient is interperable as a linear multiplicative factor of the return.  How can I interpret the meaning of $\beta_3$, the interaction term?
 A: $\beta_{3}$ is modeling non-linear effects of $r$ on the future values. Suppose $\beta_{3} > 0$ (everything below is reversed if $\beta_{3} < 0$). In this particular case some things you could interpret are that: 
(1) If $r_{t-1} > 0$ then the effect of $r_{t}$ is increased (or moved toward 0 if $\beta_{1} < 0)$. 
(2) If $r_{t-1} < 0$ then the effect of $r_{t}$ is decreased (or moved toward 0 if $\beta_{1} < 0$). 
(3) If both $r_{t-1}$ and $r_{t}$ have the same sign, then there is a positive effect on $r_{t+1}$. 
(4) If $r_{t-1}$ and $r_{t}$ have a different sign, then there is a negative effect on $r_{t+1}$. 
The simplest way to display these effects are to, for example, plot the model fitted $r_{t+1}$ against $r_{t}$ with a different line for different values of $r_{t-1}$; $-2,-1,0,1,2$ may be suitable choices (you may want to adjust this grid depending on the scale of $r$). In that case, the first line would be $\beta_{1}r_{t} - 2\beta_{2} -2\beta_{3}r_{t}$ - the second one would be $\beta_{1}r_{t} - \beta_{2} -\beta_{3}r_{t}$, and so on.  This is called an interaction plot - you have to choose a discrete grid of points for one of the interacting variables (I've chosen $r_{t-1}$ here) since it would be impossible to plot the interaction across a continuum. The interpretation may become more clear after making this plot. 
