# model specification problem

I have the following model that I would like to rebuild:

$Y_{i,t}=a+bx_{i,t}+cx_{i,t-1}+e_{i,t}$

I' am wondering now whether this is the same as the model above:

$Y_{i,t}=a+ d\Delta x_{i,t}+e_{i,t}$, where $\Delta x_{i,t}=x_{i,t} - x_{i,t-1}$?

I believe this is true but i just need to be 100% sure.

Thank you

• I replaced "delta" with "\Delta". Hopefully that's what you intended. If not, you can edit. You should include an edit or additional discussion to clarify your question (see Alexis' answer). Aug 9, 2014 at 2:09
• this is indeed the notation I had in mind. thank you for clarification Aug 9, 2014 at 6:30
• Now that you ask a clear question, it's easy to answer: The two models are not the same, but the second model is a special case of the first, with $b=1$ and $c=-1$. Aug 9, 2014 at 7:32

I would say a few things.

First, I wonder if your notation is off? In the second model, do you mean: $Y_{i,t}=a+ d\Delta x_{i,t}+e_{i,t}$, where $\Delta x_{i,t}=x_{i,t} - x_{i,t-1}$?

If that is so, then I would say that these models are not equivalent, because there are at least two kinds of short term effects $x$ can have on $Y$ (1) effects of change in $x$ (i.e. $\Delta x$), and (2) level effects (i.e. effects of $x$ itself).

Your second model includes only the effects of change in $x$.

You first model includes the effect of the level of $x$, plus an adjustment for the effect of the level of $x$ from last time, which indirectly gives an effect of change in $x$.

• thank you for your detailed answer! The problem with my data/model is that it puts too much emphasis on the these two variables although in the overall model they "should" only play a tangential role. I was hoping to circumvent this problem by including only the change but as you mentioned I'll loose the level effects. Are there any possibilities to "control" a certain emphasis on model variables? Aug 9, 2014 at 6:28
• Not sure what you mean... do mean the effect sizes for your $x$ variables are larger than for some other variables that you care about? Aug 9, 2014 at 7:29
• I was referring to the coefficients. From an economical point of view the coefficients should take on certain levels that "make sense". I fear that my model somehow overestimates the effect of xi,t−xi,t−1 (bookvalues) these values and thus the coefficients are way off values you would expect. Aug 9, 2014 at 7:57
• So it sounds like your expectations may need updating? :) Aug 9, 2014 at 8:22