# How to interpret a linear model which contains differences of an included variable?

I am unsure about the interpretation of a linear model which contains variable $x$ at time $t_1$ and the difference from $x$ at time $t_2$ and $x$ at time $t_1$:

$$y = \alpha x_{t1} + \beta x_{diff} + \epsilon$$ where $x_{diff} = x_{t2} - x_{t1}$.

At first I was tempted to interpret the model the "usual" way. I would have said, that $\alpha$ can be interpreted as the change in $y$, when $x_{t1}$ changes by $1$ and $\beta$ as the change in $y$, when $x_{diff}$ changes by $1$ holding all other variables constant. However, after rewriting the model I think that this interpretation is wrong.

$$y = \alpha x_{t1} + \beta ( x_{t2} - x_{t1}) + \epsilon$$ $$y = (\alpha - \beta) x_{t1} + \beta x_{t2} + \epsilon$$

Now I would say, that $\beta$ is the change in $y$, when $x_{t2}$ changes by $1$ holding all other variables constant. But doesn't this imply that $x_{t1}$ remains constant and therefore $\beta$ represents the influence of the effect of the changes between both points in time? If this should be true, then my first interpretation would be correct for this model (replacing $\alpha$ by $\gamma$):

$$y = \gamma x_{t1} + \beta x_{t2} + \epsilon.$$

Hence, $\beta$ was interpreted correctly but $\alpha$ was not.

Does this make sense or am I on the wrong way?

• You could decompose any linear model like this. It seems to depend on where you're looking for an effect and what you believe is meaningful. What's the reason for looking at the difference in the first place? Is it because $x_{t1}$ and $x_{t2}$ are related but you think the difference is uncorrelated with the first x? – Josh Magarick Oct 10 '16 at 16:11
• I assume that x_t1 and x_diff have different effects on y. In my case I think that a high value in x_t1 has a negative effect on y, whereas x_diff has a positive. I decomposed the value in x_t2 because I want to capture both effects. – Alex Oct 10 '16 at 16:30
• Look at lag models, autocorrelation, and autoregression for starters. – Carl Oct 10 '16 at 17:19
• @Carl: I don't think it is related to time series. – Alex Oct 10 '16 at 21:54
• The way you are expressing your equation appears to have the form of a lag model. So why would it not be related to a lag series? – Carl Oct 11 '16 at 1:48