# Cumulative Abnormal Returns (CARs) vs ARIMAX

What is the difference in using an ARIMA model with covariates ($X_i$) to estimate the shock of $X$ on time series $Y$ and using a Cumulative Abnormal Returns (CAR)?

I have limited knowledge about CAR (or AR), but from here, the procedure seems to resemble one of taking a first difference -- but we are subtracting the expected change of $X$ from the actual change of $X$ before using it to explain $Y$.

Step 1
Determine the market return for one day. This is the amount that the market increased or decreased in value.

Step 2
Determine the return on an individual stock for one day. This is the amount that the specific stock increased or decreased in value.

Step 3
Subtract the market return from the return on the individual stock. The result is the abnormal return. For example, if the market return was 10 points and the stock return was 15 points you would subtract 10 from 15 to get an abnormal return of 5 points.

Step 4
Repeat steps 1 through 3 for each of the days that fall within your chosen time-frame. For example, if you wanted to calculate the cumulative abnormal return of a stock over a period of four days you would need to repeat steps 1 through 3 a total of four times, once for each of the four days.

Step 5
Add the abnormal returns from each of the days. The result is the cumulative abnormal return. For example, if you were calculating the cumulative abnormal return for a period of four days and the abnormal returns were 2, 3, 6, and 5, you would add these four numbers together to get a cumulative abnormal return of 16.

Is this the correct way to describe the model?

$$Y_t = \alpha + \beta(\Delta x_{t}-E(\Delta x_{t})) + \epsilon_t$$

Also, I was wondering if there is an intuitive way to understand the difference between CAR and ARIMAX. I can see that the dependent variable is operationalized differently, as in ARIMAX we only explain $Y$'s ARIMA residual at time $t$, not the actual value of $Y$. But is it possible to say more about their respective strengths and weaknesses and the ideal situations to use them?