# Stupid Q: Interpretation of slope in OLS

#### Jonni

##### New Member
Hi ppl

I'm running a bunch of regressions on a series of stock returns (decimal values).

One of the models is:

r_t = a0 + b1*rm_t + e

where r_t is the stock return and rm_t is the market return (both decimal values).

My interpretation is that a0 is the mean return of r_t, and that b1 is the sensitivity to the market return.

So if my constant is: 0.00108623 i have a daily return of 0,11 pct. b1 is -0.0110737; how would I intrerpret this and is my interpretation of the constant correct?

#### Jont

##### New Member
One unit increase in market return (so rm_t + 1) will cause an increase of b1 in stock return r_t.
The constant is just the y-intercept, or what r_t would be if rm_t = 0.
In this case, however, your slope is a negative. So one unit increase in rm_t will decrease your r_t by b1... Is this what you meant?

#### noetsi

##### No cake for spunky
ao is the intercept. As noted it is the value of r_t when b1*rm is 0. It is only an average (a mean value) in the special case that you are using a dummy variable. Then it is the mean on the dependent variable for the level of the dummy variable that is coded 0 (that is the reference level). At least for simple regression. The definition gets more complex when you have more indpendent variables.

#### d21e7x11

##### New Member
Hi Jonni,

I'd like to add few words to the interpretation of the intercept in your model.

In your model, think what would happen if rm_t=0? Then r_t = a0 so the interpretaion of a0 is the value of the stock return when the market return is 0. Now, if you run your model with the variable

rmcen_t = rm_t - mean,

instead of rm_t, and obtain an intercept a0cen, the interpretation is more interesting. Same as above, a0cen is the value of the stock return when rmcen_t=0 BUT rmcen_t is 0 when rmcen_t=mean, where mean is the average market return.

So in this case, the interpretation of the intercept a0cen is the value of the stock return when the market return is equal to its average. Furthermore, an arbitrary value of rm_t can be used instead of the mean. Then, the intercept is the stock return when the market return is equal to that value.

#### noetsi

##### No cake for spunky
That is a very interesting way of modifying the model to make the intercept more meaningful.