David Black wants to test whether the estimated beta in a market model is equal to one. He collected a sample of 60 monthly returns on a stock and estimated the regression of the stock’s returns against those of the market. The estimated beta was 1.1, and the standard error of the coefficient is equal to 0.4. What should Black conclude regarding the beta if he uses a 5% level of significance?
A- equal to one is rejected
B- not equal to one cannot be rejected
C - equal to one cannot be rejected.
The answer is C.
My question is why is the null hypothesis that the beta is equal to one? I always thought that the hypothesis that basically says what you’re trying to prove isn’t true is the null? So wouldn’t that make the null hypothesis, “the estimated beta is not equal to one”?
I’m not sure if the null always has to include the = sign. A null hypothesis is by default when you say that there’s no relationship between the two variables. I think that the null could also be that beta is not equal to 1. I was also confused by that question too pmond. I thought it was the H alternative.
The null always has to have an equals sign (and either > or < if it is a one-tailed test). You could never have a null that was H0: beta !=1 or H0: beta > 1.
In this case, they null hypothesis is:
H0: beta = 1
Ha: beta != 1
The question you are attempting to answer is that if the average beta value based on 60 measurements of return is 1.1 and you know that the standard error for beta is 0.4, do you reject that the beta you measured is statistically significantly different from 1? In other words, your sample has measured an average beta of 1.1, but is that really different from 1 or is the observed difference versus the hypothesized value due to chance?
It would be a mistake to say you’re trying to “prove” anything. In fact, you can never “accept” the null hypothesis as being true (hence, we “fail to reject” it). You can reject the null, however (with a certain level of confidence).
In this case, failing to reject the null that the population beta is equal to 1 is not saying that it IS equal to 1. It is simply saying that we do not have enough evidence to demonstrate statistically that 1.1 in the sample is different from 1.
As a practical matter, the null is really asking whether the sample portfolio returns track the market portfolio returns (beta=1 means a one unit change in one associates with a 1 unit change in the other). In this case, the math shows that we cannot reject that hypothesis.
Remember that this is a one sample t-test to test a variable against a hypothesized mean NOT a two-sample means comparison test; you are not (directly) testing whether there’s a relationship betwen two variables. You are testing whether the beta you observed in your sample is equal to 1 or is different from 1.
