R15.P1.T2.STOCK & WATSON_CH 5-Topic: Interpret Hypothesis Tests about Regression Coefficients

[gargi.adhikari]

In reference to R15.P1.T2.STOCK & WATSON_CH 5 Topic: Interpret Hypothesis Tests about Regression Coefficients
For the Significance Test, why are we comparing the Slope-Value of -2.28 to zero...?Why are we expecting the Slope to be zero...? and performing the Hyp-Test by comparing against zero...?

 

[ShaktiRathore]

Hi,
The test is that: H0: populn mean(B1)=0 and Ha: populn mean(B1)!=0 where B1 is populn value that corresponds to sample vale of b1.
we know that for confidence level 95% t-stat~2 we know that for a sampling distribution(assumed t-distribution here) the sample mean must lie within B1 +/- t-crit*S.E. =B1 +/- 2*S.E. or that rearrange to get the 95% confidence interval for populn mean of sample mean +/- 2*S.E. so that we are 95% confident that the populn mean for the coefficent is within this interval sample mean +/- 2*S.E.=b1+/-2*S.E.=-2.28+/-2*.48=-2.28+/- .96=-3.24 to -1.32 as the interval does not contains 0 there fore we are 95% confident that the B1!=0.
Let t-stat be the t-value such that the interval contains the 0 thus -2.28+t-stat*.48=0 =>t-stat=(2.28-0)/.48=4.75 thus t-crit must be atleast (>=)4.75 for the interval to contain 0 and accept the null that B1=0 thus as at 95% CL as t-crit=2<4.75 we reject the null and conclude that the B1!=0.
thanks

 

[David Harper CFA FRM]

Hi @gargi.adhikari

You are correct in the sense that we do not require any particular value for the hypothetical ("true") slope parameter; "hypothetical" = value under the null hypothesis.

But two things (in addition to what @ShaktiRathore just cross-posted )

1. By convention, when we ask "is the coefficient significant?" we are implying a test of the null that the coefficient is equal to zero; i.e., zero connotes insignificance because, after all, if the coefficient is zero then the associated independent variable appears to have no influence on (association with) the dependent variable.
   
2. The t ratio computed in a typical regression output implicitly assumes the zero; for the same reason: the "default" t ratio intends to test null hypo that population param is zero.

But β=0 is not necessary. We can assume a different null. Thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much ! Exactly what I was looking for ..
So is this the takeaway...? If β=0 which in turn is our NULL HYP criteria here, then that would imply that the independent variable has no influence on the dependent variable. So if NULL HYP fails, then that confirms that the Independent and Dependent variable are indeed co-related..? So in other words, the purpose of the HYp Test with the Regression Coeff was to prove that there is correlation. Did I get that right...?

 

[David Harper CFA FRM]

@gargi.adhikari Yes, you are correct (in every word, that I can see)!

Null is: H(0): β = 0, but could be different. Our most common difference is maybe regressing a portfolio against its benchmark such that the null is (H0): β = 1.0
Therefore: H(A): β <> 0

We draw a sample, regress and observe b = 2.28, which is obviously not zero. But it's impossible to tell if it's a little or a lot different than zero. The t ratio (aka, t statistic) essentially standardizes the 2.28 with t = (b - β)/σ = (b - β)/SE = (2.28 - 0)/0.48 = 4.75. This 4.75 is a standard deviation on a student's t, which is "almost normal." This tells us 2.28 is 4.75 unit standard deviations away from zero. Luck (ie, sampling variation) should produce something non-zero but still within one or two standard deviations. 4.75 is too far for luck, if in fact ("conditional on") the true parameter being zero. So, we reject the null in favor of a belief that β must be non-zero. And if beta is non-zero, it's also correct to say that the variables are "correlated" (by the way) because correlation and beta are functions of each other, per β(X,Y) = ρ(X,Y)*σ(X)/σ(Y), so non-zero beta also implies non-zero correlation.

 

[gargi.adhikari]

@David Harper CFA FRM @ShaktiRathore Thank you guys. Y'all are the the best !