In the above exmaple - is the t-stat they provide the t-stat for testing whether or not that coeficcient is statistically different from 0? Is it always like that?
Yes, they are exactly those T-stat you can also do by hand. And the P-value is the probability of commiting type error 1 or the probability of reject the null hypothesis when in fact it was true.
You must compare the p-value with the “alpha” you chosed for your research. Alpha is commonly chosen to be 5% or 0.05, but can also be 0.001, 0.01, 0.10, or even 0.145358, etc. If p-value is lower than alpha, then you can reject the null hypothesis for that variable.
Just wanted to clarify one point-- alpha is the probability of making a Type I error. A p-value is the probability of observing a test statistic that’s at least as extreme as the result at hand, given that the null is true.
I used the t-stat but you have to calculate another t-stat… I do not understand why…
Question
3 of 6
Based on the results in Exhibit 1, the value of the test statistic relating to Hamilton’s null hypothesis about the value of the coefficient on SPREAD is closest to:
4.28.
0.24.
0.11.
Incorrect.
The null hypothesis is H0: bspread = 1.
The calculated value of the t-statistic is t = (1.0264 – 1.0)/standard error.
The standard error is 1.0264/4.28 = 0.24.
The calculated value for t = (1.0264 – 1.0)/0.24 = 0.11.
2014 CFA Level II
“Correlation and Regression,” by Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, and David E. Runkle
Section 2.6
“Multiple Regression and Issues in Regression Analysis,” by Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, and David E. Runkle
I also I’ve seen that definition in Wikipedia, lol. In the article named “p-value” It says the “equal or more extreme result observed…”. Honestly, I don’t understand this definition. As far as I remember from L1 or L2 is that this is not explained in the books nor the term told. Can you share a more detailed explanation?
The CFA curriculum isn’t a statistics text, so it’s expected for topics to be underdeveloped in comparison to a pure finance or economics topic. Wikipedia is good for many things, but if you can access a reputable textbook, that’s another good option.
For example, the QM text discusses the p-value, saying that it is the smallest significance level at which the null hypothesis can be rejected (or something like that). This doesn’t actually define a p-value-- it kind of tells you how the decision rule works (i.e. a p-value less than alpha leads you to reject Ho).
So, when you see “…p-value is the probability of seeing a result at least as extreme as the current result, given a true null,” think back to what happens when you conduct a hypothesis test.
You calculate a test statistic, and you use this calculated test statistic to look up a p-value in a table related to that distribution (t, z ,f, etc.). Let’s say you conduct a (one-tailed, mu>0) t-test with a very large sample (n=1,000) [this works similarly for a two-tailed test, but I’m simplifying slightly]. Imagine that the calculated test statistic is equal to 1. In other words, this standardized value is 1 standard deviation above the mean. If you use a t-table with the appropriate degrees of freedom, you will see that about 84% of the probability density lies to the left of this value of +1. Consequently, the p-value for this test would be the probability density that lies to the right of this value (since we were testing in the upper tail). So what you have is the remaining 16% of the probability density that includes values at least as “extreme” as +1 (from +1 to positive infintity). However, this is only correct if we assume that the null hypothesis is true (the true distribution is centered in the same manner that we implied through our test). You can now conclude that the p-value represents the probability of getting a result that is at least as extreme (or unusual) as the result we just obtained, assuming that the null is true.
The t-stats given in those tables are calculated to test for any significant relationship (positive or negative, a null of beta i = 0 and an alternative hypothesis of beta i not equal to zero). It is the same calculation: (estimated coefficient - assumed parameter value)/ standard error for that coefficient = (1.0264 - 0)/ standard error = 4.28— use this information to solve or the standard error of the spread coefficient. Once you have this, you can now recalculate the hypothesis assuming that the true value of the spread coefficient is 1. (1.0264 - 1)/0.24 = 0.11.
Just remember that that calculation is already done under the assumption of no relationship (coefficient equal to zero). You need recalculate that statistic for a different assumed value (1, in this case).
My problem also was I didn’t read properly all the statement; instead I “scanned” it to search for the relevant information an dont waste time.
"Hamilton wants to test the null hypothesis that the coefficient on SPREAD is equal to 1 against the alternative hypothesis that it is not equal to 1. She is also interested in how closely the S&P 500 predicts the electric utility index returns. Hamilton wants to use the regression results to address both of these issues. Finally, she wants to determine whether the model has serial correlation. Selected values of the t-distribution are shown in Exhibit 2."
That’s the reason I thought was Ho= 0
I am using this technique for a few months and it is the 1 mistake related.
Do you think is a good idea to scan the statement searching the info or is better to take your time reading it before doing the 6 questions?
I had the fear to discover something that I maybe lost in between so far. However, felt comfortable with your explanation. I now get that the proper definition of p-value is not what I said but the one you exposed.
I must to tell, nonetheless, that the p-value is exactly the same probability to make a type error 1. Is it correct to say this?
Glad to help. It can happen from time to time if you scan the vignette.
Are you saying the bold statement is why you thought you should test Ho: beta(spread) = 0? Or do you mean that you only concluded it because you didn’t completely read the bold statement?
I think you should find what works best for you. Scanning is okay as long as you become very effective and efficient at gathering enough information to answer the question correctly (this was my approach). You have time to practice, so see if the scanning method works for you. If it doesn’t, try reading a little more around the lines that you believe contain the answer (just to be sure).
Fear only keeps us from learning …it’s only helpful to ask questions.
It is not correct. This is where things might get a little more tricky. I will try to keep it short, but if you want more of an explanation, I’ve included some links. I’m not familiar with adding graphs, images, or math type on the forum, so I found someone who already has this on a website.
Anyway, p-values (this is calculated assuming a state of nature) and the probability of making a type I error (alpha, we select this) are not the same thing.
Recall that all of our calculations to get the p-value (test statistic) are assuming that the null is correct. In other words, we believe the null is true, but due to sampling variation, we expect to see a result different than the null (and if our results are crazy enough, we might start to disagree with the null). How often can we expect to see at least a certain result, then, based on some distribution? Think of my previous example. In the example, we believe that the mean is zero and standard deviation is 1, and we expect a result at least as extreme as +1 about 16% of the time (purely from sampling variation). This is all the p-value tells you-- how likely it is you will observe something, but it doesn’t tell you how likely that observation is right or wrong.
These links are pretty good references with images to help:
Glad you found it helpful. Certain writers on there are better than others (more clear), but it’s often helpful (minitab is a statistical software company, so that does increase the reliability of this blog). The best source of information is a well known statistics website, textbook, or a well-educated statistician.
I missed that part of the statement, so I assumed that the question was Ho=0. That’s was the reason to ask you for the effectiviness of the “scanning method”, which I will try to improve to make less mistakes like this!