Let’s say we have a regression with 3 independent variables (i.e., k = 3) and 50 observations, which has a DW statistic of 1.51.

We know from the DW table that the appropriate critical values are:

α = 0.05: dL = 1.42, dU = 1.67 => our statistic of 1.51 is in between dL and dU, implying that the test is inconclusive for positive serial correlation

α = 0.01: dL = 1.25, dU = 1.49 => our statistic of 1.51 is greater than dU, implying no serial correlation

Is this really implying that the test concludes no serial correlation with 99% confidence but the test is inconclusive with 95% confidence?

In other words, how is it possible that we are less sure of the result at a lower level of significance? It seems very counterintuitive. Or did I make a rookie mistake?

The key with this is that there really isn’t much of a discrepancy.

In the first example, they’re clearly saying “inconclusive”.

In the second case, you fail to reject H_{0}, which really is “inconclusive”.

Poor teaching of significance testing misleads students to think p > alpha means the null is true (in this case that no serial correlation exists), which just isn’t true. p > alpha means we don’t have enough evidence against the null to reject it at some pre specified alpha level. In our problem, there is some evidence against the idea of no serial correlation which is the null hypothesis, but it doesn’t surpass the threshold at the .05 level and clearly not at the .01 level-- but both are kind of “inconclusive”.

Failure to reject the null does NOT doesn’t not constitute proof or suggestion of the null’s truth. This is really where the question boils down to-- the way significance testing works (framing a conclusion) doesn’t really change from test to test.

Edit: because I was a doofus and couldn’t decide between “doesn’t” and “does not” when typing, I wrote “…doesn’t not…” which changed the written meaning. The correct meaning is that fail to reject Ho does not constitute evidence for the null.

Thanks, very helpful. This statement was the key misunderstanding on my part: “Failure to reject the null doesn’t not constitute proof or suggestion of the null’s truth.”

I incorrectly mistyped “doesn’t not” but I meant that FTR H_{0}does not constitute proof of or suggestion of the null hypothesis. It sounds like you gathered correct meaning, but just be sure. Sorry for my mistake.

Also, the misunderstanding is something that the CFAI could clear up if they had half a brain in statistics; it’s unintuitive at first to someone learning the material, but someone claiming competency to teach the material should recognize this as elementary and key for understanding (aka CFAI sucks at stats).

I noticed this received a down vote (not necessarily OP). If the post was unhelpful, be sure to let me know so I can understand what causes concern in my post. Thanks!

It’s quite alright! (Still in the hole about 20 or so from today alone, but these don’t mean much to me except if someone has somenthing valuable to address!)