Time Series Analysis - AR Model

Dear all, please advise on points below. Thanks in advance.

  1. An example of AR(2), AR(3) and AR(4) model. I presume we can’t just call it a AR(4) just because the equation has a lag 4 term.

  2. F test, Null hypothesis being R square = zero?

  1. Sure we can. AR(4) = 4 lags included.

  2. Not really (even though I understand were you are coming from). R2 is an descriptive statistic and not a theoretical parameter. Generally we only test hypothesis about theoretically expected parameters and not descriptive statistics. Therefore, in the F test is a test of the H0 that all the theoretical parameters included in the regression (like B1, B2, B3 or so) are, jointly, zero versus the alternative hypothesis that they are not.

Specifically, we’re testing that all _ slopes _ are zero, not all regression coefficients (which would also include the intercept).

Do you mean the F-test in multiple regression, or something (as the title of the thread suggests) involving AR models in time series analysis?

I would be careful in stating the test is for all slopes (in a techinal sense). Regressions are often run to fit a model with curvature (as long as the model is linear in its parameters), meaning that no slopes are estimated, only rates of curvature and shift parameters. You are right with the catch, though. It is safe to say that global (overall model utility) F-test does not involve testing the intercept, B0, but does involve every other term.

I was confining my answer to the CFA curriculum, in which we do only multiple _ linear _ regression. For nonlinear regression, you’re correct that it’s not slopes per se.

F-Statistic: H_0: All slope coefficients = 0 H_a: At least one slope coefficient != 0

Ah, yeah that makes sense! I wasn’t aware (just registered for the first time for L1) that the CFAI didn’t require knowledge beyond a linear estimation. Thanks for clarifying!