R15.P1.T2.DIEBOLD_CH7_PARTIAL_AUTO-CORRELATION

[gargi.adhikari]

Hi,
In reference to R15.P1.T2.DIEBOLD_CH7_PARTIAL_AUTO-CORRELATION :-

I am having a bit of a confusion with the verbiage circled in Red below.
What I have managed to understand on this topic is that :- the PACF ( Partial Auto Correlation) allows us to identify the "Order" of the AutoRegression(AR) Models - AR (1 ) vs. AR ( 2) vs. AR( 3) ...etc...and in PACF we remove the effects of the in between lag variables ...

But Partial Auto Correlation itself is not Auto Regression..or is it...? OLS Regression is just used to find the value of the coefficients I guess ...??

Please pardon my ignorance on this topic ...trying to understand the best I can on this one... and as always many thanks for taking the time to share insights on this...

 

[David Harper CFA FRM]

Hi @gargi.adhikari I inserted "(aka, partial autogression)" to help convey the concept. The correct term is only partial autocorrelation; you are correct that it's not itself an autoregression, but instead that regressions are one way to find the partial autocorrealtion. But I don't think it's terrible analogy here because, after all, it is an autoregressive relationship: we are talking about the relationship between y(t) and previous instance of the same variable y(t-1), y(1-2). Maybe the aka is distracting. But at a shallow (non mathematical level)

• A time series is a set of values associated with the same variable ordered in time--e.g., stock prices--and its own history: y(t), y(t-1), y(t-2) ....
• Autocorrelation is just a type of correlation where the variables happen to be the same variables but only occurring at different points in time; e.g., autocorrelation is correlation, ρ[y(t), y(t-2)] except instead of different variables they are the same variable lagging in time
• However, this autocorrelation[y(t), y(t-2)] does not control for the effects of y(t-1), y(t-3) or any other lags; partial autocorrelation does control for such effects
  
• Analogously, in the regression Y = α + β1*X1 + β2*X2 + β3*X3 ..., the autocorrelation is analogous to the ρ(Y, X2) which invokes no regression and "does not control for the other variables," but the partial autocorrelation is analogous to ρ(X2, Y) = β2*σ(Y)/σ(X2); ie, the multivariate regression coefficient does "control for the effect of the other variables" and is a function of the embedded "partial autocorrelation coefficient." Do you see why I called it that? I just wanted to liken it to the multivariate regression coefficient because the partial slope coefficient does represent the association "after controlling for the other independent variables." At the same time, you are correct that the partial autocorrelation (to my knowledge) is not determined by such a simple (auto)regression. It actually requires two regressions, one for each of the partially correlated variables. But the partial autocorrelation seems to me conceptually similar to the correlation coefficient that is embedded in the partial slope coefficient (i.e., where the multivariate regression is a time series, of course, such that independent variables are lags)

Here is a thread https://forum.bionicturtle.com/threads/partial-autocorrelation.9223 where @brian.field posted a super helpful paper with an Excel I hope to implement next year for the Diebold

and here is one my paraphrase explanations at https://forum.bionicturtle.com/thre...ationary-time-series-diebold.8391/#post-48644 I hope that's helpful!

Hi @[email protected] See below I copied Diebold's explanation for partial autocorrelation (which is excellent, in my opinion). If you keep in mind the close relationship between beta and correlation, then you can view this is analogous to the difference between (in a regression) a univariate slope coefficient and a partial multivariate slope coefficient. We can extract correlation by multiplying beta by cross-volatility; i.e., β(x w.r.t. y) = β(x, y) = ρ(x,y)*σ(x)/σ(y) such that ρ(x,y) = β(x, y)*σ(y)/σ(x). Say we regress today's (time series) value on a lagged value three days ago; ie., y(t-τ) = y(t-3). If we just regress against only the 3-day lagged values, that's a univariate regression of the sort y(t) = b0 + β[y(t),y(t-3)]*y(t-3), and if we multiply this regression's slope by the cross-volatility, we'll get the correlation coefficient, ρ. But if we regress against the variables instead of one--i.e., y(t-1), y(t-2), and y(t-3)--then the β[y(t),y(t-3)] is a partial slope coefficient in a multivariate regression, such that if we multiply it by the cross-volatility, we get the partial correlation. That's the meaning of Diebol's "partial autocorrelations, on the other hand, measure the association between y(t) and y(t-τ) after controlling for the effects of y(t-1), ..., (y-τ+1)." We're getting the correlation but presumably controlling for the correlation with the interim days. I hope that explains!

Diebold Explanation from Chapter 7:
"Finally, the partial autocorrelation function, p(τ), [dh: as opposed to ρ(τ) which denotes autocorrelation per the familiar rho] is sometimes useful. p(τ) is just the coefficient of y(t-τ) in a population linear regression of y(τ) on y(t-1), ... , y(t-τ). We call such regressions autoregressions, because the variable is regressed on lagged values of itself. It’s easy to see that the autocorrelations and partial autocorrelations, although related, differ in an important way. The autocorrelations are just the “simple” or “regular” correlations between y(t) and y(t-τ). The partial autocorrelations, on the other hand, measure the association between y(t) and y(t-τ) after controlling for the effects of y(t-1), ..., (y-τ+1) ; that is, they measure the partial correlation between y(t) and y(t-τ)."/QUOTE]

 

[gargi.adhikari]

Got it Thanks so much @David Harper CFA FRM