How to turn many variables into one for the PACF equation

Question

I have been trying to calculate the PACF manually, but I encountered some issues with the following equation:

$PACF = \frac{Covariance ([Y_{t}|Y_{t-1}, Y_{t-2},...,Y_{t-k+1}],[Y_{t-k}|Y_{t-1}, Y_{t-2},...,Y_{t-k+1}])}{\sigma_{[Y_{t}|Y_{t-1}, Y_{t-2},...,Y_{t-k+1}]} \sigma_{[Y_{t-k}|Y_{t-1}, Y_{t-2},...,Y_{t-k+1}]} } $

Let's supposed we are trying to calculate the PACF for lag 3, so k=3.

Nominator: To be able to calculate the Covariance we need two variables. The equation for covariance requires cov(x, y).

Denominator: To be able to calculate the standard deviation we need one variable.

With lag = 3 on the nominator, there would be 6 variables in total, three on the left and 3 on the right side of the comma.

For lag 2 the way we turn 4 variables into two is by plotting the two variables on each side of the formula against each other, finding a line of best fit using OLS, and then finding the residuals. This then gives us one variable, so two in total for the nominator.

How would I do this for lag 3, I could maybe plot 3 variables and find that correlation, but that clearly isn't it because what about lag 4 or lag 5.

So how do I turn the list of variables into two for the nominator (or one for each standard deviation on the denominator)?

Answer 1

The solution to your problem is multiple linear regression. You can easily regress $Y_t$ and $Y_{t-k}$ on multiple lags. For example, for lag 4 you would run $\hat Y_{t-4} = \beta_1 Y_{t-1} + \beta_2 Y_{t-2}+\beta_3 Y_{t-3}$ and use that to calculate the residuals.

You can run multiple linear regression on as many independent variables as you want, though visualising it with a plot is difficult.