Suppose that we believe that an AR(p) process is a fit for some time series. We now show how to calculate the process coefficients using the following techniques: (1) estimates based on ACF or PACF values, (2) using linear regression and (3) using Solver. We illustrate the first of these approaches on this webpage.
One approach is to use the Yule-Walker in reverse to calculate the φ0, φ1, …, φp, σ2 coefficients based on the values of μ, γ0, …, γp (ACF values). Alternatively, we use the values μ, γ0, π1…, πp (PACF values), which it turns out are equivalent.
Example 1: Use the statistics described above to find the coefficients of the AR(1) process based on the data in Example 1 of Autoregressive Processes Basic Concepts.
The first 8 of 100 data elements are shown in column B of Figure 1. We next calculate the mean, variance and PACF(1) values. From these we can estimate the process coefficients as shown in cells G8:G10. This estimate of the time series is the process yi = 4.983 + .394yi-1 + εi where σ2 = 1.421703.
Figure 1 – Estimation of AR(1) coefficients
As we can see, the process coefficients are pretty close to the original coefficients used to generate the data in column B (φ0 = 5, φ0 = .4 and σ2 = 1) with the exception of σ2, which is a little high.
Observation: We can use this approach for AR(2) processes, by noting that
Thus
and so
Example 2: Use the statistics described above, to find the coefficients of the AR(2) process based on the data in Example 1.
We show two versions in Figure 2. The lower version is based on the ACF using the formulas described in the above observation. The upper version is based on the PACF using Property 1 of Partial Autocorrelation of AR(p) Processes.
Figure 2 – Estimation of AR(2) coefficients
