The stationarity conditions for an AR(3) process are:

- The process must have a constant mean and variance over time.
- The autocovariance function must depend only on the lag, not on the time point.
- The characteristic polynomial of the process must have all its roots outside the unit circle.

The characteristic polynomial of an AR(3) process is given by:

$$
C(z) = 1 - \phi_1 z - \phi_2 z^2 - \phi_3 z^3
$$

where $\phi_1, \phi_2, \phi_3$ are the AR coefficients.

To find the roots of this polynomial, we can use the cubic formula or a numerical method. The stationarity conditions can be expressed in terms of the coefficients as follows¹:

- $\phi_1 + \phi_2 + \phi_3 < 1$
- $\phi_2 - \phi_1 - \phi_3 < 1$
- $\phi_3 > -1$
- $\phi_3 < 1 - \phi_2$
- $\phi_2 + \phi_3 > -1$

These conditions ensure that the roots of the characteristic polynomial have a modulus greater than one, which implies that the AR(3) process is stationary.

For more information on stationarity and invertibility of ARMA processes, you can refer to these sources²³⁴.

Source: Conversation with Bing, 10/7/2023
(1) Conditions for Stationarity and Invertibility. https://eml.berkeley.edu/~powell/e241b_f06/TS-StatInv.pdf.
(2) time series - Stationarity of AR(p) process - Cross Validated. https://stats.stackexchange.com/questions/402696/stationarity-of-arp-process.
(3) What are the stationarity conditions for an AR(4) process?. https://stats.stackexchange.com/questions/518136/what-are-the-stationarity-conditions-for-an-ar4-process.
(4) Stationarity Conditions for an AR(2) Process - University of Victoria. https://web.uvic.ca/~dgiles/blog/AR2.pdf.