## Stationarity Conditions for an AR(3) Process

**An AR(3) process** is a time series model where the current value depends on the previous three values. It's represented by the equation:

```
y_t = c + φ₁y_(t-1) + φ₂y_(t-2) + φ₃y_(t-3) + ε_t
```

where:

* `y_t` is the value of the series at time t
* `c` is a constant
* `φ₁`, `φ₂`, and `φ₃` are the autoregressive coefficients
* `ε_t` is a white noise error term

**For an AR(3) process to be stationary**, it must satisfy the following conditions:

1. **Mean Reversion:** The process must have a constant mean over time.
2. **No Unit Roots:** The characteristic equation of the AR(3) process must not have any roots on the unit circle.

### Characteristic Equation and Roots

The characteristic equation of an AR(3) process is:

```
1 - φ₁z - φ₂z² - φ₃z³ = 0
```

where `z` is a complex variable.

The process is stationary if and only if all the roots of this equation lie **outside** the unit circle in the complex plane.

### Inequalities for Stationarity

To ensure that all roots lie outside the unit circle, the following inequalities must hold:

1. **φ₁ + φ₂ + φ₃ < 1**
2. **-φ₁ + φ₂ - φ₃ < 1**
3. **φ₁ - φ₂ - φ₃ < 1**
4. **-φ₁ - φ₂ + φ₃ < 1**
5. **φ₃ < 1**

These inequalities can be derived from the conditions for the roots to be outside the unit circle. They are often referred to as the **Perron-Frobenius conditions** for stationarity.

**In summary**, an AR(3) process is stationary if and only if the five inequalities above are satisfied. These inequalities ensure that the process has a constant mean and does not exhibit explosive behavior.