You obtain the following estimates for an AR(2) model of...

You obtain the following estimates for an AR(2) model of some returns data
$$
y_t=0.803 y_{t-1}+0.682 y_{t-2}+u_t
$$
where $u_t$ is a white noise error process. By examining the characteristic equation, check the estimated model for stationarity.

Key Concepts

Autoregressive (AR) Models

An autoregressive model is a time series model where the current value of the series is expressed as a linear combination of its past values and a random error term. AR models are commonly denoted by AR(p), where 'p' specifies the number of lagged terms used in the model. These models help capture the dependency structure in the data over time.

Characteristic Equation

The characteristic equation is a polynomial equation derived from the coefficients of an autoregressive model, formed by setting the lag operator polynomial equal to zero. It is a key tool for analyzing the dynamic properties of the model, such as its response to shocks and its stability over time.

Stationarity and the Unit Circle Condition

Stationarity in a time series implies that its mean, variance, and autocorrelations remain constant over time. For an AR model to be stationary, all the roots of its characteristic equation must lie outside the unit circle, meaning that their absolute values are greater than one. This condition ensures that shocks to the system dissipate rather than persist indefinitely.

White Noise Process

A white noise process represents the random error term in time series analysis, characterized by a constant variance, zero mean, and no autocorrelation. It is presumed to be a sequence of uncorrelated random variables, and its presence ensures that the unpredictable component of the series does not affect the stationarity of the autoregressive process.

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