Question

An autoregressive process of order one with correlation parameter \(\rho\) is stationary only if \(|\rho|<1 .\) Discuss Bayesian inference for such a process. How might you (a) impose stationarity through the prior, (b) compute the probability that the process underlying data \(y\) is non-stationary, (c) compare the models of stationarity and non-stationarity?

Short answer

Impose stationarity by using a Uniform prior for \( \rho \) on \([-1, 1]\). Compute non-stationarity probability using the posterior distribution of \( \rho \). Use Bayes factors for model comparison.

Step-by-step solution

Understanding Autoregressive Process

An autoregressive process of order one, denoted as AR(1), can be defined by the equation: \[ y_t = \rho y_{t-1} + \epsilon_t \]where \( \epsilon_t \) represents a white noise error term and \( \rho \) is the correlation coefficient. For the process to be stationary, the absolute value of \( \rho \) must be less than 1, i.e., \(|\rho| < 1\). A stationary process has constant statistical properties over time.

Imposing Stationarity Through the Prior

To ensure the AR(1) process is stationary in a Bayesian framework, we can set a prior distribution on \( \rho \) such that the probability is zero outside the interval \((-1, 1)\). For example, choosing a Uniform prior for \( \rho \) on \([-1, 1]\) can serve this purpose. This prior reflects the belief that \( \rho \) lies strictly within this interval, embodying the stationarity condition.

Computing the Probability of Non-Stationarity

To compute the probability that the process is non-stationary, one can use the posterior distribution of \( \rho \). After observing data \( y \) and updating our beliefs using Bayes' theorem, we calculate the probability \( P(|\rho| \geq 1 \mid y) \) using the posterior distribution of \( \rho \). If this probability is near zero, it suggests little evidence for non-stationarity.

Comparing Stationarity and Non-stationarity Models

Use Bayesian model comparison techniques such as Bayes factors to compare the stationary and non-stationary models. Calculate the Bayes factor between models where \(|\rho| < 1\) (stationary) and \(|\rho| \geq 1\) (non-stationary). A Bayes factor greater than 1 favors the stationary model, while less than 1 favors the non-stationary model. This involves evaluating the marginal likelihoods of the models, integrated over all parameters but \( \rho \).

Key concepts

Autoregressive Process

An autoregressive process describes how current data points in a time series are related to past data points. Specifically, in the AR(1) model, each value is influenced by its immediate predecessor, modified by some correlation factor \( \rho \), and a randomness element \( \epsilon_t \). This is represented as \[ y_t = \rho y_{t-1} + \epsilon_t \].

• \( \rho \): Correlation parameter indicating how strongly past values influence current values.
• \( \epsilon_t \): The white noise error term, reflecting random fluctuations.

To understand how autoregressive processes operate, think of each data point adjusting slightly from the previous, guided by a balance of predictable past influence and randomness.When \(|\rho| < 1\), the process reaches a statistical equilibrium; the properties, such as mean and variance, stay consistent over time. This idea is known as stationarity.

Stationarity in Time Series

Stationarity is a key concept in analyzing time series as it allows meaningful predictions. In simple terms, a stationary time series has a constant mean, variance, and autocorrelation throughout its length. In the context of an AR(1) process, this means the series behaves predictably over time without drifting.To maintain stationarity in a Bayesian framework, we can use prior distributions to limit \( \rho \). By choosing a prior that only allows values within \([-1, 1]\), we restrict \( \rho \) to levels that encourage stationarity. A common choice would be a Uniform prior on this interval, signifying equal likelihood given our prior knowledge.Stationarity ensures our statistical analyses, like forecasting future values or testing hypotheses, remain valid over time. It is essential for maintaining the viability of models constructed using time series data.

Bayesian Model Comparison

Bayesian model comparison allows us to evaluate which model, stationary or non-stationary, better explains a given time series data set. One method for this is calculating the Bayes factor, which compares the likelihood of data under different models.

• Bayes Factor: - A ratio of the evidence for two competing hypotheses, calculated from the likelihood functions. - If the Bayes factor is greater than 1, it suggests more support for the stationary model; less than 1 supports non-stationary.
• Marginal Likelihoods: - These are integrated over all model parameters, factoring in our uncertainty of parameter estimates.

By examining the Bayes factor, we can quantify which version of our model is more credible in explaining the observed data. This not only offers insights into the behavior of the system but also provides a statistical foundation to prefer one hypothesis over another.