Question

Suppose that {X,t=0,±1,±2,…}is a stationary process satisfying the equations X t =ϕ 1 X t−1 +…+ϕ p Xt−p +ϵ t where ϵ t ∼WN(0,σ 2), and ϵ is uncorrelated with X s for each sp, is Xn+1 =1X n +…+ϕ pX n+1−p . What is the mean squared error of ?

Answer 1

The mean squared error of a stationary process, where X is defined by its past values and white noise, is equivalent to the variance of the noise term, given the uncorrelated nature of the noise with past X values.

Step-by-step explanation:

The question concerns a stationary process in which a time series X is defined by a linear combination of its past values and a white noise component εt. The mean squared error (MSE) of predictions made using this process can be thought of as the variance of the noise term σ2 since the past values of X are known and deterministic when making the forecast for Xn+1. Thus, the MSE would be the same as the variance of εt, assuming all the covariances between εt and Xs for s<p are zero (uncorrelated).

Using concepts from the Central Limit Theorem and properties of expectations, the mean or expected value (μ) of X, given a large number of samples, would approach the population mean, and the standard error of the mean would be σx divided by the square root of the sample size (n).