Titner Criterion and Variance Reduction in Differenced Stationary Time Series with High Autocorrelation

To prove that if we start with a stationary series but with first-order autocorrelation greater than 0.5, the variance of the series diminishes when it is differentiated, we can use mathematical equations and properties.

Let's denote the original stationary series as Xt, and let's assume it has a first-order autocorrelation denoted as ρ1.

The differenced series is denoted as Yt = Xt - Xt-1, where t represents the time index.

To prove that the variance of the differenced series diminishes, we need to show that Var(Yt) < Var(Xt).

First, let's calculate the variance of the differenced series Yt: Var(Yt) = Var(Xt - Xt-1)

Using the properties of variance, we can rewrite this as: Var(Yt) = Var(Xt) + Var(Xt-1) - 2Cov(Xt, Xt-1)

Since the original series Xt is stationary, it has a constant variance Var(Xt) = Var(Xt-1) = σ^2 (let's assume σ^2 is the variance of Xt).

Now, let's substitute these values into the equation: Var(Yt) = σ^2 + σ^2 - 2Cov(Xt, Xt-1) = 2σ^2 - 2Cov(Xt, Xt-1)

Now, let's consider the autocorrelation coefficient between Xt and Xt-1, which is denoted as ρ1: Cov(Xt, Xt-1) = ρ1 * σ^2

Substituting this into the equation for Var(Yt): Var(Yt) = 2σ^2 - 2(ρ1 * σ^2) = 2σ^2(1 - ρ1)

Since the autocorrelation coefficient ρ1 is greater than 0.5 (given in the problem), we know that 1 - ρ1 is negative.

Therefore, Var(Yt) = 2σ^2(1 - ρ1) < 2σ^2

Since 2σ^2 is the variance of the original stationary series Xt, we have shown that Var(Yt) < Var(Xt).

This proves that if we start with a stationary series but with a first-order autocorrelation greater than 0.5, the variance of the series diminishes when it is differentiated (i.e., the differenced series has a smaller variance than the original series).