Purely Random Time Series

A purely random time series y₁, y₂, …, y_n (aka white noise) takes the form

where

Clearly, E[y_i] = μ,  var(y_i) = σ²i and cov(y_i, y_j) = 0 for i ≠ j. Since these values are constants, this type of time series is stationary. Also note that ρ_h = 0 for all h > 0.

Example 1: Simulate 300 white noise data elements with mean zero.

Using the formula =NORM.S.INV(RAND()) we can generate a sample of 300 white noise elements, as displayed in Figure 1.

Figure 1 – White Noise Simulation

We see that there is a random pattern. Using the techniques described in Autocorrelation Function and Partial Autocorrelation Function we can also calculate ACF and PACF values, as shown in Figure 2.

Figure 2 – ACF and PACF for White Noise simulation

Although the theoretical ACF values are ρ_k = 0 for all k > 0, the sample values r_k won’t necessarily be exactly 0, as we can see from the left side of Figure 2. Based on Property 3 of Autocorrelation Function

Since n = 300, a 95% confidence interval for r_k is 0 ± NORM.S.INV(.025)/SQRT(300) = ±0.11316.

Figure 2 shows 40 values for r_k. We would expect that about 40(.95) = 2 of these values would be outside the 95% confidence interval. In fact, two ACF values are outside this range, namely r₉ = .11842 and r₁₉ = .13366.

Using the Ljung-Box test, we see that none of the 40 ACF values is significantly different from zero:

p-value = CHISQ.DIST.RT(46.2803,40) = .229 > .05 = α

We can perform similar tests for the PACF values.