Can non-stationary processes when passed through a linear or non-linear system generate stationary processes? If so, give an example.
Added by Nancy M.
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- A non-stationary process is one whose statistical properties, such as mean and variance, change over time. - A stationary process has statistical properties that do not change over time. Show more…
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Let Xt = a + bt + Yt, where a and b are nonzero constants and Yt is a stationary time series. (i) Show that {Xt} is not stationary. (ii) Let ΔXt = Xt - Xt-1. Show that {ΔXt} is stationary. (iii) Determine the autocovariances of {ΔXt} in terms of those of {Yt}. (iv) Now suppose that {Yt} is a MA(1) process so that Yt = εt + βεt-1 Write down an equation for ΔXt in terms of b, β, εt and B, the backward shift operator. (v) Show that Var(ΔXt) ≥ Var(Yt).
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We can also write the autoregressive model as a linear process. Let us consider AR(1) first: X_t = W_t + ̑_1X_{t-1} This is a special form of the linear process model if the infinite sum exists. In other words, we need ̑_1 < ∞ for the time series to be well-defined. We also have a more general result that AR(p) is stationary and causal if the corresponding linear process converges. ARMA model as a linear process: X_t = 0.5X_{t-1} + W_t + 0.5W_{t-1} Can you write X_t as a linear process? True False Is X_t causal? True
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Consider the time series xt = β1 + β2t + wt, where β1 and β2 are known constants and wt is a white noise process with variance σ^2w. (a) Determine whether xt is stationary. (b) Show that the process yt = xt − xt−1 is stationary. (c) Show that the mean of the moving average vt = (1/2q + 1) * ∑(q, j=-q) xt−j is β1 + β2t, and give a simplified expression for the autocovariance function.
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