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How do you calculate unconditional expectations?
The unconditional expectation of X, E(X), is just a number: e.g. EX = 2 or EX = 5.8. The conditional expectation, E(X |Y = y), is a number depending on y. If Y has an influence on the value of X, then Y will have an influence on the average value of X.
What is conditional variance in GARCH?
A process, such as the GARCH processes, where the conditional mean is constant but the conditional variance is nonconstant is an example of an uncorrelated but dependent process. The dependence of the conditional variance on the past causes the process to be dependent.
Which is smaller conditional or unconditional forecast error?
I have trouble showing that conditional forecast error of AR (1) has smaller variance than the unconditional one. I can show that cond. forecast error is: YT + 1 = aYT + ϵT + 1 ˆYT + 1 = E[aYT + ϵT + 1 | YT, YT − 1,…] = aYT E[YT + 1 − ˆYT + 1] = E[ϵT + 1] = 0 Var[YT + 1 − ˆYT + 1] = E[(YT + 1 − ˆYT + 1)2] = E[ϵ2T + 1] = σ2
What does this say for the variance of the unconditional?
I am not sure if this cleans up nicely, but what does this say for the variance? it seems like the unconditional is just going to blow up infinitely (given I could have made a mistake in the derivation of this question).
How to prove the variance of AR ( 1 )?
For the autoregressive AR (1) process x t = δ + ϕ x t − 1 + η t, I am trying to prove that the variance is: γ 1, x = ϕ σ x 2. I have tried many manipulations but I cannot succeed.
How to find the unconditional mean of a time series?
First you’ll need to find the unconditional mean and variance of the AR (1) process Yt = aYt − 1 + εt, εt ∼ (0, σ2): Here I have assumed that the process is stationary, i.e. |a| < 1 so I can write E[Yt] = E[Yt − 1]. V[Yt + 1 − ˆYt + 1] = E[(aYt + 1 + εt + 1 − E[Yt])]2 = X + σ2. You’ll need to find an expression for X .