8 e 9 - Arma models Flashcards
Can we estimate “non-parametrically” mu, sigma^2, gamma(h) and rho(h), without specifying a model?
If (Yt) is stationary yes, and fairly simple.
We get:
- sample mean
- (corrected) sample variance
- sample autocovariance for lag h
- sample autocorellation for lag h
What do we expect for the estimated ACF?
What if (Yt) is not stationary?
In this case mu(t) = E(Yt) depends on t so we cannot use the sample mean. And we only have one observation y(t) for estimating E(Yt).
Same issues for variance and autocorelation.
In some applications we have “replicates” (a random sample) of the time series. And (Yi,t) with i=1,…,n are i.i.d.. In this case we can use the sample mean of the observations in t of the various time series to estimate mu(t)=E(Yt).
However in many applications we do not have replicates so we would need models fro non-stationary time series that introduce a temporal dependence for mu(t).
Why are ARMA(p,q) models a popular class?
One of the reasons is that the order p,q can be understood, in principle, from the ACF.
What is an AR(1)?
An AR(1) is an autoregressive process of order 1 and is presented as the (stationary and causal) solution of a finite-difference stochastic equation, with t ∈ (-∞, +∞).
Continues in the photo.
The two cases depend on the alpha of the equation.
Check of the fact that AR(1) is the only stationary solution if a stationary solution exists.
Check taht the solution in photo is stationary
Nella covarianza la formula conteneva un “- E(Yt) * E(Yt+h)”, ma siccome E(Yt) = 0 lo abbiamo omesso direttamente.
Dal controllo risulta che (Yt) sia stazionario.
