Econometrics 6: AR(1) and ARMA Flashcards
What is an autoregressive model?
An autoregressive model of order p is defined as follows:
yt = α + Σβiy_{t-i} + ut
where ut is white noise.
Suppose an AR(1) process has |φ|<1. Does it follow that it is weakly stationary?
No. It is asymptotically weakly stationary, but unless it has the correct initial conditions its mean and variance will change over time.
What is a lag polynomial?
The lag operator L operates on time series variables such that Lyt = Ly_{t-1}.
When can we write that Φ(L)^{-1} = 1/(1-φL)?
If the roots of the polynomial Φ(L) are strictly outside the unit circle.
Does a stationary AR(1) satisfy an LLN? If so, state it.
Yes.
Let yt be covariance stationary with mean µ and autocovariances γh. If the autocovariances are absolutely summable, then the sample mean ȳT -> µ.
The conditions are satisfied for a stationary AR(1).
Does a stationary AR(1) satisfy a CLT? If so, state it.
√Tȳ -> N[0, σ^2(Ψ(1))^2]
Is the OLS estimate of phi hat in a stationary AR(1) consistent?
Yes.
What is an ARMA model?
An autoregressive moving-average model augments an AR(p) model by adding q lags of the error term.
Under what conditions can we write an ARMA model in MA form?
If φ(L) is invertible.
Do stationary ARMA models satisfy an LLN?
All covariance-stationary ARMA(p,q) models are covariance-stationary and ergodic in the mean. Therefore, yes.
When do we need to use ML to estimate an ARMA model?
If we know the distribution of the error terms, we can use ML to estimate the parameters.
How should we choose p and q when constructing ARMA models?
Either use the information criteria, or start with a large number of parameters and successively test that the highest-order term is statistically insignificant.
Irregular lag models are feasible, but should only be used if there is some a priori reason to use them, such as seasonality.
