A method of estimating parameters, given observations; it finds parameter estimates by finding the parameter values that maximise the LLF
Define strong stationarity?
Describes a stochastic process whose unconditional probability distribution does not change when shifted in time
Define weak stationarity?
Describes a stochastic process where the mean and variance are fixed through time, but the covariance can change through time
3 conditions for weak stationarity?
E(yt)=mu for t=1->infinity
E(yt-mu)(yt-mu) = sigma^2 (where sigma squared is less than infinity)
E(yt1-mu)(yt2-mu) = gamma(t2-t1) for all t1, t2
What is the invertibility condition?
Invertibility allows MA models to be written as AR models, which implies, more generally, that a stochastic process is invertible if an ARMA model can be written as AR models
When is an ARMA(p,q) model stationary?
If the roots of fi(L) polynomial lie outside the unit circle
When is an ARMA(p,q) model invertible?
If the roots of theta(L) polynomial lie outside the unit circle
Explain the key difference between a non-stationary and stationary series?
Both have unconditional means of 0 if both start a 0, therefore the key difference lies in the conditional mean (ie. the mean of the process given the most recent observation) (see notes)
A non-stationary series has at least one unit root, and therefore its probability distribution will change through time. It will have non-constant mean/variance and therefore will look different in different time periods.
Two types of non-stationarity, and how to deal with them?
1) RW with drift (ie. stochastic trend): yt=mu+y(t-1)+error(t)
2) Deterministic trend (ie. trend stationary): y(t)=mu(t)+error(t)
very important in notes: finding mean and variance of RW with drift
Key difference between RW with drift and deterministic trend?
Deterministic is non-stationary in its mean, but constant in its variance
RW with drift is non-stationary in its mean and variance
How is stationarity induced in a stochastic trend? (expand)
Differencing it; a non-stationary series that must be differenced ‘d’ times to induce stationarity is integrated of order d, where d=number of unit roots
Initial hypotheses for Dickey-Fuller unit root test?
H0: fi=1 (non-stationary)
H1: fi less than 1 (stationary)
THESE CHANGE THOUGH!!!
Explain why we cannot use a t-test to establish DF test result?
Because under the null, the process is non-stationary (ie. yt and y(t-1) etc.) and therefore CLT doesn’t apply!
Show in detail how to set up a DF test? Explain why this method works? Explain why we STILL can’t use a t-test to test this?
What did Dickey-Fuller do? What is the test statistic equation? When do we reject the null of non-stationarity?
They tabulated the asymptotic distribution of the LSE for fork under the null hypothesis of it being a unit root, therefore we can just compare the ordinary t-statistic with values of the DF distribution!
Statistic (see notes)
Reject H0 if test statistic is more negative than the CV
CVs for DF test
test types and learn them, end of DF