Arma Models Flashcards
Why do we introduce the ARMA processes into time series analysis?
Classical regression is insufficient in explaining all dynamics of a time series - we must introduce the correlation. We allow the dependent variable to be influenced by all the past values of the independent variables an/or its own past values.
When is forecasting possible and when is it not?
If the present can be modelled by the past then we can forecast - extent to which forecasting is possible can be deduced from the autocorrelation function
Whats an important qualities of the autoregressive process
Xt is stationary if it is an autoregressive process, it has mean zero for all t (we must ensure this is the case)
How can we adjust AR(p) to ensure the expected value of the time series is 0
Replace Xt with Xt - mew in the AR(P) formula where mew is the non zero mean of the time series
Explain the method and assumptions used to write Xt and AR(1) process as an infinite sum of white noise
Using recursion we can obtain that Xt can be written in terms of a sum of white noise plus phi^k by Xt-k. By taking k tending to infinity and assuming that any constants phi are less than one this suggests and infinite sum of white noise
For AR(1) model where phi = 1 what is the significance of this?
This is Xt is a random walk
What does explosive time series mean
Time series values quickly increase in magnitude
For an AR(1) process how does the value of phi affect stationarity and its usefulness in terms of predication?
If phi’s magnitude is less than 1 Xt is stationary and useful for prediction
If Phi’s magnitude is 1 Xt is a non stationary random walk
If Phi’s magnitude is greater than 1 Xt is stationary but not useful as it depends on future values
Explain R function order(a,b,c)
a = order of autoregressive model
b = amount of differences applying
c = Order of the moving average
Define causality
A process not depending on the future is known as causal
For what values of phi is AR(1) process causal?
Only for values where magnitude is less than 1
How do AR(1) process relate to a moving average?
when phi is less than 1 in magnitude we can write an AR (1) process as an infinite moving average process
What important quality do MA(q) models have
They are stationary for any values of theta
What is different about the MA(1) model and the AR(1) model’s autocorrelation
AR(1) model never has autocorrelation zero when MA(1) can have this
Key thing to note about stationary time series and their covariance? gamma h?
Gamma h = Gamma -h
