How to calculate the variance of a MA model?
The 1st order moving average model , denoted by MA (1) is x t = μ + w t + θ 1 w t − 1, where w t ∼ i i d N ( 0, σ w 2). Variance: Var ( x t) = Var ( μ + w t + θ 1 w t − 1) = 0 + Var ( w t) + Var ( θ 1 w t − 1) = σ w 2 + θ 1 2 σ w 2 = ( 1 + θ 1 2) σ w 2 ACF : Consider the covariance between x t and x t − h .
When is a MA model said to be invertible?
An MA model is said to be invertible if it is algebraically equivalent to a converging infinite order AR model. By converging, we mean that the AR coefficients decrease to 0 as we move back in time. Invertibility is a restriction programmed into time series software used to estimate the coefficients of models with MA terms.
Which is an indicator of a MA ( 1 ) model?
Thus a sample ACF with a significant autocorrelation only at lag 1 is an indicator of a possible MA (1) model. For interested students, proofs of these properties are in the appendix. Suppose that an MA (1) model is x t = 10 + w t + .7 w t − 1, where w t ∼ i i d N ( 0, 1).
What are the coefficients of the MA model?
Consider the MA (2) model x t = 10 + w t + .5 w t − 1 + .3 w t − 2, where w t ∼ i i d N ( 0, 1). The coefficients are θ 1 = 0.5 and θ 2 = 0.3.
What are the coefficients of a MA ( 2 ) model?
Consider the MA (2) model x t = 10 + w t + .5 w t − 1 + .3 w t − 2, where w t ∼ i i d N ( 0, 1). The coefficients are θ 1 = 0.5 and θ 2 = 0.3. Because this is an MA (2), the theoretical ACF will have nonzero values only at lags 1 and 2.
How are error terms calculated in an ARIMA model?
That is, the observation Y is first regressed against its previous values Y t − 1,…, Y t − n and then one or more Y − Y ^ values are used as the error terms for the MA model. But how are the error terms calculated in an ARIMA (0, 0, 2) model?
How to calculate the autocorrelation of a MA model?
Autocorrelations for higher lags are 0. So, a sample ACF with significant autocorrelations at lags 1 and 2, but non-significant autocorrelations for higher lags indicates a possible MA (2) model. Consider the MA (2) model x t = 10 + w t + .5 w t − 1 + .3 w t − 2, where w t ∼ i i d N ( 0, 1).