enforces stationarity constraints on the estimation. Unless cancellation of factors is present (see the next section), it is unlikely for X-13ARIMA-SEATS’ nonlinear estimation to approach the boundary of the stationary region, since the log-likelihood approaches -∞ as this boundary is approached. If the likelihood is defined conditionally with respect to the AR parameters, stationarity is neither assumed nor enforced by the X-13ARIMA-SEATS software. Model estimation, forecasting, etc., are not compromised by parameter values outside the stationary region in this case. Inference results, however, are affected, as noted in Section 4.5. Special techniques (as in, e.g., Fuller 1976, Section 8.5) are required for inference about AR parameters outside the stationary region. 5.4 Cancellation (of AR and MA factors) and overdifferencing Cancellation of AR and MA factors is possible when a model with a mixed ARMA structure is estimated. A model as in (1) or (3) is said to have a mixed ARMA structure if either p > 0 and q > 0, or P > 0 and Q > 0. (Technically, a model with p > 0 and Q > 0, or with P > 0 and q > 0, is also mixed, but such mixed models are unlikely to lead to cancellation problems.) The simplest example of cancellation occurs with the ARMA(1,1) model, (1 - φB ) z t = (1 - θB ) a t , when φ = θ . Cancelling the (1 - φB ) factor on both sides of the model (1 - φB ) z t = (1 - φB ) a t leaves the simplified model, z t = a t . Because of this, the likelihood function will be nearly constant along the line φ = θ . This can lead to difficulties with convergence of the nonlinear estimation if the MLEs for the ARMA(1,1) model approximately satisfy ˆ φ = ˆ θ . Analogous problems occur in more complicated mixed models when an AR polynomial and an MA polynomial have a common zero (e.g., the ARIMA (2,1,2)(0,1,1) model that is used as a candidate model for the automdl spec). For a fuller discussion of this topic, see Box and Jenkins (1976), pp. 248-250. If the X-13ARIMA-SEATS program has difficulty in converging when estimating a mixed model, cancellation of AR and MA factors may be responsible. In any case, possible cancellation can be checked by computing zeroes of the AR and MA polynomials (setting print=roots in the estimate spec), and examining these for zeroes common to an AR and an MA polynomial. If a common zero (or zeroes) is found, then the model should be simplified by cancelling the common factor(s) (reducing the order of the corresponding AR and MA polynomials), and the model should be re-estimated. Cancellation need not be exact, but may be indicated by zeroes of an AR and an MA polynomial that are approximately the same. It is also possible for estimated MA polynomials to have factors that cancel with differencing operators. This occurs when a model has a nonseasonal difference and an estimated nonseasonal MA polynomial contains a (1 - B ) factor, or the model has a seasonal difference and an estimated seasonal MA polynomial contains a (1 - B s ) factor. For example, the model (1 - B )(1 - B s ) z t = (1 - θB )(1 - Θ B s ) a t involves such cancellation
44 CHAPTER 5. POINTS RELATED TO REGARIMA MODEL ESTIMATION if either θ or Θ is estimated to be one.
- Spring '14
- Filename, Filename extension, Computer file, X-13ARIMA-SEATS