boxjenk(const,diffs=0,sdiffs=0,ar=0,sar=0,maxl,define=deq) writing
@acbjICRS(fPlot=1) deq writing
boxjenk(const,diffs=0,sdiffs=0,ar=1,sar=0,maxl,define=deq) writing
@acbjICRS(TITLE="") deq writing
boxjenk(noconst,diffs=0,sdiffs=0,ma=5,sma=0,maxl,define=deq) writing
@acbjICRS deq writing
boxjenk(noconst,ar=5,diffs=1,ma=1,sar=1,sdiffs=1,sma=1,maxl,define=deq) writing
@acbjICRS deq writing
boxjenk(noconst,ar=2,diffs=1,ma=1,sar=1,sdiffs=1,sma=1,maxl,define=deq) writing
@acbjICRS deq writing
boxjenk(noconst,ar=0,diffs=1,ma=2,sar=0,sdiffs=1,sma=2,maxl,define=deq) writing
@acbjICRS(fplot=1) deq writing
boxjenk(const,diffs=1,sdiffs=0,ma=4,sma=1,NOMAXL,define=deq) writing
@acbjICRS deq writing
boxjenk(noconst,diffs=1,sdiffs=0,ar=14,sar=0,NOMAXL,define=deq) writing
@acbjICRS deq writing
ac_1 wrote:Also. am I right in thinking the coefficients of ARIMA models are not necessarily restricted to be between -1 and +1, implying a non-explosive model; it's the inverse roots which should be inside the complex unit circle: i.e. if AR roots lie inside the complex unit circle ARIMA model is stationary, and if MA roots lie inside the complex unit circle ARIMA model is invertible?
TomDoan wrote:I'm baffled about what you're trying to do. You *know* how many differences/seasonal differences you've done so you will know exactly how many unit roots you have. The AR parts are forced to be stationary by maximum likelihood. (There is no unconditional likelihood for a non-stationary model).
The MA's are also going to be forced inside or on the unit circle. An MA with a bad root has an equivalent model with the root flipped inside. (That has nothing to do with maximum likelihood). An MA with a unit root is permitted, but in practice would never be estimated since it's a boundary value, and is usually a sign of an overdifferenced model: for instance, if x is white noise, then x=u, so (1-L)x=(1-L)u so first differencing x induces a unit root MA process.
TomDoan wrote:No, that is not correct,
ac_1 wrote:(a) Is 5 decimal places sufficient, all scenarios, for any data series?
ac_1 wrote:Is acbjICRS.src procedure not correct? If so exactly where are the errors?
ac_1 wrote:(a) Is 5 decimal places sufficient, all scenarios, for any data series?
ac_1 wrote:For a good and "stable" ARMA forecasting model, my present understanding is that the coefficients, including higher order models, should always be between +/-1 and the inverse roots should always be in (green), or on (black), but never outside (red), the unit circle, both sides: AR and MA. Correct?
ac_1 wrote:The natural question is under what circumstances (e.g. misspecification, over-parameterization) would the estimated ARMA coefficients lead to values being outside the +/-1 range with the inverse roots being outside the complex unit circle? Specifically, what is the relationship between the estimated ARMA coefficients and inverse roots?
TomDoan wrote:1. The coefficients (as you indicated above) don't have to be in the unit interval when you have more than one in a polynomial.
TomDoan wrote:2. "Stable" means that the model doesn't change over time, which is completely different from "stationary". The roots say nothing about stability. If you are looking at sequential estimates of a model (particularly if you are doing automatic model selection), it is quite possible that the form of the model will change from one interval to another (and, of course, that means that the roots will be radically different since the models are). There is nothing wrong with that---all the models are approximations. An AR(2) and an ARMA(1,1) can have very similar dynamics (and have similar log likelihoods and produce similar forecasts) despite having "root" patterns which are (by construction) completely different.
TomDoan wrote:3. If BOXJENK with MAXL fails to estimate successfully, you should probably ignore the model. @BJAUTOFIT will put in the last log likelihood whether the model converges or not, which, if the model has a major issue, will generally be much worse than the better behaved models. Remember that if an ARMA(p,q) model is adequate, an ARMA(p+1,q+1) is unindentified, and unlikely to estimate properly, but it also is clearly dominated by smaller model.
TomDoan wrote:The roots tell you nothing about misspecification. (Misspecification is usually detected when there is significant autocorrelation left in the residuals). The roots tell you nothing about overparameterization. (Overparameterization is detected when an information criterion gives you a worse value than a smaller model).
ac_1 wrote:TomDoan wrote:1. The coefficients (as you indicated above) don't have to be in the unit interval when you have more than one in a polynomial.
MWH(1997) p.340-p.342, constraints on parameters:
restrictions for AR parameters
For p = 1, −1 < phi1 < 1.
For p = 2, the following three conditions must all be met: −1 < phi2 < 1, phi2 + phi1 < 1, phi2 − phi1 < 1. But what is the constraint on phi1?
What are the restrictions for p ≥ 3?
ac_1 wrote:for MA parameters
For q = 1, −1 < theta1 < 1.
For q = 2, the following three conditions must all be met: −1 < theta2 < 1, theta2 + theta1 < 1, theta2 − theta1 < 1. What is the constraint on theta1?
Similarly, what are the conditions for q ≥ 3?
ac_1 wrote:And are the restrictions the same if estimated jointly as an ARMA(p,q) model?
ac_1 wrote:TomDoan wrote:2. "Stable" means that the model doesn't change over time, which is completely different from "stationary". The roots say nothing about stability. If you are looking at sequential estimates of a model (particularly if you are doing automatic model selection), it is quite possible that the form of the model will change from one interval to another (and, of course, that means that the roots will be radically different since the models are). There is nothing wrong with that---all the models are approximations. An AR(2) and an ARMA(1,1) can have very similar dynamics (and have similar log likelihoods and produce similar forecasts) despite having "root" patterns which are (by construction) completely different.
Yes, I am looking at sequential estimation from automated model selection using GMAUTOFIT, and for some series the form of the model does change. The point I am making here is that acbjICRS is useful for depicting the form from the inverse roots. On that note, previously:
ac_1 wrote:TomDoan wrote:3. If BOXJENK with MAXL fails to estimate successfully, you should probably ignore the model. @BJAUTOFIT will put in the last log likelihood whether the model converges or not, which, if the model has a major issue, will generally be much worse than the better behaved models. Remember that if an ARMA(p,q) model is adequate, an ARMA(p+1,q+1) is unindentified, and unlikely to estimate properly, but it also is clearly dominated by smaller model.
What does adequate mean? How to handle non-convergence in a sequential scheme from either @BJAUTOFIT or @GMAUTOFIT? And further how to modify these procedures to take into account MAXL?
ac_1 wrote:TomDoan wrote:The roots tell you nothing about misspecification. (Misspecification is usually detected when there is significant autocorrelation left in the residuals). The roots tell you nothing about overparameterization. (Overparameterization is detected when an information criterion gives you a worse value than a smaller model).
The roots exist according to The fundamental theorem of algebra.
ac_1 wrote:Back to the question: What is the relationship and interpretation regarding the estimated ARMA coefficients (parameter values) and inverse roots (being inside, on, outside) relative to the complex unit circle?
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