I am currently building a vector autoregressive (VAR) model to analyse multivariate sensor signals. During the pre-processing procedure I implemented the first-order differencing twice and the differenced sensor signals are all tested as stationary by the Augmented Dickey Fuller (ADF) test with p
Probably, you are yet to choose the appropriate lag length for the test. You may look for a lag length that would render no serial correlation in the residuals. Please go through the Autocorrelation LM Test which will give you the lag at which no serial correlation exists in the residuals. Once you avoid the serial correlation, your VAR model might become stable.
How did you detect that the built models were not stable? Please let me recommend this work on how to assess the quality of modelling:
Article Assessing Point Forecast Bias Across Multiple Time Series: M...
Regarding VAR: check the assumptions of your model. If time series features change in time then weighted estimators or sliding window estimators may prove useful. Also, I recommend the metrics presented in the above chapter in order to evaluate and compare modelling performance.
Hi Andrey! For judging the stability of VAR models, please see page 8 of the attached link. http://pareto.uab.es/lgambetti/Lecture2M.pdf
I don't understand when you write "During the pre-processing procedure I implemented the first-order differencing twice". Do you mean that working with the second difference of all your series leads to an unstable VAR model? Like Andrey Davydenko, I am interested in how you can conclude that. Using maximum likelihood estimation perhaps? I think that you should investigate the possibility of a unit root and consider an ECM.
Guy Mélard Yes, the second differenced time series are all tested as stationary by the ADF test (with p-value < 0.01) but the corresponding VAR models are not stable. I judged the stability of VAR models by seeing if all the roots of the AR companion matrix are larger than one (please see page 8 of the document attached above).
Sk. Riad Arefin Sorry for the late reply. I was wondering if this optimal lag length would be inconsistent with the optimal lag length yielded by AIC or BIC criterion.
This is what I understood, although it is surprising. Indeed, it is not frequent to use second differences. If this is not justified, to get residual autocorrelation that cannot easily be handled by a VAR model. I suppose that you obtained the VAR estimates by regression. Note that maximum likelihood will not allow you to fit an unstable model but will probably give you estimation problems (like sensitivity to initial values for the parameters, and/or estimates close to the boundary).
I repeat: I think that you should investigate the possibility of a unit root and consider an ECM (if these second differences are really justified, of course).
Hongmin Zhu, yeah, Optimal Lag Length would alter with different criteria. Kindly have a look at this video. I think this might help.
How to estimate and interpret VAR models in Eviews - Vector Autoregression model - YouTube (JD Economics)
Guy Mélard Thank you very much for your suggestions. I'll further try the ECM model.
Many thanks Sk. Riad Arefin for your suggestions. I'll have a look at the Breusch–Godfrey test.
If all the eigenvalues of a certain matrix are less than 1 a VAR process is stable. A stable VAR process is stationary. A stationary process is not necessarily stable. I am not sure that a non-stable stationary process is of interest but this probably depends on your application. (See Chapter 2 of Lutkepohl (2005), New Introduction to Multiple Time Series, Springer.)
Perhaps you should review your model. Heteroskedasticity in your data is not relevant. The covariance matrix of the disturbance vector should be constant. In econometrics I(2) models are rare. Have you considered if any of your series are cointegrated? it is difficult to make any real suggestions without understanding the nature of your sensor signals.
John C Frain Thank you very much for your suggestions. My signals are all amplitude modulation (AM) signals (with a certain frequency). For all the sensor signals I only extract part of them with a window for my research interest. The selected signals seem to have no linear trend but their amplitude values vary with time (the amplitude values peak at the centre and decrease towards two sides). Please see https://reviseomatic.org/help/2-modulation/Amplitude%20Modulation.php for an example of the AM signals.
Hongmin Zhu Beyond a basic knowledge of AM and FM for radio signals I am not familiar with your topic. I fail to see exactly what you are measuring or what you are trying to model or why the signal extracted from one sensor is dependent on the signal from another. If you are modeling the amplitude of the signal at fixed intervals then it contains a deterministic component (fixed frequency sin wave and possibly the output derived from some deterministic input). Your observations must be adjusted for these deterministic components before you do your ADF tests.
Hongmin Zhu When you say that you "extract part of them" you mean in the frequency domain? Or in the time domain? In other words, you mean that you take a subsample of your data series, or that you apply a filter? I am asking because filters may introduce serially correlated noises which may cause problems in the VAR estimation like the ones you are experiencing (lag selection and autocorrelation at high order).
Good luck!
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