Time series

(This post got published on The Statistics Forum yesterday.)

The short book review section of the International Statistical Review sent me Raquel Prado’s and Mike West’s book, Time Series (Modeling, Computation, and Inference) to review. The current post is not about this specific book, but rather on why I am unsatisfied with the textbooks in this area (and correlatively why I am always reluctant to teach a graduate course on the topic). Again, I stress that the following is not specifically about the book by Raquel Prado and Mike West!

With the noticeable exception of Brockwell and Davis’ Time Series: Theory and Methods, most time-series books seem to suffer (in my opinion) from the same difficulty, which sums up as being unable to provide the reader with a coherent and logical description of/introduction to the field. (This echoes a complaint made by Håvard Rue a few weeks ago in Zurich.) Instead, time-series books appear to haphazardly pile up notions and techniques, theory and methods, without paying much attention to the coherency of the presentation. That’s how I was introduced to the field (even though it was by a fantastic teacher!) and the feeling has not left me since then. It may be due to the fact that the field stemmed partly from signal processing in engineering and partly from econometrics, but such presentations never achieve a Unitarian front on how to handle time-series. In particular, the opposition between the time domain and the frequency domain always escapes me. This is presumably due to my inability to see the relevance of the spectral approach, as harmonic regression simply appears (to me) as a special type of non-linear regression with sinusoidal regressors and with a well-defined likelihood that does not require Fourier frequencies nor periodogram (nor either spectral density estimation). Even within the time domain, I find the handling of stationarity  by time-series book to be mostly cavalier. Why stationarity is important is never addressed, which leads to the reader being left with the hard choice between imposing stationarity and not imposing stationarity. (My original feeling was to let the issue being decided by the data, but this is not possible!) Similarly, causality is often invoked as a reason to set constraints on MA coefficients, even though this resorts to a non-mathematical justification, namely preventing dependence on the future. I thus wonder if being an Unitarian (i.e. following a single logical process for analysing time-series data) is at all possible in the time-series world! E.g., in Bayesian Core, we processed AR, MA, ARMA models in a single perspective, conditioning on the initial values of the series and imposing all the usual constraints on the roots of the lag polynomials but this choice was far from perfectly justified…