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I'm following this book with regards to modelling temperature data in a piece-by-piece manner.

On page 44, after fitting an AR(3) model to the de-trended and de-seasonalised data, the authors examine the variance of the residuals and:

calculate the daily empirical variance by averaging the values of the squared residuals of a particular day over all years.

Which they then proceed to model via a truncated Fourier series.

On the next page (page 45) they eliminate the seasonal dependency in the variance (the empirical variance) by:

dividing the residuals by the square root of the fitted variance.

My question is this:

1) Are they dividing the original residuals by stretching out the fitted variance across the length of the original residuals?

2) Why would you approach modelling the variance like this (i.e. create the empirical variance time series) instead of just fitting a truncated Fourier series to the squared residuals?

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  • $\begingroup$ when u have found your answer accept the one you like to close the question. $\endgroup$ – IrishStat Dec 19 '16 at 17:53
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I pursued your thread and read the available material (unusual for me ) because I am very interested in the subject of extracting structure from data without injecting structure and am always up for creative analytics. This book like all other books that I haven't written should be taken with a grain of salt. Let me comment on some of your comments..

  1. In the very old (but bad) days analysts would detrend/deseasonalize first clearly premising no ARIMA structure and no outliers/level shifts etc. and THEN violate their assumed premise and identify an ARIMA model. Equally flawed some researchers suggest building an ARIMA model first and then detrend/deseasonalize to deal with deterministic structure. Some researchers approach the problem simultaneously/holistically essentially crafting a hybrid model taking into account any pulses/level shifts/multiple time trends and changes in seasonal dummies along with ARIMA structure while incorporating any user specified causal series like heating degree days / cooling degree days.

  2. Averaging the values of the squared residuals for each of the 7 days and then modelling these variances with a Fourier model appears to me to be an ARCH/GARCH technique but inconsistent with ARIMA modelling of the squared residuals which is the normal ARCH/GARCH approach. It appears to be very ad hoc as Fourier methods while often suitable for physical processes leave a lot to be desired for other kinds of data and I am not sure how this fits into their playbook.

  3. Dividing the residuals by the square root of the fitted variance is an attempt to determine the weights (normalization) for a subsequent Generalized Least Squares (GLS) solution given that the 7 daily error variances are different from each other AND that the daily variance for each day is changing over time. AUTOBOX , a piece of time series software that I have helped to develop) uses a similar BUT different scheme to render the error variance to be invariant over time. http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html might be of some help here .

Their attempt to deal with error variance change is laudable but impracticable as far as I am concerned. Hope this helps unveil some of the "stuff" .

Time to get another book or some more advice.

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