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Some Concluding Remarks
Chapter 18
Some Concluding Remarks 18.1 My aim throughout this book has been to provide readers with a set of robust and useful techniques that they can call upon to analyze time series that occur in their own research fields. Clearly, considerations of how subject-specific theory may be incorporated into the analysis will be fielddependent but may typically influence the choice of variables to be included and provide constraints that could be imposed on the models being developed. My view would be that, in these circumstances, such constraints should be tested wherever possible, an opinion that reflects my (perhaps essentially British) “pragmatic” approach to statistical modeling in general.1 Related to this, my preference would always be to begin, wherever possible, with a general model and then test plausible restrictions with the aim of moving toward a simpler model that has empirical support, rather than starting with a tightly specified model whose (typically implicit) restrictions have not been subjected to statistical testing and which might, therefore, be seriously misspecified: in other words, I recommend following a general-tospecific modeling strategy of the type often associated with Hendry (1995). 18.2 I would also hope that readers will have gained an appreciation of the principle that the properties a time series displays will impact upon the behavior to be expected from it. To take an important current issue, the examples on modeling monthly global temperatures show that this series can be represented by an ARIMA(0,1,3) process that does not contain a significant drift term. That the series is I(1) rules out the possibility that temperatures exhibit reversion to a constant mean and so invalidates any theory that posits that there is an “equilibrium setting” that temperatures inexorably return to. However, the absence of a significant drift implies that there is also no underlying, albeit stochastic, trend, so that after three months forecasts converge on the current level of the series, so ruling out predictions of everincreasing future temperatures. In fact, given that innovations are persistent, and that the innovation variance is relatively small when compared to the overall variability of the series (the ratio is approximately 15%), it is, thus, no surprise that there may be extended periods in which the series is generally increasing, or indeed decreasing, and also extended periods in which the Applied Time Series Analysis. DOI: https://doi.org/10.1016/B978-0-12-813117-6.00018-1 © 2019 Elsevier Inc. All rights reserved.
311
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series appears to bounce randomly around a locally constant level, all of which are features of the observed evolution of global temperatures. The fact that temperatures are I(1) also rules out the common practice of fitting deterministic trends to the entire or, more usually, segments of the series: at best these can be interpreted as descriptive exercises to which no valid statistical inference can be attached.2 18.3 As well as incorporating and testing relevant theory considerations, “institutional” knowledge can also be important when analyzing data and time series are no exceptions to this. One, perhaps arcane, but fondly remembered example from my own experience, relates to when I was working for the Bank of England in the early 1980s on leave from my academic position. It was here that I became interested in seasonal adjustment and signal extraction and it was also when the Bank began to monitor closely the evolution of the monetary aggregates, as monetarism and control of these aggregates was popular with the Conservative government of the time. The “narrowest” of the selection of aggregates then available, M0, consisted primarily of the amount of notes and coin in circulation with the public, that is, cash. The Bank’s statisticians, who monitored this aggregate on a weekly basis, were aware of a “spike” in the aggregate during the first two weeks of July of each year. This was known as the “wakes week” effect, referring to the practice in northern towns of the United Kingdom, primarily in Lancashire and Scotland, for all factories to shut at the same time for summer holidays. Workers and their families consequently drew out much larger amounts of cash than usual to fund their holidays, so producing the aforementioned spike in M0. The Bank statisticians, adjusted the series to remove this spike and, hence, eradicate a transitory, but predictable, uptick in the trend of the data during that month which, if left in the series, might have provoked the unwary into seeing an acceleration in the growth of this monetary aggregate. 18.4 Effects such as these, often referred to as outliers, are a recurrent feature of time series and, although mentioned briefly in Chapter 5, Unit Roots, Difference and Trend Stationarity, and Fractional Differencing, have not been afforded a chapter of their own in this book: they may, in fact, be analyzed within the transfer function setup introduced in Chapter 12, Transfer Functions and ARDL Modeling. Nevertheless, they can be important and should always be looked out for when initially exploring data.3 Also omitted is any discussion of panel data, which combines time series with cross-sectional observations. This has become a popular area of research, particularly in economics, for it enables the analysis of, typically short, time series available on a set of cross-sectional variables. Panel data warrants a textbook of its own, with Baltagi (2013) being a well-known example.
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ENDNOTES 1. I use “pragmatism” here for both its vernacular meaning of indicating a “practical, matter-offact way of solving problems” and as a philosophical tradition that is centered on the linking of practice and theory, describing a process whereby theory is extracted from practice and then applied back to practice. Pragmatism as a philosophical movement begun in the United States during the 1870s and is most closely associated with Charles Sanders Peirce and William James (see Bacon, 2012). Although often referred to as “American pragmatism,” it was heavily influenced by Charles Darwin and the earlier “British empiricists” John Locke and David Hume. Statistical pragmatism has been proposed by Kass (2011) as a foundation for an eclectic statistical inference that goes beyond narrow frequentist and Bayesian positions, and emphasizes the “identification of models with data,” recognizing “that all forms of statistical inference make assumptions that can only be tested very crudely and can almost never be verified.” A link between philosophical and statistical pragmatism has been provided by the great statistician George Box. As he memorably stated in Box (1979), “all models are wrong, but some are useful,” and in Box (1984) he emphasized the importance of theory practice interaction using many examples from the development of statistical thinking. 2. The finding of no significant drift in the ARIMA model for monthly global temperatures is, of course, at odds with anthropogenic global warming theories that radiative forcings, such as those from CO2, which have pronounced upward trends, have a significant impact on temperatures, unless some of these forcings approximately cancel each other out. This cannot be examined within the context of modeling monthly temperatures, for forcing observations are not available at this frequency for a sufficiently lengthy period. The use of annual data, as in Example 12.2, does enable forcing effects to be isolated and there are found to be some offsets. 3. The seminal references to the identification and modeling of outliers, often referred to as intervention analysis, are Box and Tiao (1975); Abraham (1980); Tsay (1986b); and Chang et al. (1988); with Mills (1990, Chapter 13) providing a textbook treatment.
Some Concluding Remarks 18.1 My aim throughout this book has been to provide readers with a set of robust and useful techniques that they can call upon to analyze time series that occur in their own research fields. Clearly, considerations of how subject-specific theory may be incorporated into the analysis will be fielddependent but may typically influence the choice of variables to be included and provide constraints that could be imposed on the models being developed. My view would be that, in these circumstances, such constraints should be tested wherever possible, an opinion that reflects my (perhaps essentially British) “pragmatic” approach to statistical modeling in general.1 Related to this, my preference would always be to begin, wherever possible, with a general model and then test plausible restrictions with the aim of moving toward a simpler model that has empirical support, rather than starting with a tightly specified model whose (typically implicit) restrictions have not been subjected to statistical testing and which might, therefore, be seriously misspecified: in other words, I recommend following a general-tospecific modeling strategy of the type often associated with Hendry (1995). 18.2 I would also hope that readers will have gained an appreciation of the principle that the properties a time series displays will impact upon the behavior to be expected from it. To take an important current issue, the examples on modeling monthly global temperatures show that this series can be represented by an ARIMA(0,1,3) process that does not contain a significant drift term. That the series is I(1) rules out the possibility that temperatures exhibit reversion to a constant mean and so invalidates any theory that posits that there is an “equilibrium setting” that temperatures inexorably return to. However, the absence of a significant drift implies that there is also no underlying, albeit stochastic, trend, so that after three months forecasts converge on the current level of the series, so ruling out predictions of everincreasing future temperatures. In fact, given that innovations are persistent, and that the innovation variance is relatively small when compared to the overall variability of the series (the ratio is approximately 15%), it is, thus, no surprise that there may be extended periods in which the series is generally increasing, or indeed decreasing, and also extended periods in which the Applied Time Series Analysis. DOI: https://doi.org/10.1016/B978-0-12-813117-6.00018-1 © 2019 Elsevier Inc. All rights reserved.
311
312
Applied Time Series Analysis
series appears to bounce randomly around a locally constant level, all of which are features of the observed evolution of global temperatures. The fact that temperatures are I(1) also rules out the common practice of fitting deterministic trends to the entire or, more usually, segments of the series: at best these can be interpreted as descriptive exercises to which no valid statistical inference can be attached.2 18.3 As well as incorporating and testing relevant theory considerations, “institutional” knowledge can also be important when analyzing data and time series are no exceptions to this. One, perhaps arcane, but fondly remembered example from my own experience, relates to when I was working for the Bank of England in the early 1980s on leave from my academic position. It was here that I became interested in seasonal adjustment and signal extraction and it was also when the Bank began to monitor closely the evolution of the monetary aggregates, as monetarism and control of these aggregates was popular with the Conservative government of the time. The “narrowest” of the selection of aggregates then available, M0, consisted primarily of the amount of notes and coin in circulation with the public, that is, cash. The Bank’s statisticians, who monitored this aggregate on a weekly basis, were aware of a “spike” in the aggregate during the first two weeks of July of each year. This was known as the “wakes week” effect, referring to the practice in northern towns of the United Kingdom, primarily in Lancashire and Scotland, for all factories to shut at the same time for summer holidays. Workers and their families consequently drew out much larger amounts of cash than usual to fund their holidays, so producing the aforementioned spike in M0. The Bank statisticians, adjusted the series to remove this spike and, hence, eradicate a transitory, but predictable, uptick in the trend of the data during that month which, if left in the series, might have provoked the unwary into seeing an acceleration in the growth of this monetary aggregate. 18.4 Effects such as these, often referred to as outliers, are a recurrent feature of time series and, although mentioned briefly in Chapter 5, Unit Roots, Difference and Trend Stationarity, and Fractional Differencing, have not been afforded a chapter of their own in this book: they may, in fact, be analyzed within the transfer function setup introduced in Chapter 12, Transfer Functions and ARDL Modeling. Nevertheless, they can be important and should always be looked out for when initially exploring data.3 Also omitted is any discussion of panel data, which combines time series with cross-sectional observations. This has become a popular area of research, particularly in economics, for it enables the analysis of, typically short, time series available on a set of cross-sectional variables. Panel data warrants a textbook of its own, with Baltagi (2013) being a well-known example.
Some Concluding Remarks Chapter | 18
313
ENDNOTES 1. I use “pragmatism” here for both its vernacular meaning of indicating a “practical, matter-offact way of solving problems” and as a philosophical tradition that is centered on the linking of practice and theory, describing a process whereby theory is extracted from practice and then applied back to practice. Pragmatism as a philosophical movement begun in the United States during the 1870s and is most closely associated with Charles Sanders Peirce and William James (see Bacon, 2012). Although often referred to as “American pragmatism,” it was heavily influenced by Charles Darwin and the earlier “British empiricists” John Locke and David Hume. Statistical pragmatism has been proposed by Kass (2011) as a foundation for an eclectic statistical inference that goes beyond narrow frequentist and Bayesian positions, and emphasizes the “identification of models with data,” recognizing “that all forms of statistical inference make assumptions that can only be tested very crudely and can almost never be verified.” A link between philosophical and statistical pragmatism has been provided by the great statistician George Box. As he memorably stated in Box (1979), “all models are wrong, but some are useful,” and in Box (1984) he emphasized the importance of theory practice interaction using many examples from the development of statistical thinking. 2. The finding of no significant drift in the ARIMA model for monthly global temperatures is, of course, at odds with anthropogenic global warming theories that radiative forcings, such as those from CO2, which have pronounced upward trends, have a significant impact on temperatures, unless some of these forcings approximately cancel each other out. This cannot be examined within the context of modeling monthly temperatures, for forcing observations are not available at this frequency for a sufficiently lengthy period. The use of annual data, as in Example 12.2, does enable forcing effects to be isolated and there are found to be some offsets. 3. The seminal references to the identification and modeling of outliers, often referred to as intervention analysis, are Box and Tiao (1975); Abraham (1980); Tsay (1986b); and Chang et al. (1988); with Mills (1990, Chapter 13) providing a textbook treatment.