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Statistics. by David Freedman; Robert Pisani; Roger PurvesReview by: Gary A. SimonJournal of the American Statistical Association, Vol. 74, No. 368 (Dec., 1979), pp. 927-928Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2286430 .Accessed: 15/06/2014 05:25
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Book Reviews 927
An Introduction to Bilinear Time Series Models. C.W. Granger and A.P. Andersen. Gottingen, Germany: Vandenhoeck and Ruprecht, 1978. 94 pp. $12.00 (paperback).
This excellent monograph introduces statisticians to a poten- tially useful class of nonlinear time series models in a most ap- pealing fashion. The exposition consists almost exclusively of examples designed to capture the fancy of the time-domain analyst. A case is made for use of a simple diagnostic procedure-examina- tion of the sample autocorrelation function of the series of squares- that will spur immediate empirical experimentation by practitioners.
The authors' main points are these:
1. Fits of linear time-domain models that are judged adequate, in the sense that residual autocorrelations (or spectra) signal white- noise errors, tempt analysts to ignore the fitting of nonlinear models that can forecast consistently better than the linear models.
2. Bilinear models are nonlinear models that are intuitively appealing and can be handled readily by anyone familiar with the traditional linear models.
The authors offer persuasive anecdotal evidence for these points while admitting that many theoretical questions about bilinear models remain to be answered, especially conditions for stationarity and invertibility of the models.
Three features of the authors' case that I found appealing are
1. Presentationt of simple substantive models that are bilinear. As an illustration, suppose the true rate of return of series St} is given by the MA (1) model
Xi(X-X4t - xi) = et + be-e i -i < b < 1
where { tt} is a white-noise sequence of independent zero mean, variance a2 random variables. The model may be written
X = Xt-1 + EtXt-i + b.t-lXt-g X
which is a bilinear model. Terms involving products of the errors and the data are the distinguishing characteristic of such models. This model can be further complicated by assuming that the actual data are of the form yt = xt + ?t, where litl is a white-noise disturbance.
2. Demonstration of the time series structure of the square. Let {xei be generated by a bilinear model that enjoys suitable sta- tionarity restrictions. Let lyet be a series generated by an ARMA model that has identical second-order moment properties. The authors produce examples to show that the series lXt2 and Iyt2 I will not have identical second-order moment properties. Thus examination of the autocorrelation function of lXt2 suggests itself as a technique for identifying bilinearity in practice.
3. Analysis of both sinulated and actual series. These examples yield considerable insight. For instance, the authors analyze Series B of Box and Jenkins (1970): IBM daily common stock closing prices for 169 trading days starting 17 May 1961. The linear model fitted by Box and Jenkins to the price change, xet, is
Xe = et + .26 eet-I
with residual variance 24.8. Some autocorrelations of the residuals and the squared residuals are as follows:
rk( O) 1 2 3 4 5 6
rk(E) -.01 -.07 -.12 .16 0 -.14 rk (f2) .31 .06 -.04 -.06 -.07 -.09
Box and Jenkins judged the rkt'i) not inconsistent with the as- sumption of white-noise errors, but the large re (V2) suggests bi- linearity. Granger and Andersen fitted the bilinear model
=t .02 1e-ent-i + f t
where In{ t is assumed to be white noise with estimated variance 23.5. Forecasts from this model improved forecast MSE, relative to the linear model, by 10.7 percent over a 15-day period and 8.2 percent over a 30-day period.
Granger and Andersen emphasize the use of bilinear models as tools to improve forecasting accuracy. I would suggest another potential use. My guess is that bilinearity frequently results from excluding important variables from our models, as we almost
invariably do when we do univariate time series analysis. As an illustration, I constructed an AR(3) model for daily average flow of a certain Wisconsin river at a certain site. The autocorrelations of the residuals signaled white-noise errors, but those of the squared residuals signaled bilinearity. Then I added a simple polynomial in daily precipitation and its lags to the model. Neither the residuals nor the squared residuals signaled any problems. In this applica- tion the bilinear analysis was used as a diagnostic tool to help decide the merits of adding a new variable to the model.
While bilinear models may be useful ad hoc tools for producing better forecasts, they will have a more profound influence on time series analysis if they can assist us in thinking more deeply about variables that should enter our models. The extent to which they can do this cannot be determined until more theoretical and em- pirical work has been done.
ROBEJRT B. MILLER University of Wisconsin, Madison
REFERENCE
Box, G.E.P., and Jenkins, G.M. (1970), Time Series Analysis, Forecasting and Control, San Francisco: Holden-Day.
Statistics. David Freedman, Robert Pisani, and Roger Purves. New York: W.W. Norton and Co., 1978. xv + 506 pp. $13.95.
Consider the following statistical issue. Two exams were given in a course. The first exam had average score 50 and standard deviation 10. The second exam had average score 40 and standard deviation 15. The correlation between the two scores was +.4. If a student scored 60 points on the first exam, what score would you expect the student to have on the second exam?
The solution goes something like this. The student was one standard deviation above average on the first exam. The student should be one standard deviation above average on the second exam, except for the regression effect. The regression effect means that he or she should be predicted to be +.4 X one standard de- viation -.4 standard deviation above average on the second exam. This prediction is then 40 + .4 X 15 = 40 + 6 = 46.
Believe it or not, this solution will be produced by many students getting their basic course from Statistics by Freedman, Pisani, and Purves-a clever intuition-based introduction to the subject.
Besides regression and correlation, the student will learn about controlled studies versuis observational studies, histograms, averages, standard deviations, the normal curve for data, elementary proba- bility, expected value and standard error, sampling and associated standard errors, confidence intervals, and several hypothesis tests. There are interesting chapters on the Current Population Survey and on chance models in genetics. All the material can be covered in a one-semester course.
The regression-correlation problem, by the way, comes very early in the text. This is genuinely interesting to most students, and it can win their sympathy for the subject matter.
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The authors have clearly thought about the process of teaching statistics to mathematically disinclined college undergraduates. A valuable teaching guide accompanies the text. This guide con- tains chapter-by-chapter advice and solutions to the exercises. It also contains a diagnostic pretest, along with the results when the test was given at the University of California, Berkeley. Users of the text will be shocked when they learn how poorly the students at Berkeley performed; they will be shocked again when they administer the pretest to their own students.
The authors try to develop a feeling for data. The student looks at many scatterplots before seeing a formal method for finding correlations. There are many exercises geared toward making the stuLdent appreciate the meaning of a standard deviation; most students will learn to look at a histogram and make a reasonable guess at the standard deviation.
The statistical sampling process is conceptualized as drawing tickets from a box. This idea is not unique to Freedman, Pisani, and Purves, but here the idea is heavily emphasized. The student learns that the ticket-drawing model is the heart of statistical thinking. P. 407 reminds the reader, 'Statistical inference can be justified only in terms of an explicit chance model for the data. No box, no inference.' In many places in the text, the student
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928 Journal of the American Statistical Association, December 1979
is asked to formulate a problem in terms of a box model, telling what is on the tickets, and noting how many tickets there are. Processes leading to successes and failures are conceptualized as 0-1 boxes. Processes leading to measurements have box models that may be largely summarized by the average of the box and the SD of the box.
The reader is not overwhelmed by new symbols and exotic vocabulary. There are no Greek symbols. The word 'probability' has been replaced by 'chance.' There are no unions or intersec- tions. Mutually exclusive events are called 'incompatible.' Events themselves are called 'outcomes.' There are no summation signs. The letter n for sample size is not used-it is replaced by the ex- pression 'number of draws,' referring to a box model.
The text contains many quick-and-easy examples for which the answers are in the back of the book. These are well suited for classroom discussion, and they can be used for relief from the usual lecture routine. There are many footnotes; the inquisitive reader will find these rewarding.
An impressive innovation is the use of the SD line in regression problems. This is a line passing through the point of averages (which is (x, y) in usual notation) and having slope (sign of r) (SD of Y)/ (SD of X). The SD line is one that is instinctively drawn on a scatterplot, and the student learns why the actual regression line is different from the SD line. (My colleague Jerry Dallal has pointed out to me that the SD line need not have a slope close to that of the first principal component.)
This text asks the student to do rather little computation. I sup- port this point of view; it has been my experience that even the well-intentioned student with a calculator will not be able to find a correlation coefficient correctly. It is far better that he or she appreciate what a correlation coefficient ought to be. Moreover, on all but the smallest data sets, most statistics will be computer generated. Problems in the text usually include averages, standard deviations, and correlations as given information. The student still has to do arithmetic to find standard errors, confidence limits, and chi-squared values. These calculations are not terrible, and the computational effort does not grow with the sample size.
Even though calculations are not vital, techniques are still given. The standard deviation is found by (a) obtaining the average, (b) subtracting the average from each member of the list, (c) com- puting the rms (root mean square) of the resulting list. This is just the use of SD = (1/n _,_1 (xi-x)2)* without the symbols. There is a technical note (p. 65) to tell the reader that one can also compute
SD = (average of (entries)2 - (average of entries)2)1 .
The 1/n definition of the SD makes things easy to understand at this stage, although a bit of a problem is created later, since the t statistic requires the 1/ (n - 1) form, and the new version of the standard deviation is called SD+.
Correlation coefficients can also be calculated. The technique is startling, although it makes sense at this level. To find the cor- relation coefficient, the student is given this method (p. 124): 'Convert each variable, to standard units. The average of the products gives the correlation coefficient.' This is, of course, an appalling computational technique. The student will only perform this arithmetic on a few very small data sets with round numbers. With a data set consisting of 20 pairs, the student is virtually certain to get the wrong answer with any computational method.
The authors are very careful to give the reader only one method of doing any calculation, except perhaps with modifications in technical notes or footnotes. At least two of these situations are troubling to me. The first deals with the standard error of an aver- age. Because problems dealing with things like total number of successes use sample totals (of zeros and ones), the reader is taught that SE of sample total = estimate of SD X (sample size)*. Since other problems will deal with sample averages, the reader is later told that SE of sample average = SE of sample total/sample size. This is done on p. 325 for percentages and on p. 373 for other kinds of data. The authors explain it this way in their instructor's manual:
Section 20.2 presents our version of (pq/n)*, except that the formula doesn't appear. This may seem a bit idiosyncratic, and we would like to explain why we moved from the con- ventional formula to our version.
The students seemed to find (pq) rather hard to swallow. So we teach them to make a model with 0's and l's in the box. Since we are working in percents, the formula becomes
(SD of 0-1 box/V/n) X 100% .
We presented it this way for several years, but there was still a hitch. The students were willing to compute an SE as SD X V/n in part V (where SE of a sum was explained-G.S.). When they hit part VI (requiring SE of a percentage-G.S.), there was a tremendous shifting of gears needed to compute the SE as SD/V/n. Once they changed over, they stopped being able to compute the SE for a sum as SD X Vn: they insisted on dividing. We tried hard to explain that there was one formula to use with sums and another for averages, but they wouldn't buy it.
Eventually, we decided to have only one formula: the SE for a sum. Everything else is worked out fr..
Statistics 4th Edition Freedman
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