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Experience with using the Box-Cox transformation when forecasting economic time series
Journal
of Econometrics
14 (1980) 277-280.
0 North-Holland
Publishing
Company
EXPERIENCE WITH USING THE BOX-COX TRANSFORMATION WHEN FORECASTING ECONOMIC TIME SERIES A Comment* Dale J. POIRIER Uniwrsity Received
of Toronto, Toronto, Ont. M5S IAl,
September
1979, final version
received
Canada
November
1979
In a recent study Nelson and Granger (1979) reported on their experience with using the BoxCox transformation to forecast twenty-one actual economic time series and four simulated series. The objective of this brief note is to offer some qualifications to the results of Nelson and Granger, and to offer a few alternative interpretations.
In a recent study Nelson and Granger (1979) reported on their experience with using the BoxXox transformation to forecast twenty-one actual economic time series and four simulated series. By the inherent nature of their study, their results can only be suggestive, but this does not necessarily diminish their interest to econometricians. The objective of this brief note is to offer some qualifications to the results of Nelson and Granger, and to offer a few alternative interpretations. As the authors acknowledge (p. 57), the Box-Cox transformation presupposes that the variable to be transformed is positive. Thus, strictly speaking, the Box-Cox transformation is intended to induce a truncated normal distribution [cf. Poirier (1978)]. Whether this truncation effect is negligible (as the authors assume) is somewhat hard to determine a priori since it depends on the unknown parameters of the distribution including the Box-Cox parameter i. If, however, the researcher is able to do this, then it seems that the researcher is admitting the existence of fairly strong a priori information. Hence, a Bayesian analysis, albeit difficult due to the nonlinearities involved, may be more appropriate. The fact that ‘the majority of levels economic activity are naturally positive’ (p. 57), admits truncation, but does not alone imply that the truncation can be ignored. Ignoring the truncation implies that the authors are comparing distributions with positive densities defined over different domains. For example, if *This note was prepared while the author was a Visiting Professor at the Institute of Economic Research, Kyoto University, Japan. Special thanks are owed to Takamitsu Sawa for his valuable comments. Any errors are, however, the sole responsibility of the author.
278
D.J. Poirier,
Experience
with the Box-Cox
transformation:
Comment
the distribution is normal, then there is a positive probability that one may observe a negative value. Even if this probability is ‘negligible’, asymptotically at least one negative observation will almost surely be observed. This is all that is required to contradict many of the competing possible distributions (e.g., lognormal). While admittedly the authors’ primary objective is to compare forecasting performances, at least implicitly they are also trying to determine the correct distribution. In some unpublished work, Randall Olsen of Yale University has suggested by way of an empirical example that ignoring the truncation tends to yield an estimate of i biased toward zero. Intuitively, this may be explained by the fact that the lognormal case (2 =0) is the only member of the BoxCox family of distributions that produces a normal distribution with no truncation. Olsen also notes that ignoring the truncation results in an understatement of the variance in the estimated Box-Cox parameter. Possibly, the authors’ result that their estimated residuals do not appear to follow a normal distribution may reflect poor estimates of ,? and/or the fact that the estimated residuals should follow a truncated normal distribution. It may also be appropriate to consider various goodness-of-fit tests other than merely looking at estimated skewness and kurtosis measures. With regard to forecasting matters, it would seem that choosing the ‘correct’ point forecast is an issue over which reasonable researchers may disagree. The desirability of unbiased forecasts depends on one’s loss function. Since the mean of the original variable does not exist when - 15 2 < 0 [cf. Poirier and Melino (1978) and Huang and Kelingos (1979)], the concept of unbiased forecasts would appear to be meaningless in such cases. Since many of the twenty-one estimated ,? values reported by the authors have confidence intervals which overlap values in the range - 1 ZL
D.J. Poirier,
Experience
with the Box-Cox
transformation:
Comment
279
The reported simulation experiments appear to reflect more favorably on the Box-Cox transformation, at least when the assumptions underlying its use are approximately valid. The fact that the Box-Cox transformation may not perform very well when no value of 2 exists for which the transformed series is normal (more appropriately, truncated normal), is not too surprising. Rather than lament this fact, it seems more desirable to ask why economic time series may be distributed such that they cannot be transformed into even approximately normal distributions. One explanation may be that normality is more appropriate when a different structural relationship is considered, e.g., one involving exogenous variables. Another explanation might be that a wider family of possible distributions must be considered. The viewpoint underlying the Box-Cox transformation is one of ‘parametric robustness’ compared to the usual specifications in econometrics: normality and lognormality. ‘Robustnic’ statisticians are fond of reminding econometricians that there is nothing ‘normal’ about the Gaussian distribution, and presumably they would not be surprised by the failure of a one-parameter transformation to induce Gaussianity in all cases. Finally, so as to not sound too negative in commenting on this thoughtprovoking piece of work, I should add that I agree with the authors that the Box-Cox parameter 2 is likely to be hard to pin down in many applications to economic data.2 This has been my experience with many applications of this technique to cross-sectional data sets [e.g., Foot and Poirier (1980)]. In part this reflects the fact that the Box-Cox parameter is fundamentally different than the other parameters in the model. I doubt whether robustnics would be surprised that it is often hard to estimate 2 accurately. To econometricians of the anti-Bayesian persuasion, this difficulty is disturbing since it implies that unless prior information is provided about 2, any inferences drawn about i or other parameters in the model are likely to be highly tentative. The most dangerous conclusion that an unwary reader may draw from the study by Nelson and Granger, is that rather than deal with the robustnics’ criticisms, one should blindly assume a conventional value of i (say 0 or 1). At a minimum one should perform a diagnostic test of an assumed value for 2. This is quite simple in the case of i.=O due to the absence of truncation [cf. Poirier and Ruud (1980)]. In a favorable light, I interpret the study by Nelson and Granger as possibly motivating a Bayesian approach to parametric robustness. I fear, however, that others may interpret it as warranting ignoring the criticisms of robustnic statisticians who are finally starting to gain the attention of some econometric practitioners. ‘While I am willing to admit wide confidence intervals for i, I must, however, note that the reported value of &=6.62 for series A (average workweek of production workers, ordnance and accessories) lies in a distant tail of what I believe would be most researchers’ prior distributions for 1,. So that an overly critical reader does not conclude that the authors’ procedure for estimating their models, and parttcularly 2, is faulty, I think this point estimate deserves some comment by the authors.
280
D.J. Poirier, Experience
with the Box-Cox
transformation:
Comment
Foot, D.K. and D.J. Poirier, 1980, Public decision making in Canada: The case of the AntiInflation Board, International Economic Review 21, 4899504. Huang, C.J. and J.A. Kelingos, 1979, Conditional mean function and a general specification of the disturbance in regression analysis, Southern Economic Journal 45, 71&717. Nelson, H.L. Jr. and C.W.J. &anger, 1979, Experience with using the Box-Cox transformation when forecasting economic time series, Journal of Econometrics 10, 57-69. Poirier, D.J., 1978, The use of the Box-Cox transformation in limited dependent variable models, Journal of the American Statistical Association 73, 284-287. Poirier, D.J. and A. Melino, 1978, A note on the interpretation of regression coefficients within a class of truncated distributions, Econometrica 46, 1207-1209. Poirier, D.J. and P. Ruud, 1980, A simple Lagrange multiplier test for lognormal regression, Economics Letters, forthcoming. Spitzer, J.J., 1978, A Monte Carlo investigation of the BoxCox transformation in small samples, Journal of the American Statistical Association 73. 4888495.
of Econometrics
14 (1980) 277-280.
0 North-Holland
Publishing
Company
EXPERIENCE WITH USING THE BOX-COX TRANSFORMATION WHEN FORECASTING ECONOMIC TIME SERIES A Comment* Dale J. POIRIER Uniwrsity Received
of Toronto, Toronto, Ont. M5S IAl,
September
1979, final version
received
Canada
November
1979
In a recent study Nelson and Granger (1979) reported on their experience with using the BoxCox transformation to forecast twenty-one actual economic time series and four simulated series. The objective of this brief note is to offer some qualifications to the results of Nelson and Granger, and to offer a few alternative interpretations.
In a recent study Nelson and Granger (1979) reported on their experience with using the BoxXox transformation to forecast twenty-one actual economic time series and four simulated series. By the inherent nature of their study, their results can only be suggestive, but this does not necessarily diminish their interest to econometricians. The objective of this brief note is to offer some qualifications to the results of Nelson and Granger, and to offer a few alternative interpretations. As the authors acknowledge (p. 57), the Box-Cox transformation presupposes that the variable to be transformed is positive. Thus, strictly speaking, the Box-Cox transformation is intended to induce a truncated normal distribution [cf. Poirier (1978)]. Whether this truncation effect is negligible (as the authors assume) is somewhat hard to determine a priori since it depends on the unknown parameters of the distribution including the Box-Cox parameter i. If, however, the researcher is able to do this, then it seems that the researcher is admitting the existence of fairly strong a priori information. Hence, a Bayesian analysis, albeit difficult due to the nonlinearities involved, may be more appropriate. The fact that ‘the majority of levels economic activity are naturally positive’ (p. 57), admits truncation, but does not alone imply that the truncation can be ignored. Ignoring the truncation implies that the authors are comparing distributions with positive densities defined over different domains. For example, if *This note was prepared while the author was a Visiting Professor at the Institute of Economic Research, Kyoto University, Japan. Special thanks are owed to Takamitsu Sawa for his valuable comments. Any errors are, however, the sole responsibility of the author.
278
D.J. Poirier,
Experience
with the Box-Cox
transformation:
Comment
the distribution is normal, then there is a positive probability that one may observe a negative value. Even if this probability is ‘negligible’, asymptotically at least one negative observation will almost surely be observed. This is all that is required to contradict many of the competing possible distributions (e.g., lognormal). While admittedly the authors’ primary objective is to compare forecasting performances, at least implicitly they are also trying to determine the correct distribution. In some unpublished work, Randall Olsen of Yale University has suggested by way of an empirical example that ignoring the truncation tends to yield an estimate of i biased toward zero. Intuitively, this may be explained by the fact that the lognormal case (2 =0) is the only member of the BoxCox family of distributions that produces a normal distribution with no truncation. Olsen also notes that ignoring the truncation results in an understatement of the variance in the estimated Box-Cox parameter. Possibly, the authors’ result that their estimated residuals do not appear to follow a normal distribution may reflect poor estimates of ,? and/or the fact that the estimated residuals should follow a truncated normal distribution. It may also be appropriate to consider various goodness-of-fit tests other than merely looking at estimated skewness and kurtosis measures. With regard to forecasting matters, it would seem that choosing the ‘correct’ point forecast is an issue over which reasonable researchers may disagree. The desirability of unbiased forecasts depends on one’s loss function. Since the mean of the original variable does not exist when - 15 2 < 0 [cf. Poirier and Melino (1978) and Huang and Kelingos (1979)], the concept of unbiased forecasts would appear to be meaningless in such cases. Since many of the twenty-one estimated ,? values reported by the authors have confidence intervals which overlap values in the range - 1 ZL
D.J. Poirier,
Experience
with the Box-Cox
transformation:
Comment
279
The reported simulation experiments appear to reflect more favorably on the Box-Cox transformation, at least when the assumptions underlying its use are approximately valid. The fact that the Box-Cox transformation may not perform very well when no value of 2 exists for which the transformed series is normal (more appropriately, truncated normal), is not too surprising. Rather than lament this fact, it seems more desirable to ask why economic time series may be distributed such that they cannot be transformed into even approximately normal distributions. One explanation may be that normality is more appropriate when a different structural relationship is considered, e.g., one involving exogenous variables. Another explanation might be that a wider family of possible distributions must be considered. The viewpoint underlying the Box-Cox transformation is one of ‘parametric robustness’ compared to the usual specifications in econometrics: normality and lognormality. ‘Robustnic’ statisticians are fond of reminding econometricians that there is nothing ‘normal’ about the Gaussian distribution, and presumably they would not be surprised by the failure of a one-parameter transformation to induce Gaussianity in all cases. Finally, so as to not sound too negative in commenting on this thoughtprovoking piece of work, I should add that I agree with the authors that the Box-Cox parameter 2 is likely to be hard to pin down in many applications to economic data.2 This has been my experience with many applications of this technique to cross-sectional data sets [e.g., Foot and Poirier (1980)]. In part this reflects the fact that the Box-Cox parameter is fundamentally different than the other parameters in the model. I doubt whether robustnics would be surprised that it is often hard to estimate 2 accurately. To econometricians of the anti-Bayesian persuasion, this difficulty is disturbing since it implies that unless prior information is provided about 2, any inferences drawn about i or other parameters in the model are likely to be highly tentative. The most dangerous conclusion that an unwary reader may draw from the study by Nelson and Granger, is that rather than deal with the robustnics’ criticisms, one should blindly assume a conventional value of i (say 0 or 1). At a minimum one should perform a diagnostic test of an assumed value for 2. This is quite simple in the case of i.=O due to the absence of truncation [cf. Poirier and Ruud (1980)]. In a favorable light, I interpret the study by Nelson and Granger as possibly motivating a Bayesian approach to parametric robustness. I fear, however, that others may interpret it as warranting ignoring the criticisms of robustnic statisticians who are finally starting to gain the attention of some econometric practitioners. ‘While I am willing to admit wide confidence intervals for i, I must, however, note that the reported value of &=6.62 for series A (average workweek of production workers, ordnance and accessories) lies in a distant tail of what I believe would be most researchers’ prior distributions for 1,. So that an overly critical reader does not conclude that the authors’ procedure for estimating their models, and parttcularly 2, is faulty, I think this point estimate deserves some comment by the authors.
280
D.J. Poirier, Experience
with the Box-Cox
transformation:
Comment
Foot, D.K. and D.J. Poirier, 1980, Public decision making in Canada: The case of the AntiInflation Board, International Economic Review 21, 4899504. Huang, C.J. and J.A. Kelingos, 1979, Conditional mean function and a general specification of the disturbance in regression analysis, Southern Economic Journal 45, 71&717. Nelson, H.L. Jr. and C.W.J. &anger, 1979, Experience with using the Box-Cox transformation when forecasting economic time series, Journal of Econometrics 10, 57-69. Poirier, D.J., 1978, The use of the Box-Cox transformation in limited dependent variable models, Journal of the American Statistical Association 73, 284-287. Poirier, D.J. and A. Melino, 1978, A note on the interpretation of regression coefficients within a class of truncated distributions, Econometrica 46, 1207-1209. Poirier, D.J. and P. Ruud, 1980, A simple Lagrange multiplier test for lognormal regression, Economics Letters, forthcoming. Spitzer, J.J., 1978, A Monte Carlo investigation of the BoxCox transformation in small samples, Journal of the American Statistical Association 73. 4888495.