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CARMICHAEL’S ARCTAN TREND: PRECURSOR OF SMOOTH TRANSITION FUNCTIONS

Published online by Cambridge University Press:  12 November 2015

Terence C. Mills  and
Kerry Patterson
Terence C. Mills
Affiliation:
Loughborough University
Kerry Patterson
Affiliation:
University of Reading.
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Abstract

In an almost unreferenced article, Fitzhugh Carmichael (1928), writing of the period around the First World War, noted that “during the past twelve years many economic series have undergone what appears to be a permanent change in level.” These are prescient words that are widely applicable today. Carmichael noted that the then-standard practice of linear detrending was inappropriate in the presence of what we would now call “structural breaks”; as a result he proposed a method that would not only model a nonlinear trend, but would be suitable for situations where the transition from one regime to another was smooth. This study establishes the precedence of Carmichael’s ideas, re-examines his methods, and solves the problems that he thought would hinder wider applications of his approach, which has since become a central part of contemporary nonlinear econometric methods and for which Carmichael should be given credit.

Type
Articles
Information
Journal of the History of Economic Thought , Volume 37 , Issue 4 , December 2015 , pp. 537 - 557
Copyright
Copyright © The History of Economics Society 2015 

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References

REFERENCES

Aldrich, John. 1995. “Correlations Genuine and Spurious in Pearson and Yule.” Statistical Science 10 (4): 364–376.CrossRefGoogle Scholar
Andrews, Donald. 1993. “Tests for Parameter Instability and Structural Change with Unknown Change Point.” Econometrica 61 (4): 821–856.CrossRefGoogle Scholar
Bacon, David, and Watts, Donald. 1971. “Estimating the Transition between Two Intersecting Lines.” Biometrika 58 (3): 525–534.CrossRefGoogle Scholar
Bai, Jushan. 1997. “Estimating Multiple Breaks One at a Time.” Econometric Theory 13 (3): 315–352.CrossRefGoogle Scholar
Bai, Jushan, and Perron, Pierre. 1998. “Estimating and Testing Linear Models with Multiple Structural Changes.” Econometrica 66 (1): 47–78.CrossRefGoogle Scholar
Bai, Jushan, and Perron, Pierre. 2003. “Computation and Analysis of Multiple Structural Change Models.” Journal of Applied Econometrics 18 (1): 1–22.CrossRefGoogle Scholar
Beveridge, Stephen, and Nelson, Charles R.. 1981. “A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycle.” Journal of Monetary Economics 7 (2): 151–174.CrossRefGoogle Scholar
Biddle, Jeff. 1999. “Statistical Economics, 1900–1950.” History of Political Economy 31 (4): 607–651.CrossRefGoogle Scholar
Carmichael, Fitzhugh. 1928. “The Arc Tangent in Trend Determination.” Journal of the American Statistical Association 23 (163): 253–262.CrossRefGoogle Scholar
Chan, Kung-Sik, and Tong, Howell. 1986. “On Estimating Thresholds in Autoregressive Models.” Journal of Time Series Analysis 7 (3): 179–190.CrossRefGoogle Scholar
Cooper, Suzanne. 1998. “Multiple Regimes in U.S. Output Fluctuations.” Journal of Business and Economic Statistics 16 (1): 92–100.Google Scholar
Cramer, Jan. 2002. “The Origins of the Logistic Regression.” Tinbergen Institute Discussion Paper, TI 2002 119/4.Google Scholar
Frisch, Ragnar, and Waugh, Frederick V.. 1933. “Partial Time Regressions as Compared with Individual Trends.” Econometrica 1 (4): 387–401.CrossRefGoogle Scholar
Giovanis, Eleftherios. 2008. “Additional Smoothing Transition Autoregressive Models.” MPRA Paper No. 24657.Google Scholar
Granger, Clive, and Newbold, Paul. 1974. “Spurious Regressions in Econometrics.” Journal of Econometrics 2 (2): 111–120.CrossRefGoogle Scholar
Gray, Stephen. 1996. “Modeling the Conditional Distribution of Interest Rates as Regime-Switching Process.” Journal of Financial Economics 42 (1): 27–62.CrossRefGoogle Scholar
Hall, Lincoln. 1925. “A Moving Secular Trend and Moving Integration.” Journal of the American Statistical Association 20 (149): 13–24.CrossRefGoogle Scholar
Hall, Lincoln. 1926. “The Determination of Past and Present Secular Trends.” Journal of the American Statistical Association 21 (154): 206–212.CrossRefGoogle Scholar
Hansen, Bruce. 2000. “Testing for Structural Change in Conditional Models.” Journal of Econometrics 97 (1): 93–115.CrossRefGoogle Scholar
Hansen, Bruce. 2011. “Threshold Autoregression in Economics.” Statistics and Its Interface 4 (2): 123–127.CrossRefGoogle Scholar
Harvey, Andrew C. 1985. “Trends and Cycles in Macroeconomic Time Series.” Journal of Business and Economic Statistics 3 (3): 216–227.Google Scholar
Hooker, Reginald. 1901. “Correlation of the Marriage-rate with Trade.” Journal of the Royal Statistical Society 64 (3): 485–492.Google Scholar
Hudson, Derek. 1966. “Fitting Segmented Curves whose Join Points have to be Estimated.” Journal of the American Statistical Association 61 (316): 1097–1129.CrossRefGoogle Scholar
Karsten, Karl. 1926. “The Harvard Business Indexes—A New Interpretation.” Journal of the American Statistical Association 21 (156): 399–418.Google Scholar
Kuznets, Simon. 1928. “On the Analysis of Time Series.” Journal of the American Statistical Association 23 (165): 398–410.CrossRefGoogle Scholar
Lukkonen, Ritva, Saikkonen, Pentti, and Teräsvirta, Timo. 1988. “Testing Linearity Against Smooth Transition Autoregressive Models.” Biometrika 75 (3): 491–499.CrossRefGoogle Scholar
Mills, Terence. 2011. The Foundations of Modern Time Series Analysis. Basingstoke: Palgrave-Macmillan.CrossRefGoogle Scholar
Mitchell, Wesley. 1927. Business Cycles: The Problem and its Setting. New York: NBER.Google Scholar
Morgan, Mary. 1990. The History of Econometric Ideas. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Norberg, Arthur. 1990. “High-Technology Calculation in the Early 20th Century: Punched Card Machinery in Business and Government.” Technology and Culture 31 (4): 753–779.CrossRefGoogle Scholar
Pearl, Raymond, and Reed, Lowell. 1923. “On the Mathematical Theory of Population Growth.” Metron 3 (1): 6–19.Google Scholar
Pearson, Karl. 1896. “Mathematical Contributions to the Theory of Evolution, III: Regression, Heredity and Panmixia.” Philosophical Transactions of the Royal Society of London, Series A (187): 253–318.Google Scholar
Perron, Pierre. 1989. “The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis.” Econometrica 57 (6): 1361–1401.CrossRefGoogle Scholar
Perron, Pierre. 1992. “Testing for a Unit Root in a Time Series with a Changing Mean.” Journal of Business and Economic Statistics 8 (2): 153–162.Google Scholar
Persons, Warren. 1916. “Construction of a Business Barometer Based upon Annual Data.” American Economic Review 6 (4): 739–769.Google Scholar
Persons, Warren. 1919. “General Considerations Relating to Secular Trend, 1903–18.” The Review of Economics and Statistics 1 (1): 38–39.Google Scholar
Phillips, Peter. 2005. “Challenges of Trending Time Series Econometrics.” Mathematics and Computer Simulation 68 (5–6): 401–416.CrossRefGoogle Scholar
Quandt, Richard. 1958. “The Estimation of the Parameters of a Linear Regression Obeying Two Separate Regimes.” Journal of the American Statistical Association 53 (284): 873–880.CrossRefGoogle Scholar
Quandt, Richard. 1960. “Tests of the Hypothesis that a Linear Regression System Obeys Two Separate Regimes.” Journal of the American Statistical Association 55 (290): 324–330.CrossRefGoogle Scholar
Robison, D. E. 1964. “Estimates for the Points of Intersection of Two Polynomial Regressions.” Journal of the American Statistical Association 59 (305): 214–224.CrossRefGoogle Scholar
Sanquer, Matthieu, Chatelain, Florent, El-Guedri, Mabrouka, and Matin, Nadine. 2013. “A Smooth Transition Model for Multiple-regime Time Series.” IEEE Transactions on Signal Processing 61 (7): 1835–1847.CrossRefGoogle Scholar
Smith, Bradford B. 1926. “Combining the Advanatges of First-Difference and Deviation-from-Trend Methods of Correlating Time Series.” Journal of the American Statistical Association 21 (153): 55–59.CrossRefGoogle Scholar
Teräsvirta, Timo. 1994. “Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models.” Journal of the American Statistical Association 89 (425): 208–218.Google Scholar
Tjostheim, Dag. 1986. “Estimation in Nonlinear Time Series Models.” Stochastic Processes and their Applications 21 (2): 251–273.CrossRefGoogle Scholar
Tong, Howell. 1978. “On a Threshold Model.” In Hua Chen, Chi, ed., Pattern Recognition and Signal Processing. Amsterdam: Sijthoff and Noordhoff, pp. 575–586.CrossRefGoogle Scholar
Tong, Howell. 1982. “Discontinuous Decision Processes and Threshold Autoregressive Time Series Modelling.” Biometrika 69 (1): 274–276.CrossRefGoogle Scholar
Tong, Howell. 1983. Threshold Models in Non-linear Time Series Analysis. New York: Springer.CrossRefGoogle Scholar
Tong, Howell. 2010. “Threshold Models in Time Series Analysis—30 Years On.” Research Report No 471. Pokfulam: University of Hong Kong.Google Scholar
Van Sickle, Jenna. 2011. “A History of Trigonometric Education in the United States: 1776–1900.” PhD thesis, Columbia University.Google Scholar
Verhulst, Pierre-François. 1845. “Recherches mathématiques sur la loi d’accroissement de la population.” Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles 18: 1–42.CrossRefGoogle Scholar
Yule, G. Udny. 1897. “On the Theory of Correlation.” Journal of the Royal Statistical Society 60 (4): 477–489.CrossRefGoogle Scholar
Yule, G. Udny. 1926. “Why Do We Sometimes Get Nonsense-correlations between Time-series? A Study in Sampling and the Nature of Time Series.” Journal of the Royal Statistical Society 89 (1): 1–63.CrossRefGoogle Scholar
Yule, G. Udny. 1927. “On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer’s Sunspot Numbers.” Philosophical Transactions of the Royal Society of London. Series A (226): 267–298.Google Scholar