Abstract
A time series is defined as a set of quantitative observations arranged in chronological order. We generally assume that time is a discrete variable. Time series have always been used in the field of econometrics. Already at the outset, JAN TINBERGEN (1939) constructed the first econometric model for the United States and thus started the scientific research programme of empirical econometrics. At that time, however, it was hardly taken into account that chronologically ordered observations might depend on each other. The prevailing assumption was that, according to the classical linear regression model, the residuals of the estimated equations are stochastically independent from each other. For this reason, procedures were applied which are also suited for cross section or experimental data without any time dependence.
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References
An introduction to the history of time series analysis is given by
MARC NERLOVE, DAVID M. GRETHER and JOSÉ L. CARVALHO, Analysis of Economic Time Series: A Synthesis, Academic Press, New York et al. 1979, pp. 1 – 21.
The first estimated econometric model was presented in
JAN TINBERGEN, Statistical Analysis of Business Cycle Theories, Vol. 1: A Method and Its Application to Business Cycle Theory, Vol. 2: Business Cycles in the United States of America, 1919 – 1932, League of Nations, Economic Intelligence Service, Geneva 1939.
That autocorrelation of the residuals can cause problems for the statistical estimation and testing of econometric models was first noticed by
DONALD COCHRANE and GUY H. ORCUTT, Application of Least Squares Regression to Relationships Containing Autocorrelated Error Terms, Journal of the American Statistical Association 44 (1949), pp. 32 – 61.
In this article, one can also find the transformation to eliminate first order autocorrelation which was named after these two authors. With this transformation and the testing procedure proposed by
JAMES DURBIN and GEOFFREY S. WATSON, Testing for Serial Correlation in Least Squares Regression, I, Biometrika 37 (1950), pp. 409 – 428; II, Biometrika 38 (1951), pp. 159 – 178,
econometricians believed to cope with these problems.
However, methods of time series analysis had already been applied earlier to investigate economic time series.
WARREN M. PERSONS, Indices of Business Conditions, Review of Economic Statistics 1 (1919), pp. 5 – 107,
was the first to distinguish different components of economic time series. Such procedures are still applied today. For example, the seasonal adjustment procedure SEATS, which is used by EUROSTAT and which is described in
AUGUSTIN MARAVALL and VICTOR GOMEZ, The Program SEATS: ‚Signal Extraction in ARIMA Time Series‘, Instruction for the User, European University Institute, Working Paper ECO 94/28, Florence 1994,
is based on such an approach.
The more recent development of time series analysis has been initiated by the textbook of
GEORGE E.P. BOX and GWILYM M. JENKINS, Time Series Analysis: Forecasting and Control, Holden Day, San Francisco et al. 1970; 2nd enlarged edition 1976.
This book mainly proposes the time domain for the analysis of time series and focuses on univariate models. The theoretical basis of this approach is the decomposition theorem for stationary time series shown by
HERMAN WOLD, A Study in the Analysis of Stationary Time Series, Almquist and Wicksell, Stockholm 1938.
An argument in favour of the application of this time series approach is that shortterm predictions thus generated are often considerably better than predictions generated by the use of large econometric models. This was shown, for example, by
CLIVE W.J. GRANGER and PAUL NEWBOLD, Economic Forecasting: The Atheist’s Viewpoint, in: G.A. RENTON (ed.), Modelling the Economy, Heinemann, London 1975, pp. 131 – 148.
Besides analyses in the time domain there is also the possibility to analyse time series in the frequency domain. See, for example,
CLIVE W.J. GRANGER and MICHIO HATANAKA, Spectral Analysis of Economic Time Series, Princeton University Press, Princeton N.J. 1964.
Extensive surveys on modern methods of time series analysis are given by
JAMES D. HAMILTON, Time Series Analysis, Princeton University Press, Princeton N.J. 1994, and
HELMUT LÜTKEPOHL, New Introduction to Multiple Time Series Analysis, Springer, Berlin et al., 2005.
In JAMES D. HAMILTON’s book one can also find remarks on the relation between ergodicity and stationarity (pp. 45ff.).
Textbooks focusing on the application of these methods are
WALTER ENDERS, Applied Econometric Time Series, Wiley, New York, 3rd edition 2010, as well as
HELMUT LÜTKEPOHL and MARKUS KRÄTZIG (eds.), Applied Time Series Econometrics, Cambridge University Press, Cambridge et al. 2004.
For a deeper discussion of stochastic processes see, for example,
ARIS SPANOS, Statistical Foundations of Econometric Modelling, Cambridge University Press, Cambridge (England) et al. 1986, pp. 130ff, or
EMANUEL PARZEN, Stochastic Processes, Holden-Day, San Francisco 1962.
The test statistic for the variance of single estimated autocorrelation coefficients is given by
MAURICE STEVENSON BARTLETT, On the Theoretical Specification and Sampling Properties of Auto-Correlated Time Series, Journal of the Royal Statistical Society (Supplement) 8 (1946), pp. 24 – 41.
The statistic for testing a given number of autocorrelation coefficients was developed by
GEORGE E.P. BOX and DAVID A. PIERCE, Distribution of Residual Autocorrelations in Autoregressive Moving Average Time Series Models, Journal of the American Statistical Association 65 (1970), pp. 1509 – 1526,
while the modification for small samples is due to
GRETA M. LJUNG and GEORGE E.P. BOX, On a Measure of Lack of Fit in Time Series Models, Biometrika 65 (1978), pp. 297 – 303.
The Lagrange-Multiplier test for residual autocorrelation has been developed by
TREVOR S. BREUSCH, Testing for Autocorrelation in Dynamic Linear Models, Australian Economic Papers 17 (1978), pp. 334 – 355, and by
LESLIE G. GODFREY, Testing Against General Autoregressive and Moving Average Error Models When Regressors Include Lagged Dependent Variables, Econometrica 46 (1978), S. 1293 – 1302.
The test on normal distribution presented above has been developed by
CARLOS M. JARQUE and ANIL K. BERA, Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals, Economics Letters 6 (1980), pp. 255 – 259.
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Kirchgässner, G., Wolters, J., Hassler, U. (2013). Introduction and Basics. In: Introduction to Modern Time Series Analysis. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33436-8_1
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