A common procedure in environmental statistics is to estimate and test for
trends in time series data. Usually, this is done with the help of a
regression model including time as one of the independent variables. However,
very often, the residuals present an autocorrelation structure which may
cause a distortion in the variance of least squares estimators and, therefore,
in the tests results. Detection of climatic changes or increase in
concentration of pollutants provide good examples of such series, where
the need for the development of appropriate and rigorous tools is of the
utmost importance. In this talk we will consider, first, the linear
regression model with any set of independent variables and errors following
an autoregressive, AR(p), process. In this general case, the maximum
likelihood estimators are difficult to obtain and not much is known about
their statistical properties. On the other hand, the least squares estimators,
although loosing some of their optimal properties, are easy to evaluate and
keep some important properties, namely, they are unbiased, consistent, their
theoretical variances are known and may be obtained from the sample trough
consistent estimators. Consequently, under the assumption of normality, it
is possible to derive asymptotic tests and confidence intervals for the
regression parameters. However, under some typical cases of the design matrix
X, (like polynomial trends, seasonality or trigonometric polynomials) we will
show that the Maximum Likelihood and Least Squares estimators are
asymptotically equivalent and, in such cases, it is possible to prove
optimality properties. In particular, it is possible to derive optimal
asymptotic tests for the linear hypothesis. An application of these methods
will be made to a series of monthly average temperature measurements in
Lisbon, from January 1856 to December 1999, trough the use of a model that
includes trend and seasonality.
About the Speaker
Teresa Alpuim is a professor in the Department of Statistics and
Operations Research at the University of Lisbon, Portugal. She received
her Ph. D. from this University in 1989 working in the area of Extreme Value
Theory. Prof. Teresa Alpuim's present research interests
include time series analysis, linear models and regression analysis, spatial
statistics and the application of these methods to environmental
problems. She is a member of the Centro de Matemática e Aplicações
Fundamentais, a research unit of the University of Lisbon, where
she leads a project on Applied Stochastic Processes.
References
Background about this topic can be found in many books on regression
and time series. Some good references are
W. A. Fuller (1976) Introduction to Statistical Time Series
(chapter 9).
J. D. Hamilton (1994) Time Series Analysis.
B. L. Bowerman and R. T. O'Connor (1987) Forecasting and Time
Series: An Applied Approach.
A. Sen and M. Srivastava (1990) Regression Analysis; Theory, Methods
and Applications
