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of α becomes constant to a few decimal places. Another variation that is available in some computer packages is to estimate α and β0 to β1 simultaneously using maximum likelihood.

The validity of this type of approach depends on the simple model εt = αεt−1 + ut being reasonable to account for the autocorrelation in the regression errors. This can be assessed by looking at the correlogram for the series of residuals e1, e2, …, en calculated at step 1 of the above procedure. If the autocorrelations appear to decrease quickly to approximately zero with increasing lags, then the assumption is probably reasonable. This is the case, for example, with the correlogram for Southern Hemisphere temperatures, but is less obviously true for Northern Hemisphere temperatures.

The model εt = αεt−1 + ut is the simplest type of autoregressive model. In general these models take the form

where μ is the mean of the series, α1 to αp are constants, and εt is an error term with a constant variance that is independent of the other terms in the model. This type of model, which is sometimes called AR(p), may be reasonable for series where the value at time t depends only on the previous values of the series plus random disturbances that are accounted for by the error terms. Restrictions are required on the α values to ensure that the series is stationary, which means, in practice, that the mean, variance, and autocorrelations in the series are constant with time.

To determine the number of terms that are required in an autoregressive model, the partial autocorrelation function (PACF) is useful. The pth partial autocorrelation shows how much of the correlation in a series is accounted for by the term αp(xt−p − μ) in the model of equation (8.17), and its estimate is just the estimate of the coefficient αp.

Moving-average models are also commonly used. A time series is said to be a moving-average process of order q, MA(q), if it can be described by an equation of the form

where the values of z1, z2, …, zt are random values from a distribution with mean zero and constant variance. Such models are useful when the autocorrelation in a series drops to close to zero for lags of more than q.

Mixed autoregressive-moving average (ARMA) models combine the features of equations (8.17) and (8.18). Thus an ARMA(p,q) model takes the form

with the terms defined as before. A further generalization is to integrated autoregressive moving-average models (ARIMA), where differences of the original series are taken before the ARMA model is assumed. The usual

reason for this is to remove a trend in the series. Taking differences once removes a linear trend, taking differences twice removes a quadratic trend, and so on. Special methods for accounting for seasonal effects are also available with these models.

Fitting these relatively complicated time series to data is not difficult, as many statistical packages include an ARIMA option, which can be used either with this very general model, or with the component parts such as auto­ regression. Using and interpreting the results correctly is another matter, and with important time series, it is probably best to get the advice of an expert.

Example 8.3:  Temperatures of a Dunedin Stream, 1989 to 1997

For an example of allowing for autocorrelation with a regression model, consider again the monthly temperature readings for a stream in Dunedin, New Zealand. These are plotted in Figure 8.2, and also provided in Table 8.3.

The model that will be assumed for these data is similar to that given by equation (8.12), except that there is a polynomial trend, and no exogenous variable. Thus

where Yt is the observation at time t measured in months from 1 in January 1989 to 108 for December 1997; Mkt is a month indicator that is 1 when the observation is in month k, where k goes from 1 for January to 12 for December; and εt is a random error term.

This model was fitted to the data by maximum likelihood using a standard statistical package, assuming the existence of first-order autocorrelation in the error terms so that

Table 8.3

Monthly Temperatures (°C) for a Stream in Dunedin, New Zealand, 1989–1997

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8.8  Forecasting

There are many reasons for carrying out time series analysis. These include an interest in understanding the structure of the series, deciding whether there is evidence of a significant trend, or deciding whether the mean of the series has changed at some point in time due to some outside influence. Often, however, the analysis is conducted in order to be able to make sensible forecasts of future values of the series, preferably with some idea of the size of the forecasting errors that are likely to occur.

Forecasting of a time series using statistical methods involves extrapolating from the known past and present values into the future. Extrapolation is always dangerous, so anyone who attempts this can expect to be seriously wrong at times. With time series, the problem is that the model may change for some unexpected reason in the future. Or, indeed, the model may be wrong anyway, although it fits the existing data quite well. Actually, models will seldom be exactly correct anyway. Hence, extrapolating an approximate model into the future will probably lead to larger forecasting errors than the model itself suggests. Having said all this, the fact is that it is sometimes very useful to have some idea of what the future values of a time series are likely to be. There is therefore a large amount of literature on this topic.

Some of the methods that are often used, and are available in standard statistical packages, include the following:

1.A trend curve is fitted to the available data and then extrapolated into the future. The trend curve might in this case be linear, a polynomial, or perhaps some more complicated function.

2.Exponential smoothing is used, where the estimate of the mean of a series at time t is the weighted average of past values given by

xˆt = αxt + α(1 − α)xt−1 + α(1 − α)2xt−2 + … + α(1 − α)nxt−n

or equivalently,

where α is a smoothing constant with a value between 0 and 1. The value of α can be chosen by minimizing the prediction errors for the existing data, and equation (8.23) can then be used to predict into the future, with the actual value of xt replaced by its estimate from the equation. Exponential smoothing can be extended to situations with trend and seasonal variation using something called the HoltWinters procedure (Chatfield 2003).

3.An ARIMA model (Section 8.6) can be fitted to the data and then extrapolated forward. This requires the identification of an appropriate model, which is then predicted forward in time. There are some statistical packages that attempt to do this completely automatically.

It would be nice to be able to say which of these methods is generally best. However, this seems impossible because it depends upon the particular circumstances. A detailed discussion of the matters that need to be considered is given by Chatfield (2003).

8.9  Chapter Summary

•Time series analysis may be important because it gives a guide to the underlying process producing a series. It may be necessary to know whether apparent trends and changes in the mean of a series are real, to remove seasonal variation in order to estimate the under­ lying changes in a series or to forecast the future values of a series.

•The components that often occur in time series are a trend (a tendency to increase or decrease), seasonal variation within the calendar year, other cyclic variation, excursions away from the overall mean for relatively short periods of time, and random variation. These components are displayed in a number of examples that are presented.

•Serial correlation coefficients measure the tendency for observations to be similar or different when they are different distances apart in time. If this tendency exists, then it is also called autocorrelation. A plot of the serial correlation coefficients for different distances apart (lags) is called a correlogram.

•A random time series is one in which each of the observations is an independent random value from the same distribution. There are a number of standard nonparametric tests for randomness. The runs above and below the median, the sign test, and the runs up and down test are described and illustrated on a time series of minimum July temperatures in Uppsala from 1900 to 1981. None of the tests give evidence of nonrandomness.

•The change-point problem involves determining whether there is any evidence of a change in the mean of a time series, without knowing when the change may have occurred. This requires a proper allowance for multiple testing if all possible times of change are considered.

•The detection of trend in an environmental time series is a common problem. In testing for trend, it is important to decide what time scale is important.

•When serial correlation is negligible, regression analysis is a useful tool for detecting trend. A regression model is set up that includes an allowance for trend, and the significance of the estimated trend is tested using the usual regression methods. The Durbin-Watson test can be used to see whether there is serial correlation in the regression

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residuals. If there is evidence of serial correlation, then the analysis can be modified to allow for this.

•The Mann-Kendall test is a test for monotonic trend in a series when there is no serial correlation. It can be modified to accommodate seasonality in the series being tested, with a correction for serial correlation between seasons if necessary.

•Some more complicated approaches to time series modeling are described: regression models with serially correlated residuals, autoregressive (AR) models, moving average (MA) models, mixed autoregressive­-moving average (ARMA) models, and integrated auto­ regressive moving average (ARIMA) models.

•Frequency domain analysis (spectral analysis) is briefly described.

•Methods for forecasting the future values of a time series (extrapolating a fitted trend, exponential smoothing, and the use of ARIMA models) are briefly described.

Exercises

Exercise 8.1

The data shown in Table 8.6 are pH values read at quarterly intervals for the years 1980 to 1989 for a lake. Carry out the seasonal Mann-Kendall test to see whether there is any evidence of trend in the pH values, and report your conclusions.

Exercise 8.2

Fit a regression model to the data in Table 8.6 to test for trend, using the approach shown in Example 8.2, and report your conclusions. Allow for seasonal effects by treating the four quarters of the year as the four levels

Table 8.6

Quarterly pH Values for a Lake