Modeling asymmetric volatility in weekly Dutch temperature data
Introduction
High-frequency temperature data, like daily or weekly data, have several characteristic features. First and most obvious, the mean temperature shows substantial seasonal variation. In The Netherlands for example, daytime temperatures range between 0 and 4°C in winter, while daytime temperatures vary from 16 to 20°C in summer. Second, the volatility of temperatures is not constant within the year but appears to follow a fairly regular seasonal pattern as well. At the beginning of Dutch winters, the standard deviation of weekly temperatures is almost twice as large as at the end of summer. This implies that temperatures are less predictable in winter than in summer.
A third feature of Dutch temperature data, documented in Tol (1996), is that large (small) absolute deviations from the mean tend to cluster. As a consequence, the conditional forecastability of temperatures also varies within summer and winter. Interestingly, the same feature holds for many high-frequency financial time series (such as daily stock market returns and interest rates). To describe this volatility clustering in empirical finance one often uses the so-called autoregressive conditionally heteroskedastic (ARCH) model put forward by Engle (1982). Over the last 15 years, this model has been the subject of intensive research; see the surveys of Bollerslev et al., 1992, Bollerslev et al., 1994 and Bera and Higgins (1993), among others. A popular extension of the basic ARCH model is the generalized ARCH (GARCH) model; see Bollerslev (1986). To capture time-varying predictability, Tol (1996) fits a GARCH model to daily Dutch temperature data in winter and summer periods, and demonstrates its usefulness for describing the volatility clustering feature of the data.
In the present paper we show that Dutch temperature data have yet another feature. This fourth property is that the impact of temperatures lower than expected on conditional volatility is different from the impact of temperatures higher than expected. Furthermore, this impact is changing over the year as well. In particular, the correlation between conditional volatility and the ‘surprise’ in the temperature is negative in winter and positive in summer. Hence, in winter (summer) temperatures lower than expected lead to larger (smaller) conditional variance than temperatures higher than expected.
The aim of this paper is to develop a time series model which is capable of describing the above-mentioned four features for weekly temperatures in The Netherlands, observed over a period of 30 years. The plan of the rest of this paper is as follows. First, in Section 2 we discuss the stylized facts of the weekly temperature data in more detail. In Section 3, we introduce a variant of the quadratic GARCH (QGARCH) model of Engle and Ng (1993) and Sentana (1995) that can capture all observed features. Section 4 presents the in-sample estimation and out-of-sample forecasting results. We estimate the proposed model and two nested versions using the first 25 years of data, while we hold out the last 5 years to evaluate their forecasting performance. Both the in- and out-of-sample evidence suggests that a model with asymmetric volatility is to be preferred. In Section 5, we conclude this paper with some remarks.
Section snippets
Weekly Dutch temperature data
In this section we document the four characteristic features of Dutch temperature data mentioned in the Section 1. The time series under scrutiny, denoted yt, is the mean weekly temperature in The Netherlands, which is constructed from the daily series analyzed in Tol (1996) by simple averaging (over 7 days). The first observation in every year is taken to be the week which starts on the first day of February. This month is halfway through the Dutch winter, and usually has the lowest
The models
In this section we introduce the model which is used to describe the various features of our weekly temperature series. Summarizing the evidence presented in Section 2 and Tol (1996), the model should allow for (i) seasonal variation in the mean, (ii) seasonal variation in the variance, (iii) volatility clustering, and (iv) changing asymmetry in the relation between last week's temperature and the volatility of this week's temperature.
Given the visual evidence in Fig. 1, Fig. 2, we decide to
Empirical results
In this section we evaluate the in-sample estimation results and the out-of-sample forecasting performance of the three competing GARCH models.
Concluding remarks
In this paper we have proposed and evaluated a nonlinear GARCH model for weekly temperatures in The Netherlands. Both the in-sample estimation results and out-of-sample forecasting performance suggest that our nonlinear GARCH model is superior to a linear GARCH model, thereby confirming visual evidence on an asymmetric relation between this week's surprise in temperature and the volatility of next week's temperature. Our model implies that temperatures lower than expected lead to larger
Acknowledgements
This paper was prepared for a special issue of Environmental Modeling and Software. We thank the editor Michael McAleer for several helpful comments and Richard Tol for making the Dutch temperature data available to us.
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