The volatility of asset returns, which is viewed as a measure of uncertainty or risk, plays a crucial role in financial decision problems such as risk management, option pricing, and portfolio management. One of the most prominent features of asset return volatility is that it changes over time. In particular, periods of hectic movements in prices alternate with periods during which prices hardly change, see Section 2.4. This characteristic feature commonly is referred to as volatility clustering. In this chapter, we discuss time series models that can be used to describe this feature. In particular, we discuss (extensions of) the class of (Generalized) AutoRegressive Conditional Heteroskedasticity [(G)ARCH] models, introduced by Engle (1982) and Bollerslev (1986).

The outline of this chapter is as follows. In Section 7.1, we discuss representations of the basic GARCH model. Several extensions are briefly reviewed in Section 7.2. We emphasize which of the stylized facts of returns on financial assets can and cannot be captured by the various models. Several aspects that are relevant for implementing GARCH models in practice are discussed in some detail in Section 7.3. This includes testing for conditional heteroskedasticity, parameter estimation, and diagnostic checking. In Section 7.4 we focus on out-of-sample forecasting. Both the consequences for forecasting the conditional mean in the presence of ARCH, as well as forecasting volatility itself are discussed.

INTRODUCTION: DIVERSITY AND EVOLUTION

In planting a garden, we celebrate diversity. There are the plants that we deliberately cultivate for our own benefit, but also the vast and often unseen array of microscopic organisms as well as the obvious birds and butterflies that we may actively encourage.

If you study nature, or are just intrigued by the diversity in the garden, it can perhaps at first be puzzling. Looking more closely at the range of diversity you soon realise there are patterns that when pieced together help to create a picture of life, which we can classify into a system that can be communicated. This pattern is the result of almost 4 billion years of organic evolution on Earth from a common ancestor that we all ultimately share. In sharing common ancestry, whether from millions of years ago or more recently, we have shared characteristics, such as the details of our cellular structure and chemistry or the form of a flower. These shared characteristics are the raw data that enable us to discover these patterns and ultimately build a classification. This provides a structure within which to name organisms and recognise their evolutionary relationships, a system that can be understood worldwide and without ambiguity. This is the science of systematics, which helps bring order and sense to this diversity, and an understanding of it is key knowledge in horticulture.

In this chapter the focus is on seasonal fluctuations in business and economic time series. The graphs in Section 2.2 suggest that seasonal fluctuations can be the dominant source of variation in a quarterly or monthly observed time series, once we have dealt with the trend in the series. For example, the results of the auxiliary regression (2.3) for the first differences of quarterly UK consumption indicates that more than 90 percent of the variation in this series can be assigned to seasonal movements. Similar numbers are routinely found for a host of other (macro-)economic time series, see Miron (1996), among others. A second observation from the graphs in Section 2.2 is that often the seasonal patterns do not appear to be stable over time. Such evolving patterns may emerge because the time series behavior in one or a few seasons changes, while it may also be that total seasonal variation changes as time passes by.

There are two different, and in fact opposing, attitudes towards seasonality. The first is to consider seasonality as a ‘nuisance’, which is not of much interest in itself and only complicates the analysis of other relevant time series features, such as the trend and nonlinearity. Put differently, seasonality is regarded as a form of data contamination, suggesting that seasonal fluctuations should be removed prior to any further analysis of the time series. This is the rationale behind so-called seasonal adjustment procedures.

For many economic time series variables it can happen that one or more observations are markedly different from the other observations. This often is due to the occurrence of exceptional and usually unpredictable events. Such outliers occur rarely, are (often) unforecastable, and are (assumed to be) caused by exogenous influences. An illustrative example is given by the May 1968 uproar in France, which caused important macro-economic variables such as industrial production to have a much lower value than usual, see Figures 6.1 and 6.2.

As another example, the stock market crash on Monday October 19, 1987, may be considered as an extraordinary event, which gave rise to a return that is markedly different from the bulk of the data, see Figures 6.3 and 6.4. Of course, stock market crashes do happen once in a while, but the fact that automated trading programs were one of the main reasons for the ‘Black Monday’ crash to be that dramatic makes this return observation quite exceptional.

Sometimes aberrant observations are part of the process which you want to model, that is, they are in fact the most interesting observations in a time series. For example, the effect of substantial price discounts on product sales makes the low price data very informative. Other effects that can lead to outliers are more difficult to capture with a time series forecasting model. For example, again price discounts but now by competitors generally are impossible to predict.

This book is about how geophysics is used in the search for mineral deposits. It has been written with the needs of the mineral exploration geologist in mind and for the geophysicist requiring further information about data interpretation, but also for the mining engineer and other professionals, including managers, who have a need to understand geophysical techniques applied to mineral exploration. Equally we have written for students of geology, geophysics and engineering who plan to enter the minerals industry.

Present and future demands for mineral explorers include deeper exploration, more near-mine exploration and greater use of geophysics in geological mapping. This has resulted in geophysics now lying at the heart of most mineral exploration and mineral mapping programmes. We describe here modern practice in mineral geophysics, but with an emphasis on the geological application of geophysical techniques. Our aim is to provide an understanding of the physical phenomena, the acquisition and manipulation of geophysical data, and their integration and interpretation with other types of data to produce an acceptable geological model of the subsurface. We have deliberately avoided presenting older techniques and practices not used widely today, leaving descriptions of these to earlier texts. It has been our determined intention to provide descriptions in plain language without resorting to mathematical descriptions of complex physics. Only the essential formulae are used to clarify the basis of a geophysical technique or a particular point. Full use has been made of modern software in the descriptions of geophysical data processing, modelling and display techniques. The references cited emphasise those we believe suit the requirements of the exploration geologist.

INTRODUCTION

Plants provide the living elements in the garden, with their immense variety of life cycles, forms, colours, textures and scents. They add vitality and bring change, both within the cycles of the seasons and their own life cycles. Annuals may be used to provide a short buzz of colour in one growing season, whereas woody perennials may outlive the garden creators. An understanding of these life cycles is paramount in deciding how to place and use them within the garden.

The decision of which particular plants to select should be reached after considering overall factors of garden style and size, functional purposes, aesthetic value and environmental factors. These factors that come into play when designing gardens and landscapes are further discussed in Chapter 13. It is possible for the gardener to alter and ameliorate the environment in order to increase the range of plants that may be grown, or to enable them to thrive and develop more successfully. Container growing provides an extreme example, where the growing media can be selected or mixed specifically for the desired plants. Raised beds also offer an opportunity of providing suitable conditions, including enhancing the natural drainage of the selected site. Structures can be erected to enable climbers to be grown more easily; windbreaks allow less hardy plants a chance. Drainage can be improved, soil improvers and fertilisers added, or aquatic environments created.

An approach that should always be borne in mind is that of selecting plants suitable for existing site conditions. This may not always be possible, but does offer a more sustainable alternative. An appreciation of the natural habitats of plants is vital here, both considering their countries or regions of origin and the habitat they occupy within their regions. Naturalistic styles of gardening and wildlife gardens fit well with this theme.

INTRODUCTION

Once the foundations and adjustments of plants and their environment have been studied, and reflected upon, then our attention turns to the application of this knowledge to meet, or sometimes to exceed, our needs and expectations, however challenging, humble or extravagant they may be. Moving forwards with new products and ideas need not mean forgetting tried and tested methods but achieving a balance that we are comfortable with and that we know is robust enough to withstand any changes or challenges that may occur in the future, be they climatically, environmentally or financially induced.

Gardening for food must be one of the success stories of the last few years, with a massive increase in interest shown within this area by all ages and all interest groups. The allotment has come of age and has developed into a prized possession. Everyone is starting to ‘grow their own’ and being encouraged to do so by a number of the large UK charitable organisations, including the Royal Horticultural Society, the National Trust and Garden Organic. The supermarket chains are also providing support and impetus to the whole movement, underpinned by the government’s ‘Five a day’ campaign.

Remember that vegetables and fruit are very decorative and can be also used as focal points and garden features, such as hedges or garden dividers or to clothe arbours and pergolas.

Features such as these lead us into another area of public interest and leisure activity: designing gardens and landscapes, encompassing everything from ‘credit crunch’ DIY to the London 2012 Olympics. This chapter introduces or re-acquaints us with the art and skill of design, with themes and ideas influenced, adjusted or re-used from ancient times but with a whole plethora of emerging materials, both living and non-living, keeping in mind the call for minimum impact on the environment and all things to be sustainable (more of this in Chapter 18).

INTRODUCTION

Ideally, a garden should be attractive throughout the year. To achieve this requires an understanding of the factors that change with the changing seasons and how plants react to these changes. It is also important to recognise that the climate varies considerably throughout the UK, not only from north to south but also from the warmer, wetter west influenced by the Gulf Stream to the drier, more continental climate of the east. Finally, our climate is changing due to the influence of global warming and this too will affect the plants that can be grown successfully.

Seasonal interest in the ornamental garden can be maintained by choosing plants that are attractive at different times in the year. A succession of early spring bulbs (Galanthus, Crocus and Narcissus) is followed by spring-, summer- and autumn-flowering annuals, perennials, shrubs and trees. In winter, interest is maintained with evergreens, often with colourful berries: Pyracantha, Cotoneaster and Sorbus spp.; coloured bark: Prunus serrula and Acer griseum; variegated leaves, e.g. Ilex and Euonymus, or flowers, such as Hamamelis mollis. Autumn colour is another important item in the seasonally attractive garden.

INTRODUCTION

This chapter will show that growing plants in protective structures can be enormously beneficial for commercial growers and gardeners who are producing crops all year round or trying to extend the growing season in the spring and autumn. These extra crops can attract premium prices for producers, an important factor for those whose cash flow is reduced during the winter months.

The positioning, building and management of any structure requires careful planning and preparation of the site in order to maximise the benefits and justify the additional costs involved compared to growing in the open ground.

Protective structures are made from many materials and constructed in many shapes and sizes: from the home-made polythene lean-to on an allotment, a cedar wood amateur greenhouse in a private garden to a modern, fully automated aluminium glasshouse covering many hectares. Polythene and netting tunnels, conservatories, cold frames and cloches are also considered as protected structures. What they all have in common is the ability to create an environment that induces improved plant growth compared to those growing in an open environment outside.

The uses for protective growing structures are varied. Many are used to produce edible crops grown direct in the ground or modern methods can be used where plants are grown in artificial growing media and have no contact with the soil such as hydroponic and nutrient film units.

Growers can use protected structures for a wide range of crops throughout the complete production cycle or for a specific growing period, such as propagation and establishing newly potted plants. Some houseplants and cut flower crops require carefully controlled light regimes to ensure they have colour and flower at a certain time in order to be saleable for a specific market; other plants require protection from the extremes of the weather to ensure they grow to the desired size and quality within strict time periods.

The econometric analysis of economic and business time series is a major field of research and application. The last few decades have witnessed an increasing interest in both theoretical and empirical developments in constructing time series models and in their important application in forecasting. This book aims at reviewing several important developments within the context of forecasting business and economic time series.

A full-blown textbook on all aspects of time series analysis will cover thousands of pages. For example, the field of unit root analysis has expanded in the last three decades with such a pace and variation that a book only on this topic would take more pages than the current book does. This book is therefore not intended to be a survey of all that is available and that can be done in time series analysis. Obviously, such a selection comes with a cost, that is, the discussion will sometimes not be as theoretically precise as some readers would have liked. Merely, it is our purpose that the readers should be able to generate their own forecasts from time series models that adequately describe the key features of the data, to evaluate these forecasts and to come up with suggestions for possible modifications if necessary. In some interesting cases, though, we also recommend further reading. To attain this, we make a selection between all the possible routes to constructing and evaluating time series models, between all the possible estimation methods, and between all the various tests that can be used.

The final typical feature of business and economic time series that is treated in this book is non-linearity. Similar to the features of trend and seasonality discussed in Chapters 4 and 5, respectively, it is difficult to define non-linearity otherwise than in the context of a model. Loosely speaking, however, a time series may be said to be linear when the impact of a shock is (i) proportional to its size, (ii) independent of its sign (in an absolute sense), and (iii) independent of the current value of the time series. Whenever one of these three properties does not hold, a time series may be said to be non-linear. As an example, consider the quarterly US unemployment rate in Figure 2.19. For this series we observe that the average increase during recessions is larger in an absolute sense than the average decrease during expansions. This suggests that the US unemployment rate is non-linear, as positive shocks (leading to more unemployment) appear to have a larger impact than negative shocks. Alternatively, the observed asymmetric behavior of the unemployment rate suggests that the effects of a given shock depend on the prevailing state of business cycle, as they seem to differ across recessions and expansions.

The three properties of shocks mentioned above are implied characteristics of ARMA models, as discussed in Chapter 3. Consequently, by definition time series exhibiting non-linear characteristics cannot be adequately described by such models.

INTRODUCTION

The majority of plants may suffer with pests or disease outbreaks at some time or other. Integrated pest management (IPM) brings together all aspects of pest and disease control: initially using cultural techniques, biological control, environmental control and then finally selective pesticides as a backup. This has developed as a more sustainable method of pest and disease control, and as a pesticide-resistance management tool. The vast majority of UK-grown edible crops are produced within strict guidelines that have been developed between advisors, growers and the retail outlets (www.assuredproduce.co.uk). The non-governmental organisation GlobalG.A.P. (www.globalgap.org) sets voluntary standards for growth and production of many crops including edibles, ornamentals and livestock production. Both organisations specify the importance of using IPM techniques in crop production.

Controllable environmental conditions can influence pest and disease pressure by inhibiting plant pathogens and by providing more suitable conditions for biological control agents to work more effectively. Pesticide regulation in Europe and the impact due to loss of key active ingredients on agriculture and horticulture has recently been reviewed by Chandler (2008) with the conclusion that IPM should be at the centre of European Union (EU) crop protection policy.

Introduction

There is a wide variety of geophysical methods based on electrical and electromagnetic phenomena that respond to the electrical properties of the subsurface (Fig. 5.1). Some are passive survey methods that make measurements of naturally occurring electrical or electromagnetic fields and use rather simple survey equipment. Others are active methods that transmit a signal into the subsurface and use sophisticated multichannel equipment to derive multiple parameters related to the electrical properties of the subsurface. EM and electrical surveys are routinely undertaken on and below the ground surface, but electrical surveys require contact with the ground so only EM measurements are possible from the air.

Survey methods that involve the measurement of electrical potentials (see Sections 5.5 and 5.6), associated with the flow of subsurface current, by direct electrical contact with the ground, are collectively known as electrical methods. These include the self-potential (SP), resistivity, induced polarisation (IP) and applied potential (AP) methods. With the exception of the SP method which measures natural potential, all the others depend on the electrical transmission of the current into the ground. Electromagnetic (EM) methods use the phenomenon of electromagnetic induction (see Section 5.7) to create the subsurface current flow and measure the magnetic fields associated with it. A variant of the EM method uses high-(radio and radar) frequency electromagnetic waves (see online Appendix 5) with the results resembling the seismic reflection data described in Chapter 6.