This volume considers Rome during Late Antiquity and the Middle Ages by documenting continuity and innovation in the city's artistic, cultural, economic, intellectual, and ritual life. Admitting the influential legacy of antiquity, the essays presented here shift discourse away from episodes of renewal in Rome so as to shed light on cultural stability and the emergence of new trends, including the creative responses to adversity. To be sure, the past exerted a pervasive influence on medieval Rome. Despite ample evidence that urban population levels and governing institutions suffered after the fifth century CE, the political and social turmoil of these times frequently sparked innovation. Turning toward the cultural changes of post-classical and medieval Rome without categorizing them as desperate measures, this collection of essays valorizes the city's resilience.

The story of renewal hinges on the concept of decline; yet people feared Rome's decay long before Alaric's invasion in 410 CE and the city's fall that supposedly ensued. Notices of the city's dented pride appeared already during the imperial period, when authors sometimes pondered why the wounds appeared to have been self-inflicted. Tacitus, for example, distinguished between the capital's natural tragedies and those sparked by urban unrest by noting that Rome ‘was devastated by fires; the most ancient shrines were destroyed and the Capitol itself was burnt at the hands of the Roman people.’ Rome's integrity as an imperial capital, already vulnerable in the first century CE, began to be deemed as particularly threatened in the wake Alaric's sack of 410. For instance, Jerome perceived the assault on the city as a portent of the empire's demise. ‘The head of the Roman empire was hacked off, and, to speak more truly, the entire world perished together with that one city,’ wrote Jerome in a play on words that relies on the similarity between urbs (city) and orbis (world). Jerome's prediction of the end times from his commentary on the prophet Ezekiel drew upon his readings in the Apocalypse of John. As an exile in Palestine, Jerome railed against emperor Honorius's military policies in the western empire while advocating for the ascetic practices of elite women in Rome, whose virtues appeared as if a bulwark against invasions. Plausibly, Jerome sounded the apocalyptic alarm to galvanize potential supporters for his cause after the demise of his main patron, Pammachius.

Recent studies in the post-classical and medieval history of Rome have focused attention upon historical memory with great nuance, exploring how perceptions of the past generated innovations and new expressions of identity. This volume considers Rome's cultural developments during Late Antiquity and the Middle Ages by shifting discourse away from the concepts of decline and renewal. There is no doubt that ancient Rome generated a consequential afterlife, since long-lasting reverberations originating from the imperial era shaped the medieval city. Further, there is clear evidence of post-classical Rome suffering from terrible hardships, and there is no denying the gravity of problems that arose throughout the Middle Ages. Yet Romans responded productively and inventively to both changing conditions and dire circumstances. They formulated new modes of literary production, new responses to food shortages, new approaches to scholarly thought, new formats of music and liturgies, and new schemes for presenting monumental artworks during the periods following the so-called fall of Rome. The essays included here by no means ignore periods of renewal, nor do we dispose of the indications of decline altogether; we simply wish to call attention to the importance of cultural stability and productive responses to evolving circumstances. In sum, the authors included in this volume take the post-classical city of Rome out of the shadow of its classical past to explore its continuing transformations and enduring creativity.

Most of the research in this volume stems from a series of scholarly talks hosted by the Marco Institute for Medieval and Renaissance Studies at the University of Tennessee, Knoxville, each of which questioned the theme of renewal by analyzing an aspect of Rome's culture during the Middle Ages and the Renaissance. The Institute's support brought together scholars who considered the ways in which post-classical Rome turned away from classical norms and how Romans responded productively to harsh circumstances. Grateful appreciation is due to all who were involved for their various contributions, including helpful feedback and excellent questions.

The editors would like to thank the staff of the Marco Institute for Medieval and Renaissance Studies for generous assistance facilitating work on this publication. Thomas Burman, who now directs the Medieval Institute at Notre Dame, previously served and offered terrific guidance as the Riggsby Director of the Marco Institute. Jay Rubenstein, subsequently appointed to the same post, provided beneficial support and expert oversight.

The examples in Chapter 2 demonstrate that many time series in economics and business have a trending pattern, where for macroeconomic time series such trends typically move upwards. Although many practitioners would be able to indicate roughly what a trend is (“a general tendency for a variable to increase or decrease over time”), a formal definition of a trend cannot be given otherwise than in the context of a model. Put differently, only after we have agreed upon a time series model that (presumably) describes the data at hand, we can define a trend within this framework and discuss the “trend properties” of the time series in a meaningful way. In this book, the focus is on ARMA-type time series models, and hence the current chapter deals with trends within this model class.

For several reasons it is important to investigate the appropriate formulation of the trend in a time series prior to putting effort into modeling other data features and forecasting. First and foremost, the trend will dominate long-run out-of-sample forecasts, although also short-run forecasts can be affected, as we will see in Section 4.4. An inadequate specification of the trend possibly leads to forecasts that are biased or inaccurate in other ways. Second, a time series that displays a trend is nonstationary, in the sense that it does not have a constant mean. In addition, depending on the nature of the trend, the unconditional variance of the series is not constant over time but increases with every new observation.

Univariate time series models, as discussed in the previous chapters, can be very useful for descriptive analysis and for out-of-sample forecasting. Such univariate models are restrictive, however, in the sense that they rely purely on the history of the time series itself and do not try to exploit information present in other variables. Given that often economic variables are closely related with other variables, such information may be potentially useful and using it may lead to more realistic descriptions of the behavior of economic variables and also to more accurate out-of-sample forecasts. In fact, ignoring relationships with other variables may give rise to complications in the specification of an appropriate univariate time series model. It may be that the empirical specification of a univariate model is hampered by outliers and structural shifts, which may be attributed to one or more other variables than the yt variable under consideration. For example, the change in the trend of US industrial production around 1979.4 (see Figure 2.3) may be caused by the large oil price increase around that time. Additionally, the same oil price variable may be responsible for the large negative growth in production in 1974/1975, which seems to correspond to an innovation outlier. In other words, it may be worthwhile to include an oil price variable in a model for US industrial production to render it more realistic and more accurate.

This book concerns the construction of time series models for describing the dynamic properties of economic variables and for out-of-sample forecasting. The economic variables can originate from various subject areas in economics and business, including macro-economics, finance, and marketing. Specific examples of time series of interest are inflation rates, unemployment rates, stock market returns, and market shares. Out-of-sample forecasts for such variables are often needed to set policy targets. For example, the forecast for next year's inflation rate can lead to a change in the monetary policy of a central bank. A forecast of a company's market share in the next few months may lead to changes in the allocation of advertising budgets. The models in this book can be called econometric time series models because we use econometric methods for analysis.

Time series variables can display a wide variety of patterns. Typically, macroeconomic aggregates such as industrial production, consumption, and wages show an upward trending pattern. Industrial production, tourism expenditures, and retail sales, among many others, display a pronounced seasonal pattern, that is, tourism spending is usually largest during the summer and retail sales tend to peak around Christmas. Another feature is that certain observations on economic data look aberrant in the sense that they occur rarely and deviate strongly from the typical behavior of the variable. For example, if new car registrations are almost zero in a certain month because of a computer breakdown, this does not reflect the true sales of new cars.

In this chapter the focus is on key features of business and economic time series. It also serves to introduce several of the empirical time series that will be used throughout this book as running examples.

The five key features of economic and business data that we consider are (i) trends, (ii) seasonality, (iii) influential data points, (iv) time-varying (conditional) variance (conditional heteroskedasticity) and (v) non-linearity. Typically, an economic time series displays at least one, but usually two or more of these features. To keep matters simple, however, in this chapter each series is analyzed for only one of these five features at a time. In later chapters, the various possible models for each of these features will sometimes be combined to illustrate the practice of time series modeling.

In this chapter, each of the five data features will be illuminated using simple regression-based calculations. This should not imply that these regression models are the best we can do. They are merely helpful tools to demonstrate the properties of the data. A second important tool in this chapter is graphical analysis. In many cases it is already quite helpful just to put the data in a graph with the values of the observations (or some transformation thereof) on the vertical axis and time on the horizontal axis. However, in case of many observations or of a time series with large variance, it may sometimes be more insightful to construct scatter or correlation plots.

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.