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The Philosophy of Economic Forecasting
THE PHILOSOPHY OF ECONOMIC FORECASTING Clive W. J. Granger
1
INTRODUCTION
I believe that it is accurate to say that the typical economic forecaster does not ask herself questions about the underlying philosophy concerning what is being attempted. In fact, most could not define “philosophy” or would even attempt to do so. This is not necessarily a sign of intellectual weakness as even philosophers have difficulty defining philosophy and its objectives. “One might say that philosophy is what philosophers characteristically do” is a quotation from an article on Philosophy in “The Encyclopedia of Philosophy,” Macmillan, (1996). The same article stated that an earlier article on the same topic in the original version of the Encyclopedia “identified the distinctive feature of philosophy as its being a critical discussion of critical discussion.” It also says that “Philosophy as a commonly characterized is a multifaceted discipline that resists simple characterization.” One obvious approach is to ask a philosopher what he does but typically it will be very difficult to understand the answer as, like any other discipline, it has its own distinct terminology. However, philosophers and others will often ask interesting and penetrating questions that deserve the attention of forecasters, both to improve their understanding of what they are doing and possibly, in consequence, improve the quality of their output. Forecasting is a very ancient occupation. The earliest groups of people who gathered together in small villages would be interesting in forecasting the seasons to know when to plant crops, when to move camp to where herds would be passing by, or when the salmon were running. Then, as now, forecasting was largely a practical process, involved with statements that could produce decisions that improved the economic well-being of the group involved. 2 PREREQUISITES For convenience, it is important at this juncture to make two fundamental points that will be needed in what follows. The first is that I will consider the “economy” to consist of all the decision makers involved such as the consumers, investors, employers, and government policy makers as well as the various economic institutions, Handbook of the Philosophy of Science. Volume 13: Philosophy of Economics. Volume editor: Uskali M¨ aki. General editors: Dov M. Gabbay, Paul Thagard and John Woods. c 2012 Elsevier BV. All rights reserved.
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such as the banks, corporations, trusts, and so forth. This will be called the “actual economy.” In the literature prepared by philosophers the economy is usually taken to be the same as the constructs considered by economic theorists. This I will call the “theoretical economy.” The economic theorists construct models based on sets of assumptions and perceived rational behavior by decision makers in an attempt to represent a simplified form of the actual economy. Sometimes these models are successful, sometimes less so according to data analysis. It is certainly true to say that philosophers in their writings about economics often confuse the theoretical economy with the actual economy. It is probably also correct to say that most decision makers in the actual economy are unaware of the results in the theoretical economy, and they do not suffer very much from this, although this could be less clear in the area of finance. The second fundamental point is that it will be necessary to distinguish between “forecasting” and “prediction.” Forecasting will be limited to the extrapolations based on empirical models or data exploration, whereas a prediction will be formed from a theoretical model. These differences are further explored in Section 3. It will be helpful to use what has become the standardize “set-up” for forecasting in recent years. Let Xt be a time series measured at equal intervals of time, such as each minute, day, or month as appropriate. The requirement that the series is recorded at equal time intervals can be relaxed but is technically more difficult. The fact that months are not strictly equal in length is merely pedantic and has been considered in the literature. It will be assumed that the series is not measured continuously in time, which is true in the actual economy, but continuous time is often assumed in the theoretical economy. This assumption will be discussed further below. Xt will usually be taken to be a single series but it could be a vector, where Xt is a particular economic variable such as a price, unemployment, or production. The current moment of time is denoted n indicating “now,” so that Xn is the current value of the X series. At time n there is available some empirical data including past and present values of Xt , which is denoted XPn , so that XPn = {Xn , Xn−1 , Xn−2 , . . . , }. In practice there are only a finite number of past values available, but the impact of this fact is usually considered to be small. Other data series and their past will also be available, denoted as W Pn , where W is usually a vector. It is usual to consider the information available at time n which is used to form a forecast. An “information set” In could consist, for example, of XPn and W Pn . A wider information set Jn could consist of the contents of In and also Y Pn , where Yt is another series. We may be interested to know if the Y series contains useful information that is not in the other series, as will be seen later. The information sets can also include non-numerical information, including opinions or constraints from economic theories. When forecasting it is usual to have a specific horizon in mind, so the objective will be taken to look ahead h steps, to the value of Xn+h . As the future may well be uncertain, otherwise one would not be forecasting, Xn+h will be a random variable and thus can be described by a conditional distribution function Fx,n,h (In ) =
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Prob (Xn+h < x | In ) or its associated density function fx,n,h (In ), which is the derivative of Fx . Here the forecast will be a density function dependent on the information set used and the horizon selected. It is important to note that even if h is fixed, as n changes the information set will alter and so the predictive density will also change. Thus forecasts will change with information, with horizon, and with time. Historically, it was often too difficult to provide a predictive distribution and so simpler statistics were used instead. Typically only the forecast of the mean, denoted mn,h , was given, such as “unemployment rate next month will be 7%” or “inflation will be 5%.” Such figures only capture the middle of the predictive distribution and without some idea of the width and shape of the distribution sensible decisions are difficult to achieve. Statisticians would strongly recommend providing measures of uncertainty, such as the variance or the 95% confidence interval, although these were often difficult to interpret or were so wide that they were embarrassing! This basic setup is quite general, as it can cover several important special cases. For example, Xt can be constrained in some way, such as being positive or bounded to be between zero and one in value. It also included “event forecasts,” such as a volcano will erupt, or there will be a financial crisis, or a business cycle downturn. Such events are captured with a zero-one random variable, with zero for when the event does not occur, and the predictive distribution will consist of just a probability pn,h of the event happening and probability 1 − pn,h of it not occurring. The probability pn,h will be a function of the information set used and so will evolve over time. It might be said that economic historians just look backwards and that some economist just look sideways, but forecasters have to look back to be able to look forward. They have to select the useful pieces of information from the past from the mass of information that is available from which to form their forecasts It is clear that many forecasts could be made and so a process of evaluation is essential to learn what seems to be helpful to decision makers and what does not. At time n + h there will exist the observed values of the series Xn+h and a forecast of this quantity that was made at time n, i.e., the predictive density f (x, n, h(In )) based on the information set In . The error series, which is defined as the actual minus the forecast will have density f (Xn+h − x, n, h(In )). At time n + h all components of this density are known and so all of the properties of the error can be obtained and the evaluation can be conducted on them. A standard measure is to record the average “likelihood” of the actual, given by the average of f (Xn+h , n, h(In )) over some appropriate sample. The expectation of the error density, defined as Z ∞ xf (x, n) −∞
produces the “point forecast error” = actual - point forecast, or in notation en,h = Xn+h − mn,h . Traditionally, and certainly until the end of the twentieth century,
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forecast evaluation concentrated on these errors. If the forecast is to be used by a decision make, the cost of making an incorrect forecast might be measure by a “cost function ce ,” so that an error of the amount e results in a cost ce . Many aspects of economic forecasting, including the evaluation of forecasts, can be found in “ The Handbook of Economic Forecasting ” edited by G. Elliott, C.W.J. Granger, and A. Timmermann, Elsevier, 2006. A critical aspect of forecasting is the choice of the information set, which will include data from the past and present, any available stated plans or policies for the future and include changes in the structure of the economy, laws and institutions. A forecast will be forward looking partially based on backward information, but not entirely so. The proposition that a society, and thus an economy, should have some stability over time and some momentum together with understandable changes, is not a very surprising one and forms the basis of many forecasts. Thus, potentially at least, we learn from the lessons of history when forecasting. 3 FORECASTING, PREDICTION, AND ECONOMICS Much of the actual, as well as theoretical, economy is forward looking. Decision makers make decisions now that will have impacts in the future. Investors decide where to invest now and wait to see what return occurs; consumers buy now but consume over the net few hours or days; a house-buyer decides now and uses the purchase over several years; an employer agrees to hire a worker and makes use of his or her skill over some later period. Essentially the decision maker has to forecast the future consequences of the decision. Further, policy making is clearly about the future. It can be thought of as a number of alternative conditional forecasts. If the agency does one thing, we expect the future to be like this. But if the agency does something else, the future will be like that. The policy make chooses the better, expected future. It is thus seen that major components of both macro- and micro-economics will involve forecasting. The extensions to international economics and to finance are obvious and in fact both of the actual economies in these areas are major consumers of forecasts, sometimes called “expectations” in the media. At this point it is important to carefully distinguish between “forecasting” as discussed in Section 2 above, and “prediction.” Here prediction will mean taking a model, usually based on economic theory and assuming that it is correct, then drawing implications from it about the behavior of the economy, at least the theoretical one and possibly the actual one. Some example are given below. Prediction occurs when there is a model of the form Xt = a + bYt + et where X and Y are a pair of specific economic variables, e is a residual and the coefficients a and b have been estimated or chosen in some way. The model could be theoretical or empirical in origin. A simple example might have Y a
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basic interest rate and X inflation. A prediction of X is formed by inserting a particular value for Y into the equation, and then assuming that the model is correct. If, in the model Xt is replaced by Xt+1 then putting an observed value for Yt into the model will provide a forecasting of the next value of X. This paper will usually be considering forecasts, although usually of a more sophisticated form. A simple example of a prediction comes from price theory and says that if a company raises prices then the consequence is a reduction in sales. This is the kind of prediction that does not depend on a sophisticated theory and is observed in the actual economy to be usually true, but not always. These exceptions can be explained both using theory and empirical arguments. Blaug [1980, 2nd edition, 1992, pp. 151] suggests several other similar examples: “an increase in demand leads to a rise in both output and product prices,” “a lump sum tax on business profits will have no effect on output,” and “a rise in money wages causes a fall in employment.” Note that all of these predictions are non-specific, the amount of the change and the timing are not given. Usually forecasts would be more specific. On occasions a prediction from the theoretical economy about that economy can be evaluated by the actual economy. For example, Blaug [as before] states that the theory of the firm “predicts unequivocally that a profit maximizing firm in a perfectly competitive market will not advertise: it has no incentive to do so because it faces a perfectly elastic demand curve and can sell all that it can produce.” As many firms do advertise their differentiated products, then the assumptions upon which the theory is based have to be incorrect. As will be seen, prediction plays an important role in the topic of model evaluation and that forecasting becomes embroiled in the discussion. 4 THE PURPOSE OF ECONOMICS It is useful to know the purpose for some body of work as it give a starting point for evaluation. For some disciplines such as medicine, psychology, and law, the purpose is to be helpful to their clients. However, for areas such as history or mathematics, which are certainly important and distinguished, being immediately helpful is unlikely to be the suggested purpose. The are a number of philosopher/economists such as Lawson [1997], Redman [1991], and Blaug [1980, 2nd edition, 1992] who consider many aspects of the interaction between the two areas, and the latter two take definite views on the purpose. Blaug [1980, 2nd edition, 1992, p. 246] states “the central aim of economics is to predict and not merely to understand” although he goes on to mix up forecasting and prediction. His statement should be taken to mean that prediction should be used to evaluate an economic theory. Redman [1992, p. 120] quotes Worswick [1972] “the idea of economics as positive science makes predictability the test of its performance, the prediction of relationships is situations not previously observed, as well as the prediction of future events, which in some ways is the acid test.” This statement covers prediction in cross-sectional situations where everything occurs at the same
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time. In this paper only future events will be considered. Of course, prediction is not the purpose of economics, even of theoretical economics. It is quite easy to formulate theories, one starts with a group of reasonable assumptions, adds a few generally accepted economic concepts such as a rational market, include some institutional constraints, and then prove theorems about the “economy” so derived. The theory can be simple or it can be very complicated but it usually will not be unique. The question of evaluation will have to ask if any of the models is adequate, in some fashion, and then which is the best. The evaluation can perhaps be based on prediction and on falsification. Redman [1992, p. 24] also quotes Kuhn [1970] about a demarcation criterion without which no field is potentially a science: “(1) concrete predictions must emerge from the practice of the field; (2) for a subclass of phenomena, whatever passes for predictive success must be achieved; (3) predictive techniques must have roots in a theory which, however metaphysical, simultaneously justifies them, explains their limited success, and suggests means for their improvement in both precision and scope. Finally, the improvement of predictive technique must be a challenging task, demanding on occasions the very highest measure of talent and devotion.” Unfortunately, most of the early discussions of topics such as “the purpose of economics” and “is economics a science?” are based on viewpoints that are now generally considered outmoded. The early position taken was that physics (of a traditional form) was the standard against which one measured a field being a science. The methodology of traditional physics was the one to use for comparison. In this area the world was deterministic and not stochastic, and experiments should get the correct answer if properly conducted, or if repeated often enough the average will certainly tend to the correct value. Certainly the old theory had its clear successes, as Blaug [1992, p. 7] points out “who can deny the extraordinary predictive power of Newtonian theory particularly after the confirmation in 1758 of Edmond Halley’s prediction of the return of ‘Halley’s comet,’ topped in 1846 by Leverrier’s use of the inverse-square law to predict the existence of a hitherto unknown planet, Neptune, from the observed aberrations in the orbit of Uranus.” Since then the same theory has been extended to forecast the time and height of tides on virtually every beach in the world from now and for many years into the future. Of course all such forecasts are based on a set of assumptions and if the change so will the forecast, certainly a tsunami would change tides in the short run and if the Moon broke into two pieces the tide would be changed forever. As the twentieth century evolved, with the advent of quantum physics and areas such as meteorology and oceanography, the definition of a science changed together with the appropriate methodology. The move from dealing with just inanimate objects, as with classical physics, to animate ones as with medicine and biology, greatly extends the range of the subjects being considered and of the topic “what is a science?” The next extension is to objects (or subjects) that are individual decision makers, covered by areas such as psychology, economics, and political science, although
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some parts of biology will also fall into this category. As the type of objective being studied changes it is reasonable to expect that the methodology will evolve as will the objective of the subject area. I do not think that economists have been involved in a careful enough discussion about what is the purpose of economics. My personal view comes from the observation made above that the economy consists of many types of decision makers. It follow then that the objective of economics should be to help decision makers make better decisions. It is fairly easy to understand now to do this in some parts of economics such as finance and macro-economics, but much more difficult in those areas where decisions are not emphasized, such as parts of cross-section, micro- and theoretical economics. 5 PHILOSOPHICAL QUESTIONS A number of questions will now be considered that may be thought of as having philosophical origins.
5.1
Is the Economy Deterministic or Stochastic?
A system, such as an economy, is deterministic if its progress can be fully described without the use of probabilities (other than zero and one). Know the past will completely determine the immediate future, and then by iteration, all of the future. The basic idea that the universe could then be deterministic comes from classical physics, which has known, simple, and unchanging laws that operate everywhere and at all times. As stated before, this allows the positions of the planets to be known exactly in the future, for example, and thus the timing of tides anywhere on earth can be determined, provided that the basic assumptions continue to hold. Historians may also subscribe to their domain of interest being deterministic, as the past cannot be changed, although interpretations about why things happen are not constant: the reason for the decline of the Roman Empire or for the occurrence of the Black Death changes every decade or so. Although history is certainly unchanging, what we know about it is by no means constant and so interpretation can evolve. The theoretical economy is often assumed to be deterministic as results are usually easier to obtain under this assumptions. It should be pointed out that for a theoretical economy, ANY basic assumption can be made. the only requirement is that the participants in this economy behave in an “economically sensible” fashion given these assumptions. There is no requirement that these economies be realistic and there can exist several of them about the same topic but with different assumptions. In contrast, a stochastic system is one that has to use probabilistic concepts in describing its progress, such as “if today we are at A, then tomorrow we will be at B with probability 0.4, and at C with probability 0.6.” With this definition, anything that is not deterministic must be stochastic. However, it is more convenient to consider processes that are “purely stochastic,” which would rapidly collapse to
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a constant (or possibly a simple trend) if the stochastic component was switched off, and then to have systems which consists of both deterministic and purely stochastic components. Suppose that we could invent a measure Q which takes values between zero and one, with Q = 0 corresponding to deterministic and Q = 1 to purely stochastic. Then classical physics would have Q = 0, quantum physics Q = e, where e is a small positive number, Q = 0.3 for meteorology, and Q = 0.6 for macroeconomics with Q = 0.9 for much of finance. I suppose that history has Q = 0, but that is open to debate. Only if Q = 0 can you assume that perfect forecasting is possible, otherwise forecast errors will occur. However, if errors do occur it may be because we are not good at forecasting rather than because of the inherent stochastics. A complicating issue is the existence of a class of mathematical iterative processes known as “chaos” or chaotic A very simple example (known as the logistic map) is Xt+1 = a(Xt (1 − Xt )) with the starting value X1 chosen in the region (0,1) and the parameter a chosen to be in the region 3.6 to 4.0. Data generated from this map has the properties of white noise, with a constant mean and variance and all serial correlations that are zero, when estimated. However, the generated data does not have the properties of an independent series, although this is sometimes claimed in the chaos literature, as powers of the series are not serially uncorrelated. However, more complicated maps than the logistic shown above can generate series that have more of the properties of an independent series and will also be chaotic. Some of these maps are used in computer programs to generate “artificial random numbers” that are used in various statistical procedures such as the bootstrap and simulations. Provided the sample sizes used are not too large, these artificial series will usually work well and appear to be random. Nevertheless, a long enough series will fail a test of randomness and will suggest that they are generated from a deterministic map. One can then go and devise a yet more complicated map which will produce series that are more difficult to distinguish from strictly random, although with enough data they can be, in theory at least. It is seen that the division between deterministic white chaos and a truly stochastic random process is becoming very unclear. Whether there is any difference in the limit is truly a philosophical question; it is deep and difficult but its solution has little practical relevance. If a theoretical economy is deterministic it is due to the assumptions being used. If an actual economy is deterministic, then this is a basic property of the economy. It is possible for part of an economy to be deterministic, but not other parts, just as in the actual physical world the tides could be deterministic, but the temperature of the water involved could be stochastic. An example is economics would be the month of the year with the greatest store sales in a European country (December) and the amount sold in that month. Limitations on the causal relationships between the deterministic and stochastic parts are discussed later. It might be worth pointing out that a totally deterministic society is very boring as it is highly forecastable, at least in the short run. There will be no horse racing
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or gambling; there is little point in playing any sporting contest as everyone know the outcome and every political election can be very short and inexpensive. The winner is already known. These results require the generating map to be either know or well approximated by a neural network analysis. The degradation of the forecasts as the horizon increases is due to the accumulated effects of a round-off error. A very well known economic theorist, Sir John Hicks, stated [Hicks, 1979] that economics is “on the edge of science and on the edge of history” as it tries to use the techniques of science but its subject matter behaves differently. Hicks [1986] says “If a scientific theory is good, it is good now and would have been good a thousand years ago ... but the aspects of economic life which we need to select in order to make useful theories can be different at different times.” (Quotes from [Redman, 1991, p. 106].) Chaos theory is designed for a deterministic, unchanging physical world and has not performed well in the decision theoretic world of economics.
5.2
Can An Economic Agent Have Perfect Foresight?
Amongst the assumptions that are sometimes made within an economic theory for the sake of simplicity is that of “perfect foresight.” The purpose of such assumptions is to reach some immediate conclusions from the theory, and they are later dropped to see if the same conclusions hold in a more general situation. If an agent had perfect foresight, she could continually make optimum investments and accumulate considerable wealth. She would win every “game” situation as she would know her opponent’s choices. It is doubtful if every agent could be assumed to have perfect foresight as game situations would have no solutions as no one would be able to make a choice. This is not a very interesting assumption and is a totally unrealistic one. If everyone had perfect foresight the economy as we know it would cease to exist. There would be no markets as everyone would know the eventual price obtained, economic policies would not work as they are perfectly anticipated and everyone knows the impact, if any. Occasionally economic theorists who are used to making a perfect foresight assumption will criticize an economic forecaster for producing imperfect forecasts. Only a little thought produces reasons why many forecasts will be imperfect such as short-term and, even more clearly, long-term weather forecasts, as well as forecasts of a horse race. There are too many things that can occur. Sen [1986] points to two particular reasons for economics: the large number of individual decision makers involved in a typical economy (a hundred million households in the United States and the European Union, with even more in China and India), and also the many interactions between these decision makers.
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5.3 Is It Worth Making A Forecast That Is Not Perfect? In 1928 Oscar Morgenstern, who later became famous as a co-author of the first book on game theory, published in German a pamphlet on the irrelevance of economic forecasts. This work has become known through the attack on it by Marget [1929]. Morgenstern essentially makes the point that economic forecasts can never be perfect as they are based on an inadequate economic theory and poor data. He states that if they are not perfect, then they cannot be used for policy purposes. His arguments relate to classical, pre-quantum physics and his policy points have been largely superseded by the development of decision making under uncertainty. Marget essentially tries to weaken the arguments proposed without going to a stochastic viewpoint. It should be noted that Morgenstern makes no mention of his position in his later books on “the accuracy of economic date” and on the “forecastability of stock market prices,” the second of which is written with me. A few writers go beyond the imperfection of economic forecasts to conclude that all such forecasts are so bad that they should be disregarded. A few may even claim that economic forecasting is not possible, although there is a difficulty with semantics. Blaug [1982, p. 158] firmly disagrees saying “If prediction of human behavior were truly impossible, if none of us could predict anything about the behavior of other people, economic life itself, not to mention theories about economic life, would be unimaginable. Not only would the total incapacity to predict economic events wipe out economic theory: it would wipe out every other type of economics, as well as all pretences of offering advice to governments and business enterprises.”
5.4 Omniscience Suppose there exists some entity that is omniscient. Following the philosophical literature I will call the entity “God” and use the pronoun “He,” without there being any implications from these choices. The fact that if God has omniscience then there are important philosophical implications is not of immediate relevance for an essay about forecasting. I will not be concerned with policy questions at this moment and certainly not free-will. By omniscience I take it to mean that God could forecast any component of the economy at any time in the future if He cared to do so. In particular, he could have perfect foresight about the behavior of every decision maker. It follows that if He were rational and utility maximizing He could quickly acquire immense wealth, and would soon be testing the theories about there being a satiation level for wealth. Of course, as God would have nothing to spend his wealth on, it is irrational to expect Him to have a standard utility function. As a forecaster, should my behavior change if I am told that there exists a God who can forecast perfectly? It would imply that if I improve my technique and gather a good enough information set, then I should be able to do almost as well. Alternatively, it could imply that I would need godly abilities and resources to
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do so well, and so should be content with much less. If I add a small positive probability that God does not exist, I am back to a stochastic economy. The idea that an all-knowing God with omniscience will know the future of us all with certainty is probably placing us at too high a level of importance. For Him to foresee the exact position and behavior of every living creature, from microbes up, on every planet under his control, is quite possibly within his computing abilities, but it is very unclear why He would want to ever undertake that task, even though it only has to be done once. We have no idea of His reasoning, requirements, and desires and we certainly do not know if His time scale is the same as the one with which we are involved, or if beings on other planets thinking vastly quicker or slower than us. Without more basic knowledge, or further assumptions, it is doubtful if the possibility of omniscience has any impact on everyday economic forecasting.
5.5
Why Cannot We Forecast Perfectly?
It is general knowledge that economic forecasts are not perfect, and this is the basis for many jokes. Sen [1986] states “It is, in fact, tempting to see the economist as the trapeze performer who tends to miss the cross bar, or as the jockey who keeps falling off his horse.” What is not clear is why the economists should be singled out. Does a patient who is sick go to a doctor one week and then complain the following week if he is not completely cured? Are there articles in the press asking why the horse-racing correspondent did not pick all the winners yesterday or why the weather forecasts do not turn out to be perfect? Sen is obviously reflecting the traditional approach, taken by Morgenstern and based on the believe (or the assumption) that economics is a science, in the old fashioned sense. However, Sen does then go on to give at least two plausible reasons for the non-perfection: the difficulty in anticipating human behavior; and aggregation or size effect. 1. Anticipation of human behavior follows from the fact that many individual decision makers are involved (who are not automatons) and that their decisions will evolve as the learn, their tastes change, their choice sets evolve, and the institutions and society changes around them. Each person can react differently to these changes. 2. The size effect comes from the fact that there are many millions of families in the typical economy, with complicated interactions. As Sen says, there are “millions of human beings each with different values, objectives, motivations, expectations, endowments, rights, means, and circumstances.” This will make aggregation difficult both for theorists and for data analysts without some simplifying assumptions. Some of these assumptions may be reasonable, but others (such as having “representative agents”) are generally thought to be not useful in practice. These are certainly important and relevant reasons that would be likely to be included in a defense by an empirical forecaster, although there, more attention
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would be paid to the likelihood of the economy being stochastic. If the level of stochasticity is high, forecasts will be imperfect. Sen was concerned with the topic “Prediction and Economics Theory” and so was considering the use of a theoretical model to provide “predictions,” which are not necessarily forecasts as mentioned earlier. They are based on the assumption that the theory is correct. He discusses the relevance of topics such as “equilibrium,” “rationality,” “maximization” and, later, the use of an assumption such as “self-choice goal” in which each act of choice of a person is the pursuit of one’s own goal (such as the maximization of utility). He finds that many of these concepts are difficult to use for prediction. Currently certain equilibrium models are being used for long-term macro forecasting, but evaluation is difficult. Of course it has to be admitted that the forecasts may be imperfect because of the incompetence of the forecasters. They may be using poor quality data sets or insufficiently sophisticated forecasting techniques, or it could just be that the computing power is insufficient for the task, as the economy is just too complicated. If these were all or some of our problems, I believe that we could expect to see improvements in forecastability, either a steady progress or a series of steps as breakthroughs occur. There has been some progress, but the variables being forecast have changed in nature and it seems that the data may have declined in quality in some important cases.
5.6 Differences in Forecastability It has been observed in the actual economy that economic variables vary in their “forecastability;” that is, the extent to which they can be forecast. For example, it is quite easy to forecast the demand for electricity at every hour tomorrow in an American city. The demand is to a very large extent determined by the particular day, by the regular pattern of activities through the day of the consumers in the city, and the forecast of the temperature for the day. Since temperature is quite easy to forecast twenty-four hours ahead, and as the use-of-electricity pattern is stable given the temperature, the forecast can be made a day ahead and will usually be very accurate. This forecast is of considerable importance to the electric utility company that supplies the electricity for the region. It is worth noting that the variable essentially consists of two components: the regular daily use patter (which changes each day); and the very complicated (possibly stochastic) weather component. At the other extreme returns from stock market prices are very difficult to forecast, as was found empirically from very early work by Bachelier [1900] and later by various statisticians considering a model called “the random walk hypothesis,” which essentially says that these returns are not forecastable. Economic theorists later stated these ideas in the “efficient market hypothesis,” which notes that if you could forecast the return from any speculative asset, you would have a “money machine” which would produce unlimited amounts of money. As such a machine is impossible, the returns cannot be forecastable.
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Naturally, most economic variables lay between these two extremes. It is generally true that anything that one can easily profit from, good forecasts are difficult to make. Commodity prices, interest rates, exchange rates, and commodity rates fall into this group. However, if an interest rate is used as a control variable, such as by the Federal Reserve Bank, The Australian National Bank, or the European Bank, it does become easier to forecast. Some variables from a stock market, such as daily volume traded, or the number of stocks advancing in a day, or even daily volatility of an individual stock are all somewhat forecastable. It is generally true that the levels of variables are much more forecastable than the corresponding changes or rates of return. For example, the level of unemployment rather than the change in it, or the price level rather than inflation. All such statements are observed properties of the actual economy as observed through the lens of the present forecasting methodology. They could change as the methodology improves. It might be noted that if the economy was deterministic, then the level of a variable and its change would both be perfectly forecastable. Economic variables are included to change in value as new and relevant information accumulates. This happens very quickly in a speculative market, quite slowly for the major variables in macroeconomics, and very slowly in population economics. In general, users of forecasts prefer higher accuracy rather than lower, but not necessarily in all cases. For example, if someone offered to forecast the date of your death, most people would prefer not to have that information even though knowledge of it would lead to more rational investment decisions.
5.7
Should Forecasts Be Rational?
The “rational expectations” revolution in macroeconomics took place in the 1970’s, but the basis of the idea and the corresponding theory was developed a decade early by Muth in 1961. It was observed that economic decision makers were being assumed to be rational and that their decisions would be influenced by forecasts or “expectations” and so these also should be rational in the sense that they should not be obviously sub-optimum. A “rational expectation” should use all the relevant information that is available and also appropriate methods of forming a forecast. A given set of forecasts could be shown to be “irrational” if they were clearly sub-optimum, as shown by a statistical test. The theory could become complicated as some of the models used to form the expectations involved expectations, so some problem usually reflects the lack of subtlety about timing of occurrences within the model. If better forecasts are easily available it would certainly be irrational to use inferior ones, but finding the very best may be too expensive in terms of searching for slightly better methods and a few useful but expensive last pieces of information. An implication of the considerations of rational expectations was that the usefulness and relevance of much government policy was thrown into doubt, at least over the long run. An unexpected change in policy could still have an immediate effect, which will continue into the future.
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Many countries have important survey efforts that try to measure the plans of industrialists about their future capital investment and employment and also consumer about their buying plans. the most successful of these seems to be the investment plans, which once started are less easy to stop without considerable cost. This possibility illustrates the fact that useful forecasts can come from a non-theory or model based approach.
5.8 How Far Can One Forecast? The question of how far one can successfully forecast is tied to the topic of evaluation. A forecasting model can be run out into the indefinite future, but the relevant question is: over what horizon are the forecasts of any value? The answer is also tied to the concept of the forecastability of various types of economic variables. Some variables are slow moving and are very predictable over long horizons, such as population growth, whereas others have virtually no possibility to forecast, such as speculative returns. A deterministic variable can be (perfectly forecast) into the indefinite future without any error. In contrast, a variable with a stochastic element will steadily accumulate the stochastic, unforecastable component until no forecastable part can be detected. In macroeconomic, the longest horizons attempted are about three years, although forecasts up to ten years are sometimes presented. In finance, the longest forecast horizons are usually much shorter. In might be noted that evaluation is difficult for these long term forecasts as it takes many years for the actual value to become known to compare to the forecast. Some economic series contain what may appear to be clear trends; that is, a steadily increasing central value, such as a straight line, or a quadratic or exponential curve in time. A simple example would be Real Gross National Product, Investment or Consumption. If such a variable grows steadily ti will produce an exponential curve and such a curve is occasionally found in practice, but not in every country. However, series such as unemployment, prices, and interest rates do not contain trends, and so cannot be forecast over long periods, other than naively. 6
CAUSALITY AND CONTROL
6.1 Causality and Control This is a very important and extensive topic that is covered in a different chapter in this Handbook prepared by Kevin Hoover. Here I will just discuss some aspects of forecasting in economic causality. Most, but not all, economists accept the proposition that the cause occurs in time before the effect. The time distance between the two may be very small, but it has to be positive. See, for example White [2005]. There is also an extensive
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literature on instantaneous causality, but as it has no implications for prediction, I will not discuss it. Hume [1739] and Hicks [1979] can be included in those who specifically have the effect occurring before the effect. This temporal order does not necessarily indicate predictability of the cause by the effect. A definition suggested by the famous mathematician, Norbert Wiener (and previously discussed by Bunge [1963] and doubtless other philosophers) is the idea that I extended and made specific in Granger [1969] and [1980]. There are two requirements: 1. that the cause precedes the effect; and 2. that the cause has information about the effect that is not available in a wide group of other variables. In terms of distributions, where F (X/Y ) denotes the distribution function of the random variable X conditional on Y , then Y does not cause X (with respect to W ) if F (X/Y (W )) = F (X/W ), where here W is a vector of variables not including Y or X. If the equality is replaced with an inequality then causality is indicated. An immediate implication of the definition is that causality of Xt¯+1 by Yt means that Yt will help forecast Xt+1 in distribution. The definition can also be stated using just means rather than distributions and “tests for causality” can be easily constructed in this case. In economics, this has become the most commonly used definition of causality because it is easy to understand and to test. It has also been much misunderstood and misused. Due to the forecasting aspect of the definition, it is often used to help in the specification of empirical models that are to be used for forecasting. The link between causality and control may seem to be an obvious one and to be closely related to forecasting. The usual position is that if a causal relationship is known, or believed to be known, then it can be manipulated to provide a control mechanism and thus appropriate policy methods. Economic writers strongly disagree on the relationship between cause and control, some make it the basis of their definition of causality, such as Hoover [2001] and Pearl [2000], but other claim that there is no necessary link. This is not an appropriate place to survey such a complicated topic. My personal view can be given in terms of an example. Suppose that it is observed that a particular New York newspaper has considerable influence with its readers when recommending who to vote for in local elections. A wealthy investor decides to buy the newspaper so that it will support politicians of his choosing. If this becomes widely known, the original causality/control will be lost and a new forecasting regime will begin. the example shows that the can be causality, but when it is used as a control, the causal relationship can be broken. I believe that controllability is a deeper concept that causality and thus more difficult to test for using economic data. One topic that has received little attention is: can a deterministic process cause a stochastic one?
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6.2 Forecasting and Ethics In the traditional areas of forecasting such as predicting the weather or the timing of the next high tide, it is difficult to imagine the forecast having any impact on the variable being considered. The physical process that generates rain, for example, is unaware of the statements made by the weather forecaster. However, this will not necessarily be the case in the social sciences such as economics. If a financial journalist states that a certain stock price should increase over the next week, investors may believe the forecast, invest accordingly, and induce a change in price. The resulting ethical problem is clear as investors who know the forecast early and invest immediately, as the market will adjust to the forecast price rapidly. Those who clearly have a chance of profiting from the situation are the journalist herself, her immediate family and friends, and possibly the staff of the newspaper involved. Many economic forecasts potentially have the ability to influence the actual economy. A forecast of a turning point in the business cycle or a higher employment rate or an increase in inflation could produce a policy change by some government agency. Naturally, this will only occur if the forecaster is particularly accomplished and has a previous good performance record. However, the policy changes may ensure that the forecasts do not become correct, suggesting that the forecaster will be less successful. A forecaster is trained to produce the best forecast possible but there is no need to produce just a single value. One can make a forecast without taking into account the possible policy change and a further forecast will indicate the likely policy change and its impact. Ethical problems will largely arise if a forecaster produces values that represent a biased viewpoint or if the values are not released publicly immediately. Transparency of the data and methods used and to who and when they are issued become essential features. With sufficient information the question of determining ethics becomes one for the forecasting community who have to evaluate the forecasts produced. With the availability of superior computers and forecasting programs widespread, evaluation is easy to accomplish. Some econometric models and other techniques are deliberately biased to represent the view of some political party, on the left or right for example, and will naturally produce biased forecasts. A central bank which will forecast inflation, and will also be interested in controlling inflation, will likely produce downward biased forecasts of inflation. This can be achieved by using a non-symmetric cost function when forming the forecast. There seems to be no ethical problem if this behavior is well understood, but if kept a secret it is a problem.
6.3 Evaluation The only major topic in the area of forecasting that has not been considered here is the question of how to evaluate forecasts. Evaluation is an important component in the forecasting process as it allows one to improve techniques and to discover
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which methods are the most satisfactory in practice. This is a wholly pragmatic subject, perhaps based on basic aspects of decision theory, and is therefore not particularly relevant to the main theme of this paper. BIBLIOGRAPHY ´ [Bachelier, 1900] L. Bachelier. “Theory of Speculation.” Ann. Sci. Ecole Norm. Sup. (3), No. 1018, Gauthier-Villars, Paris, 1900. [Baranzini and Scazzieri, 1986] M. Baranzini and R. Scazzieri, eds. “Foundations of Economics.” Basil Blackwell, Oxford, 1986. [Blaug, 1980] M. Blaug. “The Methodology of Economics: or, How Economists Explain.” Cambridge University Press, New York, 1980. [Granger, 1969] C. W. J. Granger. “Testing For Causality and Feedback.” Econometrica 37. pp. 424–438, 1969. [Granger, 1988] C. W. J. Granger. “Testing for Causality: A Personal Viewpoint.” Journal of Economic Dynamics and Control 2, 329-352, 1988. [Hicks, 1979] J. R. Hicks. “Causality in Economics.” Basil Blackwell, Oxford, 1979. [Hicks, 1986] J. R. Hicks. “Is Economics a Science?” in Baranzini and Scazziero (eds.), Foundations of Economics, Basil Blackwell, Oxford, 1986. [Hoover, 2001] K. Hoover. “Causality in Macroeconomics.” Cambridge University Press, New York, 2001. [Hume, 1730] D. Hume. “A Treatise of Human Nature”, 1730. Page numbers refer to the edition by L.A. Selby-Bigge, Clarendon Press, Oxford (1888). [Lawson, 1997] T. Lawson. “Economics and Reality.” Routledge, New York, 1997. [Marget, 1929] A. W. Marget. “Morgenstern on the Methodology of Economic Forecasting.” Journal of Political Economy 37, pp. 312-339, 1929. [Redman, 1991] D. A. Redman. “Economics and the Philosophy of Science.” Oxford University Press, New York, 1991. [Pearl, 2000] J. Pearl. “Causality: Models, Reasoning, and Inference.” Cambridge Univ. Press, New York, 2000. [Sen, 1986] A. K. Sen. “Prediction and Economic Theory.” Proceedings of The Royal Society London A, 407, pp. 3-23, 1986. [White, 2005] H. White. “Causal, Predictive and Explorative Modeling in Economics.” Oxford University Press, 2005. [Worswick, 1972] G. D. N. Worswick. “Is Progress in Economic Science Possible?” Economic Journal 82, pp. 73-100, 1972.
1
INTRODUCTION
I believe that it is accurate to say that the typical economic forecaster does not ask herself questions about the underlying philosophy concerning what is being attempted. In fact, most could not define “philosophy” or would even attempt to do so. This is not necessarily a sign of intellectual weakness as even philosophers have difficulty defining philosophy and its objectives. “One might say that philosophy is what philosophers characteristically do” is a quotation from an article on Philosophy in “The Encyclopedia of Philosophy,” Macmillan, (1996). The same article stated that an earlier article on the same topic in the original version of the Encyclopedia “identified the distinctive feature of philosophy as its being a critical discussion of critical discussion.” It also says that “Philosophy as a commonly characterized is a multifaceted discipline that resists simple characterization.” One obvious approach is to ask a philosopher what he does but typically it will be very difficult to understand the answer as, like any other discipline, it has its own distinct terminology. However, philosophers and others will often ask interesting and penetrating questions that deserve the attention of forecasters, both to improve their understanding of what they are doing and possibly, in consequence, improve the quality of their output. Forecasting is a very ancient occupation. The earliest groups of people who gathered together in small villages would be interesting in forecasting the seasons to know when to plant crops, when to move camp to where herds would be passing by, or when the salmon were running. Then, as now, forecasting was largely a practical process, involved with statements that could produce decisions that improved the economic well-being of the group involved. 2 PREREQUISITES For convenience, it is important at this juncture to make two fundamental points that will be needed in what follows. The first is that I will consider the “economy” to consist of all the decision makers involved such as the consumers, investors, employers, and government policy makers as well as the various economic institutions, Handbook of the Philosophy of Science. Volume 13: Philosophy of Economics. Volume editor: Uskali M¨ aki. General editors: Dov M. Gabbay, Paul Thagard and John Woods. c 2012 Elsevier BV. All rights reserved.
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such as the banks, corporations, trusts, and so forth. This will be called the “actual economy.” In the literature prepared by philosophers the economy is usually taken to be the same as the constructs considered by economic theorists. This I will call the “theoretical economy.” The economic theorists construct models based on sets of assumptions and perceived rational behavior by decision makers in an attempt to represent a simplified form of the actual economy. Sometimes these models are successful, sometimes less so according to data analysis. It is certainly true to say that philosophers in their writings about economics often confuse the theoretical economy with the actual economy. It is probably also correct to say that most decision makers in the actual economy are unaware of the results in the theoretical economy, and they do not suffer very much from this, although this could be less clear in the area of finance. The second fundamental point is that it will be necessary to distinguish between “forecasting” and “prediction.” Forecasting will be limited to the extrapolations based on empirical models or data exploration, whereas a prediction will be formed from a theoretical model. These differences are further explored in Section 3. It will be helpful to use what has become the standardize “set-up” for forecasting in recent years. Let Xt be a time series measured at equal intervals of time, such as each minute, day, or month as appropriate. The requirement that the series is recorded at equal time intervals can be relaxed but is technically more difficult. The fact that months are not strictly equal in length is merely pedantic and has been considered in the literature. It will be assumed that the series is not measured continuously in time, which is true in the actual economy, but continuous time is often assumed in the theoretical economy. This assumption will be discussed further below. Xt will usually be taken to be a single series but it could be a vector, where Xt is a particular economic variable such as a price, unemployment, or production. The current moment of time is denoted n indicating “now,” so that Xn is the current value of the X series. At time n there is available some empirical data including past and present values of Xt , which is denoted XPn , so that XPn = {Xn , Xn−1 , Xn−2 , . . . , }. In practice there are only a finite number of past values available, but the impact of this fact is usually considered to be small. Other data series and their past will also be available, denoted as W Pn , where W is usually a vector. It is usual to consider the information available at time n which is used to form a forecast. An “information set” In could consist, for example, of XPn and W Pn . A wider information set Jn could consist of the contents of In and also Y Pn , where Yt is another series. We may be interested to know if the Y series contains useful information that is not in the other series, as will be seen later. The information sets can also include non-numerical information, including opinions or constraints from economic theories. When forecasting it is usual to have a specific horizon in mind, so the objective will be taken to look ahead h steps, to the value of Xn+h . As the future may well be uncertain, otherwise one would not be forecasting, Xn+h will be a random variable and thus can be described by a conditional distribution function Fx,n,h (In ) =
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Prob (Xn+h < x | In ) or its associated density function fx,n,h (In ), which is the derivative of Fx . Here the forecast will be a density function dependent on the information set used and the horizon selected. It is important to note that even if h is fixed, as n changes the information set will alter and so the predictive density will also change. Thus forecasts will change with information, with horizon, and with time. Historically, it was often too difficult to provide a predictive distribution and so simpler statistics were used instead. Typically only the forecast of the mean, denoted mn,h , was given, such as “unemployment rate next month will be 7%” or “inflation will be 5%.” Such figures only capture the middle of the predictive distribution and without some idea of the width and shape of the distribution sensible decisions are difficult to achieve. Statisticians would strongly recommend providing measures of uncertainty, such as the variance or the 95% confidence interval, although these were often difficult to interpret or were so wide that they were embarrassing! This basic setup is quite general, as it can cover several important special cases. For example, Xt can be constrained in some way, such as being positive or bounded to be between zero and one in value. It also included “event forecasts,” such as a volcano will erupt, or there will be a financial crisis, or a business cycle downturn. Such events are captured with a zero-one random variable, with zero for when the event does not occur, and the predictive distribution will consist of just a probability pn,h of the event happening and probability 1 − pn,h of it not occurring. The probability pn,h will be a function of the information set used and so will evolve over time. It might be said that economic historians just look backwards and that some economist just look sideways, but forecasters have to look back to be able to look forward. They have to select the useful pieces of information from the past from the mass of information that is available from which to form their forecasts It is clear that many forecasts could be made and so a process of evaluation is essential to learn what seems to be helpful to decision makers and what does not. At time n + h there will exist the observed values of the series Xn+h and a forecast of this quantity that was made at time n, i.e., the predictive density f (x, n, h(In )) based on the information set In . The error series, which is defined as the actual minus the forecast will have density f (Xn+h − x, n, h(In )). At time n + h all components of this density are known and so all of the properties of the error can be obtained and the evaluation can be conducted on them. A standard measure is to record the average “likelihood” of the actual, given by the average of f (Xn+h , n, h(In )) over some appropriate sample. The expectation of the error density, defined as Z ∞ xf (x, n) −∞
produces the “point forecast error” = actual - point forecast, or in notation en,h = Xn+h − mn,h . Traditionally, and certainly until the end of the twentieth century,
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forecast evaluation concentrated on these errors. If the forecast is to be used by a decision make, the cost of making an incorrect forecast might be measure by a “cost function ce ,” so that an error of the amount e results in a cost ce . Many aspects of economic forecasting, including the evaluation of forecasts, can be found in “ The Handbook of Economic Forecasting ” edited by G. Elliott, C.W.J. Granger, and A. Timmermann, Elsevier, 2006. A critical aspect of forecasting is the choice of the information set, which will include data from the past and present, any available stated plans or policies for the future and include changes in the structure of the economy, laws and institutions. A forecast will be forward looking partially based on backward information, but not entirely so. The proposition that a society, and thus an economy, should have some stability over time and some momentum together with understandable changes, is not a very surprising one and forms the basis of many forecasts. Thus, potentially at least, we learn from the lessons of history when forecasting. 3 FORECASTING, PREDICTION, AND ECONOMICS Much of the actual, as well as theoretical, economy is forward looking. Decision makers make decisions now that will have impacts in the future. Investors decide where to invest now and wait to see what return occurs; consumers buy now but consume over the net few hours or days; a house-buyer decides now and uses the purchase over several years; an employer agrees to hire a worker and makes use of his or her skill over some later period. Essentially the decision maker has to forecast the future consequences of the decision. Further, policy making is clearly about the future. It can be thought of as a number of alternative conditional forecasts. If the agency does one thing, we expect the future to be like this. But if the agency does something else, the future will be like that. The policy make chooses the better, expected future. It is thus seen that major components of both macro- and micro-economics will involve forecasting. The extensions to international economics and to finance are obvious and in fact both of the actual economies in these areas are major consumers of forecasts, sometimes called “expectations” in the media. At this point it is important to carefully distinguish between “forecasting” as discussed in Section 2 above, and “prediction.” Here prediction will mean taking a model, usually based on economic theory and assuming that it is correct, then drawing implications from it about the behavior of the economy, at least the theoretical one and possibly the actual one. Some example are given below. Prediction occurs when there is a model of the form Xt = a + bYt + et where X and Y are a pair of specific economic variables, e is a residual and the coefficients a and b have been estimated or chosen in some way. The model could be theoretical or empirical in origin. A simple example might have Y a
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basic interest rate and X inflation. A prediction of X is formed by inserting a particular value for Y into the equation, and then assuming that the model is correct. If, in the model Xt is replaced by Xt+1 then putting an observed value for Yt into the model will provide a forecasting of the next value of X. This paper will usually be considering forecasts, although usually of a more sophisticated form. A simple example of a prediction comes from price theory and says that if a company raises prices then the consequence is a reduction in sales. This is the kind of prediction that does not depend on a sophisticated theory and is observed in the actual economy to be usually true, but not always. These exceptions can be explained both using theory and empirical arguments. Blaug [1980, 2nd edition, 1992, pp. 151] suggests several other similar examples: “an increase in demand leads to a rise in both output and product prices,” “a lump sum tax on business profits will have no effect on output,” and “a rise in money wages causes a fall in employment.” Note that all of these predictions are non-specific, the amount of the change and the timing are not given. Usually forecasts would be more specific. On occasions a prediction from the theoretical economy about that economy can be evaluated by the actual economy. For example, Blaug [as before] states that the theory of the firm “predicts unequivocally that a profit maximizing firm in a perfectly competitive market will not advertise: it has no incentive to do so because it faces a perfectly elastic demand curve and can sell all that it can produce.” As many firms do advertise their differentiated products, then the assumptions upon which the theory is based have to be incorrect. As will be seen, prediction plays an important role in the topic of model evaluation and that forecasting becomes embroiled in the discussion. 4 THE PURPOSE OF ECONOMICS It is useful to know the purpose for some body of work as it give a starting point for evaluation. For some disciplines such as medicine, psychology, and law, the purpose is to be helpful to their clients. However, for areas such as history or mathematics, which are certainly important and distinguished, being immediately helpful is unlikely to be the suggested purpose. The are a number of philosopher/economists such as Lawson [1997], Redman [1991], and Blaug [1980, 2nd edition, 1992] who consider many aspects of the interaction between the two areas, and the latter two take definite views on the purpose. Blaug [1980, 2nd edition, 1992, p. 246] states “the central aim of economics is to predict and not merely to understand” although he goes on to mix up forecasting and prediction. His statement should be taken to mean that prediction should be used to evaluate an economic theory. Redman [1992, p. 120] quotes Worswick [1972] “the idea of economics as positive science makes predictability the test of its performance, the prediction of relationships is situations not previously observed, as well as the prediction of future events, which in some ways is the acid test.” This statement covers prediction in cross-sectional situations where everything occurs at the same
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time. In this paper only future events will be considered. Of course, prediction is not the purpose of economics, even of theoretical economics. It is quite easy to formulate theories, one starts with a group of reasonable assumptions, adds a few generally accepted economic concepts such as a rational market, include some institutional constraints, and then prove theorems about the “economy” so derived. The theory can be simple or it can be very complicated but it usually will not be unique. The question of evaluation will have to ask if any of the models is adequate, in some fashion, and then which is the best. The evaluation can perhaps be based on prediction and on falsification. Redman [1992, p. 24] also quotes Kuhn [1970] about a demarcation criterion without which no field is potentially a science: “(1) concrete predictions must emerge from the practice of the field; (2) for a subclass of phenomena, whatever passes for predictive success must be achieved; (3) predictive techniques must have roots in a theory which, however metaphysical, simultaneously justifies them, explains their limited success, and suggests means for their improvement in both precision and scope. Finally, the improvement of predictive technique must be a challenging task, demanding on occasions the very highest measure of talent and devotion.” Unfortunately, most of the early discussions of topics such as “the purpose of economics” and “is economics a science?” are based on viewpoints that are now generally considered outmoded. The early position taken was that physics (of a traditional form) was the standard against which one measured a field being a science. The methodology of traditional physics was the one to use for comparison. In this area the world was deterministic and not stochastic, and experiments should get the correct answer if properly conducted, or if repeated often enough the average will certainly tend to the correct value. Certainly the old theory had its clear successes, as Blaug [1992, p. 7] points out “who can deny the extraordinary predictive power of Newtonian theory particularly after the confirmation in 1758 of Edmond Halley’s prediction of the return of ‘Halley’s comet,’ topped in 1846 by Leverrier’s use of the inverse-square law to predict the existence of a hitherto unknown planet, Neptune, from the observed aberrations in the orbit of Uranus.” Since then the same theory has been extended to forecast the time and height of tides on virtually every beach in the world from now and for many years into the future. Of course all such forecasts are based on a set of assumptions and if the change so will the forecast, certainly a tsunami would change tides in the short run and if the Moon broke into two pieces the tide would be changed forever. As the twentieth century evolved, with the advent of quantum physics and areas such as meteorology and oceanography, the definition of a science changed together with the appropriate methodology. The move from dealing with just inanimate objects, as with classical physics, to animate ones as with medicine and biology, greatly extends the range of the subjects being considered and of the topic “what is a science?” The next extension is to objects (or subjects) that are individual decision makers, covered by areas such as psychology, economics, and political science, although
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some parts of biology will also fall into this category. As the type of objective being studied changes it is reasonable to expect that the methodology will evolve as will the objective of the subject area. I do not think that economists have been involved in a careful enough discussion about what is the purpose of economics. My personal view comes from the observation made above that the economy consists of many types of decision makers. It follow then that the objective of economics should be to help decision makers make better decisions. It is fairly easy to understand now to do this in some parts of economics such as finance and macro-economics, but much more difficult in those areas where decisions are not emphasized, such as parts of cross-section, micro- and theoretical economics. 5 PHILOSOPHICAL QUESTIONS A number of questions will now be considered that may be thought of as having philosophical origins.
5.1
Is the Economy Deterministic or Stochastic?
A system, such as an economy, is deterministic if its progress can be fully described without the use of probabilities (other than zero and one). Know the past will completely determine the immediate future, and then by iteration, all of the future. The basic idea that the universe could then be deterministic comes from classical physics, which has known, simple, and unchanging laws that operate everywhere and at all times. As stated before, this allows the positions of the planets to be known exactly in the future, for example, and thus the timing of tides anywhere on earth can be determined, provided that the basic assumptions continue to hold. Historians may also subscribe to their domain of interest being deterministic, as the past cannot be changed, although interpretations about why things happen are not constant: the reason for the decline of the Roman Empire or for the occurrence of the Black Death changes every decade or so. Although history is certainly unchanging, what we know about it is by no means constant and so interpretation can evolve. The theoretical economy is often assumed to be deterministic as results are usually easier to obtain under this assumptions. It should be pointed out that for a theoretical economy, ANY basic assumption can be made. the only requirement is that the participants in this economy behave in an “economically sensible” fashion given these assumptions. There is no requirement that these economies be realistic and there can exist several of them about the same topic but with different assumptions. In contrast, a stochastic system is one that has to use probabilistic concepts in describing its progress, such as “if today we are at A, then tomorrow we will be at B with probability 0.4, and at C with probability 0.6.” With this definition, anything that is not deterministic must be stochastic. However, it is more convenient to consider processes that are “purely stochastic,” which would rapidly collapse to
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a constant (or possibly a simple trend) if the stochastic component was switched off, and then to have systems which consists of both deterministic and purely stochastic components. Suppose that we could invent a measure Q which takes values between zero and one, with Q = 0 corresponding to deterministic and Q = 1 to purely stochastic. Then classical physics would have Q = 0, quantum physics Q = e, where e is a small positive number, Q = 0.3 for meteorology, and Q = 0.6 for macroeconomics with Q = 0.9 for much of finance. I suppose that history has Q = 0, but that is open to debate. Only if Q = 0 can you assume that perfect forecasting is possible, otherwise forecast errors will occur. However, if errors do occur it may be because we are not good at forecasting rather than because of the inherent stochastics. A complicating issue is the existence of a class of mathematical iterative processes known as “chaos” or chaotic A very simple example (known as the logistic map) is Xt+1 = a(Xt (1 − Xt )) with the starting value X1 chosen in the region (0,1) and the parameter a chosen to be in the region 3.6 to 4.0. Data generated from this map has the properties of white noise, with a constant mean and variance and all serial correlations that are zero, when estimated. However, the generated data does not have the properties of an independent series, although this is sometimes claimed in the chaos literature, as powers of the series are not serially uncorrelated. However, more complicated maps than the logistic shown above can generate series that have more of the properties of an independent series and will also be chaotic. Some of these maps are used in computer programs to generate “artificial random numbers” that are used in various statistical procedures such as the bootstrap and simulations. Provided the sample sizes used are not too large, these artificial series will usually work well and appear to be random. Nevertheless, a long enough series will fail a test of randomness and will suggest that they are generated from a deterministic map. One can then go and devise a yet more complicated map which will produce series that are more difficult to distinguish from strictly random, although with enough data they can be, in theory at least. It is seen that the division between deterministic white chaos and a truly stochastic random process is becoming very unclear. Whether there is any difference in the limit is truly a philosophical question; it is deep and difficult but its solution has little practical relevance. If a theoretical economy is deterministic it is due to the assumptions being used. If an actual economy is deterministic, then this is a basic property of the economy. It is possible for part of an economy to be deterministic, but not other parts, just as in the actual physical world the tides could be deterministic, but the temperature of the water involved could be stochastic. An example is economics would be the month of the year with the greatest store sales in a European country (December) and the amount sold in that month. Limitations on the causal relationships between the deterministic and stochastic parts are discussed later. It might be worth pointing out that a totally deterministic society is very boring as it is highly forecastable, at least in the short run. There will be no horse racing
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or gambling; there is little point in playing any sporting contest as everyone know the outcome and every political election can be very short and inexpensive. The winner is already known. These results require the generating map to be either know or well approximated by a neural network analysis. The degradation of the forecasts as the horizon increases is due to the accumulated effects of a round-off error. A very well known economic theorist, Sir John Hicks, stated [Hicks, 1979] that economics is “on the edge of science and on the edge of history” as it tries to use the techniques of science but its subject matter behaves differently. Hicks [1986] says “If a scientific theory is good, it is good now and would have been good a thousand years ago ... but the aspects of economic life which we need to select in order to make useful theories can be different at different times.” (Quotes from [Redman, 1991, p. 106].) Chaos theory is designed for a deterministic, unchanging physical world and has not performed well in the decision theoretic world of economics.
5.2
Can An Economic Agent Have Perfect Foresight?
Amongst the assumptions that are sometimes made within an economic theory for the sake of simplicity is that of “perfect foresight.” The purpose of such assumptions is to reach some immediate conclusions from the theory, and they are later dropped to see if the same conclusions hold in a more general situation. If an agent had perfect foresight, she could continually make optimum investments and accumulate considerable wealth. She would win every “game” situation as she would know her opponent’s choices. It is doubtful if every agent could be assumed to have perfect foresight as game situations would have no solutions as no one would be able to make a choice. This is not a very interesting assumption and is a totally unrealistic one. If everyone had perfect foresight the economy as we know it would cease to exist. There would be no markets as everyone would know the eventual price obtained, economic policies would not work as they are perfectly anticipated and everyone knows the impact, if any. Occasionally economic theorists who are used to making a perfect foresight assumption will criticize an economic forecaster for producing imperfect forecasts. Only a little thought produces reasons why many forecasts will be imperfect such as short-term and, even more clearly, long-term weather forecasts, as well as forecasts of a horse race. There are too many things that can occur. Sen [1986] points to two particular reasons for economics: the large number of individual decision makers involved in a typical economy (a hundred million households in the United States and the European Union, with even more in China and India), and also the many interactions between these decision makers.
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5.3 Is It Worth Making A Forecast That Is Not Perfect? In 1928 Oscar Morgenstern, who later became famous as a co-author of the first book on game theory, published in German a pamphlet on the irrelevance of economic forecasts. This work has become known through the attack on it by Marget [1929]. Morgenstern essentially makes the point that economic forecasts can never be perfect as they are based on an inadequate economic theory and poor data. He states that if they are not perfect, then they cannot be used for policy purposes. His arguments relate to classical, pre-quantum physics and his policy points have been largely superseded by the development of decision making under uncertainty. Marget essentially tries to weaken the arguments proposed without going to a stochastic viewpoint. It should be noted that Morgenstern makes no mention of his position in his later books on “the accuracy of economic date” and on the “forecastability of stock market prices,” the second of which is written with me. A few writers go beyond the imperfection of economic forecasts to conclude that all such forecasts are so bad that they should be disregarded. A few may even claim that economic forecasting is not possible, although there is a difficulty with semantics. Blaug [1982, p. 158] firmly disagrees saying “If prediction of human behavior were truly impossible, if none of us could predict anything about the behavior of other people, economic life itself, not to mention theories about economic life, would be unimaginable. Not only would the total incapacity to predict economic events wipe out economic theory: it would wipe out every other type of economics, as well as all pretences of offering advice to governments and business enterprises.”
5.4 Omniscience Suppose there exists some entity that is omniscient. Following the philosophical literature I will call the entity “God” and use the pronoun “He,” without there being any implications from these choices. The fact that if God has omniscience then there are important philosophical implications is not of immediate relevance for an essay about forecasting. I will not be concerned with policy questions at this moment and certainly not free-will. By omniscience I take it to mean that God could forecast any component of the economy at any time in the future if He cared to do so. In particular, he could have perfect foresight about the behavior of every decision maker. It follows that if He were rational and utility maximizing He could quickly acquire immense wealth, and would soon be testing the theories about there being a satiation level for wealth. Of course, as God would have nothing to spend his wealth on, it is irrational to expect Him to have a standard utility function. As a forecaster, should my behavior change if I am told that there exists a God who can forecast perfectly? It would imply that if I improve my technique and gather a good enough information set, then I should be able to do almost as well. Alternatively, it could imply that I would need godly abilities and resources to
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do so well, and so should be content with much less. If I add a small positive probability that God does not exist, I am back to a stochastic economy. The idea that an all-knowing God with omniscience will know the future of us all with certainty is probably placing us at too high a level of importance. For Him to foresee the exact position and behavior of every living creature, from microbes up, on every planet under his control, is quite possibly within his computing abilities, but it is very unclear why He would want to ever undertake that task, even though it only has to be done once. We have no idea of His reasoning, requirements, and desires and we certainly do not know if His time scale is the same as the one with which we are involved, or if beings on other planets thinking vastly quicker or slower than us. Without more basic knowledge, or further assumptions, it is doubtful if the possibility of omniscience has any impact on everyday economic forecasting.
5.5
Why Cannot We Forecast Perfectly?
It is general knowledge that economic forecasts are not perfect, and this is the basis for many jokes. Sen [1986] states “It is, in fact, tempting to see the economist as the trapeze performer who tends to miss the cross bar, or as the jockey who keeps falling off his horse.” What is not clear is why the economists should be singled out. Does a patient who is sick go to a doctor one week and then complain the following week if he is not completely cured? Are there articles in the press asking why the horse-racing correspondent did not pick all the winners yesterday or why the weather forecasts do not turn out to be perfect? Sen is obviously reflecting the traditional approach, taken by Morgenstern and based on the believe (or the assumption) that economics is a science, in the old fashioned sense. However, Sen does then go on to give at least two plausible reasons for the non-perfection: the difficulty in anticipating human behavior; and aggregation or size effect. 1. Anticipation of human behavior follows from the fact that many individual decision makers are involved (who are not automatons) and that their decisions will evolve as the learn, their tastes change, their choice sets evolve, and the institutions and society changes around them. Each person can react differently to these changes. 2. The size effect comes from the fact that there are many millions of families in the typical economy, with complicated interactions. As Sen says, there are “millions of human beings each with different values, objectives, motivations, expectations, endowments, rights, means, and circumstances.” This will make aggregation difficult both for theorists and for data analysts without some simplifying assumptions. Some of these assumptions may be reasonable, but others (such as having “representative agents”) are generally thought to be not useful in practice. These are certainly important and relevant reasons that would be likely to be included in a defense by an empirical forecaster, although there, more attention
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would be paid to the likelihood of the economy being stochastic. If the level of stochasticity is high, forecasts will be imperfect. Sen was concerned with the topic “Prediction and Economics Theory” and so was considering the use of a theoretical model to provide “predictions,” which are not necessarily forecasts as mentioned earlier. They are based on the assumption that the theory is correct. He discusses the relevance of topics such as “equilibrium,” “rationality,” “maximization” and, later, the use of an assumption such as “self-choice goal” in which each act of choice of a person is the pursuit of one’s own goal (such as the maximization of utility). He finds that many of these concepts are difficult to use for prediction. Currently certain equilibrium models are being used for long-term macro forecasting, but evaluation is difficult. Of course it has to be admitted that the forecasts may be imperfect because of the incompetence of the forecasters. They may be using poor quality data sets or insufficiently sophisticated forecasting techniques, or it could just be that the computing power is insufficient for the task, as the economy is just too complicated. If these were all or some of our problems, I believe that we could expect to see improvements in forecastability, either a steady progress or a series of steps as breakthroughs occur. There has been some progress, but the variables being forecast have changed in nature and it seems that the data may have declined in quality in some important cases.
5.6 Differences in Forecastability It has been observed in the actual economy that economic variables vary in their “forecastability;” that is, the extent to which they can be forecast. For example, it is quite easy to forecast the demand for electricity at every hour tomorrow in an American city. The demand is to a very large extent determined by the particular day, by the regular pattern of activities through the day of the consumers in the city, and the forecast of the temperature for the day. Since temperature is quite easy to forecast twenty-four hours ahead, and as the use-of-electricity pattern is stable given the temperature, the forecast can be made a day ahead and will usually be very accurate. This forecast is of considerable importance to the electric utility company that supplies the electricity for the region. It is worth noting that the variable essentially consists of two components: the regular daily use patter (which changes each day); and the very complicated (possibly stochastic) weather component. At the other extreme returns from stock market prices are very difficult to forecast, as was found empirically from very early work by Bachelier [1900] and later by various statisticians considering a model called “the random walk hypothesis,” which essentially says that these returns are not forecastable. Economic theorists later stated these ideas in the “efficient market hypothesis,” which notes that if you could forecast the return from any speculative asset, you would have a “money machine” which would produce unlimited amounts of money. As such a machine is impossible, the returns cannot be forecastable.
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Naturally, most economic variables lay between these two extremes. It is generally true that anything that one can easily profit from, good forecasts are difficult to make. Commodity prices, interest rates, exchange rates, and commodity rates fall into this group. However, if an interest rate is used as a control variable, such as by the Federal Reserve Bank, The Australian National Bank, or the European Bank, it does become easier to forecast. Some variables from a stock market, such as daily volume traded, or the number of stocks advancing in a day, or even daily volatility of an individual stock are all somewhat forecastable. It is generally true that the levels of variables are much more forecastable than the corresponding changes or rates of return. For example, the level of unemployment rather than the change in it, or the price level rather than inflation. All such statements are observed properties of the actual economy as observed through the lens of the present forecasting methodology. They could change as the methodology improves. It might be noted that if the economy was deterministic, then the level of a variable and its change would both be perfectly forecastable. Economic variables are included to change in value as new and relevant information accumulates. This happens very quickly in a speculative market, quite slowly for the major variables in macroeconomics, and very slowly in population economics. In general, users of forecasts prefer higher accuracy rather than lower, but not necessarily in all cases. For example, if someone offered to forecast the date of your death, most people would prefer not to have that information even though knowledge of it would lead to more rational investment decisions.
5.7
Should Forecasts Be Rational?
The “rational expectations” revolution in macroeconomics took place in the 1970’s, but the basis of the idea and the corresponding theory was developed a decade early by Muth in 1961. It was observed that economic decision makers were being assumed to be rational and that their decisions would be influenced by forecasts or “expectations” and so these also should be rational in the sense that they should not be obviously sub-optimum. A “rational expectation” should use all the relevant information that is available and also appropriate methods of forming a forecast. A given set of forecasts could be shown to be “irrational” if they were clearly sub-optimum, as shown by a statistical test. The theory could become complicated as some of the models used to form the expectations involved expectations, so some problem usually reflects the lack of subtlety about timing of occurrences within the model. If better forecasts are easily available it would certainly be irrational to use inferior ones, but finding the very best may be too expensive in terms of searching for slightly better methods and a few useful but expensive last pieces of information. An implication of the considerations of rational expectations was that the usefulness and relevance of much government policy was thrown into doubt, at least over the long run. An unexpected change in policy could still have an immediate effect, which will continue into the future.
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Many countries have important survey efforts that try to measure the plans of industrialists about their future capital investment and employment and also consumer about their buying plans. the most successful of these seems to be the investment plans, which once started are less easy to stop without considerable cost. This possibility illustrates the fact that useful forecasts can come from a non-theory or model based approach.
5.8 How Far Can One Forecast? The question of how far one can successfully forecast is tied to the topic of evaluation. A forecasting model can be run out into the indefinite future, but the relevant question is: over what horizon are the forecasts of any value? The answer is also tied to the concept of the forecastability of various types of economic variables. Some variables are slow moving and are very predictable over long horizons, such as population growth, whereas others have virtually no possibility to forecast, such as speculative returns. A deterministic variable can be (perfectly forecast) into the indefinite future without any error. In contrast, a variable with a stochastic element will steadily accumulate the stochastic, unforecastable component until no forecastable part can be detected. In macroeconomic, the longest horizons attempted are about three years, although forecasts up to ten years are sometimes presented. In finance, the longest forecast horizons are usually much shorter. In might be noted that evaluation is difficult for these long term forecasts as it takes many years for the actual value to become known to compare to the forecast. Some economic series contain what may appear to be clear trends; that is, a steadily increasing central value, such as a straight line, or a quadratic or exponential curve in time. A simple example would be Real Gross National Product, Investment or Consumption. If such a variable grows steadily ti will produce an exponential curve and such a curve is occasionally found in practice, but not in every country. However, series such as unemployment, prices, and interest rates do not contain trends, and so cannot be forecast over long periods, other than naively. 6
CAUSALITY AND CONTROL
6.1 Causality and Control This is a very important and extensive topic that is covered in a different chapter in this Handbook prepared by Kevin Hoover. Here I will just discuss some aspects of forecasting in economic causality. Most, but not all, economists accept the proposition that the cause occurs in time before the effect. The time distance between the two may be very small, but it has to be positive. See, for example White [2005]. There is also an extensive
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literature on instantaneous causality, but as it has no implications for prediction, I will not discuss it. Hume [1739] and Hicks [1979] can be included in those who specifically have the effect occurring before the effect. This temporal order does not necessarily indicate predictability of the cause by the effect. A definition suggested by the famous mathematician, Norbert Wiener (and previously discussed by Bunge [1963] and doubtless other philosophers) is the idea that I extended and made specific in Granger [1969] and [1980]. There are two requirements: 1. that the cause precedes the effect; and 2. that the cause has information about the effect that is not available in a wide group of other variables. In terms of distributions, where F (X/Y ) denotes the distribution function of the random variable X conditional on Y , then Y does not cause X (with respect to W ) if F (X/Y (W )) = F (X/W ), where here W is a vector of variables not including Y or X. If the equality is replaced with an inequality then causality is indicated. An immediate implication of the definition is that causality of Xt¯+1 by Yt means that Yt will help forecast Xt+1 in distribution. The definition can also be stated using just means rather than distributions and “tests for causality” can be easily constructed in this case. In economics, this has become the most commonly used definition of causality because it is easy to understand and to test. It has also been much misunderstood and misused. Due to the forecasting aspect of the definition, it is often used to help in the specification of empirical models that are to be used for forecasting. The link between causality and control may seem to be an obvious one and to be closely related to forecasting. The usual position is that if a causal relationship is known, or believed to be known, then it can be manipulated to provide a control mechanism and thus appropriate policy methods. Economic writers strongly disagree on the relationship between cause and control, some make it the basis of their definition of causality, such as Hoover [2001] and Pearl [2000], but other claim that there is no necessary link. This is not an appropriate place to survey such a complicated topic. My personal view can be given in terms of an example. Suppose that it is observed that a particular New York newspaper has considerable influence with its readers when recommending who to vote for in local elections. A wealthy investor decides to buy the newspaper so that it will support politicians of his choosing. If this becomes widely known, the original causality/control will be lost and a new forecasting regime will begin. the example shows that the can be causality, but when it is used as a control, the causal relationship can be broken. I believe that controllability is a deeper concept that causality and thus more difficult to test for using economic data. One topic that has received little attention is: can a deterministic process cause a stochastic one?
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6.2 Forecasting and Ethics In the traditional areas of forecasting such as predicting the weather or the timing of the next high tide, it is difficult to imagine the forecast having any impact on the variable being considered. The physical process that generates rain, for example, is unaware of the statements made by the weather forecaster. However, this will not necessarily be the case in the social sciences such as economics. If a financial journalist states that a certain stock price should increase over the next week, investors may believe the forecast, invest accordingly, and induce a change in price. The resulting ethical problem is clear as investors who know the forecast early and invest immediately, as the market will adjust to the forecast price rapidly. Those who clearly have a chance of profiting from the situation are the journalist herself, her immediate family and friends, and possibly the staff of the newspaper involved. Many economic forecasts potentially have the ability to influence the actual economy. A forecast of a turning point in the business cycle or a higher employment rate or an increase in inflation could produce a policy change by some government agency. Naturally, this will only occur if the forecaster is particularly accomplished and has a previous good performance record. However, the policy changes may ensure that the forecasts do not become correct, suggesting that the forecaster will be less successful. A forecaster is trained to produce the best forecast possible but there is no need to produce just a single value. One can make a forecast without taking into account the possible policy change and a further forecast will indicate the likely policy change and its impact. Ethical problems will largely arise if a forecaster produces values that represent a biased viewpoint or if the values are not released publicly immediately. Transparency of the data and methods used and to who and when they are issued become essential features. With sufficient information the question of determining ethics becomes one for the forecasting community who have to evaluate the forecasts produced. With the availability of superior computers and forecasting programs widespread, evaluation is easy to accomplish. Some econometric models and other techniques are deliberately biased to represent the view of some political party, on the left or right for example, and will naturally produce biased forecasts. A central bank which will forecast inflation, and will also be interested in controlling inflation, will likely produce downward biased forecasts of inflation. This can be achieved by using a non-symmetric cost function when forming the forecast. There seems to be no ethical problem if this behavior is well understood, but if kept a secret it is a problem.
6.3 Evaluation The only major topic in the area of forecasting that has not been considered here is the question of how to evaluate forecasts. Evaluation is an important component in the forecasting process as it allows one to improve techniques and to discover
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which methods are the most satisfactory in practice. This is a wholly pragmatic subject, perhaps based on basic aspects of decision theory, and is therefore not particularly relevant to the main theme of this paper. BIBLIOGRAPHY ´ [Bachelier, 1900] L. Bachelier. “Theory of Speculation.” Ann. Sci. Ecole Norm. Sup. (3), No. 1018, Gauthier-Villars, Paris, 1900. [Baranzini and Scazzieri, 1986] M. Baranzini and R. Scazzieri, eds. “Foundations of Economics.” Basil Blackwell, Oxford, 1986. [Blaug, 1980] M. Blaug. “The Methodology of Economics: or, How Economists Explain.” Cambridge University Press, New York, 1980. [Granger, 1969] C. W. J. Granger. “Testing For Causality and Feedback.” Econometrica 37. pp. 424–438, 1969. [Granger, 1988] C. W. J. Granger. “Testing for Causality: A Personal Viewpoint.” Journal of Economic Dynamics and Control 2, 329-352, 1988. [Hicks, 1979] J. R. Hicks. “Causality in Economics.” Basil Blackwell, Oxford, 1979. [Hicks, 1986] J. R. Hicks. “Is Economics a Science?” in Baranzini and Scazziero (eds.), Foundations of Economics, Basil Blackwell, Oxford, 1986. [Hoover, 2001] K. Hoover. “Causality in Macroeconomics.” Cambridge University Press, New York, 2001. [Hume, 1730] D. Hume. “A Treatise of Human Nature”, 1730. Page numbers refer to the edition by L.A. Selby-Bigge, Clarendon Press, Oxford (1888). [Lawson, 1997] T. Lawson. “Economics and Reality.” Routledge, New York, 1997. [Marget, 1929] A. W. Marget. “Morgenstern on the Methodology of Economic Forecasting.” Journal of Political Economy 37, pp. 312-339, 1929. [Redman, 1991] D. A. Redman. “Economics and the Philosophy of Science.” Oxford University Press, New York, 1991. [Pearl, 2000] J. Pearl. “Causality: Models, Reasoning, and Inference.” Cambridge Univ. Press, New York, 2000. [Sen, 1986] A. K. Sen. “Prediction and Economic Theory.” Proceedings of The Royal Society London A, 407, pp. 3-23, 1986. [White, 2005] H. White. “Causal, Predictive and Explorative Modeling in Economics.” Oxford University Press, 2005. [Worswick, 1972] G. D. N. Worswick. “Is Progress in Economic Science Possible?” Economic Journal 82, pp. 73-100, 1972.