The interplay of these forces, feeding off each other in a dynamic and synergisticway, led to an environment where foreign exchange trading increased rapidly. Withworld-wide daily trading at approximately three trillion US Dollars [39], the ForeignExchange market is by far the largest and most liquid market in the world, whereliquidity refers to currencies’ ability to be easily converted through an act of buyingor selling without causing a significant movement in the price and with minimumloss of value.While an exchange rate qualifies as a security because it is traded daily, it generates no cash flow like a bond, offers no dividend like a stock, and has neither adiscounted present value nor a terminal value as in any typical asset. Currencies havebeen termed a medium of exchange and thus while traded like a security, they are notan asset in the true sense of the word. Hence, one should not expect exchange ratesto have a return as they are just the grease to exchange products across differentgeographical borders. However, because there are various forces that affect the supply and demand of currencies, currency values change almost continuously, therebygenerating return opportunities (positive or negative) even though theoretically theyshould not. Hence, if currencies are in excess demand relative to a given rate because of an influx of foreign investors, the currency will appreciate, but throughchanges in international trade and investment, the assets in the foreign country willbe overvalued leading to either a reduction in demand or an increase in the quantitysupplied leading to a subsequent depreciation.Attempts to explain and predict exchange rate movements have largely remainedunsuccessful. For instance, many researchers have tried to predict future currencylevels using various structural and time series models and have come to the conclusion that currencies are extremely hard to forecast and that these models performno better than a random walk model [287]. While there is a broad consensus aboutlong term movements over several years being driven by fundamental economic indicators, there seem to be no reliable methods of forecasting exchange rates overmedium to short time horizons such as days, weeks, or even months. In fact, currency values in the short term can be very volatile and erratic for largely inexplicablereasons. Krugman and Obstfeld [235] write:If exchange rates are asset prices that respond immediately to changes in expectations andinterest rates, they should have properties similar to those of other asset prices, for example,stock prices. Like stock prices, exchange rates should respond strongly to “news,” that is,unexpected economic and political events; and, like stock prices, they therefore should bevery hard to forecast.
However, forecasting is necessary in order to give policy advice for prominent practical tasks involving trading and hedging currencies where decisions must be madewithout advance knowledge of their consequences.As a first task we take a data-driven approach in order to examine whetherthere is any predictability in the behavior of daily exchange rate data. We challenge the widely believed idea of efficient markets as offered by modern finance
1 Motivation
5
theory [124, 126] by using a new supervised learning forecasting technique that hasemerged from the fields of computational geometry and computational statistics.This technique is called support vector machines (SVM) and has proven to successfully exploit patterns in many nonfinancial applications. In addition, SVM handlesthe complexity of identifying the optimal decision function in a very interesting way.It operates in a space with possibly infinite dimensions as opposed to the well knownapproach of artificial neural networks (ANN), which is designed to approximate anonlinear function in input space. The hypothesis is, that with a different quantitative approach which has the computational power to process a sufficient amount ofinformation, behavior that was previously believed random might become predic.Second, we address the fact that as business becomes more global, more andmore nonfinancial companies are finding themselves increasingly exposed to foreigncurrency exposure. On average, firms judge between a quarter and a third of theirrevenues, costs and cashflows as being exposed to movements in exchange rates. Itis therefore not a surprise that multinational firms, according to Servaes and Tufano[364], consider foreign exchange risk as the most costly type of corporate risk on asix point scale (see Table 1.1).In the absence of any reduction in exchange rate volatility, firms must protectthemselves from the potential losses arising from unexpected changes in exchangerates provided that markets are imperfect in the sense of [298] and it is not possiblefor the company’s shareholders to hedge the risk themselves. We investigate theproblem of an industrial corporation that seeks to optimally hedge its foreign exchange exposure with a combination of linear and nonlinear financial contractsTable 1.1 “Without riskmanagement, how costlywould the following risks beto your company over thenext 5 years, considering bothlikelihood and magnitude ofloss?”
Factors
%4 or 5
N
Foreign exchange risksStrategic risksFinancing risksCompetitive risksFailure of company projectsExecution risksReputational risksCommodity price risksOperational risksInterest rate risksCredit risksRegulatory or government risksLoss of key personnelProperty and casualty risksLitigation risksNatural catastrophe risksEmployee misdeedsTerrorism risksPolitical risksPension or healthcare shortfallsWeather risks
53%47%40%39%36%35%33%32%31%31%28%26%26%22%21%17%13%13%11%10%9%
239232233237236231232238232239237238235236233237234235238228235
Scale is “Not Important” (0) to “Very Important” (5)
6
1 Motivation
given a stochastic representation about the underlying exchange rate. Despite alarge body of research, rooted in economics, international finance, portfolio theory,statistics, and operations research having explored the optimal hedging of foreignexchange risks, newspaper headlines regularly suggest, that firms have either notfound a recipe yet against adverse exchange rate fluctuations, or the applied methods and strategies do not provide sufficient protection against currency risks. Decision making in real life situations of high complexity is often fraud to mistakes,and many problems seem to be too difficult to expect correct or at least adequatedecisions. Could it be that it is too complex for decision-makers to arrive at an optimal solution from a computational complexity perspective? First, firms may beexposed to different sources of uncertainty. Not only is there uncertainty related tothe exchange rate of various currency pairs, but there usually exists uncertainty regarding the underlying exposures which must be estimated in order to determinethe appropriate volume of a potential derivative hedge. Second, derivative marketsoffer a plethora of different products with different pay-off structures, and it is notobvious which product or which combination of products is most appropriate for therisk structure of a particular firm at a certain point in time. Third, decision-makingunder uncertainty usually involves finding a good compromise between objectiveswhich may be contradictive and highly individual, random quantities with respectto the information at the moment, and the inclusion of personal expectations. As aconsequence, there are open questions concerning the relation between the theoretical benefits of hedging, reduced volatilities, corporate performance, and the appetitefor risk. For instance, it is still an open question, whether hedging can do more thanjust reducing potential losses and potential gains. It is therefore of both theoreticaland practical interest to examine how the problem of finding a particular hedgingstrategy that optimally matches a firm’s preferences towards risk and reward can besolved in a computationally efficient way and compares against static benchmarkstrategies.
Chapter 2
Analytical Outlook
2.1 Foreign Exchange Market PredictabilityResearch in the area of exchange rates is vast and any attempt to survey the exchange rate theory in its totality would be impossible. Consequently, the objectiveof Part II is more moderate. It provides a partial review of the classical conceptsof exchange rate determination theory and financial theory. In the last few decades,exchange rate economics has seen a number of developments, with substantial contributions to both the theory and empirics of exchange rate determination. Importantdevelopments in econometrics and the increasing availability of high-quality datahave also been responsible for stimulating the large amount of empirical workon exchange rates. We restrict ourselves to explore the fundamental parity conditions purchasing power parity (PPP) and interest rate parity (IRP) as prerequisitesfor the understanding of economic equilibrium along with explanations given byeconomists on why exchange rates in reality do often not behave according to theselaws. Furthermore the notions of market efficiency are highlighted, upon whichstandard finance theory is built.While our understanding of exchange rates has significantly improved, empiricalstudies have revealed a number of challenges and open questions in the exchangerate debate which have remained unsolved. For instance, while all of the potentialexplanations for deviations from equilibrium conditions appear fairly reasonableand have theoretical merit, the PPP disconnect and PPP excess volatility puzzles,along with the forward anomalie puzzle have not yet been convincingly explainedand continue to puzzle the international economics and finance profession. Furthermore, it is a well-known phenomenon that the forward rate is not an unbiased predictor for the expected exchange rate as the market efficiency hypothesis postulates.It is argued that this inexplicability has nothing to do with economic and financialtheorists being not capable enough, but rather with the complex dynamics that driveexchange rates, the computational complexity of behaving according to economictheory, and the nature of complexity itself. For these reasons, selected proveablemathematical results on the difficulty of calculating market equilibrium, and thecomputational difficulties with market efficiency are provided.C. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 2,c Springer-Verlag Berlin Heidelberg 2009
7
8
2 Analytical Outlook
2.2 Exchange Rate Forecasting with Support Vector MachinesIn Part III, we address the problem of predicting daily exchange rate directions withSupport Vector Machines (SVM). The construction of machines capable of learningfrom experience has for a long time been the object of both philosophical and technical debate. The technical aspect of the debate has received an enormous impetusfrom the advent of electronic computers. They have demonstrated that machines candisplay a significant level of learning ability, though the boundaries of this abilityare far from being clearly defined. The availability of reliable learning systems isof strategic importance. When computers are applied to solve a practical problemit is usually the case that the method of deriving the required output from a set ofinputs can be described explicitly. As computers are applied to solve more complex problems, however, situations can arise in which there is no known method forcomputing the desired output from a set of inputs, or where that computation maybe very expensive.This motivates the use of an alternative strategy where the computer is instructedto attempt to learn the input/output functionality from examples, which is generallyreferred to as supervised learning, a sub-discipline of the machine learning field ofresearch. Machine learning models are rooted in artificial intelligence which differs from economic and econometric theory since statistical inferences are madewithout any a priori assumptions about the data. Intelligent systems are thereforedesigned to automatically detect patterns, i.e., any relations, regularities or structure inherent in a given dataset, that exists despite complex, nonlinear behavior. Ifthe patterns detected are significant, a system can be expected to make predictionsabout new data coming from the same source. Thus, intelligent systems are datadriven learning methodologies that seek to approximate optimal solutions of problems of high dimensionality. Since, in contrast, theory driven approaches give riseto precise specifications of the required algorithms for solving simplified models,the search for patterns replaces the search for reasons.Our starting point is to examine the degree of randomness inhibited in the chosenEUR/USD, EUR/GBP, and EUR/USD time series. The strategy is to build econometric models in order to extract statistical dependencies within a time series thatmay be based on linear and/or nonlinear relationships. If such dependencies are significant, then the time series is not totally random since it contains deterministiccomponents. These may be important indicators for the predictability of exchangerate returns. Second, we use time series analysis methods for building empiricalmodels that will serve as benchmarks for the SVM specifications.We represent the underlying problem of forecasting exchange rate ups and downsas a classification task. In supervised learning, it is assumed that a functional relationship is implicitly reflected within the input/output pairings. The estimate ofthe target function which is learnt or output by the learning algorithm is knownas the solution of the learning problem. There exist many possible target functionsthat can separate the data. However, there is only one that maximizes the distancebetween itself and the nearest data point of each class. The problem of learningthis optimal target function is formally represented by the binary classification
2.2 Exchange Rate Forecasting with Support Vector Machines
9
problem (BCP). The BCP is a known problem in the field of computational geometry, which is the branch of computer science that studies algorithms for solving geometric problems. The input to a computational-geometry problem is typically a description of a set of geometric objects, such as a set of points, a set ofline segments, or the vertices of a polygon. The output is a response to a queryabout these objects, or even a new geometric object. In its most general form, thecase of whether two sets of points in general space can be separated by k hyperplanes, the BCP is known to be NP-complete, i.e., not solvable by a polynomial timealgorithm.In order to address the BCP, we propose the use of SVM, a state-of-the-art supervised learning system which, based on the laws of statistical learning theory [405],maps the input dataset via kernel function into a high-dimensional feature spacein order to enable linear data classification. SVM has proven to be a principled andvery powerful method that in the few years since its introduction has already outperformed many other systems in a variety of applications. The forecasting approachthat we adopted is what we call a statistical or purely data driven approach thatborrows from both fundamental and technical analysis principles. It is fundamentalsince it considers relationships between the exchange rate and other exogenous financial market variables. However, our approach also has a technical component: itis somewhat irrational in a financial context since it depends heavily on the conceptsof statistical inference.Ever, since the introduction of the SVM algorithm, the question of choosing thekernel has been considered as very important. This is largely due to the effect thatthe performance highly depends on data preprocessing and less on the linear classification algorithm to be used. How to efficiently find out which kernel is optimalfor a given learning task is still a rather unexplored problem and subject to intensecurrent research. We choose to take a pragmatic approach by comparing a range ofkernels with regards to their effect on SVM performance. The argument is that evenif a strong theoretical rational for selecting a kernel is developed, it would have tobe validated using independent test sets on a large number of problems. The evaluation procedure is twofold. Out-of-sample forecasts are evaluated both statisticallyvia confusion matrices and practically via trading simulations.The results of Part III shed light on the existence of a particular kernel function which is able to represent properties of exchange rate returns generally wellin high dimensional space. In particular, it is found that hyperbolic SVMs performconsistently well in terms of forecasting accuracy and trading performance via asimulated strategy. Moreover, we find that p-Gaussian SVMs, which have hardlybeen tested yet on real datasets, but in theory, have very interesting properties, perform reasonably well in predicting EUR/GBP and EUR/USD return directions. Theresults can be valuable for both practitioners, including institutional investors, private investors, risk managers, and researchers who will focus on SVM models andtheir technical improvements. We strongly believe, that given the shortcomings ofeconomic and financial theories in explaining real-world market behaviors, modernmethods of machine learning, pattern recognition, and empirical inference will playan essential role in coping with such complex environments.
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2 Analytical Outlook
2.3 Exchange Rate Hedging in a Simulation/OptimizationFrameworkIn Part IV, it is implicitly assumed that a way to understand corporate hedging behavior is in the context of speculative motives that could arise from eitheroverconfidence or informational asymmetries. A model for managing currencytransaction risk is developed with the objective to find a possibly optimal combination of linear and nonlinear financial instruments to hedge currency risk over aplanning period such that the expected utility at the planning horizon is maximized.We require the goal function to address the conflicting empirical finding that firmsdo like to try to anticipate events, but that they also cannot base risk managementon second-guessing the market. For this purpose, we assume the fictitious firm tohave different future expectations than those implied by derivative prices. Whenaddressing these questions, one must recognize that traditional mean–variance ormean–quantile-based performance measures may be misleading if products withnonlinear payoff profiles such as options are used. In addition, the present knowledge that underlies the field of decision making is simple principles that definerationality in decision making and empirical facts about the cognitive limits thatlead us not to decide rationally. As such, it has become a stylized fact that individuals perceive risk in a nonlinear fashion.For these reasons, it is proposed to embed a mean–variance–skewness utilitymaximization framework with linear constraints in a single-period stochastic combinatorial optimization problem (SCOP) formulation in order to find optimal combinations for forward and European style straddle option contracts given preferencesamong objectives. The advantage of using SCOPs over deterministic combinatorialoptimization problems (DCOPS) is that the solutions produced may be more easilyand better adapted to practical situations where uncertainty cannot be neglected.The use of SCOPs instead of DCOPs naturally comes at a price: first, the objectivefunction is typically much more computationally demanding. Second, for a practicalapplication of SCOPs, there is the need to assess probability distributions from realdata or subjectively, a task that is far from trivial. We show that the proposed SCOPis computationally hard and too difficult to be solved analytically in a reasonableamount of time. This suggests the use of approximate methods, which sacrifice theguarantee of finding optimal solutions for the sake of getting good solutions in asignificantly reduced amount of time. Furthermore, computation of the objectivefunction is nonconvex and nonsmooth. The solution space of the resulting optimization problem therefore becomes fairly complex as it exhibits multiple local extremaand discontinuities. In order to attack the given problem and derive near-optimaldecisions within a reasonable amount of time, a simulation/optimization procedureis suggested. Simulation/optimization has become a large area of current researchin informatics, and is a general expression for solving problems where one has tosearch for the settings of controllable decision variables that yield the maximum orminimum expected performance of a stochastic system as presented by a simulationmodel. Our approach to simulation/optimization addresses real-world complexityby incorporating the following technical features.
2.3 Exchange Rate Hedging in a Simulation/Optimization Framework
11
For modeling the exchange rate, a smooth transition nonlinear PPP reversionmodel is presented. Its key feature is a mathematical function which allows forsmooth transition between exchange rate regimes, symmetric adjustment of theexchange rate for deviations above and below equilibrium, and the potential inclusion of a neutral corridor where the exchange rate does not mean revert butmoves sideways. Apart from equilibrium which is given by PPP, only two parameters need to be estimated: the speed of PPP reversion and exchange rate volatility.We believe that the proposed exchange rate model is very attractive. It includes along-run component (PPP), a medium-term component (adjustment speed), and ashort-term component (volatility) which, depending on their estimated values, mayovershadow each other and therefore cover a variety of phenomena observed in thereal foreign exchange market.Optimization of complex systems has been for many years limited to problemsthat could be formulated as mathematical programing models of linear, nonlinear and integer types. The best-known optimization tool is without a doubt linearprograming. Linear programing solvers are designed to exploit the structure of awell defined and carefully studied problem. The disadvantage to the user is thatin order to formulate the problem as linear program, simplifying assumptions andabstractions may be necessary. This leads to the popular dilemma of whether tofind the optimal solution of models that do not necessarily represent the real systemor to create a model that is a good abstraction of the real system but for whichonly very inferior solutions can be obtained. When dealing with the optimizationof stochastic systems, obtaining optimal values for decision variables generally requires that one has to search for them in an iterative or ad hoc fashion. This involvesrunning a simulation for an initial set of values, analyzing the results, changingone or more values, rerunning the simulation, and repeating the process until asatisfactory (optimal) solution is found. This process can be very tedious and timeconsuming and it is often not clear how to adjust the values from one simulationto the next. The area of metaheuristics arose with the goal of providing somethingbetter than simple searching strategies (e.g., grid search) by integrating high-levelintelligent procedures and fast computer implementations with the ability to escapelocal optimal points. The specific metaheuristic we use is a variant of the scattersearch algorithm, which has proven to be highly successful for a variety of knownproblems, such as the Vehicle Routing Problem, Tree Problems, Mixed IntegerPrograming, or Financial Product Design.In order to show our simulation/optimization model’s applicability in a practical context, a case study is presented, where a manufacturing company, locatedin the EU, sells its goods via a US-based subsidiary to the end-customer in theUS. Since it is not clear what the EUR/USD spot exchange rate will be on future transaction dates, the subsidiary is exposed to foreign exchange transactionrisk under the assumption that exposures are deterministic. We take the view thatit is important to establish whether optimal risk management procedures offer asignificant improvement over more ad hoc procedures. For the purpose of modelvalidation, historical data backtesting is carried out and it is assessed whether theoptimized mean–variance–skewness approach is able to outperform nonpredictive,
12
2 Analytical Outlook
fixed-weight strategies such as unitary spot, forward, and straddle, as well as amixed strategy over time. We compare the alternative strategies in dynamic backtesting simulations using market data on a rolling horizon basis. The strategies areevaluated both in terms of their ex ante objective function values, as well as in termsof ex post development of net income.We find in our experiments that scatter search is a search method that is bothaggressive and robust. It is aggressive because it finds high-quality solutions early inthe search. It is robust because it continues to improve upon the best solution whenallowed to search longer. We find that our approach to hedging foreign exchangetransaction risk adds value in terms of reducing risk and enhancing income. Theoptimized mean–variance–skewness strategy provides superior risk-return results incomparison to the passive strategies if earnings risk is perceived asymmetrically interms of downside risk. Even with low levels of predictability, there is a substantialloss in opportunity when fixed-weight strategies (which assume no predictability)are implemented relative to the dynamic strategy that incorporates conditioninginformation. In particular, the pure forward strategy is found to have the lowestreturn per unit of earnings risk whereas the European style straddle strategy andthe 1/3 strategy reveal similar risk-return characteristics. Interestingly, our researchalso contrasts the finding that currency forward contracts generally yield better results in comparison to options, since a passive straddle strategy would have yieldedsuperior results compared to a forward strategy. Apart from our backtesting results,it is believed that the proposed simulation/optimization procedure for determiningoptimal solutions has important implications for policy making. Having easy accessto relevant solutions makes it easier for policy makers to explore and experimentwith the model while incorporating their own intuition before deciding on a finalplan of action. Despite the many problems that economic forecasts from economicsystems confront, these models offer a vehicle for understanding and learning fromfailures, as well as consolidating our growing knowledge of economic behavior.
Chapter 3
Equilibrium Relationships
3.1 Purchasing Power Parity TheoremPurchasing power parity (PPP) theory states that in the long run, the exchange ratebetween the currencies of two countries should be equal to the ratio of the countries’ price levels. PPP theory has a long history in economics, dating back severalcenturies, but the specific terminology was introduced in the years after World WarI during the international policy debate concerning the appropriate level for nominal exchange rates among the major industrialized countries after the large-scaleinflations during and after the war [69]. Since then, the idea of PPP has become embedded in how many international economists think about the world. For example,Dornbusch and Krugman [102] noted:Under the skin of any international economist lies a deep-seated belief in some variant ofthe PPP theory of the exchange rate.
Rogoff [342] expressed much the same sentiment:While few empirically literate economists take PPP seriously as a short-term proposition,most instinctively believe in some variant of purchasing power parity as an anchor for longrun real exchange rates.
Macroeconomic literature distinguishes between two notions of PPP: Absolute PPPand Relative PPP.
3.1.1 Absolute PPPPPP follows from the law of one price, which states that in competitive markets,individual identical goods will sell for identical prices when valued in the samecurrency at the same time. The reason why the law of one price should hold is basedon the idea of frictionless goods arbitraging: if prices were not identical, peoplecould make a riskless profit by shipping the goods from locations where the priceC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 3,c Springer-Verlag Berlin Heidelberg 2009
15
16
3 Equilibrium Relationships
is low to locations where the price is high. If the same goods enter each market’sbasket used to construct the aggregate price level – and with the same weight –then the law of one price implies that PPP exchange rate should hold between thecountries concerned.1Absolute PPP holds when the purchasing power of a unit of currency is exactlyequal in the domestic economy and in a foreign economy, once it is converted intoforeign currency at the market exchange rate. Let Pt be the price of the standardcommodity basket in domestic terms and Pt∗ the price of the same basket in a foreigncountry. Formally, absolute PPP states that the real exchange rate Rt between twocountries should beP∗(3.1)Rt = St t = 1Ptwhere St denotes the nominal exchange rate in domestic currency per unit of foreigncurrency. Taking the natural logarithm, absolute PPP can be stated as ∗Pln(St ) + ln t = 0(3.2)Ptor
st = pt − pt∗
(3.3)
Hence, the real exchange rate in its logarithmic form may be written asrt = st + pt∗ − pt
(3.4)
3.1.2 Relative PPPIn a world where market participants have rational expectations, several factorsmight doubt the existence of absolute PPP. For instance, it is unrealistic to assumethat all goods are identical, tradable, that there are no transportation costs, taxes,tariffs, restrictions of trade, or border effects and that competition is perfect. Thus,it has become common to test relative PPP, which holds if the percentage changein the exchange rate over a given period just offsets the difference in inflation ratesin the countries concerned over the same period. If transportation costs and traderestrictions are assumed to be constant (c), relative PPP can be written as1 Nevertheless, the presence of any sort of tariffs, transport costs, and other nontariff barriers and
duties would induce a violation of the law of one price. Also the assumption of perfect substitutability between goods across different countries is crucial for verifying the law of one price. Inreality, however, product differentiation across countries creates a wedge between domestic andforeign prices of a product which is proportional to the freedom with which the good itself can betraded. An example often cited in the literature is the product differentiation of McDonald’s hamburgers across countries. Examples for which the law of one price may be expected to hold are goldand other internationally traded commodities [342]. For a discussion on the empirical evidence ofthe law of one price see [357].
3.1 Purchasing Power Parity Theorem
17
ln(St ) = c + ln
PtPt∗
(3.5)
After taking first differences, price level changes determine the exchange ratedevelopment PtΔ ln(St ) = Δ ln(3.6)Pt∗The expected exchange rate change thus equals the difference in inflation of therespective countries. Thus, if the EU price level rose 10% and the US price levelrose 5%, the Euro would depreciate 5%, offsetting the higher EU inflation andleaving the relative purchasing power of the two currencies unchanged. If absolute PPP holds, then relative PPP must also hold. However, if relative PPP holds,then absolute PPP does not necessarily hold, since it is possible that commonchanges in nominal exchange rates are happening at different levels of purchasingpower.
3.1.3 Empirical EvidenceThe validity of absolute and relative PPP and the properties of PPP deviations havebeen the subject of an ongoing controversy in economic and econometric literature.Until the 1970s, early empirical studies have examined by ordinary least squares(OLS) regression analysis, whether there is a significant linear relationship betweenrelative prices and the exchange rate. In general, such tests led to a rejection ofthe PPP hypothesis except for countries exhibiting high inflation [136] which suggested that PPP may represent an important benchmark in long-run exchange ratemodeling. However, the OLS method suffers from econometric weaknesses since itignores the econometric requirement that residuals of the estimated regression equation must be stationary. If stationarity is not given, the use of nonstationary data canlead to a spurious regression, i.e., nonsense correlation [172], which makes it impossible to validly undertake conventional OLS based statistical inference procedures.Otherwise, if linear regression residuals are stationary, then although a strong longrun linear relationship exists between exchange rates and relative prices, it is stillinvalid to undertake conventional statistical inference because of the bias present inthe estimated standard errors [26, 116]. Thus, although some researchers could notreject the argument that PPP deviations are nonstationary and therefore permanent(for instance [6, 133, 343]), it does not necessarily mean that these findings must beaccepted.In 1987, the concept of cointegration emerged [116, 209, 210] which seemed tobe an ideal approach to testing for PPP. A cointegrated system requires that any twononstationary series which are found to be integrated of the same order are cointegrated if a linear combination of the two exists which is itself stationary. If thisis the case, then the nonstationarity of one series exactly offsets the nonstationarity of the other series and a long-run relationship is established between the two
18
3 Equilibrium Relationships
variables. Testing for no cointegration led to mixed results. While some studies inthe late 1990s resulted in reports about the absence of significant mean reversion ofthe real exchange rate for the recent floating experience [276], other applied workon long-run PPP among the major industrialized economies has been more favorable towards the long-run PPP hypothesis (e.g., [72, 73, 82, 228]).One important reason for the fact that statistical tests of the 1980s to examinethe long-run stability of the real exchange rate often failed was given by [133, 134](followed by [142, 268]) who argued that these tests are not powerful enough. Thispower problem cannot simply be solved by increasing the sample size from monthlyto daily data, since increasing the amount of detail concerning short-run movementscan only give more information about short-run as opposed to long-run behavior[367]. In order to get more information about the long-run behavior of a particularreal exchange rate, one approach is to use more years of data. However, long periods of data may span different exchange rate regimes, and therefore exhibit regimeswitches or even structural breaks. In an early study in this spirit, using annual datafrom 1869 to 1984 for the USD/GBP real exchange rate, [133] was able to reject thehypothesis of a random walk at the 5% level. Similar results to Frankel’s were obtained by [160], and [268] which were also unable to detect any significant evidenceof a structural break between the pre- and post-Bretton Woods period. Taylor [382]extended the long-run analysis to a set of 20 countries over the period 1870–1996and also finds support for PPP and coefficients that are stable in the long run. A different approach to providing a convincing test of real exchange rate stability, whilelimiting the time frame to the post-Bretton Woods period, is to increase the panel byusing more countries. By increasing the amount of information employed in the testsacross exchange rates, the power of the test should be increased. Abuaf and Jorion[2] examined a system of ten first-order autoregressive regressions for real dollarexchange rates over the period 1973–1987, where the autocorrelation