Please cite this article as: E. Sanidas, Four haTechnol. Forecast. Soc. Change (2013), http
a b s t r a c t
Article history:Received 27 November 2010Received in revised form 18 November 2013Accepted 20 November 2013Available online xxxx
In theory and practice, it is difficult to accept a particular model of explanation and forecastingof exchange currencies as the literature review reveals. However, the currencies that are calledcommodity currencies, such as the Australian dollar are heavily influenced by commodityprices cycles, and hence they might be easier to analyze and predict. We investigate thepossibility that the Australian dollar is primarily determined by a handful of harmonic cycleswhich in turn are based not only on commodity prices cycles but also on commodityproduction cycles and in general on economic cycles. In this way we can get a very good fit ofthe relevant data and good out-of-sample forecasts. We cross check these results by referringto the main issues involved, such as fundamentals, short and long cycles, and so on. In addition,our analysis, forecasting ability, and conclusions still hold for three more commoditycurrencies examined here: New Zealand's dollar, Canada's dollar, and Norway's krone.
Although the literature on exchange rates is alreadysubstantial, the record of having been able to properlyunderstand and in particular to forecast these rates has beenpoor.Many ‘puzzles’have beenmentioned in numerous papers,as the literature review shows further below. The aim of thepresent paper is to attempt to clarify these puzzles by followinga non-conventional approach: we want to put more emphasison the merits of forecasting even if it is not very clear how thefundamentals justify these forecasts. We will claim that aminimum number of four harmonical cycles can explain asubstantial (about 85%) of theAustralian dollar (A$), a so-calledcommodity currency; and that these harmonics can improve
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rmonic cycles explain an://dx.doi.org/10.1016/j.t
forecasts considerably. Similar conclusions are reached for theother three commodity currencies examined here: NewZealand's dollar, Canada's dollar, and Norway's krone.
However, wewill not ignore a discussion and cross checkingof the underlying fundamentals. There are some importantreasonswe chose the A$ for a full analysis in this study.2 First, itis rather the only genuine commodity currency in the sensethat it is the only one that freely floats (hence it is not pegged,or floating in amanagedway, etc.; see [1]). Second, theA$ beingstrongly associated with trade of metals, is also associated withtechnological issues that are relevant to the supply of thesegoods [2–4]. And third, the ratio of commodity exports out oftotal Australian exports is very high (about 75%).
2 From this point till the subsection 3.4 we will only examine the A$exclusively. Then in 3.4 we will briefly examine the other three currencies.
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
2 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
It becomes then necessary to briefly remind the readerabout the floating history of the A$. Up to November 1971 itwas fixed to sterling, then to the US dollar up to September1974, and then to the trade weighted index (TWI). FromNovember 1976 to December 1983 it was set in terms of theTWI under a crawling peg system [5]. The Australian dollarstarted freely floating in December 12, 1983. Fig. 1 showsthe nominal A$ in terms of US$ since December 1983. It isworth noting the wide and persistent fluctuations of thiscurrency. The same Figure also shows the commodityprices series.
It will also be useful to briefly describe the Australianeconomy. Its population, about 21 million, live in a verylarge country/continent with several big cities and a smallrural population. Australia is rich in natural resources. Themanufacturing sector never became the dominant sector inthe economy. Exports consist of about 60 to 80% naturalresources or nearly defined natural resources. Australia canbe considered as a “small” country for the purpose of thisanalysis: it cannot influence world prices, and is a price takeroverall. Thus, Australia's dollar is one of these commoditycurrencies [7] that are heavily influenced by commodityprices. Usually these currencies are those of small econom-ically defined countries hence not being able to influence theworld economy and to a considerable extent commodityprices.
Consequently “the A$ appreciates (depreciates) in bothnominal and real terms when the prices of certain com-modities exported by Australia, e.g. coal, metals, and otherprimary industrial materials, rise (fall) in internationalmarkets” ([8] p. 82). On the other hand, Hatzinikolaou andPolasek ([8] p. 83) observed that “our estimate of the longrun elasticity of the exchange rate with respect to commod-ity prices is 0.939 and statistically not different from unity,which strongly supports the commodity–currency hypo-thesis”. The main commodities having the largest percent-age in the construction of the commodity price index inAustralia are: gold, coking coal, beef and veal, steaming coal,iron ore, wool, and aluminum [6].
Source: the source of data is RBA [6].
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Fig. 1. The floating nominal A$ and commodity prices series [6].Source: the source of data is RBA.
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
This brief introduction leads us to believe that we ought tofurther examine commodity currencies such as the A$. Thiswill be carried out not only in relation to commodity pricescycles, but also in relation to their harmonics (hence indi-rectly involving economic cycles as well). Section 2 brieflyexamines commodity currencies in the context of severaltypes of relevant cycles. Section 3 introduces the proposedmodel and provides econometric evidence, which allows usto make some forecasts in the medium and long term.Section 4 concludes.
2. Commodity currencies and cycles
2.1. Links to fundamentals
De Grauwe and Grimaldi [9] have summarized some ofthe main issues and explained why the fundamentals ofexchange rates do not seem to work properly. For examplethere is the puzzle of excess volatility according to which thevolatility of the exchange rate by far exceeds the volatility ofthe underlying economic variables (this is also confirmed inour paper). It became clear that monetary instability alonecould not possibly explain the persistent exchange rate volatilitythat remains even to this date. “But the long half-lives of shocksobserved in the data are incompatible with the concept oflong-run monetary neutrality…a potential solution to this PPPpuzzle may lie in identifying a shock that is both sufficientlyvolatile and persistent…” [7]. Thus, although the purchasingpower parity (PPP) hypothesis was strongly suggested asexplaining the behavior of exchange rates, recent researchdismisses this hypothesis ([10]; for the Australian dollar see[11]).
Chen and Rogoff [7] in particular treated commodityprices as the missing link in the PPP puzzle. Nonetheless, thepuzzle remained, even after having included these prices intoregression estimations by using autoregressive (AR) models.These authors also included the Balassa–Samuelson relativeproductivity differences in their models. Their overallconclusion is very revealing: “Hence, we find the PPP puzzleto be like the Russian dolls, in that after controlling for twopromising real shocks-peeling away layers of the original PPPpuzzle—we are still faced with the identical, despite smaller,PPP puzzle” ([7] p. 25).
The speed of convergence to PPP in the long run isextremely slow; persistent deviations from PPP exist in thelong run. For the so called commodity currencies, even aftertaking into account the effect of commodity price shocks,there is still a purchasing power puzzle in the residuals,implying that there is high degree of persistence remaining[7]. If PPP holds then the real exchange rate is stationary andfluctuates around a fixed value (its mean) in the short run.PPP can be tested by testing for unit roots in a univariateframework or by applying cointegration methods in a multi-variate framework.
The massive empirical testing of PPP has generally castdoubt on long run PPP, either by rejecting PPP that follows astationary process, or by suggesting that the real exchangerate adjusts too slowly back to an equilibrium long run rate.Consequently, as Spiro [12] reminds us forecasts beyond athree-year horizon, based on PPP tend to outperform othertechniques. However, Mansur et al [13] have showed that the
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PPP hypothesis is still valid for the Australian dollar becausethey used fractional cointegration, and hence low frequencydynamics (which means longer cycles). This last point will betaken up in this present paper for further analysis.
Cashin et al [14] have shown that the Australian currency(as well as 18 other currencies) is clearly cointegrated withcommodity prices and that the PPP model is subsequently aweak model for countries like Australia. A visual examinationof the commodity price and exchange rate series (see Fig. 1)show that noticeable peaks and troughs occur during ap-proximately the same years for both series. To support this‘coincidence’ we should bring some evidence as to howimportant is the component of Australian primary productsin the world production and in the Australian total exports.As Humphreys ([15] p. 5) says: “Australia has managed toincrease its share of world iron ore production to 20% from13%, its share of nickel mine production to 17% from 8%, itsshare of copper mine production to 7% from 4%, and its shareof zinc mine production to 17% from 13%”.
This observed and confirmed link between commoditycurrencies and commodity prices should then furtheranalyzed by examining commodity prices series in moredetail. Thus, there is substantial evidence that commodityprices are governed by regular cycles. More precisely, Cashinet al ([16] pp. 282-3) have found that most commodities(rural and mineral) have price cycles of an average durationof between 6 and 8 years. Jalali-Naini and Asali [17] showedthat crude oil prices have cyclical behavior and have a cycleof an average duration of 4.29 years from 1972 to 2002, and13.5 years from 1861 to 2002. Herendeen and Hallberg [18]attempted to provide an answer to their question: “whatdrives agricultural cycles?” in the USA and “why did theagricultural boom last seven years, andwhy did the bust alsolast seven years?”
We must also note that in the long run, the real price ofnon-renewable resources, such as copper seems to bestationary (in particular there is no downward trend) asSvedberg and Tilton [19] have demonstrated. This remarkcontributes to our model because in the long run commoditycurrencies seem to be also stationary. In addition, it isimportant to stress that most commodities are cointegratedthus generating a common cycle for themselves [20,21].Furthermore, industrial commodity prices should not andcannot be absent in a more comprehensive analysis. This iseven more pertinent if we observe that recently in the last40 years or so the peaks and troughs of industrial commodityprices seem to coincide with the prices of all other com-modity prices [22]. Overall, a long run relationship betweenthe real exchange rates and commodity prices can beestablished in 40% of the commodity exporting countries[16].
2.2. Links to specific cycles
This brief foregoing review of our knowledge on commodityprice cycles solicits an answer as towhat the deeper reasons arefor these cycles. Some of these reasons are related to productionand consumption of commodities. Thus, for example, Cortazarand Casassus [23] have shown that the optimal timing of amineexpansion is intrinsically related to changes in copper prices viathe investment process. Achireko and Ansong [24] have
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
demonstrated that gold prices are required in mine valuations.The relationship between supply and demand as well asinventories in mineral production has been well evidenced(e.g. [2]). Baldursson [25] suggested a 10 year inventory cyclefor the aluminum industry and linked this cyclewith themetal'sprice. Gilbert [26] examined the aluminum market in a verydetailed way by linking demand, supply, stocks, expectations,imports, capacity, lagged responses, short and long termhorizons, according to a competitive storage model. The ap-parent cycle of about 8 years (as seen in his graphs) is thuswellreproduced through this model (although the author is notdirectly preoccupied with the determination of the cycleperiod).
Are commodity cycles linked with economic cycles orany other cycles? Modis [27] explains that research onvarious types of cycles contains both difficulties andrewards. It is not always easy to justify the use of cycles invarious disciplines and also in economics. However, severalresearchers have attempted to link cycles to economicactivities. Moore [28] provided explanations of the 8-yearcrop cycles as being linked to meteorological cycles and toVenus's 8-year cycle of motion in relation to the Earth andthe Sun. Labys and Maize [29] have found evidence thatcommodity price fluctuations impact on output, wages andemployment, interest rates, and so on, thus on businesscycles. Hua ([30] p. 775) has shown that commodity pricesand macroeconomic/monetary variables are cointegrated:“we conclude that there is only one strongly significant longrun economic relationship between the levels of non-oilprimary commodity prices and the levels of economicactivities, dollar exchange rate, and oil prices”. As Cashin etal ([16] p. 292) have remarked, “cycles in economic activityalone do not drive the evolution of commodity prices, andthat other factors, particularly supply conditions in individ-ual commodity markets, are likely to be a key determinant ofcycles in commodity prices”.
Watkins and McAleer ([31] p. 878) found that “in mostof the samples considered for the sevenmetal markets, testsfor cointegration determined the existence of one statisti-cally significant long run relationship among the futuresprice, spot price, stock level and interest rate”. Cashin andMcDermott ([32] p. 249) ascertained that when Australiawas still a non-industrialized country “the wool exportcycle was in synchronization with the business cycle priorto the First World War”. De Miguel et al [33] establishedthat oil price shocks and the business cycle in Spain havebeen interconnected in a significant way. Deaton andLaroque ([34] p. 290) have demonstrated that “the com-modity price is stationary around its supply price, andcommodity supply and world income are cointegrated”.Black et al [35] examined the economic impact of the coalboom and bust.
Regarding economic cycles, A'Hearn and Woitek ([36]p. 321) have clearly established that “in the advanced NorthAtlantic economies, a fairly regular long cyclewith a periodicityof 7–10 years is identified in all countries” (at least for the UK,Germany, France, the Netherlands, the US and Canada, thestatistical testing is very clear). This cycle explains about 40% ofoverall fluctuation in industrial output ([36], p. 341). Theseauthors also found a less regular and less powerful short cycleof 3–5 years duration. Zarnowitz ([37] p. 6) suggested that “the
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
3 Here we use the term “resonance” to indicate that even if the true cyclemight be, for example, 7.95 years, in the end, as a consequence of theinteractive process between the four cycles, the resonance effect will lead toan exact harmonic relationship between all cycles considered here (thus 4,8, 16, and 32).
4 With harmonics, it is not important whether we choose the 8-year cycleas the basis cycle or not.
4 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
average duration of peacetime cycles in the US is about fouryears”. Both these cycles (of 7–10 and 3–5 years periods)correspond to previously researched similar cycles called Juglarand Kitchin respectively ([37] p. 322), and are due toinvestment in equipment. In addition there is the constructioncycle of about 15–20 years period, usually called the Kuznetscycle ([37] p. 323).
Bodger et al [38] have clearly uncovered a 32-year cycle inthe world energy consumption series which might also berelated to Kuznets cycles. Furthermore, these authors alsoconclude that “there is clear evidence of international synchro-nization of long term fluctuations in the advancedNorth-Atlanticeconomies” ([38] p. 341). Samuelson [39] recognized theimportance of cycles of various durations as we just mentionedand attempted to explain them in terms of capital formation,saving, investment and so on. Cycles of longer duration(probably 40–50 years) such as the Kondratieff cycles have alsobeen studied (see for example, [40]). Other important articles oneconomic cycles confirming this brief review are: [41–47]. ForAustralian business cycles, see [48, 49].
How are economic cycles and commodity currenciesrelated? In general the latter are shock absorbers for somany other economic phenomena represented by economiccycles; these phenomena are, for example, related to imports,money supply, and wages. In the very short term, variablessuch as the interest rate can affect the exchange rate ofcommodity currencies, however, in the longer term econom-ic cycles are supposedly the source of interaction betweenimports, money supply, and wages. For example, if thenominal exchange rate was not allowed to depreciate toabsorb the effects of decreasing real commodity prices,domestic wages would have to decrease instead.
Kearns [50] built a model that relates the A$ with theprice of commodity exports (hence, also their cycles), theprice of imports, non-traded output, and the domesticmoney supply, and thus indirectly linked the commodityprice cycles with economic cycles. Karfakis and Phipps([51] p. 272) have similarly concluded that “movements inthe terms of trade account for much of the long-runvariation in the exchange rate” of the Australian dollar(thus having already included changes in relative pricelevels and interest rate differentials). Mills and Pentecost[52] in examining the US versus UK data found that there isa cyclical relationship between the real exchange rate andrelative outputs of these two countries, the direction ofcausality being from outputs to the exchange rate, withonly week feedback effects.
3. The proposed model and econometric results
3.1. Preliminary investigations
In this paper the Fourier analysis will indicate economet-rically how pertinent the assumption that the A$ can beoverwhelmingly approximated by 3–4 cycles harmonic toeach other is (it is efficient to be parsimonious, hence thefewer the cycles the better). Some important points cansummarize the Fourier analysis (see [53,54]) as it will beused here. A time series f(t) can be fully represented by thesum of sinusoidal functions for all harmonic frequencies.Thus it becomes significant for our paper that if we can isolate
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
only a few such sinusoidal functions that mostly explain theinitial series f(t) (for example, a high R2 will indicate howimportant these cycles are) then we can provide evidence thatour preliminary theoretical discussion lies on the correct track.More precisely mathematically we can summarize the salientpoints of Fourier functions as follows:
f tð Þ ¼ Aþ C cos ω0t þ ϑð Þ ¼ Aþ B1 cos ω0tð Þ þ B2 sin ω0tð Þ ð1Þ
where C is the amplitude, ω0 is the angular frequency (relatedto frequency f as in cycles per time by ω0 = 2πf and f = 1/T,where T is the period of the cycle); and θ is the phase shift.When Eq. (1) is expressed as a function of both cosine and sine,and estimated in an ordinary least squares (OLS) regression theamplitude and phase shift, if needed, can be indirectlycalculated as a function of B1 and B2. If we want to includemore than one angular frequency, as in harmonicswe can havemultiples of ω0 as in 2 ω0 or 3 ω0, etc. and run a multipleregression with as many independent variables as the numberof harmonics plus the fundamental frequency ω0. In practice,two consecutive harmonics introduce a change in the ampli-tude of the longer period cycle, thus allowing for variousheights in amplitude once we combine all consideredharmonics.
Following this brief sketch on Fourier analysis, in thispaper, we include harmonics into the picture. First, we candetect these harmonics according to cycles suggested earlierin the Introduction. Thus, given the basic cycle of commodityprices of about 8 years (as already shown in the literaturealready reviewed), we can also find a minimum number ofother cycles of multiple duration (hence harmonics) whichall together might explain a large proportion of the totalvariance of the A$. Thus cycles of 4 years, 12, 16, 24, etc.could be included in the explanation of the A$ time series.The rationale which will be followed in this paper is that thecommodity prices cycle of about 8 years will producethrough harmonical vibrations and resonance3 some otherharmonic cycles whose duration are multiples of 8: 16, 32,and 4 (and 2 or 64 if necessary). The mechanism of thisprocess is very simple: every alternate slowdown (or peak)of the 8-year cycle4 tends to be more severe, hence theexistence of 16 or 32-year cycles (hence also the existence ofdifferent amplitudes).
Second, the detection of the proposed cycles can also becarried out empirically. A preliminary inspection of the A$series for detection of cycles can be carried out through itsspectrum; with about 310 monthly data (starting fromDecember 1983) and by allowing a wide window we getpeaks around the 45th month (3.75 years) and 180th month(15 years); however, the data series is not long enough for adetailed and reliable analysis. A more time consumingapproach is to calculate the R2 of all regressions with variousharmonics: e.g. 2, 4, 8, and 16; or 3, 6, 12, and 24; or 3.8, 7.6,15.2, and 30.4 years, etc. Or more efficiently we can calculate
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
the R2 of all regressions based on periods 1 to 40 years andby augmenting the years by 0.25 years each time. This“approach can be viewed as a regression variant of spectralanalysis” ([55] p. 1024). The highest R2 will indicate to ussome of the most important cycles of the A$ series (seeFig. 2).
More precisely, we first regress the A$ against time andconstant, and then regress the residuals thus obtained againstthe harmonics as mentioned above (note that the non-detrended A$ series provided the same or very similar results).The R2 “periodogram” in Fig. 2 shows that important cycles havea duration of 4, 8, and 13 to 19 years. Effectively, themajor peaksare for T = 16 (which is 4 years), T = 32 (8 years), about T =52 (13 years) and about T = 76 (19 years). Smaller peaks occurat T = 24, etc., which are ignored in our present analysis asdeemed to be due to the presence of the major frequencies (wehave subsequently checked these minor cycles empirically andfound them relatively insignificant).
From Fig. 2, as expected, we have obtained cycles ofduration of 4 and 8 years given the influence of commodityprices on the A$. The 4-year cycle is just a harmonic of the8-year cycle (or the other way around). For the other cycleswe could adopt the simple principle that they should haveperiods that are double of each other: 4, 8, 16 (which agreeson average with the peaks of periods between 13 and 19 inFig. 2), and 32 years.5 The 16 and 32 year cycles can alsorepresent economic cycles such as the Kuznets cycles sincecommodity prices affect economic activities (as reviewed inIntroduction). This set of cycles has to be verified empiricallygiven some other criteria as well: first and foremost forecastsoutside sample of regressions; second, the R2 of the 4 cyclesconsidered; and third, theoretical justifications. Theoretically,we have already mentioned the importance of the around8-year cycle of commodity prices; its harmonics can justifythe longer cycles of 16 and 32 years as these are stronglyrelated to economic cycles and the interaction between the
5 With more data available in the near future, we might discover that the64 year cycle is also present. Also, with more data, we might have a differentset of harmonics, such as 3.8, 7.6, 15.2, and so on in our estimations. Inaddition, our daily data analysis seems to suggest that the true cycles are ofslightly less duration (3.9, 7.8, and so on). Nonetheless, the present set ofsuggested harmonics with monthly data provides good forecasts (out ofsample ranges) and hence we trust that these cycles are pertinent.
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
latter and the commodity prices cycle (as reviewed inIntroduction).
3.2. Empirical results
Having ‘theoretically’ determined the underlying set ofcycles we now propose to check this empirically. Table 1shows the results, the dependent variable being the A$ as US$cents per 1 A$ (ADOL). The four cycles suggested are of 4, 8,16, and 32 years as already indicated.6 We use 285 monthly7
data from December 1983 to August 2007 (arbitrarily chosenin the second half of 2007 and as a consequence we havepredictions out of sample from September 2007 to Septem-ber 2013); to check for robustness we also used severalcut-off points for prediction. The results for N = 285 (1 to285 in column 4) are as expected confirmed with the resultsfor N = 3088 (up to July 2009 in column 1), despite theexceptional fluctuations during the extra 23 months due tothe global financial crisis.
In Table 1 we also included some other variables whichtraditionally are deemed to have an impact on exchangerates. Thus, the difference between the rates of interest inAustralia and the USA (‘INTADUS’), the balance of payments(‘BAL’), and the terms of trade (‘TOT’) are included inregressions (2) and (3). These variables are also significantbut they explain a small part of the wide fluctuations of theA$. Thus, if cycles are excluded then the R2 is only 5.5% as itwill be shown further below. This result agrees with DeGrauwe and Grimaldi [9] as mentioned in the Introduction.Also note that from columns (1) and (2) of Table 1 wededuce that the constant is about 0.77. This could beconsidered as the long term trend; however, as we suggestin this paper this trend might be deemed to be the
20 years period are not included because they were found to be empiricallyinsignificant or at least not more important than the remaining cycles (thesame conclusion applies for smaller harmonics such as 2 years). Further-more, other cycles of similar fundamental duration and its harmonics suchas 5, 10, 20 and 40 years, or a combination of this set and the previous set of4, 8, 16, and 32 years did not produce better results than the one finallychosen here.
7 Daily data yield similar results.8 Other cut-off points were selected as well yielding similar results. More
cut-off points are examined in sub-section 3.4.
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
Notes: the symbols for variables mean: c4 is cosine of 4 year cycle; s4 is sine of 4 year cycle and so on; INTADUS is the difference between the rates of interest inAustralia and the USA; BAL is balance of payments; TOT is terms of trade; and ADOL is Australian dollar (US$ cents per A$).
6 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
combination of our four cycles which have a long termmeanof about 0.77.
In Table 1 we also see that the DW test for autocorre-lation (around 0.3) for regressions (1), (3), and (4) ofTable 1 clearly shows that even after the inclusion of thefour cycles, there is still serial autocorrelation of order one.This is expected because the cycles smooth out the widelyfluctuating series, leaving out a residual that fluctuatesmuch less, but still fluctuates with no apparent cycles (thecorrelation between the residuals and the original series ofA$ is about 0.38, since the R2 is about 0.85). Thus, whenADOL(-1) is included as an explanatory variable shown inregression (2) in Table 1, the coefficient of 0.82 issignificant and suggests that the half-life9 duration is of3.5 months.10
9 The formula used is log(1/2) / log(ρ) where ρ b 1 (ρ = 0.82 in our case)(see also [14,56]). This is only a rough approximation as our model here isnot a pure AR(1) model (since it also contains cycles, etc.). In any case half-life estimates should be used with caution.10 We also used 2-day data and estimated an AR(1) model that alsoincluded the four cycles; the half-life remained about 3.5 months.
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
If we adopt a simple AR(1) model (thus ADOL =0.968 ∗ ADOL(−1) + 0.0226 with OLS estimation), thenthe half-life duration is about 21 months. In this simpleAR(1) model, if we also add the fundamental variables BAL,INTADUS, and TOT,11 the half-life estimate remains the same(about 21 months). These two models provide differentresults for the half-cycle duration. Different results are alsoobserved in the current literature (see also [14,56]). We willcheck further below through our forecasts that our ownestimates seem to be closer to reality (about 3.5 months).
Having confirmed the significance of the four harmonicswenow show their individual contribution in Fig. 3. We can seethere that effectively, as already mentioned, harmonics differamongst themselves in terms of phase, and amplitude.12 Thelonger the period of the cycle, the higher the amplitude is.
11 All data are available in RBA site [6], various issues.12 Contrary to the possibility suggested by De Groot and Franses [45], ourharmonics do not reach peaks and troughs at the same time due to thesedifferences in phases and amplitudes between the four cycles as Fig. 3shows, and to the differences in coefficients of regressions relevant to eachcycle as will be seen below.
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
Note: the line that fluctuates around the zero axis is the residuals plot. -0.4-0.3-0.2-0.10.00.10.20.30.40.50.60.70.80.91.01.11.2
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Fig. 4. Forecasts according to the proposed model for the A$. Note: the line that fluctuates around the zero axis is the residuals plot.
Notes: (i) the smaller the duration of the cycle, the smaller the amplitude; (ii) each cycle is the combination of the cosine and sine components weighted by the corresponding regression coefficients of (4) in Table 1; the period shown is from December 1983 to December 2019.
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The four harmonics shown individually (for the A$)
Fig. 3. The four harmonics shown individually (for the A$). Notes: (i) the smaller the duration of the cycle, the smaller the amplitude; (ii) each cycle is thecombination of the cosine and sine components weighted by the corresponding regression coefficients of (4) in Table 1; the period shown is from December 1983to December 2019.
The crucial issue of forecasts, especially outside theestimation period can convince us much more about theabove propositions. Fig. 4 shows the raw data (up toSeptember 2013), the forecast line up to December 2014,and the residuals of the OLS regression using the cosines andsines of the 4 cycles (4, 8, 16, and 32 years). The regressionwas fit to the data from December 1983 to August 2007.13
The A$ is peaking in 2012 sometime. It predicts the “comeback” of the A$ after the recent world financial crisis verywell (in October 2010 — the A$ was about US$0.98, after it
13 Other cut-off points were also applied, for example up to July 2005 orJuly 2006, or December 2005. In all cases the out of sample forecasts aresimilarly good. More forecasts with more cut-off points will be shown insubsection 3.4.
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
dropped down to US$0.64 around January 2009). This fastrecovery is also indicated by our estimate of the ADOL(-1)coefficient of 0.82 as indicated previously. Effectively, fromthe lowest point reached (0.64 in January 2009) to the pointof reaching the 4-cycle trend again (in September 2010),it took approximately 7–8 months (thus confirming the3.5 months half-cycle).
3.3. Cross-checking results
Wewill now cross-check our results in several ways. First,we can regress the A$ (ADOL) as a function of commodityprices (COPR), or terms of trade (TOT)14 directly by applying
14 COPR and TOT are cointegrated (results not shown here).
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
Note: COPR stands for commodity prices; TOT stands for terms of trade; for other variables see Notes of Table 1.
8 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
lags that correspond to the 4, 8, and 16 years (hence 48, 96,and 192 months; the 32 years cycle is not included becauseof lack of data). Also the 2-year lag and 10-year lags areincluded. The results confirm the analysis so far (see Table 2).To check the influence of the AR (1) scheme the ADOL(−1)was also included: the high value of this variable's coefficient(around 0.82 in the case of COPR) confirms our previousresults.
Second, if we exclude the four harmonics, how do theother traditional variables perform? The OLS estimationsshow a very low R2 (about 6% when COPR is included—as inTable 3 in column 1, and about 15% when TOT is included,not shown here). Note that the R2 jumps to about 85% whenwe include the four cycles as we have shown above inTable 1. Table 3 shows that themain economic variables BAL,INTADUS, and COPR- or TOT are cointegrated with the A$.Note the ECM(−1) coefficients (see Table 3) show a veryslow speed of adjustment.
Third, should we examine the A$ in its nominal or realterms? So far we have done it in nominal terms. A graphcomparing the 2 series will provide the answer (see Fig. 5):the two curves are almost parallel and hence would yieldsimilar results in our regressions. To further check, by using
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
quarterly real A$ from 1970, we regressed the real A$ againstlags of 16 (thus 4 year cycle), 32 (thus 8 year cycle), 64 (thus16 year cycle), and 128 quarters (thus 32 year cycle) of A$;these lagged variables were found to be significant (resultsnot shown here).
Fourth, how do Australian financial authorities handlethe fluctuations of the A$? Are there any interventionsand what is their influence on the A$? The modelspresented here suggest that government intervention'simpact is very small (at the most 10%–15%). Thisconclusion agrees with other research papers. For exam-ple, Edison et al. [57] found that these interventions donot significantly impact on the A$'s volatility. In addition,Schwartz [58] observed and showed that intervention bymonetary authorities has little effect on (floating) ex-change rates. Thus, government intervention has beenrather only slightly significant. For example, the regres-sion errors (as observed in Fig. 4) seem to well coincidewith some policy measures. According to Makin ([59]p. 336), “intervention was highest in July 1986 to preventfurther depreciation by buying Australian dollars, but wasalso high in February 1989 to stem appreciation”. Thesetwo interventions agree with the magnitude and sign of
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
the residuals of our model (negative during 1986 andpositive during 1989, as seen in Fig. 4).15
Fifth, a source of endogeneity might be the impact of theUS economy on the A$ [7]. However, we also carried out asimilar analysis to the A$ vis-à-vis the US$, namely that of theA$ vis-à-vis the British Pound, and the A$ vis-à-vis theKorean Won. In both cases the R2 was high (about 72% and77% respectively) when the four harmonics were included.Consequently, and as argued by Chen and Rogoff [7] as well,the US economy has not played a distinctly significant impacton the A$.
3.4. Application to and forecasting of four commodity currencies
It is expected that the above harmonic model can also beapplied to other commodity currencies. As already men-tioned in the Introduction, the A$ was chosen to beextensively examined in this paper because it is rather theonly genuine commodity currency in the sense that it is theonly one that has been freely floating for a long time (seealso [7]), since December 1983 and hence we have sufficientnumber of data to carry out our spectral analysis (theCanadian dollar is similar to the A$ in this respect). Also, the
15 A detailed analysis of the residuals is nonetheless out of the scope of thispaper.
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
A$ represents a country whose exports of commodities ofprimary resources (such as ores, metals, raw food, etc.)constitutes a very large proportion of its total exports, that isabout 70% [60] (the Canadian dollar is not similar to the A$in this respect, see footnote 17 too).
However, some other currencies can be safely considered asbeing similar to the A$ in terms of being “genuine” commoditycurrencies and forwhichwe have data for a long period of time.These currencies are New Zealand's dollar (NZ$), which startedfloating approximately (as on March 1985) at the same timeas the A$ [7]; Norway's Krone (NKR), which nonetheless hasbeen partially regulated [7] but freely floating since December199216; and Canada's dollar (CA$),17 which has also beenfloating for a long time (since 1970/1). Unfortunately we do nothave sufficient number of data for other commodity currenciessuch as Chile's Peso, Brazil's Real, and South African's Rand. Forthese currencies, the free floating started very recently as canbeseen in Fig. 6, for example, for Chile's Peso and South Africa'sRand, it started in the middle or late 1990s.
16 Before December 1992, it was under the regime of fixed exchangeagainst a basket of trading currencies, often with small adjustments [63].17 For New Zealand and Norway, their percentage of resources exports issimilar to that of Australia (around 65% to 70%), but for Canada it is muchlower (around 35%) as this country is more industrialized than the otherthree (New Zealand, Australia, and Norway).
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
Notes: The y-axis expresses “one unit of local currency equals x units of US$”. Data for the Brazilian Real (BRR) started in 1995; those for the Chilean Peso in 1984. The sources of data are [6] for the A$, [61] for the Chilean Peso (CHPE), and [62] for all other currencies in the graph (SAR stands for the South African Rand, and NKR for the Norwegian Krone). The SAR is situated at the lower part of the graph.
Fig. 6. Commodity currencies fluctuations. Notes: The y-axis expresses “one unit of local currency equals x units of US$”. Data for the Brazilian Real (BRR) startedin 1995; those for the Chilean Peso in 1984. The sources of data are [6] for the A$, [61] for the Chilean Peso (CHPE), and [62] for all other currencies in the graph(SAR stands for the South African Rand, and NKR for the Norwegian Krone). The SAR is situated at the lower part of the graph.
Note: The source of data is RBA [6]; nominal quarterly A$ are calculated from monthly data by using averages for 3-month periods.
020406080
100120140160180
Dec-1983
Dec-1985
Dec-1987
Dec-1989
Dec-1991
Dec-1993
Dec-1995
Dec-1997
Dec-1999
Dec-2001
Dec-2003
Dec-2005
Dec-2007
Dec-2009
Dec-2011
Nom
inal
: US$
cen
ts p
er A
$R
eal:
trad
e w
eigh
ted
inde
xNominal Real
The nominal A$ and real A$ compared
Fig. 5. The nominal A$ and real A$ compared. Note: The source of data is RBA [6]; nominal quarterly A$ are calculated from monthly data by using averages for3-month periods.
10 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
Consequently, in this paper we present our resultsseparately, only for the NZ$, the NKR, and the CA$ besidesthe A$. We have chosen three samples of actual data (hencethree cut-off points) and produced forecasts beyond thesecut-off points, as shown in Table 4. The global financialcrisis of 2008/2009 and beginning of 2010 was the reasonfor including these three years in the 2nd and 3rd samples.We referred to this crisis in a previous subsection andfound that the long term trend as suggested by the modelwas not followed by that crisis and we explained that afterthe bottom of the global recession was reached aroundJanuary 2009 it took several months again before thecurrencies reached their long term trend suggested by ourmodeling.
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
The results of our model are shown in Appendix:Fig. A1a, b, and c (for the A$), Fig. A2a, b, and c (for theCA$), Fig. A3a, b, c for NZ$, and Fig. A4a, b, and c (for theNKR). Note that we also followed the same samplingcut-off procedure for the A$, for consistency reasons.Although these graphs show the performance of themodels for the four currencies and the three samplesclearly in a visual manner (for example see how thepredicted residuals are as small as the remaining residualsof the regression), Table 5 summarizes some of the maincharacteristics of these forecasts. For each currency weinclude indicators or criteria which can assist the readerappreciate the validity of the forecasting ability of the fittedmodels.
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
18 As already mentioned, a good fit on the raw data does not guarantee theadequacy of the model, hence the out-of-sample forecasts are crucial here.19 All these results and data can be available on request.
Table 5Characteristics of regressions for forecasting.
A$ (0.758 and 0.131)* CA$ (0.839 and 0.117) NZ$ (0.619 and 0.106) NKR (0.148 and 0.020)
Notes: *The two numbers in brackets are the average of each currency for the whole period available (up to September 2013) and its standard deviation. Inthe first column, N stands for the number of observations for each sample (as per Table 4); SEE stands for standard error of estimate of the regression; AFEstands for average forecasting error of the period beyond (outside) the sample: thus for the sample with cut-off point December 2011, the AFE is the squareroot of the average of squared errors for the period January 2012 to September 2013; for the sample with cut-off point December 2010, the AFE is theaverage for the period January 2011 to September 2013; and for the sample with cut-off point December 2007, the AFE (not in brackets) is the average forthe period January 2011 to September 2013 (thus, the global financial crisis period is skipped for the calculation), and the AFE in brackets is the average forthe period January 2008 to September 2013. Finally the “cycles” row indicates the duration in years of the four harmonics (e.g. for the A$, 4 years, 8 years,16 years, and 32 years).
Table 4Samples of data for forecasting.
A$ CA$ NZ$ NKR
1st sample From Dec 1983 to Dec 2007 From January 1971 to December 2007 From March 1985 to December 2007 From Dec 1983 to Dec 20072nd sample From Dec 1983 to Dec 2010 From January 1971 to December 2010 From March 1985 to December 2010 From Dec 1983 to Dec 20103rd sample From Dec 1983 to Dec 2011 From January 1971 to December 2011 From March 1985 to December 2011 From Dec 1983 to Dec 2011
Thus, the R2 indicates the high explanation of the dataaccording to the three samples shown in Table 4; this R2 isvery high, between around 80% and 94%. The number ofobservations is sufficiently high for the estimation of ourmodels based on cycles whose maximum duration reachesbetween 30 and 37 years. Especially in the case of Canada,the number of observations (between 444 months and492 months depending on the cut-off point) makes ourresults robust vis-à-vis the other currencies with a smallernumber of observations. Note that Canada's suggestedcycles' longer periods of 4.65 to 37.2 years in relation tothe other currencies are perhaps due to the availability oflonger samples used in the analysis for Canada than for theother countries, but this could also be due to other factorssuch as composition of exports, degree of industrialization,and so on.
The intercept of each regression is also important to includein Table 5 because it shows how the series is close to beingstationary aswe have assumed in our analysis. Thus, we observethat this intercept is rather steady as we use increasingly largersamples and at the same time it is very close to the average of thewhole series (from the start of each sample as per Table 4 toSeptember 2013); all this confirms the stationarity of each series.The standard error of estimate (SEE) for each regressionprovides us with a good idea as to the magnitude of error inthe regressions and also what to expect in terms of forecastingerrors. This SEE not only depends on the R2 of each regression,but also on how volatile the series is. Thus, the most volatileseries is the A$ (this can be seen by comparing the standard
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
deviation with the average; and by the magnitude of thestandard error if two averages are similar). Consequently, the A$exhibits the highest SEE given the magnitude of R2.
Finally, and confirming all previous comments, theaverage forecasting error (AFE) outside18 the sample ofestimated regression is similar to the SEE and often it islower than that! This is an excellent forecasting performancenot previously attained by any other model in the literature(on the contrary everybody knows – see earlier subsections –how difficult it is to forecast currencies and especially inthe medium and long terms). The relevant graphs (seeAppendix) show these errors in a clear way: the outside ofthe sample forecasting errors are just a continuation of theinside the sample estimated errors.19
One could also suggest that we are able to forecast onecurrency by using the model of another currency. However,as substantially explained so far, the fundamental techno-logical cycles which have about 8 to 9 years of duration aredifferent for each currency. These cycles and their 3harmonics depend on several factors such as commoditycomposition of exports of each country; composition of
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
20 Whatever we say about the A$ in this section are applied to the otherthree currencies as well.
12 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
resources and commodities exported; percentage of re-sources in total exports, degree of industrialization, neigh-bors as trading partners, economic cycles, etc. As empiricallyalready shown, the cycle's duration does not necessarilycoincide between the four countries. Thus, it is found that thefour harmonics for the A$ have cycles of 4, 8, 16, and 32 yearsduration. The NZ$ harmonics have cycles of 4.25, 8.5, 17, and34 years duration. The CA$ harmonics have cycles of 4.65,9.3, 18.6, and 37.2 year duration. Finally, the NorwegianKrone harmonics have cycles of 4.2, 8.4, 16.8, and 33.6 yearduration. Consequently, it will be very wrong to use the A$model to predict the CA$ trend, since there is a relatively bigdifference in the cycle duration. However, it will be muchbetter to use the NZ$ model to predict the Norwegian Krone(and vice versa) since the duration of their cycles almostcoincides. And so on.
Nonetheless, all these simple OLS regressions linking theoriginal data of one currency with the trend suggested by themodel of another currency are not sufficient for forecastingpurposes, since a simple OLS regression must also pick updifferences in amplitude and phase besides differences in cycleduration. In addition, one could also suggest thatwe are able toforecast one commodity currency by regressing it againstanother commodity currency. However, the same objectionsapply in this case as in the previous case. In conclusion, oneneeds to apply themodeling process suggested in this paper inorder to properly analyze and forecast for medium and longterm horizons.
Ourmodeling process has, however, some limitations. First, itworks under the assumption that the global financial andmonetary order are steady and without substantial breaks. Forexample, it cannot predict structural changes if a global financialcrisis like the recent one takes place; or if the European Union isweakened and the Euro is abolished, and so on. Second, it worksunder the assumption that the US$ continues to be the maininternational exchange and reserves currency. Third, any drasticchanges in the gold and petrol reserves might also affect thecommodity currencies in an unpredictable way. Fourth, anysubstantial changes in the balance of payments of the USA, and/or other large in terms of GDP and exports countries mightgenerate unpredictable crises. Fifth, ourmodeling suggested fourharmonics for each currency given the availability of the numberof observations. However, aswe obtainmore data for each series,we can re-estimate the regressions and we might find slightdifferences in the duration of cycles. Thus, it would seemreasonable to say that every 2–3 years the estimation procedureshould be updated if one wants to use the modeling for preciseforecasts; for example, a cycle of 8.2 years might be better than8.1 years after updating is done, and so on. Despite theselimitations, it is important to mention again that our paper ismeant to throw some light into the possibilities of some cyclesbeing very important to understanding and forecasting com-modity currencies.
4. Conclusion
We investigated the possibility that the Australian dollarand three other commodity currencies examined here, thatis, the NZ$, the CA$, and the NKR is primarily determined by ahandful of harmonic cycles which in turn are based on thecommodity prices cycles, commodity production cycles and
Please cite this article as: E. Sanidas, Four harmonic cycles explain anTechnol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.t
in general on economic cycles. The well-known Fourieranalysis according to which any time series can be recon-structed by the sum of all sinusoidal functions was thetheoretical basis used to represent these cycles. The applica-tion of such analysis to the trend and fluctuations of the A$has produced some very interesting and original results. Thefour harmonic cycles used in this respect (of 4, 8, 16, and32 year duration for the A$, and slightly different for theother three currencies) explain about 85% of the monthly ordaily changes of these commodity currencies, the remainingbeing attributed to ad hoc situations, policy measures, and soon.
Thus, one of themain conclusions is that there is no “puzzle”(as seen in the introductory subsections in reviewing literature)in the behavior of a currency such as the A$ (or the othercurrencies20) oncewe take into account non-linear relationshipsdue to the existence of commodity price (or production etc)cycles and their harmonics. The two cycles of 4 and 8 yearduration are not sufficient to explain thewide fluctuations of theA$. Thus, the 16 and 32 year cycles are also found to explainthese fluctuations. Somehow these four harmonics linked tocommodity prices and economic activities are related to themechanism of demand and supply of the A$ and othercommodity currencies vis-à-vis all other currencies, and inparticular vis-à-vis the US$ (an examination of such amechanism is beyond the scope of this paper; however, see avery relevant study by Polasek [64]).
In addition, the existence of the four cycles since 1970 forthe CA$ and since later years for the other commoditycurrencies, when the floating regime was introduced mightadd some extra evidence for the validity of the PPP theoryafter all, although the PPP version does not hold true at leastin its traditional analysis of integer cointegration. Both thispaper and Mansur et al. paper [13] seem to support the ideathat once we include low frequency dynamics and hence longperiod cycles the Australian – for example – dollar revertsback to its “mean”. This leads us to remark that we mighthave to reassess what we mean by “mean”, “trend” and“stationarity” in time series; for in our case the remaining15% of unexplained variance seems to fluctuate around the4-cycle “trend” as determined in this paper.
Furthermore, it is important to emphasize that in thispaper we also provided forecasts for all four commoditycurrencies in a systematic way. Thus, we examined at leastthree samples of data for each currency, for three differentcut-off points, and produced forecasts for medium to longperiods outside the samples. These forecasts are excellentaccording to several used criteria. They confirm thevalidity of our modeling according to the underlyinganalysis.
Acknowledgments
A grant by SNU has contributed towards the research ofthis paper. I also thank anonymous referees for their usefulcomments and suggestions.
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
Fig. A1. a. Forecasting for A$, estimated sample up to December 2007. Note: the line that fluctuates around the zero axis is the residuals plot. b. Forecasting for A$,estimated sample up to December 2010. c. Forecasting for A$, estimated sample up to December 2011.
Please cite this article as: E. Sanidas, Four harmonic cycles explain and predict commodity currencies' wide long term fluctuations,Technol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.techfore.2013.11.008
14 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
Fig. A2. a. Forecasting for CA$, estimation up to Dec. 2007. b. Forecasting for CA$, estimation up to Dec. 2010. c. Forecasting for CA$, estimation up to Dec. 2011.
Please cite this article as: E. Sanidas, Four harmonic cycles explain and predict commodity currencies' wide long term fluctuations,Technol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.techfore.2013.11.008
Fig. A3. a. Forecasts for NZ$, estimation up to Dec. 2007. b. Forecasts for NZ$, estimation up to Dec. 2010. c. Forecasts for NZ$, estimation up to Dec. 2011.
Please cite this article as: E. Sanidas, Four harmonic cycles explain and predict commodity currencies' wide long term fluctuations,Technol. Forecast. Soc. Change (2013), http://dx.doi.org/10.1016/j.techfore.2013.11.008
Fig. A4. a. Forecasts for NKR, estimation up to Dec. 2007. b. Forecasts for NKR, estimation up to Dec. 2010. c. Forecasts for NKR, estimation up to Dec. 2011.
16 E. Sanidas / Technological Forecasting & Social Change xxx (2013) xxx–xxx
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Elias Sanidas is an Associate Professor at Seoul National University, Seoul,South Korea. His main two areas of research are Economics of technology,knowledge, and organizational innovations; and International trade andeconomic development. He was educated in Greece, France, and Australia.He also taught in these countries, plus South Korea currently. He haspublished a book and several articles.
d predict commodity currencies' wide long term fluctuations,echfore.2013.11.008
