Here is a scientific paper published in 2007, which is concerned about the proof of the 5.55 years (Kitchin) business cycle in the stock exchange markets, the last 50 years. It is not concerned about the Crisis of the 2008-9 , which should be subtsracted as effect from the predictions and analysis, but only about the cyclic phenomenon of the 5.55 years, which is a 1/4 submutliple of the well known (solar cycle , or global climate cycle of 22.2 years , the dicovery of which gave the nobel prize to the Kuznets. The Kitchin cycle of 5.55 years seem to be stronger than the Kuznet cycle of 22.2 years. To prove it is used spectral analysis.
NOTE ON THE EFFECT OF THE 11 YEARS GLOBAL CLIMATE CYCLE ON THE PRICES OF THE CAPITAL MARKETS
By Kyritsis Costas , Sotiropoulos Ioannis, Gogos Christos, Kypriotelis Efstratios.
Department of Finance and Auditing , TEI of Epirus, Preveza, September 2007 Greece
Abstract
In this paper we prove and analyze the effect of the 11 years sunspot, and global climate cycle, on the volatility of the prices of Stock Exchanges. We predict an increase of the prices from the present time (2007) till 2011, a next major maximum of the prices during 2010-2011 and a fall of the prices for the period 2011-2017. We discuss sequences of causal explanations based on the balance of demand and supply and comment on the general value of such an omniscience wisdom that involves at least five different sciences.
Key words: Stock Exchanges, volatility index, 11-years sunspot cycle, spectral analysis.
Jel: C32,C4,C53, E32, E37, E41
- Introduction
In this short note we shall try to describe the effect of the 11 years sunspot and meteorological global climate cycle, on the growth and decay of the prices in the stock Exchanges. We shall also try to give a sketchy sequence of causalities based on the balance of demand and supply that may help to understand this effect. We are not of course the first to have discovered this effect. In the contrary it is an old discovery, there is extensive (although not much known) literature, and the last 20 years, it has been considered to be more and more important. Nevertheless the first of the authors re-discovered this effect during 2000-2001 while performing simulation and statistical analysis of time series of stock exchanges. The more we studied it, the more we overcome our hesitations and doubts, and the more we are scientifically convinced for its validity and reality. We must mention that the study of cycles in time series, even when they are weather data time series, is a difficult, and tricky endeavor, and only the last decades, due to the advance of computers and massive data analysis, it has started to become clear how to detect and analyze them. The idea of business cycles is of course old (e.g. see Henry, Ludwell, Moore in references) but it was never handled in statically, sufficient robust way, by the economists. The same, surprisingly, was also the situation with the idea of weather cycles, in academic and practical applications, research. (see Burroughs Williams James in references).
We shall also discuss the value of such a knowledge. If we are in search for a common systematic behavior of the capital markets, other than the assumed constant long run trend (drift), then an eternal of astronomic source cycle, of nature seems the only first good solution. If we are in search for a systematic reliable behavior that is not incidental to specific historic contingencies of the evolution of economies , and thus non-repeatable, or it is not a short-term shock-wave of non-repeatable historic situations, the resorting to the effect of eternal “pace” originating in nature (e.g. meteorology) seems to us to be the first best solution. Of course the seasonal cycles (winter-summer ) is a traditional concept of how nature may have an effect on the prices. But searching for other not so obvious cycles like the 11-years cycle, that can be also very strong and significant, seems to be an intriguing matter. In addition forecasting or risk management of investments, that is based on such eternal cycles, is entirely more reliable and scientifically clear, compared to forecasting based on news from newspapers, that either are already late , or of unknown eventual effect on the prices of the capital markets. In involving sources of causalities in macroeconomics, that go beyond classical macroeconomics, to the areas of ecology and meteorology, widens our concept of “world” in economic causalities. It creates a significant “omniscience wisdom” that may be of benefit to all, and also makes us more conscious of the overall functions of life and society. This eventually increases the awareness of ourselves and the determination we have to our fate, and the success of our goals. Each time we take advantage and go parallel to the flow of time and power of evolution, in larger time intervals, we increase our ability of success in our goals, and we also increase the awareness of our consciousness. The psychologist reassure us that increasing awareness in our activities unavoidably increases our self-esteem and success as individuals and collectively. Including nature in economics is probably an important ingredient of the philosophy contained in the newly developing trend of “green economics” that claims to transcend classical economics. The effects are also invertible. If we know the details of the sequence of how nature influences the changes in the prices of business, then we are also more aware of how business in their turn have an effect on nature. As an important development in this direction is the issuing and trading of weather financial derivatives , in the Stock Exchanges, that are based on meteorological variables, like average temperature, humidity etc of various town in USA, and Europe. The main indented value of such financial derivatives is the hedging in the prices of commodities of energy consumption (like electricity, oil, natural gas etc) to eliminate waves of prices based on seasonality, but also to hedge for the systematic increase of the average temperature of the planet, and for the extreme weather phenomena.
Finally there is, of course, the restricted financial benefit in investments for individuals, enterprises, or domestic national economies, when forecasting based on existing eternal cycles of nature. Budgeting becomes more successful. When the individuals are investing in capital markets, they are unavoidably based on faith, and even the metaphysical element is involved. Such a knowledge helps them to overcome fears, avoid bad decisions, based on public panic, control more their emotions , keep an optimistic metaphysical context, with positive hopes, helps them to avoid the worn-out of their soul, while at the same time it contributes in taking more successful financial risk management decisions.
2. The 11 years sunspot and the 11-years global climate meteorological cycle
In this paragraph we present same basic and useful to have in mind (while reading this paper) facts of the 11 years cycle. We have to enter the jurisdictions of three sciences : astronomy, meteorology, and ecology.
Figure 1
From Astronomy: There much material in the internet and some of the interesting sites and panels are mentioned in the references. The north and south magnetic poles of the sun, interchange every 11 years. After 22 years the magnetic north pole has returned back to its initial position. This phenomenon is parallel with a cycle in the number of appearing sunspots on the surface of the sun. The sun spots are observable as darker spots on the surface, but essentially they are a known effect of the “meteorology” of the solar plasma, as most probably are anti-cyclones of the solar plasma, in other words at this spot the plasma is rotating fast (like a vast tornado) generating strong magnetic field, and with a inward to the center of the sun direction. This blocks the radiation from inner layers to the surface, giving the appearance of the darker spot.
In the chart in this paragraph we see the cyclic behavior of the number of appearing sunspots when plotted in time. We must mention that although an astronomic cycle the period of 11 years, is statistically only constant with variations that maybe longer than one year! We see in the figures 1, and 2 that the last peak of the sunspots activity was during 2000, and the next peak is estimated during 2011 or starting 2012. The predictions mentions a delay of the next peak towards the end of 2011 or during 2012. In addition it is predicted that the intensity of this peak shall be much higher by a factor 30%-50% .
Figure 2
From meteorology: A very good book on weather cycles is that of Burroughs (see references) Satellite measurements since 1978 confirm that the total output of the solar energy varies in cyclic way every 11-years in exact proportionality with the number of appearing sunspots. Also measurements of the earth’s magnetic field give that the 11-years sun’s cycle produces an 20-22-years earthly magnetic field cycle (Hale cycle). Thus through radiation, and through the solar wind that reaches the earth, (and which is responsible for the polar aurora phenomenon) the meteorological phenomena are influenced by the 11-years sun-cycle, to produce the 11-years cycle in the global climate. This cycle includes cyclic behavior of successive maxima-minima of earthly surface temperature, level of rainfalls, drought severities , electromagnetic storms, frequency of appearing tornados etc. A seemingly related famous meteorological cycle of the global climate, is the QBO (quasi-biennial oscillation) that seems to have period of 2.75 years (11/2=5.5, 5.5/2=2.75) . The QBO appears as a cycle of the speed of the stratospheric wind. In this short paper we presented only one major cycle that of the 11 years of the global climate. But we have found that there are at least 5 more cycles with their harmonics, that range from the 11 years to 5 minutes. The presentation and analysis of them cannot be the content of the present little paper. We hope to enlarge on them in other papers or in a book dedicated to them. All of the above cycles have also been discovered in meteorology. Meteorology has also discovered and studied longer cycles that range from 300 years to thousands of years that cannot be related with the modern form of capital markets.
From Ecology: Finally the sequence of causalities reaches the growth of life on earth. A most impressive example are the tree-rings. If we cut vertically a tree, we observe in its main body, rings that are the growth of the thickness of the body of the tree every year. These are the tree-rings. It is impressive that every 11-years the tree-rings are thicker. This means a more intense growth of the trees (dendroclimatology) This seems to be due to increased rainfalls, increased radiation, surface temperature extremes etc. Similar cycles appear in varves (layer lines) of clay, corals , ice cores, etc. The same 11-years cyclic behavior is observed with the water level of the lakes. It is obvious that the growth of grains in agriculture is more intense and follows the same 11-years cyclic pattern.
It has been also measured an impressive coincidence of the 11-years cycle with the number of lynx furs purchased by Hudson’s Bay Company in the McKenzie river region from 1821 to 1910. This is seen in the figure 3, This reflects the periodic fluctuations of the species that traditionally is described by the Voltera’s Prey-predator equations that do create oscillations. Nevertheless the real measurements prove that the coefficients of the Voltera equations are such that the actual period coincides with the 11-years cycle of the global climate. We see the chart below. (Data from the Global Population Dynamics database, University College, London, England)
Figure 3.
- How are the cycles detected in the time series?
In searching for cycles in time series it is of key importance to know technical ways to detect them. As with statistics, so with time series too, it is very easy to be misled when analyzing with standard ways the data. For example it is very easy to miss obvious cycles, or to think you have found cycles when in fact it is only a peculiarity of a particular piece of the sample. The present state of the art in academic research is such that on the same sample data many different and often contradicting models may fit quite well. Thus, the successful research, requires not only standard fit of a model, but at first exhaustive familiarization with the data . We have to observe them many hours and make many simple calculations with a calculator, in relation with the actual situation from which they emerge, before we start analyzing them in a more sophisticated way with many different techniques, in many different areas of the sample etc. After the wide success of the theory of the random walk and the efficiency of the markets, most of the research for cycles or cycles-based systematic correlations, became the least publicly enforceable category of ideas. We must mention also that different sciences are oriented to different techniques for cycles detection in time series. For example electric engineering signal experts use FIR and IRR filters, and spectral analysis, of which the classical economist are not very familiar. Experts in seismology, use much different techniques. Physicists may, still use other techniques, like operator theory of quantum oscillators, that engineers or economists, or meteorologist may know very little. We must mention also that the Box-Jenkins parsimony of low p, q order e.g. ARMA(p,q) time series, would entirely make invisible long run cycles, that involve higher order, longer term memory.
To detect cycles in time series we may assume either stationary or non-stationary time series. But if the time series is assumed non-stationary, then the classical spectral analysis cannot be claimed to apply, as almost all of its theorems are proved for stationary time series. It is an assumption after all significant for the multitude of paths, rather than a single path. The fact that most time series models are fitted on a single path, explains why so many different models, fit to the same data. After all a stochastic process is defined on a sample of paths. Any one-element sample would be vague enough to support many different assumptions. A more reliable method is to produce, in a standard way many overlapping paths, of the same time interval (from a single long path or even better e.g. for indexes from all components securities of the index) , and then let the data decide clearly if it is stationary or not process. In other words it is best to use the non-parametric data-generated process or time series.
An other phenomenon that, any researcher that has studied extensively, time series, will come to it, and seems to escape the dominating mentality of academic researchers of time series is the next: A spectral analysis based on the finest bins of the data, would not give the same cyclic behaviors, with analysis based on many different time scales, with many different larger bins. In other words the element of time scale and time bins, almost defines a different world of phenomena in the data, not derivable by a flat spectral analysis on the finest (smallest) time steps (bins). Similarly the relative size of the sample and the suspected major period of cyclic behavior, is very important factor in extracting the necessary information.
We find that the methods that meteorologist use to detect cycles in time series data, are the most natural, simple, and the ways that they fail to detect existing cycles have been quite well studied. We mention them briefly:
- Sherman’s statistic: It is simple statistical test based on a chart, that decides of randomly spaced, in time, events, are uniformly random, or are clustered in concentrations, or have a cyclic periodicity.
- Spectral Analysis: (see Lambert H. Koopmans in references) A classical tool based on Fourier transformation, and available in many software packages. Its limitations are quite well known. We may add here new relevant techniques like the maximum entropy spectral analysis (mesa), that tries make analysis based on assumptions for the available data , that are maximally non-committal with regard to the unavailable data. Then the singular spectrum analysis that is detecting best anharmonic oscillations to the basic frequency (resorting to classical techniques of orthogonal functions and principal components).
- ARMA, SARIMA, FIR, IRR filters etc: The classical fit techniques of linear recursive models. If the characteristic roots are complex and not real, and of norm one, we detect cyclic effects. Economists usually avoid complex roots.
- Wavelet analysis: Originally used to study the way that the spectrum changes, when we move to successive intervals in the data. (see Parcival D.B. et al in the references) It reproduces the time series as superposition not of constant repeating cycles but as superposition of “shock waves” which, at a frequency start with low amplitude go to a maximum and then diminish again. Much like the sound of “vowels” in speaking. It is best for time series with many abrupt peaks, and the orthogonal base of “wavelets” can be after all, any conceivable pattern.
- Best frequency finder: This is a technique devised by us, that we have used extensively in our research. It is a technique that does not seem to be known by the majority of the researchers, or at least used in their research. . It would deserve a separate paper to describe it. Its goal is not to find all frequencies with their amplitudes, and phases, but to find the one and most strong frequency if it exist at all. It consists in fitting with ordinary least squares method, a linear recursive equation of the type Δ(xn –xn-1 )=a xn . (See Goldberg Samuel in references). The equations is so chosen that the least squares fit of it is very easily done, e.g. in Microsoft excel with the function Linest, and the coefficient a is estimated as slope of a least squares line. Such an equation, if the constant a, turns out to be negative, has complex, characteristic root, and thus models an harmonic cycle. If a is non-negative no cycle exists. If a cyclic frequency is found then it is the best frequency, of period T and afterwards we fit a classical Fourier series trigonometric term, on this period (defined by a) to find the best fit amplitude and phase.
When analyzing time series for cyclic behavior it would give more vivid results if we “detrend” the original time series. In this note we adopt a rather classical method: We apply the analysis not on the original series, but on the volatility time series of the original (volatility=standard deviation of the continuous time, one-step percentage rate of change). Even if the original time series was an “almost” (non-stationary) random walk, the detrended in this way would be an (stationary) “almost” white noise.
In this short note we shall use only the classical spectral analysis technique, to prove the existence of the 11-years global climate cycle in the prices of the Stock Exchanges.
- The 11 years global climate cycle in the stock Exchange markets.
To prove the effect of the 11-years global climate cycle on the stock exchange markets we select the most highly traded index of S&P500 of the CME (Chicago USA). We present a chart of its data from the period 1985-2006 in figure 4 . We put a vertical line in this figure at the year of most recent solar sunspots peak. In the figure 5 we present the options implied volatility index VIX of options on S&P500, of the CBOE exchange. We put a vertical line in this figure at the year of most recent solar sunspots peak. We also plot a least squares fit of a 5-order polynomial that shows its slow periodic motion. This index is a kind of “de-trending” or “pre-whitening” of the index S&P500. We prefer nevertheless to calculate directly a “de-trending” or “pre-whitening” of the index S&P500 by taking its volatility. In particular we take its 10 days volatility adjusted for year scale. In other words we take the xn=Squarer-root(252)*Stanrad-deviation-for-10-days-of-Log(Pn/Pn-1) Where Pn is the prices of the initial S&P500 index . The volatility is thus the standard deviation of the continuous time daily rate of change. We present this time series in the figure 6. We put a vertical line in this figure at the year of most recent solar sunspots peak. Then we apply spectral analysis for this volatility time series, that the periodogram by frequency is shown in the figure 7. The spectral analysis is calculated by the software SPSS. We put a vertical line at the frequency of the 11-years. We see also clearly the frequencies of harmonics of this cycle especially of the half 5.5 years cycle.
Figure 4
Figure 5
Figure 6
Figure 7
- Sequences of causalities , demand-supply and cosmic habitforce.
In this paragraph we shall discuss the causalities behind the 11-years cyclic behavior of the capital markets. We have shown this behavior only for large stock Exchanges, like that of the CME, CBOE, ECBOT (futures on the index S&P500) at Chicago, in USA. But we have discovered the same cycle in many other large exchanges, including the vast Interbank Market of the cross-currencies exchanges rates , that is operated by the banks. The techniques and the volume of the results could not be appropriate for this short note.
Let us analyze, shortly, how the seasonality in the prices e.g. of grains like Soya, are explained by the balance of demand and supply in the classical seasonal cycle.
The Soya is not collected all the weeks of the year, therefore the supply of this grain is concentrated in a single season. The supply follows a cyclic annual pattern in time as time series. Assuming an average constant demand during the year, the balance of demand-supply (on higher supply on the same demand the prices decrease) would result in a time-series of prices that is annually cyclic in time. We shall not complicate the reader with various types of equations of the balance of demand-supply (there is an extensive literature on it), but we shall be constant that almost all of the systems of equations of demand and supply would give the above mentioned result. The previous is a well known and accepted fact. It is easy to extend it for an 11-years cyclic pattern of prices, if we know that the supply of Soya follows in addition, an 11-years cyclic pattern. This is plausible from what has been mentioned in the paragraph 2 on the 11-years cycle in ecology.
Similar arguments can be produced for the energy commodities like electricity, crude oil, heating oil, gasoline, natural gas etc. This time the supply may be considered in the average constant, while the demand by cyclic in time with period 11-years. In fact all the volume of the activities follows this cyclic patterns of 11-years. In the paragraph 3 we proved it for the 10-days volatility of CME index S&P500 . It seems that it holds that the time series of volumes of transactions is very strongly directly correlated to the volatility time series of the transaction prices (see e.g. Chen, Gong-Meng, Michael Firth, Oliver M. Rui, 2001 in the references) Therefore we may safely conclude that the 11-years cycles exists also in the volume of transactions of the stock Exchanges. As the volume of transactions is directly related to the cash, and exchanges of currencies , we deduce that such a cycle is expected to be found in the currencies commodities too of the Interbank Market. In fact this 11-years cycle is not confined to the prices of the capital markets. It exists in all business activities, including real estate. It is only that in capital markets the data are easier available, and more sensitive in time.
Still we believe that the various paths of demand-supply balance in various industries, and through e.g. input-output equations among them, is not the whole system of the causalities that could explain this phenomenon. Maybe it is only one side of the coin. Probably there are deeper lines of causalities that go parallel to he demand-supply causalities. Andrew Carnegies the well know major owner of the steel industry in the beginning of the 20th century in USA, mentioned the term cosmic habitforce to discuss how the various human activities in business tune between them like precession motion, have their own momentum conservation and eventually are compiled to the human subconscious. The size of this paper would not permit us to enlarge on it.
- The value of the knowledge of the effect of nature’s cycles on the growth of the wealth of nations
It is difficult to underestimate the value of such a prognostication, as the above, that was sketchy only presented, in the present short paper. We prefer to talk about prognostication which is a rather philosophical and social concept that mere stochastic process forecasting. The reason is that, as it seems, in modern sciences what is missing is not techniques and analysis, but a strong merging with good values and goals. It is usually said that prognosticating is of two kinds objective (Ïf I do nothing such and such will happen) and subjective (if I do so and so, such and such will happen).
At first we shall mention some of the values in such a prognostication that refers not in the forecasting itself but rather on the nature of involved concepts and methods. As a scientific method the value has the next basic characteristics:
1) It is not narrow statistically phenomenological (mere technical analysis or mere econometrics)
2) It is multi-science
3) It is synergistically wise
4) It is based on inside the planet and solar system celestial physics predictable action and flow of energy.
5) It is based on the functions of life on earth
6) It is intriguing for self-knowledge and knowledge of interrelations of phenomena in different cosmic scales (astronomical, ecological, social etc)
7) It is still mainly unexplored in its full details from the scientific point of view
8) It is essentially optimistic as far as scientist’s ability to handle the random, to forecast and realize that “pure randomness” and “chaos” may be in the scientist minds rather than in the hidden reality of the observed phenomena.
9) It gives to scientists, who love their quantitative work, the opportunity to prove that knowledge may bring directly good luck in wealth creation, when there is diligence for good risk management and the character in practice is strong, following not only the laws of the intellect, but also the laws of the emotions. In this way they may avoid the classical accusation of “paralysis by analysis” from the world of business to the world of academics
10) It gives hopes, for avoiding , public panic, economic disasters, suicides of individuals etc
During 20th century lived Buckminster Fuller, (see references) a rare and fine and global thinker that had omniscience wisdom, besides being a mellontologist (futurist). He was awarded by 42 honourable titles of various universities during the 20th century. We would like to quote him as he expresses once more the value of prognostication which is neither bad news neither good news but it can become good news to those that would respond appropriately and bad news to those that would ignore it. “…I am firmly convinced that I can see clearly a number of coming events, and I am therefore vitally, eager that people should not be hurt by the coming of these events, particularly when I can see ways in which it would be possible not only for them to avoid hurt, but even to prosper by and enjoy what now seems to me to be inevitable.”
According to him wealth starts from physical energy, which is so abundant as coming from outside the planet, that humanity can consider to consists from a few billions of “billionaires” in this energy and power. To quote him again:
“…Wealth is, then, the already organized human capability and know-how to employ the fixed inanimate, planetary assets and omni-cosmically operative and only celestially emanating, natural energy income, in such a manner as to predictably cope with so many forward days of so many human lives by providing for their (1) protection, (2) comfort, and (3) nurturing, and for (4) the accommodation of the ongoing development by humans of their as-yet-untapped store of intellectual and aesthetic faculties, while (5) continually eliminate restraints and (6) increasing the range and depth of their information-accumulating experience.”
In the next, lets run through the scales from the individual to a government and a group of nations to state some of the benefits of the prognostication of the 11 years cycle.
1) The individuals in their investments and wealth creation can use such a prognostication so as to know what sequence of 5-6 years shall have almost year after year gains and which 5-6 years shall have year after year losses. Some of the successful millionaires have revealed in their books, one of their “secret rules of success in the capital markets”. The rule is: “Buy whenever you have the funds and keep the assets at least for 10 years. Then during the next 15 years, find a year that the capital gains are satisfactory and sell”. Obviously such a rule holds good because of the 11-years cycle. Furthermore this rule does not require that you know the phase of the 11-years cycle, in other words which year is the peak and which year the minimum. If they would also know the concept of the 11-years cycle and its source in nature, they could improve the rule, and reduce its duration from 25 years to just at most 11.
2) The brokers are also much interested in such a prognostication, as the volumes of their transactions follow also this cycle. We mentioned that the volatility of the index is directly correlated with the volumes of the transactions. Therefore the brokerage companies can budget reliably for their revenues and thus for the size of their human capital.
3) As the enterprises assets contain investments in the capital markets, or real estate, such a forecasting, is valuable to anticipate their revenues.
4) Insurance companies, may very well adjust the hazard rates of many insurable events. The hazard rates (not only of disasters originating from extreme weather phenomena, but also of car accidents, and human diseases) wave following the 11-years cycle. Thus insurance premiums can be in other sequence of years, cheaper, and in other sequence of years, more expensive, in all cases better handled by both insurance companies and insured customers.
5) Also for governments budget such a prognostication is valuable. The government can know when to expect capital gains or losses and also when to expect public satisfaction or discontent (usually their personal gains or losses in the capital markets are related faulty by the public to bad or good financial policies by the government)
6) Finally whole unions of nations can benefit by such a prognostication in the same way that a domestic government can. Industries production and operations, and not only capital markets are influenced by the 11-years cycle. Budgeting that takes in consideration this cycle is obviously a more realistic and successful budget.
If more investors are aware of this cycle, in their investment decisions, will have as effect by the supply-demand balance to smooth-out and reduce the severity of the wave. This is exactly what the Buy-and-Holders would wish for.
But to realize and make best use of such a prognostication in society and unions of nations, we must adopt the philosophical and practical attitude of win-win in the deals and interrelations. The creation of new wealth by new technology and procedures is unavoidably a wealth of all society not only of parts of it. This excludes the concept and belief of “zero-sum game” in the capital markets. Only the win-win attitude and principle is appropriate and true in the global economy. This win-win attitude goes parallel with a mentality of abundance of resources and technology, rather than a scarcity mentality of resources and technology in business and investments.
REFERENCES
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Oliver M. Rui, 2001 “The dynamic relation between stock returns, trading volume, and volatility.” The financial review 38, 153-174.
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[3] Buckminster R. Fuller Critical Path
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[8] Goldstein H. Multilevel Statistical Models
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A classical and old book that contains practically all the essential ideas:
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Under Adaptive Learning∗
(University of Pittsburgh)
(University of New Orleans)
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[ 12 ] Richard W. Katz (Editor), Economic Value of Weather and Climate
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[13 ] Lars Tvede Business Cycles
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[14] Lambert H. Koopmans The spectral Analysis of Time series
Academic press 1995
[15] Longford Nicholas T. Random Coefficient Models
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[16] Malliaris A.G. and Brock W.A. Stochastic Methods in Economics and Finance
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[18 ] Patrick A. Pintus 2006
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[19] Tsoukalas John D. Can a Sunspot Driven Model Replicate Recognizable Business Cycles? (University of Maryland and Bank of England) Ekonomia, 2004, vol. 7, issue 2, pages 89-120
Internet links
A link for recent forecast of the next peak of sunspots:
A link for solar influence data center:
For next solar cycle prediction
