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The Review of Economic Studies. Quantitative Economics — Journal of the Econometric Society. Review of Industrial Organization. Canadian Journal of Economics , 50 5 , December Journal of Industrial Economics , 64 4 , December Handbook on the Economics of Retail and Distribution , Chapter 9, pp Emek Basker editor.

January , Edward Elgar Publishing. Marketing Letters , Volume 26 2 June , Quantitative Marketing and Economics , Volume 12 3 September , Economic Inquiry , Volume 52 2 , Advances in Econometrics , Volume 31 , pp. Advances in Economics and Econometrics: Theory and Applications. Tenth World Congress of the Econometric Society. Journal of Econometrics , May , Marketing Letters , 19 3 December , American Economic Review , 97 2 May, , A visual analysis is required for the early recognition of common faults in experimental and real data such as breaks, drifts or rare events Kantz and Schreiber, The visual analysis provides information about variation amplitude, possible trends and pattern evolution McCullough et al.

In Fig. From the original evolution Fig. Data includes remarkable economic and financial events in the period Hong et al. During the first 18 years, prices sharply rise for 4 years until reaching the maximum value.

Then, prices fall for longer periods and remain fluctuating at low level until the next price rising attempt. The next price rally extended for 3 years whereupon prices fall dramatically to the lowest level of the time series. Low prices during the following 44 years shows that three sharp price rising trends evolving between three to 5 years. After reaching the maximum value of the rally, prices strongly decrease and remain fluctuating at low level for about eight to 10 years. This is a similar pattern exhibited during high prices period; however, it is much longer.

High price pattern repeats between and exhibiting the two price rise attempts followed by the sharp falls.

### Econometric Models of Minerals Markets Uses and Limitations

Then, between and , low price pattern comes back showing the three short prices growing periods followed by larger price decreasing and fluctuation periods. Finally, between and the high prices dynamic is repeated. The two price ascents evolved in a shorter period; however, the characteristic sudden prices decline is also displayed.

Values heuristically chosen. This behaviour is consistent with the price-demand inelasticity of the mining industry, and in particular because of the significant time delay between investment and production. From Fig. Many economic and financial time series such as exchange rates, asset prices and macroeconomic aggregates, such as real Gross Domestic Product, are non-stationary. However, they may become stationary computing the differences between consecutive observations by using one of the two differencing techniques: first differencing or transformation Hyndman and Athanasopoulos, ; Zivot and Wang, Establishing the stationarity of a system is a complex task and almost impossible to determine by a single test.

As the null hypothesis H 0 is commonly rejected for large sampling sizes Alquist et al. We performed three tests to determine the stationarity of the system. We therefore reject the null hypothesis and conclude that the series is stationary.

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We evaluate the uniqueness of the solution of the reconstructed vector in the phase space to distinguish between a deterministic chaos and the irregular random behaviour Kaplan, ; Kaplan and Glass, ; Kodba et al. We assess the deterministic behaviour by two tests. Kaplan, This test examines the continuity of the orbits of the phase space reconstructed from the delayed original series and measures the separation between E-statistics of the original time series and a set of surrogates created. We determine the level of determinism of the time series by examining the percentage of overlapping E-statistics between the original time series and its surrogates in a barplot.

We observe a noticeable separation between them which leads us to conclude that the signal is deterministic see Supplementary Fig.

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## The World Copper Market

Secondly, we use the false nearest neighbours method which uses the function E 2 d Cao, that can distinguish between deterministic and stochastic signals. Thus, while stochastic signals exhibit invariant E 2 d values approximately equal to one, for all values of m , E 2 d values fluctuate for deterministic signals and diverge from one Cao, see Supplementary Fig.

We assert the deterministic features of the time series by the high determinist level of the E-statistic and fluctuations of E 2 d values. The complexity of systems can be used to distinguish between periodic, chaotic and genuine random behaviour of the systems related to biology and social sciences. Entropy is a statistical tool to measure the complexity of nonlinear dynamics systems Balasubramanian et al. In dynamic systems often chaotic Pincus, entropy is known as the rate of new information generation Lake et al. In theory, periodic systems are less complex having the lower entropy, and random noise systems are more complex and less predictable having the higher entropy and chaotic systems present intermediate values of entropy Ferrario et al.

The capacity to characterise the behaviour of a time series without need of a previous established hypothesis regarding the genesis of the system Ferrario et al. Systems complexity has been associated to the presence of chaotic patterns. Complex systems present coordinated movements and an evolutive learning and development process Yentes et al. These are prominent features to assess on systems involving human being such as markets behaviour. The development, adaptation and evolution of human social learning and cognitive skills in the economic environment is based on experiences acquired from environmental stimulus through time.

These skills are crucial for setting perception, preferences and the decision-making process of humans interacting into the economic environment Baker et al. This pattern only reflects the design of both algorithms, while values decreases while r increases Lu et al. For both, ApEn and SampEn, periodic process reports smaller values, stochastic process higher values that can go toward infinity for the SampEn and chaotic systems reports intermediate values.

Consistency of entropy results is measure by comparing the relative distance between ApEn and SampEn values obtained from each data sets for different setting of r and m. Original copper prices show values closer to chaotic behaviour at 0. Original dataset shows values close to chaotic behaviour for 0. However, there is not possible to reach any conclusion due to the erratic behaviour of random and original time series. Values of original copper prices are closer chaotic behaviour at 0.

Original dataset values are close to chaotic behaviour for 0. Chaotic system and original times series show high similarities which are intensified by the consistency of both tests exhibiting almost the same results. Random times series exhibit divergent values for both tests and largest errors compare to chaotic and original time series.

SampEn shows higher values compare to ApEn which can be the result of limitations to assess short stochastic datasets. We selected SampEn because it is more reliable for assessing small data sets Lake et al.

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We concluded that the original copper price time series exhibits chaotic patterns similar with the chaotic Logistic Map. Thus, we assert that the time series was not generated by a stochastic process neither has periodic behaviour. This finding reaffirms our assertion that copper prices do not exhibit periodic behaviour fluctuating in super cycles and the debatable adequacy of stochastic models for representing mineral commodity prices Giles, ; Mandelbrot, ; Sanei, ; Watkins and McAleer, Using the false nearest neighbours method Becks et al. However, as this method is based on some subjective and criticisable assumptions such as the heuristic tolerance R t Cao, we conduct a second test.

Using a phase separation plot Provenzale et al. Sensitivity to the initial conditions is asserted by the rapid exponential divergence of initially close trajectories through time Becks et al.

It can be asserted that the chaotic behaviour of a data set should be assessed by generating several trajectories starting from a given point surrounded by a number of initial conditions. However, this methodology is only true for theoretical or ideal chaotic systems, theoretical stochastic dynamic models or for elementary models of well-known chaotic behaviour represented by a set of differential equations or time-delay models that reduce the complexity of their dynamic behaviour Fradkov and Evans, ; Guegan, ; Vlad et al.

In economics time series, a single time series trajectory is often the main and solely available source of information describing systems dynamics.

## The World Copper Market - Structure and Econometric Model | G. Wagenhals | Springer

Experimental series of complex systems involving biological, physics and social sciences are constrained by using short data sets, because of their technical, temporal or physiological restrictions Becks et al. Although several attempts have been made to describe the copper price dynamics, there is not yet a mathematical time delay model or a set of differential equations describing its long-term behaviour annual base in the last century. Possible combinations are highlighted in light blue Table 2.

It reflects the loss of sensitivity of the systems at high embedding dimension and long-time delay. All m values are appropriate.