**Introduction**

The presence of long-term trends, both linear and nonlinear, in the differences between various expenditure categories of producer and consumer prices has been reported in several papers [1-7]. These trends allow predicting prices of many commodities at time horizons of several years. Such predictions are of fundamental importance for all stock market participants because of their anchoring and stabilizing effect. A striking feature associated with the trends is high volatility during the transition periods between adjacent trends. At first glance, the transient processes in the differences between price indices are driven by some stochastic forces, and thus, can not be predicted. It was found, however, that during the transition to new trends, some differences mimic trajectories of a pendulum-like oscillation [3,4]. Therefore, these trajectories are predictable ones and have been actually used in forecasting crude oil and gold ore prices [4,6]. The next natural step is to compare individual trajectories in order to evaluate their similarity in shape and timing of major peaks and troughs. If a price difference lags behind others, it would be possible to predict its evolution at a time horizon of corresponding lag.

The model derived in [1,4] implies that the difference between the headline PPI, PPI, and the index for an individual commodity, iPPI, can be described by a linear time function over time intervals of several years:

**1. The model**The model derived in [1,4] implies that the difference between the headline PPI, PPI, and the index for an individual commodity, iPPI, can be described by a linear time function over time intervals of several years:

*iPPI(t) – PPI(t) = A + Bt (1)*

where A and B are empirical constants, and t is the elapsed time. Therefore, the “distance” between the PPI and the studied index is a linear function of time, with a positive or negative slope B.

As an example, Figure 1 presents the difference between the iron and steel index and the PPI (all PPI indices are retrieved from the BLS web-site: http://www.bls.gov/data on 20.08.2009). There is a transition period between 2000 and 2001. This turning point is characterized by a slightly elevated volatility. Since 2008, the difference has been also passing a turning point, but this time with a very high volatility caused by the uncertainty in the characteristics of the following trend. A naive assumption about the future trend is that it will repeat the predecessor but with an opposite sign. The green line in Figure 1 represents the hypothetical new linear trend.

From Figure 1, the fundamental feature of the difference is that all deviations from the trends were only short-term ones. This implies that the current or future deviations from the new trend, which has been under development since 2008, must be promptly compensated. This feature allows a short-term (months) price prediction along the trend. The principal task of this study is different - to predict the evolution of the difference during the current transition period.

As an example, Figure 1 presents the difference between the iron and steel index and the PPI (all PPI indices are retrieved from the BLS web-site: http://www.bls.gov/data on 20.08.2009). There is a transition period between 2000 and 2001. This turning point is characterized by a slightly elevated volatility. Since 2008, the difference has been also passing a turning point, but this time with a very high volatility caused by the uncertainty in the characteristics of the following trend. A naive assumption about the future trend is that it will repeat the predecessor but with an opposite sign. The green line in Figure 1 represents the hypothetical new linear trend.

From Figure 1, the fundamental feature of the difference is that all deviations from the trends were only short-term ones. This implies that the current or future deviations from the new trend, which has been under development since 2008, must be promptly compensated. This feature allows a short-term (months) price prediction along the trend. The principal task of this study is different - to predict the evolution of the difference during the current transition period.

Figure 1. Illustration of linear trends in the difference between the headline PPI and the producer price index of iron and steel in the U.S. There are two quasi-linear segments with a turning point near 2000. Currently, the difference passes second transition period. Two linear trends with relevant linear regression lines are also shown. A tentative future trend is shown by a green line.

**2. Price prediction**Relationship (1) describes the evolution of a given difference. The absolute value of such a difference may vary between commodities due to variation in start years. This might complicate the cross-commodity comparisons. So, it is instructive to use the differences normalized to the PPI:

*(iPPI(t)-PPI(t))/PPI(t)*. In a sense, the normalized differences represent the evolution of the rate of deviation from the PPI over years. For historical and logical reasons, we have chosen the following commodities: iron and steel, crude petroleum (domestic production), and gold ores. Figure 2 depicts corresponding time histories of the normalized deviation from the PPI. The initial inspection reveals the following apparent features: the (normalized deviation from the PPI of the) index for gold ores is characterized by a higher volatility; the index for iron and steel lags by several months behind the other two indices; the deviation of the crude oil index is comparable to the PPI itself. Figure 2. The deviation of the iron and steel price index, the index of crude oil, and the index for gold ores from the PPI, normalized to the PPI.

In order to refine the pattern we additionally normalized the curves in Figure 2 to their peak values between 2005 and 2009:

The normalization allows a direct comparison of corresponding shapes. In Figure 3, we display the normalized index for iron and steel shifted by six and eight months back in the past for the synchronization of its peak with that observed in the normalized index for crude petroleum (upper panel) and gold ores (lower panel), respectively. The (normalized) index for crude petroleum demonstrates larger discrepancies from the index of iron and steel in the overall shape and timing of the current trough. On the other hand, despite its higher volatility, the index of gold ores is very similar to that of iron and steel. Simple smoothing with a weighted MA(3) makes the curves resemblance even better. As an invaluable benefit of the resemblance, one can use the eight-month lag to predict the future of the iron and steel price index.

Figure 3. The deviation of the iron and steel price index from the PPI, normalized to the PPI and the peak value after 2005 as compared to the normalized deviations of the index for crude petroleum (upper panel) and gold ores (lower panel). The normalized index for iron and steel is shifted six and eight months back in the past, respectively.

Between 2006 and 2010, the deviation of the price index for iron and steel from the PPI in the USA repeats the trajectory of the deviation of the index of gold ores with an eight-month lag. Therefore, the prediction of price for iron and steel at this horizon is straightforward – it will be growing during the next six to eight months. It is likely that in 2010 the index of iron and steel will approach closely the level attained in August 2008. From this level, it will be declining in the long run following the new trend, as shown in Figure 1.

1. Kitov, I., Kitov, O., (2008). Long-Term Linear Trends In Consumer Price Indices, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(2(4)_Summ), pp. 101-112.

2. Kitov, I., (2009). Apples and oranges: relative growth rate of consumer price indices, MPRA Paper 13587, University Library of Munich, Germany.

3. Kitov, I., Kitov, O., (2009). A fair price for motor fuel in the United States, MPRA Paper 15039, University Library of Munich, Germany,

4. Kitov, I., Kitov, O., (2009). Sustainable trends in producer price indices, Journal of Applied Research in Finance, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. I(1(1)_ Summ), pp. 43-51

5. Kitov, I., Kitov, O., (2009). PPI of durable and nondurable goods: 1985-2016, MPRA Paper 15874, University Library of Munich, Germany

6. Kitov, I., (2009). Predicting gold ores price, MPRA Paper 15873, University Library of Munich, Germany

7. Kitov, I., (2009). Predicting the price index for jewelry and jewelry products: 2009-2016, MPRA Paper 15875, University Library of Munich, Germany

In order to refine the pattern we additionally normalized the curves in Figure 2 to their peak values between 2005 and 2009:

*(iPPI(t)-PPI(t))/[PPI(t)*max{iPPI-PPI)}]*

The normalization allows a direct comparison of corresponding shapes. In Figure 3, we display the normalized index for iron and steel shifted by six and eight months back in the past for the synchronization of its peak with that observed in the normalized index for crude petroleum (upper panel) and gold ores (lower panel), respectively. The (normalized) index for crude petroleum demonstrates larger discrepancies from the index of iron and steel in the overall shape and timing of the current trough. On the other hand, despite its higher volatility, the index of gold ores is very similar to that of iron and steel. Simple smoothing with a weighted MA(3) makes the curves resemblance even better. As an invaluable benefit of the resemblance, one can use the eight-month lag to predict the future of the iron and steel price index.

Figure 3. The deviation of the iron and steel price index from the PPI, normalized to the PPI and the peak value after 2005 as compared to the normalized deviations of the index for crude petroleum (upper panel) and gold ores (lower panel). The normalized index for iron and steel is shifted six and eight months back in the past, respectively.

**Conclusion**Between 2006 and 2010, the deviation of the price index for iron and steel from the PPI in the USA repeats the trajectory of the deviation of the index of gold ores with an eight-month lag. Therefore, the prediction of price for iron and steel at this horizon is straightforward – it will be growing during the next six to eight months. It is likely that in 2010 the index of iron and steel will approach closely the level attained in August 2008. From this level, it will be declining in the long run following the new trend, as shown in Figure 1.

**References**1. Kitov, I., Kitov, O., (2008). Long-Term Linear Trends In Consumer Price Indices, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(2(4)_Summ), pp. 101-112.

2. Kitov, I., (2009). Apples and oranges: relative growth rate of consumer price indices, MPRA Paper 13587, University Library of Munich, Germany.

3. Kitov, I., Kitov, O., (2009). A fair price for motor fuel in the United States, MPRA Paper 15039, University Library of Munich, Germany,

4. Kitov, I., Kitov, O., (2009). Sustainable trends in producer price indices, Journal of Applied Research in Finance, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. I(1(1)_ Summ), pp. 43-51

5. Kitov, I., Kitov, O., (2009). PPI of durable and nondurable goods: 1985-2016, MPRA Paper 15874, University Library of Munich, Germany

6. Kitov, I., (2009). Predicting gold ores price, MPRA Paper 15873, University Library of Munich, Germany

7. Kitov, I., (2009). Predicting the price index for jewelry and jewelry products: 2009-2016, MPRA Paper 15875, University Library of Munich, Germany

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