1987, 2001, 2008 … 2011

The essence of any quantitative model consists in the accuracy of prediction or predictive power. The higher is signal/noise ratio in a given data set the better can be the estimate of model parameters or the uncertainty of corresponding prediction. For the S&P 500 stock market index, the most prominent signals were measured during the periods of the fastest change: 1987, 2001 and 2008.

We have developed a model [1] linking the S&P 500 and its returns to the population of some characteristic age. The original model links the S&P 500 annual returns, Rp(t), to the number of nine-year-olds, N9:

Rp(t) = AdlnN9(t) + B (1)

where Rp(t) is the S&P 500 yearly return, A and B are empirical coefficients to be determined by some fitting procedure. They may change depending on the approximation used to represent N9. In the previous post on the S&P 500 returns we have approximated N9 by the number of three-year-olds, N3, six years before. Accordingly, we have obtained a prediction of the S&P return at a six year horizon, i.e. in 2010 one can foresee the returns in 2016. Relevant empirical relationship is as follows”

Rp(t+6) = 100dlnN3(t) - 0.23 (2)

Figure 1 depicts the S&P 500 returns, both actual one and that predicted by relationship (2). Both curves are coinciding in practical terms between 2008 and the middle of 2010.

In 1987 and 2001 abrupt falls in the returns were also observed. In this respect, the model also demonstrates an excellent predictive power, as Figures 2 and 3 depict. There are obvious differences between the measured S&P 500 returns. In 1987, the fall was very fast but not deep, from +0.3 to -0.1. In 2001, the returns declined gradually from +0.3 to -0.3 in 2002. In 2008, the observed curve fell from 0 to -0.5, i.e. approximately same as in 1987.

In all cases the model gives a good prediction of the timing and amplitude of the observed returns. So, the model has a good predictive power, considering that the prediction can be obtained at a nine year horizon with the birth rate used as a proxy to N9.

Therefore, one might treat our prediction of the 2011 fall as a reliable one.

During the last two weeks, the S&P 500 has been growing at a healthy pace. Currently, it exceeds the predicted level by approximately 50 to 100 points. This is a good reason to suggest that a significant force, which must eventually return the index to the trend line, has been developing in September 2010. If the growth continues into the second half of September one might expect a dramatic drop in October 2010. However, this will be just a part of the overall decrease to the level of -0.5 expected in July-August 2011.

Figure 1. The prediction of the S&P 500 annual return for the period between 2008 and 2012. We tentatively put the September's closing level at 1030.

Figure 2. The prediction of the S&P 500 annual return for the period between 1985 and 1989.

Figure 3. The prediction of the S&P 500 annual return for the period between 1998 and 2003.

1. Kitov, I., Kitov, O. (2010). S&P 500 returns revisited, http://ideas.repec.org/p/pra/mprapa/21733.html.


Economics as an iterative research program

1. We do not know how it has happened …

2. The importance of the economic profession just grows …

3. Give us more time and resources …

4. Our models has been dramatically improved since the last unpredicted event …

5. We do not how it has happened again (and again) …


On deflation, once again

Five years ago we developed an empirical model describing inflation in developed countries and published a forecast for the USA at a ten year horizon in 2006 (Kitov, 2006ab) as a linear and lagged function of labour force. For the USA the model is as follows:

DGDP(t) = 4.0dLF(t-2)/LF(t-2) – 0.03 (1)

where DGDP is the GDP deflator at time t, and LF is the level of labour force two years before, t-2. In equation (1), the slope of linear relationship is 4.0 and intercept is -0.03. With labour force constant (dLF=0), the overall price inflation in the USA would remain negative at the level of -3% per year. For inflation to be sustainable at the level of 2% (the fed’s unannounced target) the growth in labour force of 1.25% per year is required.

Eventually, the link between price inflation and labour force was successfully tested for cointegration (Kitov, Kitov, Dolinskaya, 2007) and validated using 1-D boundary elements methods (Kitov, Kitov, 2010). This links holds for other developed countries as well (Kitov, Kitov, 2010).

Figure 1 illustrates the linear and lagged relationship between inflation (GDP deflator) and the change rate of labour force in the USA for the period between 1960 and 2006. The estimates of the change rate of labour force are shifted by two years ahead in order to synchronize the predicted peaks of inflation with those actually observed in 1975 and 1984. Due to the uncertainty in the labour force (and inflation) measurements the most reliable readings correspond to the changes of the largest magnitude, as described by the BLS. Every sound model of price inflation must explain these peaks.

Figure 1. Measured inflation (GDP deflator) and that predicted from the dLF/LF (shifted 2 years ahead). An agreement is observed throughout the whole period with some short fluctuations in labour force potentially induced by the population corrections implemented by the Census Bureau in the census years.

Figure 2 presents two smoothed inflation curves – the measured and the predicted one. The smoothing was attained by a 7-year moving average, MA(7), with a one-year step. The predicted curve is shifted by 2.5 years (we used two time scales with a half-a-year shift) ahead in order to fit the inflation peak near 1978. The original (not shifted) predicted curve is also shown in order to illustrate that it is actually leading by 2.5 years.

Figure 2. MA(7) of the predicted and measured inflation. The prediction is made according to the relationship (1). The predicted curve is shifted by 2.5 years ahead.

Figure 3 displays the observed and predicted cumulative inflation curves starting from 1965. The predicted cumulative curve is obtained by a progressive summation of values from 1963 and is also shifted ahead by 2 years. There is an almost complete agreement between the cumulative curves for the whole period. The only small deviation occurred around 1993 and corresponds to a sharp drop of the rate of labour force growth as induced by the baseline working age population correction.

The predicted cumulative curve is very sensitive to free term in (1). Even the initial difference of 0.0001 results in a tangible deviation from the measured curve after 50 years. Therefore, the value of free term in (1) can be estimated with a good accuracy. With cumulative curves, one can obtain the most accurate coefficients solely on the basis of visual fit. Also, since these two cumulative curves have R2>0.999 and actually represent indexes one can replace the secular growth in the overall price as the cumulated growth in labour force two years before. In a sense, these curves are similar to conservation laws or integral equations in physics (Kitov, Kitov, 2010).
Figure 3. Comparison of the cumulative values of the observed and predicted inflation presented in Figure 1. The predicted curve starts from 1963 and is shifted by 2 years ahead. An agreement is observed with a notable change from convexity before 1980 to concavity one after 1980.

Relationship (1) can be used for a prediction at a larger time horizons using labour force projections made by various institutions. For example, the projections made by CBO (2004) and BLS (2005) undoubtedly indicate a decrease in the participation rate and a decaying growth rate of the working age population. According to these projections, staring from 2010, the annual increase in labour force will be less than 1,200,000 – the value separating inflation and deflation. Hence, the year of 2012 is likely to mark the beginning of the deflationary era in the USA (which hopefully is the global disaster the Mayans talked about) because of the two-year lag between the labour force change and inflation. Figure 4 details the prediction based on the CBO’s projection of the labour force. After peaked at 3.2% in 2007, the rate of price inflation has been at a gradual decrease, which will lead to the first red figure in 2012.

Hence, we are waiting an extended deflationary period in the USA since 2012.
Figure 4. Predicted inflation rate for the period between 2006 and 2016 according to the CBO’s (2004) labour force projection. A deflationary period starts in 2012.

Bureau of Labor Statistics. (2005a). Labor force projections to 2014: retiring boomers. Monthly Labor Review, November 2005.

Congressional Budget Office. (2004). CBO’s Projections of the labor force. September 2004.

Kitov, I. (2006a). Inflation, unemployment, labor force change in the USA, Working Papers 28, ECINEQ, Society for the Study of Economic Inequality,http://ideas.repec.org/p/inq/inqwps/ecineq2006-28.html

Kitov, I. (2006b). Exact prediction of inflation in the USA, MPRA Paper 2735, University Library of Munich, Germany

Kitov, I., Kitov, O., Dolinskaya, S. (2007). Inflation as a function of labor force change rate: cointegration test for the USA, MPRA Paper 2734, University Library of Munich, Germany

Kitov, I., Kitov, O. (2010). Dynamics of Unemployment and Inflation in Western Europe: Solution by the 1-D Boundary Elements Method, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. V(2(12)_Summer), pp. 94-113.


Does crude drive the price index of steel and iron?

This is a quarterly update.

In September 2009, we reported that the price index of crude oil had been likely evolving in sync with that of iron and steel, but with a lag of two months. In order to present both indices in a comparable form, the difference between a given index, iPPI, and the overall PPI was normalized to the PPI: (iPPI(t)-PPI(t))/PPI(t). The normalized differences represent the evolution of the rate of deviation from the PPI over years.

Figure 1 depicts the corresponding time histories of the normalized deviations from the PPI, including the most recent period since June 2010. Simple visual inspection reveals the following feature: the (normalized deviation from the PPI of the) index of iron and steel lags by two months behind the (normalized) index of crude oil.

Figure 1. The deviation of the iron and steel price index and the index of crude oil from the PPI, normalized to the PPI.

In order to reduce both deviations to the same scale we additionally normalized the curves in Figure 1 to their peak values between 2005 and 2009:


This scaling allows a direct comparison of corresponding shapes. In Figure 2, we display the normalized index of iron and steel shifted by two months ahead to synchronize its peak with that observed in the normalized index for crude petroleum. (The period between May and July 2010 is included.) The scaled index of crude demonstrates just minor discrepancies from the index of iron and steel in the overall shape and timing of the peak and trough. Simple smoothing with MA(3) makes the curves resemblance even better. As an invaluable benefit of the resemblance, one can use the two-month lag to predict the future of the iron and steel price index.
Figure 2. 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 deviations of the index for crude petroleum normalized in the same way. The normalized index for iron and steel is shifted two months ahead.

Between 2006 and 2010, the deviation of the price index of iron and steel from the PPI in the USA repeats the trajectory of the deviation of the index of crude petroleum (domestic production) with a two-month lag. Therefore, the prediction of iron and steel price for at this horizon is a straightforward one. It is likely that in the fourth quarter of 2010 the index of iron and steel will be decreasing following the observed fall in the index for crude petroleum.

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

PPI of copper ores and grains

Three months ago we revisited our prediction on the evolution of the PPI of copper and grain, which had been made in 2009. The original prediction limited the level of the copper PPI in 2010:

Therefore, copper price will likely not be growing to its peak in April 2008 (491.7), but will likely return to heights around 350.

Three months ago we suggested that:

In the short-run, the index for copper will NOT be growing too long, at least NOT till the end of 2010.

Figure 1 compares the prediction and actual behavior for the producer price index of copper ores relative to the overall PPI. All in all, the prediction was right: by the end of 20009 the price index of copper has reached the level of 350 (375) and even higher in the beginning of 2010 (443 in April). However, it has not reached the 2008 level and started to fall in the second half of 2010. It is very likely that the fall will continue, potentially with a higher volatility, in 2010 and will be stretched into 2011. At a two to four year horizon, the price index for copper ores should return to the pre-2008 level.

Figure 1. Evolution of the price index of copper ores relative to the PPI.

The price index for grains has been following our predictions as well. In 2009 we wrote:

It is instructive to compare two major spikes in the grains index in 1996 and 2008 relative to the PPI. In order to avoid comparing absolute values, which undergo secular growth, the evolution of the difference between the PPI and the price index of grains normalized to the PPI. Figure 3 presents the normalized curves. The left panel shows that the spike in the grains PPI in July 1996 is similar in relative terms to that observed in 2008. The right panel tests this hypothesis: the spikes are synchronized - for the black line is shifted forward by 142 months. From this comparison, it is likely that decline in the grains index relative to the PPI will extend into the 2010s.

Three months ago we confirmed the prediction that the index of grains would follow the path observed in 1996:

The index for grains will continue its decline relative to the PPI. As a consequence, one can expect that the index for food will be also decreasing and this decline will stretch into the 2011

Figure 2 illustrates the accuracy of our prediction of the index of grains. The index has been falling relative to the PPI and the trajectory actually repeats that observed 142 months before. We expect the normalized difference to follow up the once observed recovery path. It is interesting that this time the through was not as deep as in 1996. Supposedly, there are more efficient mechanisms counteracting the growth in the price of grains when were in 1996. Judging from Figure 2, the index of grain will not be growing during the next several years.

Figure 2. Evolution of the difference between the PPI and the price index of grains normalized to the PPI. Upper panel: Comparison of the current curve to that observed 142 months ago. Lower panel: Same as in the upper panel; the most recent period.

Short term prediction
In the short-run, the index for copper will likely be falling in 2010. The index for grains will continue its slight decline relative to the PPI.

For details see also our papers:
 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, v. 1, (in press)

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


Crude in 2010

Three months ago we presented a forecast for the PPI for crude oil and oil price in August 2010. Tentatively, we put the index at the level between 160 and 180 in August 2010. Crude oil price corresponding to this level of index should be between $62 and $70 per barrel. It’s time to revisit the price.

All our estimates are based on the existence of long-term sustainable trends in the differences between various subcategories of the producer price index (PPI). The concept is illustrated in Figure 1, where the difference between the overall PPI and that for crude petroleum is approximated by three linear trends. The most recent trend started in the beginning of 2010 and has been strengthening since then. Without loss of generality, we consider that the new developing trend will be a mirror reflection (opposite but equal slope) of the previous trend observed between 2002 and 20008. This is the long-term prediction for oil price with the PPI of crude at the level of 75 point in 2016.

Figure 2 presents the short term view elaborating on the most recent transition period from July 2008 to August 2009 and the details of the new trend. In March 2009, we presented two predictions. In (1) we presumed that, when reached the trend, the price would follow along it. Second prediction was based on a “dynamic overshoot” with oil price dropping much below the new trend, as shown by solid diamonds in Figure 2. From these two predictions, the first was right.

Figure 3 details our prediction made in June 2010 on the evolution of the oil price index between June and August 2010. We expected that the price would fall and the predicted curve rise above the trend, as shown by red circles. This is a consequence of the fluctuation around the trend: the price can not be retained just on one side of the trend line. The measured price did not touch the trend line, however. So, our prediction was not fully right despite the price actually has fallen significantly. The Augusts’ estimates of the producer price index for crude petroleum will be published in the middle of September. It will definitely show a decrease relative to July, but the price will not drop to $70 per barrel.

Therefore, we foresee two possible scenarios. A more likely one supposes that the price in September will fall below $70 per barrel and the measured difference (open circles) in Figure 3 will intercept the trend line. This scenario will be in line with our prediction of the S&P 500 level below 1000 in September.

We can not exclude that the price (and the S&P 500) will not drop in September cumulating more potential relative to the trends. Then, this must happen in October-November and the potential drop will just increase in time as the deviation between the actual curve and the trend. The trend itself seems to be well-established already since the price goes along the trend since August 2009. It might also happen that the true trend is different form the predicted one. It may have lower slope than during the previous period. Then the price will be decreasing a lower rate into the second half of the 2000s. One can better estimate the trend slope in couple years, but before actual data are available we will retain our hypothesis on the slope value.

All in all, we expect the price index of crude petroleum to follow the new trend in the long run with short-term fluctuations of various amplitude and period.

Figure 1. Sustainable linear trends in the difference between the overall PPI and that for crude oil between 1987 and 2010. Currently, we observe the emergence of a new trend, which supposedly is a mirror reflection of the previous one.

Figure 2. The evolution of the difference between the overall (all commodities) PPI and that for crude oil.

Figure 3. The measured and predicted difference between the overall PPI and the index for crude petroleum.

1. Crude Oil And Motor Fuel: Fair Price Revisited


S&P 500 in September 2010

Good news from August 2010 is that the S&P 500 market index is back on the track predicted couple years ago. Figures 1 and 2 update the previous versions published on Seeking Alpha in August with the closing level of ~1050 reported on August 31. Both predicted curves are very close to the observed ones over the whole period between March 2009 and August 2010. This prediction would have been a convincing one for everybody except market players who do believe that stock prices are unpredictable.

So, we will continue tracking the level of S&P 500 and its returns. The next move is likely below the trend to compensate for a short positive excursion in July 2010. This might be a drop by 40 to 80 points, likely to the level below 1000. It might be accompanied by a small panic. However, a positive jerk associated with local positive news is not excluded but it should not be high in amplitude.

Our concern about possible repetition of the 1987 fall, if the index would continue its deviation from the predicted trend into October 2010, has been resolved by the drop of around 50 points from the July’s level of 1101. So, there is no danger of a severe panic on the stock market.

Below we repeat a mandatory part with a bit of mathematics for the readers interested in details of our excellent (in terms of predictive power) model. The model is also presented in our working paper [1] and monograph [2].

The original model links the S&P 500 annual returns, Rp(t), to the number of nine-year-olds, N9. In order to extend the prediction in time we use the number of three-year-olds, N3, as a proxy to N9 and obtain a forecast at a six-year horizon:

Rp(t+6) = 100dlnN3(t) - 0.23 (1)

where Rp(t+6) is the S&P 500 return six years ahead (in 2010 one can foresee the returns in 2016). Figure 1 depicts germane S&P 500 returns, both actual one and that predicted by relationship (1). Both curves are coinciding in practical terms.

Because of the observed linear growth in N3 one can replace it with linear trends for the period between 2008 and 2011, as Figure 2 shows. This model predicts that the S&P 500 stock market index will be gradually decreasing at an average rate of 37 points per month. All fluctuations in N3, as observed in Figure 1, are smoothed in this linear representation.

Figure 1. Observed and predicted S&P 500 returns. The last point for the observed series is August 2010.

Figure 2. Observed S&P 500 monthly close level and the trend predicted from the number of nine-year-olds. The slope is of -37 points per month. The same but positive slope was observed between February 2009 and April 2010. The last point in the observed series is August 2010. All in all, the S&P 500 is back on the predicted trend.

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