Stock market behavior has always been a source of surprises for researchers, traders, and investors. The long term aggregate price trends have been explained as related to fundamental factors. These factor, however, are numerous and not well defined. As a result, no prediction is possible.

In [1] we presented selected results of modelling, which demonstrate the robustness of long-term (years!) prediction. In the model, only one factor drives trends and fluctuations ain S&P 500. Therefore, our results are easy to interpret and repeat. Since March 2009, S&P 500 has been growing. This fast growth was foreseen in February 2009. Since March 2009, we have been reporting comparisons of monthly returns – observed and predicted. So far, the match is excellent.

In this post, preliminary results for July 2009 are reported, as accompanied by formal introduction of the model. The model is an empirical one and needs to be assessed and (sometimes) updated when new data are available. The last assessment (published in this blog) was carried out in June 2009. Since the model links S&P 500 to the growth rate in real GDP, we made relevant forecast for the second quarter of 2009 as +5%. This value is very high compared to the consensus (Conference Board) prediction of -0.7% .

In [1] we presented selected results of modelling, which demonstrate the robustness of long-term (years!) prediction. In the model, only one factor drives trends and fluctuations ain S&P 500. Therefore, our results are easy to interpret and repeat. Since March 2009, S&P 500 has been growing. This fast growth was foreseen in February 2009. Since March 2009, we have been reporting comparisons of monthly returns – observed and predicted. So far, the match is excellent.

In this post, preliminary results for July 2009 are reported, as accompanied by formal introduction of the model. The model is an empirical one and needs to be assessed and (sometimes) updated when new data are available. The last assessment (published in this blog) was carried out in June 2009. Since the model links S&P 500 to the growth rate in real GDP, we made relevant forecast for the second quarter of 2009 as +5%. This value is very high compared to the consensus (Conference Board) prediction of -0.7% .

**1. S&P 500 vs. the number of nine-year-olds**

To begin with, in Figure 1 we present observed and predicted S&P 500 returns for the period between 1985 and 2003 and their residual. Main finding is that the observed and predicted time series are cointegrated [1], both according to the Engle-Granger tests and the Johansen approach, i.e. there exists a long-term equilibrium relation between S&P 500 returns and the number of 9-year-olds in the USA. The latter is the driving force of the stock market and real GDP.

We have been carefully tracking the evolution of S&P 500 since 2007, when predicted a sharp fall in 2008 [1]. The updated relationship between S&P 500 returns and the (extrapolated) number of nine-year-olds is as follows:

Rp(t) = 165dln[N3(t+6)] - 0.17 (1),

In (1), Rp is the 12-month cumulative return; N3 is the number of three-year-olds; N9 is the number of 9-year-olds; t+6 – time shifted by six years ahead to extrapolate the number of 3-year-olds into the number of 9-year-olds.

Figure 2 predicts S&P 500 index using the number of 9-year-olds extrapolated from the number of 3-year-olds. There are two new points, June (919) and July (~990) 2009, since the last update in June 2009. The latter figure is a preliminary one as picked on July 30, 2009. It is not likely that this figure will change dramatically during the last trading day in July.

A sudden drop in 2008 and 2009 to the level of 700 should be followed by an increase to 1800 in 2010. Notice that the start of the current growth in S&P 500 was first predicted in March 2009, when the market was very low with the close at ~735 in February. The last five points together with the turn to the growth were forecasted.

Figure 2. Evolution of S&P 500. Red line – observations; black line – prediction from the number of 9-year-olds. The prediction is obtained using (1).

**2. S&P 500 vs. real GDP per capita**The main problem for an accurate prediction consists in the fact that the number of 9-year-olds in the end of any decade is prone to high bias. Only decennial censuses (next due in 2010) allow adequate estimates. Because of high uncertainty in N9, we have proposed to use real GDP per capita, GDPpc, as a proxy to the N9 [2-3]. Originally, the link between real GDP growth rate and the change rate of the number of 9-year-olds was found by Kitov [4]. Corresponding relationship should work in both directions, i.e. one can estimate the growth rate of real GDP from population measurements, and the number of 9-year-olds from real GDP measurements.

In relationship (1), we replace N9(t) with GDPpc(t), taking into account that second term in the relationship between real GDP per capita and population is constant. Figure 3 displays the observed S&P500 returns and those obtained using real GDP, as presented by the US Bureau of Economic Analysis (http://www.bea.gov/). As before, the observed returns are 12-month cumulative values. The predicted returns are obtained from the relationship

Rp(t) = 15.0*dln(GDPpc(t)) - 0.32 (2)

where GDPpc(t)) is represented by the average (annualized) growth rate during four previous quarters. It is worth noting that there are no monthly readings of real GDP available. Hence, only quarterly figures can be compared.

The period after 2000 is well predicted including the sharp increase in 2003. Therefore, it is reasonable to assume that the 9-year-old population was not well estimated by the US Census Bureau after 2003. This conclusion is supported by the cointegration test conducted for real GDP per capita and the charge rate of the number of 9-year-olds [3], which proves the existence of a long-term equilibrium linear relation between these two variables since the early 1960s. As a result, one can use either N9(t) or GDPpc(t) for the modelling of the S&P 500 returns, where one of them is more appropriate. Obviously, the GDPpc(t) is consistent with the S&P 500 returns after 2007. The years between 2007 and 2010 should confirm or reject this statement.

Currently, relationship (2) holds. The deep fall in 2008-2009 is well described. It also predicts that real GDP will start to increase in the nearest future following the observed increase in S&P 500. It would be very strong evidence in favour of (2).

Today the BEA will announce the Q2 figure. In Figure 3, red circle is our prediction (made a month ago) for 2009Q2. The rate of growth should be +5% relative to the previous quarter. The BEA should also provide a comprehensive revision to all historical reading of GDP. We expect a positive revision to real GDP estimates during the last four years.

In June, S&P 500 did not change from its May level of 919. (In May 2009, it grew by +47 units from 872 to 919). In July, the June non-growth is likely to be compensated with a rise by approximately 80 points from 919 to ~1000. Figure 4 shows the past and future predictions using the number of 3-year-olds.

Figure 3. The link between S&P 500 returns and real GDP per capita between 2000 and 2009. Red circle is a prediction of the growth rate for 2009Q2. The rate of growth is taken at the level of +5% relative to previous quarter.

[2] Kitov, I., Kitov, O., Dolinskaya, S., (2008). Comprehensive Macro – Model For The US Economy, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(4(6)_Wint), pp. 405-418. http://www.jaes.reprograph.ro/articles/winter2008/ComprehensiveArticle8.pdf

[3] Kitov, I., Kitov, O., Dolinskaya, S., (2009). Modelling real GDP per capita in the USA: cointegration tests, Journal of Applied Economic Sciences, Spiru Haret University,Faculty of Financial Management and Accounting Craiova, vol. 4(1(7)_ Spr), pp. 80-96. http://www.jaes.reprograph.ro/articles/spring2009/KitovI_KitovO_DolinskayaS.pdf

[4] Kitov, I., (2006). GDP growth rate and population, Working Papers 42, ECINEQ, Society for the Study of Economic Inequality, http://ideas.repec.org/p/inq/inqwps/ecineq2006-42.html

Figure 4. Observed and predicted S&P 500 returns. The July level of S&P 500 index is around 1000, from its level of 919 in June 2009. The past five months are relatively well predicted.

**Conclusion**Between 1985 and 2009, the S&P 500 returns can be accurately described by population estimates and readings of real GDP per capita. The model based on the number of 9-year-olds produces a time series which is cointegrated with the S&P 500 returns, i.e. reveals a weak causality, as proved by the cointegration tests. The next event to support the presence of the link between GDP and S&P 500 is the announcement of the growth rate in Q2 2009. From Figure 3, one can expect a positive figure, which is likely to be larger than 1 or 2 percentage points.

**References**[1] Kitov, I., Kitov, O., (2007). Exact prediction of S&P 500 returns, MPRA Paper 6056, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/6056.html

[2] Kitov, I., Kitov, O., Dolinskaya, S., (2008). Comprehensive Macro – Model For The US Economy, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(4(6)_Wint), pp. 405-418. http://www.jaes.reprograph.ro/articles/winter2008/ComprehensiveArticle8.pdf

[3] Kitov, I., Kitov, O., Dolinskaya, S., (2009). Modelling real GDP per capita in the USA: cointegration tests, Journal of Applied Economic Sciences, Spiru Haret University,Faculty of Financial Management and Accounting Craiova, vol. 4(1(7)_ Spr), pp. 80-96. http://www.jaes.reprograph.ro/articles/spring2009/KitovI_KitovO_DolinskayaS.pdf

[4] Kitov, I., (2006). GDP growth rate and population, Working Papers 42, ECINEQ, Society for the Study of Economic Inequality, http://ideas.repec.org/p/inq/inqwps/ecineq2006-42.html

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