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2/29/12

Unemployment will drop to 7.8% by 2013

In 2006, we developed three individual empirical relationships between the rate of unemployment, u(t), price inflation, p(t), and the change rate of labour force, LF(t), in the United States. We also built a general relationship balancing all three variables simultaneously. Since measurement (including definition) errors in all three variables are independent it may so happen that they cancel each other (destructive interference) and the general relationship might have better statistical properties than the individual ones.   For the USA, the best fit model for annual estimates is a follows: 
u(t) = p(t-2) + 2.5dLF(t-5)/dtLF(t-5) + 0.0585       (1) 
where inflation (CPI) leads unemployment by 2 years and the change in labor force by 5 years.  We have already posted on the performance of this model several times.  
Here a model with monthly estimates of CPI, u, and labor force is presented. The time lags are the same as in (1) but coefficients are different since we use month to month a year ago rates of growth. We have also allowed for changing inflation coefficient. The best fit models for the period after 1978 are as follows: 
u(t) = 0.63p(t-2) + 2.0dLF(t-5)/dtLF(t-5) + 0.07;  between 1978 and 2003
u(t) = 0.90p(t-2) + 4.0dLF(t-5)/dtLF(t-5) + 0.30; after 2003
There is a structural break in 2003 which is needed to fit the predictions and observations in Figure 1. Due to strong fluctuations in monthly estimates of labor force and CPI we smoothed the predicted curve with MA(24). The rate of unemployment became more sensitive to the change of inflation and labor force. Alternatively, definitions of all three (or two) variables were revised around 2003, which is the year when new population controls were introduced by the BLS.  
All in all, the monthly model predicts the observed rate of unemployment which has recently dropped to 8.3%. We expect the rate to fall further to the level of 7.8% by the end of 2012.  
Figure 1. Observed and predicted rate of unemployment in the USA.

2/28/12

The FRB, the BEA, and the BLS lie on inflation

Couple days ago we posted on the federal funds rate. The interest rate is defined by the Federal Reserve as a major instrument to control price inflation. Figure 1 depicts the cumulative values of effective rate, R, and the rate of consumer price inflation, CPI, multiplied by 1.4. In the long run, these two curves evolve along the same trend and intercept every fifteen to twenty years. We presumed that the main idea to keep R above the rate consumer price inflation is that a higher funds rate should suppress price inflation due to the effect expensive money.

On the other hand, the FRB has likely to retain the interest rate at the long term level of price inflation in order to create neutral conditions for money supply. This would be a wise prerequisite for a central bank. Then why the FRB needs that factor of 1.4? Actually it does not and the answer comes from the historical GDP data. The problem of the multiplier is in wrong estimates of inflation since 1950. Essentially, the FRB, the BEA, and the BLS lie.

Figure 2 depicts the evolution of real GDP per capita in the US since 1870. As we have already mentioned in our posts, there are two trends in the historical GDP data – before and after 1950. Before 1940, the (red) regression line with a slope of ~$61 per year in Figure 2 provides a good approximation of the actual curve. After 1950, the actual curve evolves along a straight line with a slope of $387 per year, i.e. the slope rises by a factor of 6.34 after 1950. Real GDP is defined as the ratio of nominal GDP and the GDP deflator. Both values are measured and estimated (also using a subjective hedonic factor) by the BEA and BLS. Therefore, the estimates of real GDP per capita heavily depend on the definition of price inflation.

Let’s suppose that the real GDP curve evolves along the old trend after 1940, as shown in Figure 2. Then the level of real GDP per capita in 2008 would have been $10,774 instead of $31,178 as estimated by. This means that the GDP deflator was underestimated by a factor of 2.89 (=31178/10774). The reported increase in the level of consumer prices since 1960 was of 7.27, i.e. CPI(2008)/CPI(1960) =7.27. Then we expect that the actual price increase (i.e. reported plus underestimated) would have been 7.27+2.89=10.16, and the rate of CPI inflation was underestimated by 10.16/7.27=1.4 times.

A big surprise! This is exactly the factor of the federal funds rate above the rate of price inflation. Hence, the FRB retains the interest rate at the level of actual inflation and thus does not influence inflation. The BEA, BLS and FRB lie (intentionally or not) about the rate of inflation and the growth in real GDP. The current level of GDP per capita in the US should be around $11000 not $31,000.

Figure 1. Cumulative values of the monthly estimates of R and the CPI multiplied by a factor of 1.4.

Figure 2. Historical estimates of real GDP per capita.

2/27/12

PPI of durable and nondurable goods

Three years ago we presented the difference between the PPI of durable and nondurable goods and stated that one could predict its evolution at a several year horizon. This difference is characterized by the presence of sustainable mid-term trends.  Obviously, the future of both indices is of crucial importance for industries behind relevant goods, for the stock market, and for real economy. Our analysis provides a long-term prediction. Its reliability depends on the growth in real GDP in the near future.

The evolution of the producer price index of durable and  nondurable goods is reported by the BLS at a monthly rate. In 2009, we presented Figure 1 which demonstrates that the difference has two distinct quasi-linear and both positive branches: between 1988 and 2000 (June), and from 2001 to 2008 (January). Red and blue lines highlight segments between 1988 and 2000, and from 2001 to 2008, respectively. Corresponding linear regression lines in Figure 1 have slopes of +0.05 and -6.8. The former slope is a negligible one, and the latter indicates that the index for nondurables has been growing since 2001 by 6.8 units if index faster than that for durables. In January 2008, the difference reached the level of -40, descending along the trend. Then the difference dropped to -67 in June and recovered to -20 by the end of 2008. This effect has been observed for all other commodities and is related to the financial crisis and recession.
Our naïve assumption about the next move after 2009 was that the difference would develop a new positive trend which would repeat the trend observed between 2001 and 2008, but with an opposite sign. The green line in Figure 1 predicts the expected evolution of the difference after 2009. Because the green line has a positive slope, the index for durables will be catching up that for nondurables since 2009. According to our assumption, the rate of approaching to the index for nondurable goods will be +6.8 units of index per year during the next 7 years. We also suggested that the actual trend might be different but almost inevitably with a positive slope. 
Here, we revisit this prediction. Figure 2 displays the evolution of the difference between 2009 and 2012 which is characterized by a very high level of volatility. The local peak in 2009 was followed by a sharp drop in the difference to the level of -59 in April 2011 and a relatively slow increase since then. We have introduced a new tentative trend for the difference which is less steep than in Figure 1. We expect the difference to reach -10 by 2020.  In any case the index of durables has to grow at a higher rate than the index of nondurables in the 2010s with possible (large-amplitude) fluctuations around the trend. We are going to revisit this difference in 2013.

Figure 1. Evolution of the difference between the PPI of durables and nondurables between 1985 and 2009. Red and blue lines highlight segments between 1988 and 2000, and from 2001 to 2008, respectively. Green line predicts the evolution of the difference after 2008, as a mirror reflection of the linear trend between 2001 and 2008.

Figure 2. Evolution of the difference between the PPI of durables and nondurables between 1990 and 2012. Red line highlights the segment from 2001 to 2008y. Green line predicts the evolution of the difference after 2009 which is less steep than in Figure 1.  

2/26/12

Analysis: Chesapeake Energy Corporation's share price

First, we report on the performance of our pricing model linking share prices of energy companies with the difference between the headline and core CPI. In essence, we are trying to use the core CPI as an energy independent (dynamic) reference to the headline CPI index which includes energy. Then the difference might be related to the energy pricing power relative to other goods and services.  This idea has proven to be fruitful for oil (energy) companies and other categories of companies in the S&P 500 index and other consumer price indices.  
Our original pricing model states that a share price, for example, that of Chesapeake Energy Corporation, CHK(t), can be approximated by a linear function of the difference between the core CPI, CC, and headline CPI, C 
CHK(t) = A + B (CC(t) - C(t))                      (1) 
where A and B are empirical constants; t is the elapsed time. It should be noted that both indices are fixed to be contemporary to the modeled price.  Also, both linear coefficients (slopes) are equal, which might be an oversimplification. This model has proven its predictive power for many companies and we have been reporting on its performance since 2009  
In January 2011, we extended the set of defining indices by the consumer price index of energy, E, and the producer price index of crude petroleum, OIL, together with the overall PPI. We also introduced time shifts between the price and defining CPIs and varying coeffcients.  
Here we estimate three new models for the period between July 2003 and January 2012. The relevant estimates of CPI and PPI through January 2012 have been retrieved from the BLS website. Figures 1 through 3 display the observed and predicted models. The best model are defined by standard error. For CHK, the best fit models are as follows:  
CHK(t) = 3.73C(t+1)–2.05CC(t)–8.64(t-1990)-162.57    (2)
CHK(t) = 1.76CC(t+3) + 0.35E(t+1)-9.42(t-1990) – 248.05 (3)
CHK(t) = 0.91PPI(t+1) +0.032OIL(t+1) -5.30(t-1990) – 42.47  (4) 
In all models, some future estimates of the defining indices are needed to describe the current price. This means that the CHK price is likely to drive the CPI and PPI components.   
The model defined by CC and E is the best among these three models. It provides the smallest model error of $3.27.  In any case, all models have predicted the sharp fall in the price in 2008 and the following recovery in 2009. The price has been falling since July 2011. The defining indices lag behind the CHK price and one could expect these indices not to grow fast in the first quarter of 2012. A slight fall in OIL and E is not excluded.
Figure 1.  The observed CHK price and that predicted from the core and headline CPI; stdev=$3.69.
Figure 2.  The observed COP price and that predicted from the core CPI and the consumer price index of energy;  stdev=$3.27
Figure 3.  The observed CHK price and that predicted from the overall PPI and the producer price index of crude petroleum (domestic production); stdev=$4.13. 
In its most general form, our pricing model states that a share price, SP(t), can be approximated by a linear function of the difference between two CPI components with different lags behind the price:
SP(t) = A + B1CPI1(t + t1) + B2CPI2(t + t2) + C(t-t0)                                       
where A, Bi, and C are empirical constants for the studied period; t is the elapsed time; t1 and t2 are  the time delays between the share and the  CPIs, both to be determined. We seek to minimize the standard model error, RMSE, by the LSQ method in ordre to find all 6 coefficients (A,Bi,C,ti) for two CPI components among the set of 92 
This approach was also successful for Chesapeake Energy Corporation. In January 2011, we estimated a preliminary model for CHK with a smaller set of major CPI categories and found the following model: 
CHK(t)= -1.47ED(t-7) + 0.41E(t+1) +11.4(t-1990) – 12.1; RMSE=$2.68. 
where ED(t-7) is the consumer price index of education which leads the price by 7 months.  
In April 2011, we revisited the CHK model with 92 defining CPIs and found that the best-fit 2 model for CHK(t) is based on the index of tuition, other school fees, and child care (TUIT) contemporaneous with the share, and the index of energy (E) lagging by 2 months:  
CHK(t)= 0.52TUIT(t-0) + 0.43E(t+2) – 16.77(t-1990) – 21.48; stdev=$2.64, March 2011 
 In other words, the price of a CHK share defines the behaviour of the index of energy and the model with TUIT explains the overall behaviour of the CHK price much better than models (1) through (3). Figure 4 depicts the observed and predicted price.  The current version of the model is as follows: 
CHK(t)= 0.52TUIT(t-0) + 0.43E(t+2) – 16.99(t-1990) – 16.06; stdev=$2.81, January 2012    
The model error has increased since April 2011 to $2.81. This model is also depicted in Figure 4 and shows that the current price is slightly below the predicted one.  We expect a correction to both predicted and observed prices in the near future.  Figure 5 presents the evolution of the TUIT and E indices.  

Figure 4. Observed and predicted CHK share prices. Upper panel: March 2011. Lower panel:  January 2012.  
Figure 5. The index of tuition, TUIT, and the index of energy, E.

2/25/12

PPI of metals

We have been following the evolution of several price indices of metals since 2008. Our general approach is based on the presence of long-term sustainable (linear and nonlinear) trends in the evolution of the CPI and PPI in the United States. The difference between various components of these indices is not a random one but rather a predetermined process. Using these trends, one can predict consumer and producer price indices for select goods, services and commodities. 
In this post, we revisit the trends in the PPI of three commodities related to metals: steel iron, nonferrous metals, and metal containers. Originally, we reported on these items in 2008 and then revisited in 2010.  
1.               Figure 1 compares the difference between the PPI and the index for iron and steel (101). The difference is characterized by the presence of a sharp decline between 2001 and 2008. Between 1985 and 2000, the curve fluctuates around the zero line, i.e. there was no linear trend in the absolute difference. A year ago we expected the negative trend to start transforming into a positive one as the green line in Figure 2 shows. Our previous predictions were correct. For example, two years ago we wrote:
“Between March and June 2009, the difference continued to increase, and likely reached its peak in June (Figure 2). In July or August 2009, the difference will stall around its peak value and then will start to decrease. As a result, the index for iron and steel will be growing faster than the PPI. In the short run, one can expect a fast recovery of iron and steel prices to the level observed in January-March 2008, i.e. the index will reach the level 210 to 220. However, this recovery will not stretch into 2011, and the index of iron and steel will be declining in the long run to the level of 2001, as depicted in Figure 2.  In other words, the period between 2008 and 2010 is characterized by very high volatility, which will fade away after 2011. “  
2.     The index for non-ferrous metals (102) shows an example of the absence of sustainable trends in the normalized difference. The curve is rather a comb with teeth of varying width. Although varying, the distance between consecutive troughs is several years at least. Therefore, we expect this index to decrease relative to the PPI and the difference in Figure 3 to rise to the level of -10. Non-ferrous metals will be getting cheaper.   
3.             The index for metal containers (103) provides an excellent example of linear trends in the normalized difference (see Figure 4). There are two distinct periods between 1960 and 2008 with a turning point in 1987. As we predicted in 2009, the sudden drop in the difference in the end of 2008 manifested the start of transition to a new period with a negative trend. The price index for metal containers will be increasing relative to the PPI, i.e. the index will get back its price setting power.   

Figure 1. The difference updated for the period between June 2010 and January 2012. As expected, the difference has been decreasing during the reported period and sank below the new trend (green). The trajectory has to turn up in the near future and reach the new trend.  This means that the price index for iron and steel will be growing at a lower rate than the overall PPI.   

Figure 2. The evolution of the difference between the PPI and the price index of iron and steel between January 2005 and January 2012. Green line predicts the evolution of the difference after 2008. Red circles represent the difference between April 2009 and January 2012.
 Figure 3.  The evolution of the difference between the PPI and the index of nonferrous metals from 1985 and January 2012. There are no linear trends in the difference, but its behavior demonstrates a clear periodic structure with relatively deep but short troughs, which reflect the fast growth in the PPI for nonferrous metals. 
Figure 4. The evolution of the difference between the PPI and the index of metal containers from 1985 to January 2012. There are distinct linear trends in the difference. The difference has started its transition to a negative trend.

2/24/12

The rate of unemployment in the UK will likely rise to 9%


We estimated a version of Okun’s law for the UK in July 2011. We developed an integral version of Okun’s law and applied the LSQ method to estimate coefficient in :
u(t) = u(t0) + bln[G/G0] + a(t-t0)  (1)
where u(t) is the predicted rate of unemployment at time t, G is the level of real GDP per capita, a and b are empirical (LSQ) coefficients.   The best-fit (Okun’s) model minimizing the RMS error of the cumulative model (1) is as follows:
du = -0.39dlnG + 0.63 (2)
The most recent estimate on the unemployment rate in the UK is 8.4% for the fourth quarter of 2011 as reported by the NSO for the economically active population between 16 and 64 years of age. The Conference Board has also published an estimate of real GDP per capita for 2011. Figure 1 depicts the observed and predicted curves of the unemployment rate, including the rates for 2011. The overall agreement is very good but the current rate of unemployment is lower than the predicted one by 0.6%.  Figure 1 suggests that the rate of unemployment has been driven by real economic growth and one can expect the rate to grow to the level of 9.0%. The will be no decease in the rate of unemployment if the growth rate of real GDP per growth does not exceed (0.63/0.39=) 1.63% per year.  In 2011, this rate was only 0.09%.

Figure 1.  The observed and predicted rate of unemployment in the UK between 1971 and 2011. 

2/23/12

How CPI drives the federal funds rate


The interest rate defined by the Federal Reserve is an instrument to control inflation. Ignoring the heaps of quasi-economic lie around the effect of the monetary policy we just present some observations.  Figure 1 depicts the effective rate, R, and the consumer price inflation. The former has to control the latter. One can see that the rate lags behind the CPI since 1980, i.e. inflation grows at its own rate and R has to follow up. The idea of R is that a higher rate should suppress inflation due to the effect expensive money. The reaction of inflation is also expected not momentarily but with some time lag.
The cumulative influence of the interest rate should produce a desired effect in the long run and inflation should go in the direction towards bearable values. Figure 2 displays the cumulative effect, i.e. the cumulative values of the monthly estimates of R and CPI multiplied by 1.4. This is an intriguing plot. In the long run, the R curve fluctuates around the CPI one and returns to it every 15 to 20 years. It seems that the sign of deviation of R from the 1.4CPI curve does not affect the behavior of the CPI. Therefore, the influence of monetary policy is under strong doubt. The Feds have tried all means to return the CPI to R without any success and have to return R to the CPI.  

Figure 1. The federal funds rate, R, and the rate of consumer price inflation, CPI, between 1956 and 2012.

Figure 2. Cumulative values of the monthly estimates of R and CPI multiplied by a factor of 1.4.

A paradox of income inequality among young people



We have already presented the evolution of Gini coefficient in the USA since 1994 as measured by the Census Bureau. Figure 1 depicts time series in all age groups including the youngest ones. The difference between the age group between 15 and 24 and 25 to 34 is astonishing. In the youngest age group, the level of income inequality is the highest. Gini coefficient fluctuates around 0.51. In the second youngest group, Gini is rock solid at the level of 0.42. And both coefficients are calculated only for people with income, i.e. the portion without income does not influence the Gini estimates.
Figure 2 shows the Gini as a function of age for some selected years. The lowermost Gini belongs to the group between 24 and 34 and then starts to grow to the level of the youngest group. There should be two competitive mechanisms to produce this minimum.


Figure 1. The evolution of Gini coefficient in various age groups between 1994 and 2009.


Figure 2. Gini coefficient as a function of age

2/21/12

Modeling Avery Dennison's share price

This is a regular revision of our stock pricing model as applied to Avery Dennison Corporation (AVY). We decompose the time series of a given share price into a weighted sum of two individual consumer price indices plus linear time trend and free term over an extended period of time. The best fit model has to provide the lowermost RMS residual and also do not change with time.

In April 2011, we presented an original model for Avery Dennison Corporation based on the consumer price index of food (F) and that of new and used motor vehicle (NUMV). In the original model, the former CPI component led the share price by 4 months and the latter one led by 2 months. Figure 1 depicts the evolution of both (NSA) CPIs through January 2012. The upper panel of Figure 2 displays the predicted and observed monthly closing prices (adjusted for splits and dividends) as of March 2011.
In September 2011, we revisited the model using the prices and CPIs for the period through September 2011. (The CPIs were available only for August 2011.) The principal result was that the underlying model is practically the same as six months before with the same time lags but slightly different coefficients. In March 2011, we predicted a fall in the price which actually happened. In September, we expected that AVY stocks would be falling by the end of 2011 down to $16 per share from the September closing level $25.08 (see the middle panel in Figure 2). This was not a good prediction.

Here we revisit the model using data through January 2012 (see the lower panel in Figure 2). The three best-fit models for AVY(t) are as follows

AVY(t) = -4.24F(t-4) – 3.23NUMV(t-2) + 23.29(t-1990) + 799.24 , March 2011
AVY(t) = -3.92F(t-4) – 2.70NUMV(t-2) + 21.60(t-1990) + 710.60 , September 2011
AVY(t) = -3.61(t-5) – 2.10NUMV(t-3) + 19.86(t-1990) + 622.01 , January 2012

where AVY(t) is a share price in US dolalrs, t is calendar time. Relevant coefficients are both negative. The slope of time trend is positive. There is some drift in all coefficients caused by the uncertainty in measurements of both the stock prices and CPIs and by a slight collinearity between the CPI difference and linear time trend. Nevertheless, all models provide a prediction at a two- to three-month horizon.

The currently predicted curve in Figure 2 leads the observed price by 3 months with the residual error of $2.81 ($2.57 in September) for the period between July 2003 and January 2012. The model residual for the same period is shown in Figure 3. We expect the actual price to have a slight negative correction in the near future and the predicted curve to grow up in order to close the gap between the predicted and observed prices. The residual in Figure 3 has to return to 0. This might be accompanied by a slight fall in food prices.

Figure 1. Evolution of the price of F and NUMV.




Figure 2. Observed and predicted AVY share prices. Upper panel – March 2011; middle panel – September 2011; lower panel – January 2012.


Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $2.81. Currently, the price is slightly overestimated.

2/20/12

Modeling Exxon Mobil's share price

In this post, we describe the pricing model for Exxon Mobil’s (XOM) share as based on our concept of stock dependence on consumer price index. Unlike in the simplified model for ConocoPhillips’ share, which included only the core and headline CPI, here we use a set of 92 individual consumer price indices to select the best two.  

Exxon Mobil provides an example of a company with share price leading defining components of the CPI.  Our model is seeking two CPI components from a large number of pre-selected ones, which minimize the difference between observed (monthly closing price adjusted for dividends and splits) and predicted prices for the period between July 2003 and January 2012. A two-component (2-C) model also includes free term (constant) and linear time term, which compensates well know linear (time) trends between various CPI components. The best-fit 2-C model for XOM(t) is as follows: 

XOM(t)= -1.70OFH(t-3) – 2.98RRM(t-10) + 22.73(t-2000)  + 581.17 

where OFH in the index other food at home lagging  the stock price by 3 months, RRM is the index of recreation reading materials leading by 10 months, (t-2000) is the elapsed time. Figure 1 depicts the evolution of both CPIs.  
Figure 2 depicts the observed and predicted prices, the latter shifted three months ahead for synchronization, i.e. the predicted curve leads the observed price by 3 months. The model residual error shown in Figure 3 has standard deviation of $4.41 for the period between July 2003 and January 2012.   
The estimated model shows that Exxon Mobil’s share will be growing in 2012Q1.
Figure 1. The evolution of CPIs.

Figure 2. Observed and predicted XOM share prices.

Figure 3. The model error.

Modeling Computer Science Corporation’s share price

Two years ago, we first presented a share price (monthly closing price adjusted for splits and dividends) model for Computer Science Corporation (CSC) as based on the decomposition into a weighted sum of two consumer price indices (NSA borrowed from the BLS database). Approximately a year ago we revisited the original model using all data available through March 2011. The defining indices were obtained three years ago: the index of motor vehicle parts (MVP) and the index of sporting goods (SPO). The CPI components were leading by 0 and 5 months, respectively. Figure 1 depicts the evolution of both indices which provide the best fit model, i.e. the lowermost RMS residual error, between July 2008 and March 2011:

CSC(t) = -3.83MVP(t-0) + 3.16SPO(t-5) +16.31(t-1990) – 137.20, March 2011

where CSC(t) is the share price in US dollars, t is calendar time. In April, we predicted the curve in the upper panel of Figure 2 which is synchronized with the observed one. The residual error was of $3.28 for the period between July 2003 and March 2011. Since the MVP index has been growing since 2002 and the SPO index had a slight negative trend, we predicted that the share price would not be growing in 2011.

In reality, it has fallen slightly from $50 in March to the level of $24 per share in December 2011. Such a dramatic fall is difficult to describe with stochastic price models but our deterministic model has survived the crisis in CSC. There was a period of intensive growth in the MVP index (see Figure 1) which was converted in the share price drop. Currently, the defining indices are the same as three years ago:

CSC(t) = -3.81MVP(t-1) + 3.35SPO(t-7) +15.97(t-1990) – 158.37, January 2012

with slightly increased time delays of 1 month and 7 months, respectively. In the lower panel of Figure 2 the predicted and observed prices are depicted. The error term of the model between July 2003 and January 2012 is displayed in Figure 3 with stdev=$3.51. The residual was negative during the past quarter and we expect the price to grow in 2012Q1 for the model error to return to 0.

 
Figure 1. Evolution of the price indices MVP and SPO.



Figure 2. Observed and predicted CSC share prices.

Figure 3. Model error, i.e. the difference between the observed and predicted price; stdev = $3.51

Modeling Pepco Holdings' share price

In April 2011, we introduced a new model for Pepco Holdings (POM). The defining CPI indices were as follows: the index of food away from home (SEFV) and the index of owners' equivalent rent of residence (ORPR). (See model details in Appendix). Figure 1 depicts the evolution of these CPIs which lead the POM share price by 4 and 5 months, respectively. The best fit model, i.e. the lowermost RMS residual error, between July 2010 and March 2011:

POM(t) = -2.66SEVF(t-4) +1.06ORPR(t-5) +11.83(t-1990) + 101.35, March 2011

where POM(t) is the share price in U.S. dollars, t is calendar time. The upper panel in Figure 2 displays the observed monthly closing price and that predicted by the above relationship.

In April, we predicted that “In the second quarter of 2011, the model foresees a rise by $1.5.” Actual monthly closing price has increased from $18.55 in March to $19.63 in June 2011. The predicted price is well within the high/low monthly bounds, i.e. practically within the uncertainty bounds of the POM price.

Here we revisit the initial model with new data available through January 2012. The model is stable and is defined by the same CPIs with similar coefficients. Time delays are also similar but the SEVF leads the share price by 5 months:

POM(t) = -2.19SEVF(t-5) +0.76ORPR(t-5) +10.57(t-1990) + 95.97, January 2012

In the lower panel of Figure 2, we show the current model and the uncertainty bounds as presented high and low monthly prices. The overall fit is good and we expect the current price to fall in the 2012Q1. Figure 3 demonstrates that the model error in January 2012 is positive and it must fall back to 0 in the near future.

Figure 1. The evolution of defining CPI.


Figure 2. Observed and predicted POM share prices.

Figure 3. The model error, stdev= $0.89

Appendix
In its general form, our pricing model is as follows:

sp(tj) = Σbi∙CPIi(tj-Di) + c∙(tj-2000 ) + d + ej (1)

where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-Di) is the i-th component of the CPI with the time lag Di, i=1,..,I; bi, c and d are empirical coefficients of the linear and constant term; ej is the residual error, which statistical properties have to be scrutinized. By definition, the bets-fit model minimizes the RMS residual error. The time lags are expected because of the delay between the change in one price (stock or goods and services) and the reaction of related prices. It is a fundamental feature of the model that the lags in (1) may be both negative and positive. In this study, we limit the largest lag to eleven months. Apparently, this is an artificial limitation and might be changed in a more elaborated model.

System (1) contains J equations for I+2 coefficients. For POM we use a time series from July 2003 to January 2012, i.e. 105 monthly readings. Due to the negative effects of a larger set of defining CPI components their number for all models is (I=) 2. To resolve the system, we use standard methods of matrix inversion. Usually, solutions of (1) are stable with all coefficients far from zero. In the POM model, we use 92 CPI components. They are not seasonally adjusted indices and were retrieved from the database provided by the Bureau of Labor Statistics.

Due to obvious reasons, longer time series guarantee a better resolution between defining CPIs. In general, there are two sources of uncertainty associated with the difference between observed and predicted prices. First, we have taken the monthly close prices (adjusted for splits and dividends) from a large number of recorded prices: monthly and daily open, close, high, and low prices, their combinations as well as averaged prices. Second source of uncertainty is related to all kinds of measurement errors and intrinsic stochastic properties of the CPI and its components. One should also bear in mind all uncertainties associated with the CPI definition based on a fixed basket of goods and services, which prices are tracked in few selected places. Such measurement errors are directly mapped into the model residual errors. Both uncertainties, as related to stocks and CPI, also fluctuate from month to month.