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. 

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

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.

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.