The rate of unemployment in the U.S. will fall to 6.2% by 2014

On March 1, 2012 we predicted (in a Seeking Alpha post) the rate of unemployment in the U.S. to fall down to 7.8% by 2013. The BLS announced 7.8% for September 2012. Here we present our basic model and predict the evolution of unemployment in 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 was a follows:

u(t) = p(t-2.5) + 2.5dLF(t-5)/dtLF(t-5) + 0.0585 (1)

where inflation (CPI) leads unemployment by 2.5 years (30 months) and the change in labor force leads by 5 years (60 months). We have already posted on the performance of this model several times.

For the model in this post, we use monthly estimates of the headline CPI, u, and labor force, all reported by the US Bureau of Labor Statistics. 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-2.5) + 2.0dLF(t-5)/dtLF(t-5) + 0.07; between 1978 and 2003

u(t) = 0.90p(t-2.5) + 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 structural break in 2003 may be associated with the change of sensitivity of the rate of unemployment 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. The Census Bureau also reports major revisions to the Current Population Survey, where the estimates of labor force and unemployment are taken from.

On March 1, 2012 the monthly model predicted a drop from 8.3% in February to 7.8% by the end of 2012. Figure 1 depicts the original prediction (upper panel) and the observed fall in the rate of unemployment (lower panel). Figure 2 shows that the observed and predicted time series are well  correlated (Rsq.=0.81). This is a good statistical support to the model.

Figure 3 depicts the predicted rate of unemployment for the next 12 months. The model shows that the rate will fall to 6.2% by September 2013. For 105 observations since 2003, the modelling error is 0.4% with the precision of unemployment rate measurement of 0.2% (Census Bureau estimates in Technical Paper 66).

 

Hence, one may expect 6.2% [±0.4%].

 

 

Figure 1. Observed and predicted rate of unemployment in the USA as obtained in March and October 2012.

Figure 2.  Observed vs. predicted rate of unemployment between 1967 and 2012. The coefficient of determination   Rsq=0.81.

Figures 3. The predicted rate of unemployment. We expect the rate to fall down to 6.2% in September 2013.

Employment in Japan

We continue modeling the evolution of the employment rate in developed countries with Japan. In this study we use the trade-off between the change in unemployment and employment and Okun’s law. Figure 1 compares the change in the rate of employment (the employment/population ratio), de, and the rate of unemployment, du, in Japan. The change in the rate of unemployment is as volatile as that of unemployment and they differ drastically compared to the synchronized evolution of these variables in the U.S. That’s why we have failed to obtain a reasonable Okun’s law for Japan. As before, all data sets on unemployment and employment have been retrieved from the U.S. Bureau of Labor Statistics. The estimates of real GDP per capita have been retrieved from the database provided by the Conference Board.

Figure 1. The (negative) change in the rate of unemployment compared to the change in the rate of employment in Japan.

In this blog, we have already presented several empirical relationships predicting the employment/population ratio from the growth rate of real GDP per capita. This was a natural extension of Okun’s law for unemployment.

Here we estimate an employment/GDP model for Japan similar to Okun’s law. For Japan, the best-fit model has been obtained by the least-squares (applied to the cumulative sums):

det = 0.02dlnGt – 0.53, t<1978

det = 0.14dlnGt – 0.42, t>1977 (1)

 

where dlnGt is the change rate of real GDP per capita at time t. Figure 2 shows the cumulative curves for the time series in (1). There is a structural break near 1978 which is expressed by a dramatic shift in slope and a slight break in intercept. The employment/population ratio varies between from 64%% in 1970 and 56% in 2010. The agreement is excellent. Figure 3 present results of a linear regression with R2=0.95 for the period between 1971 and 2010. We consider both variables as stationary ones over the long run despite the obviously negative trend since 1970.

Figure 2. The cumulative curves for the observed and predicted change in the employment/population ratio, de.

Figure 3. Linear regression of the measured and predicted curves in Figure 2.

Employment in France

There is a trade-off between the change in unemployment and employment. Figure 1 compares the change in the rate of employment (the employment/population ratio), de, and the rate of unemployment, du, in France. As expected, the change in the rate of unemployment is more volatile except the shift in the employment rate near 1982. This is a completely artificial break from 53.2% in 1981 to 55.3% in 1982, and we do not need to model it. All data sets on unemployment and employment have been retrieved from the U.S. Bureau of Labor Statistics.

Figure 1. The (negative) change in the rate of unemployment compared to the change in the rate of employment in France.

In this blog, we have already presented several empirical relationships predicting the employment/population ratio from the growth rate of real GDP per capita. This was a natural extension of Okun’s law for unemployment.

Here we estimate an employment/GDP model for France similar to Okun’s law. For France, the best-fit model has been obtained by the least-squares (applied to the cumulative sums):

de = 0.155dlnG– 0.65, t<1994

de= 0.25dlnG – 0.30, t>1993 (1)

where dlnG is the change rate of real GDP per capita at time t. Figure 2 shows the cumulative curves for the time series in (1). There is a structural break near 1994 which is expressed by significant shifts in slope and intercept. The employment/population ratio varies between from ~56%% in 1970 and 50.4% in 1992. The agreement is very good. Figure 3 present results of a linear regression with R2=0.91 for the period between 1971 and 2010.

Figure 2. The cumulative curves for the observed and predicted change in the employment/population ratio, de.

Figure 3. Linear regression of the measured and predicted curves in Figure 2.

Employment in Canada

There is a trade-off between the change in unemployment and employment. Figure 1 compares the change in the rate of employment (the employment/population ratio), de, and the rate of unemployment, du, in Canada. As expected, the change in the rate of unemployment is more volatile. We have retrieved all data on unemployment and employment from the U.S. Bureau of Labor Statistics.

Figure 1. The (negative) change in the rate of employment compared to the change in the rate of unemployment in Canada.  

In one our previous posts we have estimated Okun’s law for Canada. It is instructive to estimate a model similar to Okun’s law for the employment/population ratio, e. For Canada, the best-fit model has been obtained by the least-squares (applied to the cumulative sums):  

de_t = 0.40dlnG_t – 0.70, t<1984

de_t = 0.56dlnG_t – 0.76, t>1983    (1)  

where dlnG_t is the change rate of real GDP per capita at time t. Figure 2 shows the cumulative curves for the time series in (1). There is a structural break near 1984 which is expressed by a significant shift in slope and a minor change in intercept.  The employment/population ratio varies between from ~54.5% in 1971 and ~64.1% (!) in 2008. The agreement is very good. Figure 3 present results of a linear regression with R²=0.84 for the period between 1971 and 2010.

Figure 2. The cumulative curves for the observed and predicted change in the employment/population ratio, de. 

Figure 3. Linear regression of the measured and predicted curves in Figure 2.

On the absence of structural unemployment in Canada

We have estimated a version of Okun’s law for the USA, France and Spain. As beforfe, we have apply a LSQ technique to the integral version of Okun’s law:


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 coefficients.

For Canada, we have estimated a similar model with a structural break somewhere between 1980 and 1990. The best-fit (dynamic) model minimizing the RMS error of the cumulative model (1) is as follows:

du = -0.28dlnG + 1.16, t<1983
du = -0.28dlnG + 0.30, t>1982 (2)

This model suggests no shift in the slope and a bigger change in the intercept around 1983. Figure 1 depicts the observed and predicted curves of the unemployment rate. The agreement is very good. Figure 2 shows that when the observed time series is regressed against the predicted one, R2=0.87. Here we do not test both time series for stationarity but presume that the rate of unemployment has to be a stationary time series in the long run.

The integral form of the dynamic Okun’s law (1) is characterized by a standard error of 0.68% for the period between 1971 and 2010. The average rate of unemployment for the same period is 8.2% with a standard deviation of the annual increment of 0.94%.

One can suggest that the rate of unemployment has been driven by real economic growth and there is no much room for structural unemployment.

Figure 1. The observed and predicted rate of unemployment in the Canada between 1970 and 2010.

Figure 2. The measured time series is regressed against the predicted one. R2=0.87 with both time series likely to be stationary.

When the rate of unemployment will fall to 5%? Likely never

Update: A working paper is available with more technical details.

The intuition behind Okun’s law is very simple.  Everybody can feel that the rate unemployment is likely to rise when real economic growth is very low or negative. An economy needs fewer employees to produce the same or smaller real GDP because of permanent productivity growth. Thus, Okun’s law describes quantitatively the negative correlation between real economic growth and the change in unemployment rate.

We have rewritten Okun’s law using the growth rate of real GDP per capita instead of GDP. For the USA we have already obtained the following empirical relationship:     

dw = -0.406dlnG + 1.113, t<1979

dw = -0.465dlnG + 0.866, t>1978     (1) 

where dw is the predicted annual increment in the rate of unemployment, dlnG=dG/G is the relative change rate of real GDP per capita per one year. By definition, for a discrete form of Okun’s law one has: du_i=dw_i+e_i, where e_i is the model residual error at discrete time i. We have estimated all coefficients and the beak year in (1) by minimizing the cumulative sum of e_i squared.

In (1), the rate of real GDP growth has a threshold of (0.866/0.465=) 1.86% per year for the rate of unemployment to be constant. When dlnG is larger than 1.86% per year the rate of unemployment in the U.S. starts to decrease. Figure 1 displays the evolution of dlnG since 1979. On average, the rate of growth was 1.65% per year, i.e. slightly lower than the threshold and the rate of unemployment has been increasing since 1979.

Figure 1. dlnG as a function of time. Also shown is the threshold of 1.86% per year, the mean growth rate of 1.65% per year.            

When integrated between 1951 and t, equation (1) can be rewritten in the following form:

w_t = 3.3 – 0.406ln[G_t/G₁₉₅₁] + 1.113(t-1951)  + c₁  , t<1979
w_t = w₁₉₇₈ – 0.465ln[G_t/G₁₉₇₈] + 0.866(t-1978)  + c₂ ,  t>1978 (2)

where w_t is the predicted rate of unemployment. The intercept c₁=c₂≡0, as is clear for t=t₀.  Instead of using the continuous form (2), we calculate cumulative sums of the annual estimates of dlnG with appropriate initial conditions. By definition, the cumulative sum of the observdd du’s is the time series of the unemployment rate, u_t. Figure 2 depicts the measured and observed curves. 

The agreement is excellent and has been obtained by a formal statistical method. The integral form of the dynamic Okun’s law (2), i.e. w_t=f(lnG_t), is characterized by a standard error of 0.55% for the period between 1951 and 2010. The average rate of unemployment for the same period is 5.75% with the average annual increment of 1.1%.  All in all, this is a very accurate model of unemployment. And this fact is the most intriguing one.

Figure 2.  The observed and predicted rate of unemployment in the USA between 1951 and 2010. 

Our empirical model suggests a tangible shift in the slope and a significant change in the intercept around 1979. This is a very important finding. There are two terms in (2) which define the evolution of the unemployment rate: real economic growth, as expressed by the relative change in real GDP per capita, counteracts the positive linear time trend. Figure 3 depicts both components. The difference or the distance between a(t-t₀)  and –bln(G_t/G₀)-u₀ in Figure 3 is the rate of unemployment. 

The importance of the structural break in 1979 is obvious when we extend the trend a(t-t₀)  observed before 1979. The distance would be much larger with the old trend after 1979, i.e. the rate of unemployment would have been also higher than that actually measured. If to extend the current time trend and the dependence on G through 2050 one can project the rate of unemployment as Figure 3 also depicts. Without a new structural break, the rate of unemployment in 2050 will be near 25%. This is grim news. It might happen that the U.S. is currently struggling through a transition to a new relation in (2) which will keep the rate of unemployment below 10%. In any case, the growth rate of real GDP per capita has to be much higher than 2% per year in order to reduce the current rate of unemployment to the level of 5%.  Such a rate is not expected in the near future. 

As an alternative, we have tried a logarithmic time trend instead of the linear one. The logarithmic trend easily follows from our model of economic growth which has an inertial component inversely proportional to the attained level of real GDP per capita:

dlnG/dt = 0.5dlnN₉/dt + C/G (3)

where dlnN₉/dt is the change rate of the number of 9-year-olds and C is an empirically estimated constant. The term C/G represents the inertial rate or growth, i.e. the rate of growth corresponding to no changes in the age pyramid.  Figure 4 demonstrates the observed evolution of G since 1950 and gives two projections: a linear one with an annual increment C=$591.5 and an exponential growth following the trend before 2010.  The deviation between these projections is fast and the next few years should distinguish between them. Figure 5 provides some examples of developed countries with linear trend in real GDP per capita. 

We have introduced a similar trend term in the original Okun’s law and obtained:

dw/dt  = A/G_t + bdlnG/dt  (4)

By integrating (4) one obtains

w_t = u₀ + bln[G_t/G₀] + A∫dt/G_t  (5)

In the long run, the evolution of G_t is linear over time. Observations show that the change in the specific age population over the period of 50 and more years is negligibly small, ∫dlnN₉/dt ~ 0. Then dlnG/Gdt=C/G and G_t=G₀+C(t-t₀). Therefore, both terms in (5) have a logarithmic trend in time and w_t may vary around u₀. For equation (2), these trends are different (linear and logarithmic) and w_t must grow with time if there are no structural breaks. Figure 3 illustrates this divergence and the necessity of structural breaks. 

We have checked the predictive power of (5) relative to (2) and found no improvement. On the contrary, (5) does not allow to describe the whole period between 1951 and 2010 with one constant A.  Figure 6 depicts a model with A=28000 and b=-0.45. The model is very accurate between 1970 and 1990, overestimates the rate before 1970, and underestimates the observed rate after 2000. 

Figure 3. The evolution of two components in (2) defining the unemployment rate. 

Figure 4. The evolution of G over time with a projected linear trajectory for C=$591.5 and an exponential trajectory G=G₀exp(0.0209t), where the exponent corresponds to that obtained for the period between 1950 and 2010.

Figure 5. Some examples of linear evolution of real GDP per capita in developed countries.

Figure 6. The observed rate of unemployment and that predicted by (5) with A=28000 and b=-0.45. 

There is a fundamental concern about the excellent performance of Okun’s law in the U.S. The rate of unemployment is measured as a portion of labor force with the fluctuating rate of participation.   This means that the sensitivity of unemployment to real economic growth, expressed by Okun’s law, does not depend on the rate of employment itself.  Figure 7 compares the change in the rate of employment (the employment/population ratio), e, and the rate of unemployment. These two variables have been evolving in sync. Before 1980, the change in the rate of unemployment is relatively higher. After 1980, their amplitudes are very close. 

Figure 7. The (negative) change in the rate of employment compared to the change in the rate of unemployment. 

We have estimated a model similar to Okun’s law for the employment/population ratio, e:

de = 0.277dlnG – 0.457, t<1983

de = 0.496dlnG – 0.87, t>1982     (6) 

Figure 8 compares the observed and predicted change in the employment/population ratio. Figure 9 shows the cumulative curves for the time series in Figure 8 and explains the structural break near 1982.  The employment/population ratio grew from ~57% in 1982 and ~63% in 1989. This break also explains a similar break in the unemployment rate near 1980. The change in slope in (2) and (6) is rather similar: both the rate of employment and unemployment is more sensitive to the rate of change in GDP. 

This is the effect we have already reported and modeled for the rate of participation in labor force, l_t. To account for the effect of varying rate we introduced a factor, f_t, exponentially depending on the difference between some reference rate, l₀, and current rate, l_t: f_t=f₀exp[g(l_t-l₀)], where f₀  and g are empirical constants. The intuition behind the model is simple. The employment/population ratio and thus labor force increases with real GDP. When the rate of labor force participation undergoes a, say, 1% increase almost all new employees enter the workforce at the level of marginal personal income. Observations show that personal incomes are distributed exponentially in the low income range, i.e. the number of people with a given income decreases exponentially with increasing income. Accordingly, the input of the newcomers into the increasing GDP decreases exponentially with increasing labor force.  Thus, the sensitivity of employment/population ratio to real GDP increases with the ratio.   

This factor should be applied to Okun’s law as well. We will address this topic in the next post on employment. 

Figure 8. The observed and predicted change in the employment/population ratio, de.

Figure 9. The cumulative curves for the observed and predicted change in the employment/population ratio, de.

On the healthy growth in employment

After the employment  report for June, a number of bloggers (e.g. Mark Thoma and David Leonhardt ) suggested that a healthy monthly  increment in employment would be between 100,000 and 150,000. I have addressed this issue in my previous post as  well. The current employment/population ratio is aproximately 58%. One should notice that only people of 16 years of age and over are counted in.
On avearge, this civilian population has been growing since January 2010 by 143,00 per month. With the rate of 0.58 one would expect 83,000 per month. Actually, the increment since January 2010 is 76,000 per month . This estimate is from household employment data, i.e. the same source as for the civilian population.  The difference of 6,000 can not be considered as a dramatic one.

A responsible commenter should check actual data.