**This is a very important methodological post. It presents principal results of tests for cointegration of the long-term linear lagged relationship between inflation and the rate of labor force change. The latter is the driving force of price inflation in developed countries.**

**For the United States, the link has been tested by the Johansen and Engle-Granger techniques. Both demonstrate the presence of a cointegrating relation between these measured variables. So, there exists a long term equilibrium relation between these two variables in the USA. Later on we will demonstarte the presence of the same link in France and Austria.**

**Same tests for cointegration were applied to the link between real GDP per capita and the rate of change in the number of nine-year-olds. Also successfully.**

Complete working paper with all relevant figures and tables is available at the RePEc:

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, http://mpra.ub.uni-muenchen.de/2734/01/MPRA_paper_2734.pdf**Abstract**

Previously, a linear and lagged relationship between inflation and labor force change rate, p(t)= A1dLF(t-t1)/LF(t-t1)+A2 (where A1 and A2 are empirical country-specific coefficients), was found for developed economies. The relationship obtained for the USA is characterized by A1=4.0, A2=-0.03075, and t1=2 years. It provides a root mean square forecasting error (RMFSE) of 0.8% at a two-year horizon for the period between 1965 and 2002 (the best among other inflation forecasting models) and has a perfect parsimony - only one predictor. The relationship is tested for cointegration. Both variables are integrated of order one according to the presence of a unit root in the series and its absence in their first differences. Two methods of cointegration testing are applied - the Engle-Granger one based on the unit root test of the residuals including a variety of specification tests and the Johansen cointegration rank test based on the VAR representation. Both approaches demonstrate that the variables are cointegrated and the long-run equilibrium relation revealed in previous study holds. According to the Granger causality test, the labor force change is proved to be a weakly exogenous variable - a natural result considering the time lead and the existence of a cointegrating relation. VAR and VECM representations do not provide any significant improvement in RMSFE. There are numerous applications of the equation: from purely theoretical - a robust fundamental relation between macroeconomic and population variables, to a practical one - an accurate out-of-sample inflation forecasting at a two-year horizon and a long-term prediction based on labor force projections. The predictive power of the relationship is inversely proportional to the uncertainty of labor force estimates. Therefore, future inflation research programs should start from a significant improvement in the accuracy of labor force estimations.

JEL Classification: C32, E3, E6, J21

Key words: cointegration, inflation, labor force, forecasting, USA, VAR, VECM

**Introduction**

Inflation forecasting plays an important role in modern monetary policy in developed countries. There is a trade-off between theoretical and practical aspects of the forecasting –. Any practical application has to be based on a sound theoretical background, and every theory must concentrate empirical facts and everyday practice in a few quantitative relations. In reality, monetary authorities more often use expert judgments instead of theoretical models. These judgements are often based on some concepts empirically proved to be wrong such as the conventional Phillips curve, “expectation augmented” Phillips curve or NAURU approach. As a matter of fact, there is no unique and comprehensive model in mainstream economics and econometrics, which is able to explain observations and predict inflation in developed countries.

A study of inflation as an economic variable driven solely by labor force change has been carried out by Kitov (2006a-c, 2007) for the USA, Japan, France, and Austria. This study has confirmed the existence of a linear relationship between inflation, unemployment and labor force. In the USA, the inflation-labor force relationship is characterized by a two-year time lag. Regression analysis (Kitov, 2006b) has shown a high level of correlation (R2>0.9) between the variables and the lowermost, among available forecasting models, out-of-sample root-mean square forecasting error (RMSFE) at a two-year horizon, which corresponds to the observed lag. An important advantage of the model consists in its perfect parsimony - single predictor is used to fit observations in several countries during the last 25 to 45 years. Moreover, autoregressive properties, extensively used in many economic and econometric models of inflation, are not necessary because the relation between labor force change and inflation is strict. It does not allow any deviation between observed and predicted inflation when both variables are measured precisely.

As a consequence, the existence of a strict link between inflation and labor force change denies the concept of rational expectation thoroughly adopted in economics, i.e. future values of inflation are known to the extent some past values of labor force level are known. At the same time, the strict relation allows for a partial inflation control through labor force control and provides clear foundations for a sound economic policy.

Both variables are non-stationary, however, implying a possibility for the regression results to be spurious ones despite the existence of a theoretical background (Kitov, 2006b, 2007) based on the fact that personal income distribution in the USA does not change with time when normalized to the total population and GDP. Therefore, econometric tests are necessary whether the relationship between inflation and labor force is a cointegrating one. The cointegration between inflation and labor force change would mean that the results of the previous regression analysis hold from the econometric point of view. For the USA, cointegration analysis is essentially a bivariate one. This significantly simplifies the implementation of relevant analysis and statistical inferences. However, in the case of modeling inflation using labor force change rate, a straightforward application of econometric methods potentially leads to wrong conclusions.

The rest of the paper is organized in four sections and one appendix. Section 1 briefly introduces the model and presents some results obtained for the USA in the previous study. Section 2 is devoted to the estimation of the order of integration in measured inflation and labor force change rate. Unit root tests are carried out for original series and their first differences. GDP deflator represents inflation in the study.

In Section 3, the existence of a cointegrating relation between the measured and predicted inflation is tested. The latter is represented by four time series reflecting improvements in the accuracy of prediction associated with suppression of random noise in the labor force readings. The presence of a unit root in the difference (residual) between the measured and predicted inflation implies the absence of cointegration between the variables and a strong bias in the results of the previous regression analysis. The residuals obtained from regressions of the measured inflation on the predicted ones are also tested for the unit root presence. This approach is in line with that proposed by Engle and Granger (1987).

The maximum likelihood estimation procedure developed by Johansen (1988) is used to test for the number of cointegrating relations in a vector-autoregressive representation (VAR). The existence of a cointegrating relation is studied as a function of the predictor smoothness – from the original predictor to its three-year moving average. VAR and vector-error-correction models (VECM) are estimated for the forecasting purposes.

Section 4 discusses the principal results and their potential importance for economic theory and practical application, and concludes. Appendix introduces a pre-determined variable (from linear to a quasi-stochastic one) trend, which is of a crucial importance for cointegration study of time series related to population estimates such as labor force level.

**The static model**

There is a unique link between inflation and labor force change rate in the USA as obtained for the last 45 years (Kitov, 2006c). In econometric sense, it is a long-run equilibrium relation or static model which uses only original values of the involved parameters not their first differences. It is advisable to use the rate of labor force growth (not increment) as a predictor in order to match the dimension of inflation – [t-1]. An implicit assumption of the model is that inflation does not depend directly on parameters associated with real economic activity. Moreover, inflation does not depend on its own previous and/or future values because it is completely controlled by a force of different nature – by the dynamics of the portion of population participating in labor force.

As defined by Kitov (2006a), inflation is a linear and potentially lagged function of labor force change rate:

π(t)=A1dLF(t-t1)/LF(t-t1)+A2 (1)

where π(t) is the inflation at time t, LF(t) is the labor force level at time t, t1 is the time lag between the inflation and labor force, A1 and A2 are country-specific coefficients, which have to be determined empirically. The coefficients may vary through time for a given country as different units of measurements (or definitions) of the studied variables are used, but for true values of inflation and labor force they are constant.

Linear relationship (1) defines inflation separately from unemployment. This representation is an adequate one for the purpose of the paper. However, inflation and unemployment are two indivisible features of a unique process (Kitov, 2007). The process is the growth of labor force level, which is accommodated in real developed economies via two channels. The first channel is the increase in employment and corresponding change in personal income distribution (PID). All persons obtaining new paid jobs or their equivalents presumably change their incomes to some higher levels. There is an ultimate empirical fact, however, that the US PID practically does not change with time in relative terms, i.e. when normalized to the total population and total income (Kitov, 2005a,b). The increasing number of people at higher income levels leads to a certain disturbance in the PID. This over-concentration (or over-pressure in physical terms) of population in some income intervals above its neutral value must be compensated by such an extension in corresponding income scale, which returns the PID to its original density. Related stretching of the income scale is called inflation (Kitov, 2006a). The mechanism responsible for the income scale stretching, obviously, has some relaxation time, which effectively separates in time the source of inflation, i.e. the labor force change, and the reaction, i.e. the inflation itself.

The second channel is inherently related to those persons in the labor force who tried but failed to obtain new paid jobs. These people do not leave the labor force joining unemployment. Supposedly, they do not change the PID because they do not change their incomes. Therefore, the net labor force change equals the unemployment change plus the employment change, the latter process expressed in economies as the lagged inflation.

Kitov (2006c) proposed to estimate coefficients in (1) not by a linear regression of inflation on labor force change rate. Such a regression balances measurement errors, i.e. the deviations from a true relationship, through the entire period of observations (between 1960 and 2004) and has two important drawbacks: estimates of linear regression coefficients are usually biased due to errors in the predictor and the influence of larger fluctuations is overestimated. The latter might be of a higher importance in our case, considering the practice of revisions carried out by the US Bureau of Labor Statistics (BLS) and the Census Bureau (CB).

An alternative to regression arises from the fact that the labor force level is actually measured in monthly surveys not the change rate. Absolute uncertainty associated with annual measurements of the labor force level is constant or improving with time. Therefore, relative uncertainty associated with the net change in the labor force level during a given period is inversely proportional to the net change. When the net change in labor force level is large enough, as observed in the USA between 1960 and 2004, one obtains a significant improvement in the accuracy of the estimates of cumulative values. Considering measurement errors, a simple summation of annual estimates is always less accurate than a direct measurement of corresponding levels. (In the case of labor force, however, the revisions conducted by the BLS effectively cancel corresponding measurement errors out through time and provide practically the same accuracy of cumulative change for both the summation and the net level change techniques.) Therefore, the best-fit coefficients in (1) have to provide the same cumulative values for the measured and predicted inflation at the end of period, the latter having a decreasing uncertainty with time. This procedure should result in somewhat more accurate estimates, which are also different from those obtained by a standard linear regression. The estimated long-run relationship between inflation and labor force for the USA is as follows (Kitov, 2006c):

p(t)=4.0*dLF(t-2)/LF(t-2)-0.03075 (2)

The left-hand side of the equation is called below the measured inflation, pm(t) , and the right-hand side – the predicted one, pp(t). A statistical assessment of the linear relationship carried out for the USA indicates that RMSFE of inflation forecasts at a two-year horizon for the period between 1965 and 2002 is only 0.8%. This value is superior to that obtained with any other inflation model, including the random walk approach, by almost a factor of 2. The entire period was also split into two segments before and after 1983. The forecasting superiority is retained with RMSFE of 1.0% for the first (1965-1983) and 0.5% for the second (1984-2002) sub-period.

There is one aspect not analyzed in the previous papers - both variables are non-stationary. Hence, validity of the regression results is under doubt (Granger & Newbold, 1967). When regressed, two I(1) processes may show unreasonably high correlation or goodness-of-fit and severely biased estimates of corresponding regression coefficients. This might happen to the studied relationship despite the sound theoretical foundation linking such macroeconomic variables as real GDP growth and inflation to population parameters only (Kitov, 2005ab, 2006ad). These links are treated as “mechanical” ones; i.e. any change in defining population parameters is always fully transmitted into a strictly proportional change in corresponding macroeconomic variables.

There are several predicted inflation time series obtained by Kitov (2006c) for GDP deflator and CPI measurements using corrected labor force readings. Corresponding corrections in labor force level include redistribution of step-like revisions to population controls carried out by the US CB and the BLS. These corrections smooth original labor force series (SA) and effectively removes the largest spikes from its first difference. Probably, the only exception is the years between 1990 and 1993, when an unprecedented number of revisions to population estimates and labor force survey methodology were carried out. The standard procedure of redistribution failed to iron the spikes out. So, the corrected labor force time series still contains an artificial spike near 1990. (The revision carried out by the BLS in July 2006 effectively resolved the problem of 1993, however.)

The set of the predicted time series also includes the one based on half-year shifted labor force measurements. This time series is better synchronized with inflation measurements because it averages monthly readings from July to June of the next year with December being in the middle of the period. The shifted series gives a larger R2 than the original one presented by the BLS (Kitov, 2006c).

**The unit root tests**

It is clear from the behavior of the measured and predicted (in fact, labor force change rate) inflation curves displayed in Figure 1 that both time series are potentially characterized by the presence of unit roots because they demonstrate some clear non-stationary features. Moreover, the cumulative measured and predicted inflation curves are definitely non-stationary with a strong (stochastic or deterministic) trend component. In such a situation, a spurious regression is probable as modern econometric research shows. Therefore, some specific tests have to be carried out in order to prove validity of the results obtained in Kitov (2006b,c). In particular, one has to prove that the measured and predicted time series are integrated of order one, I(1), and are cointegrated, i.e. the residuals of their regression or VAR representation create a stationary, I(0), series which meets a number of specific requirements (Hendry&Juselius, 2001).

From statistical point of view, the principal problem of the econometric analysis related to the relationship between inflation and labor force change consists in the balance between the length of the series and the uncertainty of corresponding readings. Standard statistical tests work relatively well with time series of several hundred readings. When the size of a sample is small, statistical inferences based on asymptotic distributions provided with econometric packages may be biased. The replacement of asymptotic distributions by those obtained specifically for small samples reduces the power of the tests. Increasing the number of readings via sampling rate one observes decreasing accuracy. When measurement uncertainty is high no useful signal (i.e. long-term relationship) can be retrieved from the noise. There is no general solution of the problem and one always has to compare the outcome of statistical tests with the bulk of external information - theoretical and empirical.

As a first step in the analysis, we have to prove that the measured and predicted inflation are I(1) processes. There is a number of standard unit root tests provided with many econometric packages – the Dickey-Fuller, the Augmented DF (ADF), the modified DF t-test using a generalized least-squares regression (DF-GLS), Phillips-Perron and others. From this set, we use only the ADF and DF-GLS because the DF has a skewed distribution with a long left tail complicating discrimination the null of a unit root, and the PP test works better asymptotically, i.e. is not reliable at short time series. Standard programs of Stata9 statistical package are used in the current study.

There are several time series modeled in (Kitov, 2006c), which may be potentially tested for the unit root presence: the measured inflation (GDP deflator and CPI inflation) for the period between 1960 and 2004 (45 readings); numerous time series predicted from the labor force change rate (dLF/LF) – original, shifted by half a year, and those smoothed by moving average of various width. All the predicted time series are also present in two intervals: from 1958 to 2002 and from 1960 to 2004. The latter series are the versions of the former ones but shifted by two years ahead in order to synchronize the predicted inflation readings with corresponding measured values because of the two-year lag. The original predicted series retain the actual lead ahead of the measured values important for causality tests. Because the original and the two-year shifted time series differ by four values (each of the series contain two readings different from its counterpart) one may potentially expect a slightly different result of the unit root tests. From all available time series, measured and predicted, only those associated with the annual readings of GDP deflator are studied here. Table 1 lists results of the ADF and DF-GLS tests for a unit root in the measured GDP deflator and six predicted inflation time series including MA(2) and MA(3) of the predicted time series which are shifted by two years ahead. The series measured represents the GDP deflator time series between 1960 and 2004, the series predicted and shifted are synchronized (shifted by two years ahead) with the measured one, and the original (not shifted) predicted series are titled as predicted2 and shifted2. The last raw in the Table lists 1% critical values for corresponding tests. For the DF-GLS test, only values for lags 1 and 4 are presented because results for lags 2 and 3 are very close to those for lag 4.

One can derive an overall conclusion from Table 1 that no one series is a stationary one because the null of the unit root presence can not be rejected. It is worth noting that the predicted series shifted by two years, such as the predicted and predicted2, actually give different but close results. It does not affect the general conclusion of non-stationarity, however.

The series displayed in Table 1 are non-stationary. But it does not mean that these series are I(1) and some additional efforts are necessary to prove the assumption. For an I(1) process, the first difference has to be a I(0) process. Therefore, the same tests are repeated on the first differences of the two principal time series – dmeasured and dpredicted, which are displayed in Figure 2. Corresponding test results are presented in Table 2.

A standard unit root test usually contains many specifications related to various features potentially present in time series. The tests listed in Table 2 use two different assumptions on the presence of constant and trend in the first differences. First, we allow for a deterministic linear trend in the processes and test for a single unit root in an AR(p) representation, where p=0,1,2,3 for the ADF test and p=1,2,3 for the DF-GLS test. All the tests for the dmeasured, except those with lag 3, imply the rejection of the null of the unit root presence at the 1% level. For the dpredicted, only tests with lags 0 and 1 reject the null.

The presence of a constant term is a more reasonable hypothesis for the first differences of such processes as inflation and labor force change. When a constant specification is used in the ADF unit root tests, results are more favorable for the rejection of the null for both dmeasured and dpredicted. For the DF-GLS, the null is rejected for all lags in the dmeasured and only for lag 1 in the dpredicted. Because of the mixed results for the dpredicted, more information is necessary for rejection or acceptance of the null.

Here we have to draw a bold separation line between the procedures for estimation of GDP inflation and labor force change rate. Inflation is measured by acquisition of information on prices. There are some short-term corrections related to the inflation measurements which incorporate late information and new methodology. There is no correction extended far in the past, however, which is dependent on new inflation readings. Thus, one can consider subsequent readings of inflation as relatively independent in view of the measuring procedures.

This is not the case with the labor force measurements, however. The labor force level is estimated in monthly surveys using some limited population samples. Simultaneous estimates of age-gender-race distribution, then converted into population controls, are used for mapping the limited sample results onto the entire population. Such a construction of the labor force estimates has two drawbacks – the sample changes are slow in order to study the dynamics of the inter-sample evolution, and the population controls are determined both from the monthly-quarterly-yearly population estimates and decennial censuses.

The sample stability implies the presence of a link between subsequent labor force estimates. Corrections to the population controls are of even larger influence on labor force readings, however. After every decennial census, the BLS conducts severe (sometimes several per cent) revisions of the population age structure, which are extended long in the past. These revisions effectively include synchronized changes in the population controls over the entire period. Figure 3 demonstrates a remarkable effect of such revisions on the change in the number of people in a one-year-wide population cohort – the difference between 15-year-olds and 14-year-olds a year before. This age is of importance for the labor force measurements because statistics starts at 15 years of age. Therefore their influx is one of the defining components of the labor force increase along with net migration and total deaths. The difference between the numbers is everything but of a natural shape. Considering the disturbances introduced into the labor force time series by the BLS one should not be surprised that the DF-GLS tests does not reject the null for the first difference of the predicted inflation, which is based on the labor force change rate. Some statistical properties of population time series important for the cointegration analysis are discussed in Appendix.

The dmeasured time series is characterized by a very low average (0.00019±0.011) compared to the "min" and "max" values of -0.036 and +0.031. Slope and intercept of the regression on time are -0.00012±0.00012 and 0.24±0.24, respectively, i.e. the null of zero slope and intercept can not be rejected. This result implies that the dmeasured has no trend and intercept and the assumption of constant specification is applicable to the previous tests. The overall conclusion is that the dmeasured series has no unit root and the dpredicted series would not have a unit root if not the artificial distortion induced by the labor force measurement procedure. Therefore, the original series measured and predicted are integrated of order one.

**The cointegration tests**

The assumption that inflation and labor force change in the USA are two cointegrated non-stationary time series is equivalent to the assumption that the difference between the measured and predicted inflation, e(t)=pm(t)- pp(t), is a stationary or I(0) process. It is natural to start with a unit root test in the difference. If e(t) is a non-stationary variable having a unit root, the null of the existence of a cointegrating relation can be rejected. Such a test is associated with the Engle-Granger's approach, which requires pm(t) to be regressed on pp(t) as the first step, however. Since the predicted variable is obtained by a procedure similar to a linear regression and provides the best fit for cumulative values, we skip the regression and start with an analysis of the difference.

The hypothesis of a unit root is tested by the same procedures as before – the ADF and DF-GLS. If the null of a unit root is rejected, the hypothesis of no-cointegration is also rejected. In this case, the equilibrium relationship between the measured and predicted inflation is valid and a vector error-corrected model can be estimated using the results of the first stage.

Table 3 presents some results of the unit root tests. There are three differences tested: diff1=(pm(t) - pp(t)), diff2=(pm(t) -MA(2)), and diff3=(pm(t) -MA(3)). The ADF and DF-GLS results indicate the absence of a unit root in the diff1 residual series, except may be for lag 3 in the DF-GLS tests. The diff1 series of the largest importance because it is based on the inflation predicted from the original labor force series. As discussed in Section 2 and Appendix, the labor force measurements are potentially biased by a strong autocorrelation introduced by the revisions to the population controls. One can expect the enhancement of the autocorrelation in the differences (Chiarella&Gao, 2002). Moreover, this autocorrelation has to be more prominent in the smoothed series, where random noise in measurements is suppressed and piece-wise systematic deviations are of relatively higher amplitude. The series diff2 and diff3 illustrate these effects – the DF-GLS tests produce the results which are higher than corresponding 1% critical values also listed in the Table. So, smoothing leads to deterioration in the test results compared to those for the original series.

It is worth noting that the moving average time series provide a better description of the observed inflation as represented by standard deviation in the residual series. The best fit is obtained for the MA(3) series with standard deviation twice as low as that provided by the original predicted series: from 0.017 for diff1 to 0.009 in diff3. The price paid for the better description is higher values in the unit root tests. Therefore, one can obtain a wrong statistical inference by over-suppressing of the stochastic component in real data, as shown in Appendix. Concluding the discussion, we reject the null of the unit root presence in the three difference series.

The next step is to use the Engle-Granger's approach and to study statistical properties of the residuals obtained from linear regressions of the measured inflation on various versions of the predicted inflation. Table 4 summarizes principal results of the regression analysis – coefficients with their standard deviations, R2, RMS(F)E (F stays for forecasting because of the two year lag), and t for the null of zero coefficients. The dependent variable is always the measured series and predictor varies from the predicted to MA(3). Results of specification tests on heteroskedasticity, omitted variables, ARCH effects, and serial correlation, also listed in Table 4, indicate that the residuals of the regressions, except those related to MA(3), meet the requirements defining I(0) process and have properties of a white-noise realization necessary for a VAR representation of the measured and predicted inflation. Therefore, one cannot reject the null that the dependent variable and the predictors are cointegrated. The reasons for the failure of the specification test for MA(3) have been already discussed.

The Johansen's (1988) approach is based on the maximum likelihood estimation procedure and tests for the number of cointegrating relations in the vector-autoregressive representation. The above analysis has shown that the VAR representation is an adequate one due to the properties of the regression residuals. The Johansen's approach allows simultaneous testing the existence of cointegrating relations and determining their number (rank). For two variables, only single cointegrating relation is possible. When cointegration rank is 0, any linear combination of the two variables is a non-stationary process. If rank is 2, both variables have to be stationary. Thus, when the Johansen test results in rank 1, a cointegrating relation does exist.

Table 5 lists trace statistics, eigenvalues, LL and other information obtained from the cointegration rank tests for a number of predictors and trend specifications. The maximum number of lags included in the underlying VAR model is 4 in all tests. No null of cointegration rank 1 can be rejected. So, there is a cointegrating relation between the measured and predicted inflation.

Because of the two-year lag behind the labor force change and the existence of the cointegrating relation or long-run equilibrium relationship (2), a test on causality direction is a trivial one. In this test, the variables predicted and shifted are replaced with their original version predicted2 and shifted2, which leads the measured variable by two years. Results of the causality tests are presented in Table 6, which demonstrates that the predictors are weakly exogenous variables.

Now we are sure that the measured and predicted inflation series are cointegrated and the principal results of the previous research hold. Therefore, the estimates of the goodness-of-fit and RMSFE are accurate. The estimates were obtained in a simplified regression procedure, which does not use autoregressive properties of noise. A standard moving average smoothing provides a substantial suppression of the noise, as the increase in R2 from 0.7 for the annual readings to 0.9 for MA(3) demonstrates (Kitov, 2006c).

A VAR representation may potentially provide a further improvement due to additional noise suppression. In practice, AR is a version of a weighted moving average, which optimizes noise suppression throughout the whole series. In the VAR model, we use the shifted2 series as an exogenous predictor and the maximum lag 4 instead of the pre-estimated one maximum lag 3. The level of correlation between the measured and predicted series does not allow any past values of inflation to influence the present ones and, hence, the past values of labor force change. The equilibrium relationship is a strict one and converges to a unique curve when measurement errors are eliminated. At any time, inflation does not contain information important for labor force change. So, the labor force change rate is an exogenous variable. One cannot deny the influence of current inflation on current labor force level. This influence is very weak, however, as this study shows.

The distribution of the VAR error term is close to the normal distribution with skewness=-0.05 and kurtosis=3.99, the former being of much higher importance for the Jarque-Bera normality test and validity of statistical inference. The VAR stability is guaranteed by the eigenvalues of the companion matrix lower than 0.54. There is practically no autocorrelation at lags from 1 to 4 as follows from the LM statistics. Therefore the VAR model accurately describes the data and satisfies principal assumptions on the residuals.

A VECM representation uses additional information to that provided by the VAR models due to separation of noise and equilibrium relationship. So, it potentially provides an improvement on the VAR models. Tables 7 and 8 list some results of the VAR and VECM for a number of predictors. There is only a marginal improvement in RMSFE on that obtained by the linear regressions. The smallest RMSFE is 0.0076 in the VAR representation (MA(3) as a predictor) and 0.0073 in the VECM representation (shifted2 as a predictor). The best RMSFE from the linear regressions for the period between 1965 and 2002 is 0.008.

One can conclude that these powerful statistical methods fail to improve on the simple predictions. A significant decrease in the forecasting uncertainty is possible only through a major increase in the accuracy of inflation and labor force measurements. This conclusion is in line with a standard research cycle in hard sciences.

**Conclusion**

The core result of the analysis consists in formal statistical confirmation of the existence of a unique linear and lagged (two years for the USA) relationship between inflation and labor force change. Hence, the two I(1) variables are cointegrated in statistical sense; i.e. their residual time series is proved to be a stationary one with white noise characteristics - no unit root, normal underlying distribution, no omitted variables, no autocorrelation, no heteroskedasticity. The absence of such cointegration test was a weak point in the previous papers. It exposed the results of modeling and theoretical consideration to criticism.

Observed inflation is always a combination of two components – a pre-determined variable trend (demonstrating stochastic features), which one-to-one repeats the behavior of some true labor force change, and a noise component related to measurement errors in both inflation and labor force level. These errors include random and piece-wise systematic parts.

In terms of physics, there is a process "mechanically" linking true inflation to true labor force change. Observed deviations from the relationship describing the mechanical link are measurement errors. The long-run equilibrium relationship implies the absence of any structural breaks necessary for the explanation of the changes in the US inflation observed during the last 50 years. Such structural breaks are a common feature of a majority of models developed in the econometric framework, where inflation is considered as a stochastic process.

The noise component demonstrates some artificial features related to the procedures used by the US BLS for the labor force estimation. As in hard sciences, improvements in the measurement procedures, both for inflation and labor force, can potentially provide any desired accuracy of inflation forecast at a two-year horizon and even further if to improve the accuracy of labor force projections. It is worth noting that there is no error-correction mechanism associated with the deviations of true inflation values from the true predicted ones, but there exist error-correction mechanisms, which compensate random and systematic errors through time. For the labor force series, this mechanism is obviously associated with multiple revisions, which reshape the error pattern, in amplitude and timing, at very long time horizons.

The long-run relationship between inflation and labor force change rate is explained by the reaction of personal income distribution on the disturbance induced by those persons who enter employment and change their incomes correspondingly. This consideration is a part of the new economic concept, which relates macroeconomic variables to quantitative characteristics of population only.

**Appendix. Variable deterministic trend**

In hard sciences, situations where a non-stationary time series practically repeats the behavior of another series with some time lag are common. Finding of such a resemblance is a first step of a variety of research programs. The next natural step is to reduce the level of noise associated with measurement errors. For a scientist, this step is necessary for obtaining a higher confidence in the existence of a strict link between two variables or in the absence of such a link. So, improving measurement accuracy results in the acceptance of a true relationship or in the rejections of a false one.

In econometrics, however, the presence of such a pre-determined variable trend often leads to deterioration of the performance of standard statistical procedures associated with non-stationary processes. The danger to wrongly reject the presence of a true link in standard econometric procedures is associated with a general rejection of the existence of real deterministic trends in economic time series except the simple ones as connected to economic growth. Such simple trends are easily removed by detrending procedures. Otherwise, there exist only stochastic trends in econometrics, which represent the central problem in analysis of non-stationary time series. In some cases, this assumption leads to a biased result. For example, when correlation between two series in strong and random measurement errors are significantly suppressed, even small systematic errors induced by estimation procedures become defining for rejection of cointegration, i.e. for rejection of the true link between the variables.

There are several key assumptions, which lay in the basement of the cointegration concept. One of them is associated with the properties of residuals necessary for two or more non-stationary processes to be cointegrated. The existence of a cointegrating relation means that the residual of some linear combination of non-stationary variables is stationary. This residual time series has to be an independent and identically distributed with zero mean and constant variance, IID(0,Ω2), where IID is not necessary the normal distribution. Such residuals guarantee unbiased estimates of relevant coefficients.

The mainstream research of cointegration phenomenon allows for some deterministic trend in non-stationary variables which is very often considered as a linear or quadratic one. The inclusion of such trend terms sometimes helps to distinguish between purely stochastic non-stationary processes and stationary processes with deterministic trends. There are some studies of more complicated cases with nonlinear deterministic trends. But most of them consider analytic functions. Here we extend the notion of deterministic trend to the level of deterministic variable (potentially stochastic) process, i.e. to such a process, which has stochastic statistical properties but is fully pre-determined. In physics, this notion is similar to decomposition of a measured time series into true and noise components.

The importance of deterministic variable trend for econometrics can be illustrated by a simple but principal example associated with the discussion in Section 2 of growth in the number of 15-year–olds in the USA, N15(t). The evolution of this number is definitely described by a stochastic time series, supposedly, by I(2) process. In the framework of econometrics, this implies that the expected value of the change in N15 is zero (if to exclude a small deterministic trend):

N15(t)-N15(t-1)=dN15(t)=e(t) (A1)

where e(t) some stochastic ID(0,s2) process. (Actual dN15 average value between 1960 and 2004 is 29900 and standard deviation is 193000. If to exclude the years between 1960 and 1962, the mean=14500 and stdev=115000.) Thus, dN15(t) must have very low level of autocorrelation for the small time series under consideration. Asymptotically, autocorrelation at any lag must be zero – E[(e(t)*e(t-s)]=0 for any s≠0. In fact, autocorrelation associated with the actual series of dN15(t) between 1960 and 2004 is below 0.25 at lags from 1 to 15. Therefore, the annual increment in the N15(t) can hardly be predicted from its previous values. This is a standard situation in econometrics, which deals mainly with economic and financial variables. A general convention dictates that there is no way to exactly predict the forthcoming change in stock prices or CPI inflation. Only statistical estimates are available. This is not the case with the dN15(t), however.

First, one has to learn some statistical properties of the N15. Figure A1 shows the annual N15 readings, the first difference, dN15, and also the second differential d2N15. (There are also N0 and N14 and their differentials shown in the Figure.) Due to strong variations in the number of 15-year-olds in the earlier 1960s we limit the N15 series to the period between 1963 and 2004 which contains 42 readings.

The ADF test rejects the null of a unit root in the d2N15 time series and accepts the null for N15 (for lag 0 and larger) and dN15 (for lag 1 and larger). The DF-GLS tests for the same series assume the acceptance of the null in the d2N15, however (see Table A1). Thus, the problem of a unit root in the d2N15 is a controversial one and needs more attention to the population estimation process. For our purposes, we reject non-stationary in the d2N15.

Regardless of the unit root presence in the N15 and its stochastic properties, one can accurately predict its value at a one (and more) year horizon using the change in the number of 14-year-olds one year ago, dN14(t-1)=N14(t-1)-N14(t-2), as a proxy to the dN15(t)=dN14(t-1)+w(t), where w(t) is associated with the differences in death and migration rate between 14-year-olds and 15-year-olds in the given year. Another way to obtain an accurate estimate of the N15(t) is to update N14(t-1) by the inflation-deflation method used by the Census Bureau in the population estimates. Actually, the N15(t) contains the same population cohort, by the year of birth, as the N14(t-1) plus net migration, m(t), and less total deaths, r(t), in the population of the given age. Therefore, one can rewrite relationship (A1) in the following forms:

N15(t)=N15(t-1)+dN14(t-1)+w(t) (A2)

N15(t)=N14(t-1)+r(t)+m(t) (A3)

Relative importance of w(t) can be estimated from the difference between dN15(t) and dN14(t-1) normalized to the dN14(t-1), as presented in Figure A2. The shape of the curve is not a surprise because of multiple revisions made by the USA Census Bureau. The years between decennial censuses are characterized by low values (<0.1) t="2007,…,2010." r2="0.98," t="1,…,45." a="0.005" a="0.1.">

**References**

Kitov, I., (2005) A model for microeconomic and macroeconomic development. Working Paper 05, ECINEQ, Society for the Study of Economic Inequality

Kitov, I. (2005b). Modeling the overall personal income distribution in the USA from 1994 to 2002. Working Paper 07, ECINEQ, Society for the Study of Economic Inequality

Kitov, I. (2006a). Inflation, Unemployment, Labor Force Change in the USA. Working Paper 28, ECINEQ, Society for the Study of Economic Inequality.

Kitov, I. (2006b). The Japanese economy. Available at SSRN: http://ssrn.com/abstract=886663

Kitov, I. (2006c). Exact prediction of inflation in the USA. Available at SSRN: http://ssrn.com/abstract=916060

Kitov, I. (2006d). GDP growth rate and population. Working Paper 42, ECINEQ, Society for the Study of Economic Inequality.

Kitov, I., (2007). Inflation, Unemployment, Labor Force Change in European countries. In T. Nagakawa (Ed.), Business Fluctuations and Cycles. Hauppauge NY: Nova Science Publishers. (in press)

Kitov, I. (2005b). Modeling the overall personal income distribution in the USA from 1994 to 2002. Working Paper 07, ECINEQ, Society for the Study of Economic Inequality

Kitov, I. (2006a). Inflation, Unemployment, Labor Force Change in the USA. Working Paper 28, ECINEQ, Society for the Study of Economic Inequality.

Kitov, I. (2006b). The Japanese economy. Available at SSRN: http://ssrn.com/abstract=886663

Kitov, I. (2006c). Exact prediction of inflation in the USA. Available at SSRN: http://ssrn.com/abstract=916060

Kitov, I. (2006d). GDP growth rate and population. Working Paper 42, ECINEQ, Society for the Study of Economic Inequality.

Kitov, I., (2007). Inflation, Unemployment, Labor Force Change in European countries. In T. Nagakawa (Ed.), Business Fluctuations and Cycles. Hauppauge NY: Nova Science Publishers. (in press)

*Figure 1. The measured and predicted time series between 1960 and 2004. The labor force estimates are obtained at the BLS web-site in July 2006. Therefore the strong spike near 1993 discussed in (Kitov, 2006c) has disappeared.*

*Figure 2. The series dmeasured and dpredicted obtained as the first differences of the measured and predicted in Figure 1.*

*Figure 3. The difference between N15(t) and N14(t-1) in the USA for the period between 1960 and 2004*

a)

b)

c)

*Figure A1.*

a) The number of 15-, 14-, and 0-year-olds as a function of time between 1900 and 2004. Notice a large step in the earlier 1960s in the 15- and 14-year-olds corresponding to the burst in birth rate after the WWII.

b) The first differences of the time series presented in panel a). The step near 1960 is transformed into a spike, which is associated with natural causes. The spikes near the years of decennial censuses are of artificial character.

c) The second differentials of the series N15 and N0.

Figure A2. The ratio [dN15(t)-dN14(t-1)]/dN14(t-1) as a function of time. The years between decennial censuses are characterized by low (<0.1)>

a) The number of 15-, 14-, and 0-year-olds as a function of time between 1900 and 2004. Notice a large step in the earlier 1960s in the 15- and 14-year-olds corresponding to the burst in birth rate after the WWII.

b) The first differences of the time series presented in panel a). The step near 1960 is transformed into a spike, which is associated with natural causes. The spikes near the years of decennial censuses are of artificial character.

c) The second differentials of the series N15 and N0.

*Figure A3. The original (open circles) and corrected (solid diamonds) difference between N15(t) and N14(t-1). The latter is obtained by elimination of the linear trend and non-zero constants from the former. There are two low-amplitude spikes survived near 1990 and 2000, however.*

*Figure A4. Comparison of abs(dN15(t)-dN14(t-1)) and abs(dN14). The average values are 2200 and 95000, respectively. Their ratio is ~0.023.*

*Figure A5. Labor force participation rate in the USA between 1960 and 2004. The years between 1965 and 1996 are characterized by a linear growth. Linear trend explains more than 97% of the participation rate variation during this period. In 2000, a period of slight decrease in the participation rate started.*

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