Patents as a measure of technology shocks and the aggregate U.S. labor market.

Rossana, Robert J.

I. INTRODUCTION

One of the dominant approaches to macroeconomic research in the past several decades has been the island paradigm of Friedman (1968) and Phelps (1970). In this framework, economic agents operate under incomplete information regarding relative and absolute prices when making economic decisions. This paradigm is used in the rational expectations literature (see McCallum 1980 and the references therein) in a market-clearing framework to study, among other things, the effects of fiscal and monetary policies in aggregate economic systems. This market-clearing model is sufficiently popular that it is rare to find a textbook treatment of macroeconomics that omits this "new classical" model of the aggregate economy.

One component of this aggregate economic model is a labor market where nominal wages are flexible. (1) This approach to the labor market was developed by Lucas and Rapping (1969); in their model, it is assumed that labor suppliers operate under incomplete information about the aggregate price level, thereby raising the possibility that output produced in the economy will vary with the level of price expectations. (2) Despite its central role in explaining economic fluctuations in a popular model of the economy, there is surprisingly little evidence available in previous research testing the implications of this model of the labor market. One objective of the present study is to provide some empirical evidence on this market-clearing model of the labor market.

In carrying out such a test, it will be necessary to provide an explanation for the nonstationary nature of labor market series, such as real wages and employment. One possible explanation for this apparent nonstationarity would be the technology shocks frequently used in the real business cycle literature (e.g., Long and Plosser 1983), where it has often been assumed that these shocks are I(1) unobservable stochastic processes. Thus these shocks to the production function may be used as a way of explaining why macroeconomic time series are also I(1). A similar mechanism involves the possibly nonstochastic scalar preceding the aggregate production function. In the economic growth literature, this magnitude, called the Solow residual, measures the growth in aggregate output that cannot be explained by the growth of quality-adjusted factor inputs in production. This residual is sometimes thought to be measured by the stock of knowledge used by economic agents (Barro and Sala-i-Martin 1995, p. 351). Increases in the stock of knowledge enable the aggregate economy to produce more goods and/or new goods from quality-adjusted resource levels when knowledge increases. If the stock of knowledge is nonstationary, then so, too, would be output and other magnitudes in an aggregate economy. (4) Thus the nonstationary nature of aggregate data can be explained by a nonstationary element of the aggregate production function (whether stochastic or deterministic), however that variable is interpreted.

If the sources of the apparent nonstationarity in macroeconomic data are assumed to be unobservable I(1) stochastic processes, an implication of this fact is that the inability to measure technology shocks renders it impossible to test the implications of economic models that would arise if these shocks could be measured. For example, if technology shocks are I(1), then optimizing behavior by households and firms will result in cointegrating relationships between technology shocks and choice variables set by the public. If it were possible to measure shocks to the production function, one could test for these long-run cointegrating relationships and the parameters of these cointegrating vectors could be estimated. (5) In addition, it is desirable to explicitly incorporate technical change into economic models, if it could indeed be measured, because doing so may result in more reliable estimates of structural parameters of economic interest, and it may permit improved tests of economic hypotheses. For example, structural changes can impair tests for unit roots in time series and there is a substantial body of literature which develops tests for unit roots in the presence of structural breaks in economic data. (6)

In this article, it will be assumed that technology shocks or the stock of knowledge are a function of utility patent data. These data measure patents issued to individuals and firms, and because technological progress is at least in part a consequence of new knowledge available to the public as a result of research and development activities, this seems a sensible measure of increments to the stock of knowledge. These data series have been used in previous work to study the extent to which expenditures on research and development are related to patents. (7) Here patents will be used as the motivating force for the nonstationary or I(1) characteristics of labor services and real wages in the aggregate labor market. As a result, estimates of the structural parameters from the labor market will be obtained from the long-run relationships arising in the labor market between real wages, labor services, and patents.

There is measurement error associated with using patents to measure the stock of knowledge in an aggregate economy or to justify the I(1) characteristics of labor market data. (8) For example, the model used in this article presumes that new patents lead to measurable increases in output produced, yet not all patents lead to commercially viable products. In addition, there are changes in the stock of knowledge that are unrelated to new ideas and patents. The stock of knowledge may expand through learning-by-doing and there is evidence that this occurs (see Lucas 1993, pp. 259 62, and the references cited there). Efficiency increases of this sort would seem to have little to do with patent flows. But if patents are correlated with the stock of knowledge, then this measure of new ideas should be a sensible measure of the impact of new knowledge on the industrial composition of aggregate economies. At least some of the observed changes in industrial structure are related to the invention of new ideas, leading to the birth and death of firms and the reorganization of existing industries. But it may also be true that changes in the organizational structure of industries may be caused by forces unrelated to the development of new knowledge.

An equilibrium model of the labor market implies that, if the shock to technology is I(1), there will be two cointegrating vectors in a trivariate vector autoregression (VAR) for real wages, labor services, and the technology shock, and this implication will be tested using aggregate U.S. data. The cointegrating vectors from the labor market will be explicitly derived, and it will be seen that estimating the elements of these cointegrating vectors delivers exactly identified estimates of three structural parameters from the optimization problem solved by the representative firm, conditional on an assumed value for the discount rate. These parameters measure the severity of the training or adjustment costs attached to the labor force, the speed of adjustment of labor, and the degree of diminishing returns in production. The labor market model is built on the assumptions of diminishing returns in production and rising marginal adjustment costs for labor; given these assumptions, the adjustment speed for labor is between zero and unity. The empirical estimates will thus enable us to determine if these assumptions are evident in labor market data.

Regarding the empirical results herein, there are reasons (discussed later) suggesting that patents may be subject to a structural break associated with changes in the legal system, so unit root tests are applied to the patent series with and without endogenous breaks in the series. It is found that the patent series is indeed I(1), whether or not a structural break is permitted in the patent series. Aggregate data provides consistent evidence that technology shocks are cointegrated with real wages and labor services and there is a substantial amount of evidence that there are two cointegrating vectors in the data. However, full-sample estimates of the elements of the cointegrating vectors and their implied estimates of structural parameters provide relatively little support for this model of the labor market. Qualitative parameter assumptions from the model are quite different from estimated values of various structural parameters and the magnitudes of estimated parameters often appear implausible. But there are at least two regime changes in the sample (the formation of the Organization of Petroleum Exporting Countries [OPEC] cartel in October 1973 and the policy regime change by the Federal Reserve System in October 1979) that render parameter estimation problematic, and so subsample estimates are investigated to see if there is evidence of instability in the rank of the cointegrating matrix or in estimated parameters. The evidence observed is that the number of cointegrating vectors is robust to the choice of sample, despite the fact that a reduction in sample size should reduce the power of cointegration tests, but there is clear evidence of parameter instability. Substantial differences in some parameter estimates are apparent over various subsamples. Thus it seems clear that although one source of possible instability in empirical work has been accounted for by the use of a measure of the stock of knowledge, there seem to be other forms of structural or regime changes that may still contaminate parameter estimates.

Furthermore, in all subsamples but one, the predictions of the model are not supported by the data. However there is one subsample where the model provides a good description of the market for labor. In the period prior to the formation of OPEC, estimated structural parameters match up quite well with the assumptions of the model. The estimated speed of adjustment for labor is high, an intuitively appealing result, and other structural parameter estimates are found to be qualitatively consistent with the assumptions made on the demand side of the labor market: in addition, estimated parameters other than the speed of adjustment seem to be of plausible magnitude. The evidence suggests that new forms of knowledge, as measured by the stock of patents, do not provide a complete description of the source of nonstationarity evident in macroeconomic data subsequent to the formation of the oil cartel.

The article is organized as follows. The next section provides a discussion of the demand and supply sides of the labor market model. The labor market error-correction VAR is derived there, and its time-series properties will be discussed. The cointegrating matrix is derived, and it will also be shown that the cointegrating matrix does not contain parameters from the stochastic process obeyed by any unobservable shocks in the model. Section III provides diagnostic statistics for the patent data, including tests for unit roots with and without structural breaks in the patent data, to discover if the patent data are an I(1) series. This section also presents the estimates of the elements of the cointegrating vectors and structural parameters of the model over the entire sample and various subsamples. Section IV offers concluding remarks.

II. A MODEL OF THE LABOR MARKET

In this section, an equilibrium model of a labor market will be formulated. The model is a version of one described in Rossana (1998, pp. 437M2). (9) Functional forms are used that permit the derivation of an error-correction VAR in labor services, real wages, and the stock of knowledge. The model presumes that wages clear the market, and it is simplified so that its implications may be obtained with a minimum of economic detail. For example, constants are omitted throughout, and the household choice problem is simplified so that the dynamics evident in the VAR will come only from the demand side of the market. The objective of the analysis will be to derive the long-run relationships for the three time series in the VAR, which will serve as the basis for the empirical work reported later.

The Behavior of the Firm

The firm is assumed to solve the following cash flow maximization problem.

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In (1), [gamma] is the discount rate (0 < [gamma] < 1), [[zeta].sup.d] is a technology shock to the firm's production function, w is the real wage rate (the firm's output price is normalized to unity), n refers to labor services, and [[alpha].sub.i] > 0[for all]i. [E.sub.0] is a conditional expectation operator where expectations are formed at time zero. The firm produces a nonstorable output using a technology that is a quadratic form. Payments to labor are given by the product of the real wage and the flow of labor services used in production. The last term in the cash flow expression measures the training costs attached to changes in the level of labor services. Because the parameters [[alpha].sub.0] and [[alpha].sub.1] are assumed positive, the firm has diminishing returns in production associated with its use of labor services and costs of adjustment, or the training costs of labor, rise at the margin. Because of these adjustment costs, the firm will have an incentive to gradually adjust its labor usage in response to shifts in the determinants of labor use in production (see later discussion). Regarding the technology shock, [[zeta].sup.d], it will be assumed that [[zeta].sup.d] ~ I(1). Further discussion about the stochastic process obeyed by this shock to the production function will be provided later.

The optimal choice of labor services in production leads to the Euler equation,

(2) [gamma][[alpha].sub.1][E.sub.t][n.sub.t+1] - [[alpha].sub.0] + [gamma](1 + [[alpha].sub.1])][n.sub.t] + [[alpha].sub.1][n.sub.t-1]

= [w.sub.t] - [[zeta].sup.d.sub.t].

The firm is assumed to have contemporaneous and lagged information at its disposal when choosing the optimal level of labor services at each instant of time. Thus the expectation operator is applied only to [n.sub.t+1]. Using standard methods, it may be shown that (2) implies that the expected discounted marginal product of labor will be set equal to the marginal cost of labor, where the latter includes the wage rate and the marginal cost of training an additional unit of labor.

The Behavior of the Household

Household behavior is very simple. The optimization problem to be solved is

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where c refers to consumption, n is labor supply, [[zeta].sup.s] is a preference shock, and [[mu].sub.i] > 0[for all]i. The relationship between the household's sources and uses of income is given by [c.sub.t] = [w.sub.t][n.sub.t] + [D.sub.t] where [D.sub.t] refers to an exogenous dividend stream (saving is assumed fixed and is thus ignored). Eliminating consumption from (3) by replacing it with the income of the household from its labor supply and dividends leads to the Euler equation,

(4) [[mu].sub.0][w.sub.t] - [[zeta].sup.s.sub.t] - [[mu].sub.1][n.sub.t] = 0.

Because there are no lags appearing in the household's objective function, this optimality condition is simply a static one. This condition merely implies that the household sets the marginal rate of substitution between consumption and labor supply equal to the real wage rate.

The Labor Market VAR

The labor market vector autoregression may be derived by using the Euler equations for the firm and household. Use (4) to eliminate the wage rate from (2). Suppressing the expectation operator for convenience, the resulting expression is

[n.sub.t+1] - ([[mu].sub.0][[alpha].sub.1](1 + [gamma]) + [[alpha].sub.0]] - [[mu].sub.1]/[gamma][[alpha].sub.1][[mu].sub.0])[n.sub.t]

+ [[gamma].sup.-1][n.sub.t-1] = ([[mu].sup.-1.sub.0][[zeta].sup.s.sub.t] - [[zeta].sup.d.sub.t])/[gamma][[alpha].sub.1],

This equation may be factored to give

(1 - [[lambda].sub.1]L)(1 - [[lambda].sub.2]L)[n.sub.t+1] = ([[mu].sup.-1.sub.0][[zeta].sup.s.sub.t] - [[zeta].sup.d.sub.t])/[gamma][[alpha].sub.1],

where [[lambda].sub.1] and [[lambda].sub.2] are eigenvalues (0 < [[lambda].sub.1] < 1, [[lambda].sub.2] > 0), and L is the lag operator. Reintroducing the expectation operator and solving forward with the unstable root, [[lambda].sub.2], gives

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Expression (5) displays the well-known property that quasi-fixed state variables are determined by all future sequences of exogenous variables, which in this model are the shocks to household preferences and the technology of the firm. The parameter [[lambda].sub.1] gives the speed of adjustment to equilibrium for labor services. This parameter is between zero and unity because there are costs of adjustment attached to labor which rise at the margin (which follows if [[alpha].sub.1] > 0).

The optimal investment demand schedule in (5) contains the expected levels of the two exogenous variables in the model. To derive the labor market VAR, a decision must be made about the manner in which agents forecast these discounted series of shocks in the expression. Agents will be assumed to forecast these shocks using the Weiner-Kolmogorov prediction formula (Sargent 1978, p. 304), and for the preference shock [[zeta].sup.s.], this is

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [theta](L) is a polynomial in the lag operator appearing in the stochastic process for [[zeta].sup.s]. To complete the derivation of the labor market VAR, assumptions must be made about the time-series processes obeyed by the shocks in (5). For the shock to household preferences, [[zeta].sup.s], it will be assumed that [[zeta].sup.s] ~ iid(0, [[sigma].sup.2.sub.s]). Applying (6) to the discounted infinite series for this shock causes the discounted infinite series to reduce to [[zeta].sup.s.sub.t]. For the shock to the firm's production function, [[zeta].sup.d.sub.t], it will be assumed that this is an observable driftless random walk (measured by the stock of patents). (10)

Applying the Weiner-Kolmogorov prediction formula yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using the stochastic process for each shock and (6), I obtain

(7) [n.sub.t] = [[lambda].sub.1][n.sub.t-1] + [[[lambda].sub.1]/[[alpha].sub.1](1 - [gamma][[lambda].sub.1])][[zeta].sup.d.sub.t] - [[lambda].sub.1]/[[alpha].sub.1][[mu].sub.0]][[zeta].sup.s.sub.t],]

an expression that is one part of the labor market VAR.

The next step in constructing the VAR for the labor market is to derive an expression for real wages. Using the Euler equations for the firm and household and a procedure similar to the derivation of(5), I can obtain the following expression for the real wage:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using this result and (6) yields

(8) [w.sub.t] = -[[alpha].sub.0] + [[alpha].sub.1] (1 + [gamma][1 - [[lambda].sub.1]])][n.sub.t] + [[alpha].sub.1][n.sub.t-1] + [(1 - [gamma][[lambda].sub.1].sup.-1][[zeta].sup.d.sub.t] - [gamma][[lambda].sub.1][[mu].sup.-1.sub.0][[zeta].sup.s.sub.t].

Finally, using (7)-(8) and the stochastic process for the technology shock, I obtain the VAR for the labor market which can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[kappa].sub.1t] = -[[lambda].sub.1][([[alpha].sub.1] [[mu].sub.0]).sup.-1], [[kappa].sub.2t] = - [gamma][[lambda].sub.1 [([[alpha].sub.1][[mu].sub.0]).sup.-1][[zeta].sup.s.sub.t], and [[kappa].sub.3t] is assumed to be a serially uncorrelated, iid disturbance from the stochastic process obeyed by [[zeta].sup.d.sub.t]. It is further assumed that [[zeta].sup.s.sub.t] and [[kappa].sub.3t] are mutually uncorrelated. Using the definitions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [[??].sub.it] = [[GAMMA].sup.-1][[kappa].sub.it], this VAR may be rewritten in its error-correction form as

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[OMEGA].sub. = [[lambda].sub.1][[alpha].sub.0] - [[alpha].sub.1] (1 - [gamma][[lambda]s.ub.1])(1 - [[lambda].sub.1]), and [[OMEGA].sub.2] = 1/[[alpha].sub.1](1 - [gamma][[lambda].sub.1]) > 0. It is clear that the matrix of coefficients attached to the lagged levels of the series in the trivariate VAR has reduced rank. (11) The time-series system is therefore cointegrated; as a result, the matrix of coefficients preceding the lagged levels of time series can be written as [PI] = [alpha][beta]' where [alpha] is known as the adjustment matrix and [beta] is the cointegrating matrix.

The cointegrating matrix may be normalized by setting some of its elements equal to unity, and this normalization causes the elements of the adjustment matrix to have useful economic content (i.e., the elements of the adjustment matrix contain parameters of economic interest as a consequence of this normalizing transformation of the cointegrating matrix). In optimizing models with one state variable, such a normalization is arbitrary on statistical grounds, but on economic grounds it only makes sense to attach a unit coefficient to the state variable. Here, normalization is arbitrary on economic and statistical grounds. The choice of normalizing transformation is a matter of convenience, and it is easily shown that normalization has no impact on the information revealed by estimates of the cointegrating vectors. (12)

It is convenient to set the coefficients attached to labor services to unity. Doing so gives the following adjustment and cointegrating matrices:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Having obtained these matrices, one could proceed to estimate their elements using any of the available estimation methods. Before doing so, it is instructive to consider the determinants of the lag length in (9).

This lag length is just one with no lagged difference terms appearing in (9), and this single lag is a consequence of assuming, in the typical manner, that there are costs of adjustment attached to labor services. (13) Estimation of this VAR with data at monthly frequency, as will be done shortly, will almost certainly yield lag lengths exceeding one with statistically significant lagged difference terms. The question then arises as to how one might rationalize the discrepancy between the lag length implied by the model and the lag length uncovered empirically when this VAR is estimated. In the context of this model, two possibilities can be given to explain these empirical lag lengths.

One is that there may be lagged difference terms in the time-series process obeyed by the technology shock experienced by the firm. Univariate methods are used below to test this series for unit roots, and it was found that the lag length for this series was greater than one. A second possibility is that the unobservable shock to household preferences may obey a time-series process containing autoregressive terms, thereby displaying serial persistence. For example, suppose that the household preference shock obeys the stochastic process [phi](L)[[zeta].sup.s.sub.t] = [[kappa].sub.4t] where [phi](L) is a finite-order polynomial in the lag operator with all of its roots outside the unit circle and [[kappa].sub.4t] is an lid error term with mean zero and constant variance. Then it may be shown that the adjustment matrix will be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

illustrating the fact that the adjustment matrix now contains both structural parameters and parameters from the autoregressive polynomial attached to the shock to household preferences. Thus estimation of the elements of the adjustment matrix does not directly yield estimates of structural parameters. But the cointegrating matrix does not contain the parameters in [phi](L), so this matrix can be estimated even when long lags are found in the estimated VAR, thereby providing estimates of structural parameters of interest. (14) In the context of the model, the only way to explain long lags for real wages and labor services within the estimated VAR is that preference shocks are serially correlated and lags beyond one are apparent in the estimated VAR (see later discussion). As a result, estimates of the elements of the adjustment matrix will not be given because these estimates would not deliver information on structural parameters of economic interest.

Finally, conditional on an assumed value for the discount rate, three structural parameters ([[lambda].sub.1], [[alpha].sub.0], [[alpha].sub.1]) are exactly identified from estimation of the elements of the cointegrating matrix so that an estimate of this parameter matrix yields estimates of these underlying structural parameters from the demand side of the labor market. The parameter [[alpha].sub.0] determines if there are diminishing returns in production associated with the use of labor. This parameter is important in part for its role in determining the sufficiency of intertemporal optimality criteria because sufficiency depends on the curvature of the technology and other functions within the economic model. The parameter [[alpha].sub.1] determines if there is partial adjustment of labor services in production for adjustment costs must rise at the margin to provide an incentive for the firm to adjust its labor inputs slowly over time. If there are adjustment costs rising at the margin, then [[lambda].sub.1] must lie between zero and unity because partial adjustment of the labor force is optimal for the firm when adjustment costs rise at the margin. Thus, if the model is a good description of the labor market, the estimated values of these latter two parameters should be consistent; a finding of a positive estimated value for [[alpha].sub.1] should be associated with a value of [[lambda].sub.1], reflecting partial adjustment of the labor force in each period.

III. TESTING THE IMPLICATIONS OF THE LABOR MARKET MODEL

Data and Tests for Unit Roots

The data used to test the labor market model were obtained from two U.S. government agencies, and these data are publicly available. Labor services are measured by man-hours of production workers in the private sector. Data on average weekly hours and the number of production workers were obtained from the Bureau of Labor Statistics. The same agency provided the data on average hourly earnings of production workers and the Consumer Price Index for all urban workers, series used to construct a measure of the real wage rate. Data on utility patents issued was provided by the U.S. Patent and Trademark Office. The data are unadjusted for seasonality, they are measured in natural logs at monthly frequency, and they cover the period 1964:1 to 1998:12.

The stock of knowledge will be measured by the cumulative sum of patents granted by the U.S. Patent and Trademark Office. (15)

If knowledge does not depreciate and if the initial stock of patents is zero, then knowledge in any time period is given by

k(t) = [t.summation over([tau]=1)]p([tau]),

where k denotes the stock of knowledge and p prefers to the flow of patents. The empirical work to follow uses the stock of knowledge in the estimated VAR but the analysis of this section examines the time series characteristics of the flow of patents. The reason for doing so is that there is reason to believe (see note 18) that there has been a structural change in the environment in which patents are granted. This structural instability is most easily seen in the patent flow series (see Figure 1) and is obscured in the stock series because the cumulative sum of the patent flow series yields a stock series that is very smooth. Because a unit root cannot be rejected in either the stock or flow series using standard Dickey-Fuller tests (more will be said later on unit root testing of the stock data), it is useful to use the flow series in this context because it permits us to address the possible effects of structural change on tests for unit roots.

The utility patent data measures the monthly flow of patents issued in the United States; it includes patents issued to individuals and corporations, both foreign and domestic. Patents must be filed in whatever country individuals or firms wish to receive patent protection. The industrial structure in the United States can be affected by new ideas developed domestically or in foreign countries, so it does not seem reasonable to distinguish between foreign and domestic patents. (16)This monthly patent data (not measured in logs) has a sample mean of about 6245, with a standard error of approximately 1735. Figure 1 contains a plot of this patent data in natural logs over the estimation period. It seems clear on the basis of this graph that the series does not have a constant mean; it appears that the last decade has seen patents granted per year at an increasing rate as compared to earlier years in the sample Indeed, an unprecedented increase in the rate at which patents are issued has been documented elsewhere (see Kortum and Lerner 1998). (17)

Test results, indicating that real wages and labor services are I(1), may be found elsewhere. (18) The patent data must be tested for nonstationarity. The presence of a unit root in this patent data is the motivating force that explains the presence of unit roots in real wages and labor services. Table 1 provides a variety of diagnostic statistics that reveal information on the stationarity of the data. The table contains sample autocorrelations and partial autocorrelations, denoted respectively by [[rho].sub.l] and [r.sub.l] where l denotes lag length. The table contains Bartlett (1946) standard errors for the sample autocorrelations, derived from

var([[rho].sub.l]) = [T.sup.-1] [1+2[summation over(j<l)][[rho].sup.2.sub.j],

where the sample size is denoted by T(T = 420). The standard error for the partial autocorrelations is [T.sup.-1/2]. Ljung and Box (1978) Q-statistics are also provided. Autocorrelations and partial autocorrelations are contained in Table 1 for the patent data in levels and for first and second differences of the data.

These statistics suggest that the levels data are nonstationary because the autocorrelations are more than two standard errors away from zero, even at 12 lags. This is also true when the data are differenced once and even twice. The Q-statistics decisively reject the null hypothesis that all 12 sample autocorrelations are zero. Similarly, the partial autocorrelations are more than two standard errors away from zero at all lags, irrespective of the degree of differencing. Similar results can be observed at lag lengths considerably beyond 12. Thus I would conclude from these diagnostic statistics that the patent data are nonstationary, containing at least one unit root and possibly more.

Unit root tests must be applied to the patent data. In view of the results from Table 1, it is reasonable to test for more than one unit root in the data, especially because the methods in Johansen (1991) will be used later, and these procedures are inappropriate to data that are I(2). Consequently, the test of Dickey and Pantula (1987) will be used. Their test requires estimating

(11) (1 - L)[y.sub.t] = [[theta].sub.0] + [[theta].sub.1][y.sub.t-1] + [l.summation over (i=1)] [[zeta].sub.i] (1 - L)[y.sub.t-i] + [[phi].sub.t]

Assuming a maximum of three unit roots in the patent series, the procedure requires testing sequentially for fewer unit roots until we can no longer reject the null hypothesis [H.sub.0]:[[theta].sub.1] = 0. Begin by defining [y.sub.t] to be the second difference of the patent series and estimate (11), thereby testing for a third unit root. If the null hypothesis is rejected using the t-statistic on [[theta].sub.1] and the critical values for the [[tau].sub.[micro]] statistic in Fuller (1976, p. 371), then repeat the test with [y.sub.t] as the first difference of the patent series and so on. Table 1 reports the results of these tests. The lag length l must be chosen and it was selected by using the BIC criterion of Schwarz (1978). (19) This criterion selects l = 9. At the 1% level, the test results indicate that three and two unit roots are rejected by the data but one unit root cannot be rejected, even at the 10% level. Thus I conclude, subject to one qualification, that the patent data are I(1). (20)

That qualification concerns the possibility of structural breaks in the patent data. There has been a considerable amount of research effort devoted to the issue of tests for unit roots in the presence of structural breaks. Perron (1989) argued that time series may be I(0), having structural breaks either in their levels or growth rates, and that time series may appear to be I(1), even if they are in fact I(0), when using conventional Dickey-Fuller tests if these structural breaks are ignored. Perron (1989) provides evidence that macroeconomic time series appear to have these structural breaks and may be stationary, given these exogenous breaks. His work uses exogenous breakpoints, and there have been a number of studies developing tests for unit roots when breakpoints are unknown, such as the test of Zivot and Andrews (1992). It is reasonable to ask if these tests should be applied to the patent data.

The analysis of section II treats technology shocks in the manner of much of the economic growth literature, namely, that technical progress is of the disembodied, exogenous type, unrelated to economic events. Clearly this is not the case. New products are frequently developed in response to movements in relative prices or perceived profit opportunities. For example, it is plausible to argue that energy-efficient products were developed in response to the increase in the relative price of oil in October 1973. Research and development activities leading to those new products may have generated new patents. Thus the formation of the OPEC cartel could be viewed as a structural break in the patent series. In addition, Kortum and Lerner (1998) document institutional changes in the legal system that may have increased the rate at which patents are issued. For these reasons, it seems a good idea to apply unit root tests that permit the possibility of a structural break in the patent data to determine if the series is I(1) even if allowance is made for a structural break in the series. The test of Zivot and Andrews (1992) (hereafter Z-A) will be used for this purpose.

Structural breaks may occur in either the levels or growth rates of a time series. Inspection of Figure 1 does not suggest any distinct changes in the level of the patent series, but there seems to be the possibility of a growth rate change in the neighborhood of 1984. Therefore I use the version of the Z-A test that allows for a structural break in the growth rate of the series that is being tested for unit roots. To carry out this test, it is necessary to estimate

(12) [y.sub.t] = [[theta].sub.0] + [[theta].sub.1]t + [[theta].sub.2][DT.sub.t]([omega]) + [[theta].sub.3][y.sub.t-1] + [l.summation over (i=1)] [[zeta].sub.i](1 - L)[y.sub.t-i] + [[phi].sub.t]

The circumflex indicates that a magnitude corresponds to the estimated break fraction [omega]([omega] = [T.sub.B]|T where [T.sub.B] refers to the breakpoint in the data. In (12), if [DT.sub.t]([omega]) = t - T[omega] if t > T[omega], 0 otherwise. The testing strategy proposed by Z-A involves using a conventional t-statistic to test [H.sub.0]:[[theta].sub.3] = 1, choosing the breakpoint that yields a value of the test statistic that is least favorable to the null hypothesis. At each value of the break fraction [omega], the lag length l must be chosen. Here I follow the model specification procedure used by Perron (1989) and Z-A. Beginning with the maximum lag [l.sub.max] = 36 and working backward, choose the first value of l such that the t-statistic on [[zeta].sub.1] is greater than or equal to 1.6 and where the t-statistic on [[zeta].sub.m] for m < l was less than 1.6. (21) Table 1 reports the statistic required for this test. The breakpoint, chosen by the selection procedure, is 1982:6, the year in which legislation was passed that is thought to have had a substantial impact on the patent approval process. (22) Using critical values provided by Z-A (p. 257), a unit root cannot be rejected at the 10% significance level. Thus the patent series is I(1), whether or not I allow for the possibility of structural change. Tests for cointegration may now be conducted.

Cointegration Tests and Estimated Cointegrating Vectors

The maximal eigenvalue statistic of Johansen (1991) is used to test for cointegration. (23) Critical values for the test are taken from Hansen and Juselius (1995). The test statistic is adjusted by the small-sample correction factor found in Boswijk and Franses (1992, 372). (24) Centered seasonal dummies were included in the estimated VAR because none of the time series in the system have been adjusted for seasonality (real wages, for example, clearly contain seasonal fluctuations due to the evident seasonal component in the Consumer Price Index). Intercepts are included in each cointegrating vector, but for the sake of brevity, estimates of these parameters are not provided. (25) The lag length of the VAR must be determined prior to carrying out the tests, and using the BIC criterion, this lag length was determined to be l=4. This lag length is used in all estimated equations to follow.

The full-sample estimates suggest that it may be informative to examine subsample parameter estimates. The subsamples contained in Table 2 were chosen because they have been repeatedly used in studies concerned about the effects of structural change on parameter estimation methods or hypothesis tests. (26) The first subsample concerns the formation of the OPEC cartel in October 1973, raising the relative price of energy, and it is one event that has been widely used in studies on the effects of structural change. The second subsample is related to the October 1979 change in the manner of conducting monetary policy, an example of a monetary policy regime change. At this time, the Federal Reserve switched from pegging the Federal Funds interest rate to a policy of reserve targeting, resulting in more variability in interest rates. Both of these break points will be used in examining the issue of rank instability in the cointegrating matrix and parameter instability in the estimated cointegrating vectors. Table 2 reports the results from the cointegration tests over the entire sample and for these two selected breakpoints in the data.

Let r denote the number of cointegrating vectors. The maximal eigenvalue test is a test of [H.sub.0]: r = [r.sub.0] against [H.sub.A]: r = [r.sub.0] + 1 and I begin with [r.sub.0] = 0, testing sequentially until the null hypothesis cannot be rejected. For the entire data sample, the null that r = 0 can be rejected at the 90%, significance level. One cointegrating vector is also rejected by the test statistic at the same significance level. As for the presence of two cointegrating vectors, the null cannot be rejected at the 90% level in the entire sample. Thus it seems to be the case that the data provide fairly strong support for the idea that endogenous labor market variables are cointegrated with unit root technology shocks, measured here by the stock of patents. This aspect of the evidence provides support for this model of the labor market.

Estimates of the elements of the two cointegrating vectors evident in the data are given in Table 2 using the maximum likelihood estimator provided in Johansen (1991). As stated earlier, normalization of the cointegrating vectors is arbitrary in the context of a model of the labor market, but this choice has no impact on the information about structural parameters that can be obtained from estimated elements of the cointegrating vectors. In what follows, parameters associated with labor services will be normalized to unity. For convenience, the cointegrating matrix from (10) is restated here, conditional on this normalizing transformation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the above matrix, [[lambda].sub.1] is the parameter that may be used to infer the speed at which labor services approach their equilibrium level, [[omega].sub.1] = [[lambda].sub.1][[alpha].sub.0] - [[alpha].sub.1] (1 - [gamma][[lambda].sub.1]) (1 - [[lambda].sub.1]), and [[omega].sub.2] = 1/[[alpha].sub.1](1 - [gamma][[lambda].sub.1]) > 0. Inspection of these results shows that once the elements of the cointegrating vectors are estimated, it is possible to uniquely recover estimates of the structural parameters ([[lambda].sub.1], [[alpha].sub.0], [[alpha].sub.]), given a value for the discount rate [gamma]. It will be assumed that the discount rate is 2% per year, converted to a monthly value. The precise value for the discount rate can be varied with little effect.

Inspection of the results in Table 2 reveals that full sample estimates of the cointegrating vectors provide little support for this equilibrium model of the labor market. Among the parameters where qualitative predictions arise from the model, only the estimated value of [[alpha].sub.0] has the qualitative sign assumed in the labor market model (it implies that there are diminishing returns in production), but its magnitude is implausibly large. The estimated value of [[alpha].sub.1] implies that there are costs of adjustment that decline at the margin, a result implying that there is no well-defined finite investment demand schedule for labor. The estimated value of [[lambda].sub.1] is particularly poor, with a negative value that is smaller than -1. In the presence of adjustment costs, this parameter is expected to be positive and lie between zero and unity. Overall these parameter estimates provide little support for an equilibrium model of the labor market. But a difficulty with these results is that there are at least two structural regime changes in the sample, rendering hypothesis testing and parameter estimation problematic. For this reason, subsample estimates using these two breakpoints will now be examined.

Across subsamples of the data, there could be changes in the number of cointegrating vectors apparent in the data and/or there may be changes in parameter values contained within the cointegrating vectors. The Johansen (1991) test was applied to all of the subsamples with the appropriate correction factor applied to the test statistics. The results consistently show that there are two cointegrating vectors in all subsamples. Thus there is no evidence of changes in the number of cointegrating vectors in the VAR in any subsample, despite the presumed loss of power in the test due to declining sample size, which could conceivably alter the implications of the test statistic. Rank instability in the cointegrating matrix does not seem to be a major concern in these results.

However the subsample results do provide striking evidence of substantial changes in parameter estimates in various subsamples. Consider estimates of the parameter [[lambda].sub.1], the parameter that can be used to infer the speed of adjustment for labor. This parameter is between zero and unity in the model of the labor market, and this is found to be the case for the subsample preceding the formation of the OPEC cartel. This parameter estimate is particularly plausible, for it implies that labor is adjusted relatively rapidly to its equilibrium level, a result that is intuitively appealing because the marginal training costs of labor should be much lower than marginal installation costs attached to quasi-fixed inputs, such as capital goods. (27) But note how different the estimates of this parameter are within other subsamples. For example, the period subsequent to the formation of the OPEC cartel yields an estimate that is very strange. The estimated value is negative, contrary to the labor market model, and it is of enormous size and thus totally implausible. In other subsamples, we observe equally different results. Prior to the policy regime change by the Federal Reserve, the estimated value of this parameter implies very rapid adjustment of labor to its equilibrium level within the month. Adjustment is complete within the month. The data subsequent to the regime change by the Fed produces another parameter estimate of no intuitive appeal; the parameter estimate is far above unity, and it implies unstable behavior in the market for labor. Viewed from the perspective of this parameter estimate, an equilibrium model of the labor market provides a good description of the aggregate U.S. labor market only in one historical period, namely, that preceding the formation of OPEC.

Much the same story is observed in other parameter estimates. The parameters [[alpha].sub.0] and [[alpha].sub.1] are structural parameters from the optimization problem solved by the firm. These parameters are assumed positive in the model so that the firm's technology displays diminishing returns to labor services and to ensure that there are costs of adjustment attached to labor that rise at the margin. Comparing subsample results, estimated values of these parameters often vary enormously. Estimates of [[alpha].sub.0] are consistent in the sense that in three of four subsamples the estimated value for this parameter is qualitatively consistent with the assumption of diminishing returns in production. But the subsample results reveal enormous fluctuations in the estimated magnitude of this parameter. Over the entire sample and in the subsample after the formation of OPEC, estimates of this parameter are implausibly large. Regarding estimates of [[alpha].sub.1], we observe estimates that are often qualitatively different from what is assumed in the labor model (note the subsamples before and after the period of the Federal Reserve policy regime change) or they have estimated values of enormous size (see the period prior to the Fed policy change for the most extreme example) and thus offer little economic appeal. But prior to the formation of the OPEC cartel, the estimated values of these parameters are qualitatively consistent with the labor market model. Strictly speaking, it is only prior to the OPEC shock that this model is valid because it is only in this period when the estimated value of [[alpha].sub.1] is consistent with adjustment costs rising at the margin. Elsewhere, training costs are falling at the margin so that labor should not be regarded as quasi-fixed. Rather, it should be assumed to be a variable factor input in production. Overall, these parameter estimates provide little support for an equilibrium view of the aggregate labor market, but once again, that conclusion seems to be conditional on the choice of the data sample.

There is also the possibility of model misspecification here. As already stated, the labor market model is constructed so that the dynamics in the labor market come entirely from labor demand. This occurs because the labor supply optimization model had no lags appearing in the objective function used by the household. That simplification does prevent supply-side parameters, contained in the household's objective function, from appearing in the estimated VAR, and this omission may have a role to play in the empirical performance of this labor market model.

As already stated, the period prior to the formation of OPEC is the only subsample where the model is found to be valid because it is the only in this sample data where we find that there are diminishing returns in production ([[alpha].sub.0] > 0), costs of adjustment rise at the margin ([[alpha].sub.1] > 0), and where the speed of adjustment for labor is empirically found within the bounds implied by the model (0 < [[lambda].sub.1] < 1). It is interesting to note that the adjustment cost literature was developed in the 1960s as an attempt to explain the serial persistence in investment data, and the evidence in this article indicates that this is indeed a plausible way to explain data up through the early 1970s. It is also the subsample closest to the time period originally used by Lucas and Rapping (1969), who first advocated a market-clearing model of the labor market. Furthermore, this subsample data excludes the period of the 1980s, a period when the personal computer was developed and when there were many reorganizations of industries due to mergers and acquisitions. (28) It also excludes the acceleration in technical progress in the 1990s as documented by Basu et al. (2001).

This subsample also excludes any innovations in technology that arose in response to the increase in the relative price of oil. One way to interpret the subsample findings is that the patent data ceases to be an adequate measure of technical progress at some point in the data sample, possibly because it is a poor measure of technical progress in relation to the OPEC cartel and the industrial reorganizations of the 1980s.

To summarize, the empirical results provide support for the idea that I(1) technology shocks are at least one reason why economic time series are also I(1) and that these technology shocks are cointegrated with choice variables set by economic agents. But it also evident that there is substantial parameter variability in the data, which may be due to other forms of structural change that are not adequately captured by increments to the stock of knowledge.

IV. CONCLUDING REMARKS

New classical models of the economy are frequently used in macroeconomic research, and a component of these models is a labor market where money wages are flexible. There is little evidence testing this model of the labor market, and one purpose of this article is to provide some empirical evidence on the suitability of this model as a description of the aggregate U.S. labor market.

In doing this test, one issue that must be confronted is the nonstationary nature of labor market time series, and an explanation must be provided for this nonstationarity. Technology shocks are often a component of macroeconomic models, and these shocks are frequently assumed to be unobservable I(1) stochastic processes. Although this assumption does explain why other economic magnitudes are I(1), a consequence is that it is impossible to test economic hypotheses that could be tested if shocks to the production function could be measured. However one possible interpretation of these shocks is that they are measured by the stock of knowledge available to the public, so that increases in knowledge enable the economy to produce more and/or new goods from a given resource base. Under this interpretation, there is a measure of new knowledge that can be used for testing economic hypotheses.

In this article, increments to the stock of knowledge are assumed to be measurable by the number of utility patents granted by the U.S. government. Patents are a reasonable measure of innovations that lead to new products in the market and, if patents are I(1), then economic theory implies that they will be cointegrated with choice variables set by firms and households. Within an equilibrium model of a labor market, this exogenous nonstationary process will be the cause of the apparent unit roots in labor market series, such as labor services and real wages. It is shown that an error-correction VAR for labor services, real wages, and the technology shock will arise in this market, and there will be two cointegrating vectors within the cointegrating matrix describing the long-run relationship between the endogenous labor market series (labor services and real wages) and the exogenous I(1) process measured by the stock of utility patents. This implication is tested using aggregate U.S. labor market series. Unit root tests are applied to patent data and the series is found to be I(1), whether or not an endogenous structural break is allowed in the patent series. There is a substantial body of evidence provided that there are two cointegrating vectors in a trivariate time-series system containing man-hours, real wages, and patents issued.

Estimation of the cointegrating vectors permits estimation of structural parameters arising from the demand side of the labor market, given an assumed value for the real interest rate. One of these parameters determines whether there are training costs attached to labor services and how severe these costs are to the firm. Parameter estimates provide little support for this model of the labor market over the entire data sample, but there are at least two regime changes in the sample data (the formation of the OPEC cartel and the switch in operating procedures by the Federal Reserve) that may account for the poor performance of the model. Subsample estimates using these structural breakpoints are examined. The number of cointegrating vectors apparent in the data is found to be quite robust to the data sample chosen, but substantial differences in estimated structural parameters are observed in different subsamples. One subsample (prior to October 1973) provides strong empirical support for the market-clearing model.

The performance of the model deteriorates after the early 1970s. Thus it may be useful to consider extensions to the analysis that might improve the performance of the model. One possible extension would be to assume that producers use intermediate materials, such as energy, in the production of output, thereby incorporating energy prices into the VAR arising in the labor market. This extension would have the advantage of explicitly incorporating the relative price of energy, measuring the effects of the formation of the OPEC cartel, directly into the labor market model. A second possible extension would be to permit labor market dynamics to emerge from the labor supply decision by allowing lagged labor supply to enter into the utility maximization of the household. Although these extensions may be attractive on theoretical grounds, they will also introduce additional structural parameters into the cointegrating matrix, thus complicating the identification of structural parameters from estimation of the elements of the cointegrating vectors. These extensions are left for future investigation.

ABBREVIATIONS

VAR: Vector Autoregression

OPEC: Organization of Petroleum Exporting Countries

TABLE 1
Diagnostic Statistics for Utility Patent Data

Autocorrelations

 [[rho].sub.1] [[rho].sub.2] [[rho].sub.3]

d = 0 0.698 0.676 0.722
SE 0.049 0.069 0.083
d = 1 -0.491 -0.100 0.318
SE 0.049 0.060 0.060
d = 2 -0.628 -0.127 0.346
SE 0.049 0.065 0.065

Partial autocorrelations

 [r.sub.1] [r.sub.2] [r.sub.3]

d = 0 0.698 0.369 0.376
d = 1 -0.491 -0.449 0.045
d = 2 -0.628 -0.673 -0.273
SE 0.049 0.049 0.049

Tests for unit roots without structural breaks

 [H.sub.0] d = 3 -13.369 *

Test for a unit root with a structural break

Autocorrelations

 [[rho].sub.4] [[rho].sub.5] [[rho].sub.6] [[rho].sub.7]

d = 0 0.574 0.602 0.611 0.505
SE 0.097 0.105 0.113 0.120
d = 1 -0.292 0.038 0.177 -0.258
SE 0.064 0.067 0.067 0.068
d = 2 -0.315 0.061 0.195 -0.252
SE 0.070 0.073 0.073 0.074

Partial autocorrelations

 [r.sub.4] [r.sub.5] [r.sub.6] [r.sub.7]

d = 0 -0.114 0.113 0.105 -0.061
d = 1 -0.152 -0.152 0.006 -0.146
d = 2 -0.229 -0.308 -0.111 -0.072
SE 0.049 0.049 0.049 0.049

Tests for unit roots without structural breaks

 [H.sub.0]: d = 2 -8.881 *

Test for a unit root with a structural break

 [H.sub.0]: d = 1 -2.509

Autocorrelations

 [[rho].sub.8] [[rho].sub.9] [[rho].sub.10]

d = 0 0.556 0.579 0.442
SE 0.125 0.131 0.137
d = 1 0.061 0.260 -0.350
SE 0.070 0.071 0.073
d = 2 0.037 0.274 -0.364
SE 0.076 0.076 0.079

Partial autocorrelations

 [r.sub.8] [r.sub.9] [r.sub.10]

d = 0 0.097 0.155 -0.174
d = 1 -0.204 0.145 -0.087
d = 2 -0.349 -0.050 -0.021
SE 0.049 0.049 0.049

Tests for unit roots without structural breaks

 [H.sub.0]: d = 1

Test for a unit root with a structural break

Autocorrelations

 [[rho].sub.11] [[rho].sub.12] Q(12)

d = 0 0.519 0.527 1784.3
SE 0.140 0.145
d = 1 0.123 0.198 334.44
SE 0.077 0.077
d = 2 0.132 0.209 420.08
SE 0.083 0.083

Partial autocorrelations

 [r.sub.11] [r.sub.12]

d = 0 0.087 0.107
d = 1 -0.128 0.083
d = 2 -0.197 0.005
SE 0.049 0.049

Tests for unit roots without structural breaks

 -1.628

Test for a unit root with a structural break

N.B. Standard errors are denoted by SE, d refer to the degree of
differencing, and Q(12) is the statistic of Ljung and Box (1978)
testing the null that all 12 autocorrelations are zero. Tests for
unit roots with no structural breaks are the test statistics proposed
by Dickey and Pantula (1987). The unit root test with a structural
break is that of Zivot and Andrews (1992). An asterisk (*) indicates
rejection of the null hypothes at the 1% level.

TABLE 2
Cointegration Tests and Estimated Cointegrating Vectors

 1964:1-1998:12 1964:1-1973:9

Cointegration test statistics
[H.sub.0]: r = 0 35.47 [H.sub.0]: r = 0 34.42
[H.sub.0]: r = 1 14.62 [H.sub.0]: r = 1 21.14
[H.sub.0]: r = 2 3.45 * [H.sub.0]: r = 2 6.82 *

Parameter estimates
[[OMEGA].sub.1] -0.474 [[OMEGA].sub.1] -0.122
[[OMEGA].sub.2] -0.642 [[OMEGA].sub.2] 1.032
[[alpha].sub.0] 41.840 [[alpha].sub.0] 0.052
[[alpha].sub.1] -1.547 [[alpha].sub.1] 0.968
[[lambda].sub.1] -4.570 [[lambda].sub.1] 0.066

 1973:10-1998:12 1964:1-1979:9

Cointegration test statistics
[H.sub.0]: r = 0 21.28 [H.sub.0]: r = 0 23.99
[H.sub.0]: r = 1 21.17 [H.sub.0]: r = 1 11.62
[H.sub.0]: r = 2 3.06 * [H.sub.0]: r = 2 3.57 *

Parameter estimates
[[OMEGA].sub.1] -1.236 [[OMEGA].sub.1] 1.922
[[OMEGA].sub.2] -0.543 [[OMEGA].sub.2] -0.228
[[alpha].sub.0] 2578.5 [[alpha].sub.0] 0.061
[[alpha].sub.1] -1.736 [[alpha].sub.1] -4.389
[[lambda].sub.1] -36.59 [[lambda].sub.1] -0.023

 1979:10-1998:12

Cointegration test statistics
[H.sub.0]: r = 0 25.46
[H.sub.0]: r = 1 10.29
[H.sub.0]: r = 2 2.29 *

Parameter estimates
[[OMEGA].sub.1] -1.684
[[OMEGA].sub.2] -0.514
[[alpha].sub.0] -0.746
[[alpha].sub.1] -1.951
[[lambda].sub.1] 1.630

The maximal eigenvalue test statistic for cointegration, as well as
the maximum likelihood estimator for cointegrating vectors, are
obtained from Johansen (1991). Test statistics are adjusted by the
correction factor in Boswijk and Franses (1992). An asterisk (*)
indicates that the null hypothesis cannot be rejected at the 90%
significance level.

(1.) See McCallum (I980, p. 720) for an explicit derivation showing that the famed Lucas supply function, a critical component of rational expectations models, can be derived from a labor market with flexible wages when labor suppliers operate under incomplete information about the price level.

(2.) Variants of this model may be found in Sargent (1978, pp. 472 78), Kennan (1988), and Rossana (1998). Many would be skeptical that market clearing occurs in the labor market, but there is some evidence showing that nominal wages are more flexible than might be commonly believed. See McLaughlin (1994) for evidence on money wage flexibility.

(3.) Kennan (1988) provides estimates of structural parameters arising from this type of labor market model but does not address the issue of unit roots in labor market data. The present study will provide estimates of long-run equilibrium relationships obeyed by real wages and employment using an explicit measure of technical progress as the source of the unit roots in the labor market.

(4.) A third possibility would be mergers and acquisitions of firms, activities that would cause permanent changes in the levels of factor inputs used in production.

(5.) If there are I(l) unobservable shocks in economic models, these shocks would appear in the disturbance vector of any VAR derived from an economic model containing these shocks (thus the covariance matrix for such a model would no longer be finite as the sample size tends to infinity). As a result, linear combinations of observable economic magnitudes would not be I(0). Rather they would be I(1). See Rossana (1995, p. 10) for more on this point.

(6.) See Perron (1989), Zivot and Andrews (1992), and other studies in the July 1992 issue of the Journal of Business & Economic Statistics for examples of research dealing with statistical inference in the presence of structural change.

(7.) See Griliches (1990) for further discussion of patent data and its uses. In addition, Gardner and Joutz (1996) use patent data in a model of economic growth.

(8.) Measurement error does not necessarily invalidate the asymptotic validity of the statistical methods used if the errors are uncorrelated with the time series used in the analysis.

(9.) The version of this model in Rossana (1998) was not empirically tested, and no discussion was provided of the connection between I(1) technology shocks and labor market time series. The discussion in this earlier study focused on the issue of normalizalion within the cointegrating matrix.

(10.) There is no loss of generality in assuming that the stochastic process for the technology shock is simply a driftless random walk because the focus of the analysis will be on long-run relationships between patents and labor market magnitudes. Let the technology shock obey a finite order ARI(p, 1) process [zeta](L)(1 - L)[[zeta].sup.d.sub.t] = [[kappa].sub.3t] where [zeta](L) has all of its roots outside the unit circle and [[kappa].sub.3t] is an iid disturbance. All of the parameters in the autoregressive polynomial [zeta](L) would appear in the error-correction VAR, but they would only appear in the lagged difference terms in that VAR. Such autoregressive parameters would not be present ill the parameter matrix attached to the lagged levels of the time series in the system, which is the parameter matrix of interest for investigating cointegrating relationships. Intuitively, the reason for this is that the autoregressive polynomial has an impact on the transitory dynamics of the time-series system. It does not influence the long-run relationships maintained by the series in the error-correction VAR. An explicit analysis, illustrating the fact that the cointegrating matrix is independent of the parameters in such an autoregrcssive polynomial, may be found in Rossana (1998, pp. 430 32).

(11.) It has rank two as long as [[lambda].sub.1] [not equal to], which is the case in this model.

(12.) In models with one state variable, economic theory indicates that state variables are functions of exogenous parameters in equilibrium. Thus it is quite natural to normalize the coefficient attached to the state variable to unity. In a market context, theory does not indicate how to normalize within the cointegrating matrix, but as long as a structural model is available to be used in interpreting parameter estimates, there is no danger that misleading inferences will be drawn from estimated cointegrating vectors. Rossana (1998) provides further discussion on these issues.

(13.) It is also possible that adjustment costs affect cash flow for more than one period, an assumption that would be plausible with high-frequency data. If this is true, then there would be additional lagged levels of labor appearing in the error-correction VAR.

(14.) Put differently, the transient effects of serial correlation in unobservable disturbances do not affect estimates of the cointegrating matrix because this matrix describes the long-run equilibrium of the market for labor. See Rossana (1998, pp. 433-34) for more on this point. It should also be noted that strictly speaking, the speed of adjustment is not a structural parameter because it is well known that this is an eigenvalue that is a function of exogenous variables in the model.

(15.) Gardner and Joutz (1996) use patent applications in their study of economic growth because data on patents granted were unavailable. Patents granted, rather than patent applications, would seem to be a better measure of increments to the stock of knowledge because there has been at least some attempt to screen patent applications for legitimacy by a disinterested agency.

(16.) In practice, the distinction between a domestic and foreign patent does not reveal the location where the research leading to the patent was actually done. The patent holder can generally assign the patent to a U.S. subsidiary or its foreign parent firm as it wishes.

(17.) The study by Kortum and Lerner(1998) attempts to determine why this structural change (or trend break) occurred. There is an interesting discussion contained there documenting institutional changes in the U.S. legal system that seem to account for the increase in the rate at which patents have been issued.

(18.) Rossana and Seater (1992) show that real wages cannot reject a unit root null hypothesis using standard Dickey-Fuller tests. Unit root tests were, in fact, used on the real wage and labor services series, and although not reported, these test results generally do not reject the unit root null hypothesis that they are I(1). Furthermore, the testing strategy of Dickey and Pantula (1987) was carried out and the results indicate that these series are not 1(2) or 1(3). Finally, neither series appears to contain a unit root at lag one and at the seasonal lag using the test statistics of Hasza and Fuller (1982).

(19.) Hall (1994) shows that the augmented Dickey-Fuller test retains its validity when data-based procedures, such as the BIC criterion, are used to choose the lag l, in (11). There can be a substantial gain in the power of this unit root test when the data are used to choose this lag length.

(20.) There is also no evidence of a unit root at the seasonal lag in this series using the test procedures of Hasza and Fuller (1982). Standard augmented Dickey-Fuller test results for the stock of patents are as follows: for d = 2, the test statistic is -6.532 and, for d = 1, the test statistic is -1.634 using lag lengths chosen by the Schwarz criterion. The null hypothesis of double unit roots is rejected, but a single unit root cannot be rejected.

(21.) This manner of choosing the lag length for computing the test statistic produces lag lengths that are much greater than those selected by use of the BIC criterion. Whereas the conclusions drawn from the Dickey and Pantula (1987) test are unaffected by the precise value of the lag length, the results from the Z-A test are much more sensitive to the precise value of the lag length chosen.

(22.) This is the year in which the Federal Courts Improvement Act was passed, consolidating all patent appeals cases into a newly created appeals court with jurisdiction over all such cases. This appellate court is believed to have caused an increase in the rate of patent approvals, and the formation of this court can be viewed as a structural change that streamlined the patent approval process. See Kortum and Lerner (1998) for further details.

(23.) Version 4.31 of RATS (Doan 1996) and the CATS procedure of Hansen and Juselius (1995) were used to carry out the computations in this paper. Johansen (1991) also provides a trace statistic for cointegration testing. This test statistic generates identical results regarding the number of cointegrating vectors in the data.

(24.) The Johansen (1991) statistic is adjusted by the correction factor (T- (n + 1)l)/T where n refers to the dimensionality of the time series system (n = 3). There is evidence in Gregory (1994) that this cointegration test lacks power in sample sizes ranging from T = 50 to T = 200, far below the sample size used in this article for the entire sample, where T = 420. Subsample sizes used below are closer in size to those in Gregory (1994).

(25.) It seems reasonable to include intercepts in virtually any cointegrated time series system because economic models would generally have constant terms in their long-run equilibrium relationships unless an explicit decision has been made to use functional forms that might rule them out. Although the elimination of constant terms may be convenient for theoretical work, there seems little justification for omitting intercepts from an applied point of view.

(26.) I also looked at subsample parameter estimates using the 1982:6 breakpoint chosen by the Z-A test for a unit root. The parameter estimates from this subsample are quite similar to the results using the October 1979 Fed breakpoint, so these estimates are not given for the sake of brevity.

(27.) The speed of adjustment of any state variable is defined in the adjustment cost literature to be the partial derivative of the change in the state variable with respect to its own level within an optimal investment demand relationship. Thus the adjustment speed can be measured by merely subtracting one from the estimated value of [[lambda].sub.1].

(28.) See Kaplan and Stein (1993) for some evidence on the acceleration of merger and acquisition activity in the 1980s.

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* My thanks to Jim Hirobayashi of the U.S. Patent and Trademark Office for providing the utility patent data used in this article and for helpful information about its characteristics. I am grateful to David A. Dickey, Adrian R. Fleissig, and Alastair R. Hall for comments and useful discussions. Two anonymous referees provided many valuable suggestions. The usual disclaimer applies regarding responsibility for errors and omissions.

Rossana: Professor of Economics, Department of Economics, 2074 F/AB, Wayne State University, Detroit, MI 48202. Phone 1-313-577-3760, Fax 1-313-577-9564, E-mail r.j.rossana@wayne.edu