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NBER WORKING PAPER SERIES ON THE COVAR lANCE STRUCTURE OF EARNINGS AND HOURS CHANGES John M. Abowd David Card Working Paper No. 1832 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 February 1986 We are grateful to Gary Chamberlain, Zvi Griliches, and Whitney Newey for comments and suggestions. An earlier version of this paper was circulated under the title "The Covariance Structure of Earnings and Hours Changes in Three Panel Data Sets." We thank the Department of Economics at the Massachusetts Institute of Technology for their hospitality while this paper was completed. The research reported here is part of the NBER's research program in Labor Studies. Any opinions expressed are those of the authors and not those of the National Bureau of Economic Research. ER brking Paper #1832 Fdruary 1986 On the Covariance Stricture of Earnings and HOUrS Changes ABSTRPCT This paper presents an empirical analysis of changes in individual earnings and hours ov&r time. Using longitudinal data from three panel surveys, we catalogue the main features of the covariance structure of changes in earnings and hours. We then present an interpretation of these features in terms of both a life—cycle labor supply model and a fixed—wage labor contract nDdel. Our major findings are: (1) there is a remarkable similarity in the covariance structure of earnings and hours changes across the three surveys; and (2) apart from simple measurement error, the major component of variance in earnings and hours affects earnings and hours equi—proportionately. John M. Abowd Department of Economics MIT Cambridge, MA 02139 (617) 253—1526 David Card Department of Economies Princeton University Princeton, NJ 08544 (609) 452—4045 —1— Recent empirical studies of longitudinal earnings and hours data focus on the contemporaneous correlation between hours of work and average hourly earnings.L" This focus arises naturally from the life— labor supply model, which explains movements in hours over time in cycle terms of changes in the value of work and unanticipated changes in wealth)' Nonetheless, there is widespread agreement that most of the observed variation in hours over time is not explained by contemporaneous movements in wages)' correlation On one hand, the cross—sectional between percentage changes in annual hours and percentage changes in average hourly earnings is apparently dominated by measure- ment error.' On the other hand, state—of—the—art estimates of the life—cycle labor supply model yield small and often statistically insignificant elasticities between hours variation and wage movements.-" In this paper, we present a more general analysis of the relation between movements in earnings and movements in hours over time.' Using longitudinal data from three panel surveys, we catalogue the main features of the covariance structure of earnings and hours. We then present an interpretation of these features in terms of both a life— cycle labor supply model and a model of fixed—wage labor contracts. Our major findings are: (1) there is remarkable similarity in the covariance structure of earnings and hours changes across the three data sources and (2) apart from simple measurement error, the major component of variance in earnings and hours affects earnings and hours equi— proportionately. In the first section of the paper, we describe the covariance —2— matrix of changes in annual earnings and annual hours of work for male household heads in three panel surveys: the Panel Study of Income Dynamics (PSID); the National Longitudinal Survey of Older Men (NLS); and the control group of the Seattle and Denver Income Maintenance Experiment (SINE/DIME). Using method—of—moments estimation techniques, we test for parsimonious representations of the data from all three sur- veys. We find that changes in earnings and changes in hours are uncorrelated at lags in excess of two years in the PSID and the NLS, and at lags in excess of one year in the SINE/DIME. We also find evidence of nonstationarity In the covariances of earnings and hours from all three data sources. In the second section of the paper, we present a life cycle labor supply model and derive its Implications for the data described in Section I. We also discuss the implications of a labor contracting model in which hours of work are varied by employers with no corresponding change in average hourly earnings. In the third section of the paper, we present estimation results for a simple two—factor model of earnings and hours generation. Th.e model for earnings and hours consists of a systematic component (reflecting individual productivity in the labor supply model or changes in labor demand in the alternative model) and an unsystematic or measurement error component. The estimation results reveal that both cnponents are important in the data, and that the systematic component generally influences earnings and hours equi—proportionare].y. This leads us to conclude that most observed changes In earnings and hours —3— that are not attributable to measurement error occur at fixed average hrly wage rates. I. Data Description The longitudinal earnings and hours data in this paper are drawn from the Panel Study of Income Dynamics, the National Longitudinal Survey of Older Men, and the nonexperimental families in the Seattle and Denver Income Maintenance Experiment. From the PSID, we have drawn 1448 male household heads whose records indicate nonzero earnings and hairs in each year from 1969 to 1979 (the of the survey). A brief summary contained of third through thirteenth waves the characteristics of the sample is in Table 1. We have included only those male household heads who were between 21 and 64 years of age in each of the sample years. Average annual hours (at all jobs) are more or less constant throughout the eleven—year period, while average hourly earnings (adjusted for grow erratically. Our data set includes the Survey of inflation) Economic Opportunity subsample of the PSID and consequently overrepre— sents nonwhite and eleven proportionately more low income and nonwhite households from sample. Fran 1318 low income households. The requirement of continuous years of earnings and hours data, on the other hand, eliminates the relatively the National Longitudinal Survey of Older Men we have drawn men who where less than 65 years old in 1975 and who reported non- zero earnings and hours in each of the survey years 1966, 1967, 1969, 1971, 1973 and l975.' Table 2 summarizes the characteristics NLS of the sample. Average annual hours for this older sample of men decline —4— throughout the nine—year sample period, particularly between 1973 and 1975. In interpreting the NLS data, it is important to keep in mind that the later waves of the survey were administered biennally. As a consequence, the changes in earnings and hours from 1969 to 1975 refer to changes in annual totals over two—year intervals. From the Seattle and Denver Income Maintenance Experiment we have drawn a sample of 560 male household heads who enrolled in the control group of the experiment.-" These male heads were between 21 and 64 years of age during the first four years of the experiment, and reported nonzero earnings and hours in each of the first eight experimental half— year periods. We excluded a small number of heads who reported more than 2500 hours of work in any one of the six month periods. The size of the SINE/DINE sample reflects the relatively small initial survey and the reduction in sample size associated with the requirement of eight periods of labor market data. The demographic characteristics of the SINE/DIME sample and time series information on earnings and hours are recorded in Table 3. Labor market information in the income experiment was collected every four months and aggregated into six—month data by the experiment's contrac- tors. Since enrollment dates differ by household, the experimental periods correspond to different periods of calendar tine for different households. The first experimentalperiod, for example, contains data from calendar periods between early 1971 and late 1972. Like the other two samples the SINE/DIME control group over repre- sents low—income and nonwhite households. Average hours of work — 5— (recorded at an annual rate in Table 3) were essentially constant throughout earnings the first six of the SIME/DB4E periods of the expe rinnnt. control Hours and real group fell significantly in the eighth and ninth experimental periods. These periods contain labor market information hours in from Late 1974 arid early 1975. changes in mean earnings and the SIME/DIME sample are therefore consistent with observed in the other two changes samples. The complete covariance matrices oF changes in the logarithms of annual earnings and annual hours for the three samples are recorded in Tables 4, 5, and 6. In order to partially control for differences in labor force experience in the three samples, we have computrl LiLen covariances using the residuals from unrestricted multivariate regressions of changes in earnings and hours on time period indicator variables and potential characteristics experience (age rnLrtus education minus 5). The of the data are not signt[icantly affected by removing the predicted effects of experience, since the explanatory power of the experience regressions is negligible in all these data sets. The covariances of the changes in earnings and hours with potential experience are recorded in the final rows of the Tables. To control for the fact that the SIME/DIME data are drawn from different calendar periods, depending on the date of assignment into the experiment, changes in earnings and hours from the SIME/DIME were regression adjusted using potential experience and a series of indicator variables for month—of—assignment into the experiment. In none of the sixteen regressions for the eight changes in earnings and hours were —6-- these indicator variables jointly statistically significant at conven- tional significance levels, however. Table 4 contains the covariances of changes in the logarithms of annual earnings and annual hours for our sample of household heads from the PSID. The upper left hand triangle of the table presents the covariances of changes in earnings and their associated standard errors. The lower right hand triangle of the table presents the covariances of changes in annual hours. The lower left hand block contains the cross— covariances between changes in earnings and hours. The cross—sectional variation in percentage changes in annual earnings and hours is large: the standard deviation of the change in the logarithm of earnings is at least 35 percent, while the standard deviation of the change in the logarithm of annual hours is at least 25 percent. The variances and covariances of changes in annual earnings and hours also vary over time. Cross—sectional dispersion in earnings and hours growth is relatively small in 1972—73 and 1973—74, and rela- tively large in 1975—76. In contrast to aggregate time series data, consecutive changes in individual earnings and hours are strongly nega- tively correlated. The first—order serial correlation coefficients of the changes in earnings and hours range from —.25 to — .45. The first— order serial cross—correlations of earnings and hours are also negative, although smaller in absolute value (between —.15 and —.25) than the corresponding autocorrelations. The contemporaneous covariances of changes in earnings and hours for household heads in the PSID are significantly positive, although too —7— small to generate a positive correlation between changes in hours and changes hourly in average hourly earnings. Since the logarithm of average earnings is the difference in the logarithms of annual earnings and annual hairs, the covariance between changes in annual hours and changes in average hourly earnings is just the difference between the covariance in hours. of changes in earnings and hours and the variance of changes For all ten changes in Table 4 this difference is negative and well determined. A final important feature of the covariance matrix in Table 4 is the absence of any large or statistically significant autocovariances at lags greater than two years. Year—to—year changes in earnings and hours in the p510 are apparently well—represented as a nonstationary bivariate second—order moving average process. Table 5 presents the covariance matrix of five experience—adjusted changes in earnings and hours from the NLS. Overall, the data are very similar to the corresponding data from the PSID, although there is more evidence of nonstationarity in the NLS data. On the second—order autocovariances the other hand, none of or cross—covariances in Table 5 are large or statistically significant, so that the NLS data may perhaps be adequately suinnarized as a nonstationary bivariate first—order moving average (MA(l)) process. If changes in annual earnings and hours between conseaitive years are represented by a second—order moving average process (NA(2)) then changes in hours and earnings at two—year gaps are represented by a first—order moving average.2' Thus the data in Tables 4 and 5 are potentially consistent with the se underlying model of earnings and hours generation. —8— Table 6 presents the covariances of experience—adjusted changes in earnings and hours between consecutive six—month intervals in the SINE/DIME survey. Again, these covariances are similar to the covari— ances of the PSID and NLS samples. There is less evidence of nonsta— tionarity in the SINE/DINE than in the other two surveys, although this may reflect the fact that changes in earnings and hours in the SINE/DIME survey are averaged over several different calendar periods. The first— order autocorrelations of earnings and hours are similar in all three surveys. The covariances between contemporaneous changes in earnings and hours, on the other hand, are relatively higher in the SINE/DINE. This implies that the simple regression coefficient of changes in the logarithxo of annual hours on changes in the logarithm of average hourly earnings is smaller in absolute value in the SINE/DINE (—.17) than in the other surveys (—.31 in the PSID and —.32 in the NLS). None of the third—order autocovariances or cross—covariances is statistically significant in Table 6, suggesting that semi—annual changes in log earnings and log hours in the SINE/DIME are close to a bivariate MA(2) process. Since a bivariate MA(2) representation of semi—annual changes in earnings and hours implies only a first—order moving average representation of annual changes, the SINE/DIME data exhibit lower order serial correlation than the PSID data. Tables 7, 8, and 9 present formal test statistics for several restrictive models of the covariance matrices of earnings and hours from the three samples. Details of the method—of—moments estimation frame— —9— work that we use to obtain these test statistics are presented in Appendix A. Table 7 presents the test results for the PSID sample. The first rw of the Table reports the goodness—of—fit statistic for a nonstationary bivariate MA(2) representation of the PSID data. This is a test that the third and higher—order autocovariances and cross— covariances of in Table 4-are jointly equal to zero. The probability value the test statistic is about six percent. Given the absence of any large or individually significant covariances of third order or higher, we conclude that a second order moving average provides an adequate representation of changes in earnings and hours in the PSID sample. The next three rows of Table 7 present goodness—of—fit statistics associated with further restrictions on the general MA(2) model. These tests are performed on the subset of autocovariances and cross— covariances up to second order, utilizing the sample variance matrix of the selected covariances..!Q! The first test is for a symmetric MA(2) model of earnings and hours. This model imposes symmetry on the block of cros s—covariances in the lower left—hand corner of Table 4. statistic has The test a marginal significance level of 2 percent, indicating some evidence against the symmetry hypothesis. The next test statistic reported in Table 7 is a test for an MA(l) representation of earnings and hours changes in the P519. Against the MA(2) alternative, this is a test that the second—order autocovariances and cross—covariances Table in 4 are jointly equal to zero. Again, the test statistic indicates some evidence against the null hypothesis. The final row of Table 7 presents the test statistic for the hypothesis that the covariance —10— structure of earnings and hours changes is time—stationary (i.e. , that all 10 variances of earnings are equal, that all 10 variances of hours are equal, etc.). This hypothesis is strongly rejected by the data. Tables 8 and 9 present an identical sequence of test statistics for the NLS and SIME/DIME samples. In both samples, the data are consistent with a bivariate MA(2) representation of earnings and hours changes. In the NLS, the second—order covariances and cross—covariances of earnings and hours are also approximately equal to zero. Like the PSUD, the SIME/DThIE and NLS covariances exhibit significant nonstationarity. The SIME/DIME and NLS covariance matrices are more nearly consistent with the symmetry restriction than the PSID covariance matrix. In light of the data presented in Tables 4—6, and the test results in Tables 7—9, we conclude that changes in individual earnings and hours are adequately summarized as a nonstationary bivariate second—order moving average. This description holds for annual data (PSID), semiannual data (SIME/DIME), and annual data at two year gaps (NLS). More restrictive models of the covariance structure of earnings and hours changes are not generally valid. In particular, nonstationarity is an important feature of the covariances of earnings and hours from all three panel surveys. II. Alternative Models of the Covariance Structure of Earnings and Hours thanges In this section we present a simplified life—cycle labor supply model and derive its implications for the covariance matrix of changes in earnings and hours in panel data. We also present an alternative —11— of earnings and hours determination that corresponds to a labor model contract with fixed average hourly earnings and employer—determined hours of work. We discuss the econometric identification of these models in the presence of survey measurement error and outline an esti— ination strategy that decomposes changes in earnings and hours into two components: a systematic component, reflecting individual productivity (in the labor supply model) or employer demand (in the contract nvdel); and a measurement error component. A. A Life—Cycle Labor Supply Model. The starting point for our model of life—cycle labor supply is an additively separable utility function of the form T (1) [log c1+. — a.+. tqhere T is the time horizon of individual I , is a fixed discount factor, c1+. represents the consumption of individual i in period represents a preference—shift variable, n is a strictly positive parameter, and h.÷. represents the hours of work of indivi- dual i in period t+j . supply and consumption We assume that individuals choose their labor in each period in order to maximize the expected value of (1), given the Information available in that period to and subject the following equation describing the evolution of their nominal assets: (2) A it-I-i =(pt 9 t it it hit —p tc.it -4-A)(l+R). —12— In this equation Alt represents the nominal value of assets held by individual i in period t Rt represents the nominal interest rate, represents the price level of consumption goods, and Pt repre- sents the real wage rate available to individual I in period t Maximization of the expected value of (1) subject to the wealth constraint (2) implies the Bellman equation: l+ii (3) where max Vi(A.) log c1 — alt i-:hi h.11 + $E denotes the expectation operator conditional on information available to individual i In period t The first—order conditions for consumption and labor supply in period t are (4a) — Sp(l + R) EV!1(A) 0 (4b) it a th1 — 80 p (1 + R ) E it t t V' (A t it+l it+1 ) 0 In addition, the derivatives of the value function V. (A ) it it follow the stochastic difference equation: (5) Alt E V(A1) = 8(1 + R) E A11 There is no exact solution to equations (4) and (5) in general. We proceed by using a well—known approximation to equation (5); namely —13— logA = log (6) loX1 + S +Rt + Et d. where d1 is the error of approximation and is assumed to be time sta- The labor supply equation can be derived from (4b) tionary. definition it = where related tion of of A1 in (5): log hi = (7) using the —n n log + ii log Pt + p log it + According to this equation, hours of work are log alt . to the contemporaneous wage rate by the intertemporal subs titu— wealth Hours n elasticity , and the of work also depend on the marginal utility preference parameter Making use of the approximation in equation (6), equation (7) can be differenced to yield the following description of the change in the logarithn of hours: (8) A log hi log hj — log — p A log 0ir + + where p n(log Alt — r1) Ei log A.) + A is the discount rate defined by 5 E 11(1 — A log Pt real Interest rate defined by r it +p) and r is the Since labor earn- ings are the product of the wage rate 8it and hours of work hit the labor supply model implies a similar specification for the change in the logarithm of earnings (g): (9) A log gif E A log hj + A = (l+p)A + ii (log X1 — log log O + fl(P-r1) Etl log Ai) + Ai —14— Changes in earnings and hours share common components associated with the revision of the marginal utility of incane; the change in tastes for leisure; and the difference between the discount rate and real rate. In addition, an individual—specific wage component interest enters the equations for earnings and hours changes with relative factor loadings of (1-i-n) and n , respectively. To complete this model of earnings and hours, we need to specify a stochastic process for the wage rate. We assume that individual wages contain a permanent component, an aggregate component, an experience— related component, and an idiosyncratic component. Specifically, we a Ssume: (10) log e1 = a1 + + 11x1 + 2 + is an economy—wide time effect, x is the number of years the individual where a1 is an individual—specific permanent wage effect, has been This in the labor force, and is an idiosyncratic wage shock. specification explicitly rules out both individual—specific respon- ses to the aggregate shock and individual—specific experience effects. We defer a detailed discussion of these assumptions to Appendix B. Using equation (10), the change in the wage rate in period t may be expressed as: (11) A log where c — = 6t—l + + yx0 — + Az ÷ 2y2t , force experience at calendar date t = 0. y , and is labor —15— Substituting equation (11) into equations (8) and (9) and appending a pair of measurement errors to log g and log hj yields a theoretical model for the bivariate stochastic process governing observed changes in individual earnings and hours (l2a) A log = (l-1-n)c +n(—r1) + (l+n)yx10 (l2b) A log hj = + (l+) — Ai ÷ Mit, flK + + rryx0 + nAzi where 6it E log + Az +161 + + Av, Etl log Alt uli E measurement error in observing log gft , and measurement error in observing log hit E The implications earnings of equations (12) for the covariance structure of and hours depend on the serial correlation properties of the productivity shock z1 , and income the taste shock , the measurement errors the unexpected change in the marginal utility of assume that tastes for leisure contain a permanent vi •, and • We component, a hoimgeneous quadratic age effect, and an individual—and— period—specific component. These assumptions imply that the error com- ponent Mit associated with the change in tastes for leisure can be written as (13) = + 2x0 + Avi 4 —16— where Uft represents We assume that an individual— and period—specific taste shock. is serially uncorrelated with a constant variance. The change in tastes for leisure is therefore a first—order moving average with a serial correlation coefficient of —.50. The measurement errors u and v' have a special structure. They are intended to represent systematic and random deviations of the survey measures of earnings and hours from their theoretical analogues. To capture these ideas, we assume that (14) rut1 tii [v7tJ '2iJ + [cut Lc2it where and •2i represent individual— and survey—specific permanent response biases, and and 21 represent transitory measurement errors. We assume that c1 and are serially uncorrelated with an arbitrary time—stationary covariance matrix. Since the changes in the measurement errors of earnings and hours reflect only the changes in the transitory components of these errors, and since the transitory measurinent errors are serially uncorrelated, Au* and v* are first—order moving averages with serial correlation coefficients equal to — .50. Notice that we are unable to separately identify the variance contribution of the random shock to preferences and the variance contribution of the transitory measurement errors, since we have assumed that they have the same serial correlation properties, and we have not restricted the correlation between the measurement errors. —17— We therefore combine these components into an unsystematic component of the changes in earnings and hours: r.it11 1 rAU it (15)) +it11 i r lit it L itJ Ltt v.J L"21t ÷ Avit + which is a stationary bivariate MA(l) process with an arbitrary contemporaneous covariance and with first—order autocorrelations equal to —.50. On the basis of the evidence that changes in earnings and hours are adequately represented as a bivariate MA(2) process, we assume that changes in individual productivity shocks in all three surveys follow a nonstationary second order moving average. The parameters of this process include the variance of the change in individual productivity shocks in period t (var Az.) , the covariance of changes in individual productivity shocks in periods t and t—l (cov(Az. , Azj)), and the covariance of changes in individual productivity shocks in periods it t and t—2 (cov(Az, Az )). The final corrrponent of variance in equations (12a) and (12b) is the unexpected change in the logarithm of the marginal utility of wealth. In general, is a function of the unanticipated component of the individual productivity shock Az11 and the unanticipated corn— —18-- ponent of the aggregate productivity shock. Consider the linear projection (across individuals at a point in time t): (16) Since + bjAz — Eti Azj) + the forecast error of AZft is serially uncorrelated, the serial correlation propertiesmf c1 depend only on the serial correlation properties of the projection error e This error depends on both individual and aggregate productivity shocks, since individuals can vary in their response to an idiosyncratic shock and in their response to an aggregate shock. For simplicity, we assume that the projection errors are serially uncorrelated (in the sense that plim This assumption is satisfied if ei e11 = 0 ) depends only on individual produc- tivity shocks, or equivalently, if unanticipated aggregate shocks generate a homogeneous shift in the marginal utility of income that is incorporated into the cons tant term in (16) We are now in a position to catalogue the implicationsof equations (l2a) and (l2b) for the covariance structure of earnings and hours in panel data. We note that in the absence of prior information these equations place no restrictions on the mean changes in earnings and hours observed in a cross—section)-! Depending on the specification of life—cycle preferences for leisure, however, equations (12) may restrict the regression coefficients of changes in earnings and hours on poten- tial experience. Equation (iZa) implies that the regression coefficient of the change in earnings on potential experience is (l-e-)y + —19— while (l2b) implies that the regression coefficient of the change in hours on potential experience is + If = life—cycle preferences exhibit no systematic curvature, then 0, and all the curvature in life—cycle hours is due to curvature in wages. In that case, an estimate of ii is available frcm the ratio of the covariance of the change in earnings with experience to the covariance of the change in hours with experience. This is precisely the instru- mental variables estimate of i fran the labor supply equation (l2b), using potential experience as an instrument for wages.--' Table 10 presents the average covariances of changes in earnings and hours with potential experience from the three data sets. important to keep earnings in ft is mind that the timing intervals of the changes in and hours are different in the three surveys. If the underlying parameters were the same in all three samples, then one would expect to observe covarlances in the SINE/DIME roughly one—half as large as those in the PSID, and covariances in the NLS roughly twice as large as those in th PSID.!±" The first row of the Table presents the average covariances of the change in earnings with potential experience. In all three data sets this covariance is negative, although in the NLS the average covariance is relatively small. The second row of the Table presents the average covariances of changes in hours with experience. In the PSID and SINE/DIME samples the covariance of hours changes with experience is smaller in absolute value than the covariance of earnings changes with experience. In the NLS sample, however, the covariance of —20— hours changes with experience is larger in absolute value than the covariance of earnings changes with experience. Under the assumption that labor supply preferences exhibit no systematic curvature, the instrumental variables estimate of n is ChI(Cg_Ch) where and Cg represent the average covariances of experience with hours and earnings changes, respectively. The instrumental variables estimates of i for n the PSID and SIME/DIME are therefore positive, while the estimate of for the NLS is negative and significantly different from zero.!i The fourth r of Table 10 contains the goodness—of—fit statistics associated with the restriction that the covariances of earnings and hours with potential experience are stable over time. The hypothesis of stationarity is marginally accepted in the PSID and SIME/DIME samples, and strongly rejected in the NLS sample. In view of this finding, for the remainder of our empirical analysis we leave the regression coefficients of the changes in earnings and hours on potential experience unrestricted, and concentrate on the covariance structure of the experience—adjusted changes. Denote the residuals of individual changes in earnings and hours from a multivariate regression on potential experience and time—specific means by A log and A log , respectively. Equations (12a) and (12b) imply (17a) A log it = (lfn)Azi + — ÷ Au. and (17b) A log hit = Experience + — + Av1 adjusted changes in earnings and hours consist of three components: a productivity component (Azit) that enters earnings and —21— hours with coefficients (l+) and n , respectively; a serially uncorrelated component associated with the unanticipated change in the marginal utility of income; and a restricted bivariate MA(l) component reflecting survey measurement error and random shocks to preferences. Table 11 summarizes the implications of equations (l7a) and (17b) for the covariance structure of experience—adjusted changes in earnings and hours. Measurement error components contribute negative first—order autocovariances in direct proportion to their variance contribution. Unanticipated changes in the marginal utility of income contribute a time—specific component to the variances and covariance of earnings and hwrs, but do not contribute to the autocovariances. Individual— specific productivity shocks contribute a time—varying variance and covariance component, and represent the only source of second—order autocovariance. As indicated in the table, the labor supply model im- symmetry on the cross—covariances of earnings and hours——a poses restriction that is satisfied in the NLS and STtW/DThIE samples but marginally rejected in the PSID sample. The labor supply model also restricts the four second—order autocovariances in each year to have the same sign. Inspection of Tables 4, 5 and 6 suggests that this restriction is typically satisfied, althcxigh the second—order covariances are estimated with relative imprecision. It is clear from Table 11 that the labor supply elasticity ii is generally overidentified. First, ratios of the second—order autoco— variances provide estinates of l+/ , the relative factor loading of productivity shocks in earnings as compared to hours. Second, if the —22-- first—order covariances vary over time, then the non—stationary com- ponents of the first—order autocovariance matrix are restricted by the Let ratio l+n/n . c'(t) denote the estimate of the first—order autocovariance matrix of earnings and hours in petiod t. Equations (17a) and (17b) imply (1+1)2 n(l+n) (18) c'(t) — C'(s) = {cov(Azi,Azii) — cov(AzjAzi [ 2 p Provided that the first—order autocovariances of the productivity shock are not equal, equation (18) Identifies the relative contribution of productivity shocks in earnings as individual provides thus In (17a) compared to hours, and an estimate of the labor supply elasticity i the stochastic model of labor supply represented by equations and (l7b), variances the zero—order covariances of earnings and hours (the of earnings and and the contemporaneous covariance) are hours unrestricted. This is a consequence of the fact that there are three free paraneters (var (Azit) var it' and coy (A z., €ft2) asso- ciated with the three zero—order covariances in each time period. In a perfect foresight model in which the marginal utility of income is constant, on the other hand, c is zero in every period and the labor supply model restricts the zero—order covariances in the same manner as the first—order autocovariances. Even in a perfect foresight model, however, the covariances of the measurement error component of earnings and hours are not separately identifiable fran the variances and first—order autocovariances of the —23— productivity variance shodc)-" It is therefore impossible to provide a unique deccmposition of the changes in earnings and hours into systema- tic and unsystematic components. One solution is to impose a prior estimate of the correlation coefficient between the measurement errors in earnings and hours. The estimate of (l-t-)/ is invariant to the choice of correlation coefficients. In this paper we do not compute variance decompositions of the earnings and hours data. B. A Fixed Wage Contract Model. The institutional structure of many employment situations is not easily reconciled with a model of the labor market in which employees unilaterally determine their hours of work subject to their current wage rate. In contrast, a wide variety of contractual models emphasize fixity in the labor market and resulting discrepancies between observed average hourly earnings and the marginal rate of substitution of consurnp— tion for leisure)2_" A prototypical model is one outlined by Abowd and Ashenfelter (1981). In their model, employers offer job packages con- sisting of a fixed average hourly wage rate and a probability distribu- tion over hours. Equilibrium in the labor market is obtained by equating expected the utilities of job packages. In any particular realization of random demand shodc driving employers' demands for hours, however, employees may prefer to work more contract. or less than required in their The implied correlation earnings and between changes in average hourly changes in labor supply is zero: earnings move in direct proportion to employer—determined hours with no corresponding change in wage rates. —24— To derive the implications of this model for the covariances of earnings and hours in panel data, assume that the wage rate received by individual i in period t contains only a permanent component, a deterministic experience component, and a period—specific component: (19) log 9it = a. + + y1xft + where, as before, x1 indicates the potential experience of i in period t . consist Assume that hours of work of individual i in period t of a permanent component ' ,a deterministic experience com- and a stochastic component ponent, a period—specific effect Pt reflecting employer demand for hours: (20) + log hi = + - + -Yx + yit Adding measurement errors u and vj to observed earnings and hours, respectively, this fixed—wage contract model implies that the residuals of individual changes in earnings and hours from a regression on potential experience are given by: (2la) log g (21b) A log = AYit + and As it hit = Ayft happens, guishable fran + this simple fixed wage contract model is indistin- a labor supply model in which the ratio (l+)/ is unity, or equivalently, in which the elasticity of labor supply is arbitrarily large. To see this, let —25— p = (l+n)/n = nAzi , c= — e) and rewrite (l7a) and (17b) as (22a) A log = i.i Az' ÷ + and (22b) A log hj = If p = 1 , var(Ay1) = Az + ÷ (22a) and (22b) are identical to (2la) and (Zlb) with var(Azr) ÷ var(ct) + 2 coy (Azt, e) and coy (Ay, Ay) = coy (Azr. Az) for • It is therefore impossible to reject the labor supply model in favor of the fixed wage contract model. If estimates of p in equation (22a) are close to unity, however, our interpretation of the data is not that small changes in wage rates are necessarily driving large changes in hours, but rather that hours and earnings are moving proportionately at constant average hourly wage rates.--' III. Estimates of a Two—Factor Model of Earnings and Hours. In this section we present estimates of the two—factor model of earnings and hours generation derived fran the theoretical models in the previous section. We report estimation results for two versions of the labor supply model: a perfect foresight version that ignores changes in the marginal utility of income; and a stochastic version that allas for unanticipated changes in the marginal utility of income. For each version of the model, and each of the three panel surveys, we pre— —26— sent sumnnry statistics for the goodness—of—fit of the model, as well as estimates of the relative contribution of productivity shocks in earnings as compared to hours (the parameter p in equation (22a)), and the implied elasticity of intertemporal substitution. The estimates are obtained by method—of—moments techniquS, using as data the covariances of experience—adjusted changes in earnings and hairs. On the basis of the preliminary data analysis in Section I, we fit the model to the contemporaneous covariances of earnings and hours and their first— and second—order autocovariances, ignoring the higher— order autocovariances. Two alternative method—of--moments estimates are presented. The first, whidi we call the equally weighted minimum distance (EW?e) estimator, minimizes the quadratic form [m — f(b)J' [m — f(b)] where m represents the vector of adjusted covariances to be fit by the model, b represents the vector of parameters of the model, and f(b) represents the vector of predicted values of the fitted moments, con- ditional on the parameter vector b • The second, which we call the. optimal minimum distance (Oft)) estimator, minimizes the quadratic form [m - f(b)J' V[m - f(b)] where V denotes the sample variance matrix of the data vector in. Chamberlain (1984) shows that the latter estimator minimizes the asymptotic covariance matrix of the estimated parameters among the class of minimum distance estimators. Table 12 sunmarizes the structural estimation results for the PSID. —27— Columns (1) and (2) report equally weighted minimum distance and optimal minimum distance estimates, respectively, for the perfect foresight model, while columns (3) and (4) report EWFID and 01W estimates of the uncertainty model. The individual productivity process (Az1) is estimated as an arbitrary nonstationary MA(2) process. In the PSID, this introduces 27 parameters: 10 variances, 9 first—order covariances, and 8 second—order covariances. The measurement error/preference shock component of variance introduces 3 additional parameters (the variance of the measurement error component in earnings, the variance of the measurement error component in hours, and the correlation of the two measurement errors). The final structural parameter is the relative contrihitlon of the productivity component to earnings (1'). In the uncertainty specification of the model there are 20 additional para- meters corresponding to the additional unrestricted zero—order covar— lances. Since the measurement error components and the productivity components are not separately identifiable, for purposes of estimation we have arbitrarily fixed the correlation of the measurement errors in and hours. The goodness—of—fit of the models and the estimates earnings of u are invariant to the choice of normalizing assumptions. first r of Table 12 shows the goodness—of—fit of the structural models relative to an unrestricted model (98 unrestricted The moments). The test statistics in the first two columns of the table suggest that the perfect foresight model is not supported by the data. The test statistics in the last two columns of the Table, on the other hand, indicate that the uncertainty model is consistent with the data at —28— conventional significance levels. The second row of the table reports All the estimated values of the productivity contribution parameter i.t . of these estimates are within sampling error of one for the PSID. Even though the perfect foresight model does not fit well, the estimates of p from this model are essentially the same as the estimates of i.t from the uncertainty model. As row 3 of Table 12 shows the implied estimates of the labor supply elasticity n are extremely variable and impre— 19/ cisely estimated.— The finding that productivity shocks influence earnings and hours proportionately suggests that systematic changes in individual hours occur at fixed hourly wage rates. There are a variety of possible interpretations of this finding. One possibility is that changes in hcnrs are determined by employer preferences, and are uncorrelated with changes in average hourly earnings. A second possibility is that the estimate of p is biased downward by changes in the marginal utility of that enter earnings and hours proportionately. If we income incorrectly fit a perfect foresight model, assuming eft = 0 in estimate of p. , equation (l7a) is biased toward one. , then the If cross— sectional variability in e is the source of downward bias in the perfect foresight estimate of p , however, then we would expect the estimate of p to increase between the perfect foresight and uncer- tainty specifications of the labor supply model Since the perfect foresight and uncertainty versions of the labor supply model yield very similar estimates of utility, of income ji , we conclude that changes in the marginal are not a likely source of bias in the estimates of —29-- p . Year—to—year changes in earnings and hours are consistent with either a large substitution elasticity, or alternatively with employer determined hours at fixed average hourly wage rates. The results in Table 12 nonetheless reveal a problem for the fixed— wage interpretation of the estimates of p . Under a stochastic labor supply model, the contemporaneous covariances of earnings and hours are not necessarily consistent with a two—factor model. Under the simple contract model outlined in Section 118, on the other hand, the zero— order covariances of earnings and hours are consistent with the same two—factor model as the higher—order autocovariances. The underlying goodness—of--fit statistics for the two—factor model when the contemporaneous covariances are restricted (in columns (I) and (2)) and unrestricted (in columns (3) and (4)) suggest that this is not the case. Whether the contanporaneous covariances of earnings and hours are restricted by the model or not, the estimate of p and the corresponding goodness—of—fit statistics are very similar between the OMD and EWMD estimation methods. Differences between the OMD and EWMD parameter estimates have the interpretation of a specification test (Hausman (1978)). Under the hypothesis that the model is correctly specified, both the EWHD and OMD estimates are consistent, and the OND estimates are efficient. It is straightforward to show that the covariance parameter matrix of the vector of differences between the EWMD and 01W estimates is equal to the difference between the OMD parameter covariance matrix and the EWMO parameter covariance matrix.--' The test statistics associated with the difference between the 01W and EWMD esti— —30— mates of p are reported in the fourth row of the Table. For both the perfect foresight and uncertainty versions of the labor supply model, the OMD and EWMD estimates of j are not significantly different. Table 13 summarizes the estimation results for the NLS. In fitting the labor supply model to the NLS, the covariances of the underlying productivity shock (Az.) contribute 12 free parameters (five variances, four first—order autocovariances, and three second—order autocovariances). Columns (1) and (2) of the table present estimation results for the perfect foresight version of the labor supply model. Columns (3) and (4) contain the results for the stochastic labor supply model, which adds 10 extra parameters corresponding to the additional unrestricted zero—order covariances. The equally weighted minimum distance estimates in columns (1) and (3) suggest that the parameter i is greater than unity, but not significantly so. The associated estimates of ii are large and Impre- cise. The goodness—of—fit of the perfect foresight model is poor: the chi—squared test statistic is 124.1 with 28 degrees of freedom. The fit of the stochastic labor supply model, on the other hand, is considerably better. The optimal minimum distance estimates in columns (2) and (4) of Table 13 provide about the same goodness—of—fit as the corresponding EWMD estimates, but yield very different estimates of the labor supply elasticity. The (ThD estimates of the perfect foresight model, in particular, earnings indicate a remarkably stronger productivity component in as compared to hours, and a relatively small estimate of i The OMD estimates of the stochastic labor supply model are closer to the —31— corresponding equally weighted minimum distance estimates, but yield a negative point estimate of ii. Our interpretation of the difference in the estimates in columns (1) and (2) of Table 12 is that the perfect foresight labor supply model provides a poor fit to the data, and as a consequence the parameter estimates are sensitive to the relative weight assigned to the deviations of particular moments from their predicted values. The fourth row of Table 4 presents specification tests based on the difference between the EWND estimates of p and the corresponding OND estimates. For the perfect foresight model the statistic cannot be computed because the estimated standard error under OMD is larger than under EWMD.?' In any case, the goodness—of—fit statistics providefairly strong evidence against the perfect foresight labor supply model. For the uncertainty model, on the other hand, the difference in the EWMD and OMP estimates of p is .71, with a standard error of .51. This result and the goodness—of—fit statistics in the third and fourth columns of the Table suggest that the stochastic labor supply model provides a reasonable description of the data. Table 14 contains goodness—of—fit statistics and parameter esti- mates obtained from the SIME/DIME sample. The conclusions from this sample are essentially the same as the conclusions from the other data sets. The perfect foresight labor supply model fits relatively poorly. The alternative model that frees up the zero—order covariances of the data is only marginally rejected. The estimates of p are within sampling error of unity in all cases. In the SIME/DII{E data, as in the PSID data, the specification tests based on the difference between the —32— EWMD and aMP estimates of p are not significant at conventional signi- ficance levels. IV. Conclusions We have presented evidence on the autocovariance structure of ear- nings and hours changes in three panel data sets: the PSID, which measures hours and earnings annually; the NLS Survey of Older Men, which measures annual hours and earnings at two year intervals; and the SIME/DIME, which measures semi—annual earnings and hours. In spite of these timing differences, and some differences across the surveys in the questionnaire used to measure hours and earnings, the autocovariance structure of all three data sets is remarkably similar. Individual changes in earnings and hours have strong negative first order auto— correlation, weak second order autocorrelation, and negligible higher order autocorrelations. Contemporaneous changes in earnings and hours are positively correlated in all three data sets. This cross—covariance is too small, however, to generate a position covariance between changes in hours and changes in average hourly earnings in any of the data sets. The sparse autocovariance structure of earnings and hours changes makes it difficult to identify models that include general specifications of measurement error, tastes, and two economic components. We develop models of hours and earnings that distinguish between an unsystema- tic component (attributable to measurement error and/or taste changes), and a systematic component. In the first model, the systematic component is interpreted as an underlying shock to individual productivity —33— that influences wages directly and hours indirectly through an intertem— poral substitution effect. In the second model, the systematic component is interpreted as a shock to employer demand for hours that influences results earnings and hours equi—proportionately. Our empirical suggest that the systematic component of variance effects ear- nings and hours equi—pifoportionately in all three panel data sets. Apart from the zero—order covariances of earnings and hours (the variances of earnings and hours and their contemporaneous covariance), a simple two—factor model provides an adequate description of the covariance structure of earnings and hours changes in all three data sets. A two—factor model is apparently too restrictive as a model of the complete covariance matrix of earnings and hours changes. In the context of a labor supply model, this finding is interpreted as evidence against perfect—foresight. In the context of a fixed—wage contract model the interpretation of this finding is unclear. More work is required to distinguish between the two models, and to develop alternative models of the covariance structure of earnings and hours. —34— Footnotes See for example Ashenfelter and Ham (1977), MaCurdy (1981), Altonji (1984), and the surveys by Killingsworth (1983) and Pencavel (1985). VThe life—cycle labor supply model is described in Ghez and Becker (1975) and Heckman (1976). Browning, Deaton and Irish (1985) provide a useful summary of the theory of consumer behavior over time. 1"Pencavej (1985, p. 151) concludes that . . . the focus of most economists' research [on labor supply] has been on a behavioral response for men appears to be of a relatively small order of magnitude. that 'Altonji (1984) compares the correlation between changes in hours and changes in average hourly earnings (about — .35 in his sample and in the PSID and NLS data sets used in this paper) with the correlation between changes in hours and changes in reported wage rates of hourly rated workers (about .01 in Altonji's sample). Altonji interpretes this difference as evidence that masurement error in hours induces a strong negative correlation between changes in hours and changes in average hourly earnings. 'HaCurdy (1981) and Altonji (1984) both arrive at estimates of the intertemporal substitution elasticity in the neighborhood of .10 — .40. -"Our empirical analysis is closely related to work by Hause (1980) and MaCurdy (1982). Hause and MaCurdy both model the serial correlation —35— structure of earnings in longitudinal data. We extend the analysis to the bivariate process of earnings and hours. 1The NLS surveys survey were administered in July through October of the years, and asked questions on earnings and hours in the previous twelve months. Unlike the PSID survey, the NLS collects no auxilliary information on overtime hours or hours of work on secondary jobs. For this and perhaps other reasons, a large fraction (30 percent) of NLS respondents report exactly 2,000 hours per year. !"our SINE/DIME data is drawn from the so—called Work Impact File assembled by SRI from the underlying survey data. The SIME/Dfl1E survey contains detailed questions on overtime and secondary jobs, and is conceptually more similar to the PSID survey than the NLS survey. Denote the change in earnings between period t—l and period t .2! by Ax . If Ax is MA(2), then Ax example, where between = s + b11 ÷ b2e is serially uncorrelated. The change in period t—2 and period t is x — x2 = x2 = Ax for 2 earnings + + (l+bi)e 1 + (b1 + b2)e2 + , but not with Notice that x — x2 is correlated with x2 — Therefore, x — x4 — x6 (or Ax4). !Q!There are at least two alternative ways to test restrictions on subsets of the covariances in Tables 4, 5, and 6, depending on whether we incorporate the restrictions that the third— and higher—order covariances are jointly equal to zero when we test the properties of the lower—order —36— covariances. Our procedure is to ignore the higher—order covariances in estimating and testing restrictions on the lower—order covariances. i2/The error aggregate e it may be serially correlated, for example, if shocks have a systematically larger effect on some individuals' marginal utility of income than others'. In this case, however, we would expect to observe nonvanishing covariances between changes in earnings and hours from years with large aggregate shocks, irrespective of the gap between these years. There is no evidence of this phenomenon in Tables 4—6. il/There are two time—varying components of the mean changes in earnings and hours in each year: the shift in the aggregate productivity shock ; and the average revision in the marginal utility of income !'MaCurdy (1981) and Altonji (forthcoming) present instrumental variables estimators of the intertemporal labor supply elasticity, using polynomials of age and experience as instruments. ---"Write equation (l2a) as Alog gft = where + io ÷ represents the regression coefficient of the change in ear- nings on experience, is a period—specific constant, and includes the stochastic components of earnings changes. If time is measured in years, then the change in annual earnings over the two year interval from t—l to t+l is Alog g.1 ÷ Alog = 1t+l + + —37— 28x1 + + 6. . Thus the covariances of changes in earnings and hours with potential experience should be roughly twice as large in the NLS as the PSID. The comparison between changes at six month intervals and changes at annual intervals is similar. instrumental variables estimate of n for the PSID is substantially larger than the estimates reported by MaCurdy (1981) or Altonji (forthcoming). Both authors use overidentified estimators that make use of several instrumental variables for wage changes. Since our primary interest is not in the instrumental variables estimates themselves, we have not explored the estimates of ii when other exogenous variables are used as instruments for changes in wages. is straightforward to show that each of the variances of the productivity shock can be increased by a fixed factor, the measurement error variances can be decreased, and the first—order autocovariances of the productivity shock can be increased in such a way as to hold constant the predicted covariance matrix of earnings and hours. !ZiHart (1983) and Rosen (1985) survey most of the theoretical literature on labor contracts. !JIn other work (Abowd and Card (1984)) we have considered the implications of labor contracts that smooth individual earnings. Suppose, for example, that observed earnings in period t represent a geometric average of earnings in the absence of contracts (g°) and desired consumption in t (c) , which we assume is constant. Then log g = (1—y) log g° + y log c , where 'y' represents the extent of —38— earnings stabilization. Taking first differences and using the notation of equation (22a), log = ILAz*. + E*i + p = (1—y) (l+n)/n and Az*. and C*1 , where are suitably defined. In the presence of earnings smoothing contracts, the parameter may fall i' below unity. 12!The labor supply elasticity n U by the formula ii = (p—l)1 is obtained from the estimate of Standard errors for . in Ti Tables 12, 13, and 14 are calculated by the delta—method. derive the covariance matrix of the difference between the OMD estimate b° and the EWND estimate be , note that under the hypothesis that the model is correct, /N (b — b*) a /nbe - b*) - —1 —l —1 /r(F'V F) FTV (m_f(b*)) /F(F'F)1 F!(m_f(b*)) where V is the covariance matrix of the vector of moments is the predicted vector of moments given the true value b* sample size, and F represents the Jacobian matrix of f b* • , in , , f(b) N is the evaluated at These equations follow from the first order conditions for and be , respectively, and a series of regularity assumptions. Using the fact that V is the variance matrix of (m_f(b*)) , it is straight- forward to derive the asymptotic variance matrix of the difference (ho_be). li/The only when variance of the OND parameter estimate is necessarily smaller both variances are estimated at the same parameter values. —39— References Abowd, John H. and Orley Ashenfelter. "Anticipated Unenployment, Temporary Layoffs and Compensating Wage Differentials." Studies in Labor Markets, ed. Sheiwin Rosen. Chicago: Univ. of Chicago Press, 1981. Abowd, John N. and David Card. Presence "Interteniporal Labor Supply in the of Long—Term Contracts." Unpublished manuscript, Princeton University, June 1984. Alt onji, Joseph A. "Intertenporal Substitution in Labor Supply: Evidence fran Micro Data." The Journal of Political Economy, forthcoming. (Draft paper dated February Ashenfelter, 1984.) Orley and John C. Ham. "Education, Unenployment, and Earnings," JcErnal of Political Economy, 87 (october 1979): S9 9—S116. Braining, Martin, Angus Deaton, and Margaret Irish. "A Profitable Approach to Labor Supply and Commodity Demand Over the Life—Cycle." Econonetrica, 53 (May 1985): 503—543. Chanberlain, Gary. "Panel Data." Chapter 22 in The Haudbook of Ecmanetrics, New York: Ghez, Gilbert Vol. 2, ed. by Zvi Griliches and Michael Intrilligator. North Holland, 1984. and Gary S. Becker. The Allocation of Time and Goods Over the Life Cycle. New Bureai York: Columbia of Economic Research), 1975. Univ. Press (for the National —40— Hart, Oliver D. "Optimal Labor Contracts Under Asymmetric Information: An Introduction." Review of Economic Studies, 50 (January 1983): 3—35. Hause, J.C. "The Fine Structure of Earnings and the On—the—Job Training Hypothesis." Econometrica (May 1980): 1013—1030. Hausnan, J.A. "Specification Tests in Econometrics." Econometrica, 46 (September 1978): 1251—1272. Heckman, James J. "A Life—Cycle Model of Earnings, Learning and Consumption." Journal of Political Economy, 84 (August 1976): 811—544. Killingsworth, Mark. Labor Supply. New York: Cambridge Univ. Press, 1983. MaCurdy, Thomas E. "An Empirical Model of Labor Supply in a Life—Cycle Setting." Journal of Political Economy, 89 (December 1981): 1059—1085. __________________ Structure "The Use of Time Series Processes to Model the Error of Earnings in a Longitudinal Data Analysis." Journal of Econometrics, Newey, Whitney IC. 18 (Special Issue 1982): 83—114. "Generalized Method of Moments Specification Tests." Unpublished manuscript, Princeton University, 1985. Pencavel, John. "Labor Supply of Men: A Survey." In The Handbook of Labor Economics, ed. by Orley Ashenfelter and Richard Layard. Forthcoming, 1985. Rosen, Sherwin, "Implicit Contracts". Journal of Economic Literature, 23 (September 1985): 1144—1176. —41— Appendix A In this appendix we summarize the estimation and -inference proce- - dures used throughcxit the paper. The basic unit of data for each individual in a particular data set is the vector of experience—adjusted changes in earnings and hours. If we denote these by A log and A log hj , then the data vector is A log V Ji a A log A log ll A 1g "iT which has dimension 2T , where T is the number of changes observed in the data set. Let " let C=-t — = 4f '—' these changes.. Estimates of C represent the mean vector of changes and represent the covariance matrix of are presented for the three data sets in Tables 4, 5, and 6. The models we estimate are models for C. Let m represent a vector whose elements are the distinct elements of C • Since C is symntric, there are only 2T(2T-i-l)/2 elements in in • In the PSID, this corresponds to 210 elements; in the NLS, 55 elements, and itt SIME/DD4E, 136. Conformably with in, let m represent the distinct elements of the individual cross—products matrix in = E in the (—)' . Then —42— The variance matrix V of the vector of covariance elements is estimated by computing the cross—sectional variance of m1 in the usual manner: —in)' v=- (m —in) (m A typical elanent of V is V cov(A in log , A cov(m m) , If = cov(A log g1 log hj.) and , A log then = j [(A — A (A log where A log j1 A log it — represents of earnings in period t • — A log — log is—k — A log — log XA log log ')(A m] x the sample average change in the logariti-in Let Q represent the matrix of fourth moments of Q=ji mimi' Q and V are related by V = Q — in in' Under fairly general conditions, independence of the y1 implies that the sample mean of hF Cm — ii) where p m has an asymptotic normal distribution: N(O , V*) is the population value of m (i.e. , the true covariance matrix of earnings and hours changes) and V* is the population value of V. Consider a model for the vector of covariance elements that depends on a lcwyer—djmensional parameter vector b , say m = f(b). Several —43— estimators are available for b: among these are the optimal minimum distance (OIID) estimator b° , which minimizes (m—f(b)) V1 (m—f(b)), and the equally weighted minimum distance (EWMD) (or least squares) estimator former be , which minimizes (m—f(b) ) '(m—f(b)). Optlinality of the estimator is discussed in Chamberlain (1984). For the OMU estimator, inference is based on the fact that under the hypothesis of a correct N • (in — specification, the minimized f(b°fl' V1 (m — f(b°)) has an asymptotic chi—squared distribution with to quadratic form degrees of freedom equal the difference between the dimension of m and the rank of the Jacobian evaluated at b* , matrix F(b) the true value of b. (See Chamberlain (1984)). If the model for in has = f(b) = H in which the special form = L°i only the last k elements of straightforward N• in are restricted to zero, it is to show that the minimized quadratic form reduces to —l in2 V22 m2 where V22 is the block of V corresponding to the elements in that are restricted to zero. This is the chi—squared test for in2 =0 in —44-- For an estimator bA that minimizes an arbitrary quadratic form (m — f(bfl' ACm — f(b)) where A is a positive definite matrix, Newey (1985) shows that the quadratic form (in - f(b R (mS- f(bA)) has an asymptotic chi—squared distribution. Here, C is a generalized inverse of the matrix R = pVp' , where — F(F'F)1 p = F'A , and F represents the Jacobian matrix of f evaluated at hA. Newey (1985) suggests a generalized inverse of R of the form S(S'RS)1 S , where S is a selection matrix of rank equal to the difference between the dimension of in (the number of movements fit) and the rank of F. In general, for a nonlinear model, the value of the quadratic form depends the exact generalized inverse selected for R. In case of a linear on model, however, the value of the quadratic form is invariant, and in the simple case of testing the restrictions in2 restrictions on m1 , equal N the 0 , with no additional value of the quadratic form can be shown to in2' V221 in2. Tests of zero restrictions are invariant to the choice of OMD or any arbitrary minimum distance estimator. Our procedure is to first test for zero restrictions and then to work with the nonzero covariances of earnings and hours. In the notation of the previous discussion, once we have accepted the hypothesis = 0 V11 , we estimate ndels for m1 and use the covariances matrix for inference. This procedure is not fully efficient, since better estimates of the unrestricted elements of m can be formed by taking into those in in2) account the fact that certain other elements of at (i.e. are zero. —45— Appendix B In this appendix, we comnnt on the assumption underlying equation (10) of the text that aggregate shocks and experience effects enter homogeneously into individual productivity. More generally, consider the alternative specification: (ba) log a1 + r16 + +2x + z Here, we have permitted the aggregate productivity shock individual ficient that r1 to affect productivity through an individual—specific response coefCross—section dispersion in r reflects the possibility individuals have permanent cyclical attributes: individuals with higher values of r1 are more responsive to aggregate shocks. We have also permitted the linear experience coefficient to have an indi- vidual component of variance. Equation (l0a) implies that the first difference of wages can be written as: (ha) A log where K = rA6 — = K+ + 2y2t + Y2x0 , and + (r —r) Aót + Az1 r is the average value of Individual heterogeniety in the experience slope and the response to aggregate shocks introduces two additional components of variance into the first differences of earnings and hours. Both of these components contribute to covariances between lagged changes in earnings and hours. First consider the component of variance associated with (r — r) If this component is large, then changes in earnings and hours in years with a negative product [vity shock should be more highly correlated with —46— changes in other years with negative shocks, and less correlated with changes in years with aggregate productivity improvements. By most measures aggregate productivity fell with business cycle recessions in 1969—70, 1973—74, and 1980—81. In the PSID data, however, covariances of earnings and hours changes between these years are very similar to covariances between arbitrary pairs of years at the same gap. We there- fore conclude that the component of variance due to individual—specific responses to aggregate shocks is small. Next, consider the component of variance in (ha) associated with cross—section dispersion in earnings growth rates. This component contrihites a fixed positive element to earnings and hjrs autocovar— iances and cross—covariances of all orders. In the data, autocovariances of third order and higher are all negligible. Again, we conclude that cross—section dispersion in experience growth rates contrihites a relatively small component of variance to wage growth. Table 1 Characteristics of the PSID Sample of Male Household Heads Average Hourly Earnings Average Annual Hours 1969 1970 1971 1972 1973 1974 1975 3.62 3.71 3.85 4.00 4.13 4.10 4.02 2308 2276 2266 2302 2324 2246 2220 1976 1977 1978 1979 4.19 4.26 4.26 4.25 2231 2236 2244 at Year— 1. Annual Hours and Average Hourly Earnings (1967 dollars) Change 2. Changes in Log Real Annual Earnings and Log Annual Hours (x 100) . Change Earnings in . 1969—70 1970—71 19 71—72 1972—73 1973—74 1974—75 1975—76 1976—77 1977—78 1978—79 2186 Change in Hours 2.5 3.0 —0.8 -0.3 6.9 4.7 2.0 1.9 —5.5 —4.2 —4.1 —2.4 4.1 2.5 0.2 —5.5 0.6 0.3 0.5 —4.2 3. Demographic Characteristics Average Age in 1969 Percent Nonwhite Average Potential Experience in 1969 4. Sample Size 35.8 27.3 18.9 1448-" NOTES: a/Data are for the calendar years listed. .1Eight outliers with reported average hourly earnings greater than $100/hour (1967 dollars) were excluded. Table 2 Characteristics of the NLS Sample of Older Men Year 1. Annual Hours and Average Hourly Earnings (1967 dollars) 1966 1967 1969 1971 1973 1975 . Change 2. Changes in Log Real Annual Earnings and Log Annual Hours (x 100) Average Hourly Earnings 3.50 3.46 3.55 3.66 3.63 3.50 Change in Earnings 1966—67 1967-69 1969—71 4.5 4.0 1971—73 1973—75 —0.2 —16.8 3.1 Average Annual Hours 2209 2190 2190 2161 2160 2003 Change in Hours 0.0 0.0 —0.1 —1.5 —11.6 3. Demographic Characteristics Average Age in 1969 49.2 Percent Nonwhite 29.8 Average P.otential Experience in 1969 34.4 4. Sample Size 1318 NOTE: Data are for twelve—month periods preceeding the interview date. Changes in earnings and hours over the two year intervals are not at annual rates. Table 3 Characteristics of the SINE/DIME Sample of Male—Reacts of Dual—Headed Households Experimental Period 1. Mm..] Hours and Average Hourly Earnings (1971 dollars) i 3.47 2093 3.53 3 4 3.63 3.73 3.83 3.88 3.88 3.85 3.88 2098 2087 2117 2135 7 8 9 Change Earnings and Log Semi—Annual Hours (z 100) Average Annual Hours 2 5 6 2. Changes in Log Real Semi—Annual Average Hourly Earnings Change in Earninjs 2104 2131 2074 2059 Change Hours 1—2 1.2 2-3 —1 .3 2.8 5.8 1.6 4.0 5—6 6—7 7—8 —1.3 0.7 —3.5 0.6 8—9 —3.0 —2.7 3—4 4—5 3.9 3. Demographic Characteristics Average Age at start of experiment 34.7 Percent Nonwhite 48.8 Average Potential Experience at start 18.2 of experiemnt 4. Sample Size NOTE: Data 560 are for six—month periods following assignment into the income experiment. The changes in earnings and hours between consecutive six—month intervals are not at annual rates. 73—06 Alog A L.a a 76—79 70—71 Lloj Ii .00) .003 (.004) 2). )0006 (.939) —.506 (.00)) .001 79-79 l0.Abo(h (.004) —.002 .004 (dl?) .!10 .000 (.003) (.975) 4900 .004 (.906) (.006) (.004) (.006) —.006 —-009 ILAljh 77—79 76—77 1.61096 75—76 17.81n9k (.006) .006 76—75 6.61096 65. —.00) (.006) (-00)) .006 (.004) (.0(0) -.031 .07) (.01)5 (.010) —.9)1 (.006) 6 Alo 73—li .003 (.00)) — —-001 - —.001 (-007) (.011) —.022 (.016) .067 .000 .136 NOTflt (.1 (7) 323 -.00] (.003) .00) (.00)) (.00)) —.001 —.002 .001 (.005) (.099) .224 -009 (.119) —.01! I92 (-III) (.006) —.006 (.005) —.091 (.021) —.046 .106 (.021) (.013) —.0)1 (.007) —.008 (.003) —.001 (.006) —.1)0 (.117) (.00)) —.004 033 (.013) 4967 (.027) (.010) —.067 .002 (.009) (.003) —.0)0 (4905) .007 —.003 (.003) (.964) .209 .003 (4906) .088 (.020) .017 (.014) -00] (.010) (.009) —.011 .001 (.004) —.00' (.005) —.00! (-005) .005 (.004) .001 (.005) (.006) -.009 (.004) —.001 .046 (.013) — .191 (.0?)) (9) *009 a 7779 (.005) —.004 .003 (.060) (-0)9) —.064 .996 (.034) 76—77 AIg (9) (.006) .006 .00' (.006) (.019) .0l3 (.009) —.087 (.032) - 75—76 âIog (7) (.006) —.002 490? (.003) (.006) —.006 (OIl) —.068 .071 (.093) (.079) —.008 —.003 (.004) —.009 (.006) (.0052 .00) (.005) —.001 .000 (.006) .006 (.007) —OIl (.007) (.01)) —.088 .111 ((II) AThg g 76—75 (6) (.11]) .217 .075 (.016) (-060) —.006 .009 (.009) (.0063 —.007 (.006) 4906 .001 (.906) (.004) .00) (.00)) .001 (4906) —.001 (.004) —.007 .17' (.0)0) 79.79 Abs 0)0) of: —.311 (.118) (.005) —.00! (.005) —.001 .005 (.00)) (.003) —.002 (.006) —.006 .006 (.006) .000 (.00)) —.002 (.007) —.060 (.0(0) (.013) .1)8 (II) Slog 6 69—70 C.art.nt. .115 —.123 (.096) (-006) .00) 000 (.006) (.004) —.002 .000 (.006) .006 (.006) (.004) —.006 (.00)) -M05 (.010) —.068 (.015) Ii (12) &I 70-71 ii, p.rev,the.ea).a' .119 —.953 (.096) (.006) —-006 .002 (.006) (.006) —.001 .00) (.003) (.003) —.005 (.003) —.090 —.016 (.090) 900r7 of the an. t oeot .atrtx. an given potential eocpertence and year onple fourth tonal are ba.ed on the coodi of ohaerva —-III (402) 002 (.006) (.003) —-003 (.006) —.010 (.01]) —.058 .110 (.013) (16) 6Io 6 767) tb,. and the scsi.) 'curs re calculated See Table I for (.07)) —.05) (.00)) 430! (-004) —.001 0.00,0 —.001 (.006) —.006 (.005) —.010 .073 (.001) 73—16 (I)) 61086 .Vcovartance of potential labor force exper(ence with the change (9 log earnings (column. (1)—fib)) ch.nge (9 log annual hours (column. (0l)—(20)h i/standard error. a sfter.dJoetbOg (or to.. log (.091) .076 (-001) —.001 (-006) —.006 .008 (.005) (.003) —.006 (.003) —.00' (.001) —.016 .090 (.0(9) 71—71 (14) hog h 71—11 (II) 61086 (.015) Il -The data ste change. to log annual e.rnlng. (A log g I, roes .nol cclii.,. (1)—flu)) end (A log I. to row! .nd cclii.,,. (tl)—(20)) 4) Iferenced between the year. indlc.ted. Ca 0_Ott) 4906 (.001) —450 —490? -003 (.004) —.007 .000 (.00)) (.004) —.001 (.007) —.016 .036 (.007) (.003) .00) (.004) —.00) (.0(1) .002 (.001) —.001 .000 (-(06) (.006) —.00! (.006) —.OIO (.006) —.001 (.015) —.050 (.016) .129 9 73—76 Ito) 0) (.003) .005 (.005) (.006) —.00) (490)) (.006) —.009 .006 (.003) .039 (.010) (.00)) .000 —.0)! (.006) (4911) —.021 (.015) —4922 (.010) (.007) .079 —.00) (.011) .001 (.004) .000 (.00)) .006 (.007) .011 (.009) .000 (.00!) (.006) —.711 (.007) —.036 (.014) —.026 (.007) —.001 .001 (.005) (.009) (-007) (.006) —.003 .006 (.007) -00] .009 (.009) .000 (.0070 6l 73-34 II 79—02 I3A0o)t II. 69—70 II. 60056 In. 77—79 9.ilogg 76—77 9.460I .010 (.008) (.005) (.005) (.006) (.0076 —.010 —.002 —.007 —.007 .00) (.005) (.005) —.007 (.0") —.041 .000 ).h!o.e 75—76 .170 (.0:9) (.005) (.00)) .003 .003 (.006) 76—75 6. Alog L (.009) —.002 .000 (.0060 72—73 70—7) 6.hlogg .959 —.066 (.093) —.097 (III) —.116 .161 (.011) (.019) 72—72 71—71 70.71 (.007) 3. 6)0(6 70—It 1. how 69—10 1. Aba ilo g 660!. aloe! (4) g 69—70 6t (3) (2) II) error. to " 165 4 N81fl 1969—1970 (Itanitard rot £010 Y.bla .070 CIII) -.003 (.005) (.006) —.006 (.0)8) —06A .167 (On) 73—76 (II) loa I, (.60)1 I9' (.00)) —.00) (.015) —493) 483 (.026) 76—77 (08) Mo 6 —.151 (.121) (.008) —.02! .1)4 (.020) 7779 (I!) t08 It (.129) —.29! .016 (.019) II 79—79 (20) 6lfl vitb: 66—67 1. Alog g Table 5 NLS (3) (4) (5) Alog h 66—67 (6) Alog h 67—69 (7) 8log h 69—71 (8) Alog h 71—73 (9) (.039) .132 (.040) .178 (.024) .330 (.036) 69—71 3. Alog Data l966—75' Covariances of Experience—Adjusted Changes in Earniis and Hours: (standard errors in parentheses) (2) Alog g Covariance of: (1) Alog g 71—73 —.068 (.038) —.040 (.008) —.073 (.018) 7375 Alog g 69—71 —.032 (.010) .000 (.004) —.011 (.010) (.006) (.007) —.017 .013 —.026 .198 —.225 . blog g 67—69 —.000 (.004) .007 —.003 (.004) h Experience1 73—75 67—69 7. Alog 11. (.086) (10) —.036 (.012) —.038 (.008) (.015) —.028 (.007) .066 (.011) —.050 (.010) (.014) —.291 (.078) .235 (.028) 7375 A1.og h —.002 (.003) .007 (.005) .011 (.005) —.158 (.04!) .081 —.001 (.003) —.013 (.006) .052 (.036) , .005 (.005) .000 (.048) .091 —.018 (.049) .114 (.023) 8log g 66—67 . —.005 (.004) (.011)) .002 (.003) .001 (.005) (.005) .109 (.016) .004 (.006) .002 (.005) —.004 (.004) .002 (.006) (.009) .133 4. tlog g 71—73 .038 (.012) .016 (.004) .015 (.004) .060 (.014) (.027) 2. 5. Alog g 73—75 —.007 (.004) —.007 (.004) -.007 (.005) —.034 (.010) tb8 g 6. Alog h 66—67 .002 (.003) —.001 (.004) —.006 (.009) —.160 (.056) 67—69 8. Alog h 69—71 .001 (.003) .009 (.009) .016 (.057) g 9. Alog h 11—13 .002 (.007) .078 (.063) .005 10. Alog h .062 (.051) NOTES -11The data represent in changes and log earnings hours over log the intervals indicated, adjusted for the conditional mean given potential experience and year of observation. See Table 2 for means. 21Covariance of potential labor market experience with the change in log earnings (colunns (l)—(5)) and the change in log hours (columns (6)—(10)). with: in (5) Table 6 Change in parentheses) / Semi—Annual Data (roe. SINE/DIME1 Covariances of Experience—Adjusted F.arninga and Hours (standard errors (10) Period 2 .153 Slog h (.028) (.024) .130 .119 .130 (3) —.050 —.052 (.027) (.025) (.026) cnarinc. of (7) (8) (9) Slog g Slog g Slog I' Period I Period 7 PerIod 8 (.018) (.018) (.020) —.050 —.038 (.038) .140 .134 6 (6) Slog g —.022 .006 (.012) .171 (.032) —.006 (.025) (.026) (2) Slog g —.046 (.014) (.010) .009 —.076 (.023) —.086 (.024) (.007) (.017) —.038 PerIod5 Period (.022) —.003 (.008) .010 (.012) .019 (.017) .006 .002 4 (4) Slog g —.007 (.008) (.009) —.01] —.013 (.010) .000 (.012) .008 (.008) (.010) Period (.013) —.001 .005 (.013) .000 (.007) (.014) .009 Slog g Slog g PerIod 2 Period 3 —.016 (.007) —.006 (.010) —.013 (.012) (.007) —.002 1 (1) Slog g Period .155 (.014) .004 (.013) .015 (.015) (.012) —.001 —.DLl (.030) .002 (.013) .000 (.010) —.007 —.022 .011 (.008) 1. Slog g Period I —.005 6. Slog g Period 6 (.008) —.036 (.012) (.013) g PerIod 4. Slog Period 4 I Period 5 5. Slog g 7. Period Slog h (.010) (.013) (.013) —.005 (.017) —.042 (.022) .120 .119 (12) Slog h (.008) Period 4 (.009) (.023) 3 (.017) (.017) —.052 (11) Alog h PerIod (.022) —.020 .145 .010 (.010) (.018) —.036 —.049 2 —.006 (.008) .130 .119 (.017) 2. Slog g Period .133 (.026) (.026) II. 3 Slog g Period 7 —.045 —.046 3. Slog g 8. Slog g Period 8 .001 (.021) (.011) (.01]) .217 9. Slog h 2 (.014) (.011) —.011 .005 ID. Slog Ii Period 3 .005 (13) Slog ii Period S (14) Slogh .100 .111 (.013) —.040 (.016) (.023) (.010) (.013) (.0L2) —.036 —.009 .010 .004 .005 (.009) (.008) (.009) —.003 —.010 —.041 (.007) (.008) .116 —.003 .000 —.055 (.009) (.006) —.017 .001 .006 —.014 .003 (.012) (.008) —.031 .020 .100 (.010) (.0'S) (.015) (.012) (.021) —.071 (.024) .113 —.042 (.019) (.024) .010 —.040 (.017) (.015) (.012) (.013) .007 —.006 (.009) (.008) (.013) —.011 —.002 (.008) .001 (.006) (.007) .004 (.008) (IS) Slog (16) Slog h Period .157 .002 —.069 —.077 .000 .155 —.009 —.077 —.003 .008 (.03!) —.010 (.031) .008 .006 of .208 (.011) (.022) .065 (.022) .017 (.009) .008 (.014) (.010) (.00!) (.015) i.lel) .002 (.011) (.010) .174 .002 (.013) (.006) (.187) .009 (.013) (.018) —.056 .000 (.023) (.234) .004 (.214) —.346 (.185) —.126 .198 .061 (.211) —.244 (.16]) .039 —.087 (.214) (.022) (.017) .014 (.012) .010 .002 .00! (.013) (.151) .139 (.031) (.013) .016 (.016) 100 (.138) (.024) —.00! (:169) — 235 (:165) — (.160) (.011) (.009) (.010) 007 (:195) —.168 (.010) (.007) — 127 (:230) —.005 156 (.195) — —.040 —.004 (.010) (.010) .001 Period 12. Slog Ii 8 7 6 S PerIod 4 13. Slog h Period Period 14. Slog K 15. Slog h Period — Period Ex e 16. Slog h 17 NOTES: experience and period—specific shocks by regreeeing changes in log earnings and i/The data have been adjusted for conditional means given potential log hours on potential experience and dummy variablea for the date of entry into the experimental group. The data are aligned in experimental tine: Period 1 refers to the first sixth—month nerind after enrollment into the control group. See Table I for means. in log hours (columns earnings (columns (l)—(8)) nnd the change potential labor market experience with thearechange in log control group. adjusted for date of entry into the experimental Experience and the change in log e.nrnlnp.e and hours covariancem (9)—(16)). Vpartial Table 7 Summary of Estimated Bivariate MA(2) Representation of Experience—Adjusted Changes in Earnings and Hours: Annual Data from the PSID 1. Goodness—of—fit statistic — MA(2) (probability value — 112 degrees of freedom) 137.19 2. Test statistic for symmetric MA(2)' 31.02 (.020) (probability value — 17 degrees of freedom) (.062). 3. Test statistic for MA(l)--" (probability value — 32 degrees of freedom) (.020) 4. Test statistic for stationary MA(2)1' (probability value — 87 degrees of freedom) 143.69 (.000) 50.39. NOTE: "Against the alternative hypothesis of a nonstationary bivariate MA(2). Table 8 Summary of Estimated Bivariate MA(2) Representation of Experience—Adjusted Changes in Earnings and Hours: Annual Data from NLS 1. Goodness—of—fit statistic — MA(2) (probability value — 12 degrees of freedom) 12.89 (.377) 2. Test statistic for symmetric MA(2)uI' (probability value — 7 degrees of freedom) (.047) 3. Test statistic for MA(l)" 14.47 14.64 (probability value — 12 degrees of freedom) (.263) 4. Test statistic for stationary MA(2)11 (probability value — 21 degrees of freedom) 68.14 (.000) NOTE: !iAgainst the alternative hypothesis of a nonstationary bivariate MA(2). Table 9 Summary of Estimated Bivariate MA(2) Representation of Experience—Adjusted Changes in Earnings and Hours: Semi—Annual Data from SIME/DIME 1. Goodness—of—fit statistic — MA(2) (probability value — 60 degrees of freedom) 2. Test statistic for symmetric tfA(2)L" (probability value — 2 degrees of freedon) 74.98 (.092) 5.90 (.434) 39.95 3. Test statistic for MA(1)(probability value — 24 degrees of freedom) (.022) 4. Test statistic for stationary MA(2)" (probability value — 65 degrees of freedom) 100.70 (.000) NOTE: VAgainst the alternative hypothesis of a nonstationary bivariate MA(2). Average Covariance of Change in Earnings with Experience Average Covariance of Change in Hours with Experience (C1) Implied Estimate of Substitution Elasticity (ii) Goodness—of—fit Statistic (probability value) [degrees of freedom] 2. 3. 4. NOTE: PSID NLS, and SIME/DIME [14] [8] [18] 1"Changes in earnings and hours between 1966 and 1967 are expressed at two—year rates for comparability with later changes. 20.78 (.107) .56 (.47) 40.96 (.000) (.61) (.04) —.05 (.04) —.13 25.80 (.104) —1.62 (.44) (.02) 1.52 —.09 (.02) (.02) (.03) —.13 —.03 NLE4" SIME/DIME (six—month changes) (annual changes over two—year gap) —.21 (annual changes) 1. (Cg) PSID, (standard errors in parentheses) with Potential Experience: Covariances of Changes in Log Earnings and Changes in Log Hours Table 10 Table 11 Implied Covariances of Experience — Adjusted Changes in Earnings and Hours: Labor Supply Model 1. var (A log gft) + 2. var (A log hit) 2n(l+n) coy (Azj, jt + 202 var (Azi) + + 21,2 coy 3. coy (A log var (1+1)2 var (Azj) + A log — 4. coy (A log A 5. coy (A log A log 1itl 6. coy (A log log 11) A (cit) (Azi, e) + log hii), 202 var n(1+n) var (Azi) + + - var n(l+2n) coy (Azj. 12 + 2 (l+ri) coy (Azj, Azii) — (cit 2paa 2 coy (Azj, Azit_1) — n(l+n) coy (Azi, Azi1) — uv0u0v coy (A log it' A log j1) 7. coy (A log A log gj_) 8. coy (A log hi A log h2) 9. coy (A log A log hi2), (1+1)2 coy (Azt, Azi_2) coy (Azj, Az.2) n(l+n) coy (Azi, Az12) coy (A log hi A log Notation: A log g and A log h refer to the residuals of the changes in the logarithm of earnings and hours from regressions on potential experience. represents the variance of the measurement error of earnings, 2 represents the variance of the measurenent error of hours, and uy represents their correlation. Goodness—of—Fit Statistic (probability value) [degrees of freedoml NOTES: is (0.772) 0.29 8,83 (2.63) 11.30 (14.59) a unit contribution to hours. fit to autocovariances of changes in earnings and hours up to second order. i (4.81) 7.27 (0.11) (0.03) (0.09) 1.68 (0.093) —19.11 (28.45) 0.95 (0.08) [48] [48] 1.09 [68] [681 61.38 (.027) Optimal Minimum Distance 62.26 (.026) 1.11 261.12 (.000) 261.15 (.000) Equally Weighted Minimum Distance uncertainty Model (Zero—Order Covariances Unrestricted) 1.14 Optimal Minimum Distance Equally Weighted Minimum Distance VProductivity component is normalized to have A"Model ON!) estimate of (probability value) vs. 4. Specification test of EWMT) Substitution Elasticity (standard error) 3. Implied Intertemporal 2/ Component to Earnings(p)— (standard error) MA(2) Productivity 2. Relative Contribution of 1. - Perfect Annual Data from Estimated Two—Factor Model of Changes in Foresight Model Covariances Restricted) (Zero—Order - of Earnings and Hours: Summary Table 12 estimate of p of ——— 5.86 (4.11) 3/ n/a— 0.15 (0.06) 7.71 (2.79) ——— (1.53) 1.61 1.56 (0.59) [161 [28] 1.17 31.38 (0.026) 1.38 (0.167) —6.80 (13.45) 0.85 (0.29) 116] 31.21 (0.027) Optimal Minimum Distance order. Equally Weighted Minimum Distance Uncertainty Model (Zero—Order Covariances Unrestricted) 94.49 (0.000) Optimal Minimum Distance (0.12) [281 124.13 (0.000) Equally Weighted Minimum_Distance Perfect Foresight Model (Zero—Order Covariances Restricted) and Hours: Annual Data front NLSU Estimated Two—Factor Model of Changes in Earnings Sumtnary Table 13 'Model is fit to autocovariances of changes in earnings and hours up to second VProductivity component is normalized to have a unit contribution to hours. --Test statistic cannot he computed. (probability value) OMI) Specification test of EWMD vs. NOTES: 4. Implied Intertemporal Substitution Elasticity (standard error) Relative Contribution of MA(2) Productivity 2/ Component to Earnings(p)—(standard error) 2. 3. Goodness—of—Fit Statistic (probability value) [degrees of freedom] 1. "C Goodness—of—Fit Statistic (probability value) [degrees of freedom] is 227.19 (0.000) [52] 229.36 (0.000) [52] Changes in . (0.124) 1.54 (7.23) 16.67 ——— 0.002 (0.998) 123.00 (438.74) (0.03) (0.07) 125.00 (1140.63) 1.01 1361 1361 1.01 (0.009) 56.98 (0.014) 58.86 Optimal Minimum Distance Equally Weighted Minimum Distance Uncertainty Model (Zero—Order Covariances Unrestricted) fit to autocovariances of changes in earnings and hours up to secondorder. ——— —55.55 (178.98) 1.06 (0.03) Optimal Minimum Distance Equally Weighted Minimum Distance Perfect Foresight Model (Zero—order Covariances Restricted) 0.98 (0.06) of Semi—Annual Data from SIME/DINEII' Estimated Two—Factor Model FProductiviry component is normalized to have a unit contribution to hours. !"Model Specification test of EWMD vs. OMD estimate of p (probability value) 4. NOTES: Implied Intertemporal Substitution Elasticity (standard error) 3. MA(2) Productivity 2/ Component to Earnings(p)— (standard error) 2. Relative Contribution of 1. of Earnings and Hours: Summary Table 14