Estimating Micro-Data Measurement Errors in Hours Worked and Hourly Wages Zvi Hercowitz∗ January 2009
Abstract Hourly wages in survey micro data is usually computed as the ratio of salary to the reported number of hours worked. This is the procedure used, for example, in the PSID (Panel Study of Economic Dynamics). Because any error in the reported number of hours translates into an opposite error in the hourly wage, it has been noted that the available micro data is likely to contain an artificial negative correlation between hours worked and hourly wages. In the labor supply literature, the usual procedure to circumvent this problem is to instrument the wage by the lagged wage– assuming that the error is serially uncorrelated. Here, a direct procedure for estimating the measurement error is proposed and implemented with PSID data. The procedure is based on a signal extraction calculation using the available data on hours and wages in two subsequent years.
∗ Contact information: Berglas School of Economics, Tel Aviv University, Tel Aviv 69978, Israel. Email:
[email protected], Tel: 972-3-6409916, Fax: 972-3-6409908. I thank Itay Saporta for excellent research assistance, and the Pinhas Sapir Center for Development and the Foerder Institute at Tel Aviv University for financial support. JEL Codes: C81, D12. Key words: measurement errors, micro-data hourly wages.
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1
Introduction and Conclusion
In the PSID (Panel Study of Income Dynamics) and in other micro data sources, the main hourly wage variable is computed as the ratio of labor income to hours worked. Hence, possible errors in the number of hours worked reported induce errors in the opposite direction in the measured hourly wages. Abowd and Card (1989) find that the cross-section correlations between changes in hours worked and hourly wages are negative, and associate this finding to the artificial negative correlation generated by the measurement errors in computing hourly wages. The possibility of such artificial negative correlation between hours worked and the hourly wage presents a problem, for example, for estimating the elasticity of labor supply to the wage. Altonji (1984) deals with this problem by instrumenting the computed wage using the hourly wage of hourly rated workers. This variable is unavailable for salaried workers and self employed; hence, the sample is largely reduced. Furthermore, given that hourly rated workers are likely to be more temporary and/or unskilled than the other workers, the sample is then biased. Blundell and MaCurdy (1999) discuss the implications of this type of measurement error for the estimation. They stress the particular form of the empirical error term–which includes the measurement error–for its distribution and for choosing instrumental variables. For example, if the measurement error is serially uncorrelated, the lagged observed wage is a valid instrument because it is orthogonal to the measurement error. Another context where the wage/hours measurement errors is problematic is the estimation of an autoregressive process for the real wage. These errors should make the measured wage appear less persistent and more volatile than the true wage. Unlike for labor supply estimation, however, using the lagged wage as instrument is obviously not a viable route in this case. This note suggests a procedure for inferring the measurement errors directly. The main purpose is to measure the magnitude of these errors. One may also correct the reported wage with the estimated errors and use the adjusted data for estimation. This note, however, does not address the properties of the estimated coefficients when using the corrected wages. The proposed procedure is based on a signal extraction calculation which exploits the sharp features of the variance-covariance structure of hours and wages generated by this measurement error in estimated AR(1) processes. Assuming that this error is white noise with constant percentage variance, the procedure is based on the following considerations: An error of percentage ξ should affect the current AR(1) residuals of log wage and log hours by ξ and −ξ, and–as shown later–the next period residuals by −ˆ ρξ and γˆ ξ, where ρ ˆ and γˆ are the estimated AR(1) coefficients obtained from the available wage and hours data. The procedure formulated here applies to a partial equilibrium setup of differentiated households. The wage process can be thought of as either exogenous or endogenous. The framework is applicable to data with a panel structure as the PSID, where both current and lagged values of wages and hours are available. 2
The implementation of the procedure with PSID data on married household heads for the period 1968-1997, for which the data is annual, indicates that the hours worked/hourly wage errors are substantial. The variance of the estimated errors as a fraction of the variance of reported wages–around levels determined by age, age squared, years of schooling and other deterministic variables–is 15 percent. The implications of correcting the wage data with the estimated errors are then illustrated by two examples. One is the AR(1) estimation of the individual wage process, and the other is computing the cross section correlations of changes in wages and hours–as in Abowd and Card (1989). In both examples, the results using corrected data differ substantially from those with the original data, and in the expected directions. The AR(1) estimates from corrected data show higher persistence and lower volatility of the innovations. The cross-section correlations, which are negative in all years in the sample with the original data, become positive in all years. The discussion proceeds as follows. Section 2 describes the procedure proposed. Then its implementation with PSID data is reported in Section 3. Section 4 concludes illustrating the implications of correcting the wage data with the estimated ion of
2 2.1
The Procedure for Estimating the Measurement Error The data
The basic assumption about the available data is that dividing total labor earnings into hours worked and hourly wages is subject to an error of percentage ξ t , which is white noise with a constant variance σ 2ξ . Total labor earnings are ∗ assumed to be measured precisely. Hence, defining w ¯it and n ¯ ∗it as the logs of the true hourly wage and true hours worked, and w ¯it and n ¯ it as the observed counterparts, we have ∗ w ¯it +n ¯ ∗it w ¯it n ¯ it
2.2
= w ¯it + n ¯ it , ∗ = w ¯it + ξ it , = n ¯ ∗it − ξ it .
(1)
The Framework
Assume that the solution to the dynamic optimization problem of individual i includes the following processes for the hourly wage and hours worked: ∗ w ∗ w ¯it = αw 0 + α1 zit + wit ,
n ¯ ∗it = αn0 + αn1 zit + n∗it ,
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∗ ∗ wit = ρwit−1 +ω ˜ it ,
(2)
n∗it = γn∗it−1 + ν˜it ,
(3)
where zit represents deterministic variables measured without error, the α1 ’s are the corresponding vectors of coefficients, ω ˜ it , ν˜it are white noise shocks, and γ, ρ represent the persistence of these shocks. The wage process can be thought of as exogenous, or part of the solution given exogenous shocks. The autoregressive structure of the “detrended” wages and hours–or deviations from the systematic components, determined by age, experience, etcetera–is important for the current purposes, as we see below. In any event, the procedure allows for the special case where γ and/or ρ are zero. This procedure can in principle be extended to a richer time-series specification of the true dynamics of the model. Given the available data, however, any such extension will reduce the sample significantly. Let us consider first the wage equation, where instead of the true variables we use the observed variables as in the standard errors-in-variables case. Using ∗ (1), the process for wit in (2) can be expressed as ¡ ¢ wit − ξ it = ρ wit−1 − ξ it−1 + ω ˜ it , or
wit = ρwit−1 + ω it ,
ω it = ω ˜ it − ρξ it−1 + ξ it .
(4)
Using (1) and (3), a similar equation applies to hours worked: nit = γnit−1 + ν it ,
ν it = ν˜it + γξ it−1 − ξ it .
(5)
Note that the current and the lagged measurement errors enter (4) and (5) with opposite signs within each equation, and with opposite signs across equations. The opposite signs across equations follow directly from the basic problem addressed here of dividing labor income into hours worked and hourly wages. The opposite signs on the current and lagged errors within each equation follow from the assumed autoregressive structure of the model: In the current period, a positive error naturally increases the current wage residual. In the following period, the same error decreases the residual because lagged variable on the right hand side is too large. The opposite holds for the residual in hours. w The OLS estimates of αw 0 and α1 should be unbiased because zit is measured precisely. However, the measurement error does affect the estimates of ρ and γ. Given that the error is white noise, the OLS estimates of ρ and γ are ρ ˆ=ρ
∗ V ar (wit ) , ∗ V ar (wit ) + V ar (ξ it )
γˆ = γ
V ar (n∗it ) , V ar (n∗it ) + V ar (ξ it )
which reflect the classical error-in-variables bias problem. Correspondingly, the
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OLS residuals from (4) and (5) can be written as1 ω it = μit − ρ ˆξ it−1 + ξ it ,
(6)
ν it = υit + γˆ ξ it−1 − ξ it ,
(7)
where μit υ it
∗ = (ρ − ρ ˆ) wit−1 +ω ˜ it , ∗ = (γ − γˆ ) nit−1 + ν˜it .
In (6) and (7), the terms μit and υit come from the model–the true lagged detrended variables and true innovations –and the other two terms appear because of the measurement errors–current and lagged. The structure of equations (6) and (7) is the following. The left-hand sides, ω it and ν it , are observed given that they are the residuals from the AR(1) equations (4) and (5) estimated with available data. On the right-hand sides, the three terms appearing in each equation are mutually independent. However, the composition of the right-hand sides among the model’s terms and the measurement errors is unknown. Across equations, the residuals should be correlated given that the errors ξ it and ξ it−1 appear in both, and because of the theoretical correlation between the model’s μit and υ it . Now we move to the main point in this note, which is to use the structure above to solve the signal extraction of ξ it , i.e., to separate the measurement error from the model’s terms in (6) and (7). The measurement error enters these two residuals in a very specific form: (1) It has opposite signs in wage and in hours, (2) Given serial correlation, also the lagged error enters. These features imply the following general criteria for detecting measurement errors from movements in measured wages and hours worked: (a) Wage and hours worked move simultaneously in opposite directions. (b) These movements are partially undone in the following period. The signal extraction problem consists in computing the coefficients of the least-squares forecast of the measurement error given the observed ω it , ν it , ω it+1 and ν it+1 . This forecast has the form ˆξ = θ1 ω it + θ2 ν it + θ3 ω it+1 + θ4 ν it+1 , it
(8)
where, eliminating the subscript i from now onwards, the coefficients are the 1 The
wage residual is ω it
=
ω it
=
ω it
=
ω it
=
wit − ρ ˆwit−1 ,
ρwit−1 + ω ˜ it + ξ it − ρξ it−1 − ρ ˆwit−1 , ∗ ∗ ρwit−1 + ω ˜ it + ξ it − ρ ˆ wit−1 + ξ it−1 ,
∗ (ρ − ρ ˆ) wit−1 +ω ˜ it − ρ ˆξ it−1 + ξ it .
A similar calculation applies to the residuals of hours.
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solution to the system ⎡ ⎤ ⎡ θ1 V ar(ω t ) Cov(ω t , ν t ) Cov(ω t , ω t+1 ) Cov(ω t , ν t+1 ) ⎢ θ2 ⎥ ⎢ Cov(ν t , ω t ) V ar(ν ) Cov(ν , ω ) Cov(ν t , ν t+1 ) t t t+1 ⎢ ⎥ ⎢ ⎣ θ3 ⎦ = ⎣ Cov(ω t+1 , ω t ) Cov(ω t+1 , ν t ) V ar(ω t+1 ) Cov(ω t+1 , ν t+1 ) Cov(ν t+1 , ω t ) Cov(ν t+1 , ν t ) Cov(ν t+1 , ω t+1 ) θ4 V ar(ν t+1 )
or, in a symmetric form, ⎡ ⎤ ⎡ θ1 V ar(ω t ) Cov(ν t , ω t ) Cov(ω t , ω t+1 ) Cov(ω t , ν t+1 ) ⎢ θ2 ⎥ ⎢ Cov(ω t , ν t ) V ar(ν t ) Cov(ω t+1 , ν t ) Cov(ν t , ν t+1 ) ⎢ ⎥ ⎢ ⎣ θ3 ⎦ = ⎣ Cov(ω t , ω t+1 ) Cov(ω t+1 , ν t ) V ar(ω t ) Cov(ω t , ν t ) Cov(ω t , ν t+1 ) Cov(ν t , ν t+1 ) θ4 Cov(ω t , ν t ) V ar(ν t )
⎥ ⎥ ⎦
Cov(ω t , ξ t ) = σ 2ξ , Cov(ω t+1 , ξ t ) = −ˆ ρσ 2ξ , Cov(ν t+1 , ξ t ) =
(10)
γˆ σ 2ξ .
Hence, to complete the signal extraction calculation in (9)-(10) all we still need is σ 2ξ . Of course, σ 2ξ is unknown until we compute the errors. Hence, we have a simultaneity problem that we need to resolve in order to identify the system. Let us proceed by looking at the covariance structure of the observed ω t and ν t according to (6) and (7): Cov(ω t , ν t ) = Cov(μt , υt ) − (1 + ρ ˆγˆ ) σ 2ξ .
(11)
Note that this expression treats ρ ˆ and γˆ as known. A refinement of the current procedure is to consider them as estimates with standard errors. In any event, in the implementation reported below, these parameters are estimated with high precision.2 As we can see in these expressions, identifying σ 2ξ requires knowing the stochastic structure of the true variables. Given the basic problem that these variables are unobservables, we adopt an iterative procedure. We start by assuming temporarily that Cov(μt , υ t ) = 0, i.e., that the true innovations to hours and wages are uncorrelated. Then, from (11) we obtain an initial value for σ 2ξ . With this value, an initial signal extraction calculation can be completed to produce an initial set of ξ t values. Using these values in (6) and (7), we obtain first estimates of μt and υt , from which their covariance can be computed. We then replace the initial zero covariance by this estimate, and recompute the values of ξ t . This procedure is repeated until convergence is reached. The details of this iterative procedure are presented in the next subsection. 2 As reported in Tables 3 and 4, the t-statistics of these coefficients, using robust standard errors, are 133 and 59.
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⎥ ⎥ ⎦
⎤−1 ⎡
(9) The residuals from regressions (4) and (5) can be used to estimate the matrix of moments in (9). Then, the assumption that the measurement error is uncorrelated with the true innovations can be used to express the vector at the right of (9) as Cov(ν t , ξ t ) = −σ 2ξ
⎤−1 ⎡
⎤ Cov(ω t , ξ t ) ⎢ Cov(ν t , ξ t ) ⎥ ⎢ ⎥ ⎣ Cov(ω t+1 , ξ t ) ⎦ , Cov(ν t+1 , ξ t )
⎤ Cov(ω t , ξ t ) ⎢ Cov(ν t , ξ t ) ⎥ ⎥ ⎢ ⎣ Cov(ω t+1 , ξ t ) ⎦ . Cov(ν t+1 , ξ t )
2.3
Estimation Steps
1. “Detrending” regressions. Regress w ¯t and n ¯ t on a constant and zt to obtain the “detrended” variables wt and nt (the first part of equations (4) and (5)). 2. AR(1) regressions. Regress wt and nt on their own lagged values without a constant to obtain ρ ˆ and γˆ and the residuals ω t and ν t . (the second part of equations (4) and (5)). Clearly, this and the previous regression can in principle be estimated simultaneously.3 3. Initial signal extraction calculation. To start with, assume that Cov(μt , υ t )0 = 0. Then, from (11) it follows that ¡ 2 ¢0 −Cov (ω t , ν t ) + Cov(μt , υ t )0 . σξ = 1+ρ ˆγˆ 4. Compute θ01 , θ02 , θ03 , θ04 using (9) and (10). 0 5. Compute the initial set of values ˆξ t using (8), (4) and (5). This set of values has a panel structure as the other data. 0 6. Updating the signal extraction calculation. Using the resulting series ˆξ t – as well as ρ ˆ, γˆ , and ω t , ν it from (4) and (5)–in (6) and (7), compute
μ0t υ 0t
= ωt + ρ ˆξ 0t−1 − ξ 0t , = ν t − γˆ ξ 0t−1 + ξ 0t .
7. Compute Cov(μt , υ t ) using the resulting values from 6. This bring us back to item 3. Replace the previous value of this covariance with the new one, and update σ 2ξ : ¡ 2 ¢1 −Cov (ω t , ν t ) + Cov(μt , υ t )1 σξ = . 1+ρ ˆγˆ 8. Back to 4. Update the signal extraction coefficients to θ11 , θ12 , θ13 , θ14 . 1 9. Continue to 5. Recalculate the measurement errors ˆξ t , and repeat the steps from 6 to 9 until σ 2ξ converges. In the implementation reported below, convergence was fairly fast. 3A
practical reason for sequential estimation is given below in Section 3.
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3
Implementation
The estimation proposed above is implemented using PSID data for married household heads, i.e., for households composed at least of a "head" and a "wife", for the period 1968-1997 for which the data is annual. The wage corresponds to the hourly wage of the household’s head, computed dividing the nominal value by the consumption price index from NIPA. The set of deterministic variables in z are: age, age squared, school years, race dummies, and year dummies. Extreme wage and hours observations are not included in the sample. Observations of more than 6000 annual hours worked, or with unreasonably high or low hourly wages are deleted. The procedure for selecting extreme wage observations is the following. First, the mean and standard deviations of the middle 50 percentiles of log wages, (i.e. observations above the 25 percentile and below the 75 percentile) are calculated for each year in the sample. Then, observations with more than 10 standard deviations above or below the mean are dropped. Applying this procedure to the middle 50 percentiles avoids including the unreasonable observations in the selection criterion.
3.1
“Detrending” Regressions
The advantage of estimating sequentially the detrending and the AR regressions is to use the largest possible sample in each step. For the detrending step only the current wage–and of course the z variables as well–is needed, while for the AR(1) estimation also the lagged wage for the same households is needed. The sample with two sequential wages is, as shown below, significantly smaller. The results for the detrending regressions are shown in Tables 1 and 2. Table 1 shows the typical inverted-U shape for life-time wages. According to the coefficients of age and age squared, the maximum wage corresponds to age 46. The coefficient on school years of 7.7 percent matches the usual size of the return to schooling found in the labor literature. Table 2 shows also a U-shape pattern for hours worked–here the maximum corresponds to 38 years–and a 1.2 percent effect of school years on hours worked.
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Table 1 Dependent variable: log(wage) Sample: 1968-1997 — Married household heads∗ Robust Variable Coefficient t-statistic st. error Age 0.0737 0.0025 28.9 Age2 −0.0008 0.0000 −25.1 School years 0.0767 0.0021 37.2 Race and year dummies included # of obs: 83596 # of clusters: 9379 Root MSE: 0.56 ∗ In the current and previous year
Table 2 Dependent variable: log(hours) Sample: 1968-1997 — Married household heads∗ Robust Variable Coefficient t-statistic st. error Age 0.0687 0.0029 23.8 Age2 −0.0009 0.0000 −24.6 School years 0.0120 0.0014 8.4 Race and year dummies included # of obs: 89255 # of clusters: 9647 Root MSE: 0.52 ∗ In the current and previous year
3.2
AR(1) Estimation
Tables 3 and 4 show the estimates of the AR(1) coefficients of the deviations from the deterministic wage profiles. In particular, the coefficient of the lagged wage is 0.72, which, according to the error-in-variables presumption should be biased downwards. Also, the standard error of 39 percent should be biased upwards. Hence, if the wage evolves exogenously, this presumption implies that households face less short-run uncertainty about wages than reflected in Table 3. The process for hours in Table 4 should reflect the same presumptions regarding persistence and volatility.
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Table 3 Dependent variable: detrended log(wage) Sample: 1969-1997 — Married household heads∗ Robust Variable Coefficient t-statistic st. error Lagged detrended log(wage) 0.721 0.0054 133.2 # of obs: 69523 # of clusters: 8045 Root MSE: 0.39 ∗ In the current and previous year
Table 4 Dependent variable: detrended log(hours) Sample: 1969-1997 — Married household heads∗ Robust Variable Coefficient t-statistic st. error Lagged detrended log(hours) 0.561 0.0094 59.4 # of obs: 77715 # of clusters: 8234 Root MSE: 0.41 ∗ In the current and previous year
3.3
Signal Extraction of the Measurement Error
The residuals from the regressions in Table 3 and 4, ω t and ν t for wages and hours, are used next to compute the covariance matrix of ω it , ν it ω it+1 and ν it+1 , required for the signal extraction calculation in equation (9). The resulting covariance matrix is ωt νt ω t+1 ν t+1
ωt 0.151 −0.038 −0.027 0.012
νt
ω t+1
ν t+1
0.152 0.023 −0.013
0.151 −0.038
0.152
νt
ω t+1
ν t+1
1 0.152 −0.085
1 −0.251
.
The corresponding correlation matrix is ωt ωt νt ω t+1 ν t+1
1 −0.251 −0.179 0.079
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1
The negative contemporaneous correlation between the residuals of wages and hours, and the negative serial correlations of residuals, suggest the presence of the measurement errors. In the absence of these errors we would expect positive signs. The contemporaneous correlation would be positive from a positive relationship between the true fluctuations of the wage and hours worked. We would also expect positive serial correlation of both variables given the likelihood of persistence in wages and hours worked.4 The iterative procedure described in Subsection 2.3 convergences in about 10 steps to a high degree of precision. The estimated variance of the measurement error is σ 2ξ = 0.0467, corresponding to a standard deviation of 21.6 percent. Compared to the standard deviation of the measured detrended wage of 56 percent, the measurement errors are quite large. The resulting signal extraction coefficients are θ1 = 0.222, θ2 = −0.224, θ3 = −0.123, θ4 = 0.105. Hence, the numerical counterpart of equation (8) for estimating the measurement error is ˆξ = 0.222ω t − 0.224ν t − 0.123ω t+1 + 0.105ν t+1 . t
(12)
The next step is to illustrate how the current procedure will perceive: (i) a measurement error and (ii) a true innovation in the wage. The calculation involves (12), the structure of the observed residuals in (6) and (7), ωt νt
= μt − ρ ˆξ t−1 + ξ t , = υ t + γˆ ξ t−1 − ξ t ,
and the AR(1) estimated coefficients ρ ˆ = 0.72, γˆ = 0.56. (i) Assume a measurement error of 1 percent in period t, i.e., ξ it = 1, while μt = υ t = 0. From (12) and (??) we have ˆξ = 0.222 + 0.224 + 0.123 × 0.72 + 0.105 × 0.56 = 0.593. it Hence, about 60 percent of the error will be detected as such. (ii) Assuming it is exogenous, consider now a 1 percent true increase in the wage in period t, i.e., μt = 1, while ξ t = 0. Theoretically, an exogenous change in the wage would induce an endogenous change in hours worked. To assign a value to such change in hours, υt , one can use the estimated structural residuals, μ ˆt, υ ˆ t , resulting from the signal extraction calculation as follows. The covariance matrix of these structural residuals is μ ˆt υ ˆt
μ ˆt 0.062 0.027
υˆ t 0.064
4 The positive sign of the cross lagged correlations is also consistent with the presence of the measurement errors–as it can be seen from (6) and (7)–but the sign could also be due to the a positive cross lagged correlation among the true variables.
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Note that these moments make possible to compute the least-squares regression coefficient of detrended hours worked on the detrended wage–or a labor supply elasticity. This coefficient is 0.027363/0.061647 = 0.44. The size of this coefficient is within the range of the labor supply elasticities estimated by MaCurdy (1981). However, unlike in that study which focused on the compensated elasticity, this coefficient corresponds to a gross elasticity. The signal extracted error is then ˆξ = 0.222 − 0.224 × 0.44 = 0.123. it Accordingly, about 88 percent of the wage increase is perceived correctly.
4 4.1
Applications Estimation of an AR(1) Wage Process
This application consists in comparing the AR(1) estimates obtained with the original and the corrected wage data. The sample used in both regressions is the one available for the corrected data, which is smaller because of the data requirements for the signal extraction calculation. Table 5 reports the results with the original data as Table 3 above. The difference between the two tables is the sample, which in Table 5 is smaller. Table 6 presents the results from the corrected data. Table 5 Dependent variable: detrended log(wage) — Original data Sample: 1969-1997 — Married household heads∗ Robust Variable Coefficient t-statistic st. error Lagged detrended log(wage) 0.741 0.0061 122.5 # of obs: 49105 # of clusters: 5754 Root MSE: 0.36 ∗ In the current and previous year Table 6 Dependent variable: detrended log(wage) — Corrected data Sample: 1969-1997 — Married household heads∗ Robust Variable Coefficient t-statistic st. error Lagged detrended log(wage) 0.890 0.0034 261.5 # of obs: 49105 # of clusters: 5754 Root MSE: 0.23 ∗ In the current and previous year 12
Using the corrected wages, therefore, increases the persistence of wage innovations from 0.74 to 0.89, and lowers their standard deviation from 0.36 to 0.23.
4.2
Cross Section Correlations of Changes in Hours and Wage
Table 7 presents the cross-section correlations of the rates of change in hours and the wage, for the years for which these rates of change are available. The column with the original data reproduces the findings in Abowd and Card (1989): the correlations are negative. Remarkably, correcting the wage data yields instead positive correlations for all years in the sample. Table 7 Cross-Section Correlations of Changes in Hours and Wages Year Original Data Corrected Data # of obs 1971 −0.394 0.243 1916 1972 −0.288 0.351 1944 1973 −0.304 0.379 1989 1974 −0.423 0.181 2019 1975 −0.375 0.179 2059 1976 −0.301 0.358 2118 1977 −0.252 0.400 2207 1978 −0.324 0.282 2257 1979 −0.368 0.244 2323 1980 −0.394 0.189 2317 1981 −0.371 0.206 2337 1982 −0.357 0.272 2358 1983 −0.317 0.374 2364 1984 −0.235 0.409 2322 1985 −0.288 0.311 2307 1986 −0.312 0.277 2357 1987 −0.302 0.305 2389 1988 −0.285 0.283 2433 1989 −0.302 0.232 2436 1990 −0.311 0.259 2450 1991 −0.283 0.267 2497 1996 −0.366 0.147 1706
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References [1] Abowd, John M. and David Card (1989) “On the Covariance Structure of Earnings and Hours Changes,” Econometrica 57. [2] Altonji, Joseph, G. (1986) “Intertemporal Substitution in Labor Supply: Evidence from Micro Data,” Journal of Political Economy, 94. [3] Blundell, Richard and Thomas MaCurdy (1999) “Labor Supply: A Review of Alternative Approaches,” Handbook of Labor Economics, edited by O. Ashenfelter and D. Card, 3. [4] MaCurdy, Thomas (1981) “An Empirical Model of Labor Supply in a LifeCycle Setting,” The Journal of Political Economy, 89.
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