The Extended Semiparametric (ESP) Model AER , 2018 by Robert Moffitt and Sisi Zhang James J. Heckman University of Chicago Econ 312, Spring 2019 Heckman Appendix – (ESP) Model
III. Some New Results on Trends in Male Earnings Volatility Heckman Appendix – (ESP) Model
• Instead, we specify the transitory component to be a − 1 � v ia = ε ia + ψ a , a − s ε i , a − s (1) s =1 • T ( T + 1) / 2 − T parameters ψ a , a − s . Heckman Appendix – (ESP) Model
• Figures 1 and 2 show the trends in a and β , respectively. Heckman Appendix – (ESP) Model
Figure 1: Extended Semiparametric (ESP) Model Estimates of Alpha 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Heckman Appendix – (ESP) Model
Figure 2: Extended Semiparametric (ESP) Model Estimates of Beta 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Heckman Appendix – (ESP) Model
• Figures 3 shows trends in the percentile points of the distribution of the 2-year change, showing that the increasing volatility reflects a widening out at all percentile points but with the largest widening occurring at the top and bottom of the change distribution. Heckman Appendix – (ESP) Model
Figure 3: Percentiles of 2-Year Difference in Male Log Earnings Residuals 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 90th Percentile 75th Percentile 50th Percentile 25th Percentile 10th Percentile Heckman Appendix – (ESP) Model
• Letting y iat be the log earnings residual for individual i at age a in year t , our model is y iat = a t µ ia + β t v ia (2) where µ ia is the permanent component for individual i at age a , v ia is the transitory component for individual i at age a , and a t and β t are calendar time shifters for the two components. • We shall maintain the usual assumption in these models that the permanent and transitory components are additive and independently distributed, an assumption that can be partially relaxed. • We also adopt the common specification that calendar effects do not vary with age, although this could be relaxed by allowing the calendar time shifts to vary with age (but we will not do that here). Heckman Appendix – (ESP) Model
• To make this definition operational, we will assume that the permanent component at the start of the life cycle is µ 0 and that an individual experiences independently distributed permanent shocks ω 1 , ω 2 , . . . , ω T through the end of life at time T . • We let the permanent component at age a be some function of these shocks: µ ia = f ( ω i 1 , ω i 2 , . . . , ω ia , µ 0 ). • We define a permanent shock ω is to be one for which ∂µ ia /∂ω ia = 1 and we assert that the only function f which satisfies this condition is the unit root process a � µ ia = µ i 0 + ω is . (3) s =1 Heckman Appendix – (ESP) Model
• Let y iat be the earnings residual from a Mincer equation for individual i at age a in year t , the model is: y iat = α t µ ia + β t υ ia (4) a � µ ia = µ i 0 + ω is (5) s =1 a − 1 � υ ia = ε ia + ψ a , a − s ε i , a − s for a ≥ 2 (6) s =1 v i 1 = ε i 1 for a = 1 (7) a = 1 , ..., A and t = 1 , ..., T . • Shocks ω ia and ε ia are independently distributed from each other and over time. Heckman Appendix – (ESP) Model
• The autocovariances implied by this model, • Fit to the autocovariances in the data: Var ( y iat ) = α 2 t Var ( µ ia ) + β 2 t Var ( υ ia ) (8) a � Var ( µ ia ) = Var ( µ i 0 ) + Var ( ω is ) (9) s =1 a − 1 � ψ 2 Var ( υ ia ) = Var ( ε ia ) + a , a − s Var ( ε i , a − s ), for a ≥ 2 (10) s =1 Var ( υ i 1 ) = Var ( ε i 1 ), for a = 1 (11) Cov ( y iat , y i , a − τ, t − τ ) = α t α t − τ Cov ( µ ia , µ i , a − τ ) (12) + β t β t − τ Cov ( υ ia , υ i , a − τ ) . . . Heckman Appendix – (ESP) Model
Cov ( µ ia , µ i , a − τ ) = Var ( µ i , a − τ ) a − τ (13) � = Var ( µ i 0 ) + Var ( ω is ) s =1 Cov ( υ ia , υ i , a − τ ) = ψ a , a − τ Var ( ε i , a − τ ) a − τ − 1 � + ψ a , a − τ − s ψ a − τ, a − τ − s Var ( ε i , a − τ − s ), for a ≥ 3 s =1 (14) Cov ( υ ia , υ i , a − τ ) = ψ a , a − τ Var ( ε i , a − τ ) (15) = ψ 21 Var ( ε i 1 ), for a = 2 , τ = 1 Heckman Appendix – (ESP) Model
• Allow the variances of the permanent and transitory shocks to be nonparametric functions of age, • Allow the ψ parameters to be nonparametric functions of age and lag length ( τ or τ + s ). Heckman Appendix – (ESP) Model
• Identification. Consider first the identification of the parameters of the age-earnings process under the stationary model α t = β t = 1, • Note that a data set of age length a = 1 , ..., A has an autocovariance matrix of the y ia with A ( A + 1) / 2 elements. • The unknown parameters in the model are σ 2 µ 0 , the A parameters σ 2 ω a ( a = 1 , ..., A ), the A ( A − 1) / 2 parameters ψ a , a − r ( r = 1 , .., a − 1), and the A parameters σ 2 ε a ( a = 1 , ... A ), for a total of [ A ( A + 1) / 2] + A + 1 parameters. • Stationary model nonparametrically not identified without A + 1 restrictions. Heckman Appendix – (ESP) Model
• Allow restrictions by imposing smoothness on the nonparametric functions σ 2 ω , ψ , and σ 2 ε as described below. • Our estimation shows that the number of parameters needed to fit the data allow the model to be heavily overidentified. • The α t and β t parameters are identified, subject to a normalization and conditional on the identification of the parameters of the age-earnings process, from the change in the autocovariance matrix elements at the same age and lag position but at different points in calendar time, which therefore requires multiple cohorts. • Since α t and β t constitute two parameters, any two elements of the matrix observed at two calendar time points is sufficient for identification. Heckman Appendix – (ESP) Model
• For example, using the variances at ages a and a ′ observed at times t and t + 1, we have Var ( y iat ) = α 2 t σ 2 µ a + β 2 t σ 2 (16) υ a Var ( y ia ′ t ) = α 2 t σ 2 µ a ′ + β 2 t σ 2 (17) υ a ′ Var ( y ia , t +1 ) = α 2 t r 2 α σ 2 µ a + β 2 t r 2 β σ 2 (18) υ a Var ( y ia ′ , t +1 ) = α 2 t r 2 α σ 2 µ a ′ + β 2 t r 2 β σ 2 (19) υ a ′ where r α = α t +1 /α t and r β = β t +1 /β t . Heckman Appendix – (ESP) Model
• We normalize the calendar shifts at t = 1 by setting α 1 = β 1 = 1. Equations (13)-(16) can be solved for α t and β t for t = 2 , ..., T . Heckman Appendix – (ESP) Model
• Nonparametric Estimation . To estimate the functions σ 2 ω a , σ 2 ε a , and ψ , specify the functions as series expansions in basis functions and use a generalized cross-validation (GCV) statistic, which has a penalty for the number of parameters, to choose the degree of the expansion. • Specific functional forms are: Var ( ω ir ) = e Σ δ j ( r − 25) j (20) Var ( ε ir ) = e Σ γ j ( r − 25) j , for r ≥ 2 (21) Var ( ε i 1 ) = ke Σ γ j (1 − 25) j , for r = 1 (22) ψ A , A − b = [1 − π ( A − 25)][Σ w j e − λ j b ] + Σ η j D ( b = j ) (23) Heckman Appendix – (ESP) Model
• The variances use exponential functions of polynomial expansions in age minus 25 (the approximate minimum age), with the initial transitory variance allowed to differ by factor k for an initial conditions adjustment. • The ψ parameters are allowed to expand in a weighted sum of exponentials, which force the parameters to asymptote to 0 as the lag length goes to infinity, and with a linear age-function factor in front of that weighted sum. • Deviations from the smooth exponential expansions are allowed at each lag length. • The unknown parameters in the model are Var ( µ i 0 ) , δ j , γ j , k , π, λ j , w j , and the η j as well as the α t and β t . • The parameters are fit to the second-moment matrix of the data using minimum distance. Heckman Appendix – (ESP) Model
• As is often the case using the PSID, only a small number of basis functions in the expansion improve the parameter-adjusted fit. • The initial variance of the permanent component is significant but the variances of the permanent shocks do not vary with age. • The transitory variance is also weakly positive in a linear function of age. Heckman Appendix – (ESP) Model
• The initial transitory variance is over twice the size as subsequent transitory shocks (as expected) but the transitory autocovariance curve is only weakly (and negatively) correlated with age and with only a single exponential. Heckman Appendix – (ESP) Model
• The λ parameter confirms that autocovariances decline with lag length and the η parameters indicate that the most recent three lags have a different impact on the current transitory component than the age-adjusted smooth exponential curve indicates. • The estimates of the α and β parameters are also shown; the figures in the text are plots of these estimates. • The second column in the Table shows the estimates of the parameters if a model stationary in calendar time is estimated (i.e., constraining α t = β t = 1). Heckman Appendix – (ESP) Model
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