On-line Appendix for the paper: Sticky Wages. Evidence from Quarterly Microeconomic Data. Appendix A. Weights used to compute aggregate indicators

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1 Hervé LE BIHAN, Jérémi MONTORNES, Thomas HECKEL On-line Appendix for the paper: Sticky Wages. Evidence from Quarterly Microeconomic Data Not intended for publication Appendix A. Weights ud to compute aggregate indicators The raw obrvations are quences of wages w i;s;k;t, where: i is an index for establishments, i = 1; :::; I s s is an index for ctors, s = 1; :::; S k is an index for reprentative occupation, k = 1; :::; 12 t is a time index, t = 1; :::; T Weights are derived from a strati ed sampling scheme, following the approach ud by the Ministère du Travail to produce an aggregate wage index. Within each ctor, establishments are weighted according to the number of employees. The ctors are in turn weighted according to their respective weights in terms of employees in the economy. Weights are de ned as follows: with! i;s;k = i;s;k s;k i;s;k = n i;s;k n s;k ; the share of employees of the category k in an establishment i; in the ctor s: n s;k denotes the total number of employees of the category k in the ctor s in the sample of the ACEMO survey (averaged over the sample period). And: s;k = N s;k N k ; the share of employees of the category k; in the ctor s at the aggregate level. N k denotes the total number of employees of the category k at the national level. This auxiliary information comes from the national register of establishments Sirene. 1

2 The weights are then ud to produce statistics for the frequency of wage changes. First we de ne a frequency of wage changes at the level of a reprentative occupation F i;s;k = 1 T P T t=1 I i;s;k;t where I i;s;k;t an indicator for a wage change of a job category k in a establishment i at date t: We can compute the average unweighted frequency of wage changes F = 1 P 1 1 Is P S P 12 I s S 12 i=1 s=1 k=1 F i;s;k Now, using weights we compute the weighted average frequency of wage changes F W = P I s P S P 12 i=1 s=1 k=1! i;s;kf i;s;k Similar formulas were ud to produce aggregate indicators on the size of wage changes Appendix B. Alternative indicators of synchronization This appendix complements ction 5.3 of the paper. It investigates a less restrictive de nition of wage change synchronization to compute an indicator of synchronization within establishments. Consider an establishment for which the number of reprentative occupations reported is N cat : We consider as a benchmark the null hypothesis of wage changes being independent across reprentative occupations, and each occurring with a probability p. For a given quarter, the number of wage changes n in that establishment then follows a binomial distribution with parameters (p; N cat ). We then consider that the hypothesis of independence can be rejected, and thus that wage changes can be considered as synchronized within the rm, whenever the obrved n is larger than the 95th percentile of the binomial distribution. The threshold n above which the wage changes are considered as synchronized depends on p and N cat. For instance with p =35%, and a number of reported occupations N cat = 12 we lect n = 7. Thresholds are n = 2 for N cat = 2 or 3; n = 3 for N cat = 4 or 5; n = 4 for N cat = 6 or 7; n = 5 for N cat = 7; 8; n = 6 for N cat = 9; 10; n = 7 for N cat = 11; 12. With this criteria, the share of synchronized wage changes within establishments is much larger than with the restrictive criteria ud in ction 5.3. (where we impod n = N cat to de ne synchronization): it is as large as 73% 2

3 as compared to 44%. We have constructed yet another alternative indicator by considering that wage changes are synchronized if all employees within the same broad occupational category experience a change in wage. Recall that in each establishment 1 to 4 broad occupational category are considered, each of them including a number of 1, 2 or 3 reprentative occupations. Only occupational categories with at least two (and at most 3) occupations were considered. The share of synchronized wage changes we obtain is 74%, hence quite large as well. Note that when restricting the de nition of synchronization to the ca of obrving exactly 3 wages change when 3 reprentative occupation are reported in a broad category, the indicator is equal to 65%. As a benchmark, under the null hypothesis, hence under a binomial distribution with parameter 38% (the average sample frequency), the share of synchronized wage change should be equal to only 38% 3 =(1 (68% 3 )) 7% (the probability of obrving 3 simultaneous wage changes, conditional on obrving one wage change). Appendix C Estimation of the two-threshold sample lection model This appendix derives our estimation method for system (2)-(3)-(4a)-(4b) of the paper. This system is replicated below for convenience: yit = z it + i + " it (1) wit = x it + i + u it (2) where w it = 0 is obrved if v it 1 < yit < v it 2 and w it = wit otherwi. We u a two-step approach following Verbeek and Nijman (1992 a,b), who have extended the Heckman (1979) procedure to models of panel data with random e ects. Given our sample size, the computational advantages implied by using a two step approach rather than a maximum likelihood strategy are substantial, while the loss of e ciency is presumably very limited. We rst consider a model with 2 outcomes that matches Verbeek and Nijman s (1992 a,b) framework, by assuming that w it is only obrved in ca of wage increas, that is when yit > v it 2. The rst step in the procedure consists in estimating the vector of parameters 2 and through maximum likelihood using a Probit model. Conditional on unobrved heterogeneity i ; the probability that wage of individual i is incread at date t is: 3

4 P (w it > 0j i ) = P (wit > v it 2 j i ) = P (" it > v it 2 z it i j i ) = P (" it < z it v it 2 + i j i ) = ((z it v it 2 + i )= " ); where (:) is the cdf of the gaussian distribution. The probability of not obrving a wage increa is P (w it 0j i ) = 1 (( v it 2 + z it + i )= " ): Integrating over unobrved heterogeneity, the likelihood of a trajectory for individual i is: L i = Z +1 1 t=1 TY [(( vit 2 + z it + i )= " )] dit [1 (( v it 2 + z it + i )= " )] 1 dit 1 p e di where T is the number of periods in the sample, and the indicator variable d it is de ned as d it = 1 if w it > 0 and d it = 0 if w it 0: We impo the identifying restriction " = 1: The above integral is computed by Gauss-Hermite quadrature. Estimators of 2 ; ; are then obtained by maximizing the likelihood of the whole sample: NX ln L = ln L i : i=1 The log-likelihood is maximized using a Newton-Raphson algorithm as implemented in the SAS-IML software. The vector of rst derivatives of the likelihood function with respect to parameters and the Hessian matrix are computed numerically. The cond step of the procedure of Verbeek and Nijman (1992 a,b), consists in estimating equation (2) by least squares, with, in the spirit of Heckman (1979), added regressors that corrects for the sample lection bias. Unlike in the cross-ctional ca, there are here two correction terms, which we will denote A 1i and A 2it ; corresponding to the conditional expectations of i and u it given lection. The coe cients on the correction terms re ect the covariances between i and i and between u it and " it : For convenience, we note y it = 2d it 1 so that y it = 1 when w it > 0 and y it = 1 when w it < 0: 4

5 A 1i = A 2it = 1 2 " + T i 2 " TX E f i + " is = y it g (3) s=1 2 E f i + " it = y it g 2 " + T i 2 # TX E f i + " is = y it g (4) s=1 The conditional expectation E f i + " is = y it ; i g is given by: E f i + " is = y it ; i g = Z +1 where E f" it =y it ; i g = y it " ( zit v it 2+ i ) 1 [ i + E f" it = y it ; i g] f( i =y it )d i (5) (y it z it v it 2 + i ) (6) T z it s=1(y it and f( i =y it ) = R +1 1 T s=1 (y it zit v it 2 + i )( 1 )( i ) vit 2 + i )( 1 )( i )d i (7) The vector of parameters can then be consistently estimated by the following linear regression restricted to obrvations with only wage increas (w it > 0): w it = x it + c 1A1i b + c 2A2it b + e it While the Verbeek and Nijman procedure assumes one threshold and two regimes, in our application, we have two thresholds and obrvations from three regimes: wage changes, stability, and decreas. Our strategy to u obrvations both from the wage decreas and wage increas regime is the following : we implement a random e ect probit model to estimate parameter t 1 (as well as an alternative parameter estimate for ) for wage decreas, similar to that above with wage increa. We recover of counterparts of the above auxiliary variables, which we label A 3i and A 4it in the ca w it < 0: vector is nally consistently estimated using the least square estimator in the following extended model on all obrvations with wage changes (w it 6= 0) : w it = x it + c 1 d + it b A 1i + c 2 d + it b A 2it + c 3 d it b A3i + c 4 d it b A4it + e it where the dummy variables d + it and d it are de ned as follow: d+ it = 1 when w it > 0; d+ it = 0 otherwi; d it = 1 when w it < 0; d it = 0 otherwi: 5

6 Appendix Tables (Tables A1 to A6) Table A1: Frequency of Wage Change Excluding obrvations at times of legal working time reduction Mean frequency Number of obrvations Total Manufacturing Construction Services Manual workers Clerical workers Intermediate occupations Managers to 19 employees to 49 employees to 149 employees to 499 employees more than 500 employees

7 Table A2: Frequency and size of hourly ba wage changes Restricting for trajectories of occupation that are obrved for full sample period Number of obrvations all increa decrea Frequency all % 31% 5% 0 to 19 employees % 28% 7% 20 to 49 employees % 28% 6% 50 to 149 employees % 29% 5% 150 to 499 employees % 33% 4% more than 500 employees % 37% 4% Size all % 3.2% -3.7% 0 to 19 employees % 4.3% -4.5% 20 to 49 employees % 4.1% -4.4% 50 to 149 employees % 3.6% -3.8% 150 to 499 employees % 3.0% -3.3% more than 500 employees % 2.2% -2.0% 7

8 Table A3: Frequency and size of monthly wage changes Number of obrvations all increa decrea Frequency all % 30% 5% 0 to 19 employees % 26% 8% 20 to 49 employees % 26% 6% 50 to 149 employees % 27% 5% 150 to 499 employees % 30% 5% more than 500 employees % 34% 5% Size all % 2.5% -3.0% 0 to 19 employees % 3.9% -4.1% 20 to 49 employees % 3.6% -4.1% 50 to 149 employees % 2.8% -3.5% 150 to 499 employees % 2.3% -3.0% more than 500 employees % 1.7% -1.6% 8

9 Table A4: Conditional probability of wage decreas Panel A Standard errors Impact on probability of wage decrea Panel B Standard errors Impact on probability of wage decrea γ 2 (γ2 ) 2 γ 2 (γ2 ) Variable category Variable Intercept *** *** "Time dependent" variables First quarter *** Seasonal dummies Second quarter *** Third quarter *** quarter *** Duration 2 quarter *** quarter *** quarter *** μ (μ) μ "State dependent" variables Cumulated inflation *** *** Cumulated unemployment variation *** *** Cumulated productivity *** *** Clerical workers *** *** Socio-occupational group Intermediate occupations *** *** Managers *** *** Industry dummies (NES) Manufacture of cars Manufacture of capital goods *** *** Manufacture of intermediate goods *** *** Construction *** *** Trade *** *** Transports *** *** Financial activities *** *** Real estate activities *** *** Services to business *** *** Personal and domestic rvices Size of the firm dummies 10 to 19 employees *** *** to 49 employees *** *** to 149 employees *** *** to 499 employees *** *** Working Time Reduction *** *** Rho Sigma² Number of obrvations Log Likelihood : (μ) Note: The model is estimated by maximum likelihood. The reference category is : Manual Worker, Manufacturing goods, Firm with more than 500 employees, Fourth quarter, Duration > 4 quarters. The impact on the probabilities of wage decrea are the impact of the change of one unit evaluated for the reference category, and for cumulated changes in inflation and unemployment equal to zero. 9

10 Table A5: Conditional probability of wage increas without random effect Variable category Variable Intercept *** *** "Time dependent" variables First quarter *** *** Seasonal dummies Second quarter *** *** Third quarter *** *** quarter *** Duration 2 quarter quarter *** quarter *** μ (μ) μ (μ) "State dependent" variables Cumulated inflation *** *** Cumulated unemployment variation *** Cumulated productivity *** *** Socio-occupational group Clerical workers *** *** Intermediate occupations *** *** Managers *** *** Industry dummies (NES) Manufacture of cars *** *** Manufacture of capital goods Manufacture of intermediate goods Construction *** *** Trade *** *** Transports *** *** Financial activities * * Real estate activities Services to business *** *** Personal and domestic rvices *** *** Size of the firm dummies 10 to 19 employees *** *** to 49 employees *** *** to 149 employees *** *** to 499 employees *** *** Working Time Reduction Number of obrvations Log Likelihood : Panel A Panel B Standard errors Impact on probability Standard errors of wage γ 1 (γ 1 ) increa γ 1 (γ 1 ) Impact on probability of wage increa Note: The model is estimated by maximum likelihood. The reference category is : Manual Worker, Manufacturing goods, Firm with more than 500 employees, Fourth quarter, Duration > 4 quarters. The impact on the probabilities of wage increa of one unit evaluated for the reference category, and for cumulated changes in inflation and unemployment equal to zero. 10

11 Table A6: Sample lection model : size of wage changes Variable category Variable Intercept Estimation method Panel A Panel B Backward-looking Backward and forward-looking OLS Two-step Two-step (excluding zeros) (sample lection) (sample lection) Intercept *** *** TV Covariates Inflation (backward, elapd spell ) *** *** *** Inflation (backward, elapd 2 spells) Inflation (backward, lagged, elapd 2 spells) Inflation (forward current spell) *** Inflation (forward, 2 spell) Inflation (forward, lagged forecast, 2 spell) Unemployment (backward, elapd spell ) *** *** *** Unemployment (backward, elapd 2 spells) Unemployment (backward, lagged, elapd spell) Unemployment (current spell forward) *** Unemployment (forward, 2 spells) Unemployment (forward, lagged forecast, 2 spell) Productivity (backward, elapd spell ) ** Productivity (backward, elapd 2 spells) Productivity (backward, lagged, elapd spell) Productivity (current spell forward) ** Productivity (forward, 2 spells) Productivity (forward, lagged forecast, 2 spell) Socio-occupational group Clerical workers *** *** *** Intermediate occupations *** *** *** Managers *** *** *** Time dummies First quarter *** *** *** Second quarter *** *** Third quarter *** *** Industry dummies (NES) Manufacturing of cars (D) * *** Manufacturing of capital goods (E) *** Manufacturing of intermediate goods (F) *** * Construction (H) *** *** *** Trade (J) *** *** *** Transports (K) *** *** * Financial activities (L) ** * Real estate activities (M) *** ** Services to business (N) *** *** *** Personal and domestic rvices (P) *** ** Firm size dummies 10 to 19 employees *** *** *** to 49 employees *** *** *** to 149 employees *** *** *** to 499 employees *** ** *** Working Time Reduction *** *** *** Corrections terms A *** A *** *** A *** *** A *** *** Number of wage obrvations RMSE Variable category Variable Intercept Panel C Panel D Backward-looking with predetermination Forward-looking with predetermination Estimation method Two-step Two-step (sample lection) (sample lection) Intercept *** *** Inflation (backward, *** TV Covariates elapd spell ) Inflation (backward, elapd 2 spells) ** Inflation (backward, lagged, elapd 2 spells) *** Inflation (forward current spell) *** Inflation (forward, 2 spell) Inflation (forward, lagged forecast, 2 spell) ** Unemployment (backward, elapd spell ) Unemployment (backward, elapd 2 spells) Unemployment (backward, lagged, elapd spell) *** Unemployment (current spell forward) Unemployment (forward, 2 spells) Unemployment (forward, lagged forecast, 2 spell) *** Productivity (backward, elapd spell ) Productivity (backward, elapd 2 spells) Productivity (backward, lagged, elapd spell) Productivity (current spell forward) Productivity (forward, 2 spells) Productivity (forward, lagged forecast, 2 spell) Socio-occupational group Clerical workers *** *** Intermediate occupations ** Managers *** *** Time dummies First quarter *** *** Second quarter *** ** Third quarter *** *** Industry dummies (NES) Manufacturing of cars (D) ** * Manufacturing of capital goods (E) *** ** Manufacturing of intermediate goods (F) Construction (H) *** *** Trade (J) *** *** Transports (K) ** Financial activities (L) *** ** Real estate activities (M) ** ** Services to business (N) *** *** Personal and domestic rvices (P) *** Firm size dummies 10 to 19 employees *** *** to 49 employees *** *** to 149 employees *** *** to 499 employees ** *** Working Time Reduction *** *** Corrections terms A *** *** A *** *** A *** *** A *** *** Number of wage obrvations RMSE Note: The model is estimated using a two-step procedure adapted from Verbeek and Nijman (1996) and prented in the on-line appendix. The dependent variable is wage change computed as 100.ln(w it/w it-1). Control variables and terms correcting for sample lection are included but not reported. Paremeters values for control variables are reported in on-line Appendix Table A5. 11

12 Table A6 (continued): Sample lection model : size of wage changes Variable category Variable Intercept Panel C Backward-looking with predetermination Estimation method Two-step Two-step (sample lection) (sample lection) Panel D Forward-looking with predetermination Intercept *** *** TV Covariates Inflation (backward, elapd spell ) *** Inflation (backward, elapd 2 spells) ** Inflation (backward, lagged, elapd 2 spells) *** Inflation (forward current spell) *** Inflation (forward, 2 spell) Inflation (forward, lagged forecast, 2 spell) ** Unemployment (backward, elapd spell ) Unemployment (backward, elapd 2 spells) Unemployment (backward, lagged, elapd spell) *** Unemployment (current spell forward) Unemployment (forward, 2 spells) Unemployment (forward, lagged forecast, 2 spell) *** Productivity (backward, elapd spell ) Productivity (backward, elapd 2 spells) Productivity (backward, lagged, elapd spell) Productivity (current spell forward) Productivity (forward, 2 spells) Productivity (forward, lagged forecast, 2 spell) Socio-occupational group Clerical workers *** *** Intermediate occupations ** Managers *** *** Time dummies First quarter *** *** Second quarter *** ** Third quarter *** *** Industry dummies (NES) Manufacturing of cars (D) ** * Manufacturing of capital goods (E) *** ** Manufacturing of intermediate goods (F) Construction (H) *** *** Trade (J) *** *** Transports (K) ** Financial activities (L) *** ** Real estate activities (M) ** ** Services to business (N) *** *** Personal and domestic rvices (P) *** Firm size dummies 10 to 19 employees *** *** to 49 employees *** *** to 149 employees *** *** to 499 employees ** *** Working Time Reduction *** *** Corrections terms A *** *** A *** *** A *** *** A *** *** Number of wage obrvations RMSE Note: The sample lection model is estimated using the two-step procedure suggested by Verbeek and Nijman (1996). s for the lection equation are very clo to tho reported in table 5 and are not reported. The dependent variable is wage change computed as 100.ln(w it/w it-1). The reference category is : Manual Worker, Manufacturing goods, Firm with more than 500 employees, Fourth quarter, Duration > 4 quarters. 12

13 Figure A1. Distribution of (non zero) wage changes, with monthly wages 13