A bivariate approach to the Mincerian earnings equation

Cunha, Danúbia R.; Saulo, Helton; Monsueto, Sandro E.; Divino, José Angelo

doi:10.5935/0034-7140.20230008

Danúbia R. Cunha Danúbia R. Cunha : danubiarodriguess@hotmail.com

Universidade Católica de Brasília, Brasil and Universidade Federal de Goiás, Brasil

https://orcid.org/0000-0003-3542-982X

Helton Saulo Helton Saulo: heltonsaulo@gmail.com

Universidade de Brasília, Brasil

https://orcid.org/0000-0002-4467-8652

Sandro E. Monsueto Sandro E. Monsueto: monsueto@ufg.br

Universidade Federal de Goiás, Brasil

https://orcid.org/0000-0002-2155-012X

José Angelo Divino José Angelo Divino: jangelo@p.ucb.br

Universidade Católica de Brasília, Brasil

https://orcid.org/0000-0001-7359-7539

Danúbia R. Cunha : danubiarodriguess@hotmail.com

Helton Saulo: heltonsaulo@gmail.com

Sandro E. Monsueto: monsueto@ufg.br

José Angelo Divino: jangelo@p.ucb.br

Data	Sample Size	Median	Average	SD	CV	CA	CK
Wages	167,271	1344.54	2278.51	3087.26	135.49%	7.06	116.51
Hours worked	167,271	40.00	39.91	11.55	28.95%	−0.55	3.44
log(Wages)	167,271	7.20	7.20	0.80	10.89%	0.41	2.01
log(Hours worked)	167,271	3.69	3.61	0.51	14.59%	−4.13	21.87

		Univariate distributions
		Normal	Student t	BS
AIC	Wages	272,072.00	255,565.00	2,753,495.00
AIC	Hours worked	249,342.00	60,438.00	1,539,515.00
BIC	Wages	272,423.00	255,936.00	2,753,846.00
BIC	Hours worked	249,693.00	60,809.00	1,539,866.00

		Univariate distributions
		Normal	Student t	BS
AIC	Wages and hours worked	515,257.00	451,026.00	4,281,716.00
BIC	Wages and hours worked	515,969.00	451,738.00	4,282,428.00

Response	Explanatory var.	Coefficient	Standard error	Wald stat.	p-value
	σ ₁	0.55	0.0019	294.05	<0.0001
	σ₂	0.51	0.0018	292.05	<0.0001
	ρ	0.19	0.0131	14.53	<0.0001
Wage	(Intercept)	5.63	0.0194	290.41	<0.0001
	Gender	0.30	0.0031	96.25	<0.0001
	Age	0.05	8.0000×10⁻⁴	66.78	<0.0001
	Age²	−6.00×10⁻⁴	8.0000×10⁻⁶	−57.56	<0.0001
	Race	0.10	0.0031	31.85	<0.0001
	Marital status	0.01	0.0069	0.94	0.3498
	Education	0.05	5.0000×10⁻⁴	97.66	<0.0001
	Belém-PA	−0.32	0.0063	−50.81	<0.0001
	Fortaleza-CE	−0.35	0.0059	−59.20	<0.0001
	Recife-PE	−0.35	0.0057	−61.28	<0.0001
	Salvador-BA	−0.31	0.0058	−53.88	<0.0001
	Belo Horizonte-MG	−0.09	0.0055	−17.04	<0.0001
	Rio de Janeiro-RJ	−0.08	0.0052	−15.34	<0.0001
	Curitiba-PR	0.01	0.0065	1.85	0.0643
	Porto Alegre-RS	−0.09	0.0051	−17.23	<0.0001
	Brasília-DF	0.09	0.0063	13.57	<0.0001
	2014	0.00	0.0033	0.72	0.4690
	2015	−0.06	0.0034	−17.51	<0.0001
	High	0.89	0.0084	105.53	<0.0001
	High mean	0.37	0.0073	50.53	<0.0001
	Mean	0.19	0.0069	26.85	<0.0001
	Low mean	0.05	0.0068	7.49	<0.0001
	Agricultural	−0.43	0.0180	−23.97	<0.0001
	Industry	−0.20	0.0087	−23.38	<0.0001
	Construction	−0.08	0.0093	−8.38	<0.0001
	Commerce, food and others	−0.21	0.0083	−25.46	<0.0001
	Education, health and soc. serv.	−0.25	0.0079	−32.17	<0.0001
	Other services	−0.21	0.0083	−25.86	<0.0001
	No employment record card	−0.20	0.0041	−47.21	<0.0001
	Autonomous	−0.14	0.0042	−32.68	<0.0001
	Civil servant	0.24	0.0072	33.29	<0.0001
	Labor Union	0.12	0.0038	32.46	<0.0001
	Social Security	0.36	0.0072	49.34	<0.0001
	Time in job	0.01	2.0000×10⁻⁴	59.78	<0.0001
Hours worked	(Intercept)	3.28	0.0166	196.99	<0.0001
	Gender	0.08	0.0027	30.55	<0.0001
	Age	0.01	7.0000×10⁻⁴	18.74	<0.0001
	Age²	−1.00×10⁻⁴	<0.0001	−15.15	<0.0001
	Race	2.00×10⁻⁴	0.0027	0.09	0.9318
	Marital status	−0.01	0.0061	−1.51	0.1312
	Education	−9.00×10⁻⁴	4.0000×10⁻⁴	−2.23	0.0255
	Belém-PA	0.00	0.0055	0.66	0.5106
	Fortaleza-CE	0.02	0.0052	3.33	9.0000×10⁻⁴
	Recife-PE	−0.01	0.0050	−2.99	0.0028
	Salvador-BA	−0.04	0.0051	−8.68	<0.0001
	Belo Horizonte-MG	0.01	0.0047	2.55	0.0109
	Rio de Janeiro-RJ	−0.09	0.0045	−19.59	<0.0001
	Curitiba-PA	−0.01	0.0057	−2.29	0.0222
	Porto Alegre-RS	0.02	0.0044	3.85	1.0000×10⁻⁴
	Brasília-DF	−0.03	0.0054	−5.01	<0.0001
	2014	0.01	0.0029	4.14	<0.0001
	2015	−0.04	0.0029	−13.45	<0.0001
	High	0.12	0.0073	16.89	<0.0001
	High mean	0.10	0.0066	15.07	<0.0001
	Mean	0.15	0.0063	23.30	<0.0001
	Low mean	0.16	0.0061	25.27	<0.0001
	Agriculture	0.16	0.0152	10.32	<0.0001
	Industry	0.02	0.0072	2.82	0.0048
	Construction	0.04	0.0078	5.47	<0.0001
	Commerce, food and others	0.07	0.0069	9.49	<0.0001
	Education, health and soc. serv.	−0.07	0.0065	−10.07	<0.0001
	Other services	−0.01	0.0069	−2.14	0.0323
	No employment record card	−0.16	0.0036	−45.28	<0.0001
	Autonomous	−0.17	0.0034	−50.85	<0.0001
	Civil servant	−0.04	0.0060	−6.86	<0.0001
	Head	0.02	0.0027	6.72	<0.0001
	Minor	0.00	0.0026	−1.10	0.2709
	Inactivity	−0.01	0.0037	−1.50	0.1348

Response	Explanatory var.	Coefficient	Standard error	Wald stat.	p-value
Wage per hour	(Intercept)	0.943	0.0229	41.23	<0.0001
	Gender	0.203	0.0037	54.42	<0.0001
	Age	0.035	0.0009	36.99	<0.0001
	Age²	0.000	0.0001	−30.31	<0.0001
	Race	0.101	0.0037	26.91	<0.0001
	Marital status	0.010	0.0084	1.16	0.2477
	Education	0.048	0.0006	85.13	<0.0001
	Belém-PA	−0.316	0.0076	−41.37	<0.0001
	Fortaleza-CE	−0.367	0.0072	−51.15	<0.0001
	Recife-PE	−0.332	0.0069	−47.83	<0.0001
	Salvador-BA	−0.267	0.0070	−38.33	<0.0001
	Belo Horizonte-MG	−0.105	0.0067	−15.73	<0.0001
	Rio de Janeiro-RJ	0.006	0.0062	0.97	0.3332
	Curitiba-PR	0.021	0.0080	2.57	0.0101
	Porto Alegre-RS	−0.110	0.0062	−17.58	<0.0001
	Brasília-DF	0.114	0.0075	15.23	<0.0001
	2014	−0.009	0.0040	−2.15	0.0312
	2015	−0.016	0.0041	−3.90	0.0001
	High	0.764	0.0097	78.59	<0.0001
	High mean	0.275	0.0088	31.24	<0.0001
	Mean	0.040	0.0083	4.79	<0.0001
	Low mean	−0.106	0.0081	−13.04	<0.0001
	Agriculture	−0.583	0.0200	−29.11	<0.0001
	Industry	−0.224	0.0101	−22.24	<0.0001
	Construction	−0.119	0.0109	−11.00	<0.0001
	Commerce, food and others	−0.278	0.0095	−29.25	<0.0001
	Education, health and soc. serv.	−0.188	0.0090	−20.87	<0.0001
	Other services	−0.200	0.0094	−21.24	<0.0001
	No employment record card	−0.034	0.0049	−7.00	<0.0001
	Autonomous	0.040	0.0047	8.51	<0.0001
	Civil servant	0.288	0.0083	34.59	<0.0001
	Labor union	0.120	0.0047	25.71	<0.0001
	Social security	0.332	0.0081	40.86	<0.0001
	Time in job	0.010	0.0002	42.06	<0.0001
	Head	0.064	0.0036	17.92	<0.0001
	Minor	0.012	0.0036	3.35	0.0008
	Inactivity	−0.062	0.0052	−11.93	<0.0001

\begin{matrix} E [\log (Y_{1 i}) | X_{i} = x_{i}] = μ_{1 i} = x_{i} β_{1}, & i = 1, \dots, n, \\ E [\log (Y_{2 i}) | X_{i} = x_{i}] = μ_{2 i} = x_{i}^{Τ} β_{2}, & i = 1, \dots, n; \end{matrix}

\begin{matrix} E [\log (Y_{1 i}) | X_{i} = x_{i}] = μ_{1 i} = x_{i} β_{1}, & i = 1, \dots, n, \\ E [\log (Y_{2 i}) | X_{i} = x_{i}] = μ_{2 i} = x_{i}^{Τ} β_{2}, & i = 1, \dots, n; \end{matrix}

\begin{matrix} E [Y_{1 i} | X_{i} = x_{i}] = μ_{1 i} = \exp (x_{i} β_{1}), i = 1, \dots, n, \\ E [Y_{1 i} | X_{i} = x_{i}] = μ_{2 i} = \exp (x_{i}^{Τ} β_{1}), i = 1, \dots, n . \end{matrix}

f (y_{1}, y_{2}; μ_{1}, μ_{2}, σ_{1}, σ_{2}, ρ) = \frac{1}{2 π σ_{1} σ_{2} \sqrt{1 - ρ^{2}}} \exp {- \frac{1}{2 (1 - ρ^{2})} [\frac{{(y_{1} - μ_{1})}^{2}}{σ_{1}^{2}} + \frac{{(y_{2} - μ_{2})}^{2}}{σ_{2}^{2}} - \frac{2 ρ (y_{1} - μ_{1}) (y_{2} - μ_{2})}{σ_{1} σ_{2}}]} .

ϕ_{2} (z_{1}, z_{2}; ρ) = \frac{1}{2 π \sqrt{1 - ρ^{2}}} \exp {- \frac{1}{2 (1 - ρ^{2})} (y_{1}^{2} + y_{2}^{2} - 2 ρ y_{1} y_{2})} .

Φ_{2} (z_{1}, z_{2}; ρ) = \frac{1}{2 π \sqrt{1 - ρ^{2}}} \int_{- \infty}^{z_{1}} \int_{- \infty}^{z_{2}} \exp {- \frac{1}{2 (1 - ρ^{2})} (u^{2} + v^{2} - 2 ρ u v)} .

\begin{matrix} x_{i}^{(j) Τ} β_{j} = β_{j 1} x_{i 1}^{(j)} + β_{j 2} x_{i 2}^{(j)} + \dots + β_{j l} x_{i l}^{(j)}, & j = 1, 2 & i = 1, \dots, n, \end{matrix}

\begin{matrix} (\begin{matrix} Y_{1 i} \\ Y_{2 i} \end{matrix}) = (\begin{matrix} x_{i}^{(1) Τ} β_{1} \\ x_{i}^{(2) Τ} β_{2} \end{matrix}) + (\begin{matrix} ε_{1 i} \\ ε_{2 i} \end{matrix}) & i = 1, \dots, n, \end{matrix}

f (u_{1}, u_{2}; μ_{1}, μ_{2}, σ_{1}, σ_{2}, ρ) = \frac{1}{2 π σ_{1} σ_{2} \sqrt{1 - ρ^{2}}} \times {[1 + \frac{1}{v (1 - ρ^{2})} (\frac{{(u_{1} - μ_{1})}^{2}}{σ_{1}^{2}} + \frac{{(u_{2} - μ_{2})}^{2}}{σ_{2}^{2}} - \frac{2 ρ (u_{1} - μ_{1}) (u_{2} - μ_{2})}{σ_{1} σ_{2}})]}^{- (v + 2) / 2} .

F (u_{1}, u_{2}; v, ρ) = \frac{1}{2 π \sqrt{1 - ρ^{2}}} {\int_{- \infty}^{u_{1}} \int_{- \infty}^{u_{2}} [1 + \frac{1}{v (1 - ρ^{2})} (u^{2} + v^{2} - 2 ρ u v)]}^{- (v + 2) / 2} .

\begin{matrix} (\begin{matrix} U_{1 i} \\ U_{2 i} \end{matrix}) = (\begin{matrix} x_{i}^{(1) Τ} β_{1} \\ x_{i}^{(2) Τ} β_{2} \end{matrix}) + (\begin{matrix} ε_{1 i} \\ ε_{2 i} \end{matrix}) & i = 1, \dots, n, \end{matrix}

F (t_{1}, t_{2}; μ_{1}, μ_{2}, δ_{1}, δ_{2}, ρ) = Φ_{2} (\sqrt{\frac{δ_{1}}{2}} [\sqrt{\frac{(δ_{1} + 1) t_{1}}{δ_{1} μ_{1}}} - \sqrt{\frac{δ_{1} μ_{1}}{(δ_{1} + 1) t_{1}}}], \sqrt{\frac{δ_{2}}{2}} [\sqrt{\frac{(δ_{2} + 1) t_{2}}{δ_{2} μ_{2}}} - \sqrt{\frac{δ_{2} μ_{2}}{(δ_{2} + 1) t_{2}}}], ρ),

f (t_{1}, t_{2}; μ_{1}, μ_{2}, δ_{1}, δ_{2}, ρ) = ϕ_{2} (\sqrt{\frac{δ_{1}}{2}} [\sqrt{\frac{(δ_{1} + 1) t_{1}}{δ_{1} μ_{1}}} - \sqrt{\frac{δ_{1} μ_{1}}{(δ_{1} + 1) t_{1}}}], \sqrt{\frac{δ_{2}}{2}} [\sqrt{\frac{(δ_{2} + 1) t_{2}}{δ_{2} μ_{2}}} - \sqrt{\frac{δ_{2} μ_{2}}{(δ_{2} + 1) t_{2}}}], ρ) \times \frac{\sqrt{δ_{1}}}{2 \sqrt{2} t_{1}} [\sqrt{\frac{(δ_{1} + 1) t_{1}}{δ_{1} μ_{1}}} + \sqrt{\frac{δ_{1} μ_{1}}{(δ_{1} + 1) t_{1}}}] \frac{\sqrt{δ_{2}}}{2 \sqrt{2} t_{2}} [\sqrt{\frac{(δ_{2} + 1) t_{2}}{δ_{2} μ_{2}}} + \sqrt{\frac{δ_{2} μ_{2}}{(δ_{2} + 1) t_{2}}}],

F (t_{1 i}, t_{2 i}; μ_{1 i}, μ_{2 i}, δ_{1}, δ_{2}, ρ) = Φ_{2} (\sqrt{\frac{δ_{1}}{2}} [\sqrt{\frac{(δ_{1} + 1) t_{1 i}}{δ_{1} μ_{1}^{i}}} - \sqrt{\frac{δ_{1} μ_{1}^{i}}{(δ_{1} + 1) t_{1 i}}}], \sqrt{\frac{δ_{2}}{2}} [\sqrt{\frac{(δ_{2} + 1) t_{2 i}}{δ_{2} μ_{2}}} - \sqrt{\frac{δ_{2} μ_{2}^{i}}{(δ_{2} + 1) t_{2 i}}}]; ρ)

[1] Danúbia R. Cunha : danubiarodriguess@hotmail.com

Brasil

Brasil

A bivariate approach to the Mincerian earnings equation

Resumo