LOBEL, ROBERT EUGENE; KLOTZLE, MARCELO CABUS; SILVA, PAULO VITOR JORDÃO DA GAMA; PINTO, ANTONIO CARLOS FIGUEIREDO

doi:10.1590/S0034-759020170507

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ARTICLES • Rev. adm. empres. 57 (5) • Sep-Oct 2017 • https://doi.org/10.1590/S0034-759020170507 copiar

Teoria do prospecto: Uma análise paramétrica de formas funcionais no Brasil

Autoria SCIMAGO INSTITUTIONS RANKINGS

RESUMO

Este estudo teve o objetivo de analisar as preferências ao risco no Brasil seguindo os preceitos da Teoria do Prospecto. Para tal, foi estimado o parâmetro de aversão ao risco da Teoria da Utilidade Esperada para uma amostra selecionada, e foram sugeridos os parâmetros da função e probabilidade, supondo diversas formas funcionais e uma nova função de valor - a log modificada. Este foi o primeiro estudo realizado no Brasil para a estimação de tais valores. Os resultados mostraram parâmetros ligeiramente diferentes daqueles encontrados em estudos realizados em outros países, apontando que, no caso da amostra estudada, os indivíduos são mais avessos ao risco e exibem uma menor aversão à perda. A distorção de probabilidade é o único elemento semelhante ao de outros países. Como esperado, o estudo constatou a superioridade dos modelos comportamentais em relação à Teoria da Utilidade Esperada (TUE). Além disso e correspondente às expectativas, o desempenho de modelos baseados na Teoria do Prospecto, TK, Função de Ponderação de Prelec e Função Valor Potencia foi superior aos demais. Por fim, a função de log modificada sugerida no estudo encaixa-se bem nos dados e pode assim ser aplicada em futuros estudos no Brasil.

PALAVRAS-CHAVE
Finanças comportamentais; teoria do prospecto; função valor; função peso; Brasil

Gender			Age
Men	12	16%	18 - 24	24	32%
Women	63	84%	25 - 31	26	35%
Total	75	100%	32 - 38	15	20%
			39 - 45	5	7%
Economic class			46 or >	5	7%
Class E	1	1%	Total	75	100%
Class D	2	3%
Class C	29	39%	Marital status
Class B	14	19%	Single	43	57%
Class A	29	39%	Divorced	7	9%
Total	75	100%	Married	25	33%
			Total	75	100%
Profession
Assistant	17	23%	Education
Intern	10	13%	High School	1	1%
Analyst	19	25%	Bachelor’s degree	42	56%
Sr. Analyst	6	8%	MBA	12	16%
Supervisor or Coordinator	8	11%	Master’s degree	16	21%
Director or Manager	15	20%	Doctoral degree	4	5%
Total	75	100%	Total	75	100%

Lottery	Result A ($)	Prob. (A)	Result B ($)	Prob. (B)	Average amount ($)
1	10	0.1	100	0.9	91
2	0	0.4	100	0.6	60
3	0	0.1	100	0.9	90
4	0	0.4	10,000	0.6	6,000
5	0	0.9	100	0.1	10
6	0	0.4	400	0.6	240
7	- 80	0.6	0	0.4	- 48
8	- 100	0.6	0	0.4	- 60
9	- 25	0.5	-	0.5	-
10	- 100	0.5	-	0.5	-

Model	Theory	w(p)	v(x)
TUE
111	PT	TK	Power
112			Exponential
113			Modified Log
121		Karmarkar	Power
122			Exponential
123			Modified Log
131		Prelec	Power
132			Exponential
133			Modified Log
211	CPT	TK	Power
212			Exponential
213			Modified Log
221		Karmarkar	Power
222			Exponential
223			Modified Log
231		Prelec	Power
232			Exponential
233			Modified Log
311	NPT	TK	Power
312			Exponential
313			Modified Log
321		Karmarkar	Power
322			Exponential
323			Modified Log
331		Prelec	Power
332			Exponential
333			Modified Log

Theory	Value Function
PT/NPT	Power	$λ = \frac{1}{2} [\frac{x_{9}^{α}}{{\|A_{9}\|}^{β}} + \frac{x_{10}^{α}}{{\|A_{10}\|}^{β}}]$
	Exponential	$λ = \frac{1}{2} [\frac{1 - e^{- {ax}_{9}}}{1 - e^{- β A_{9}}} + \frac{1 - e^{- {ax}_{10}}}{1 - e^{- β A_{10}}}]$
	Modified Log	$λ = \frac{1}{2} (\frac{β}{α}) [\frac{\ln (1 + {ax}_{9}^{α})}{\ln (1 - β A_{9})} + \frac{\ln (1 + {ax}_{10}^{α})}{\ln (1 - β A_{10})}]$
CPT	Power	$λ = \frac{1}{2} [\frac{w (0.5) x_{9}^{α}}{(1 - w (0.5)) {\|A_{9}\|}^{β}} + \frac{w (0.5) x_{10}^{α}}{(1 - w (0.5)) {\|A_{10}\|}^{β}}]$
	Exponential	$λ = \frac{1}{2} [\frac{w (0.5) (1 - e^{- {ax}_{9}})}{(1 - w (0.5)) (1 - e^{- β A_{9}}))} + \frac{w (0.5) (1 - e^{- {ax}_{10}})}{(1 - w (0.5)) (1 - e^{- β A 10}))}]$
	Modified Log	$λ = \frac{1}{2} (\frac{β}{α}) [\frac{w (0.5) (\ln (1 + α x_{9}))}{(1 - w (0.5) (\ln (1 - β A_{9}))} + \frac{w (0.5) (\ln (1 + α x_{10}))}{(1 - w (0.5) (\ln (1 - β A_{10}))}]$

Average	Median	Standard deviation	Error
0.55	0.54	0.22	4.89

Model	Average					Median					Standard deviation
Model	α	β	γ	λ	ε	α	β	γ	λ	ε	α	β	γ	λ	ε
111	0.52	0.76	0.50	1.01	0.36	0.50	0.79	0.40	0.68	0.30	0.21	0.21	0.23	1.09	0.26
112	0.10	0.08	0.73	1.61	0.36	0.08	0.04	0.78	1.28	0.33	0.10	0.19	0.25	0.93	0.24
113	0.30	0.12	0.51	1.40	0.37	0.19	0.03	0.42	0.76	0.32	0.29	0.24	0.22	1.45	0.26
121	0.36	0.47	0.25	1.16	0.49	0.28	0.40	0.11	0.91	0.47	0.23	0.23	0.34	1.06	0.21
122	0.11	0.10	0.70	1.38	0.38	0.09	0.05	0.74	1.17	0.35	0.08	0.19	0.26	0.83	0.24
123	0.65	0.37	0.26	1.27	0.46	0.89	0.26	0.16	0.90	0.44	0.41	0.35	0.33	1.35	0.22
131	0.43	0.59	0.26	1.12	0.37	0.37	0.55	0.11	0.82	0.32	0.20	0.22	0.34	1.13	0.25
132	0.12	0.09	0.76	1.40	0.38	0.09	0.04	0.88	1.20	0.35	0.10	0.19	0.24	0.84	0.25
133	0.43	0.20	0.30	1.30	0.37	0.33	0.09	0.16	0.75	0.33	0.34	0.28	0.31	1.46	0.25
211	0.23	0.33	0.59	0.64	0.72	0.12	0.24	0.49	0.42	0.73	0.27	0.26	0.19	0.65	0.25
212	0.25	0.11	0.82	1.10	0.42	0.14	0.07	0.89	1.04	0.39	0.32	0.19	0.21	0.69	0.24
213	0.83	0.61	0.73	1.01	1.00	1.00	0.78	0.69	0.93	1.13	0.35	0.40	0.11	0.86	0.36
221	0.36	0.47	0.25	1.16	0.49	0.28	0.40	0.11	0.91	0.47	0.23	0.23	0.34	1.06	0.21
222	0.11	0.10	0.70	1.38	0.38	0.09	0.05	0.74	1.17	0.35	0.08	0.19	0.26	0.83	0.24
223	0.65	0.37	0.26	1.27	0.46	0.89	0.26	0.16	0.90	0.44	0.41	0.35	0.33	1.35	0.22
231	0.28	0.37	0.32	0.76	0.58	0.19	0.28	0.13	0.57	0.57	0.25	0.25	0.35	0.64	0.24
232	0.14	0.11	0.91	1.28	0.40	0.15	0.06	0.91	1.08	0.38	0.10	0.19	0.08	0.84	0.25
233	0.81	0.64	0.40	0.93	0.84	1.00	0.92	0.25	0.78	0.90	0.36	0.40	0.28	0.85	0.31
311	0.36	0.47	0.25	1.16	0.49	0.28	0.40	0.11	0.91	0.47	0.23	0.23	0.34	1.06	0.21
312	0.11	0.10	0.70	1.38	0.38	0.09	0.05	0.74	1.17	0.35	0.08	0.19	0.26	0.83	0.24
313	0.65	0.37	0.26	1.27	0.47	0.90	0.26	0.16	0.90	0.45	0.41	0.35	0.33	1.35	0.22
321	0.36	0.47	0.25	1.16	0.49	0.28	0.40	0.11	0.91	0.47	0.23	0.23	0.34	1.06	0.21
322	0.11	0.10	0.70	1.38	0.38	0.09	0.05	0.74	1.17	0.35	0.08	0.19	0.26	0.83	0.24
323	0.65	0.37	0.26	1.27	0.46	0.89	0.26	0.16	0.90	0.44	0.41	0.35	0.33	1.35	0.22
331	0.36	0.47	0.24	1.15	0.49	0.28	0.40	0.08	0.93	0.47	0.23	0.23	0.35	1.05	0.21
332	0.10	0.10	0.65	1.38	0.38	0.09	0.05	0.69	1.16	0.35	0.06	0.19	0.27	0.83	0.25
333	0.65	0.37	0.24	1.28	0.46	0.89	0.26	0.12	0.91	0.44	0.41	0.35	0.33	1.36	0.22

Average
	Summary by theory			Summary by weighting			Summary by value
	PT	CPT	NPT	TK	KAR	PRC	PWR	EXP	LOG
alpha	0.33	0.41	0.37	0.37	0.37	0.37	0.36	0.13	0.63
beta	0.31	0.35	0.31	0.33	0.31	0.33	0.49	0.10	0.38
gamma	0.48	0.55	0.40	0.57	0.40	0.45	0.32	0.74	0.36
lambda	1.29	1.06	1.27	1.17	1.27	1.18	1.03	1.37	1.22
errors	0.40	0.59	0.45	0.51	0.45	0.48	0.50	0.39	0.54
Median
	Summary by theory			Summary by weighting			Summary by value
	PT	CPT	NPT	TK	KAR	PRC	PWR	EXP	LOG
alpha	0.31	0.43	0.42	0.37	0.42	0.38	0.29	0.10	0.78
beta	0.25	0.34	0.24	0.30	0.24	0.30	0.43	0.05	0.35
gamma	0.42	0.49	0.32	0.52	0.34	0.37	0.18	0.79	0.26
lambda	0.94	0.87	1.00	0.90	0.99	0.91	0.78	1.16	0.86
errors	0.36	0.60	0.42	0.50	0.42	0.46	0.47	0.36	0.54
Standard deviation
	Summary by theory			Summary by weighting			Summary by value
	PT	CPT	NPT	TK	KAR	PRC	PWR	EXP	LOG
alpha	0.22	0.26	0.24	0.25	0.24	0.23	0.23	0.11	0.38
beta	0.23	0.27	0.26	0.25	0.26	0.26	0.23	0.19	0.34
gamma	0.28	0.24	0.31	0.24	0.31	0.28	0.31	0.23	0.29
lambda	1.13	0.86	1.08	0.99	1.08	1.00	0.98	0.83	1.27

Model	Parameters				Authors
Model	α	β	γ	λ	Authors
211	0.12	0.24	0.49	0.42	This study
	0.88	0.88	0.61	2.25	Tversky & Kahneman (1992)Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297-323.
	0.22	-	0.56	-	Camerer & Ho (1994)Camerer, C. F., & Ho, T.-H. (1994). Violations of the betweenness axiom and nonlinearity in probability. Journal of Risk and Uncertainty, 8(2), 167-196. doi:10.1007/BF01065371 https://doi.org/10.1007/BF01065371...
	0.50	-	0.71	-	Wu & Gonzalez (1996)Wu, G., & Gonzalez, R. (1996). Curvature of the probability weighting function. Management Science, 42(12), 1676-1690.
	0.89	0.92	0.60	-	Abdellaoui (2000)Abdellaoui, M. (2000). Parameter-free elicitation of utility and probability weighting functions. Management Science, 46(11), 1497-1512.
	-	-	-	1.43	Schmidt & Traub (2002)Schmidt, U., & Traub, S. (2002). An experimental test of loss aversion. Journal of Risk and Uncertainty, 25(3), 233-249. doi:10.1023/A:1020923921649 https://doi.org/10.1023/A:1020923921649...
	0.72	0.73	-	2.54	Abdellaoui et al. (2007)Abdellaoui, M., Bleichrodt, H., & Paraschiv, C. (2007). Loss aversion under prospect theory: A parameter-free measurement. Management Science, 53(10), 1659-1674.
	0.86	1.06	-	2.61	Abdellaoui et al. (2008)Abdellaoui, M., Bleichrodt, H., & L’Haridon, O. (2008). A tractable method to measure utility and loss aversion under prospect theory. Journal of Risk and Uncertainty, 36(3), 245-266. doi:10.1007/s11166-008-9039-8 https://doi.org/10.1007/s11166-008-9039-...
	0.71	0.72	0.91	1.38	Harrison & Rutström (2009)Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38(1), 129-166. doi:10.1006/cogp.1998.0710 https://doi.org/10.1006/cogp.1998.0710...
	0.25	-1.24	0.46	1.18	Attema et al. (2013)Attema, A. E., Brouwer, W. B. F., & L’Haridon, O. (2013). Prospect theory in the health domain: A quantitative assessment. Journal of Health Economics, 32(6), 1057-1065. doi:10.1016/j.jhealeco.2013.08.006 https://doi.org/10.1016/j.jhealeco.2013....
	0.73	0.86	-	1.31	Abdellaoui et al. (2013)Abdellaoui, M., Bleichrodt, H., & L’Haridon, O. (2013). Sign-dependence in intertemporal choice. Journal of Risk and Uncertainty, 47(3), 225-253. doi:10.1007/s11166-013-9181-9 https://doi.org/10.1007/s11166-013-9181-...
221	0.28	0.40	0.11	0.91	This study
	0.49	-	0.44	-	Gonzalez & Wu (1999)Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38(1), 129-166. doi:10.1006/cogp.1998.0710 https://doi.org/10.1006/cogp.1998.0710...
	0.91	0.96	0.83	-	Abdellaoui et al. (2005)Abdellaoui, M., Vossmann, F., & Weber, M. (2005). Choice-based elicitation and decomposition of decision weights for gains and losses under uncertainty. Management Science, 51(9), 1384-1399.
222	0.09	0.05	0.74	1.17	This study
	0.28	0.09	0.91	-	Abdellaoui et al. (2005)Abdellaoui, M., Vossmann, F., & Weber, M. (2005). Choice-based elicitation and decomposition of decision weights for gains and losses under uncertainty. Management Science, 51(9), 1384-1399.
	0.86	0.83	0.62	1.58	Booij et al. (2010)Booij, A., van Praag, B., & van de Kuilen, G. (2010). A parametric analysis of prospect theory’s functionals for the general population. Theory and Decision, 68(1-2), 115-148. doi:10.1007/s11238-009-9144-4 https://doi.org/10.1007/s11238-009-9144-...
231	0.19	0.28	0.13	0.57	This study
	0.48	-	0.74	-	Wu & Gonzalez (1996)Wu, G., & Gonzalez, R. (1996). Curvature of the probability weighting function. Management Science, 42(12), 1676-1690.
	0.68	0.74	1.00	3.20	Tu (2005)Tu, Q. (2005). Empirical analysis of time preferences and risk aversion. Tilburg University: CentER, Center for Economic Research.
312	0.09	0.05	0.74	1.17	This study
312	0.009	0.002	0.40	-	Scholten & Read (2014)Scholten, M., & Read, D. (2014). Prospect theory and the “forgotten” fourfold pattern of risk preferences. Journal of Risk and Uncertainty, 48(1), 67-83. doi:10.1007/s11166-014-9183-2 https://doi.org/10.1007/s11166-014-9183-...
313	0.90	0.26	0.16	0.90	This study
313	0.03	0.003	0.45	-	Scholten & Read (2014)Scholten, M., & Read, D. (2014). Prospect theory and the “forgotten” fourfold pattern of risk preferences. Journal of Risk and Uncertainty, 48(1), 67-83. doi:10.1007/s11166-014-9183-2 https://doi.org/10.1007/s11166-014-9183-...

Notes: Values refer to the median. Tversky and Kahneman (1992)Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297-323.: Estimated value of γ in the domain of losses (γ-): 0.69. Gonzalez and Wu (1999)Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38(1), 129-166. doi:10.1006/cogp.1998.0710
https://doi.org/10.1006/cogp.1998.0710... : used the GE model (Goldstein & Einhorn, 1987Goldstein, W. M., & Einhorn, H. J. (1987). Expression theory and the preference reversal phenomena. Psychological Review, 94(2), 236-254. doi:10.1037/0033-295X.94.2.236
https://doi.org/10.1037/0033-295X.94.2.2... ) based on the probability function. The Karmarkar (1978Karmarkar, U. S. (1978). Subjectively weighted utility: A descriptive extension of the expected utility model. Organizational Behavior and Human Performance, 21(1), 61-72. doi:10.1016/0030-5073(78)90039-9
https://doi.org/10.1016/0030-5073(78)900... , 1979)Karmarkar, U. S.. (1979). Subjectively weighted utility and the Allais Paradox. Organizational Behavior and Human Performance, 24(1), 67-72. doi:10.1016/0030-5073(79)90016-3
https://doi.org/10.1016/0030-5073(79)900... model is a special case of the GE model, in which elevation parameter δ = 1. Abdellaoui (2000)Abdellaoui, M. (2000). Parameter-free elicitation of utility and probability weighting functions. Management Science, 46(11), 1497-1512.: Estimated value of γ in the domain of losses: 0.65. Tu (2005)Tu, Q. (2005). Empirical analysis of time preferences and risk aversion. Tilburg University: CentER, Center for Economic Research.: Estimated value of γ in the area/DOMAIN of losses: 0.77. Abdellaoui et.al. (2005)Abdellaoui, M., Vossmann, F., & Weber, M. (2005). Choice-based elicitation and decomposition of decision weights for gains and losses under uncertainty. Management Science, 51(9), 1384-1399.: used the GE model in both model 221 (elevation: δ+ = 0.98, δ- = 1.35 and curvature in domain of losses: γ- = 0.84) and model 222 (elevation: δ+ =0.98, δ-= 1.32 and curvature in the domain of losses: γ- = 0.87). Abdellaoui et al. (2007)Abdellaoui, M., Bleichrodt, H., & Paraschiv, C. (2007). Loss aversion under prospect theory: A parameter-free measurement. Management Science, 53(10), 1659-1674.: estimation of λ was based on Köbberling and Wakker’s (2005)Köbberling, V., & Wakker, P. P. (2005). An index of loss aversion. Journal of Economic Theory, 122(1), 119-131. doi:10.1016/j.jet.2004.03.009
https://doi.org/10.1016/j.jet.2004.03.00... definition of risk aversion. Booij et al. (2010)Booij, A., van Praag, B., & van de Kuilen, G. (2010). A parametric analysis of prospect theory’s functionals for the general population. Theory and Decision, 68(1-2), 115-148. doi:10.1007/s11238-009-9144-4
https://doi.org/10.1007/s11238-009-9144-... : used the GE model (δ+ = 0.77, δ- = 1.02, γ- = 0.59). Attema et al. (2013)Attema, A. E., Brouwer, W. B. F., & L’Haridon, O. (2013). Prospect theory in the health domain: A quantitative assessment. Journal of Health Economics, 32(6), 1057-1065. doi:10.1016/j.jhealeco.2013.08.006
https://doi.org/10.1016/j.jhealeco.2013.... : study performed of healthcare professionals, with γ- = 0.46. Abdellaoui et.al. (2013)Abdellaoui, M., Bleichrodt, H., & L’Haridon, O. (2013). Sign-dependence in intertemporal choice. Journal of Risk and Uncertainty, 47(3), 225-253. doi:10.1007/s11166-013-9181-9
https://doi.org/10.1007/s11166-013-9181-... : study performed of financial sector professionals, with λ based on/using Köbberling and Wakker’s (2005)Köbberling, V., & Wakker, P. P. (2005). An index of loss aversion. Journal of Economic Theory, 122(1), 119-131. doi:10.1016/j.jet.2004.03.009
https://doi.org/10.1016/j.jet.2004.03.00... definition = 1.00. Scholten and Read (2014): exponential function based on/using Köbberling and Wakker’s (2005)Köbberling, V., & Wakker, P. P. (2005). An index of loss aversion. Journal of Economic Theory, 122(1), 119-131. doi:10.1016/j.jet.2004.03.009
https://doi.org/10.1016/j.jet.2004.03.00... definition/model, γ- = 0.63, and logarithmic function based on Scholten and Read (2014)Scholten, M., & Read, D. (2014). Prospect theory and the “forgotten” fourfold pattern of risk preferences. Journal of Risk and Uncertainty, 48(1), 67-83. doi:10.1007/s11166-014-9183-2
https://doi.org/10.1007/s11166-014-9183-... , γ- = 0.67.

Model	Parameters				Authors
Model	α	β	Γ	λ	Authors
231	0.28	0.37	0.32	0.76	This study
	0.59	0.59	0.49	1.20	Nguyen & Leung (2009)Nguyen, Q., & Leung, P. (2009). Do Fishermen have Different Attitudes Toward Risk? An Application of Prospect Theory to the Study of Vietnamese Fishermen. Journal of Agricultural and Resource Economics, 34(3), 518-538.
	0.61	0.61	0.74	2.63	Tanaka et al. (2010)Tanaka, T., Camerer, C. F., & Nguyen, Q. (2010). Risk and time preferences: Linking experimental and household survey data from Vietnam. American Economic Review, 100(1), 557-571. doi:10.1257/aer.100.1.557 https://doi.org/10.1257/aer.100.1.557...
	0.72	0.72	0.76	2.06	Nguyen & Leung (2010)Nguyen, Q., & Leung, P. (2010). How nurture can shape preferences: An experimental study on risk preferences of Vietnamese fishers. Environment and Development Economics, 15(5), 609-631. doi:10.1017/S1355770X10000203 https://doi.org/10.1017/S1355770X1000020...
	0.48	0.48	0.69	3.47	Liu (2012)Liu, E. M. (2012). Time to change what to sow: Risk Preferences and technology adoption decisions of cotton farmers in China. Review of Economics and Statistics, 95(4), 1386-1403. doi:10.1162/REST_a_00295 https://doi.org/10.1162/REST_a_00295...
	0.11	0.11	0.13	1.35	Liebenehm & Waibel (2014)Liebenehm, S., & Waibel, H. (2014). Simultaneous estimation of risk and time preferences among small-scale cattle farmers in West Africa. American Journal of Agricultural Economics, 96(5), 1420-1438. doi:10.1093/ajae/aau056 https://doi.org/10.1093/ajae/aau056...

Model	Description	%	Quantity	PT	CPT	NPT	TK	KAR	PRC	PWR	EXP	LOG
111	pt-tk-pwr	8.3%	54	54			54			54
131	pt-prc-pwr	8.3%	54	54					54	54
133	pt-prc-log	7.3%	48	48					48			48
112	pt-tk-exp	6.6%	43	43			43				43
113	pt-tk-log	6.3%	41	41			41					41
132	pt-prc-exp	5.5%	36	36					36		36
332	npt-prc-exp	5.2%	34			34			34		34
122	pt-kar-exp	4.9%	32	32				32			32
222	cpt-kar-exp	4.9%	32		32			32			32
312	npt-tk-exp	4.9%	32			32	32				32
322	npt-kar-exp	4.9%	32			32		32			32
232	cpt-prc-exp	4.3%	28		28				28		28
212	cpt-tk-exp	2.9%	19		19		19				19
123	pt-kar-log	2.6%	17	17				17				17
223	cpt-kar-log	2.6%	17		17			17				17
313	npt-tk-log	2.6%	17			17	17					17
323	npt-kar-log	2.6%	17			17		17				17
333	npt-prc-log	2.4%	16			16			16			16
331	npt-prc-pwr	1.8%	12			12			12	12
121	pt-kar-pwr	1.8%	12	12				12		12
221	cpt-kar-pwr	1.8%	12		12			12		12
321	npt-kar-pwr	1.8%	12			12		12		12
311	npt-tk-pwr	1.8%	12			12	12			12
231	cpt-prc-pwr	1.5%	10		10				10	10
211	cpt-tk-pwr	1.4%	9		9		9			9
213	cpt-tk-log	0.5%	3		3		3					3
233	cpt-prc-log	0.5%	3		3				3			3

Summary by theory			Summary by weighting			Summary by value
PT	CPT	NPT	TK	KAR	PRC	PWR	EXP	LOG
337	133	184	230	183	241	187	288	179
51.53%	20.34%	28.13%	35.17%	27.98%	36.85%	28.59%	44.04%	27.37%

Model	Type	Median error	Average error	Standard deviation error
111	pt-tk-pwr	0.30	0.36	0.26
112	pt-tk-exp	0.33	0.36	0.24
113	pt-tk-log	0.32	0.37	0.26
121	pt-kar-pwr	0.47	0.49	0.21
122	pt-kar-exp	0.35	0.38	0.24
123	pt-kar-log	0.44	0.46	0.22
131	pt-prc-pwr	0.32	0.37	0.25
132	pt-prc-exp	0.35	0.38	0.25
133	pt-prc-log	0.33	0.37	0.25
211	cpt-tk-pwr	0.73	0.72	0.25
212	cpt-tk-exp	0.39	0.42	0.24
213	cpt-tk-log	1.13	1.00	0.35
221	cpt-kar-pwr	0.47	0.49	0.21
222	cpt-kar-exp	0.35	0.38	0.24
223	cpt-kar-log	0.44	0.46	0.22
231	cpt-prc-pwr	0.57	0.58	0.24
232	cpt-prc-exp	0.38	0.40	0.25
233	cpt-prc-log	0.90	0.84	0.30
311	npt-tk-pwr	0.47	0.49	0.21
312	npt-tk-exp	0.35	0.38	0.24
313	npt-tk-log	0.44	0.47	0.22
321	npt-kar-pwr	0.47	0.49	0.21
322	npt-kar-exp	0.35	0.38	0.24
323	npt-kar-log	0.44	0.46	0.22
331	npt-prc-pwr	0.47	0.49	0.21
332	npt-prc-exp	0.35	0.38	0.24
333	npt-prc-log	0.44	0.46	0.22
EUT		4.89	4.89	1.37

Fundação Getulio Vargas, Escola de Administração de Empresas de S.Paulo Av 9 de Julho, 2029, 01313-902 S. Paulo - SP Brasil, Tel.: (55 11) 3799-7999, Fax: (55 11) 3799-7871 - São Paulo - SP - Brazil
E-mail: rae@fgv.br

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[TFN01] Notes: According to the Brazilian Institute for Geography and Statistics, known as Instituto Brasileiro de Geografia e Estatística or simply IBGE in Portuguese:

[TFN02] Classes A and B: usually composed of those who have completed higher education, composed of bankers, investors, business owners, major landowners and people with extraordinary skills for the industry they operate in, directors and high managers, judges, prosecutors, highly educated professors, doctors, well qualified engineers, lawyers, etc.

[TFN03] Class C: most people in this class have finished high school and there is also a significant quantity of people who have completed higher education or have at least a technical level degree. Composed of those who provide services directly to the wealthier groups, such as teachers, managers, mechanics, electricians, nurses, etc.

[TFN04] Class D: people who tend not to finish high school. Composed of people who provide services to Class C, such as housemaids, bartenders, bricklayers, people who work for civil construction companies, small store owners, low-paid drivers, etc.

[TFN05] Class E: people who do not attend or finish elementary school and illiterate people. Composed of people who earn minimum salaries, such as cleaners, street sweepers and also unemployed people.