Speed of Reversion of Deviations of the Purchasing Power Parity for Brazilian Cities

Bastos, Felipe de Sousa; Ferreira, Roberto Tatiwa; Arruda, Elano Ferreira

doi:10.5935/0034-7140.20180002

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Articles • Rev. Bras. Econ. 72 (1) • Jan-Mar 2018 • https://doi.org/10.5935/0034-7140.20180002 copy

Speed of Reversion of Deviations of the Purchasing Power Parity for Brazilian Cities

Authorship SCIMAGO INSTITUTIONS RANKINGS

This study estimates the rate of reversion of deviations from the PPP for Brazilian cities considering three possible sources of bias: i) Nickell, ii) the heterogeneity of the autoregressive coefficients, and iii) generated by the temporal aggregation of price indices. The values of the estimated half-lives were approximately 4.41 and 3.18 years when considering the price index of Brazil and the average of the indices of cities as references, respectively. When using the price index for each city as the numeraire, the median half-life is 3.13 years.

Keywords:
Purchasing Power Parity; Speed of Reversion; Nickell Bias; Temporal Aggregation Bias

Author(s)	Index	Period	Cities/Countries	Numeraire	Methodology	H(p̂)years
Parsley & Wei (1996)Parsley, D. C., & Wei, S.-J. (1996). Convergence to the law of one price without trade barriers or currency fluctuations. The Quarterly Journal of Economics, 111(4), 1211-1236. doi: 10.2307/2946713 https://doi.org/10.2307/2946713...	51 Prices of Commodities	1975:01-1992:04	USA Cities	New Orleans and New York	Levin e Lin (LL) test	1.25, 1.00 and 3.75^vii vii For non-perishable, perishable and non-tradable goods, respectively.
Culver & Papell (1999)Culver, S. E., & Papell, D. H. (1999, June). Panel evidence of purchasing power parity using intranational and international data. Retrieved from http://www.uh.edu/~dpapell/citypap.pdf http://www.uh.edu/~dpapell/citypap.pdf...	Consumer Price Index	USA (1978:05-1997:04) CAN (1978:09-1997:06)	USA and Canada Cities	All Cities	Feasible GLS	USA: 3.82 CAN: 1.83ⁱ i Mean.
Nenna (2001)Nenna, M. (2001). Price level convergence among italian cities: Any role for the Harrod-Balassa-Samuelson Hypothesis? (Working Paper No. 64). Rome: Ministero dell'Economia e delle Finanze, Consiglio degli Esperti e Faculty of Economics, University of Rome La Sapienza.	Consumer Price Index	1947-2000	Italian Cities	Rome and Cross-section Average	LL test	1.97
Cecchetti et al. (2002)Cecchetti, S. G., Mark, N. C., & Sonora, R. J. (2002). Price index convergence among United States cities. International Economic Review, 43(4), 1081-1099. Retrieved from http://www.jstor.org/stable/826961 http://www.jstor.org/stable/826961... ^ Used the same cities. (-) Unavailable.	Consumer Price Index	1918-1995	USA Cities	Cross-section Average	From LL and Im, Pesaran e Shin (IPS) unit root tests with adjustments for the Nickell bias	8.50-9.70ⁱⁱⁱ iii Interval tests LL (inferior) and IPS (superior).
L. L. Chen & Devereux (2003)Chen, L. L., & Devereux, J. (2003). What can US city price data tell us about purchasing power parity? Journal of International Money and Finance, 22(2), 213-222. doi: 10.1016/S0261-5606(02)00102-X https://doi.org/10.1016/S0261-5606(02)00... ^ Used the same cities. (-) Unavailable.	Absolute Price Level built from the CPI	1918-2000	USA Cities	USA	Augmented Dickey-Fuller (ADF) test	4.98ⁱⁱ ii Median.
Carrion-i-Silvestre et al. (2004)Carrion-i-Silvestre, J. L., Barrio, T. d., & López-Bazo, E. (2004). Evidence on the purchasing power parity in a panel of cities. Applied Economics, 36(9), 961-966. doi: 10.1080/0003684042000233186 https://doi.org/10.1080/0003684042000233...	Consumer Price Index	1939:07-1992:12	Spanish Cities	Spain	From unit root tests	3.60ⁱⁱ ii Median.
Sonora (2005)Sonora, R. J. (2005). City CPI convergence in Mexico. Review of Development Economics, 9(3), 359-367. Retrieved from 10.1111/j.1467-9361.2005.00281.x https://doi.org/10.1111/j.1467-9361.2005...	Consumer Price Index	1982:01-2000.12	Mexican Cities	México DF	Mean of ρ̂ ADF with bias adjustment estimated through Kendall (1954)Kendall, M. G. (1954). Note on bias in the estimation of autocorrelation. Biometrika, 41(3-4), 403-404. doi: 10.1093/biomet/41.3-4.403 https://doi.org/10.1093/biomet/41.3-4.40...	2.00-3.00^iv iv Minimum and Maximum intervals.
Choi & Matsubara (2007)Choi, C.-Y., & Matsubara, K. (2007). Heterogeneity in the persistence of relative prices: What do the Japanese cities tell us? Journal of the Japanese and International Economies, 21(2), 260-286. doi: 10.1016/j.jjie.2006.03.002 https://doi.org/10.1016/j.jjie.2006.03.0...	36 Disaggregated CPI Items	1970:01-2002:12	Japanese Cities	All Cities	Sum of autoregressive coefficients (SARC), Impulse Response Function (IRF), RGLS and non-linear	<2.00^v v Inferior median 2 years for most numeraires.
Das & Bhattacharya (2008)Das, S., & Bhattacharya, K. (2008). Price convergence across regions in India. Empirical Economics, 34(2), 299-313. doi: 10.1007/s00181-007-0123-8 https://doi.org/10.1007/s00181-007-0123-...	Consumer Price Indices for Industrial Workers	1995:01-2004:06	Indian Regions	Cross-section Average and Nagpur	Moon-Perron (MP) test, direct Dickey-Fuller (DDF) and Robust developed by Breitung & Das (2008)Breitung, J., & Das, S. (2008). Testing for unit roots in panels with a factor structure. Econometric Theory, 24(1), 88-108. doi: 10.1017/S0266466608080067 https://doi.org/10.1017/S026646660808006... with adjustment to the Nickel bias	0.68/1.91^vi vi For common component and the idiosyncratic factor, respectively.
Nath & Sarkar (2009)Nath, H. K., & Sarkar, J. (2009). Unbiased estimation of the half-life to price index convergence among U.S. cities. Journal of Money, Credit and Banking, 41(5), 1041-1046. Retrieved from http://www.jstor.org/stable/40271593 http://www.jstor.org/stable/40271593... ^* * Used the same cities.	Consumer Price Index	1918-2006	USA Cities	Cross-section Average	GLS with FE with bias correction for Nickell and temporal aggregation	7.50
Nagayasu & Inakura (2009)Nagayasu, J., & Inakura, N. (2009). PPP: Further evidence from Japanese regional data. International Review of Economics & Finance, 18(3), 419-427. doi: 10.1016/j.iref.2008.09.003 https://doi.org/10.1016/j.iref.2008.09.0...	CPI Aggregated and Disaggregated	1990-2003	Japanese Cities	Tokyo	Moon & Perron (2004)Moon, H. R., & Perron, B. (2004). Testing for a unit root in panels with dynamic factors. Journal of Econometrics, 122(1), 81-126. doi: 10.1016/j.jeconom.2003.10.020 https://doi.org/10.1016/j.jeconom.2003.1... unit root test	≈ 2.00
Chmelarova & Nath (2010)Chmelarova, V., & Nath, H. K. (2010). Relative price convergence among US cities: Does the choice of numeraire city matter? Journal of Macroeconomics, 32(1), 405-414. doi: 10.1016/j.jmacro.2009.09.006 https://doi.org/10.1016/j.jmacro.2009.09... ^* * Used the same cities.	Consumer Price Index	1918-2007	USA Cities	All Cities	GLS with FE and Nickel bias correction	9.54ⁱ i Mean.
Mohsin & Gilbert (2010)Mohsin, H. M., & Gilbert, S. (2010). The relative city price convergence in Pakistan: Empirical evidence from spatial GLS. The Pakistan Development Review, 49(4), 439-448. Retrieved from http://www.pide.org.pk/pdf/PDR/2010/Volume4/439-448.pdf http://www.pide.org.pk/pdf/PDR/2010/Volu...	Consumer Price Index	2001:07-2008:06	Pakistani Cities	Karachi and Lahori	Spatial GLS	<0.42
Frankel & Rose (1995)Frankel, J. A., & Rose, A. K. (1995, February). A panel project on purchasing power parity: Mean reversion within and between countries (Working Paper No. 5006). National Bureau of Economic Research (NBER). doi: 10.3386/w5006 https://doi.org/10.3386/w5006...	Consumer Price Index	1948-1992	150 países	USA	OLS with Standard errors corrected through Huber/White	≈ 4.00
Wei & Parsley (1995)Wei, S.-J., & Parsley, D. C. (1995, February). Purchasing power disparity during the floating rate period: Exchange rate volatility, trade barriers and other culprits (Working Paper No. 5032). National Bureau of Economic Research (NBER). doi: 10.3386/w5032 https://doi.org/10.3386/w5032...	Sector Price Indices	1973-1986	14 OECD countries	-	-	4.25 and 4.75^viii viii For countries belonging and not belonging to the European Monetary System, respectively.
Culver & Papell (1999)Culver, S. E., & Papell, D. H. (1999, June). Panel evidence of purchasing power parity using intranational and international data. Retrieved from http://www.uh.edu/~dpapell/citypap.pdf http://www.uh.edu/~dpapell/citypap.pdf...	Consumer Price Index	1978:01-1997:02	European Union	All Countries	Feasible GLS	2.19ⁱ i Mean.
Choi et al. (2006)Choi, C.-Y., Mark, N. C., & Sul, D. (2006). Unbiased estimation of the half-life to PPP convergence in panel data. Journal of Money, Credit and Banking, 38(4), 921-938. Retrieved from https://www3.nd.edu/~nmark/wrkpaper/PPP%20Bias_JMCB.pdf https://www3.nd.edu/~nmark/wrkpaper/PPP%...	Consumer Price Index	1973-1998	21 OECD coutries	All Countries	GLS with FE with bias correction for Nickell and temporal aggregation	3.00ⁱⁱ ii Median.
Carvalho & Nechio (2011)Carvalho, C., & Nechio, F. (2011). Aggregation and the PPP puzzle in a sticky-price model. The American Economic Review, 101(6), 2391-2424. doi: 10.1257/aer.101.6.2391 https://doi.org/10.1257/aer.101.6.2391...	Artificial Data Generated by the Model	-	US Economy and Rest of the World	-	Computed directly from the impulse response functions implied by the solution of the models	3.25 and 1.17^ix ix Multisector, two-country, sticky-price model and counterfactual one-sector world economy, respectively.

Year	Numeraire
	Belém	Belo Horizonte	Brasília	Curitiba	Fortaleza	Goiânia	Rio de Janeiro	Salvador	São Paulo	Porto Alegre	Recife	Brazil	Mean
1991	0.050	0.046	0.051	0.050	0.051	0.051	0.046	0.049	0.050	0.052	0.041	0.049	0.049
1992	0.042	0.040	0.042	0.042	0.037	0.039	0.036	0.037	0.042	0.042	0.038	0.040	0.040
1993	0.033	0.032	0.034	0.034	0.026	0.034	0.030	0.032	0.034	0.034	0.033	0.033	0.033
1994	0.033	0.033	0.034	0.035	0.027	0.035	0.029	0.034	0.035	0.034	0.035	0.033	0.033
1995	0.033	0.037	0.037	0.038	0.034	0.037	0.032	0.037	0.037	0.037	0.038	0.036	0.036
1996	0.033	0.035	0.033	0.035	0.033	0.033	0.031	0.034	0.029	0.032	0.034	0.033	0.033
1997	0.033	0.033	0.032	0.034	0.033	0.032	0.032	0.033	0.021	0.031	0.033	0.032	0.032
1998	0.031	0.031	0.029	0.031	0.030	0.029	0.031	0.031	0.021	0.029	0.031	0.030	0.030
1999	0.025	0.026	0.025	0.026	0.025	0.023	0.026	0.026	0.016	0.025	0.025	0.025	0.025
2000	0.021	0.022	0.021	0.021	0.021	0.018	0.021	0.021	0.013	0.021	0.022	0.020	0.020
2001	0.017	0.017	0.017	0.016	0.017	0.015	0.017	0.017	0.011	0.017	0.017	0.016	0.016
2002	0.015	0.014	0.014	0.014	0.015	0.013	0.015	0.014	0.012	0.015	0.015	0.014	0.014
2003	0.013	0.011	0.013	0.013	0.013	0.012	0.013	0.011	0.013	0.012	0.013	0.013	0.013
2004	0.008	0.008	0.009	0.009	0.009	0.008	0.009	0.008	0.009	0.008	0.009	0.009	0.009
2005	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
2006	0.009	0.007	0.008	0.008	0.008	0.009	0.009	0.008	0.007	0.008	0.008	0.008	0.008
2007	0.014	0.012	0.014	0.013	0.014	0.014	0.014	0.014	0.014	0.013	0.014	0.013	0.013
2008	0.017	0.019	0.021	0.019	0.020	0.020	0.020	0.020	0.019	0.020	0.019	0.020	0.020
2009	0.017	0.022	0.023	0.021	0.023	0.023	0.023	0.023	0.022	0.022	0.022	0.022	0.022
2010	0.018	0.024	0.025	0.024	0.025	0.024	0.025	0.025	0.024	0.024	0.025	0.024	0.024
2011	0.017	0.021	0.022	0.022	0.022	0.021	0.023	0.022	0.021	0.020	0.022	0.021	0.021
2012	0.020	0.024	0.026	0.025	0.026	0.023	0.026	0.026	0.024	0.024	0.025	0.024	0.024
2013	0.023	0.029	0.031	0.030	0.031	0.028	0.031	0.031	0.028	0.028	0.030	0.029	0.029
Δ% per year	-3.33	-2.01	-2.25	-2.26	-2.23	-2.62	-1.78	-2.04	-2.48	-2.66	-1.38	-2.27	-2.27

	No Bias Corrections		Nickell Bias Corrected		Time Aggregation Bias Corrected		Nickell and Time Aggregation Bias Corrected
Numeraire	ρ̂ _FGLS	H(ρ_FGLS)	ρ̂ _GMUE	H(ρ̂_GMUE)	ρ̂ _GTAU	H(ρ̂_GTAU)	ρ̂ _GNTAU	H(ρ̂_GNTAU)
Belém	0.829	3.70	0.921	8.42	0.722	2.13	0.830	3.72
Belo Horizonte	0.836	3.87	0.886	5.73	0.681	1.80	0.785	2.86
Brasília	0.826	3.63	0.868	4.90	0.661	1.67	0.762	2.55
Curitiba	0.836	3.87	0.925	8.89	0.726	2.16	0.835	3.84
Fortaleza	0.835	3.84	0.876	5.24	0.670	1.73	0.772	2.68
Goiânia	0.817	3.43	0.899	6.51	0.697	1.92	0.802	3.14
Rio de Janeiro	0.846	4.14	0.894	6.19	0.691	1.88	0.795	3.02
Salvador	0.817	3.43	0.908	7.18	0.708	2.01	0.814	3.37
Porto Alegre	0.814	3.37	0.919	8.21	0.720	2.11	0.828	3.68
Recife	0.849	4.23	0.898	6.44	0.696	1.91	0.801	3.13
São Paulo	0.845	4.12	0.878	5.33	0.672	1.74	0.774	2.71
Brasil	0.939	11.01	0.94	11.2	0.744	2.34	0.855	4.41
Mean	0.832	3.77	0.901	6.65	0.699	1.94	0.804	3.18
Minimum	0.814	3.37	0.868	4.90	0.661	1.67	0.762	2.55
Maximum	0.849	4.23	0.925	8.89	0.726	2.16	0.835	3.84
Mean	0.832	3.76	0.897	6.41	0.695	1.90	0.800	3.15
Median	0.835	3.84	0.898	6.44	0.696	1.91	0.801	3.13

H(ρ̂_GNTAU) < 3	3 ≤ H(ρ̂_GNTAU) ≤ 5	H(ρ̂_GNTAU) > 5
33.33%	66.67%	0.00%

Numeraire	Δ̃	p-value
Belém	-0.255	0.799
Belo Horizonte	2.65	0.008
Brasília	1.279	0.201
Curitiba	1.753	0.080
Fortaleza	-1.041	0.298
Goiânia	-0.678	0.498
Rio de Janeiro	1.883	0.06
Salvador	0.036	0.971
Porto Alegre	-0.645	0.519
Recife	1.573	0.116
São Paulo	-1.784	0.074

	No Bias Corrections		Nickell Bias Corrected		Time Aggregation Bias Corrected		Nickell and Time Aggregation Bias Corrected
Numeraire	ρ̂ _FGLS	H(ρ̂_FGLS)	ρ̂ _GMUE	H(ρ̂_GMUE)	ρ̂ _GTAU	H(ρ̂_GTAU)	ρ̂ _GNTAU	H(ρ̂_GNTAU)
Nordeste	0.796	3.04	0.915	7.80	0.716	2.07	0.823	3.55
Norte	0.783	2.83	0.856	4.46	0.646	1.59	0.746	2.36
Sul	0.764	2.57	0.859	4.56	0.650	1.61	0.750	2.41
Sudeste	0.795	3.02	0.909	7.26	0.708	2.01	0.814	3.38
Centro-Oeste	0.768	2.63	0.919	8.21	0.720	2.11	0.828	3.67
Brasil	0.798	3.07	0.896	6.31	0.693	1.89	0.798	3.07
Minimum	0.764	2.57	0.856	4.46	0.646	1.59	0.746	2.36
Maximum	0.796	3.04	0.919	8.21	0.720	2.11	0.828	3.67
Mean	0.781	2.82	0.892	6.46	0.688	1.88	0.792	3.07
Median	0.783	2.83	0.909	7.26	0.708	2.01	0.814	3.38

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[TFN1] Notes:

[TFN2] ^*
Used the same cities.

[TFN3] ^**
Used the same cities. (-) Unavailable.

[TFN4] ⁱ
Mean.

[TFN5] ⁱⁱ
Median.

[TFN6] ⁱⁱⁱ
Interval tests LL (inferior) and IPS (superior).

[TFN7] ^iv
Minimum and Maximum intervals.

[TFN8] ^v
Inferior median 2 years for most numeraires.

[TFN9] ^vi
For common component and the idiosyncratic factor, respectively.

[TFN10] ^vii
For non-perishable, perishable and non-tradable goods, respectively.

[TFN11] ^viii
For countries belonging and not belonging to the European Monetary System, respectively.

[TFN12] ^ix
Multisector, two-country, sticky-price model and counterfactual one-sector world economy, respectively.