Price convergence, reversal speed and purchasing power parity: Stylized facts for Brazilian cities

Arruda, Elano Ferreira; Castelar, Ivan; Guimarães, Daniel Barboza; Barbosa, Rafael Barros Article Price convergence, reversal speed and purchasing power parity: Stylized facts for Brazilian cities EconomiA Provided in Cooperation with: The Brazilian Association of Postgraduate Programs in Economics (ANPEC), Rio de Janeiro Suggested Citation: Arruda, Elano Ferreira; Castelar, Ivan; Guimarães, Daniel Barboza; Barbosa, Rafael Barros (2018) : Price convergence, reversal speed and purchasing power parity: Stylized facts for Brazilian cities, EconomiA, ISSN 1517-7580, Elsevier, Amsterdam, Vol. 19, Iss. 2, pp. 219-235, https://doi.org/10.1016/j.econ.2018.01.001 This Version is available at: https://hdl.handle.net/10419/266920 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Available online at www.sciencedirect.com ScienceDirect HOSTED BY EconomiA 19 (2018) 219–235 Price convergence, reversal speed and purchasing power parity: Stylized facts for Brazilian cities Elano Ferreira Arrudaa,∗, Ivan Castelarb, Daniel Barboza Guimarãesc, Rafael Barros Barbosad aApplied Economics Department (CAEN/MAER/UFC), Federal University of Ceará, Brazil bFinance Department (CAEN/UFC), Federal University of Ceará, Brazil cBusiness Department (DA/UFC), Federal University of Ceará, Brazil dApplied Economics Department (DEA/UFC), Federal University of Ceará, Brazil Received 16 July 2014; received in revised form 24 December 2017; accepted 12 January 2018 Available online 1 February 2018 Abstract This paper analyzes the price dynamics of Brazilian cities between 1995 and 2012 to identify stylized facts about price convergence, the reversal speed of deviations between relative prices and purchasing power parity (PPP). There is evidence of a strong reduction in the absolute dispersion of prices of Brazilian cities and in the variability of relative prices. The estimated half-life of deviations from PPP reversal proved to be lower than those found for cross country data and American cities. The results also indicate that the stationarity of the real exchange rate among the cities is rejected for all the series that presented a reversal speed to deviations from the PPP smaller than the average for each numerarie considered. It is argued that the evidence of price convergence associated with a process of slow reversal speed of deviations from the PPP have influence on the non-rejection of a unit root in the real exchange rate series for some cities, however, this fact does not constitute in itself evidence against the validity of the PPP. JEL classifications: F31; R10; E31 Keywords: Purchasing power parity; Reversal speed; Real exchange rate; Price convergence © 2018 The Authors. Production and hosting by Elsevier B.V. on behalf of National Association of Postgraduate Centers in Economics, ANPEC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction In recent years, several researchers have observed flaws in the relationship called purchasing power parity, heretofore PPP, for international data. Such observations motivated several studies1to investigate the behavior of the real exchange rate using within country data. ∗Corresponding author. E-mail address: [email protected] (E.F. Arruda). 1See for example Engel and Rogers (1996), Culver and Papell (1999), Cecchetti et al. (2002) and Chen and Devereux (2003). Peer review under responsibility of National Association of Postgraduate Centers in Economics, ANPEC. https://doi.org/10.1016/j.econ.2018.01.001 1517-7580 © 2018 The Authors. Production and hosting by Elsevier B.V. on behalf of National Association of Postgraduate Centers in Economics, ANPEC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 220 E.F. Arruda et al. / EconomiA 19 (2018) 219–235 These flaws, represented by an incomplete adjustment of the level of international relative prices, can be explained by factors such as: (i) commercial barriers, as tariffs and quotas; (ii) barriers of bureaucratic origin in the establishment or creation of a distribution system of goods; (iii) failure in the adjustment of the real exchange rate to shocks in the relative prices; (iv) market failures, such as the presence of firms with monopoly power, with differentiated prices in segmented markets; (v) transport costs, associated with transferring goods from one region to another, and (vi) possible differences in the composition of the price indexes between countries.2 The empirical papers which analyze the validity of the PPP test if the series of deviations of the PPP, called relative prices, or the real exchange rate between countries are stationary. Authors such as Breuer (1994) and Froot and Rogoff (1995) present a survey of the literature which corroborates that the series in question have unit roots. Such evidence implies that the inflation differentials between countries, measured in the same currency, can persist indefinitely, or deviations from the PPP converge to a common average at a very low speed. Furthermore, the literature points to a consensus about the speed of convergence, or the reversal of PPP deviations, with a half-life3between three to five years (Abuaf and Jorion, 1990; Frankel and Rose, 1996; Wu, 1996; Papell, 1997; Lothian, 1997). The need to comprehend the persistence in PPP deviations for international data and the existence of great economic regions with a single currency, such as the European Union, encouraged specialists to analyze if countries with continental dimensions, with great diversity and regional disparities, satisfy the PPP regularity conditions and if the reversal speed is shown to be lower using intranational data. Furthermore, this approach is relevant because it shows that excessive variations in relative prices and, hence, on inflation differentials lead to the inefficient allocation of resources among economic sectors, and determine the differences in real wages and real interest rates that, in turn, influence the flows of labor and capital. Therefore, the shifting of relative prices involves substantial losses of welfare to society, besides being useful in the investigation of the degree of integration and regional growth (Nath and Vargas-Silva, 2012; Hegwood and Nath, 2013). Also, with the use of intra data, it is possible to come to a better understanding of sources of persistent deviations from the PPPs found in papers using cross-country information (Cecchetti et al., 2002). The first effort in that direction was made by Engel and Rogers (1996), which compared the variability in relative prices with disaggregated data of price indexes for the United States, Canada and between the two countries. The authors showed that the distance between cities located in the same country substantially explains the variation in prices of similar goods. Also, oscillations in prices at cities located in different countries were shown to be greater than those in equidistant cities within the same country. Yet Culver and Papell (1999), investigated PPP regularity in the post-Bretton Woods period, using international and within country data for the United States, Canada and European countries. The authors found poor evidence of the validity of PPP with international data, when compared to what was found for European countries. Only Canada presented a clear evidence when compared to that of European countries. The price convergence speed was shown to be slower in the United States than those found for Canada and the European countries. In other words, even without problems arising from trade barriers, exchange rate volatility, monetary policy differences and other factors4which restrict arbitrage in the goods market, the authors found a slow price convergence process in the United States. Using information for the 19 more populated American cities5between 1918 and 1995, Cecchetti et al. (2002) analyzed if the price indexes of those cities followed a common trend, and estimated the reversal speed of possible shocks to local prices. The authors found evidence of a temporary divergence in the price indexes of American cities, with a slow reversal process representing an estimated half-life of approximately nine years. The authors argue that the main reason for those findings are the presence of transportation costs and market failures. In another important paper, Chen and Devereux (2003) analyzed the dispersion of absolute prices in American cities between 1918 and 2000, measured by the coefficient of variation of the price level, and found strong evidence 2For a review of these aspects, see Rogoff (1996) and Taylor and Taylor (2004). 3Half-life is a measure of the velocity of adjustment in some variable. It indicates how fast the variable reverts its trajectory to the mean. 4These factors refer to problems to construct the consumer price indexes, as consumer’s preferences, weights of each item, the existence of non-tradable goods, etc. 5Atlanta, Baltimore, Boston, Chicago, Cincinnati, Cleveland, Detroit, Houston, Kansas City, Los Angeles, Minneapolis, New York City, Philadelphia, Pittsburgh, Portland, San Francisco, Seattle, St. Louis and Washington D. C. E.F. Arruda et al. / EconomiA 19 (2018) 219–235 221 that it is smaller between American cities than between OECD6countries. The authors also found indication of price convergence, and argue that the existence of convergence or divergence in the indicator between cities yields a nonstationary bilateral real exchange rate7; thus, for those cases, the non-stationary status of the variable is not evidence against PPP. The estimated average half-life of PPP deviations was of five years. Authors such as Culver and Papell (1999), Cecchetti et al. (2002) and Chen and Devereux (2003) attribute the difficulty to reject the presence of a unit root in the real exchange rate between cities to factors such as reduced speed of reversion/adjustment and the existence of convergence or divergence in prices. In fact, stationarity requires that relative prices/real exchange rate between two cities occur at a constant level. In addition, Simões and Marc¸al (2012) point out that, high frequency data favor the appearance of anomalies in the time series, such as more intense changes in the variance of the shocks and, according to Cavaliere (2005), in this situation, tests would tend to indicate more frequently the presence of unit roots. In recent papers, authors as Hegwood and Nath (2013) and Basher and Carrion-i-Silvestre (2011) point out the relation between structural breaks and slow mean reversion of relative price across cities. Others as Nakamura and Steinsson (2013) and Gorodnichenko and Weber (2016) argue that price stickness8can be one possible source of nominal price rigidity in the micro level and contribute to slow mean reversion of relative prices. Thus, the presence of structural breaks and price rigidity may also affect the results of the unit root tests. Therefore, although stationarity of relative prices to analyze the validity of PPP is the standard technique in the literature, a slow reversal speed associated with a price convergence process can contribute to the rejection of stationarity of real exchange series between cities, which does not represent evidence against the validity of PPP. Although the analysis of the validity of the PPP hypothesis is quite common in international literature, with various papers considering American, Canadian and European cities practically there are no papers dealing with the regional evolution of relative prices, with evidence of regional price convergence or the validity of PPP for Brazilian cities. Motivated by that, this paper intends to fill the aforementioned gap by investigating price convergence and reversal speed in deviations to the PPP using data for Brazilian cities. Therefore, the intention of the paper is to analyze three aspects of this theme; that is, (i) is there evidence of dispersion reduction, whether absolute or relative, in prices for the considered cities? (ii) what is the adjustment, or reversal, speed for deviations from the PPP among Brazilian cities?, and (iii) does the evidence of convergence and of a slow reversal to the PPP process contribute to the non-rejection of a unit root in the real exchange series between cities? Apart from this introduction, this paper is composed of five more sections. The following section presents the PPP Theory. The section third presents a review of literature on PPP using data for Brazil. The following section promotes a unified discussion over unit root, co-integration and structural break approaches. The section fifth section presents the stylized facts observed in this study and, lastly, in the sixth section, the concluding remarks are made. 2. PPP theory The purchasing power parity (PPP) theory states that the difference among prices in two countries is not permanent if these prices are measured in the same exchange rate. This implies that prices converge towards a price of equilibrium. The PPP theory is derived from the Law of One Price (LOP), that asserts the existence of a price of equilibrium between two countries if there are no trade barriers, legal barriers and transaction costs. In the presence of these trade barriers the LOP, and consequently the PPP, are not valid. 6Organization for Economic Co-operation and Development (OECD). Member countries: Germany, Australia, Austria, Belgium, Canada, South Korea, Denmark, Spain, United States of America, Finland, France, Greece, Holland, Hungary, Ireland, Iceland, Italy, Japan, Luxembourg, Mexico, Norway, New Zealand, Poland, Portugal, United Kingdom, Czech Republic, Slovakia, Sweden, Switzerland, Turkey. 7The following expression is used for the real exchange rate between cities, θ =EP∗ P(where P* is the level of external prices, P is the domestic price level and E is the nominal exchange rate, as it concerns economies with the same currency, since the nominal exchange rate, in this case, equals 1). 8Sticky prices refer to problems associated to physical adjustment costs, as menu costs, and informational frictions. 222 E.F. Arruda et al. / EconomiA 19 (2018) 219–235 There are two versions of the PPP theory, the absolute and the relative version. Let Pi,tand Pf i,t be the price of the good i at time t in the local and in the foreign country, respectively. In the absolute version, the PPP theory suggests that nominal exchange rate between two countries is the ratio between the local and foreign prices; that is, Ei,t =Pi,t Pf i,t (1) Usually the PPP is tested using price indexes that represent a basket of goods. The logarithm version of (1) is also frequently used to test the PPP: et− pt+ pf t= 0 (2) where et= ln Ei,t , pt= ln Pi,t and pf t= ln Pf i,t . According to this version, the PPP is valid if etis stationary or do not present permanent changes. Many factors can impair the validity of the absolute version of PPP such as governmental trade restrictions, nontradable goods, differences between consumption baskets etc. In the analysis of intranational prices, as proposed in this paper, the PPP is not affected by these barriers, because there are not governmental trade restrictions within the same country and the existence of nontradable goods are minimized. Then, the investigation about the validity of PPP is rather easier to test in the intranational context. On the other hand, the relative version of PPP states that the differences between inflations in both countries are the same as the rate of currency devaluation. To test this version the Eq. (2) can be used in a difference version. et− pt+ pf t= 0 (3) 3. Evidence for Brazil Concerning national data for the Brazilian economy, the discussion on the validity of PPP is extensive, and various authors tested the validity of such an approach for various periods in absolute or relative form. In one of those papers, Zini and Cati (1993) applied unit root and cointegration tests to verify if the hypothesis that the PPP explains the real exchange rate in Brazil is valid, and if changes in the terms of trade of the economy can explain the changes in the real exchange rate for the 1855–1990 period. The tests rejected the absolute PPP hypothesis, indicating the need to search for other factors, such as changes in the terms of trade, to explain oscillations in the long run real exchange rate. Using quarterly data between January 1980 and June 1994, and the Johansen cointegration test, Marc¸al et al. (2003) tested the PPP condition in its absolute form, and the uncovered interest rate parity (UIP). The results point out to the insufficiency of the PPP hypothesis in its absolute version. The results are more favorable when joining PPP and UIP. In an attempt to make price comparisons between various regions simultaneously, without the need to establish anyone of then as the basis, Azzoni et al. (2003) presented two procedures for the construction of within region PPP indexes. Firstly, the authors built a multilateral index from a neoclassical aggregation function. The second method is developed from an econometric perspective, and has the property of estimating prices when information is not available. The results found suggest the existence of great disparity in relative prices between Brazilian cities, something which was expected due to the size of the Brazilian territory and to its cultural and income diversity. According to Kannebley (2003) and Palaia and Holand (2010), very few authors take into account the relationship between structural breaks, such as change in the exchange rate regimen, monetary policy shocks and supply shocks, that could interfere in the path of the real exchange rate measure, which can hamper the applicability of econometric techniques for estimating the validity of PPP. Thus, Kannebley (2003) discusses the relationship between test results for the validity of PPP in Brazil and the relevant economic factors in the 1968–1994 period, which was characterized by many economic policy changes, as well as changes in the macroeconomic cenaries of the country. This factors can generates structural breaks in exchange rate series. Therefore, unit root tests are performed for the existence of a structural break according to the Perron and Vogelsang (1992) and Perron (1989) formulation, and with two breaks, according to the formulation presented in Lee and Strazicich (2003). In all the performed tests for the relative PPP, the validity of that relationship was not rejected for the 1968–1998 period. As for the absolute PPP hypothesis, it was valid only for the real exchange rate based on E.F. Arruda et al. / EconomiA 19 (2018) 219–235 223 the Wholesale Price Index — Internal Availability (Índices de Pre¸cos por Atacado – Disponibilidade Interna – IPAs), RIPA, for the 1968–1978 period. In order to test the absolute and relative version of the PPP for Brasil, Alves et al. (2001), used data that goes from 1855 to 1990, and fractional cointegration technique, which allows a smooth reversion to the mean. The authors found empirical evidence that favors the relative version only. Later on, Gamboa and Fava (2008), extended the previous paper using data until 2005 and introduced structural breaks into the analysis. The authors confirmed the validity of PPP in its relative version, even in the presence of structural breaks, and rejected the validity of PPP in its absolute version. Palaia and Holand (2010) tested the PPP in its absolute form for Brazil through econometric procedures which contemplate the possibility of the existence of structural breaks in the time series considering the 1980–2006 period. The unit root tests with structural breaks did not favor the absolute PPP hypothesis, regardless of the price index used. A cointegration test with structural break was also performed using the Gregory and Hansen (1996) methodology. The test results were not sufficiently low to reject the null hypothesis of non-cointegration. The conclusion is that it is not possible to accept the validity of absolute purchasing power parity between 1980 and 2006. Considering a period characterized by low inflation rates and commercial openness, Feijó and Morales (2008) analyzed the validity of PPP in Brazil, using monthly data for the 1994–2006. The authors adopted methodologies to test for unit root and cointegration, using price index and exchange rate series for Brazil and the United States of America. For the period as a whole, the PPP was not observed due to a structural break which occurred due to a change in the exchange rate regime in Brazil in January 1999. In period before the system change, there was cointegration; already for the period afterwards, no long run equilibrium relationship was found. According to the authors, these results can be, respectively, a reflection of the minidevaluations (band realignments) practised by the Brazilian authorities in the period which preceded the “break”, and of the short period analyzed. Simões and Marc¸al (2012) analyzed the real exchange rate for Brazil using consumer’s price indexes and 21 trade partners from 1957 to 2010. The aim of their paper was to evaluate the validity of PPP by using standard unit root tests (ADF and PP) and some modern unit root tests, as Bierens (1997) and KPSS. The PPP was rejected by the standard tests, as ADF and PP. However, when the Bierens’ and KPSS tests were applied to price indexes without temporal aggregation the PPP was not rejected. Moreover, the authors evaluated Taylor’s hypothesis, according to which the half-life of reversal is overestimated in the presence of temporal aggregation. Using different time frequency data, they found that quarterly aggregation of monthly data was 18.92% above the monthly half-life and the annual aggregation of the same time series yielded a half-life 31.15% above the monthly half-life. Thus, in general terms, the average of 21 countries yielded a half-life 35%–56% greater than price indexes without temporal aggregation. Vasconcelos et al. (2014) examined the PPP hypothesis using linear and non-linear unit root tests for the effective exchange rate of Argentina, Brazil, Chile, Colombia, Mexico, Peru and Venezuela. They firstly applied the linearity test of Harvey et al. (2008) and, secondly, after identifying the presence of linearity, they applied the Ng and Perron (2001) and Lee and Strazicich (2003, 2013) unit root tests. These tests consider the presence of structural breaks in the time series. In the non-linear time series, they used the Kruse (2011) test. The effective exchange rate was obtained from BIS (Bank of International Settlements) and data run from 1994 to 2014. The results indicated the presence of linearity in the series for Brazil, Argentina, Chile, Colombia and Peru, and non-linearity for Mexico and Venezuela. PPP was not reject for Chile, Peru and Mexico, however was rejected for the other countries. At last, Wanzeller and Gadelha (2014) tested the PPP for Brazil using data from 1994 to 2013 and unit root tests that allow exogenous and endogenous forms of structural breaks. They found evidence to reject the PPP hypothesis. Therefore, one can observe the great number of papers with national data for Brazil, and the almost complete lack of papers using within country data, although, as discussed in the previous section, the analysis of the validity of PPP using city data, or within country data, is wide spread in the international literature for American, Canadian and European cities (Engel and Rogers, 1996; Culver and Papell, 1999; Cecchetti et al., 2002; Chen and Devereux, 2003; Faber and Stokman, 2009). 4. Unit root, cointegration, structural breaks and PPP Empirically, the PPP is a relationship among exchange rates, domestic (or local) and foreign prices, where the exchange rate differences stem from domestic and foreign inflation changes (Enders, 2015). Let ptand pf trepresents 224 E.F. Arruda et al. / EconomiA 19 (2018) 219–235 the local and foreign prices at date t, respectively, and etdenote the exchange rate at time t, all in logarithm terms, then a PPP implies that et= pt− pf t+ ut(4) where utis the deviation from PPP at date t. For example, if the domestic and foreign inflation are, respectively, 15% and 20%, the price, at domestic currency, should reduce in approximately 5%. This imply that the term utallows transitory deviations from PPP. Thus, the validity of PPP consists of examining the existence of an exchange rate of equilibrium. If there is no reversion of price towards an equilibrium, one can conclude that does not exist a relationship among nominal exchange rates. In the standard approach, the PPP is tested using stationarity tests in the real exchange rates or through a cointegration test between the local and the foreign prices levels. As this paper analyzes the intranational price data from Brazil, then et= 0. This paper will choose the approach of stationary of the relative prices, or real exchange rate between cities. In this paper we test the stationarity of Brazilian city prices using the Augmented Dickey-Fuller (ADF) (1979) and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) (1992). Following Davidson and MacKinnon (1993), let us consider an autoregressive process of order 1: yt= ρyt−1+ z tδ + t(5) where ytis the time series and ztare the regressors that can contain or not an intercept, or the intercept and trend, ρ and δ are parameters and εtis a white noise process. If |ρ| > 1, ytis a nonstationary series and the its variance increases with time and approaches infinity. If |ρ| < 1, it is a stationary series. Thus, the hypothesis of stationarity can be evaluated by testing whether the absolute value of ρ is strictly less than one. The standard Dickey–Fuller (DF) test is carried out by estimating Eq. (5) after subtracting yt−1from both sides of the equation; that is, yt= αyt−1+ z tδ + t(6) where α = ρ − 1. The null and alternative hypotheses may be written as: H0: α = 0 and H1: α < 0 (7) The simple Dickey–Fuller unit root test described above is valid only if the series is an AR(1) process. If the series is correlated at higher order lags, the assumption of white noise disturbances εtis violated. The Augmented Dickey–Fuller (ADF) test constructs a parametric correction for higher-order correlation by assuming that the y series follows an AR(P) process and adding p lagged difference terms of the dependent variable y to the right-hand side of the test regression: yt= αyt−1+ z tδ + θ1yt−1+ θ2yt−2+ . . . + θpyt−p+ t(8) This augmented specification is then used to test (7) using the t-ratio. The KPSS (1992) test differs from the other unit root tests in that the series is assumed to be stationary under the null. The KPSS statistic is based on the residuals from the OLS regression of yton the exogenous variables zt. Note that in the presence of structural breaks the conclusions about those tests should be taken carefully, because the majority of the stationarity tests looses power in the presence of instabilities. Thus, the structural breaks may affect the validity of PPP, as found in Hegwood and Nath (2013), Basher and Carrion-i-Silvestre (2011). Bierens (1997) states that if a time series is stationary around a nonlinear trend, the null hypothesis of stationarity cannot be reject by the same reason; that is, the stationary tests loose power. Finally, Simões and Marc¸al (2012) point out that in the presence of time aggregation in low frequency data (as annual, for instance) the distorting effects of the nonlinearities on the stationarity tests are reduced. It is not the aim of this paper, however, to investigate at a deeper level the stationarity of relative prices in Brazil. The sole purpose of this paper is otherwise to understand the stylized facts about convergence and mean reversion to PPP. A broader and more profound approach of this problem is left to future research. E.F. Arruda et al. / EconomiA 19 (2018) 219–235 225 5. Stylized facts To extract the stylized facts investigated in this paper, monthly data will be used, between January 1995 and August 2012, through the Broad Consumer Price Index — IPCA9(Índice de Pre¸cos ao Consumidor Amplo) for the major 11 Brazilian cities10 and the Brazilian IPCA obtained through the National System of Consumer Prices — SNIPC (Sistema Nacional de Índice de Pre¸cos ao Consumidor), of the Brazilian Institute of Geography and Statistics IBGE (Instituto Brasileiro de Geografia e Estatística). In order to identify possible differences in the speed of reversal between tradable and non-tradable goods and verify that the evidence gathered for the general indexes exhibit regularity in a disaggregated context, the disaggregated price indexes for the food and drink, housing and clothing categories will be used. Therefore, food and drink and clothing will be used to incorporate such evidence on tradable goods, while housing is used for the non-tradable goods. It is expected that tradable goods present a lower price dispersion and, consequently, a higher convergence rate (or shorter half-life) in relative prices, since these are markets which are considered to be more integrated. Tradable goods, which have a higher convergence rate, tend to present a lower volatility and goods with more rigid prices display a slower reversion to the average and a more persistent volatility (Nakamura and Steinsson, 2012). Crucini and Yilmazkuday (2014) point out that the price dispersion of specific goods should be considered in the evaluation of the PPP hypothesis; once sectoral traits, as market oscillation and sectoral mark up, can interfere with price volatility. The methodology used in this paper will be performed in three stages: (i) analyze if there is evidence for a price convergence process; (ii) estimate the speed of adjustment in deviations from the PPP, or its half-life, and (iii) verify if, in fact, processes (i) and (ii) influence the unit root tests, as warned in the literature (Cecchetti et al., 2002; Chen and Devereux, 2003). 5.1. Price convergence Initially, an analysis of price indexes dispersion was done, in its absolute and relative forms, to comprehend the dynamic behavior of prices in the Brazilian cities, as well as the exam of the existence of convergence or divergence between them. The dispersion measurement for absolute prices was defined as the coefficient of price index variation for the cities, which is represented by: cv (pt)= cv pjt/t=⎛ ⎝1 nj=1 npjt − ¯p2 ¯p⎞ ⎠× 100 (9) where cv (pt)= cv pjt/tis the variation in absolute prices coefficient in period t; pjt is the price level of city j in time t and ¯p is the average price level, all in logarithms. The Graph 1 presents the evolution of this indicator. As one analyzes the figure above, it can be observed that there is a strong reduction in price absolute dispersion among Brazilian cities. The greater part of such a decrease occurs between January 1995 and January 2006, where a 91.5% reduction can be noticed. In general terms, it can be said that the IPCA absolute dispersion presented a 53.78% decrease in the analyzed period. Graph 2 presents the same analysis for the food and drink, clothing and housing categories. The sectoral analysis also points to a decrease in the absolute price dispersion for Brazilian cities. The major part of the decrease occurred between January 1995 and August 2005, in the housing and clothing sectors, with reductions of 89.68% and 96.22, respectively, while for the food and drink sector, the reduction was, until December 2005, of 90.37%. At the end of the period, the reductions were respectively 41.34%, 60.30% and 62.27% for the food and drink, housing and clothing sectors. Therefore, as expected, what is considered tradable goods showed less dispersion in their prices in the period 9In all the indexes analyzed, an average for the year 2005 is used as the base value. 10 SNIPC — IBGE provides information for Rio de Janeiro, São Paulo, Belo Horizonte, Porto Alegre, Recife, Brasília, Belém, Fortaleza Salvador, Curitiba and Goiânia. 226 E.F. Arruda et al. / EconomiA 19 (2018) 219–235 Graph 1. Evolution of price absolute dispersion among Brazilian cities. Source: Elaborated by the authors. Graph 2. Evolution of price absolute dispersion among Brazilian cities by sector. Source: Elaborated by the authors. analyzed, i.e. after September 1996, the prices of food and drink and clothing were less dispersed than those related to housing. This result, therefore, is an initial indication of the existence of a convergence process among price levels in Brazilian cities, even considering geographically dispersed cities and places which present significant regional differences. Such a fact can be justified by the absence of commercial barriers, exchange rate volatility and asymmetries in monetary policy. Afterwards, the evolution of the dynamic behavior of relative prices and its dispersion over time was analyzed. The relative price, or real exchange rate, of city j in time t, was defined as, rjt =pjt − ¯p× 100 (10) where pjt is the price level in city j in time t, and ¯p is the numeraire, both in logarithm terms. It is worth noting that, as this measure refers to the same country, as shown by Chen and Devereux (2003), this indicator can be interpreted as the real exchange rate in relation to the numeraire. In order to identify whether the results are sensitive to the choice of numeraire with regard to dispersion and speed reversal of deviations from the PPP, this study will use as numeraire the Brazilian IPCA (real rates — Brazil), the average IPCA for the cities considered (real exchange rate — Average) and the IPCA for the city of São Paulo (real rates — São Paulo), all in logarithmic terms. It is worth noting that in the disaggregated analysis, it was used the average as a reference. The city of São Paulo is used as numeraire, because it is the most important market at state level in Brazil and because price movements in this market spill over economic activity of the rest of the country (Arruda and Tatiwa, 2014). E.F. Arruda et al. / EconomiA 19 (2018) 219–235 233 Table A1 Descriptive statistics of the price indices. City Index Mean Median Std. Dev. Max Min Belo Horizonte General 89.59 87.87 30.52 147.11 40.68 Food and drink 93.83 93.44 32.18 161.28 52.54 Housing 84.36 83.33 34.34 147.33 22.74 Clothing 95.12 87.01 27.74 154.36 65.84 Belém General 90.51 90.16 31.07 149.26 44.46 Food and drink 94.65 93.03 35.08 169.23 54.91 Housing 85.26 94.35 30.11 135.20 30.11 Clothing 97.11 91.68 25.72 151.80 67.19 Brasília General 88.52 89.47 29.99 143.56 40.88 Food and drink 91.87 94.79 31.85 158.50 52.18 Housing 87.38 86.36 34.39 154.84 27.79 Clothing 93.12 87.10 28.15 153.66 61.27 Curitiba General 88.21 88.84 28.20 140.92 41.74 Food and drink 92.81 94.75 30.34 162.02 53.29 Housing 86.52 87.69 29.40 144.47 28.37 Clothing 91.58 87.23 26.85 144.88 58.43 Fortaleza General 88.85 89.47 28.17 141.96 44.74 Food and drink 92.88 95.26 30.24 159.00 53.64 Housing 85.25 87.38 33.32 139.08 25.67 Clothing 96.85 90.51 30.01 172.40 67.26 Goiânia General 87.19 88.36 28.91 138.25 41.17 Food and drink 91.83 94.49 32.40 159.66 51.23 Housing 83.04 85.63 29.09 135.34 27.22 Clothing 90.98 84.14 32.96 152.63 51.41 Porto Alegre General 87.19 87.88 28.63 139.09 40.24 Food and drink 92.91 93.49 31.37 159.86 50.05 Housing 83.83 88.83 30.03 135.83 25.79 Clothing 91.38 85.89 28.96 151.13 55.71 Recife General 89.59 89.83 29.60 144.71 42.18 Food and drink 93.49 96.17 29.97 157.56 52.49 Housing 86.27 86.85 33.17 144.00 24.33 Clothing 94.95 88.22 27.33 152.11 65.05 Rio de Janeiro General 88.33 89.74 29.69 143.88 39.50 Food and drink 91.66 95.09 30.07 157.33 52.24 Housing 84.99 90.03 32.42 144.34 23.01 Clothing 94.51 86.75 30.44 160.12 62.48 Salvador General 89.52 91.68 29.29 143.75 43.06 Food and drink 93.34 98.61 29.68 157.58 52.95 Housing 86.67 88.34 32.44 148.34 27.54 Clothing 93.56 87.15 27.50 149.79 63.87 São Paulo General 89.32 89.72 26.80 138.76 42.59 Food and drink 93.58 93.46 31.05 162.04 54.94 Housing 88.61 92.45 25.26 134.17 28.45 Clothing 93.30 86.19 27.46 152.22 61.96 Brazil General 88.89 89.37 28.50 141.76 41.72 Source: Elaborated by the authors. 234 E.F. 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