DEPARTAMENTO DE ECONOMÍA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DE?L PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DEPARTAMENTO DE ECONOMÍA PONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA DOCUMENTO DE TRABAJO N° 357 A NOTE ON THE SIZE OF THE ADF TEST WITH ADDITIVE OUTLIERS AND FRACTIONAL ERRORS. A REAPRAISAL ABOUT THE (NON) STATIONARITY OF THE LATIN-AMERICAN INFLATION SERI S Gabriel Rodríguez y Dionisio Ramirez DOCUMENTO DE TRABAJO N° 357 A NOTE ON THE SIZE OF THE ADF TEST WITH ADDITIVE OUTLIERS AND FRACTIONAL ERRORS. A REAPRAISAL ABOUT THE (NON) STATIONARITY OF THE LATIN-AMERICAN INFLATION SERIES Gabriel Rodríguez y Dionisio Ramirez Julio, 2013 DEPARTAMENTO DE ECONOMÍA DOCUMENTO DE TRABAJO 357 http://www.pucp.edu.pe/departamento/economia/images/documentos/DDD357.pdf © Departamento de Economía – Pontificia Universidad Católica del Perú, © Gabriel Rodríguez y Dionisio Ramirez Av. Universitaria 1801, Lima 32 – Perú. Teléfono: (51-1) 626-2000 anexos 4950 - 4951 Fax: (51-1) 626-2874 econo@pucp.edu.pe www.pucp.edu.pe/departamento/economia/ Encargado de la Serie: Luis García Núñez Departamento de Economía – Pontificia Universidad Católica del Perú, lgarcia@pucp.edu.pe Gabriel Rodríguez y Dionisio Ramirez A Note on the Size of the ADF Test with Additive Outliers and Fractional Errors. A Reapraisal about the (Non) Stationarity of the Latin-American Inflation Series. Lima, Departamento de Economía, 2013 (Documento de Trabajo 357) PALABRAS CLAVE: Outliers aditivos, Errores ARFIMA, Test ADF. Las opiniones y recomendaciones vertidas en estos documentos son responsabilidad de sus autores y no representan necesariamente los puntos de vista del Departamento Economía. Hecho el Depósito Legal en la Biblioteca Nacional del Perú Nº 2013-11339. ISSN 2079-8466 (Impresa) ISSN 2079-8474 (En línea) Impreso en Cartolán Editora y Comercializadora E.I.R.L. Pasaje Atlántida 113, Lima 1, Perú. Tiraje: 100 ejemplares A Note on the Size of the ADF Test with Additive Outliers and Fractional Errors. A Reapraisal about the (Non)Stationarity of the Latin-American In‡ation Series Gabriel Rodríguez Dionisio Ramirez Ponti…cia Universidad Católica del Perú Universidad Castilla La Mancha Abstract This note analyzes the empirical size of the augmented Dickey and Fuller (ADF) statistic proposed by Perron and Rodríguez (2003) when the errors are frac- tional. This ADF is based on a searching procedure for additive outliers based on …rst-di¤erences of the data named d. Simulations show that empirical size of the ADF is not a¤ected by fractional errors con…rming the claim of Perron and Rodríguez (2003) that the procedure d is robust to departures of the unit root framework. In particular the results show low sensitivity of the size of the ADF statistic respect to the fractional parameter (d). However, as expected, when there is strong negative moving average autocorrelation or negative au- toregressive autocorrelation, the ADF statistic is oversized. These di¢ culties are …xed when sample increases (from T = 100 to T = 200). Empirical applica- tion to eight quarterly Latin-American in‡ation series is also provided showing the importance of taking into account dummy variables for the detected additive outliers. Keywords: Additive Outliers, ARFIMA Erros, ADF Test JEL: C2, C3, C5 Resumen En esta nota se analiza el tamaño empírico del estadístico Dickey y Fuller au- mentado (ADF), propuesto por Perron y Rodríguez (2003), cuando los errores son fraccionales. Este estadístico se basa en un procedimiento de búsqueda de valores atípicos aditivos basado en las primeras diferencias de los datos denomi- nado d. Las simulaciones muestran que el tamaño empírico del estadístico ADF no es afectado por los errores fraccionales con…rmando el argumento de Perron y Rodríguez (2003) que el procedimiento d es robusto a las desviaciones del marco de raíz unitaria. En particular, los resultados muestran una baja sensibil- idad del tamaño del estadístico ADF respecto al parámetro fraccional (d). Sin embargo, como es de esperar, cuando hay una fuerte autocorrelación negativa de tipo promedio móvil o autocorrelación autorregresiva negativa, el estadístico ADF tiene un tamaño exacto mayor que el nominal. Estas di…cultades desapare- cen cuando aumenta la muestra (a partir de T = 100 a T = 200). La aplicación empírica a ocho series de in‡ación latinoamericana trimestral proporciona evi- dencia de la importancia de tener en cuenta las variables …cticias para controlar por los outliers aditivos detectados. Palabras Claves: Outliers Aditivos, Errores ARFIMA, Test ADF. Classi…cación JEL: C2, C3, C5 A Note on the Size of the ADF Test with Additive Outliers and Fractional Errors. A Reapraisal about the (Non)Stationarity of the Latin-American In‡ation Series1 Gabriel Rodríguez2 Dionisio Ramirez Ponti…cia Universidad Católica del Perú Universidad Castilla La Mancha 1 Introduction Additve outliers a¤ect inference of parameters in di¤erent circunstances. For example they a¤ect inference of the autoregressive and moving average estimates of ARMA(p; q) models; see Cheng and Liu (1993), Chan (1992, 1995). They also a¤ect other topics like causality tests (see Balde and Ro- dríguez (2005)), fractional estimates (see Fajardo et al. (2009), Chareka et al. (2006)). In the context of a unit root, additive outliers have been also analyzed since the contribution of Franses and Haldrup (1994). These authors show that additive outliers contaminate the limiting distribution of the unit root statistics; see also Vogelsang (1999) and Perron and Rodríguez (2003). Vogelsang (1999) suggests to use M-tests based on GLS detrended data because they are robust to the presence of negative moving average au- tocorrelation which is induced by the presence of additive outliers. Another alternative procedure is to estimate an ADF statistic corrected for dummy variables related to the identi…ed additive outliers in a preliminary step. Ro- dríguez (2004) used four Latin-American in‡ation series and show that even the M-tests indicate a rejection of the null hypothesis of a unit root. When applying an ADF corrected for dummy variables, some countries show a non rejection of the null hypothesis of a unit root indicating nonstationarity of the in‡ation series which is an opposite results obtained from the standard unit root tests. 1We want to thank Carmen Armas Montalvo and Vanessa Belapatiño for excellent research assistanship. I acknowledge …nancial support from the Department of Economics of the Ponti…cia Universidad Católica del Perú and useful comments of Patricia Lengua Lafosse. 2Address for Correspondence: Gabriel Rodríguez, Department of Economics, Pon- ti…cia Universidad Católica del Perú, Av. Universitaria 1801, Lima 32, Lima, Perú, Telephone: +511-626-2000 (4998), Fax: +511-626-2874. E-Mail Address: gabriel.rodriguez@pucp.edu.pe. 1 The procedure mentioned above needs the location of the additive out- liers. Perron and Rodríguez (2003) have suggested a powerful test, denoted by d, which works with …rst-di¤erenced data3. This procedure is more powerful than other based on levels of the data, for example; see Perron and Rodríguez (2003) for a detailed discussion. These authors claim that d is powerful even for departures from the unit root case4. The purpose of this note is to show that this claim is correct. we do it analyzing the empirical size of the ADF statistic (using d to locate additive outliers) when the DGP contains ARFIMA(p; d; q) errors. The experiment deals with di¤erent val- ues of the fractional parameter (d) to observe di¤erent departures from the unit root hypothesis. Also di¤erent structure of autocorrelation is analyzed (moving average and autoregressive). The Monte-Carlo simulations show that the ADF statistic corrected for dummy variables associated to the additive outliers su¤ers of size distortions in only few cases. For example, when the moving average parameter is close to -1 empirical size is greater than nominal size. Negative autoregressive autocorrelación has impact on the size of the ADF statistic too. However, most of these issues are …xed when sample size increases from T = 100 to T = 200 in simulations. In general, the ADF test appears to be slightly undersized. When fractional parameter is higher, distortions appear but at the same time when correlation is higher. Therefore, fractional parameter itself does not cause problems or distorions on the size of the ADF test. After simulations, we present an empirical application using quarterly in‡ation series ranging from 1970:1 until 2010:4 of 8 countries. We use a sample of eigth countries and the spirit of this exercise is very similar to Rodríguez (2004) where four countries were only used. In this note, we add more countries and more observations. In particular, the Phillips and Perron (1988) statistic shows a strong rejection of the null hypothesis of a unit root which is not rare given the sensitivity of this statistic to the presence of strong negative moving average correlation which is the case here because additive outliers are clearly present and literature has shown that they are related to this type of correlation. Similar results are obtained 3Of course, there are many other procedures to identify outliers, for example, those proposed in Tsay (1986), Chang, Tiao and Chen (1988), Shin, Sharkar and Lee (1996), Chen and Liu (1993) and Gómez and Maravall (1992a, 1992b). Another interesting ap- proach is proposed by Lucas (1995a, 1995b), and Hoek, Lucas and van Dijk (1995). See Rodríguez (2004) for a comparison with other approaches. 4However, this procedure is not robust to departures from the assumption of normality in the errors. It is mentioned and discussed by Perron and Rodríguez (2003). See also Burridge and Taylor (2006) for more evidence about this drawback and the correction they propose based on the extreme value theory. 2 with the ADF statistic and even with theM -tests andMPT tests using GLS detrended data as suggested by Elliott, Rothenberg and Stock (1996) and Ng and Perron (2001), respectively. Only Uruguay and Venezuela show non rejection of the null. However, when applying the ADF test augmented by dummy variables related to the location of the addittive outliers identi…ed by the procedure d, none of the countries reject the null hypothesis of a unit root. This note is organized as follows. Section 2 presents the model, dis- cusses the issue of outlier detection and brie‡y revises the method proposed by Perron and Rodríguez (2003). In section 3, I present the results from the simulations. Section 4 shows the empirical application and Section 5 concludes. 2 The Issue of Outlier Detection and Testing for Unit Roots with Additive Outliers The issue of outlier detection in the unit root framework is the approach taken by Perron and Rodríguez (2003) which is based on Vogelsang (1999)5. The data-generating process entertained is of the following general form: yt = dt + mX j=1 jD(Tao;j)t + ut (1) where D(Tao;j)t = 1 if t = Tao;j and 0 otherwise. This permits the presence of m additive outliers occurring at dates Tao;j (j = 1; :::;m): The term dt speci…es the deterministic components. In most cases, dt =  if the series is non-trending or dt =  + t if the series is trending. The noise function is integrated of order one, i.e, ut = ut1 + vt; where vt is a stationary process. While Perron and Rodríguez (2003), use an ARMA(p; q) for the process vt, in this paper, we assume that vt is an ARFIMA(p; d; q) process. 5Let tb(Tao) denote the t-statistic for testing  = 0 in (1). Following Chen and Liu (1993), the presence of an additive outlier can be tested using  = sup Tao j tb(Tao): Assuming that  = Tao=T remains …xed as T grows, Vogelsang (1999) showed that as T ! 1; the limiting distribution of tb(Tao) is non-standard. More precisely, tb(Tao) ) H() = W ()=( R 1 0 W (r)2dr)1=2; where W () denotes a demeaned standard Wiener process: If (1) also includes a time trend, W () will denote a detrended Wiener process. Further- more, from the continuous mapping theorem it follows that,  ) sup 2(0;1) jH()j  H: This distribution is invariant with respect to any nuisance parameters, including the correlation structure of the noise function. 3 As shown in Perron and Rodríguez (2003), the original procedure of Vogelsang (1999) has severe size distortions when applied in an iterative fashion to search for additive outliers. The reason for this is that the limiting distribution of the statistic is only valid in the …rst step of the iterations as speci…ed in Theorem 1 of Perron and Rodríguez (2003). In subsequent steps, the asymptotic critical values used need to be modi…ed. Perron and Rodríguez (2003) have proposed a more powerful iterative strategy using a test based on …rst-di¤erences of the data. Consider data generated by (1) with dt = , and a single outlier occurring at date Tao with magnitude . Then, yt = [D(Tao)t D(Tao)t1] + vt; (2) where D(Tao)t = 1, if t = Tao (0, otherwise) and D(Tao)t1 = 1; if t = Tao1 (0, otherwise). If the data are trending, a constant should be included. In this case, we are interested in d = supTao jtb(Tao)j; where tb(Tao) = ^=(2(R^u(0) R^u(1)) and Ru(j) is the autocovariance function of vt at delay j.6 To detect for multiple outliers, we can follow a strategy similar to that suggested by Vogelsang (1999), by dropping the observation labelled as an outlier before proceeding to the next step. The important feature is that, unlike for the case of the test based on levels, the limit distribution of the test d is the same as each step of the iterations when dealing with multiple outliers. The disadvantage of this procedure, compared to that based on the level of the data, is that the limiting distribution depends on the speci…c distribution of the errors vt, though not on the presence of serial correlation and heteroskedasticity7. This problem is exactly the same as that for …nding outliers in stationary time series. In this note, we analyze the empirical size of the ADF test corrected for detected additive outliers when errors vt are ARFIMA(p; d; q) process. It is equivalent to using the t-statistic for testing that = 1 in the following regression: yt = + yt1 + k+1X j=0 jD(Tao;j)tj + kX i=0 diyti + vt ; (3) 6 R^u(j) = T 1PTj t=1 v^tv^tj with v^t the least-squares residuals obtained from regression (2). Then, R^u(j) is a consistent estimate of Ru(j). 7The dependence of the distribution or departures of the normality of vt has been mentioned by Perron and Rodríguez (2003). However, Burridge and Taylor (2006) deals with this issue using extreme value theory. 4 where D(Tao;j)t = 1 if t = Tao;j and 0 otherwise, with Tao;j (j = 1; 2; :::;m) being the dates of the outliers identi…ed using the statistic d. Notice that k + 2 one-time dummy variables have to be included in (3) to remove all possible in‡uences of the additive outliers. 3 Monte Carlo Results In order to analyze the empirical size of the ADF statistic, we consider the following experiment. Let yt follow (1) where ut = ut1 + vt (a unit root process) and vt is an ARFIMA(p; d; q) process, that is (L)(1 L)dvt = (L)t, where t is an i:i:d. N(0; 1). More exactly, in one case we consider p = 1 ((L) = 1 L) and q = 0, that is (L)(1 L)dvt = t, while in the other p = 0 case and q = 1 ((L) = 1 + L); that is (1 L)dvt = (L)t. The fractional parameter d 2 [0:48 to 0:48] with a step of 0:12. Each Table and each value of  or  present three rows named “without”, “with”, and “total”. The row named “without” indicates the size of the ADF statistic when no additive outliers has been found. The word “with” indicates the size of the ADF statistic when additive outliers have been identi…ed. Therefore, the row entitle “total”means simply the sum of the two previous rows. If size is correct we expect that this row should be close to the nominal size of 5.0%. In order to save space we present only selected Tables. Each experiment is performed using 10,000 replications, nominal size at 5.0% and we use tabulated critical values (Table 1 of Perron and Rodríguez (2003)) for T = 100 and T = 200. Other extensive Tables are available upon request. In all Tables, the total iterative procedure is applied, that is, we search for all outliers and procedure …nish when no outliers are found. Two sets of Tables are presented. In one case, the lag lenght of (3) is …xed to be k = 1 while in the other case, we use the procedure t-sig proposed by Campbell and Perron (1991) for k 2 [0; 5]. In each Table, three cases are presented. In the …rst case, no outliers are in the process, that is, i = 0 for i = 1; 2; 3; 4. In the second case, we consider medium sized additive outliers: i = 5; 3; 2; 2. The …nal case is for high sized addtive outliers, that is, i = 10; 5; 5; 5. In summary, the design of the experiment follow closely Perron and Rodríguez (2003). When there are outliers a maximum of four additive outliers is considered and they are located at positions 0:20T , 0:40T , 0:60T and 0:80T , respectively. Table 1 shows the results for the case where errors are ARFIMA(0; d; 0). The …rst set of columns are the case where no outliers are present in the 5 data. The other columns shows medium and high sized additive outliers, respectively. The results show that the size of the ADF is oversized for every d < 0. More negative values of d imply more oversized ADF tests. This is true for the case where no outliers are found and when they are present in the data. For other values of d, the ADF is slighthly undersized but close to the nominal size of 5%. Given these results, in what follows, we do not consider cases where d < 0. Tables 2a-2c show size of the ADF test for ARFIMA(0; d; 1) errors, that is when there exists moving average correlation. In order to save space, we only show results for d = 0:00; 0:24; and 0:48. Table 2a indicates that ADF test is oversized for  = 0:8 and for  = 0:4. Small distortion is also found for  = 0:8. In all other cases of  and for cases where there are or not additive outliers, exact size is close to 5%. Table 2b shows the case for d = 0:24. Again, ADF test is oversized for  = 0:8 but distortions are smaller than before. In all other cases, size is better although slightly undersized. When d = 0:48 (Table 2c), that is, when memory of the errors is large the size of the ADF test is very close to the nominal size of 5%. It is true when there are or not additive outliers and for both sample sizes. In summary, there is some di¢ culties when  goes to -1 but the performance is better when d goes to 0.5. Tables 3a-3c show size of the ADF test for ARFIMA(1; d; 0) errors, that is, when there exists autoregressive autocorrelation. Again, in order to save space, we only show results for d = 0:00; 0:24; and 0:48. Table 3a indicates that ADF test has good exact size except for the case where  = 0:8 and when the process is contaminated for medium and high sized additive outliers. It is worth to mention that the distortions are smaller compared to the previous Tables and we observe that size is better when sample size is higher. Table 3b shows the case for d = 0:24. In this case, the ADF test has exact size close to the 5% although we observe small oversized results when  = 0:8. This results is more evident when d = 0:48 (Table 3c) even when there is no outliers in the process. This problem is not …xed when sample size is higher. It is more evident for extreme values of  (0:8 and 0:8). Previous results (undersized or oversized results) may be due to the se- lection of the lag length which has been …xed to unity. In order to observe if this issue is important, we present similar simulations as in the previous Ta- bles but now the lag length is selected using the procedure t-sig as suggested by Campbell and Perron (1991) considering a k 2 [0; 5]. Table 4 presents results for ARFIMA(0; d; 0) errors and for d  0. The message is that ADF test has exact size close to the 5%. In some cases, it presents slight smaller exact size. 6 Table 5a-5c are similar to Tables 2a-2c but selecting lag length with the procedure t-sig. In all Tables, the ADF test is oversized but clearly di¤erent or smaller compared to the case where k = 1. we found oversized ADF test only when d = 0:0. In other cases, when memory is larger, results are better in particular for T = 200. Finally, Tables 6a-6c are similar to Tables 3a-3c but using the procedure t-sig. The results indicate that ADF test has good size for almost every case. More clearly, the exact size is under nominal size for  < 0 but is closer to 5% when   0. The conclusion suggests that using a data dependent rule to select the lag length …xes the problems detected before8. 4 Empirical Application The Latin-American in‡ation series o¤er a good example of the strong pres- ence of big sized additive outliers in a possible nonstationary time series. Figure 1 shows quarterly in‡ation series for eight Latin-American coun- tries: Argentina, Bolivia, Chile, Colombia, Ecuador, Peru, Uruguay and Venezuela. The frequency is quarterly and the sample spans 1970:1 until 2010:4. Many or all these countries have experimented with di¤erent stabi- lization programs to stop high in‡ation episodes. Intervention of this kind, in most of these cases, has introduced additive outliers in the evolution of their in‡ation series. For example, the periods of high in‡ation in Argentina and Peru were located between 1985 and 1990, where the most important stabilization programs were applied. For example, in the case of Argentina, the most known governmental plans were the Austral Program (June 1985), the program of February of 1987, the Austral II Program (October 1987), The Spring Program (August 1988), the BB Program (1989), The Bonex 8 It is worth to mention that I simulated data using another DGP. Let fytg, t 2 Z be a weakly stationary process. Let fztg, t 2 Z be a process contaminated by additive outliers, which is described by zt = yt + mX j=1 !jXj;t ; (4) where m is the maximum number of outliers and the unknown parameter !j indicates the magnitude of the jth outlier. The Xj;t is a random variable with probability distribution Pr(Xj = 1) = Pr(Xj = 1) = pj=2 and Pr(Xj = 0) = 1pj . Therefore, Xj is the product of a Bernouilli (pj) and a Rademacher random variables; the latter equals 1 or 1, both with probability 1=2. Furthermore, yt and Xj are independent random variables. The model (4) is based on the parametric models proposed by Fox (1972). This DGP is also used by Franses and Haldrup (1994), FAjardo et al. (2009), among others. In order to save space, results from this model are not included but they indicate very similar conclusions as in the previous DGP. Such Tables are available upon request. 7 Program (January 1990) and the Cavallo’s Program (March 1991) where the dates in parenthesis correspond to the start date of the programs. In the Peruvian case, we can mention two principal stabilization programs. These are the Salinas’Program (September 1988) and the Fujimori’s Pro- gram (July-August 1990). In the Bolivian case, the episode of high in‡ation was in the middle of 1980’s. Many small stabilization programs were applied during the period between 1982 and 1984 but it was the program applied in August 1985 which stopped the high in‡ation. High in‡ation in Chile was located around 1975. Diverse programs were applied between 1975 and 1977 until the shock plan applied at the end of 1977 until 1979. A related research to this note is Rodríguez (2004) where four Latin-American countries were analyzed. In this note, we add more countries and more observations. For more details related to the in‡ationary process in some of these countries, see Rodríguez (2004). One important issue from Figure 1 is the following. Observing the ver- tical axis, four countries (Argentina, Bolivia, Chile and Peru) show huge additive outliers. The other countries show also presence of additive out- liers but their magnitudes are very di¤erent (smaller) in comparison with the above mentioned four countries. It implies that the procedure d will iden- tify more additive outliers in the former countries and less additive outliers in all other countries. Table 7 shows results from the application of standard and new unit root tests. we apply two standard unit root statistics: the Phillips and Perron and the Augmented Dickey and Fuller statistics; see Phillips and Perron (1988) and Said and Dickey (1984), respectively. Other tests are the M- tests based on GLS detrending data as suggested by Ng and Perron (2001). In all cases, lag length has been selected using the dependent recursive rule named t-sig as proposed by Campbell and Perron (1991)9. Almost in all cases, all statistics suggest a rejection of the null hypothesis of a unit root. This result is particularly clear for the Phillips and Perron (1988) test where the rejection is strong. It is not surprising if we remember that this statistic is very oversized when there are strong negative moving average correlation. Given the evidence that this type of correlation implies the presence of additive outliers (see Franses and Haldrup (1994), Vogelsang (1999)), the results of the PP test is not rare. The results are more clear in countries like Peru where the size of the additive outliers is huge. Even the robust 9 I also use the MAIC approach as suggested by Ng and Perron (2001). Results are very similar and conclusions are not modi…ed. I present the results using the t-sig method to be coherent (in terms of comparison) with the method used in simulations. 8 M-tests proposed by Stock (1999) indicate a rejection of the null hypothesis of a unit root except for the cases of Uruguay and Venezuela. Table 8 shows the results of the ADF statistic corrected for the presence of the additive outliers. The results indicate a non rejection of the null hypothesis of a unit root for all countries implying that Latin-American in‡ation series are nonstationary. Rodríguez (2004) found a similar result but only for Argentina and Peru using a shorter sample size. Charemza et al. (2005) also …nd results in favour of nonstationarity of a big set of in‡ation series when the innovations are treated as draws from a symmetric stable Paretian distribution with in…nite variance. This suggest that an appropriate treatment of extreme values is important in this context. 5 Conclusions This note analyzes the empirical size of the ADF statistic when there are additive outliers and ARFIMA(p; d; q) errors. Results indicate that a few cases implies oversized ADF tests. In most cases, the statistic is slightly undersized or very close to the nominal size of 5%. There are some di¢ culties when  goes to -1 or when  goes to j1j: An empirical application for eigth Latin-American countries indicates the di¢ culties that standard and new unit root tests have to verify if there is or no a unit root in the in‡ation time series. An application of an ADF test with dummies associated to the location of the identi…ed additive outliers con…rms that all in‡ation time series are nonstationary. It is a similar result as obtained by Rodríguez (2004) but using larger sample size and more countries. Results also are consistent with those found by Charemza et al. (2005) where results are in favour of nonstationarity of a big set of in‡ation series when the innovations are treated as draws from a symmetric stable Paretian distribution with in…nite variance. It is equivalent to say that an appropiate treatment of extreme values is important. References [1] Burridge, P. and A. M. R: Taylor (2006),.“Additive Outlier Detection Via Extreme-Value Theory,”Journal of Time Series Analysis, . 27(5), 685-701 9 [2] Campbell, J. Y. and P. Perron (1991), “Pitfalls and Opportunities: What Macroeconomists Should Know About Unit Roots,” in O. J. Blanchard and S. Fisher (eds.), NBER Macroeconomic Annual, Vol. 6, 141-201. [3] Chan, W. 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Size of the ADF Test; ARFIMA(0,d,0) Errors* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200 d = 0:48 Without 0.792 0.940 0.075 0.051 0.000 0.000 With 0.038 0.048 0.779 0.938 0.737 0.979 Total 0.830 0.988 0.854 0.989 0.737 0.979 d = 0:36 Without 0.487 0.767 0.032 0.021 0.000 0.000 With 0.023 0.037 0.514 0.792 0.428 0.761 Total 0.510 0.805 0.546 0.813 0.428 0.761 d = 0:24 Without 0.229 0.385 0.009 0.005 0.000 0.000 With 0.011 0.018 0.242 0.403 0.188 0.361 Total 0.240 0.403 0.251 0.408 0.188 0.361 d = 0:12 Without 0.088 0.122 0.002 0.001 0.000 0.000 With 0.004 0.007 0.092 0.129 0.076 0.116 Total 0.092 0.129 0.094 0.130 0.076 0.116 d = 0:00 Without 0.037 0.036 0.000 0.000 0.000 0.000 With 0.001 0.002 0.038 0.038 0.037 0.038 Total 0.038 0.038 0.038 0.038 0.037 0.038 d = 0:12 Without 0.022 0.022 0.000 0.000 0.000 0.000 With 0.001 0.001 0.023 0.023 0.023 0.022 Total 0.023 0.023 0.023 0.023 0.023 0.022 d = 0:24 Without 0.021 0.026 0.000 0.000 0.000 0.000 With 0.001 0.002 0.022 0.028 0.022 0.026 Total 0.022 0.028 0.022 0.028 0.022 0.026 d = 0:36 Without 0.023 0.032 0.000 0.000 0.000 0.000 With 0.001 0.002 0.024 0.033 0.022 0.030 Total 0.024 0.034 0.024 0.033 0.022 0.030 d = 0:48 Without 0.022 0.030 0.000 0.000 0.000 0.000 With 0.001 0.001 0.026 0.033 0.025 0.035 Total 0.023 0.031 0.026 0.033 0.025 0.035 *Lag lenght …xed at one 12 Table 2a. Size of the ADF Test; ARFIMA(0,d,1) Errors with d=0.00* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.819 0.846 0.250 0.221 0.000 0.000 With 0.043 0.045 0.651 0.692 0.845 0.891 Total 0.862 0.891 0.901 0.913 0.845 0.891  = 0:40 Without 0.094 0.103 0.012 0.006 0.000 0.000 With 0.005 0.005 0.110 0.114 0.086 0.101 Total 0.099 0.108 0.122 0.120 0.086 0.101  = 0:00 Without 0.037 0.036 0.000 0.000 0.000 0.000 With 0.001 0.002 0.038 0.038 0.037 0.038 Total 0.038 0.038 0.038 0.038 0.037 0.038  = 0:40 Without 0.059 0.060 0.000 0.000 0.000 0.000 With 0.002 0.002 0.045 0.053 0.054 0.056 Total 0.061 0.062 0.045 0.053 0.054 0.056  = 0:80 Without 0.094 0.098 0.000 0.000 0.000 0.000 With 0.002 0.003 0.064 0.079 0.082 0.090 Total 0.096 0.101 0.064 0.079 0.082 0.090 *Lag lenght …xed at one 13 Table 2b. Size of the ADF Test; ARFIMA(0,d,1) Errors with d=0.24* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.264 0.223 0.048 0.027 0.000 0.000 With 0.013 0.001 0.266 0.223 0.246 0.220 Total 0.277 0.234 0.314 0.250 0.246 0.220  = 0:40 Without 0.023 0.025 0.001 0.000 0.000 0.000 With 0.001 0.002 0.023 0.025 0.021 0.024 Total 0.024 0.027 0.024 0.025 0.021 0.024  = 0:00 Without 0.021 0.026 0.000 0.000 0.000 0.000 With 0.001 0.002 0.022 0.028 0.022 0.026 Total 0.022 0.028 0.022 0.028 0.022 0.026  = 0:40 Without 0.023 0.024 0.000 0.000 0.000 0.000 With 0.001 0.001 0.024 0.024 0.025 0.026 Total 0.024 0.025 0.024 0.024 0.025 0.026  = 0:80 Without 0.033 0.028 0.000 0.000 0.000 0.000 With 0.001 0.001 0.029 0.029 0.042 0.035 Total 0.034 0.029 0.029 0.029 0.042 0.035 *Lag lenght …xed at one 14 Table 2c. Size of the ADF Test; ARFIMA(0,d,1) Errors with d=0.48* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.059 0.041 0.004 0.001 0.000 0.000 With 0.003 0.002 0.058 0.040 0.053 0.040 Total 0.062 0.043 0.062 0.041 0.053 0.040  = 0:40 Without 0.033 0.052 0.000 0.000 0.000 0.000 With 0.002 0.003 0.030 0.053 0.030 0.049 Total 0.035 0.055 0.030 0.053 0.030 0.049  = 0:00 Without 0.022 0.030 0.000 0.000 0.000 0.000 With 0.001 0.001 0.026 0.033 0.025 0.035 Total 0.023 0.031 0.026 0.033 0.025 0.035  = 0:40 Without 0.018 0.020 0.000 0.000 0.000 0.000 With 0.001 0.001 0.032 0.041 0.048 0.055 Total 0.019 0.021 0.032 0.041 0.048 0.055  = 0:80 Without 0.024 0.019 0.000 0.000 0.000 0.000 With 0.001 0.001 0.044 0.048 0.078 0.079 Total 0.025 0.020 0.044 0.048 0.078 0.079 *Lag lenght …xed at one 15 Table 3a. Size of the ADF Test; ARFIMA(1,d,0) Errors with d=0.00* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.037 0.037 0.080 0.053 0.018 0.004 With 0.001 0.001 0.026 0.018 0.139 0.095 Total 0.038 0.038 0.106 0.071 0.157 0.099  = 0:40 Without 0.035 0.037 0.009 0.004 0.000 0.000 With 0.002 0.002 0.046 0.044 0.041 0.039 Total 0.037 0.039 0.055 0.048 0.041 0.039  = 0:00 Without 0.037 0.036 0.000 0.000 0.000 0.000 With 0.001 0.002 0.038 0.038 0.037 0.038 Total 0.038 0.038 0.038 0.038 0.037 0.038  = 0:40 Without 0.035 0.036 0.000 0.000 0.000 0.000 With 0.002 0.002 0.033 0.035 0.037 0.038 Total 0.037 0.038 0.033 0.035 0.037 0.038  = 0:80 Without 0.040 0.040 0.000 0.000 0.000 0.000 With 0.001 0.001 0.035 0.040 0.049 0.046 Total 0.041 0.041 0.035 0.040 0.049 0.046 *Lag lenght …xed at one 16 Table 3b. Size of the ADF Test; ARFIMA(1,d,0) Errors with d=0.24* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.031 0.044 0.010 0.016 0.001 0.000 With 0.001 0.001 0.012 0.019 0.022 0.031 Total 0.032 0.045 0.022 0.035 0.023 0.031  = 0:40 Without 0.025 0.034 0.001 0.000 0.000 0.000 With 0.001 0.002 0.022 0.033 0.025 0.033 Total 0.026 0.036 0.023 0.033 0.025 0.033  = 0:00 Without 0.021 0.026 0.000 0.000 0.000 0.000 With 0.001 0.002 0.022 0.028 0.022 0.026 Total 0.022 0.028 0.022 0.028 0.022 0.026  = 0:40 Without 0.022 0.024 0.000 0.000 0.000 0.000 With 0.001 0.002 0.025 0.025 0.027 0.026 Total 0.023 0.026 0.025 0.025 0.027 0.026  = 0:80 Without 0.072 0.056 0.000 0.000 0.000 0.000 With 0.001 0.002 0.064 0.073 0.096 0.092 Total 0.073 0.058 0.064 0.073 0.096 0.092 *Lag lenght …xed at one 17 Table 3c. Size of the ADF Test; ARFIMA(1,d,0) Errors with d=0.48* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.062 0.096 0.013 0.025 0.000 0.000 With 0.001 0.002 0.031 0.063 0.045 0.085 Total 0.063 0.098 0.044 0.088 0.045 0.085  = 0:40 Without 0.038 0.058 0.000 0.000 0.000 0.000 With 0.002 0.004 0.034 0.059 0.033 0.057 Total 0.040 0.062 0.034 0.059 0.033 0.057  = 0:00 Without 0.022 0.030 0.000 0.000 0.000 0.000 With 0.001 0.001 0.026 0.033 0.025 0.035 Total 0.023 0.031 0.026 0.033 0.025 0.035  = 0:40 Without 0.020 0.020 0.000 0.000 0.000 0.000 With 0.001 0.002 0.041 0.047 0.057 0.064 Total 0.021 0.022 0.041 0.047 0.057 0.064  = 0:80 Without 0.177 0.124 0.000 0.000 0.000 0.000 With 0.001 0.002 0.095 0.116 0.146 0.141 Total 0.178 0.126 0.095 0.116 0.146 0.141 *Lag lenght …xed at one 18 Table 4. Size of the ADF Test; ARFIMA(0,d,0) Errors* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200 d = 0:00 Without 0.043 0.037 0.000 0.000 0.000 0.000 With 0.003 0.002 0.047 0.040 0.040 0.036 Total 0.046 0.039 0.047 0.040 0.040 0.036 d = 0:12 Without 0.029 0.024 0.000 0.000 0.000 0.000 With 0.002 0.001 0.028 0.021 0.028 0.023 Total 0.031 0.025 0.028 0.021 0.028 0.023 d = 0:24 Without 0.028 0.023 0.000 0.000 0.000 0.000 With 0.001 0.001 0.026 0.022 0.024 0.021 Total 0.029 0.024 0.026 0.022 0.024 0.021 d = 0:36 Without 0.027 0.023 0.000 0.000 0.000 0.000 With 0.001 0.001 0.025 0.022 0.022 0.020 Total 0.028 0.024 0.025 0.022 0.022 0.020 d = 0:48 Without 0.023 0.021 0.000 0.000 0.000 0.000 With 0.001 0.001 0.021 0.021 0.019 0.021 Total 0.024 0.022 0.021 0.021 0.019 0.021 Lag length selected using the sequential t sig method 19 Table 5a. Size of the ADF Test; ARFIMA(0,d,1) Errors with d=0.00* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.356 0.282 0.130 0.087 0.000 0.000 With 0.018 0.012 0.302 0.233 0.396 0.285 Total 0.374 0.294 0.432 0.320 0.396 0.285  = 0:40 Without 0.072 0.057 0.005 0.003 0.000 0.000 With 0.005 0.003 0.078 0.056 0.072 0.053 Total 0.077 0.060 0.083 0.059 0.072 0.053  = 0:00 Without 0.043 0.037 0.000 0.000 0.000 0.000 With 0.003 0.002 0.047 0.040 0.040 0.036 Total 0.046 0.039 0.047 0.040 0.040 0.036  = 0:40 Without 0.051 0.039 0.000 0.000 0.000 0.000 With 0.002 0.001 0.044 0.039 0.045 0.039 Total 0.053 0.040 0.044 0.039 0.045 0.039  = 0:80 Without 0.048 0.041 0.000 0.000 0.000 0.000 With 0.001 0.001 0.050 0.040 0.038 0.039 Total 0.049 0.042 0.050 0.040 0.038 0.039 Lag length selected using the sequential t sig method 20 Table 5b. Size of the ADF Test; ARFIMA(0,d,1) Errors with d=0.24* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.115 0.057 0.019 0.008 0.000 0.000 With 0.007 0.003 0.112 0.054 0.114 0.054 Total 0.122 0.060 0.131 0.062 0.114 0.054  = 0:40 Without 0.029 0.023 0.000 0.000 0.000 0.000 With 0.001 0.001 0.029 0.022 0.026 0.022 Total 0.030 0.024 0.029 0.022 0.026 0.022  = 0:00 Without 0.028 0.023 0.000 0.000 0.000 0.000 With 0.001 0.001 0.026 0.022 0.024 0.021 Total 0.029 0.024 0.026 0.022 0.024 0.021  = 0:40 Without 0.030 0.023 0.000 0.000 0.000 0.000 With 0.001 0.000 0.028 0.020 0.026 0.018 Total 0.031 0.023 0.028 0.020 0.026 0.018  = 0:80 Without 0.033 0.024 0.000 0.000 0.000 0.000 With 0.001 0.001 0.027 0.022 0.029 0.021 Total 0.034 0.025 0.027 0.022 0.029 0.021 Lag length selected using the sequential t sig method 21 Table 5c. Size of the ADF Test; ARFIMA(0,d,1) Errors whit d=0.48* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=150 T=100 T=200  = 0:80 Without 0.059 0.045 0.002 0.001 0.000 0.000 With 0.004 0.002 0.055 0.044 0.054 0.043 Total 0.063 0.047 0.057 0.045 0.054 0.043  = 0:40 Without 0.027 0.024 0.000 0.000 0.000 0.000 With 0.001 0.001 0.024 0.022 0.024 0.022 Total 0.028 0.025 0.024 0.022 0.024 0.022  = 0:00 Without 0.023 0.021 0.000 0.000 0.000 0.000 With 0.001 0.001 0.021 0.021 0.019 0.021 Total 0.024 0.022 0.021 0.021 0.019 0.021  = 0:40 Without 0.023 0.021 0.000 0.000 0.000 0.000 With 0.000 0.001 0.020 0.021 0.022 0.019 Total 0.023 0.022 0.020 0.021 0.022 0.019  = 0:80 Without 0.029 0.021 0.000 0.000 0.000 0.000 With 0.000 0.000 0.021 0.020 0.029 0.022 Total 0.029 0.021 0.021 0.020 0.029 0.022 Lag length selected using the sequential t sig method 22 Table 6a. Size of the ADF Test; ARFIMA(1,d,0) Errors eith d=0.00* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.045 0.040 0.061 0.043 0.008 0.002 Whit 0.001 0.001 0.021 0.011 0.089 0.055 Total 0.045 0.041 0.082 0.054 0.087 0.057  = 0:40 Without 0.044 0.037 0.006 0.005 0.000 0.000 Whit 0.002 0.002 0.045 0.037 0.045 0.040 Total 0.047 0.039 0.051 0.042 0.045 0.040  = 0:00 Without 0.043 0.037 0.000 0.000 0.000 0.000 Whit 0.003 0.002 0.047 0.040 0.040 0.040 Total 0.046 0.039 0.047 0.040 0.040 0.040  = 0:40 Without 0.046 0.037 0.000 0.000 0.000 0.000 Whit 0.002 0.001 0.039 0.034 0.040 0.041 Total 0.048 0.038 0.039 0.034 0.040 0.041  = 0:80 Without 0.053 0.040 0.000 0.000 0.000 0.000 Whit 0.001 0.001 0.048 0.036 0.045 0.044 Total 0.054 0.041 0.048 0.036 0.045 0.044 *Lag length selected using the sequential t sig method 23 Table 6b. Size of the ADF Test; ARFIMA(1,d,0) Errors with d=0.24* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.026 0.023 0.011 0.013 0.000 0.000 Whit 0.001 0.000 0.012 0.010 0.021 0.024 Total 0.027 0.023 0.022 0.022 0.021 0.024  = 0:40 Without 0.026 0.022 0.000 0.000 0.000 0.000 Whit 0.001 0.002 0.026 0.022 0.025 0.024 Total 0.027 0.023 0.026 0.022 0.025 0.024  = 0:00 Without 0.028 0.023 0.000 0.000 0.000 0.000 Whit 0.001 0.001 0.026 0.022 0.024 0.022 Total 0.029 0.024 0.026 0.022 0.024 0.022  = 0:40 Without 0.030 0.022 0.000 0.000 0.000 0.000 Whit 0.001 0.001 0.029 0.020 0.028 0.024 Total 0.031 0.023 0.029 0.020 0.028 0.024  = 0:80 Without 0.057 0.037 0.000 0.000 0.000 0.000 Whit 0.001 0.001 0.048 0.040 0.054 0.044 Total 0.058 0.038 0.048 0.040 0.054 0.044 *Lag length selected using the sequential t sig method 24 Table 6c. Size of the ADF Test; ARFIMA(1,d,0) Errors with d=0.48* 1= 0; 2= 0; 1= 5; 2= 3; 1= 10; 2= 5; 3= 0; 4= 0 3= 2; 4= 2 3= 5; 4= 5 T=100 T=200 T=100 T=200 T=100 T=200  = 0:80 Without 0.024 0.022 0.006 0.007 0.000 0.000 Whit 0.001 0.000 0.015 0.014 0.021 0.021 Total 0.025 0.022 0.021 0.021 0.021 0.021  = 0:40 Without 0.023 0.021 0.000 0.000 0.000 0.000 Whit 0.001 0.001 0.023 0.020 0.020 0.019 Total 0.024 0.022 0.023 0.020 0.020 0.019  = 0:00 Without 0.023 0.021 0.000 0.000 0.000 0.000 Whit 0.001 0.001 0.021 0.021 0.022 0.020 Total 0.024 0.022 0.021 0.021 0.022 0.020  = 0:40 Without 0.025 0.021 0.000 0.000 0.000 0.000 Whit 0.000 0.001 0.023 0.021 0.026 0.021 Total 0.026 0.022 0.023 0.021 0.026 0.021  = 0:80 Without 0.061 0.032 0.000 0.000 0.000 0.000 Whit 0.001 0.000 0.055 0.041 0.065 0.055 Total 0.062 0.032 0.055 0.041 0.065 0.055 *Lag length selected using the sequential t sig method 25 Table 7. Standard and New Unit root Tests Phillips-Perron ADF MZGLS MZ GLS t MSB GLS PGLST k Value b k Value b k Value Value Value value Argentina -9.475a 1 -4.976a 1 -29.142a -3.817a 0.131a 0.841a 1 Bolivia -6.594a 3 -4.615a 3 -54.306a -5.211a 0.096a 0.451a 3 Chile -2.812b 12 -2.641c 12 -11.509b -2.378b 0.206a 2.211b 12 Colombia -3.125b 8 -0.683 8 -1.720 -0.822 0.477 12.672 8 Ecuador -2.702c 8 -2.243 8 -7.339c -1.911b 0.260c 3.354c 8 Peru -9.380a 5 -2.769c 5 -10.101b -2.247b 0.222b 2.426b 5 Uruguay -2.404 8 -1.450 8 -3.452 -1.292 0.374 7.092 8 Venezuela -3.186b 5 -2.199 5 -4.089 -1.414 0.345 6.011 5 Lag length selected using the recursive method t-sig; a;b;c indicate statistically signi…cancy at 1.0%, 5.0%, and 10.0%, respectively 26 Table 8. ADF Test corrected for Additive Outliers usinf d Country Value Coe¢ cient k Outliers Argentina -1.723 0.880 7 3 Bolivia -0.131 0.977 13 16 Chile -2.353 0.865 12 14 Colombia -0.329 0.986 8 3 Ecuador -0.899 0.949 12 2 Peru 1.423 1.098 19 19 Uruguay -1.378 0.958 10 3 Venezuela -1.469 0.904 7 5 Lag length selected using the recursive method t-sig; a;b;c indicate statistically signi…cancy at 1.0%, 5.0%, and 10.0%, respectively 27 -200 0 200 400 600 800 1975 1980 1985 1990 1995 2000 2005 2010 Argentina -100 0 100 200 300 400 500 1975 1980 1985 1990 1995 2000 2005 2010 Bolivia -40 0 40 80 120 160 1975 1980 1985 1990 1995 2000 2005 2010 Chile -4 0 4 8 12 16 1975 1980 1985 1990 1995 2000 2005 2010 Colombia -5 0 5 10 15 20 25 30 1975 1980 1985 1990 1995 2000 2005 2010 Ecuador -200 0 200 400 600 800 1975 1980 1985 1990 1995 2000 2005 2010 Peru 0 4 8 12 16 20 24 28 1975 1980 1985 1990 1995 2000 2005 2010 Uruguay 0 10 20 30 40 1975 1980 1985 1990 1995 2000 2005 2010 Venezuela Figure 1. Quarterly Latin-American In‡ation Series 28 ÚLTIMAS PUBLICACIONES DE LOS PROFESORES DEL DEPARTAMENTO DE ECONOMÍA Libros Cecilia Garavito e Ismael Muñoz (Eds.) 2012 Empleo y protección social. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Félix Jiménez 2012 Elementos de teoría y política macroeconómica para una economía abierta (Tomos I y II). Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Félix Jiménez 2012 Crecimiento económico: enfoques y modelos. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Janina León Castillo y Javier M. Iguiñiz Echeverría (Eds.) 2011 Desigualdad distributiva en el Perú: Dimensiones. Lima, Fondo Editorial, Pontificia Universidad Católica del Perú. Alan Fairlie 2010 Biocomercio en el Perú: Experiencias y propuestas. Lima, Escuela de Posgrado, Maestría en Biocomercio y Desarrollo Sostenible, PUCP; IDEA, PUCP; y, LATN. 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Diciembre, 2012. No. 348 “Endogenous Altruism in the Long Run”. Alejandro Lugon. Diciembre, 2012. No. 347 “Introducción al cálculo de Malliavin para las finanzas con aplicación a la elección dinámica de portafolio”. Guillermo Moloche. Diciembre, 2012. Departamento de Economía - Pontificia Universidad Católica del Perú Av. Universitaria 1801, Lima 32 – Perú. Telf. 626-2000 anexos 4950 - 4951 http://www.pucp.edu.pe/economia