Price dispersion and optimal inflation: the spanish case

1 OPTIMAL INFLATION, PRICE DISPERSION AND INFLATION EXPECTATIONS: THE SPANISH CASE Mª ÁNGELES CARABALLO Universidad de Sevilla CARLOS DABÚS CONICET and Universidad Nacional del Sur Submitted December 2010 This paper studies the relation between inflation and relative price variability (RPV) in Spain during the 19872009 period. We find that this relation presents a U-shape profile, and that the optimal annual inflation rate (defined as the one that minimizes RPV) is around 4%, higher than the 2% inflation target proposed by the European Monetary Union. More importantly, this result does not depend on whether the monetary regime is before or after the euro. Hence, the main policy implication is that disinflation efforts to achieve the 2% inflation target result in welfare losses. The key link between inflation and RPV is unexpected inflation, whose optimal level is around zero. This suggests that monetary policy matters: the welfare costs associated with higher RPV can be minimized with a credible and predictable inflation targeting policy set at the appropriate level. JEL classification codes: E30, E31 Key words: inflation expectations, inflation uncertainty, monetary policy, optimal inflation, relative price variability, semiparametric estimation. I. Introduction This paper provides new evidence on the key features of the relationship between inflation and relative price variability (RPV), focusing on Spain during the 1987-2009 period. We additionally try to determine if the inflation target proposed by the European Monetary Union (EMU) is a good guideline for monetary policy. Unlike previous studies, we consider all the features of the inflation-RPV relationship for two different monetary regimes, the first of higher inflation from the beginning of the period to the entry of Spain in the EMU in 1999, and the second sub-period of lower inflation thereafter. This allows us to investigate not only the functional form but also the stability of the relationship. Secondly, we estimate the optimal inflation rate, which depends heavily on the shape of the inflation-RPV relationship: if such a relation is linear, then the lower the inflation, the lower the RPV, so the optimal inflation rate that minimizes the welfare costs of price dispersion is zero; but this reasoning is no longer true if the inflation-RPV relationship shows a U-shape, because in this case the inflation  Carlos Dabús (corresponding autor): Departamento de Economía, Universidad Nacional del Sur, 12 de Octubre y San Juan, (8000) Bahía Blanca, Argentina; e-mail [email protected]du.ar. Mª Ángeles Caraballo: Departamento de Economía e Historia Económica, Universidad de Sevilla, Avda. Ramón y Cajal, nº 18, 41018, Sevilla, Spain; e-mail [email protected]. We thank Daniel Heymann, Fernando Navajas, Jorge Streb and two anonymous referees for their valuable comments, as well as financial support from the Junta de Andalucía (CICE, Proyecto de Excelencia SEJ-4546). The usual disclaimer applies. 2 rate that minimizes RPV could be positive. 1 In the third place, we study the role of inflation expectations and uncertainty as the linkages between inflation and RPV. Our central findings are that the relation between inflation and RPV presents a U-shape profile and, more importantly, the annual optimal inflation rate is around 4%, which is higher than the EMU’s 2% inflation target. This result is robust: it holds for different time periods. The main policy implication is that disinflation efforts to achieve the 2% target level result in welfare losses. The key link between inflation and price dispersion is unexpected inflation, whose its optimal level is around zero. Hence, our results suggest that monetary policy matters: the welfare costs associated with higher price dispersion can be minimized with a credible and predictable inflation targeting policy set at the appropriate level. A huge body of empirical evidence has found a positive inflation-RPV relationship. Whilst traditional works like Glejser (1965) and Parks (1978) show that such a relationship is linear, more recently Caglayan and Filiztekin (2003) for Turkey and Caraballo et al. (2009) for Argentina, Brazil and Peru show that it is both nonlinear and unstable across different inflationary regimes. Similarly, Fielding and Mizen (2008) for the USA and Choi (2010) for the USA and Japan find a U-shape functional profile, as well as a positive optimal inflation rate. Meanwhile, Bick and Nautz (2008), in a panel threshold model for several US cities, point out that the annual optimal inflation, i.e., the inflation that minimizes RPV, is in the range of 1.8%-2.8%. This result has important monetary policy implications: if the optimal inflation rate is positive, reducing inflation below it should increase RPV, and the welfare costs associated with higher price dispersion. Finally, Nautz and Scharff (2005) for Germany, and Nautz and Scharff (2006) for the Euro-area find that RPV is increasing with inflation, even in a low inflation environment. Several theoretical approaches try to disclose the links underlying such a relation: menu cost models emphasize the role of expected inflation, while the Lucas-type incomplete information approach argues that non-neutrality is explained by uncertainty and unexpected inflation. 2 In sum, the empirical results suggest a changing inflation-RPV relationship and support the idea of nonneutrality, regardless of the inflation environment. Thus, inflation increases price dispersion, and thus welfare costs. Nevertheless, the literature has focused only on some of the key features of that relationship. For instance, Choi (2010) studies the shape and the stability across different inflation regimes for the USA, while Choi et al. (2011) extend the study to a wider sample with inflation targeting countries. However, these papers do not 1 In the literature focused on the inflation-RPV relationship, where this paper is included, the optimal inflation rate is defined as the one that minimizes RPV. Obviously, the term “optimal inflation” could refer to many other different meanings –such as the inflation rate related to the optimal monetary rule proposed by Friedman (1969), or the optimal output-inflation trade-off assessed in Tobin (1972) and Lucas (1973), among others– but they are not going to be taken into account in this paper. 2 For additional explanations see Sheshinski and Weiss (1977), Rotemberg (1983), Caplin and Spulber (1987) and Caplin and Leahy (1991). In turn, early developments of the signal-extraction model can be found in Lucas (1973) and Barro (1976), Hercowitz (1981) and Cukierman (1983). 3 consider the crucial role played by the channels that connect inflation and RPV, in particular uncertainty and inflation expectations in different monetary and inflationary regimes. In order to explore the key features of the inflation-RPV relationship for the case of Spain, the paper is organized as follows. In Section II we describe the price data and variables. Section III contains an Ordinary Least Square (OLS) estimation of the inflation-RPV relationship for the full sample and for the two sub-samples, before and after the entry of Spain into the EMU. This is not stable across both periods, which suggests a nonlinear relation between inflation and RPV. Thus, in Section IV we check the stability of coefficients and carry out semiparametric estimations, which allow us to obtain the optimal inflation rate. Section V studies the role of inflation expectations and uncertainty. Finally, Section VI concludes. II. Price Data and Variables In this study we employ monthly price data of the Spanish Consumer Price Index (CPI), disaggregated into 57 categories, over the 1987.01-2009.09 period. 3 Data were extracted from the Instituto Nacional de Estadística (National Institute of Statistics). For each category and for the average, the inflation rate is the monthly logdifference of the CPI. RPV is a measure of the non-uniformity: it captures the inflation rate of individual prices in relation to the average inflation rate. In order to avoid spurious correlation between the mean (the average inflation) and the variance (in this case RPV) we use a modified version of the coefficient of variation (CV), as follows:   /IN/1 )IN(INw RPV t i 1/2 2 titit t    , (1) where wit is the weight of price i in the price index, INit the inflation rate of the price i, and INt the average inflation rate at time t. We consider that expression RPV in equation (1) is the best option to define RPV because it avoids two important problems. On the one hand, instead of the simple variance or standard deviation, it is not spuriously correlated with the mean of the distribution, i.e., the inflation rate. On the other hand, and more important in low inflation economies, for inflation rates near zero, the traditional formula of CV generates values of RPV tending 3 The 57 categories appear in Appendix A. 4 to infinite, which implies an “artificial” negative inflation-RPV relationship. Both series, IN and RPV, were deseasonalized by the TRAMO-SEATS method. Since we study in Section V the role of inflation expectations and uncertainty in explaining the relation between inflation and RPV, we need to decompose inflation (IN) into its components: expected inflation (EIN), unexpected inflation (UIN) and uncertainty (UN). In order to do this, as Elliott and Timmermann (2008) point out, univariate time series models seem to be appropriate to forecast inflation, especially when data are monthly. Thus, we have chosen a univariate autoregressive moving-average model for the mean inflation and we have specified a GARCH equation for the variance of the inflation model error term, which allows us to estimate a proxy for UN. 4 Since the dynamics of inflation has changed during the period, to obtain the estimations of EIN and UN the parameters of the ARMA-GARCH model have been estimated by means of recursive regression in which we used monthly inflation data from December 1979 to August 2009. EIN is derived as the one-periodahead inflation forecast and UIN is the resulting forecast error: UIN=IN-EIN. We take into account the updating information process for CPI inflation in Spain. Following the standard model of inflation forecasting, it was assumed that the available information in t-1 to forecast inflation in period t is the actual inflation until t-2 and the expected inflation for t-1, given that the actual inflation for t-1 is known about the middle of period t. Therefore, EIN was obtained from a two-step procedure. In the first stage, in order to select the appropriate number of lags, inflation for the total period has been modelled as an ARMA process using the standard BoxJenkins methodology. As is well known, the first step to model uncertainty with the variance of the error terms of the inflation model is to test if inflation is stationary (see Section III). If this is not the case, the variance of errors explodes and it makes no sense to use such a variance as a proxy of uncertainty. As usual, the Akaike information criterion has been applied to determine the optimal lag structure, from which an ARMA (1,6,12)(12) was selected as the best fitting ARMA model. Nonetheless, the forecast errors of this model were heteroskedastic, so that the inflation model could indicate uncertainty. To estimate a proxy for UN, we have specified a GARCH equation for the variance of the inflation model error term. A GARCH (1,1) 4 The GARCH model implies that uncertainty changes slowly over time. Given the role played by the monetary policy in Spain during the period analyzed in this paper, we can assume such behaviour for uncertainty. The Spanish economy reached the highest inflation rates in 1977 and 1978, due to the political transition process and the oil crises. Since then, the monetary policy aimed at reducing inflation through different restrictive measures. In fact, inflation dropped from 26% in 1977 to rates below 8% in 1987, the first year of the period under study. Monetary policy has maintained its goal of stabilizing inflation, especially with the independence law of the Bank of Spain in 1994, the government’s commitment to meet the inflation targets for the incorporation into the euro, and after the EMU, with the 2% target set by the European Central Bank. Thus, inflation uncertainty should change slowly due to the credibility of the commitments of monetary policy to reduce and stabilize inflation. 5 minimizes the Akaike criterion and, by the simultaneous estimate of the ARMA process for the mean inflation and the GARCH equation, the following new inflation model with homoscedastic forecast errors is obtained: tttttt aINaINaINaIN    1241236211 , (2) 21,2 211 2  ttt bb   , (3) where σεt2 is inflation uncertainty (UN). In the second step, once we have selected the optimal lag structure for the ARMA-GARCH model, we have calculated EIN and UN using the recursive coefficients technique. As expectations are based on past information, EIN was derived as follows: the expected value for January 1987 is calculated with the actual value from December 1979 to November 1986 and with the expected value for December 1986, and for the rest of the period we estimate the following model from December 1979 until t to derive EINt+2: tttttttttt aINaINaINaIN    12,412,36,21,1 , (4) 21,,2 21,1 2   ttttt bb   . (5) III. Monetary Regimes, Inflation and RPV: Basic Regression Prior to the regression analysis, the stationarity of IN and RPV is checked by applying the Augmented DickeyFuller (ADF), Dickey-Fuller with GLS Detrending (DF-GLS) and Phillips-Perron (PP) unit root null tests to the seasonally adjusted series for the total period. The results are presented in Appendix B, Table A2. A unit root is rejected for inflation, even though only at 10% for the ADF and DF-GLS tests when we apply the Akaike criteria to select the optimal lag length. On the contrary, the results for RPV are ambiguous: both the ADF and DF-GLS tests fail to reject a unit root, while PP rejects it. Such differences can be due to the presence of structural breaks in the series. Hence, we apply the unit root tests proposed by Perron (1997) and Vogelsang and Perron (1998), which allow for a break in the series at an unknown time. The results are presented in Appendix B, Tables A3 and A4. Both IN and RPV present possible breaks from 1997.04 to 1998.05, and a unit root is rejected only for inflation. 6 In order to check the robustness of our results, in this section we employ the seasonally adjusted monthly core inflation (CIN), i.e., inflation obtained by excluding unprocessed food and energy prices, from which the corresponding RPV has been calculated (CRPV). The ADF, DF-GLS and PP tests show different results for CIN and CRPV. Once we apply the tests proposed by Perron (1997) and Vogelsang and Perron (1998), a unit root with a break cannot be rejected for both variables. A possible break appears between 1997.08 and 1999.01 (see Appendix B for details). In all cases the breaks are associated with a change in the monetary policy regime, and with different inflation behaviour: the annualized monthly inflation rate slumped from 7.4% to 1.4% before the entry of Spain into the EMU, while it has been fluctuating between 5.3% and -0.8% thereafter. A. Basic Regression Analysis A first approach to the relation between inflation and RPV is obtained from OLS regression analysis. Taking into account the breaks mentioned above, we have run the OLS regression for the full period and for two sub-periods: before and after the EMU. In order to avoid distortions, the months in which the variables may present breaks were left out. Therefore, for IN and RPV we drop the period from 1997.04 to 1998.05. Hence, the first sub-period spans from 1987.01 to 1997.03, and the second from 1998.06 to 2009.09. For CIN and CRPV, we leave out the period from 1997.09 to 1999.01, so that we have two sub-periods: 1987.01-1997.08 and 1999.02-2009.08. Moreover, to capture the impact of inflation and deflation on price dispersion, RPV is regressed on the absolute value of inflation (AIN) and CRPV on the absolute value of core inflation (ACIN). The estimations include the number of lags of AIN, ACIN, RPV and CRPV that minimize the Akaike criterion. Thus, the resulting regression equations are:    12 1 1hththtt RPVAINRPV  , (6)      10 1 2 1 1hthth iititt CRPVACINACINCRPV  . (7) Table 1 presents the results of estimations of (6) and (7). They show that the coefficients of AIN and ACIN are positive and significant for the first period, the pre-EMU stage, while they are negative and not significant 7 for the second period, the post-EMU stage. 5 These results can hide structural changes in the inflation-RPV relationship. Since the parametric model seems to be too restrictive to capture a changing relation, in the next section we undertake a stability test and a semiparametric approach. [TABLE 1 HERE] IV. Coefficient Stability and Non-Linearities A. Coefficient Stability Additional precisions on previous evidence on a time-varying pattern of the inflation-RPV relationship were obtained by employing rolling and recursive regression equations, which allow us to capture variations of the explanatory variables coefficients (in this case AIN), without imposing any prior to the timing of break points. Hence, it is flexible in detecting structural changes over time, by allowing each rolling sample to have a completely different estimation. A parametric model is used, where RPV is the dependent variable, and the explanatory variables are AIN and the lags of RPV and AIN that minimize the Akaike criterion. Therefore, we estimate:    12 1,,1 hthtthtttt RPVAINRPV  . (8) Thus, changes in the inflation-RPV relationship can be outlined by the instability of the parameters over rolling samples. The results for β1,t, our parameter of interest, which is obtained from different rolling and recursive regression windows, are reported in Figure 1. Recursive coefficients have been estimated by successive additions of one month to the 1987.01-1991.12 sub-sample and rolling equations have been estimated for fixed windows of six, eight and ten years. Figure 1 shows that in all cases β1,t is strongly unstable and decreasing in the second half of the total period. In the case of recursive coefficients estimation, this result indicates a changing marginal impact of AIN on RPV when new months are incorporated into the estimation. In turn, the rolling 5 The same conclusions are achieved when inflation and core inflation, instead of their absolute values, are taken as explanatory variables. 8 regressions for fixed windows present a lower step of such coefficient for the post-EMU period, i.e., since 1998, approximately, and this result is robust for different sizes of the windows. 6 [FIGURE 1 HERE] In short, the empirical evidence shows an unstable relation between inflation and RPV. This varies significantly with the monetary policy regime. More precisely, coefficients are sensitive to the addition of years from 1998, and they drop and lose significance in the post-EMU period. Even more, β1,t reaches negatives values, which implies a negative relation between inflation and RPV. This negative value and, in general, the less influence of inflation on RPV, is associated with the fact that, after the entry of Spain into the EMU, the inflation rate was stabilised but RPV increased sharply. This was due to the fact that the existence of a single currency lowered the inflation of tradable goods and services compared to inflation of non-tradable ones. Hence, the lower the mean inflation the higher the difference between the inflation of tradable and non-tradable goods and services, and, therefore, RPV is high for both high and low inflation, and low for medium inflation. This explains that a linear specification for the inflation-RPV relationship does not seem to be adequate. We mean to tackle this issue in the next section. B. Semiparametric Approach and Optimal Inflation In order to find out additional information about the relation between inflation and RPV, we apply a partially linear model. As a preliminary step, we try to approximate the shape of such a relation by including a squared inflation term, as follows:    12 1 2 21 hththttt RPVININRPV  . (9) 6 Similar results for core inflation were obtained from rolling and recursive equations. The coefficient for core inflation starts to decline very sharply from 1998-1999, depending on the window size. In contrast to the inflation-RPV relationship, the coefficient for CIN is strongly significant in the pre-EMU stage for all cases. These results are available from the authors upon request. 9 The results displayed in Table 2 show evidence of a U-shape for such a relationship for the total period and the second sub-period (the squared inflation is significant) but not for the first one (the squared inflation is not significant). [TABLE 2 HERE] The next step is to carry out a semiparametric analysis, which in turn allows us to obtain the optimal inflation. To compare this with our previous findings, the same number of lags for RPV and IN of equation (6) are included, so that the following equation is estimated:      12 112 htttht INgRPVRPV  , (10) where g(INt) is an unknown non-linear smooth differential function, which relates inflation and price dispersion at time t. Therefore, our goal is the estimation of g(INt) in (10). The g(INt) function has been estimated semiparametrically in two stages. In the first one we estimate λk from the following regression equation:    12 1 12 ht t ht RPVRPV  , (11) where ht RPV  are the residual series from a non-parametric regression of each lag of RPVt on INt. In the second stage we estimate g(INt) non-parametrically from the regression:   ttt vINg  ˆ , (12) where     12 1 12 ˆ h t htt RPVRPV  . In both stages we estimate the non-parametric regressions by applying a Nadaraya-Watson kernel regression estimator. Since the results of nonparametric regression are very sensitive to the bandwidth parameter (h), h has 16 On the other hand, we check for the existence of a unit root with structural breaks by applying the tests proposed by Vogelsang and Perron (1998). These tests allow us to distinguish two key properties: i) if the break affects the constant, the trend or both of them in the series, and ii) if the rupture impact on the variable is immediate (additional outlier) or gradual (innovational outlier). Taking into account the evolution of IN and RPV, we consider that additional outlier model must fit better to check structural breaks and unit root, because the entry into the euro affects IN and RPV once and for all. In turn, we select two models, one includes breaks in the constant and the trend, and the other one considers changes only in the trend. Following Vogelsang and Perron (1998), testing for a unit root test in the additional outlier framework includes two steps. In the first one, the following equation is estimated: tt i t i tDTgDUvtuy   , (A1) where yt is the variable under study (in our case inflation and RPV), u is a constant, t is the trend, and DUt and DTt are dummies for the constant and the trend respectively. Three models can be distinguished: i) if i=A the break only affects the constant, and g=0, ii) i=C indicates a rupture in the trend, and then v=0, and iii) i=B corresponds to the case that the rupture is in both the constant and the trend. In turn, calling TB the breakpoint, DUt=1 and DTt=t-TB if t>TB, and zero otherwise. In a second stage, and from the residuals of the regression of equation (A1), we estimate by OLS (A2) if i=A, B, and (A3) if i=C. t k j ijt k jjt t i tuDTB        10 1  , (A2) t k j ijt i tt u    1 1  , (A3) where DTB =1 for t=TB+1 and 0 otherwise. According to Vogelsang and Perron (1998), two data dependent methods can be applied to detect the breakpoints. The first one (method I) selects TB that minimizes tα (t-statistics corresponding to the estimated α in equations (A2) and (A3)). In this case, the choice of TB corresponds to the break date which is most likely to reject the unit root hypothesis. The second method (method II) can be used for model A and B. In this case we 17 pay attention to tv and tg (t-statistics associated to v (model A) or g (model B) in equation (A1)). We choose the breakpoint that maximizes (minimizes) the t-statistics when the direction of the break is known a priori to be positive (negative) or the absolute value of the t-statistics when the direction of the break is unknown. Once TB is determined, the corresponding tα in equation (A2) allows us to accept or reject unit root. On the other hand, to choose the lag length k of εi t-j in (A2) and (A3) we apply two criteria. The first one consists of choosing a fixed value for k, we have considered k =5 (as in Vogelsang and Perron (1998)).The second one is based on selecting a value of k (k =k*) in such a way that in regressions (A2) and (A3) the coefficient corresponding to k* is significant, while it is not significant for k >k*. Results of applying the above methodology to IN, RPV, CIN and CRPV are presented in Tables A3 y A4. These series show a change in the trend during the period, therefore in the paper we have taken into account the results obtained by model C. Nevertheless, we have also included results for model B. Trimming is slightly different in each case but in all of them the first twelve months and the last twenty four months have been removed. As can be seen from Table A3 (model C), the unit root is rejected only for IN. With model B results are not conclusive with respect to IN, CIN and CRPV. The unit root cannot be rejected in all cases only for RPV. [TABLES A3 AND A4 HERE] C. Rolling and recursive equations for unexpected inflation We estimate the following equation for different window sizes: tkt ktktttttttttt RPVUNAUINUINEINRPV       12 1,,4,3,2,0 . Figure A1 presents the results for β2,t and β3,t. As usual, recursive coefficients have been obtained by successive additions of one month to the 1987.01-1991.12 sub-sample and rolling regressions have been estimated for a window of 8 years. As can be seen from Figure A1, both coefficients decline during the period: β2,t seems to be significant in the pre-EMU stage and β3,t is more sensitive to the sample considered. Similar results are obtained for windows of 6, 10 and 12 years (they are available from authors upon request). [FIGURE A1 HERE] 18 References Barro, Robert J. (1976), Rational expectations and the role of monetary policy, Journal of Monetary Economics 2: 1–32. 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(1978), Inflation and relative price variability, Journal of Political Economy 86: 79–95. Perron, Pierre (1997), Further evidence on breaking trend functions in macroeconomics variables, Journal of Econometrics 80: 355–385. Rotemberg, Julio J. (1983), Aggregate consequences of fixed costs of price adjustment, American Economic Review 73: 433–463. Sheshinski, Eytan, and Yoram Weiss (1977), Inflation and costs of price adjustment, Review of Economic Studies 44: 287–303. Tobin, James (1972), Inflation and unemployment, American Economic Review 62: 1–18. Vogelsang, Timothy J. and Pierre Perron (1998), Additional tests for a unit root allowing for a break in the trend function at an unknown time, International Economic Review 39: 1073–1100. 19 Figure 1. Recursive and rolling regressions -.1 .0 .1 .2 .3 .4 1992 1994 1996 1998 2000 2002 2004 2006 2008 beta1,t -.2 -.1 .0 .1 .2 .3 .4 .5 .6 1992 1994 1996 1998 2000 2002 2004 2006 2008 beta1,t -.1 .0 .1 .2 .3 .4 1992 1994 1996 1998 2000 2002 2004 2006 2008 beta1,t -.1 .0 .1 .2 .3 .4 1992 1994 1996 1998 2000 2002 2004 2006 2008 beta1,t Rolling regression: six - year window Rolling regression: eight - year window Rolling regression: ten - year window Recursive regression with fixed start date (1987.01) Notes: the significance of coefficients is for 10% of confidence intervals, and the months for which they are significant are marked in grey lines. The numbers on the horizontal axis represent the ending month of each window. For example, for a sixyear window, the value of beta1,t in 1992.12 captures the estimation of the parameter in (8) for 1987.01-1992.12, and so on. 20 Figure 2. Derivatives of g(INt) -4 -3 -2 -1 0 1 2 3 4 -.004 .000 .004 .008 .012 h=0.0007 h=0.0013 g'(INt) Gaussian kernel -4 -3 -2 -1 0 1 2 3 4 -.004 .000 .004 .008 .012 h=0.0019 h=0.0025 g'(INt) Gaussian kernel -4 -3 -2 -1 0 1 2 3 4 -.004 .000 .004 .008 .012 h=0.0013 h=0.0019 g'(INt) Epanechnikov kernel -4 -3 -2 -1 0 1 2 3 4 -.004 .000 .004 .008 .012 h=0.0025 h=0.0031 g'(INt) Epanechnikov kernel INt INt INt INt 21 Figure 3. Derivatives of g(UINt) -6 -4 -2 0 2 4 6 -.010 -.005 .000 .005 .010 h=0,0029 h=0,0042 h=0,005 -6 -4 -2 0 2 4 6 -.010 -.005 .000 .005 .010 Derivative for Gaussian kernel Derivative for Epanechnikov kernel UINt g'(UINt) g'(UINt) UINt 22 Table 1. Basic regression analysis DEPENDENT VARIABLE: RPVt DEPENDENT VARIABLE: CRPVt PERIOD 1987.012009.08 1987.011997.03 1998.062009.08 PERIOD 1987.012009.08 1987.011997.08 1999.022009.08 α 0.0007 (0.18) -0.0001 (0.88) 0.001* (0.08) α 0.0004 (0.27) 0.003*** (0.00) 0.0005 (0.44) AINt 0.07 (0.29) 0.20* (0.08) -0.03 (0.65) ACINt 0.11 (0.43) 0.47*** (0.00) -0.15 (0.49) RPVt-1 0.18*** (0.00) 0.16** (0.03) 0.29*** (0.00) CRPVt-1 0.86*** (0.00) 0.77*** (0.00) 0.90*** (0.00) R2adj. 0.81 0.78 0.85 R2adj. 0.93 0.56 0.95 Notes: *, **, *** denote that the coefficients are significant at 10%, 5% and 1% levels respectively. The t-statistics are based on standard errors computed according to the Newey-West (1987) procedure to allow for residuals that exhibit both autocorrelation and heteroskedasticity of unknown form. Terms in parentheses are the p-values associated with t-statistics. To simplify the presentation only the first lag of RPV appears in the table. 23 Table 2. Inflation, squared inflation and RPV DEPENDENT VARIABLE: RPVt PERIOD 1987.01-2009.08 1987.011997.03 1998.06-2009.08 α 0.001** (0.03) 0.0008 (0.53) 0.001*** (0.01) INt -0.21** (0.03) -0.30 (0.45) -0.25*** (0.00) INt2 41.88*** (0.00) 58.32 (0.15) 40.08** (0.03) RPVt-1 0.18*** (0.00) 0.16** (0.03) 0.28*** (0.00) R2adj. 0.81 0.78 0.86 Notes: *, **, *** denote that the coefficients are significant at 10%, 5% and 1% levels respectively. The t-statistics are based on standard errors computed according to the Newey-West (1987) procedure to allow for residuals that exhibit both autocorrelation and heteroskedasticity of unknown form. Terms in parentheses are the p-values associated with t-statistics. To simplify the presentation only the first lag of RPV appears in the table. 24 Table 3. MSFE for different values of the bandwidth parameter Bandwidth parameter Gaussian kernel Epanechnikov kernel 0.0005 7.149*10-5 4.11*10-5 0.0007† 7.134*10-5 4.097*10-5 0.0013‡ 7.182*10-5 4.018*10-5 0.0019 7.252*10-5 4.065*10-5 0.0025 7.348*10-5 4.140*10-5 0.0031 7.457*10-5 4.190*10-5 Notes: †Optimal bandwidth for Gaussian kernel. ‡Optimal bandwidth for Epanechnikov kernel. 25 Table 4. Optimal annual inflation rate Bandwidth Gaussian kernel Epanechnikov kernel 0.0007† 4.17% 0.0013‡ 4.83% 4.17% 0.0019 5.31% 4.64% 0.0025 5.88% 5.04% 0.0031 4.91% Notes: †Optimal bandwidth for Gaussian kernel. ‡Optimal bandwidth for Epanechnikov kernel. 32 Table A2. ADF, DF-GLS and PP unit root tests (1987.01-2009.08) Variable Criteria to select lags Constant and trend Constant No constant, no trend ADF DF-GLS PP ADF DF-GLS PP ADF PP INt Akaike -3.27* -2.84* -11.03*** -1.93 -1.56 -9.78*** -1.24 -3.88*** Schwarz -10.69*** -10.32*** -9.11*** -5.08*** -1.24 CINt Akaike -3.09 -2.50 -6.68*** -1.06 -0.54 -4.43*** -1.18 -2.41** Schwarz -3.09 -2.50 -1.19 -0.70 -1.13 RPVt Akaike -1.36 -1.09 -4.61*** -1.57 -0.93 -4.60*** -0.54 -1.01 Schwarz -1.36 -1.09 -1.73 -0.93 -0.54 CRPVt Akaike -1.33 -1.09 -3.45** -0.35 -0.02 -1.90 1.10 0.10 Schwarz -2.31 -1.78 -0.95 -0.74 0.22 Notes:*,**,*** denote rejection of the null at 10%, 5% and 1% level of significance. A Bartlett kernel-based estimator of the frequency zero spectrum is used for the Phillips Perron test. 33 Table A3. Unit root tests with structural breaks. Method I Model B: break in trend and constant Statistics IN k=5 IN k(t-sig) CIN k=5 CIN k(t-sig) RPV k=5 RPV k(t-sig) CRPV k=5 CRPV k(t-sig) Min tα TB 1999.03 1997.08 1994.09 1995.03 1996.07 2000.12 2002.01 2002.01 tα -5.084** -4.820* -3.493 -4.82** -3.908 -4.005 -6.396*** -5.008* Model C: break in trend Statistics IN k=5 IN k(t-sig) CIN k=5 CIN k(t-sig) RPV k=5 RPV k(t-sig) CRPV k=5 CRPV k(t-sig) Min tα TB 1997.07 1997.07 1997.08 1998.02 1997.08 1997.06 1999.01 1998.09 tα -4.748** -4.070* -3.052 -4.269 -3.708 -3.630 -3.643 -2.895 Notes: *,**,*** indicate significance at the 10%, 5% and 1% levels respectively. tα: critical values in Vogelsang and Perron (1994). k(t-sig): the lag length k of εi t-j has been selected (k=k(t-sig)) in such a way that in regressions (B2) and (B3) (for models B and C respectively) the coefficient corresponding to k(t-sig) is significant, while it is not significant for k >k(t-sig). 34 Table A4. Unit root tests with structural breaks. Method II Statistics IN CIN RPV CRPV Model B. Break in trend and constant Max tg TB 1997.07 1998.06 1998.02 1997.06 tg 1.859 3.283 18.783 11.742 tα (k=5) -4.771 -3.009 -3.678 -3.643 tα (k(t-sig)) -4.698 -4.269** -3.556 -2.699 Model C: Break in trend Max tg TB 1997.05 1998.05 1998.05 1998.02 tg 1.840 3.270 18.949 10.451 Notes: *,**,*** indicate significance at the 10%, 5% and 1% levels respectively. tα: critical values in Perron (1994). k(t-sig): the lag length k of εi t-j has been selected (k=k(t-sig)) in such a way that, in regression (B2), the coefficient corresponding to k(t-sig) is significant, while it is not significant for k >k(t-sig). Obviously for model C, max tg gives some information about TB but it cannot be used to test unit root.