Testing Regime Non-stationarity of the G-7 Inflation Rates: Evidence from the Markov Switching Unit Root Test

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1 Journal of the Chinese Statistical Association Vol. 47, (2009) 1 18 Testing Regime Non-stationarity of the G-7 Inflation Rates: Evidence from the Markov Switching Unit Root Test Shyh-Wei Chen 1 and Chung-Hua Shen 2 1 Department of Finance, Da-Yeh University 2 Department and Graduate Institute of Finance, National Taiwan University ABSTRACT Using G-7 data, this study employs a Markov Switching unit root regression to investigate the issue of the non-stationarity and non-linearity of inflation rates. The results convincingly support the view that the inflation rates in the G-7 nations are characterized by a two-regime Markov Switching unit root process. For Italy the inflation rates are characterized by a unit root process, consistent with the accelerationist hypothesis that the inflation rates are either in the high-volatility regime or in the low-volatility regime. For Canada, France, Germany, Japan, the UK and the USA, the shocks to the inflation rates are highly persistent in one regime, but have finite lives in the other regime. The high-volatility regime arises in most of the nations considered and it tends to prevail over a relatively long period. Key words and phrases: Unit root, Markov switching model, inflation, non-linearity. JEL classification: C22, E Introduction Testing for a unit root in the inflation rate has attracted substantial interest ever since the studies conducted by Nelson and Plosser (1982) and Rose (1988). This is

2 2 SHYH-WEI CHEN AND CHUNG-HUA SHEN because if there is a unit root in inflation, then, first, this is in line with the view of the accelerationist hypothesis which implies a non-stationary inflation rate. Therefore, any shock to inflation has a permanent effect. Second, the testing of the Fisher hypothesis will be heatedly discussed if inflation rate and interest rate series are not integrated of the same order. Third, if inflation rates do not follow a random walk but a mean reverting process instead, then the rational expectations version of Cagan (1956) indicates that the stable growth of money supply implies stationary inflation unless there exist bubbles. A wealth of research has been devoted to this issue. 1 Three important features or problems characterize such studies. First, the findings are mixed, if not contradictory, which means there is no corroborative conclusion vis-à-vis the stationarity property for inflation rates. Second, the majority of the researchers employ the traditional method in testing the null hypothesis of a unit root in the inflation rate. It is well-known that the power of the traditional unit root test is significantly reduced if the true data generating process of a series exhibits structural breaks (Perron, 1989). Therefore, Culver and Papell (1997), Lee and Wu (2001), Costantini and Lupi (2007), and Westerlund and Basher (in press) adopt a newly-developed (panel) unit root test with or without a break (Im et al., 2005; Westerlund, 2005) to investigate the stationarity property of inflation rates. Third, despite the abundance of studies on the behavior of inflation rates, the specification of volatility is commonly time-invariant. Recent studies, however, find that inflation rates tend to be specified as non-linear data generating processes, implying that the volatility may not be constant over time (Hassler and Wolters, 1995; Philips et al., 1999; Chen, 2000 and Charles et al., 2002). This paper attempts to re-investigate the issue of the non-stationarity of inflation rates by using the Markov Switching augmented Dickey-Fuller (hereafter MS-ADF) regression, pioneered by Hall et al. (1999), via G-7 countries. The merit of this approach is that there is no need to split the sample period into different sub-periods or to pre- 1 For example, to name a few, Crowder and Phengpis (2007) found that there is a single common stochastic trend in inflation in G7 countries; Henry and Shields (2004) found the two-regime threshold unit root process in Japan and the UK; and Malliaropulos (2000) found that the US inflation rate was better characterized by the trend stationarity with a structural break. See also Barsky (1987), Brunner and Hess (1993), Choi (1994), Baillie et al. (1996), Crowder and Hoffman (1996), and Rapach and Weber (2004) for a discussion of the time series properties of the inflation rate.

3 TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES 3 impose regime dates. Thus, no prior knowledge of the dates of structural breaks or the number of breaks is needed. In addition, this approach endogenously identifies each volatility regime, which may not be constant. The unit root test is then conducted for each regime separately. Finally, the model does not need to assume the stationarity or non-stationarity of either regime. It is possible for both regimes to be locally stationary or one to be locally stationary and the other locally non-stationary. The remainder of this paper is organized as follows. Section 2 introduces the econometric methodology that we employ, and Section 3 describes the data and the empirical test results. Section 4 presents the conclusions that we draw from this research. 2. The MS-ADF Unit Root Test Let p t denote the consumer price index. We then calculate the annualized inflation rate q t, in percentage terms, as follows: q t = log ( pt p t 1 ) 400. (1) The Markov Switching ADF regression is obtained by running the following regression: q t = a(s t ) + b(s t )q t 1 + p γ k (S t ) q t k + u t, u t NID(0,σ 2 (S t )), (2) k=1 where q t denotes the first difference of the inflation rate q t, a(s t ), b(s t ) and γ 1 (S t ),...,γ p (S t ) are regime-varying parameters, and u t is the innovation process with a regime-dependent variance-covariance matrix σ 2 (S t ). The unobservable state variable S t follows a first-order, two-state Markov Chain with a transition probability as follows: p(s t = j S t 1 = i) = p ij, (3) where i, j = 1 or 2. The unconditional probabilities for state 1 and state 2 are w 1 = 1 p 22 2 p 11 p 22 and w 2 = 1 p 11 2 p 11 p 22, respectively. The MS-ADF regression has two features. Firstly, it allows the volatility of the inflation rate to switch across regimes following a first-order Markov chain. Secondly, the autoregressive parameters in the ADF regression are also allowed to change as the volatility regimes shift, and hence they are regime-varying. In short, model (2) endogenously permits the volatility to

4 4 SHYH-WEI CHEN AND CHUNG-HUA SHEN switch as the date and regime changes. An interesting feature of this model is that no assumption is needed to impose the (non)stationarity of either regime. That is, this model allows both regimes to be locally stationary or one to be locally stationary and the other locally non-stationary. Because the estimation procedure for the Markov Switching model is well documented in the literature, we omit any discussion of the estimation and refer readers to Hamilton (1989) and Kim and Nelson (1999). 3. Data and Results We use the inflation rate data of the G-7, i.e., Canada, France, Germany, Italy, Japan, the UK and the USA, in our empirical study. The data set is obtained from the OECD Main Economic Indicators at For all countries the data are quarterly from 1955Q1 and they all end with 2007Q2. The basic summary statistics, including of sample, standard errors, maximum, minimum, skewness and excess kurtosis, for the inflation rates of the G-7 are displayed in Table 1. The data displays evidence of non-normality with all series displaying statistically significant skewness and excess kurtosis. Moreover the null hypothesis of the normality test, that the data in question are normally distributed, is overwhelmingly rejected for all seven series. The corresponding time series plot of the log of price indices and the inflation rates are graphed in the Figure 1 to Figure 3. Visual inspection of Figures 1 3 clearly highlights the explosion in inflation experienced during the 1970s and early 1980s. We begin by applying the augmented Dickey and Fuller (1981) unit root test to ascertain the order of integration of the variables. The key here is to account for serial correlation; we set p max = 12, which is the maximum lagged order of the lagged difference, and use the Modified Akaike Bayesian Criterion (MAIC), proposed by Ng and Perron (2001), to select the optimal lag length. 2 We summarize the data description and the ADF unit root test results in Table 2. We find no additional evidence against the unit root hypothesis for the price indices based on the ADF test to their level data. 2 It is by now well established that the AIC and BIC tend to select lag lengths for ADF regressions which are too small, resulting in unit-root tests with undesirable size and power properties. Information criteria with superior properties are discussed in Ng and Perron (2001).

5 TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES 5 When we apply the ADF test to the first difference of these series, we cannot not reject the null hypothesis of a unit root at the 5% level or better. This implies that the G-7 inflation rates have a unit root. Next, we examine whether we can reject the linear autoregressive model (H 0 ) in favor of a conventional Markov Switching model (H A ), which assumes that each coefficient and variance are affected by the regime in which they remain. Accordingly, this hypothesis is equivalent to testing the homoskedasticity of variance and the equality of all autoregressive parameters across regimes. It is also similar to the test of the standard ADF regression, as compared with the MS-ADF regression. As shown in Table 2, the likelihood ratio (LR) statistics for the G-7 nations are overwhelmingly greater than the χ (12) = critical value, thus rejecting the null at the 5% level or better of significance. The degree of freedom for the LR test should at least greater than 6 if we set the order of the lagged difference is equal to 1. This is because in this case there are 6 unidentified parameters {a(2), b(2), γ(2), σ(2), p 11, p 22 } under the null hypothesis of linear model. For conservative purpose, we use χ (12) = critical value because we set the maximum lagged orders p max = 12. From Table 2 we can infer that the MS-ADF model is preferable to the linear, single-regime autoregressive model with a constant conditional variance. That is, the conventional ADF test is less powerful in the presence of switching coefficients and variance. In short, we cannot reject the MS-ADF model for the G-7 countries, inflation rates. There is one econometric issue in the use of LR, and that concerns LR(H 0 H A ) reported in the last column of Table 2. Because the parameters p 11 and p 22 are not identified under the null hypothesis, the conventional LR test does not yield the standard asymptotic distribution. 3 Most researchers, however, still use the LR test to obtain valuable supporting evidence. The LR by itself, however, may not be suitable as a safe source of evidence with which to reject or not reject the null hypothesis. Throughout this paper, our LR tests are considered in the same way. Given that the MS-ADF model is not rejected for the sample, we next test for the 3 The problem comes from two sources: under the null hypothesis, some parameters are not identified, and the values are identified as zero. Hansen (1992, 1996) proposed a bounds test that addressed these problems, but its computational difficulty has limited its applicability. See Hansen (1992, 1996) and Garcia (1998) for a detailed explanation of these problems.

6 6 SHYH-WEI CHEN AND CHUNG-HUA SHEN presence of a unit root in each regime. Table 3 reports the variances, the ADF test results and the durations for each regime. The estimated value of σ 1 is substantially larger than that of σ 2, and thus regime 1 corresponds to the high-volatility regime while regime 2 corresponds to the low-volatility regime. We conduct a Monte Carlo simulation to obtain critical values for the unit root test in the MS-ADF model since the distribution under the null hypothesis is not known. 4 The p values corresponding to the t-statistics of the null hypothesis of non-stationarity in both regimes b(s t = 1) = 0 and b(s t = 2) = 0 against the respective one-sided alternatives of stationarity b(s t = 1) < 0 and b(s t = 2) < 0 are obtained by estimating Equation (2) under the null hypothesis b(s t ) = 0, S t = 1, 2, and then generating 5,000 samples of size T that follow this estimated DGP. To this end, the estimated transition probabilities are used to simulate a single series S t. Then, 5,000 series for u t are drawn from a N(0, ˆσ 2 (S t )) and the aforementioned estimates of the parameters under the null are used to generate data for q t. We next fit (2) to each realization of q t, thus obtaining two series of t-statistics for the parameter b, one for the high volatility regime and the other for the low. The resulting p-values are then the percentage of the generated t-ratios that are below the t-values from the estimated model. The results presented in Table 3 show that, first, in regime 1 (the high-volatility regime), the ADF statistic for Italy fails to reject the null hypothesis of non-stationarity because the simulated p-value is greater than In regime 2 (the low-volatility regime), the ADF statistic also fails to reject the null hypothesis of a unit root because the simulated p-value is greater than The results indicate that the inflation rates for Italy are characterized by local non-stationarity in both regimes. These findings support the fact that the inflation rate series are characterized by a unit root process, that is consistent with the accelerationist hypothesis where the inflation rates are either in the high-volatility regime or in the low-volatility regime. Second, in the case of the USA, the ADF statistics must reject the null hypothesis of a unit root in the highvolatility regime because the simulated p-value is smaller than In the low-volatility regime, the ADF statistic fails to reject the null hypothesis of a unit root because the simulated p-value is greater than 0.74, indicating that the US inflation rates are locally 4 Readers are referred to Hall et al. (1999), Kanas and Genius (2005) and Kanas (2006) for details.

7 TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES 7 mean-reverting in the high-volatility regime but are locally non-stationary in the lowvolatility regime. Turning to the cases of Canada, France, Germany, Japan and the UK, we find that the ADF statistics must reject the null hypothesis of a unit root in the low-volatility regime because the simulated p-values are smaller than 0.06 or ever less, but the ADF statistics fail to reject the null hypothesis of a unit root in the high-volatility regime because the simulated p-value is greater than 0.19 or ever more. The test results imply that the inflation rates of the five countries are locally mean-reverting in the low-volatility regime but are locally non-stationary in the high-volatility regime. In other words, for Canada, France, Germany, Japan and the UK, inflation rates are found to be highly persistent in the high-volatility regime, but to have finite lives in the low-volatility regime. Table 3 also reports the estimated durations of each regime, which show the length of each regime s occurrence. The average duration of each regime i is calculated using the formula d i = (1 p ii ) 1, where p ii is the probability that regime i will prevail over two consecutive years, i.e., the transition probability from regime i to regime i. The results reveal that, except for Japan, the high-volatility regime prevails for a longer period than the low-volatility regime for the all countries for which the high-volatility regime occurs more frequently, namely, Canada, France, Germany, Italy, the UK and the USA. For Japan, the low-volatility regime prevails for a period equal to the highvolatility regime. Therefore, the high-volatility regime arises in most of the G-7 nations considered and it tends to prevail over a relatively long period. The maximum likelihood estimation of Equation (2) yields the filter probabilities, representing the inference that the inflation is in regime i at date t. Furthermore, one could date the regime switches. 5 The figures of the filtered probabilities estimated by the MS-ADF model are summarized in the lowest panel in Figure 1 3. For periods from 1973 to 1991, these dates are classified into the high volatility regime for the G-7. Moreover, the high volatility regime (the low volatility regime) persists for quite a 5 The filtered probabilities, collected in a (T 1) vector denoted as ξ t t (ξ t t = p(s t = j Ψ t ), t = 1,..., T, Ψ t is the information set) denote the conditional probability that the analyst s inference about the value of S t is based on information obtained through date t. It is also possible to calculate smoothed probabilities, ξ t T = p(s t = j Ψ T ), which are based on the full sample.

8 8 SHYH-WEI CHEN AND CHUNG-HUA SHEN long time before switch into the low volatility regime (the high volatility regime). The graphs are consistent with the high estimated value of the transition probabilities p 11 and p 22 as shown in Table Concluding Remarks The purpose of this study is to re-investigate the issue of the non-stationarity of inflation rates for the G-7 nations by using a recent non-linear unit root test. The MS-ADF test has the advantage of neither splitting the sample period into different sub-periods nor pre-imposing regime dates. In addition, it endogenously identifies each volatility regime, and the unit root test is conducted for each regime separately. Three important results emerge from our empirical analysis. First, the results convincingly support the view that the inflation rates in the G-7 nations are non-linear series, a finding that is consistent with the evidence reported by Evans and Wachtel (1993), Kim (1993), Evans and Lewis (1995) and Henry and Shields (2004). Second, for Italy the inflation rates are characterized by a unit root process that is consistent with the accelerationist hypothesis that the inflation rates are either in the high-volatility regime or in the low-volatility regime. For Canada, France, Germany, Japan, the UK and the USA, shocks to inflation rates are highly persistent in one regime, but have finite lives in the other regime. Third, the high-volatility regime arises in most of the G-7 nations considered and it tends to prevail over a relatively long-term period. The policy implications of this paper are in line with Henry and Shields (2004), who test a unit root for the inflation rates of Japan, the UK and the USA by employing Caner and Hansen s (2001) non-linear threshold modeling technique. Our empirical evidence implies that shocks to inflation rates may have differing effects depending on the initial regime of inflation rates, the signs and sizes of the shocks, and whether or not the a shock causes a transition across regimes. A shock to inflation rates in the low persistence regime may have less of an effect than a shock of similar magnitude in the high persistence regime. Therefore, an adequate assessment of the persistence of shocks to inflation should account for the inherent non-linearity in the data.

9 TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES 9 Acknowledgements We would like to thank two anonymous referees of this journal for helpful comments and suggestions. Financial support from the National Science Council (NSC H ) is gratefully acknowledged. The usual disclaimer applies. References Baillie, R. T., Chung, C.-F., and Tieslau, M. A. (1996). Analysing inflation by the fractionally integrated ARFIMA-GARCH model. Journal of Applied Econometrics, 11, Barsky, R. B. (1987). The Fisher hypothesis and the forecastability and persistence of inflation. Journal of Monetary Economics, 19, Brunner, A. D. and Hess, G. D. (1993). Are higher levels of inflation less predictable? A state-dependent conditional heteroskedasticity approach. Journal of Business and Economic Statistics, 11, Cagan, P. (1956). The monetary dynamics of hyperinflation. in Studies in the Quantity Theory of Money, edited by Milton Friedman, Chicago: University of Chicago Press. Caner, M. and Hansen, B. (2001). Threshold autoregression with a unit root. Econometrica, 69, Charles, B. S., Philip, F. H., and Marius, O. (2002). Inflation, forecast intervals and long memory regression models. International Journal of Forecasting, 18, Chen, S.-W. (2000). Time series analysis of inflation rates of eight Pacific Basin countries. Taiwan Journal of Political Economy, 4, Choi, S. (1994). Is the real interest rate really unstable? Journal of Financial Research, 17,

10 10 SHYH-WEI CHEN AND CHUNG-HUA SHEN Costantini, M. and Lupi, C. (2007). An analysis of inflation and interest rates: New panel unit root results in the presence of structural breaks. Economics Letters, 95, Crowder, W. J. and Hoffman, D. L. (1996). The long-run relationship between nominal interest rates and inflation: the Fisher equation revisited. Journal of Money, Credit and Banking, 28, Crowder, W. J. and Phengpis, C. (2007). A re-examination of international inflation convergence over the modern float. Journal of International Financial Markets, Institutions and Money, 17, Culver, S. E. and Papell, D. H. (1997). Is there a unit root in the inflation rate? Evidence from sequential break and panel data models. Journal of Applied Econometrics, 12, Dickey, D. A. and Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49, Evans, M. D. and Lewis, K. K. (1995). Do expected shifts in inflation affect estimates of the long run Fisher relation? The Journal of Finance, 50, Evans, M. D. and Wachtel, P. (1993). Inflation regimes and the sources of inflation uncertainty. Journal of Money, Credit and Banking, 25, Garcia, R. (1998). Asymptotic null distribution of the likelihood ratio test in Markovswitching models. International Economic Review, 39, Hall, S. G., Psaradakis, Z., and Sola, M. (1999). Detecting periodically collapsing bubbles: A Markov-switching unit root test. Journal of Applied Econometrics, 14, Hamilton, J. D. (1989). A new approach to the economic analysis of non-stationary time series and the business cycle. Econometrica, 57,

11 TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES 11 Hansen, B. E. (1992). The likelihood ratio test under nonstandard conditions: Testing the Markov-switching model of GNP. Journal of Applied Econometrics, 7, S61 S82. Hansen, B. E. (1996), Erratum: The likelihood ratio test under nonstandard conditions: Testing the Markov-switching model of GNP. Journal of Applied Econometrics, 11, Hassler, U. and Wolters, J. (1995). Long memory in inflation rates: International evidence. Journal of Business and Economic Statistics, 13, Henry, O. T. and Shields, K. (2004). Is there a unit root in inflation? Journal of Macroeconomics, 26, Im, K.-S., Lee, J., and Tieslau, M. (2005). Panel LM unit-root tests with level shifts. Oxford Bulletin of Economics and Statistics, 67(3), Kanas, A. (2006). Purchasing power parity and Markov regime switching. Journal of Money, Credit and Banking, 38, Kanas, A. and Genius, M. (2005). Regime (non)stationarity in the US/UK real exchange rate. Economics Letters, 87, Kim, C. J. (1993). Unobserved component time series models with Markov-switching heteroscedasticity: Changes in regime and the link between inflation and inflation uncertainty. Journal of Business and Economic Statistics, 11, Kim, C. J. and Nelson, C. R. (1999). State-Space Models with Regime Switching, MIT Press. Lee, H.-Y. and Wu, J.-L. (2001). Mean reversion of inflation rates: Evidence from 13 OECD countries. Journal of Macroeconomics, 23, Malliaropulos, D. (2000). A note on nonstationarity, structural breaks, and the Fisher effect. Journal of Banking and Finance, 24,

12 12 SHYH-WEI CHEN AND CHUNG-HUA SHEN Nelson, C, R. and Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics, 10, Ng, S. and Perron, P. (2001). Lag length selection amd the construction of unit root tests with good size and power. Econometrica, 69, Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis. Econometrica, 57, Philips, F. H., Marius, O., and Charles, S. B. (1999). Long memory and level shifts: Re-analyzing inflation rates. Empirical Economics, 24, Rapach, D. E. and Weber, C. E. (2004). Are real interest rates really nonstationary. New evidence from tests with good size and power? Journal of Macroeconomics, 26, Rose, A. K. (1988). Is the real interest rate stable? Journal of Finance, 43, Westerlund, J. (2005). A panel unit root test with multiple endogenous breaks, Technical Report, Lund University. Westerlund, J. and Basher, S. (in press) Is there really a unit root in the inflation rate? More evidence from panel data models. Applied Economics Letters. [ Received September 2008; accepted December 2008.]

13 Table 1 Summary Statistics for inflation rates Series Mean Std Error Minimum Maximum Skewness Excess Kurtosis Normality Canada [0.000] [0.696] [0.000] France [0.000] [0.000] [0.000] Germany [0.000] [0.385] [0.007] Italy [0.000] [0.000] [0.000] Japan [0.000] [0.000] [0.000] UK [0.000] [0.000] [0.000] US [0.000] [0.000] [0.000] Figures in square brackets are p-values. TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES 13

14 Table 2 Testing for the Markov Switching in the ADF regression 14 SHYH-WEI CHEN AND CHUNG-HUA SHEN Country Sample ADF statistic LL(H 0 ) LL(H A ) LR(H 0 H A ) Canada 1955Q1 2007Q *** France 1955Q1 2007Q *** Germany 1965Q1 2007Q *** Italy 1955Q1 2007Q *** Japan 1955Q1 2007Q *** UK 1955Q1 2007Q *** USA 1955Q1 2007Q *** The estimated model under the H 0 is q t = a + bq t 1 + p k=1 γ k q t k + u t. The estimated model under the H A is q t = a(s t ) + b(s t )q t 1 + p k=1 γ k(s t ) q t k + u t. LL denotes the log-likelihood value. LR denotes the likelihood ratio test. *, ** and *** denote significance at the 10%, 5% and 1% levels, respectively.

15 ADF statistic Table 3 The Markov Switching ADF Unit Root Test Average regime duration Country Regime 1 Regime 2 σ 1 σ 2 p 11 p 22 Regime 1 Regime 2 Canada ** 2.133** 1.988** 0.993*** 0.981*** [0.129] [0.038] (0.184) (0.139) (0.007) (0.015) France *** 3.932** 1.362** 0.986*** 0.981*** [0.945] [0.000] (0.317) (0.089) (0.009) (0.014) Germany * 2.302** 1.892** 0.982*** 0.958*** [0.190] [0.057] (0.212) (0.126) (0.013) (0.028) Italy ** 1.685** 0.985*** 0.953*** [0.509] [0.424] (0.415) (0.120) (0.011) (0.030) Japan *** 4.697** 2.153** 0.994*** 0.994*** [0.850] [0.000] (0.330) (0.148) (0.006) (0.006) UK ** 5.753** 2.689** 0.982*** 0.956*** [0.284] [0.021] (0.553) (0.187) (0.013) (0.034) USA 2.481* ** 1.360** 0.986*** 0.974*** [0.061] [0.743] (0.245) (0.095) (0.010) (0.021) *, ** and *** denote significance at the 10%, 5% and 1% levels, respectively. Figures in parentheses are standard errors. Figures in square brackets are simulated p-values of the unit root tests. TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES 15

16 16 SHYH-WEI CHEN AND CHUNG-HUA SHEN 5.0 Log of Price Index 5.0 Log of Price Index CANADA Inflation FRANCE Inflation 1.00 Filter Probability 1.0 Filter Probability Figure 1 Time series plot of log of price index, inflation rate and filtered probabilities for the case of Canada and France, respectively.

17 TESTING REGIME NON-STATIONARITY OF THE G-7 INFLATION RATES Log of Price Index 5.0 Log of Price Index GERMANY Inflation 30 ITALY Inflation Filter Probability 1.0 Filter Probability Figure 2 Time series plot of log of price index, inflation rate and filtered probabilities for the case of Germany and Italy, respectively.

18 18 SHYH-WEI CHEN AND CHUNG-HUA SHEN Log of Price Index Log of Price Index Log of Price Index JAPAN Inflation UK Inflation US Inflation 1.0 Filter Probability 1.0 Filter Probability 1.0 Filter Probability Figure 3 Time series plot of log of price index, inflation rate and filtered probabilities for the case of Japan, the UK and the USA, respectively.