Structural breaks and their trace in the memory

Int. Fin. Markets, Inst. and Money 14 (2004) 117–134

Structural breaks and their trace in the memory Inflation rate series in the long-run Mar´ıa Dolores Gadea∗ , Marcela Sabaté, José Mar´ıa Serrano Department of Applied Economics, Facultad de Económicas, University of Zaragoza, Gran V´ıa 4, 50005 Zaragoza, Spain Received 1 October 2002; accepted 8 May 2003

Abstract The aim of this paper is to illustrate the risks of neglecting the potential presence of structural changes in economic series when estimating their long memory parameters. To that end, our empirical exercise is carried out for the inflation series of the UK, Italy and Spain during the period of 1874–1998. The persistence of the inflation rates is analysed using an AutoRegressive Fractionally Integrated Moving Average (ARFIMA) model, and we find that the memory parameter is significantly reduced when structural changes are taken into account. Having identified these changes in an endogenous way, the fact that they correspond to war time eras, energy shocks or changes in monetary policy, allows us to think in terms of the probable existence of such economic structural changes. Thus, it seems plausible that these events have altered the mean of the process in the three series, introducing an upward bias in the estimated memory parameters. © 2003 Elsevier B.V. All rights reserved. JEL classification: C22; E31 Keywords: Inflation rate; Fractional integration; Structural breaks

1. Introduction Whilst the integration order of the inflation series has been the subject of extensive study in applied economics during recent years, the results that have emerged from the analysis are not clear, with inflation being considered as I(1) or I(0) according to the country or period chosen and to the type of test employed (Hassler and Worlters, 1995). This ambiguity ∗

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could explain the success of the AutoRegressive Fractionally Integrated Moving Average (ARFIMA) models—to the extent that these admit intermediate situations between shocks of a permanent or transitory character—when seeking to describe the behaviour of inflation rates. Their main advantage is that they allow us to analyse the long-run behaviour of the series with their own memory parameter, and without the estimation being contaminated by the short-run parameters that are habitually used in other measures of persistence (Baillie et al., 1992, 1996; Hassler and Worlters, 1995; Baum et al., 1999). However, recent contributions have shown that fractional integration can be the equivalent of structural breaks and that the tests do not distinguish between long memory and changes in the mean. Consequently, if a series registered any genuine breaks and these were not taken into account, the results of its analysis would be biased towards fractional alternatives of a spurious type, in the same way that a series with genuine long memory properties could be wrongly characterized as a process with apparent changes in mean. This is a problem that has been recently identified in the literature (Cheung, 1993; Granger, 1966; Granger and Hyung, 1999; Bos et al., 1999; Barkoulas et al., 2001; Diebold and Inoue, 2001; Choi and Zivot, 2002). Nevertheless, although various efforts have been made to obtain tests capable of estimating the memory parameter in the presence of breaks—and thus empirically distinguishing the truth of long memory or/and shifts in the mean—has research yet not been able to provide conclusive results.1 Meanwhile, it would be interesting to offer more evidence on the risk of breaks not being taken into account when estimating the long memory parameter of a series. To that end, we use the UK, Italian and Spanish inflation series covering the years 1874–1998, a type of economic phenomenon and a sufficiently large period for us to suppose the more than likely presence of such breaks. An additional attraction of selecting these series is the debated link between inflation persistence and exchange rate regime. In this regard, Alogoskoufis and Smith (1991), Alogoskoufis (1992) and Obstfeld (1995) have related the monetary autonomy afforded by floating regimes to the longer memory inflation series in these periods. This idea is discussed by Bleaney (1999) and Burdekin and Siklos (1999) on the basis that the greater persistence is due to the presence of structural breaks during the floating periods, which would bias upward their memory parameter. What we wonder in this paper is whether, in the same way that not considering breaks in the inflation series of a certain country can distort the comparison of persistence between periods, neglecting the presence of breaks can equally distort the comparison of the degrees of persistence of inflation series between different countries during the same period. In an attempt to supply answers to these questions, the rest of the paper is organised as follows. In Section 2 we determine the integration order of these series. The memory parameter is estimated in Section 3. Section 4 is devoted to analysing the potential influence of structural breaks, that will be detected both endogenously and separately for each series. Section 5 closes the paper with a review of the main conclusions and a reflection on the implications of considering, or not, the influence of breaks when comparing the degree of inflation persistence between different countries. 1 From amongst the still limited number of papers related to this problem, attention should be drawn to those of Hidalgo and Robinson (1996), Bos et al. (1999), Hsu and Kuan (2000) and the works of Gil-Alana (2000a,b, 2001).

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2. Unit root tests and long memory signs As mentioned earlier, the analysis of the integration order is applied to the UK, Italian and Spanish inflation series covering the years 1874–1998. That is to say, we are considering the period that runs from the liquidation of the Latin Monetary Union and the generalization of the Gold Standard as the international financial system in the last quarter of the 19th century, up to the beginning of the third stage of Economic and Monetary Union (EMU), when the Italian lira and the Spanish peseta lost their domestic exchange rate identity. The selection of such a lengthy period represents an attempt to ensure the potential presence of structural breaks in the series of all three countries, whose currencies, we should add, have not suffered monetary conversions such as those experienced by the German mark and the French franc. The inflation rate is defined as dlpjt = lpjt − lpjt−1 (with j = gb, i, sp), where lpgb, lpi and lpsp, are the respective GDP deflators in logarithms. Please note we have replace deflector by deflators. The series are represented in Fig. 1. In our analysis we have chosen to use a wide range of tests which, for the case of the unit root null hypothesis, include the traditional Dickey–Fuller (DF) and Phillips–Perron (PP) tests, as well as that proposed recently by Perron and Ng (1996, 1998) and Ng and Perron (2001), which offers better qualities of size and power. This latter test (MZt) uses the GLS method of Elliot et al. (1996), which presents high local power when using different specifications of deterministic elements and modifies the PP test by estimating the asymptotic variance on the basis of the autoregressive spectral density function. We have also used 1.00

0.80

0.60

0.40

dlpuk dlpi dlpsp

0.20

87

80

73

66

59

52

45

38

31

24

17

10

03

96

89

82

94 19

19

19

19

19

19

19

19

19

19

19

19

19

19

18

18

18

18

75

0.00

-0.20

-0.40

Fig. 1. Inflation rate evolution. Source: UK and Italy, Mitchell (1998) and OCDE (2000); Spain, Prados de la Escosura (1995) and Instituto Nacional de Estad´ıstica (1985) .

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Table 1 Unit roots and stationarity tests ADF

PP

MZt-GLS

KPSS

UK Model with constant and trend AIC −4.91∗∗ −3.53

−4.95∗∗ −3.53

−4.50∗∗ (1) −3.53

ητ

0.05

Model with constant AIC −4.52∗∗ −3.52

−4.50∗∗ −3.52

−4.11∗∗ (1) −3.53

ηµ

0.80∗∗

Italy Model with constant and trend AIC −4.18∗∗ (3) −1.84

−4.43∗∗ −1.83

−5.16∗∗ (3) −1.85

ητ

0.30∗

Model with constant AIC −4.11∗∗ (3) −1.85

−4.40∗∗ −1.84

−5.14∗∗ (3) −1.85

ηµ

0.31

Spain Model with constant and trend AIC −4.39∗∗ (1) −2.97

−6.96∗∗ −2.94

−3.77∗∗ (1) −2.98

ητ

0.10

−5.89∗ −2.88

−3.37∗∗ (1) −2.95

ηµ

1.12∗∗

Model with constant AIC −3.73∗∗ (1) −2.95

The number of lags of ADF and MZt-GLS that appear in brackets has been selected in accordance with the method of Ng and Perron (2001). In the PP test the Bartlett’s window has been used as a kernel estimator, choosing the lag truncation parameter li = [i(T/100)]1/4 with i = 4. In the KPSS test the null hypothesis is stationarity. The critical values for ηµ are 0.463 at the 5% level and 0.739 at the 1% level; for ητ they are 0.146 and 0.216, respectively. We have used the Bartlett’s window with l = 4. ∗ Significant at the 5% level. ∗∗ Significant at the 1% level.

the Kwiatlowski et al. test (KPSS), whose null hypothesis is of stationarity.2 The possible existence of long memory in the series has also been explored through the graphic analysis of the autocorrelation function and of the periodogram, and finally through the application of the Lobato and Robinson (1998) test for fractional alternatives, a test of the Lagrange multiplier (LM-LR) type, which under the null hypothesis, d = 0, is distributed as an χ2 (1). The first results are set out in Table 1. We can note that the unit root tests (DF, PP and MZt-GLS) clearly reject the null hypothesis with respect to all three countries and, in accordance with the AIC criterion, select the model with trend for Spain and UK, and with drift for Italy. As regards the KPSS test, this rejects the null hypothesis of stationarity in the case of the models with constant for the Spanish and UK cases, and with trend for the Italian case. Thus, it is possible for us to accept that inflation in Spain and the UK, is a 2

On the properties of the different unit root and stationarity tests in the presence of fractional integration, see Sowell (1990), Hassler and Worlters (1995) and Lee and Schmidt (1996).

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0,45 0,4 0,35 0,3 0,25 dlpe-hpf

0,2

dlpgb-hpf 0,15

dlpi-hpf

0,1 0,05

03 19 10 19 17 19 24 19 31 19 38 19 45 19 52 19 59 19 66 19 73 19 80 19 87 19 94

89

96

19

18

82

18

18

-0,05

18

75

0

-0,1

Fig. 2. Trend-cycle inflation rate evolution. Note: Hodrick–Prescott filter with smoothing parameter λ = 100.

stationary process around a deterministic trend, while contradictory results are offered in the case of the model with drift. With respect to inflation in Italy, we can appreciate strong evidence of stationarity around a mean, as well as ambiguity with respect to the presence of a unit root when we introduce a trend. Results of this type—when it is not clear whether the inflation rate is a stationary or unit root process—have been interpreted in the literature as an indication that the series possess long memory which can be found at mid-point between I(0) and I(1) (Diebold and Rudebusch, 1989; Baillie et al., 1992, 1996; Lee and Schmidt, 1996). The analysis of the autocorrelation function and spectral density also offers indications in favour of the existence of long memory, particularly in the Spanish case. The sample autocorrelations of dlpsp present a slow rate of decay. The autocorrelations of UK inflation fall at a faster rate, whilst those corresponding to the Italian case demonstrate sinusoidal behaviour that decays slowly.3 As regards the sample periodogram, this always tends to infinity when it draws closer to the zero frequency, thus demonstrating what Granger (1966) describes as the typical spectral shape of many economic time series, indicative of long memory. Furthermore, all three series present a peak, one that is particularly marked in the Italian case, around the frequency w(4) = 0.20268. This corresponds to a period of 3 In fact, the presence of long memory could have been intuitively anticipated from the sight of the wave-like behaviour of the three inflation rates in Fig. 1, a wave-like behaviour that is even more clear in Fig. 2, where the trend-cycle series evolution are represented, reflecting the non-periodical cycle to which the inflation rates seem to adjust their dynamics. It is precisely this graphical evidence that leads Barkoulas et al. (2001) to model the merger and acquisition activity as a process with long-term dependence.

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Table 2 LR non-parametric test m

UK 8.25∗∗

20 30 40

22.37∗∗ 42.79∗∗

Italy 5.88∗∗

21.84∗

42.00∗∗

Spain 20.03∗∗ 41.76∗ 66.81∗∗

Note: We test H0 : d = 0, against the alternative d > 0. The LR test is distributed as a χ2 (1). ∗ Significant at the 5% level. ∗∗ Significant at the 1% level.

approximately 30 years and can be related with the inflationary episodes of the First and Second World Wars and of the 1970s. Finally, the application of the Lobato and Robinson (1998) test confirms the indications of persistence in the inflation series. Again, in all three cases, and particularly so in that of Spain, we reject the I(0) hypothesis against the long memory alternative (Table 2). 3. Estimating the long memory parameter The ARFIMA models were first introduced by Granger and Joyeux (1980) and Hosking (1981) in order to model the strong persistence presented by a significant number of economic series. An ARFIMA model (p, d, q) can be defined as follows: φ(L)(1 − L)d (yt − µ) = θ(L)ut where φ(L) = 1 −

p


φj L j

j=1

and θ(L) = 1 +

q


θj L j

j=1

are polynomials of lags of order p and q, respectively, whose roots lie outside the unit circle and ut is iid (0, σ 2 ). By contrast with a covariance-stationary and short memory process, whose autocorrelation function ∞


|ρ(j)| < ∞

j=−∞

is absolutely summable and decays at an exponential rate ρj ≈ cj (with c constant and |c| < 1), if d > 0 the autocorrelation functions decay at a slower hyperbolic rate ρj ≈ j (2d−1) . This implies that ∞
j=−∞

|ρ(j)| = ∞

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and the spectral density function will not be bounded in the zero frequency. If 0 < d < 0.5 the series is stationary, with finite variance and long memory; if 0.5 ≤ d < 1 the series is not stationary, with infinite variance and permanent memory, but registering a reversion mean; finally, if d ≤ 1 the series does not revert to its mean.4 From amongst the different methods used to estimate ARFIMA models, we have opted to use that of Geweke and Porter-Hudak (GPH) (1983), of the semiparametric type in the frequency domain, as well as the Conditional Sum Squares (CSS) estimator proposed by Chung and Baillie (1993) and Chung et al. (1992, 1996), based on the parametric approach and the time domain of Sowell (1992b). The use of both methods will allow us to compare the results in terms of robustness and efficiency.5 The first method, probably the most widely used in the empirical literature, is based on a regression of the periodogram in a band of low frequency ordinates near zero. Given a (1 − L)d Yt = ut process, where ut is stationary with spectral density function fu (w) continuous and bounded in the zero frequency, the memory parameter d can easily be estimated using ordinary least squares (OLS) as the negative slope of a regression of the following expression:  w   j ln {I(wj )} = β0 − β1 ln 4sin2 + ηj 2 where the spectral density function has been substituted by the sample periodogram, evaluated in a harmonic frequencies band wj = 2πj/T , j = 1, . . . , m close to 0, where m = g(T).6 One advantage of this estimator is that its properties do not, in principle, depend on the correct specification of the short-run parameters, with Geweke and Porter-Hudak (1983) maintaining that, under certain specific conditions of the m = g(T) function, the OLS estimator of d is consistent and robust in the presence of non-normality conditions.7 However, it can give rise to problems in the case of strong autocorrelation in ut . In this regard, Agiakloglou et al. (1992) have demonstrated that when Yt includes high values of the autoregressive or moving average parameters, the OLS estimator of d can present a significant bias that is not independent of the number of frequencies selected.8 The second method, CSS, proposed by Chung and Baillie (1993) and Chung et al. (1992, 1996), offers an estimator that is asymptotically equivalent to MLE and has the advantage of easier programming. These authors start from a very general ARFIMA(p, d, q)–GARCH(P, 4

A complete review of the concepts of fractional integration in economic series can be found in Baillie (1996). Other alternatives would be the Gaussian estimator of Robinson (1995) or the extension of the Dickey–Fuller test, recently carried out by Dolado et al. (2003). 6 In this way, the frequency 0 where the parameter d is not defined—it would be infinite—as well as the medium and low frequencies that might contaminate the result, are eliminated. 7 Here, Geweke and Porter-Hudak (1983) and Sowell (1992a) recommend choosing a reduced memories band when the estimator is unstable, in such a way that the selected m should move in the inverse direction of the short-run dependency of the data. However, Hassler and Worlters (1995) advise that the range of m to be modified until it is approximated to T/2, in order to evaluate the bias and stability of the estimation. 8 A number of Monte Carlo studies for finite samples, such as those of Cheung (1993) and Smith et al. (1997), have shown that the GPH method is sensitive to large-sized AR and MA parameters. 5

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Table 3 Estimation of the long memory parameter d (GPH method) τ

UK

0.5 0.6 0.7 0.8

Italy

Spain

d

t

d

t

d

t

0.37 0.42 0.56 0.72

0.33 (3.03) 0.27 (5.28) 0.44 (7.33) 0.79 (8.98)

0.21 0.39 0.67 0.64

1.00 (0.99) 0.98 (3.10) 1.52 (4.77) 1.17 (6.65)

0.66 0.56 0.54 0.63

2.07 (2.16) 1.41 (2.57) 1.36 (4.07) 1.77 (5.26)

Note: Tτ is the frequencies band used in the log-periodogram regression. The t statistics estimated with the theoretical variance π2 /6 appear in brackets.

Q) model: φ(L)(1 − L)d (yt − µ − δσt ) = θ(L)εt εt ≈ D(0, σt2 ) β(L)σt2 = ω + α(L)ε2t where the Granger and Joyeux (1980) ARFIMA model with δ = 0 has been extended in order to allow the volatility of the inflation to influence its mean. The variance has time-dependent heteroskedasticity following the GARCH(P, Q) model of Engle (1982) and Bollerslev (1986), and it is assumed that the innovations follow a density function D, which can be normal or Student’s t-test. Cheung and Diebold (1994) for the ARFIMA(0, d, 0) case and Smith et al. (1997) for the ARFIMA(p, d, q) case prove that the procedure is satisfactory, with the main biases appearing when the mean is unknown and is estimated jointly with the other parameters.9 The results of estimating the memory parameter by way of the GPH method are presented in Table 3, whilst those derived from the CSS approximated maximum likelihood are set out in Table 4. In the first case, the estimation has been carried out for Tτ , with τ = 0.5, 0.6, 0.7 and 0.8. We can note that for the case of Spanish inflation we obtain estimated parameters (around 0.5–0.6), that are quite robust to the choice of the number of frequencies. However, both UK and Italian inflation demonstrate a marked sensitivity. Undoubtedly, this result could be biased by the high autoregressive parameters (above 0.7) that are identified in the ARMA model and which, in the Spanish case, could have been nullified by the negative MA component. Furthermore, the existence of a long cycle around j = 4 in the UK case and, above all, in that of Italy, could also contaminate the results.10 In this situation, the selection of a low number of frequencies would reduce the bias although, given the small number 9 Better results would be offered by the median, because of its lower sensitivity to the presence of outliers. Moreover, Smith et al. (1997) find that the bias is always negative and, furthermore, increases when the MA coefficient is close to −1, although it does not appear to be related with the AR parameter. In this sense, it is shown that the SBIC (which tends to select the most parsimonious model) would be the best criterion for choosing the ARFIMA specification, given that increasing the number of parameters could significantly increase the bias. 10 The elimination of the cycle suggested by Sowell (1992a) requires that we have a much higher number of observations. In the case of Spain, the peak in the frequency w(j = 4) is less marked. A possible explanation for this is the extension of the inflationary cycle associated with the Civil War and post-Civil War era, capable of diluting that of the 1970s. Furthermore, for the case of Italy, we can note a small cycle for w(j = 15).

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Table 4 Estimation of the ARFIMA model with approximate maximum likelihood UK dˆ µ ˆ φˆ θˆ ωˆ αˆ βˆ υˆ Q(10) m3 m4 L SBIC

Italy

Spain

0.40 (1.73) 0.05 (1.01)

0.38 (4.00) 0.02 (1.02)

0.54 (3.86) 0.08 (0.56)

0.26 (2.34) −0.00 (−0.04)

0.35 (1.88) 0.00 (2.91)

0.34 (2.31) 0.00 (1.68) 1.00 (1.63) 0.14 (0.84) 2.76 (4.71)

0.28 (1.84) 0.01 (2.83)

0.42 (3.91) 0.00 (1.21) 0.82 (2.47) 0.45 (3.80) 3.96 (3.13)

3.35 −1.17 15.14 210.29 200.65

4.08 0.88 9.13 258.02 238.74

7.33 1.12 15.81 108.86 99.22

16.18 1.14 5.66 167.33 148.05

0.50 (4.11) 0.04 (0.89) −0.51 (−2.63) 0.38 (1.83) 0.00 (4.66)

7.81 −0.42 6.55 176.96 164.91

0.42 (5.23) 0.04 (1.27)

0.00 (4.65)

13.03 −0.39 6.58 176.10 168.87

Note: Q, Ljung-Box; SBIC, Schwarz Bayesian Information Criterion. We use the 5 or 10 first observations to initialization. In the GARCH models we suppose that the conditional density function follows a Student’s t-test given the high kurtosis.

of resulting observations, at the cost of also reducing the precision of the estimation.11 An intermediate position would appear to be more reasonable, thereby obtaining a dˆ of around 0.4 for both UK and Italian inflation. We encounter similar problems when applying the CSS method for various ARFIMA(p, d, q)–GARCH(P, Q) models. Thus, whilst in the Spanish case the estimations of the parameter d (around 0.5) are very robust with respect to different ARFIMA specifications, they vary significantly in the UK case and to an extreme extent in that of Italy.12 The selection of the model has been carried out in accordance with the significance of the parameters, the SBIC and also the Ljung-Box, which guarantees that the model captures the serial autocorrelation of the residuals. Applying these criteria, we find that Spanish inflation could be modelled with an ARFIMA(1, 0.50, 1) or (0, 0.42, 0) and UK inflation with an ARFIMA(0, 0.40, 1) or an (0, 0.38, 1)–GARCH(1, 0). Such results are similar to those obtained for the Spanish case with the GPH, as well as to the UK case when the number of frequencies chosen is τ = 0.6. When considering Italian inflation, the selected models, ARFIMA(0, 0.54, 1) and (0, 0.26, 1)–GARCH(1, 1), maintain a very broad range over the value of d, similar to that obtained by the GPH method if the frequencies band is extended. We can also 11

The results are especially lacking in robustness for the Italian case. Undoubtedly, the hyperinflationadjustment of the 1940s is giving rise to an excessive cycle in a low frequency that contaminates the long-run parameter. This impression is confirmed if we re-estimate the Italian model considering the years of hyperinflation (1936–1947) as missing data; then, the parameter stabilises slightly above 0.50 for any value of t. Another possibility is that we are faced with what Cati et al. (1999) describe as inliers, that is to say, large scale interventions in the contrary direction to the stochastic behaviour of the series which can artificially reduce the degree of persistence, up to the point of nullifying it. For this reason, the Italian inflation series would exhibit a very low d by considering a reduced number of frequencies. 12 As was the case with the GPH method, the estimation of the d parameter is much less sensitive to the specification of the model if the years 1936–1947 are eliminated, with d being situated slightly above 0.50.

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note a process whereby the memory parameter is substituted by large-sized AR elements (above 0.8) leading to over-differentiation. Similarly, we can observe how the introduction of conditional heteroskedasticity in the variance of the errors (very significant in the Italian case) reduces the value of d.13 Furthermore, if we substitute the estimation of the mean estimated in the model by the sample median, the results hardly change in the case of Spain and the UK. However, in the Italian case, the parameter d increases to 0.72 in the model with GARCH elements, suggesting the existence of some form of relationship between the mean and the variance of the inflation rate.14 With the aim of better guaranteeing the robustness of the results, we have filtered the series using the binomial expansion of the operator (1 − L)d with values of d of between 0.3 and 0.5, before going on to apply the GPH. The results demonstrate that in the case of Spanish inflation the series is slightly over-differentiated with d = 0.5, and even shows a certain degree of persistence with d = 0.4. UK inflation is clearly over-differentiated with d = 0.4 and maintains long memory with d = 0.3, whilst, the Italian series follows the same pattern as in the UK case, although with a lower level of over-differentiation for d = 0.4. In summary, from the results of the two methods considered, we could conclude that Spanish inflation exhibits a higher degree of persistence (the shocks are of a longer duration), with a value of d close to 0.5, whilst this value is below 0.4 in the UK case and slightly above it in that of Italy. In this sense, the UK parameter would be located at an intermediate point between those obtained in other studies (between 0.2 and 0.5) for post-Second World War periods and from the 1970s onwards. For its part, the Italian parameter is very similar (around 0.5) with respect to both periods. As regards the Spanish parameter, this reaches a higher value than that estimated for the post-Civil War period (0.3–0.4).15

4. Long memory or structural breaks? In the above section, we have estimated the fractional integration parameter of the UK, Italian and Spanish inflation rate series using two alternative methods, finding that all three series present long memory, with this being more marked in the case of Spain. We should nevertheless recall that this result can be distorted if we do not consider the possibility of structural breaks and they really exist; with the latter being very likely when speaking about inflation. Thus, the next step will be to test, in a separated and endogenous way, for the 13 However, the results do not change substantially if we substitute the assumption of likelihood based on a Student’s t-test, more realistic given the high kurtosis, for that of a normal distribution which, in principle, would give consistent estimators even though the assumption of the distribution was incorrect. The estimators based on the Student’s t-test would only be consistent if, furthermore, the conditional distribution was really t, in which case they would also be efficient (Fiorentini et al., 2002). 14 This hypothesis has been tested by Baillie et al. (1992, 1996) for the post-Second World War era with negative results. However, this conclusion can be modified by introducing the period of hyperinflation and abrupt stabilisation through which Italy passed between 1936 and 1947. Indeed, Italy is the country that presents the highest variance between 1874 and 1998, as well as the greatest difference between the sample mean and sample average. 15 See Hassler and Worlters (1995), Baillie et al. (1996) and Delgado and Robinson (1994), respectively.

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possible presence of changes in each inflation series. To that end, we have used the method proposed by Bai and Perron (BP) (1998a,b) for multiple structural breaks, which is based on the principle of global minimizers of the sum of squared residuals and is capable of consistently determining the number of break points over all possible partitions.16 Using this method, Bai and Perron (1998a,b) carry out three types of tests. The first, supFT (k), tests the null that there are no breaks (m = 0), as against the alternative that there are m = k breaks. The second, supFT (l + 1/l), tests the null hypothesis of the existence of a number l of breaks, as against the alternative of l + 1 changes. Finally, the so-called “double maximums” tests, Dmax and WDmax , test the null of the non-existence of structural breaks, as compared to the alternative of an unknown number of breaks.17 Moreover, the authors propose a double procedure to select the number of breaks: the use of the SBIC information criterion, which is posed as a model selection problem and a sequential method based on the application of supFT (l + 1/l).18 Thus, using the various tests and procedures outlined above as a basis, it is necessary to define a strategy in order to select the number and location of the breaks. In this regard, we take into account that BP suggest starting with VDmax and WDmax tests to see if at least one break exists. The procedure is applied using a constant as regressor zt = {1} and xt = {0}, in such a way that consideration is given to sharp changes in the average level of the inflation series. Account is also taken of the possible serial correlation of the residuals, which is corrected by way of non-parametric adjustments. Furthermore, allowance is made for heterogeneity in the data and for errors between the different segments delimited by the breaks. We consider a maximum number of eight breaks which, in accordance with the sample size T = 124, supposes a trimming ε = 0.10. Under these assumptions, the results of applying the multiple breaks technique to the UK, Italian and Spanish inflation series, together with the specifications employed, are set-out in Table 5. In the UK series, the supFT (k) appears as highly significant for the eight possible break points, whilst this is not the case for any of the supFT (l + 1/l). However, both the Dmax and the WDmax confirm the existence of breaks and the information criterion selects five break points corresponding to six inflation regimes, although the sequential procedure only detects one. Similarly, for the Spanish case, six break points (seven regimes) are selected in accordance with supFT (k), the Dmax , WDmax and the SBIC. As regards the Italian series, the information criterion selects two break points, although the supFT (k) is maximised for 16

They consider m breaks (m + 1 regimes) in a general model of the type: yt = xt β + zt δj + ut

where yt is the dependent variable; xt (p × 1) and zt (q × 1) are vectors of independent variables, where the former is univariate and where the latter can change, respectively; β and δj (j = 1, . . . , m + 1) are the corresponding vectors of coefficients; and Ti , . . . , Tm are the break points treated endogenously in the model. 17 In principle, these tests and procedures achieve a consistent estimation of breaks, still if the simulations made by Bai and Perron (1998b) and Bai (1997) illustrate how the power of the tests and procedures vary with the type and size of the breaks, with their sequence and with the distribution and characteristics of the errors. All these aspects are considered in our analysis. 18 This method is proposed by Bai (1997), who shows that the asymptotic distribution of this test is the same when using the global minimisation procedure or the sequential one at a time procedure. However, in finite samples this latter method tends to underestimate the number of breaks.

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Table 5 Multiple structural breaks in the inflation rates (Bai–Perron method) UK

Italy

Spain

SupFT : no breaks vs. m = k breaks k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8

5.30∗∗ 9.62∗∗ 6.49∗∗ 11.94∗∗ 15.94∗∗ 15.95∗∗ 14.25∗∗ 12.01∗∗

7.15∗ 2.09 2.95 2.36 3.84 50.79∗∗ 43.07∗∗ 37.64∗∗

28.46∗∗ 21.94∗∗ 18.26∗∗ 17.90∗∗ 23.35∗∗ 24.23∗∗ 21.29∗∗ 19.32∗∗

No breaks vs. a known number of breaks Dmax WDmax at 5%

15.95∗∗ 30.65∗∗

28.46∗∗ 49.10∗∗

28.46∗∗ 49.10∗∗

SupFT : l breaks vs. l + 1 breaks (SupFT (l + 1/l)) l=1 l=2 l=3 l=4 l=5 l=6 l=7

1.24 2.89 9.20 12.08† 9.20 0.85 0.00

8.81† 0.83 8.81 3.20 53.65∗∗ 3.20 1.11

6.61 6.61 5.16 6.61 3.89 5.16 0.00

Selection with the sequential method

1

Selection with the SBIC information criterion (SBIC) k=0 −5.64 k=1 −5.78 k=2 −5.80 k=3 −5.96 k=4 −5.96 k=5 −6.09 k=6 −6.03 k=7 −5.96 k=8 −5.88

0

0

−3.85 −3.89 −4.16 −4.12 −4.13 −4.09 −4.09 −3.98 −3.91

−5.36 −5.74 −5.76 −5.73 −5.79 −5.76 −5.82 −5.76 −5.59

Note: The critical value appears tabulated in Bai and Perron (1998a,b). Changes in the mean are tested selecting a trimming ε = 0.10 with a maximum number of eight structural breaks. Serial correlations in the errors is allowed for. The consistent covariance matrix is constructed using the Andrews (1991) method. † Significant at the 10% level. ∗ Significant at the 5% level. ∗∗ Significant at the 1% level.

k = 6 and the supFT (l+1/l) finds the inclusion of a sixth break point to be highly significant. The sequential procedure does not select any break, which should come as no surprise given the low significance of the first supFT ; however, both the Dmax and the WDmax are highly significant.19 The scale of the inflation reached in the period 1936–1947 (delimited with 19

When the breaks are difficult to detect, for example when there is a marked increase in the mean and this returns rapidly to its earlier level, the power of the supF(1) and of the sequential procedure is very low. In this

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Table 6 Inflation chronology Period

UK

Italy

Spain

19th century First World War Interwar period Second and post-Second World War period 1950s and 1960s The crisis of the 1970s Adjustment and nominal convergence

1875–1907 1908–1919 1920–1933 1934–1969

−0.004 0.072 −0.023 0.039

1875–1907 1908–1919 1920–1935 1936–1947

0.002 0.106 0.010 0.334

1875–1907 1908–1919 1920–1935 1936–1950

0.003 0.049 −0.010 0.115

1970–1981 1982–1998

0.124 0.045

1948–1971 1972–1983 1984–1998

0.035 0.154 0.059

1951–1971 1972–1983 1984–1998

0.066 0.138 0.058

the SBIC) can undoubtedly dilute the significance of the other structural changes.20 For this reason, in the case of Italy, we have chosen to maintain the six breaks indicated by the supFT (l + 1/l) and the supFT (k). The periods finally selected are presented in a synthetic form for the three countries in Table 6.21 We can first note how the chronology of the changes that took place in the behaviour of inflation is very similar in all three. This temporal coincidence of breaks across countries supports their likely existence; a likelihood reinforced by the fact that the change points correspond to the war time eras, energy shocks and regime shifts in monetary policy that have taken place between 1874 and 1998. Thus, over this whole period, the changes in the mean inflation rates appear to make economic sense. First, during the 19th century, prices enjoyed a long period of stability in the three countries in question, with an average rate of inflation of around zero. This era was brought to an end by the outbreak of the First World War, which triggered the first inflationary period, especially in the UK, where the growth in domestic prices averaged 7%. There followed a number of years of adjustment, until the mid-1930s, when a new inflationary period began, and with some disparities appearing between these three countries. Thus, whilst for the UK, the effect of the Second World War was moderate and the period extended until the beginning of the 1970s, with an average level of 4%, Italy suffered a period of hyper-inflation between 1936 and 1947, with an average inflation rate of 30%. However, once this period had passed, and following an intense adjustment, the evolution of prices in Italy was similar to that of the UK until the 1970s, with an average inflation rate of 3.5%. As regards Spanish prices, the inflationary shock caused by the Spanish Civil War (1936–1939) continued during the first decade of Francoist autarky, with an average inflation rate of 11%. The next break comes in 1951 when, in a timid attempt at liberalisation, certain market controls were eliminated; later, in 1957 and 1959, the economic liberalisation programmes were repeated. As a consequence, during the 1950s and 1960s, the Spanish rate reduces case, BP indicate that a useful strategy is to employ the UDmax and the WDmax to determine if at least one break is present, and then to decide with the supFT (l + 1/l). 20 It should be noted that the SBIC cannot take into account heterogeneity between segments. 21 Other works that analyse the presence of structural breaks in the inflation rate series are those of Culver and Papell (1997), Bai and Perron (1998b) and Kim (2000).

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Table 7 Estimation of the parameter d with a change in mean (GPH method) τ

UK

Italy

Spain

0.5 0.6 0.7 0.8

−0.27 (−1.03) −0.04 (−0.25) 0.24 (1.97) 0.49 (4.47)

0.06 (0.31) 0.32 (1.97) 0.57 (4.45) 0.60 (6.62)

0.17 (0.73) 0.19 (1.35) 0.26 (2.51) 0.42 (4.00)

Note: If we introduce the break of 1936–1950, the Spanish values of d would be −0.12 (−0.42), −0.004 (−0.025), 0.11 (0.89) and 0.29 (2.63) for τ = 0.5, 0.6, 0.7 and 0.8, respectively.

to a mean of 6%. The following period, again for all three countries, is delimited by the increases in the oil price of the 1970s, where we can note inflation rates of 12, 15 and 14% for UK, Italy and Spain, respectively. Finally, from the 1980s until the end of the period, the three inflation rates follow the same profile of convergence towards greater price stability. For our purpose, the economic sense of this chronology gives a certain degree of legitimacy to the procedure for taking account of breaks—filtering the series with the mean estimated in each period—and then re-estimating the memory parameter. When implementing this procedure, we will make two assumptions. First, we will consider one break in order to discriminate between the behaviour of inflation before and after the Second World War; secondly, we will include all the changes selected in accordance with the Bai and Perron (1998a,b) method. In both cases, we carry out the estimation of the fractional integration parameter using the GPH method (Tables 7 and 8) and the CSS method (Tables 9 and 10). On the basis of the results, we can clearly appreciate that the inclusion of changes in the mean reduces the memory of the series. When we consider a single break (the Second World War for the UK and Italy, the Civil War for Spain) and with a τ = 0.6 for the GPH method, the UK inflation memory disappears and that of Spain falls to d = 0.19. Italian inflation shows a lower sensitivity in the face of structural change. When including all the breaks, the series appear to be clearly over-differentiated, particularly that of the UK, and to a lesser extent, that of Italy. With the CSS method we find that the results agree, although there is a high sensitivity with respect to the selected specification. We can note a clear trade-off between high and negative θ parameters and the long memory parameter and, above all, between φ coefficients with values estimated around 0.8 and antipersistence.

Table 8 Estimation of the parameter d with all the changes (GPH method) τ

UK

Italy

Spain

0.5 0.6 0.7 0.8

−0.69 (−4.78) −0.35 (−2.82) −0.05 (−0.48) 0.19 (0.11)

−0.57 (−2.63) −0.01 (−0.03) 0.18 (0.17) 0.27 (2.40)

−0.25 (−1.23) −0.14 (−1.08) −0.05 (−0.52) 0.13 (1.22)

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Table 9 Estimation of the parameter d with a change in mean (CSS method) Model ARFIMA(p, d, q)–GARCH(P, Q)

UK

Italy

Spain

(1, d, 1)–(0, 0) (1, d, 0)–(0, 0) (0, d, 1)–(0, 0) (0, d, 0)–(0, 0) (1, d, 1)–(1, 1) (1, d, 0)–(1, 1) (0, d, 1)–(1, 1) (0, d, 0)–(1, 1)

−0.39 (−1.72)p −0.06 (−0.25)p 0.32 (2.89)q 0.62 (2.65) −0.67 (−5.98)pq −0.29 (−1.35)p 0.39 (3.98)q

−0.15 (−0.48)p −0.05 (−0.28)p 0.50 (3.57)q 0.74 (3.46) −0.64 (−5.76)pq −0.54 (−0.40)p 0.63 (3.10) 0.86 (0.83)

−0.31 (−1.72)pq 0.28 (0.55) 0.27 (1.93) 0.30 (2.40)

Note: If we introduce the break of 1936–1950, the Spanish values of d would be 0.57, 0.13, 0.22 and 0.26, respectively. The letters p, q denote the significance of the AR and MA terms.

Table 10 Estimation of the parameter d with all the changes (CSS method) Model ARFIMA(p, d, q)–GARCH(P, Q)

UK

Italy

Spain

(1, d, 1)–(0, 0) (1, d, 0)–(0, 0) (0, d, 1)–(0, 0) (0, d, 0)–(0, 0) (1, d, 1)–(1, 1) (1, d, 0)–(1, 1) (0, d, 1)–(1, 1) (0, d, 0)–(1, 1)

−0.71 (−4.61)pq −0.39 (−2.50)p −0.11 (−0.97)q 0.34 (3.51) −0.78 (3.55)pq −0.38 (−1.39)p 0.09 (0.84)q 0.28 (5.49)

−0.58 (−2.41)pq −0.24 (−1.60)p 0.19 (1.51)q 0.55 (2.51) −0.62 (−2.03)p −0.37 (3.09)p 0.26 (2.19) 0.41 (2.55)

−0.56 (−1.72)p −0.76 (2.99)p −0.11 (−0.67)q 0.01 (0.09)

5. Concluding remarks When considering the UK, Italian and Spanish inflation series, we have found that the fractional integration parameters corresponding to the period 1874–1998 point to long memory, with values of around 0.4 for the UK and Italy and 0.5 for Spain. However, when we include the structural breaks that have been endogenously detected for each series, basically the impact of the World Wars (the Civil War in case of Spain) and the oil shock of the 1970s, the memory is significantly reduced. Thus, even if the empirical exercise does not allow us to accept this decrease in the memory parameter as being true, it at least serves as an illustration—credible to the extent that the chronology of shifts itself appears credible—of how neglecting breaks could exaggerate the memory of a series. Obviously, the greater the intensity and number of breaks detected in a series, the bigger is the potential distortion in the calculation of the persistence. This would explain why if we add the 1950 break to the Spanish series, the only one absent from the UK chronology, the parameter of Spain behaves in a way very similar to that of the UK. In this sense, our study would also seem to illustrate the risks of comparing persistence parameters between countries without considering breaks, and then, on the basis of the differences that emerge in the comparison, drawing conclusions about the different degrees of monetary accommodation

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afforded by different exchange rate regimes across countries. For instance, by ignoring the mean shift experienced by the Spanish inflation rate in 1951, the longer memory of this series could have been wrongly related—if the shift was genuine—to the similarly longer monetary autonomy that prevailed in Spain during the period 1874–1998.22 Against this background, our results would offer new empirical evidence on the risks of confusing structural breaks with persistence, when starting from the comparison of the persistence parameters between countries—as other papers do between periods—conclusions want to be drawn on the national peculiarities of monetary and exchange rate policies.

Acknowledgements The authors would like to thank Jesús Gonzalo, Antonio Montañés, Laura Mayoral, Enrique Sentana, an anonymous reviewer and Ike Mathur for their helpful comments. We are grateful to Chin-Fan Chung and Francisco Goerlich for the computing of different test. This research has received the financial support of a CYCIT (SEC 98-0623) Project and the SEIM research group (SEC 269-62) Programme.

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