Testing for a Break in the Memory Parameter

Transcription

1 Testing for a Break in the Meory Paraeter Fabrizio Iacone Econoics Departent London School of Econoics Houghton Street London WCA AE April 5, 005 Abstract It is often of interest in the applied analysis to assess if changes in policies result in di erences in the persistence of a process too. We propose a test for the break in the eory paraeter of a tie series and a procedure to consistently estiate its location. The perforance of the test in sall saples is studied in a Monte Carlo siulation. We apply the procedure to the in ation of the G7 countries. Keywords: Long eory; persistence; break; Local Whittle estiator. JEL classi cation: C. Introduction The plot of any acroeconoic tie series often shows a tendency for the data to revert to a long ter reference, usually a ean or a trend, and the shocks see to reain in the syste for a relatively long tie. The corresponding saple autocorrelation function does indeed fall to zero, but only very slowly, a feature that translates in the frequency doain in a peak of the periodogras at the lowest frequencies: this was already observed by Granger (966) for econoic tie series, while anecdotal epirical evidence in other disciplines is tracked by Robinson (003) as back as 6. Paraetric odelling this phenoenon was discussed, aong the others, by Mandelbrot and Van Ness (96) and by Adenstedt (974). Despite sall di erences, both the odels generate processes with long eory: ean reversion with hyperbolically decaying autocorrelation function and spectral density with a peak at the lowest frequency, and such that these long ter properties can in both the cases be suarized with one single paraeter. I thank P.M. Robinson and S. Lazarova for any helpful coents. This research was supported by ESRC Grant R and the Denis Sargan Meorial Fund. E-ail address: f.iacone@lse.ac.uk

2 Interest in econoetrics largely arose later, after the works of Dickey and Fuller (979) and of Nelson and Plosser (9), and even then it was con ned to the I (), I (0) cases, especially in the applied works. This very restrictive odel has the advantage of a siple autoregressive representation, and it allows a rough description of persistence because I () processes do not revert to the ean at all. Clearly though no coparison in this case is possible on the basis of d between two ean reverting processes: in any applied analyses the su of the coe cients in the autoregressive speci cation is discussed, but this alternative easure does not see to be satisfactory because any I (0) process is nonetheless characterised by the exponential decline of the autocorrelation function, which is uch faster than the epirical ndings, and changes of the weights of the autoregressive structure do not odify this feature. Long eory odels are interediate between the I () and the I (0) representation, and indeed the structure proposed by Adenstedt is an integrated process with non integer order d (0; ). The paraeter d is then a ore appropriate way to describe persistence, because changes in d have large ipacts in the ediu ter e ects of the innovations in the syste, while the su of the autoregressive coe cients is not inforative because it is still for any nonzero d. Consistent estiation of a possibly non integer eory paraeter followed fro the proof of Hannan (973) for the Whittle pseudo ML estiator, but his arguent could not be directly extended to the properties of the liit distribution; this was treated a few years later by Fox and Taqqu (96) for gaussian processes, and by Giraitis and Surgalis (990) for possibly non noral ones too. Potential nonstationarity ay be addressed tapering the data as in Velasco and Robinson (000), while a tie doain procedure based on the pseudo ML for estiation of the eory paraeter was advocated by Beran (995). A seiparaetric estiator of the eory paraeter is already in Mandelbrot and Wallis (969), who based it on the R/S introduced by Hurst (95) to detect long range dependence. Yet even when the correction introduced by Lo (99) to odel nonparaetrically alternative fors of short range dependence is taken into account, the estiator based on the R/S statistic only provides a broad indication of the e ective degree of persistence. Other seiparaetric estiators could be realised fro the approxiation of the autocovariances at high lags or of the spectral density at low frequencies: focusing on these eleents sees especially sensible when the researcher is only interested in the eory paraeter, because the inference is then conducted only on those aspects that are ore indicative of the persistence and it is also robust to potential isspeci cation of the short ter dynaics. Two particularly popular estiators aong those based on the approxiation of the spectral density at the lowest frequencies are the log-periodogra regression and the local Whittle ones. The design of the log-periodogra regression estiate was already considered by Granger and Joyeux (90) and then discussed by Geweke and Porter- Hudak (93); Robinson (995a) showed its consistency and liit noral distribution under gaussianity and Velasco (000) extended it to ore general linear processes: the log-periodogra regression estiate has a closed for solution and it is very easy and fast to copute. Since it is very intuitive and it does not require the design of a speci c estiation routine, it is uch appreciated by the practitioners, and it is also iportant

3 because it provides an ideal starting value for the optiisation procedure to copute those estiators for which no closed for exists. The local Whittle estiator is based on the seiparaetric analogue of the axiu likelihood principle, at least if the process is gaussian: proposed by Künsch (97) and studied by Robinson (995b), this is ore e cient than the log-periodogra regression one because it has a lower variance. Both the standard unit root tests and the paraetric and seiparaetric estiators of the eory paraeter are designed at least under the assuption of stability of the paraeter of interest: as Lucas (976) rearked, this is actually debatable in tie series odels because of the policy regies shift that took place over the years. A change in persistence should then be regarded as one possible case of odel instability, and indeed the evidence is wide albeit often anedoctal. Ki (000) reports soe exaples in which a shift in d is perceived: the GDP in the US and in several European countries, the federal budget de cit and the in ation rate in the US; he also discussed the iplications for the acroeconoic and the policy analysis; an extensive survey dedicated to changes in the su of the autoregressive coe cients for UK data is in Benati (004). Even ore iportant, fro a policy point of view, is the discussion on changes to the persistence of in ation generated by the progress of the European Monetary Union, and indeed the Eurosyste even established the In ation Persistence Network to study and onitor it. Angeloni et al. (004) concluded their survey associating in ation persistence with unstable onetary regies, and O Reilly and Whelan (004) argued that the introduction of the euro did not per se reduce the persistence of in ationary shocks. Stability in these epirical analyses was usually checked with the test statistic described by Andrews when d = 0 and the focus is on the su of the autoregressive weights, while several tests were designed to detect shifts between d = 0 and d =. For this situation Ki introduced a ratio-based statistic under the null that d = 0 on the whole process, and Ki, Belaire-Franch and Badilli-Aador (00) (KBB hereafter) and Busetti and Taylor (004) proposed soe corrections and further developents. They anyway did not alter the original structure, and Harvey et al. (004) rearked that with that design the case d = on the whole process was confused with the presence of a change in persistence, thus aking the tests of Ki, KBB and Busetti and Taylor not very inforative. Harvey et al. then proposed a di erent test statistic that is characterised by the sae critical values (but not by the sae liit distribution) regardless of whether d = 0 or d =. But if the restriction to integer d sees too strong already in the basic description of the long run properties of a tie series, this is uch ore the case when a potential change of the easure of the persistence is discussed because iportant variations ay in fact be represented with relatively sall changes in d. A siple extension to include fractional d could obviously be designed using the axiu likelihood estiator, either in the Whittle approxiation or in the tie doain, because the supreu of the likelihood ratio or of the Wald test follow the liit distributions of Andrews, irrespective of the true value of d (although tapering ay be necessary for Whittle estiation of large d). Paraetric odels, though, require the speci cation of the short eory dynaics, and are also sensitive to breaks in that coponent only: this of course applies to the standard stability test, in which all the paraeters are discussed, but also to the Dickey 3

4 and Fuller statistic (and then to siilar type of tests), as Haori and Tokihisa (997) showed for the case of a volatility shift. Evidence of shifts in the short ter dynaics only is for exaple in Ki and Nelson (999), who discussed the change in volatility of the GDP, and in Hansen (003), who found instability in the short ter dynaics of the interest rates. We then advocate a seiparaetric procedure, rather than a paraetric one, to test for the stability of the long ter properties of a tie series. Local Whittle Estiation of a Meory Paraeter subect to a Break We foralise our odel introducing the process x t, observed at t = ; :::; n, and we describe it as the su of two unobservable processes x t and x t, such that < t if t x t = : 0 otherwise x t = x t + x t () < t if t >, x t = : 0 otherwise for a constant 0 (0; ), where [:] indicates the integer part of a nuber. The process t is stationary and invertible and has spectral density f () de ned as s = E( t t+s ) = Z f () cos sd, a siilar de nition holds for and f () and, introducing the covariance s = E( t t+s ), for the cross-spectru f (). We consider long eory processes, that we characterise as f () s G d, ; f () s G d when! 0 +. (3) Notice that we do not ake any other assuption on t and on t, thus encopassing several cases: for exaple they ay be independently distributed, but they ay also be generated by the sae process (in which case there is no break). We next introduce the Fourier transfor and the periodogra w x () = p n np t= x t e it I x () = w x () w x ( ) ; in a siilar way we de ne w x () and w x () as the Fourier transfors of x t and of x t, I x () and I x () as the respective periodogras, and nally we introduce the crossperiodogra I x () = w x () w x ( ). Although the de nition is forulated for the generic frequency, in the rest of the analysis we will often consider the particular set = n 4 () (4)

5 for = 0; ; :::; n. Notice that the processes x t, x t are not stationary, and the bound for the expectation of the periodogra provided Robinson (995a) can not be directly applied. Yet we nd that the sae result can be quickly derived (the Assuptions and the proofs are organised in the Appendix): introduce we then have the following # = [n]=n, Lea Let Assuptions A., A. and A.3 hold and introduce v x () = w x () q, v x () = # 0 G d r # 0 w x () G d For a, b f; g, for any positive integer sequence with =n! 0, ln E (v a ( ) v b ( )) = (a = b) + O +, n where (:) is the indicator function, and, with > k, k positive integer, ln E (v a ( ) v b ( k )) = O. k Since I x = I x + Re I x + I x, it follows fro Lea that, for =n! 0, E (I x ( )) = # 0 f ( )+ # 0 f ( )+O (f ( ) + f ( ) + f ( )) ln : (5) when the process x t is not subect to any break, clearly f = f = f and the orders of agnitude in (5) are the sae ones given in Robinson (995a), while if a break on d took place, so that for exaple d > d (6) (which we can assue without loss of generality), the process x t behaves like a long eory process with paraeter d but subect to an unobservable disturbance with eory of order d, and the stochastic order of agnitude of the periodogra, on =n! 0, is E (I x ( )) = # 0 G d + O d ln + d n + d. (7) Notice that the ter d in the bound in (7) is not necessarily negligible: indeed, considering proportional to n for a certain, it is of bigger order than d ln for large enough ( > ( + (d d )) (d d ) ). 5

6 Let s now de ne < x t if t [n] z t = x t, z t () = : 0 otherwise < x t if t > [n], z t () = : 0 otherwise : Fro Lea we already have a bound for E (I z ( )) and for E (I z ( )), E (I z ( )) when # = # 0. Using the sae arguents it is also iediate to show that, under (6) and for =n! 0, if # < # 0, >< >: E (I z ) = and if # > # 0 ; >< >: E (I z ( )) = # G d h + O d ln + i n h #0 # G d + # 0 G d + O d ln + i () n E (I z ( )) = # 0 G d + E (I z ) = # G d h # # 0 G d + O d ln + i n h + O d ln + i : n (9) We analyse the e ects of the break using the Local Whittle estiator as in Robinson (995b): this is coputed iniising the loss function ( ) R(d; ; I) = ln d I( ) d ln( ) (0) = where I ( ) is the periodogra and an user chosen bandwidth paraeter, and we are interested in bd = arg in R(d; ; I z ) bd () = arg in R(d; ; I z ) bd () = arg in R(d; ; I z ). Following Robinson we can then show that Proposition under (6) and assuptions B., B., B.3, B.4, then and = bd! p d () if < 0 : b d ()! p d, b d ()! p d ; () if = 0 : b d ()! p d, b d ()! p d ; (3) if > 0 : b d ()! p d, b d ()! p d ; (4) 6

7 The assuptions are the sae ones as in Robinson, but allowing for a break in the eory paraeter. The proposition states that the Local Whittle estiator converges in probability to the largest of the two eory paraeters, con ring the intuition fro Lea that the process with lower order of integration acts like a disturbance. Notice that this result depends on the fact that the estiator is seiparaetric and only uses the frequencies where the features of the spectral density are doinated by the long ter coponent: should all the frequencies be uses, as in the Whittle estiator, the estiator would converge to a point interediate between d and d. Also notice that here and after we forulate the theore for because # =! when n!. Robinson also showed that the distribution of the Local Whittle estiator is asyptotically noral, a result that we can con r even if a break in d took place. Yet in that case two additional eleents ust be taken into account: the fact that only a fraction of the observations are indexed with the highest eory paraeter, and the potential disturbance caused by the ters with lower d. We then add in the original assuptions the condition that += n! 0 where = d d. Proposition 3 Under (6) and Assuptions B., B., B.3, B.4, B.5, and if < 0 : p 0 ( b d d )! d N(0; ) (5) p ( b d () d )! d N(0; ), p ( 0 )( b d () d )! d N(0; ); (6) if = 0 : p 0 ( b d () d )! d N(0; ), p ( 0 )( b d () d )! d N(0; ); (7) if > 0 : p 0 ( b d () d )! d N(0; ), p ( )( b d () d )! d N(0; ); () This is the sae result of the original paper, siply replacing with 0 (or with, ( 0 ), ( 0 ), ( ) in (6) - () ) to take into account the fact that only a fraction of observations are considered or have the high eory paraeter. To appreciate the eaning of the condition on, notice that, setting in particular proportional to n, as in _ n for soe (0; 4=5), standard asyptotic norality then requires > 4 : (9) the lower order bias ay be reoved if the gap d d is large enough, because the potential bias depends on f and its contribution in the lowest frequencies is less iportant the saller it is with respect to f. Notice that a bigger gap is needed if 7

8 ore frequencies are used, that is, the larger, because the ratio between f and f is saller at higher frequencies, and indeed that if the liit = 4=5 is set, then no cobination of d, d in a closed subset of ( =; =) would be available to eliinate the bias. If on the other hand Assuption B.5 is not et, the lower order bias becoes relevant: Corollary 4 under (6) and Assuptions B., B., B.3, B.4, B.5 n b d d 0 G! p () 0 G ( + ) (0) and if < 0 : n d b 0 () d! p 0 if > 0 : n d b 0 () d! p 0 G G () G G () ( + ) ; () ( + ) : () Corollary 4 is very interesting, not only for its potential application to testing, but also as a guide to interpret the results of the estiation procedure when a break takes place: we can notice that the potential bias gets larger the larger, so it is possible that in sall saples the fact that a large ay deliver n (+=)! 0 ay well be countered by the larger (). This result is only apparently counterintuitive: (+) indeed, setting proportional to n the bias would then be bigger the larger the gap: the case is ruled out by the assuption B.4 but it still provides an interesting benchark because it is the type of result we can expect for the fully paraetric odel. 3 Tests for paraeter instability and detection of a break. Fro Corollary 4 we see that the order of agnitude of the bias decreases with so one could estiate d using di erent bandwidths and then see if the one associated to the sall bandwidth is uch saller. More forally, de ning bd = arg in R (d; ; I z ( )) and then e = p 4 bd b d

9 Corollary 5 setting =, =, for 4=5 > > ax f,4 ( under Assuptions B., B., B.3, B.4, < e!d N (0; ) if = 0 : e! if > 0. ) g and The statistic e can then detect the presence of a break even when is so little that both b d and b d are subect to the lower order bias, because that ter is saller in the rst case. Since anyway this test is only consistent because of the lower order bias, it is not likely to be very good in nite saples. For a ore powerful test, we rst show that Corollary 6 under Assuptions B., B., B.3, B.4, d = d s ( ) bd ( ) d b ( )! N (0; ) ;. + Therefore, considering b () = 4 ( ) bd () d b (), (3) under the sae regularity conditions, b ()! (4) for given. Since, fro Proposition, it also follows that b ()! when > 0 and d > d or when < 0 and d < d, the statistic b () can then detect the presence of a break. This is a siple Wald test, and it only requires the estiation d b () and d b (). By focussing on the seiparaetric design it has the advantage of being robust to any for of instability in the short eory coponent. Knowledge of is required, as indeed in any standard Chow test, so it is probably ore appropriate to analyse change in persistence originated by di erent policy regies because in that case the potential breakpoint is known. If the location of the break is unknown, the test statistic should be analysed in any potential, and we consider in particular and then = n for integer, and [ l ; h ] (0; ), (5) b = sup = 4 ( ) b (). Introducing ) to indicate weak convergence in the Skorohod J topology of D[0,], and B (), [0; ], a brownian otion on [0; ], then 9

10 Corollary 7 setting =, for 4=5 > > 4= (4 + ) and under Assuptions B., B., B.3, B.4, >< >: b ) (B() B()) 4( ) if = 0 b! if > 0.. The liit process is a standardised tied down Bessel process and references for that are already in Andrew s work, where he also discussed what happens when [ l ; h ] = [0; ]. Critical values can be tabulated, and indeed Andrews provided the, while a general forula is in De Long (9). We conclude proposing an estiator of the location of the break when there is indeed one: letting b = arg in bq n () = d b () + ( ) d b () (6) [ l ; h ](0;) Corollary Under Assuptions B., B., B.3, B.4 with d 6= d, b! p 0. 4 A Monte Carlo exercise. We investigate the validity in sall saples of the theoretical results with a little Monte Carlo exercise. We considered the odels: Model (M): no break in d, x t I (0:4); Model (M): no break in d; the variance doubles in the second part of the saple, x t I (0:4); Model 3 (M3): break 0 = =: x t I (0:4), x t I (0); Model 4 (M4): break 0 = =3: x t I (0:4), x t I (0); Model 5 (M5): break 0 = =: x t I (0:), x t I (0). Model is the standard design, and we use it as a benchark. It also provides us with a reference for d b () and d b (), that are not discussed in the original paper of Robinson, and for the statistics based on the, including the ones to test for the presence of the break. Model is presented to verify that changes in the short ter coponent do not a ect the quality of the estiation in sall saples either. Model 3 is the odel with a break: we intend to evaluate the precision of the estiates d b and d b, thus appreciating the sensitivity of the bias to the bandwidth, so we take a 0

11 cobination of, and such that standard liit norality holds for d b but not for bd. We also copare the perforance of the two tests for the break when 0 is unknown; nally we estiate the location of the break using b. Model 4 is analogue to Model 3, we introduced it to observe the sensibility of the bias to the change of the location, while with Model 5 we can observe the if the saller value of the bias or its slower rate of convergence prevails in the sall saple. We generated 64,, 56, 5 and 04 observations, using the Davies and Harte algorith; for each cell we siulated 00 runs. We set = 0:75n 0:79 and = 0:5n 0:49 and estiated d b, d, b d b (=4), d b (=4), d b (3=4), d b (3=4), and for each estiator we coputed the average of the di erence between the estiates and the theoretical liit value, indicating it as "bias" in Table. We also coputed the saple standard deviation as a easure of the dispersion, presenting it in Table, while in Table 3 we report the standard deviation prescribed by the asyptotic theory in Proposition. Coparing the bias and the standard deviation gives a preliinary indication of the the reliability of liit noral approxiation, but we also analyse it as in Table 4 by counting the nuber of occurrences in which the standardised t statistic exceeds the critical value of a two sided 5% test: these t statistics are unfeasible in the case of a break, because its location is actually unknown, but a feasible version could have been derived replacing 0 with b. In the sae way in Table 5 we analyse the liit noral approxiation in Corollary 6 for d b (=4) d b (=4), d b (=4) d b (3=4), d b (3=4) d b (3=4) when there is no break. Finally, in Table 6 we analyse the perforance of the two tests to detect a potential break in [=; 7=] (for the test based on d b d b we use a one sided alternative in order to increase the power); we also present the bias and dispersion of b in the odels with a break. Suarising the possible outcoes, three di erent situations are possible: bd d b d b (=4) d b (=4) d b (3=4) d b (3=4) M A A A A A A M A A A A A A M3 A A B M4 A A B M5 A B A: consistent and asyptotically noral estiation of d = 0:4; B: consistent and asyptotically noral estiation of d = 0; : consistent estiation of d = 0:4 or fo d = 0:, lower order bias. Analysing the results, we nd that when there is no break then d b is ore precise than the other estiators, having siilar bias and a saller saple standard deviation, as it was fair to expect given ore inforation (either in ters of ore frequencies or of ore observations) is used. The dispersion is fairly in line with the theory, except than for d b in the sallest saple, a result that we attribute to the very sall diension of the bandwidth in that case. The standardised t statistics of d, b d b () and d b () are coparable, and exceed the threshold in 0% to 5% of the cases: the precision is apparently increasing with the saple, and we consider these results reliable: this result is particularly interesting for d b (=4) and d b (3=4) because these ones use less

12 observations and were then potentially ore at risk of poor approxiation. Once again bd is quite an exception in the sallest saples, but the precision of the approxiation sees to iprove with larger n. In Table 5 we report the approxiation of d b ( ) bd ( ), the building block of the Andrews type of test statistic: the excessive size if fairly oderate, ranging between an additional 0% in very sall saples to a oderate 5% in the larger saples; nally, in Table 6 we report the size of the tests to detect the presence of a test at an unknown point, and those too range between 5 to 0%, the accuracy increasing with the saple size. These results are copletely atched by those obtained under the speci cation of Model, thus verifying that the seiparaetric speci cation has the great advantage of being robust to short ter instability. In Model 3 and 4, only d b (=4) and d b (3=4) are una ected by the break, the forer estiating d = 0:4, the latter d = 0. In the other cases we can appreciate the e ect of the instability of d on the estiates: the lower order bias is ore iportant the saller is the saple, the larger the percentage of frequencies with d = 0 copared to d = 0:4 and the bigger the bandwidth. Notice in particular that the lower order bias has still a strong e ect on d b even with 04 observations, and its reduction proceeds only rather slowly when n increases. When ust the lowest frequencies are used, as in d b, the bias is fairly negligible, but only if the gap d d is large enough: in the results for Model 5 the conditions for Proposition 3 are not et even by the sallest bandwith, and it is interesting to observe that the bias is apparently larger than in Model 3. This di erent patterns directly a ect the liit noral approxiation (which, according to Proposition 3 and Corollary 4 does not hold for d, b d b (3=4), d b (=4) in Model 3, 4 and 5, and in the latter case not even for d b ): while the accuracy for d b in Model 3 and 4 is iproved with the saple size, the reverse happens for occurrences above the threshold for the reaining ones. The inforation in the gap d b d b anyway is not su cient for test reliable in sall saples: even for 04 observations and a large gap the power is still around 35%, contrary to perforance of the Andrews type of test, which goes fro 40% in the sallest saple to 00% in the largest one. The detection of the presence of the break is of course less satisfactory when is saller, and indeed the power of the test e is so little to ake that statistic virtually useless. We then think that a test based on d b d b should be only preferred in very large saples, when the power can be reliable and the burden to copute the test b ay be excessive. We conclude rearking that b is indeed a satisfactory estiator even in relatively sized saples, as we can see in the last part of table 6. 5 Epirical application and discussion We applied these results to analyse the persistence of onthly in ation in the G7 countries over the years The starting date depended on the availability of data for France on Datastrea: we preferred to have all the saples covering the sae period to ake coparison easier.

13 The plots of in ation are in Figure and Figure. Coentators often describe these data allowing for two regies: a period of higher and ore volatile growth rate of prices in the beginning, usually associated to the oil shocks, and a slow return to a lower and ore stable in ation in the late 0; across countries, the UK and Italy are characterised by the highest volatility, Gerany by the lowest one. We analysed the data estiating d b and d, b coputing the test statistics to detect a break and eventually estiating b. We kept [=; 7=], so we searched for a break between 976 and 000; we also estiated d b (=4), d b (=4), d b (3=4), d b (3=4), where = =4 corresponds to 90 and = 3=4 to 996. Contrary to the Monte Carlo exercise, we set a rather conservative bandwidth, = 0:3n 0:79 and = 0:3n 0:49, in order to keep the in uence of short eory coponents extreely low. Since the estiated order of integration is in soe cases potentially high, we also analysed the rst di erences of the data. All the results are in Table 7. The estiates d b of the orders of integration ranged between 0.66 for France and 0. for Gerany, while d b reached for Italy and France and 0.4 for the UK: we then based the rest of the analysis on the rst di erences for these three countries. The gap d b d b was signi cant at the 5% level for the UK and Italy, and hit the critical value (Pvalue of approxiately 0.055) for France, while the Andrews type of test indicated a break for all the countries except Gerany. The breaks were estiated to be around 90 for the US, the UK and Japan and around 94 for Canada, France and Italy. The pattern of d b (), d b () is very useful to describe the outcoe: d b () decreased with, possibly re ecting a reduction in the persistence over tie (the higher with respect to 0, the larger the lower order bias); the drop in d b () was even larger, as if the change in d rather than the lower order bias caused it, suggesting then that the break took place between 979 and 995 and that the estiates d b (3=4) are also inforative about the order of persistence after the break. We con red this by repeating the analysis on the part of the saple following the break as selected by b (we report the results in Table ): we found no evidence of another break, although a certain instability could be suspected fro d b d b for France and Canada. We concluded the analysis addressing the question of wether the accession to the euro area resulted in a reduction of the in ation persistence: we estiated d b () and d b () for France and Italy in the saples setting in order to test for a break in 999. Notice that the corresponding to January 999 was already in the set of points for which we considered a potential break, but we treated it di erently because in this case 0 was supposed to be known, so the liit distribution is a siple standard noral and the critical value was uch saller. The hypothesis of stability was not reected anyway, thus supporting the conclusion of O Reilly and Whelan that the institution of the Eurosyste is not associated to with a reduction of the persistence in the shocks to in ation. Appendix Assuption A.. For a f; g there exists G a (0; ), d a ( =; =), and 3

14 (0; ] such that f a () = G a da + O da as! 0 +. Assuption A.. In a neighborhood (0; ") of the origin f is di erentiable a = O = O d d as! 0 +. Assuption A.3. Letting R = f () = p f () f (), then for soe (0; ], R () R (0) = O as! 0 +. Proof of Lea. Consider E (I ( )) rst. The expectation is 0 E (I ( )) = E 4@= p n t= = t s e i (t s) n t;s= 0 t e i p n s= 3 s e i sa5 where k is the covariance, and then Z f () e i(t n t;s= s) d e i (t s) which we rewrite as 0 The ter in (A.) is f ( ) f ( ) f ( ) n Z 0 Z 0 (f () f ( )) f ( ) e i( n t;s= )(t s) da e i( )(t s) da : n t;s= ( r) e i( )r d = r Z e i( r n [ 0n] = # 0 f ( ) )r d r Z r e i( )r d (A.) (A.) A = 4

15 where we used R e i( )r d = 0 for any r 6= 0. To show that A. is O ln d we notice that, since the Dirichlet kernel is bounded by O at any nonzero frequency, we can still follow the proof of Robinson (995a). The sae arguent can be applied for E (I ( )). Finally, for E (I ( )), this is 0 E (I ( )) = E 4@= p n which we rewrite as = n Z n t= Z 0 n n n t= t= s=+ n s=+ n t= s= 0 n+ 0 t e i p n t s e i (t s) n s=+ 3 s e i sa5 (f () f ( )) e i( )(t s) d + (A.3) f ( ) e i( )(t s) d (A.4) and notice that (A.4) is 0 because f ( ) P 0 n P n R t= s= 0 n+ ei( )(t s) d = 0 since t > s. The result then follows using P s=u eis = O as for exaple in Robinson and Marinucci (00), Lea 3. and following again the proof in Robinson. Assuption B.. For a f; g, as! 0 + ; f a () G a da where G a (0; ) and d a [ ; ], where = < < < =: Assuption B.. For a f; g, in a neighbourhood (0; ) of the origin, f a () is di erentiable and d d ln f a() = O( )as! 0 + : Assuption B.3. For a f; g, the sequence a t is such that a t = a "a t, t=0 a < t=0 where E("a t F t ) = 0, E("a t F t ) =, a.s., t = 0; ; ::: in which F t is the eld generated by "a s, s t, and there exists a rando variable such that E() < and for all > 0 and soe K > 0, P ("a t > ) KP ( > ). 5

16 Assuption B.4. As n!, + n! 0. Proof of Proposition. Consider P li b d () when < 0 rst. We follow the proof in Robinson (995b) replacing H =, H 0 and G 0 with d, d and #0 # G respectively; we refer to the original article for a de nition of and and of S (d). Introduce g ( ) = #0 # G d, and consider the set rst: we can follow the proof in the original paper up to Robinson s equation (3.3): when indeed = d b ()! if r ( d )+ r Iz ( ) 6 (A.4) r g ( ) is o p (). We then Rewrite (3.4) of Robinson as where I z ( ) g ( ) = g ( ) f ( ) r Iz ( ) g ( ) + f ( ) I z ( ) I " ( ) + I " ( ) (A.5) A #0 # = P l=0 le il ; f () = #0 # f () and I " () = w " () ; < " t if [n] + t < " t = : 0 otherwise, and notice that f () is not actually a spectral density because t is not stationary. The result E I z ( ) g ( ) C = ; :::; for a generic, positive nite constant C still follows using (), so r ( d )+ r g ( ) Iz ( ) 6 r f ( ) g ( ) C ( d ) + r for any > 0. Next rewrite E I z ( ) I " ( ) E I ( ) I " ( ) (A.6) +E Re I ( ) + E I ( ) (A.7) 6

17 where < t if [n] + t < t () = : 0 otherwise (A.) and I () = w () w ( ). The contribution of I ( ) I " ( ) can be discussed as in equation (3.7) fro Robinson, using the sae arguent as in Lea to show that I ( ) I " ( ) = O p f ( ) (ln ( + ) =) = and E ( 6 r r ( d )+ r r ( ) I ( ) I " ( ) ) = o () following the sae steps of Robinson. We are then left with r ( d )+ r 6 f r ( ) ( Re I ( ) + I ( )). Notice fro Lea that r f E I ( ) = O the order of agnitude of the rst ter is = 6 O >< >: r r r O O O ( d )+ r r r ( d )+ r d n! d ln f ( ) E Re I ( ) = r d d d ln! = n n d d ( d )+ n if d d < 0 d d ln if d n d = 0 d if d d > 0 n d ln which are all o p () using d < d and =n! 0 (also notice that d < + =); in a siilar way 6 r r ( d )+ r To deal with the nal contribution, we notice that r f ( ) E I ( ) = O (=n) (d d ). #0 # I " ( ) = #0 # n t [n] " t + #0 # n s6=t [n] [cos (s t) ] " s " t 7

18 and a law of large nuber arguent delivers #0 # n t [n] " t! p 0 while for the second ter 0 [ r 0 n] [cos (s n s6=t s6=t [n] r [cos (s t) ] t) ] " s " t A! = s6=t [n] r [cos (s t) ]! which is exactly the ter in the proof of Robinson so the rest of his arguent applies without odi cations. If is epty, this iplies that d b ()! p d. If is not epty, we have also to show that P inf S (d) 0! 0. Following the proof in Robinson, P inf S (d) 0 P = (a ) d I z ( ) 0! (A.9) with and, Since choosing < d 0 p = exp( >< a = >: p ( d ) p ( d ) p p < ln ) so that p s =e when! :. = (a ) = e ( ( d 0 ) + ) = + = (4e) there is > 0 such that >, (a ) +. = thus strenghtening slightly the original result.

19 We then rewrite the bound in (A.9) as P = = (a ) d I z ( ) 0; (A.0) (a ) d (I ( ) + Re (I ( ))) <! +P =l = (a ) d I z ( ) 0; (A.) (a ) d (I ( ) + Re (I ( ))) Clearly, the probability in (A.0) can be bounded by P P = (a ) d (I ( ) + Re (I ( ))) < = (a + ) d I ( ) + Re (I ( )) >!!! : (A.) which goes to zero because the arguent of (A.) converges in probability to 0. We show this discussing each ter separately: rst, d I ( ) = O (d d ) p = o p () ; d ln Re (I n ( )) = O d d p = o n = while for the two reaining ters, = a p ( = O p p = d d O p n >< d = O p >: O p ( Re I ( )) = d ) d d d ln + ( n n p =p ( d )+ + d d ln if d n d < 0 d ln if d d = 0 d if d d > 0 n n d ln d ) d d n n! d ln 9

20 and = = O p a I ( ) = p = ( p = O p (=n) (d d ) = o p (). d ) d d + n n =p ( p d ) d! d n n by The probability in (A.) then goes to 0. The probability in (A.) can be bounded P = which Robinson showed tends to 0. (a ) d I z ( ) 0 Assuption B.. For a f; g and soe (0; ] f a () G a da ( + O( )) as! 0 + ; where G a (0; ) and d a [ ; ], where = < < < =: and Assuption B.. In a neighbourhood (0; ) of the origin, a() is di erentiable where a() = P l=0 a le il. Assuption for soe nite constants c 3 and c 4. d a() d a() = O as! 0 + B.3. Assuptions B.3 holds and also E("a 3 t F t ) = c 3, E("a 4 t ) = c 4, a.s., t = 0; ; ::: Assuption B.4. As n!, + + ln! 0: n Assuption B.5. As n!, if d 6= d, then letting = d d, += n! 0.!, Proof of Proposition 3. As before, we discuss b d () when < 0 considered the expansion based on the ean value theore R () () d= d e d=d Robinson 0

21 with d e d d b : () d following the sae arguent of the original proof, and where () R p 4 d= d e = = d=d = ln Iz ( ) g ( ) ln. ( + o p ()) (A.3) Decoposing I z ( ) = I ( ) + Re I ( ) + I ( ) as before, we can rewrite (A.9) as = = = I ( ) g ( ) Re I ( ) g ( ) I ( ) g ( ) ( + o p ()) (A.4) ( + o p ()) (A.5) ( + o p ()) (A.6) Making use of (A.5) and of the decoposition following (4.) in Robinson, (A.4) is 0 #0 # t [n] q t + o p () A ( + o p ()) P where q t = " t t s=[n] " sc t s replaces z t in the original proof of Robinson but the rest follows in the sae way so P = t [n] q t converges in distribution to a noral N (0; ( 0 )). The result then holds if the two reaining ters are negligible: using Lea the ter in (A.5) is O p (=n) ln = = = o p (), while the other one is O p (=n), = which is only negligle under Assuption B.5. Proof of Corollary 6. Clearly b (, ) is asyptotically noral, being it the su of two norally distributed rando variables; the variance is V ar b (, ) = V ar bd ( ) + V ar bd ( ) + Cov bd ( ), d b ( ).

22 To copute the last eleent, using the sae decoposition in Proposition 3, = bd ( ) = bd ( ) d d = = o p = [ n] q # t + o p () A ( + o p ()) t o p = n q # t + o p () A ( + o p ()), t [ n]+ then Cov bd ( ), d b ( ) = 0 because and then V ar b (, ) = ( ) = + 4 ( ). Proof of Corollary. The proof follows fro the standard arguent for iplicitly de ned extreu estiates as for exaple in Newey and Mc Fadden (994); as before we only discuss the case d > d. Letting Q 0 () = d + ( ) (d ( < 0 ) + d ( 0 )), clearly 0 = arg in Q 0 (), Q 0 () is lower sei-continuous and [ l ; h ] is a copact set. Fro Proposition we also have that Q b n ( 0 )! p Q 0 ( 0 ), so we only have to show that, for any " > 0, for any [ l ; h ], P bqn () + " < Q 0 ()! 0 when n!. We show this dividing [ l ; h ] in [ l ; 0 ), [ 0 ; h ] and then considering P bqn () + " < Q 0 (), [ l ; 0 ) (A.7) +P bqn () + " < Q 0 (), [ 0 ; h ]. (A.) Notice that Q b n ()+" < Q 0 () when [ l ; 0 ) is bd () d +( ) bd () d ; fro Proposition we can nd < " such that P d b () d >, [ l ; 0 )! 0 and < " such that P d b () d >, [ l ; 0 )! 0, so we can bound (A.7) with P ( ( ) + " < 0)+P d b () d >, [ l ; 0 ) +P d b () d >, [ l ; 0 ) which indeed goes to 0; with a siilar arguent (A.) goes to 0 too.

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