1 Deparmen of Economics Discussion Paper Muliple Srucural Breaks in he Nominal Ineres Rae and Inflaion in Canada and he Unied Saes Frank J. Akins, Universiy of Calgary Preliminary Draf February, 00 Deparmen of Economics Universiy of Calgary Calgary, Albera, Canada TN 1N4 This paper can be downloaded wihou charge from hp://www.econ.ucalgary.ca/research/research.hm
2 MULTIPLE STRUCTURAL BREAKS IN THE NOMINAL INTEREST RATE AND INFLATION IN CANADA AND THE UNITED STATES by Frank J. Akins Deparmen of Economics The Universiy of Calgary Calgary, Albera Canada TN 1N4 Preliminary Draf February, 00 ABSTRACT In his paper I es for muliple srucural breaks in he nominal ineres rae and he inflaion rae using wo mehods. Firs, I apply he sochasic muliple srucural break model, developed by Bai and Perron (1998(a) and (b)). Second, I use he sequenial raio esimaion mehod due o Banerjee, Lumsdaine and Sock (199), Zivo and Andrews (199) and exended by Lumsdaine and Papell (1997). I exend his mehodology furher o he consideraion of hree possible breaks in each series. I use quarerly daa on he Canadian and U.S. 90 day reasury bill raes and CPI inflaion rae over Using he Bai and Perron mehodology I find very lile evidence of mean breaks in he ineres rae series, and evidence of 4 breaks in he Canadian inflaion rae and hree breaks in he U.S. inflaion rae. Using he sequenial raio mehodology, I find ha he daa is consisen wih hree breaks in he mean and he growh rae of he Canadian ineres rae, and wo breaks in he U.S. ineres rae. There appears o be very lile gain in esimaing a breaking rend model over he mean break model for boh inflaion raes.
3 1. Inroducion Since he lae 1980s, here has been a large increase in he applicaion of coinegraion echniques o uncover he relaionship beween nominal ineres raes and inflaion. The ypical requiremen is ha he nominal ineres rae and he inflaion rae each conain a uni roo. If his requiremen is me, hen co-inegraion beween he nominal ineres rae and inflaion implies ha he real ineres rae is I(0), which is heoreically defensible. In an ineresing paper, Crowder (1997) found wo breaks in he Canadian Fisher relaion, in approximaely 1971Q and 198Q. Crowder uncovers he breaks under he mainained hypohesis ha he ineres rae and he inflaion rae each have a uni roo, and he wo variables are co-inegraed. Tha is, he breaks are no direcly aribuable o he sochasic properies of he individual variables, bu raher o changes in he naure of he co-inegraing relaionship beween he variables. Following Perron (1989), i is now well known ha apparen persisence in macroeconomic daa could be he resul of unmodeled srucural breaks in he underlying daa process. Therefore, a series ha appears o be well modeled as an I(1) process, could acually be a saionary process wih one or more srucural breaks. This is he moivaion for a recen paper by Malliaropulous (000), who found evidence consisen wih one break in he mean and rend funcion for he nominal ineres rae and he inflaion rae using U.S. daa from 1960 o Each of hese breaks was found o be in he early 1980s. Hamilon (1988) uses a wo sae Markov model and finds a change in he regime of he U.S. nominal ineres rae beween lae 1979 and lae 198. Garcia and Perron (1996) use Hamilon s mehodology o analyze he quesion of wheher here are breaks in he U.S. real ineres rae series for he U.S. over he period 1961 o Their resuls are consisen wih he real ineres rae obeying hree regimes: ; ; and Their resuls are consisen wih hree regimes for he U.S. inflaion rae: a low sae from ; a high sae beginning in 1973; and a low sae beginning in he early 1980s. In his paper I apply wo classes of ess o uncover srucural breaks in he nominal ineres rae and he inflaion rae for Canada and he Unied Saes over he
4 period Firs, I apply he sochasic muliple srucural break model, developed by Bai and Perron (1998(a) and (b)). Second, I use he sequenial es esimaion mehod due o Banerjee, Lumsdaine and Sock (199), Zivo and Andrews (199) and exended by Lumsdaine and Papell (1997). I exend his mehodology furher o he consideraion of hree possible breaks in each series.
5 . Esimaing Breaks in he Mean of he Series The Bai-Perron (BP) mehodology considers he following muliple srucural break model, wih m breaks (m+1 regimes) (1) y y = x β + z δ = x β + z δ + u + u,, = 1,...,T, = T + 1,... T,... y = x β + z δ + u, = T + 1,... T 1 m+ 1 1 m 1 Where is he observed dependan variable a ime ; x is px1 and z is qx1, and y β and ( j 1,...,m + 1) are he corresponding vecors of coefficiens; and u is he δ j = disurbance erm a ime. The break poins (T,...,T 1 m ) are reaed as unknown, and are esimaed ogeher wih he unknown coefficiens when T observaions are available. In he erminology of BP, his is a parial srucural change model, in he sense ha β does no change, and is effecively esimaed over he enire sample. If β =0, his becomes a pure srucural change model where all coefficiens are subjec o change. In his paper, we search for breaks in he Canada and U.S. nominal ineres rae and inflaion using his mehodology. For each of he individual variables, he model is characerized as: () y = c j + ρy 1 + u = T j 1 + 1,... T j Therefore, he breaks are assumed o be in he consan of he regression, while he auoregressive parameer, ρ, is esimaed over he full sample, based on he opimal pariion. For he pure break model ρ = 0 in equaion (). The procedure for deecing srucural breaks, suggesed by Bai and Perron, is he following. Firs, calculae he UDMAX and WDMAX saisics. These are double maximum ess, where he null hypohesis of no srucural breaks is esed agains he alernaive of an unknown number of breaks. These ess are used o deermine if a leas
6 one srucural break is presen. In addiion, he SupF(0 l) is a series of Wald ess for he hypohesis of 0 breaks vs. l breaks. In his paper he maximum number of breaks (l) is chosen o be 5. If hese ess show evidence of a leas one srucural break, hen he number of breaks can be deermined by he SupF(l+1 l). If his es is significan a he 5 percen level, hen l+1 breaks are chosen. Finally, he number of breaks can also be chosen by he Bayesian Informaion Crieria (BIC). The Canadian and U.S. Inflaion Raes Table 1 shows he resuls of following his procedure for he Canadian and U.S. inflaion raes.
7 Table 1 Srucural Break Tess Canadian and U.S. Inflaion Raes Parial Break Model y c + ρy + u = j 1 Pure Break Model y = c + u j Canadian U.S. Canadian U.S. Inflaion Rae Inflaion Rae Inflaion Rae Inflaion Rae Udmax 9.86* 13.76* 56.63*.7* Wdmax 39.60* 18.11* 88.64* 9.63* SupF(0 1) * 17.08* SupF(0 ) 9.86* 13.77* 56.63* 15.43* SupF(0 3) 7.4* 13.61* 54.05*.7* SupF(0 4) 6.5* 10.68* 49.63* 18.01* SupF(0 5) 1.78* * 16.80* SupF( 1) 7.09* 14.66* 17.58* 1.43* SupF(3 ) 7.77*.30* 35.31* 30.6* SupF(4 3) 13.09* * 7.93 SupF(5 4) BIC * denoes significan a 95% On Table 1, he UDMAX, WDMAX and he SupF(0 l) sugges a leas one 1 break in each series, for boh he pure and parial break models. Given his, he number of breaks can be chosen by he SupF(l l+1) es or he BIC. For he Canadian inflaion rae, boh he SupF(l l+1) and he BIC es choose 4 breaks, and for he U.S. inflaion rae, each of hese ess chooses 3 breaks. The esimaed break daes and sandard errors are presened on Table, and he break daes are illusraed on Figure 1
8 Canadian Inflaion Rae Table Esimaed Mean Break Daes Canada and U.S. Inflaion Rae U.S. Inflaion Rae Pure Break Parial Break Pure Break Parial Break Model Model Model Model c s.e. (0.34) (0.34) (0.30) (0.4) c s.e (0.46) (0.54) (0.40) (0.36) c s.e (0.53) (0.73) (1.1) (0.74) c s.e. (0.3) (0.4) (0.6) (0.8) c s.e (0.9) (0.6) - - ρˆ s.e. - (0.07) - (0.068) T 1 65:04 65:04 65:04 67:01 95% c.i. 6:04-68:03 61:03-69:01 63:04-67:03 65:04-69:04 T 7:04 7:04 73:01 73:01 95% c.i 71:0-73:0 7:01-73:04 67:01-73:0 70:01-73:03 T 3 8:03 8:03 81:03 81:03 95% c.i 81:04-84:01 81:04-83:0 81:01-85:0 80:04-83:0 T 4 91:01 91: % c.i 89:03-9:0 89:04-9:03 - -
9 Figure 1 Esimaed Mean Breaks Canadian and U.S. Inflaion Raes Canadian Inflaion Rae Inflaion (%) U.S. Inflaion Rae Inflaion (%)
10 For he Canadian inflaion rae he four esimaed break daes are idenical for he pure and parial models. The laer hree break daes have sandard errors of -3 years, and do no overlap. However, he sandard error of he firs break is somewha larger, suggesing ha he firs break may no be precisely esimaed. In he U.S. he firs break dae is esimaed o be jus over one year laer in he parial break model. However, i is difficul o disinguish he firs and second breaks in eiher he pure or parial break model for he U.S., as he sandard errors are somewha large. o be For he parial break model, he auoregressive parameer for Canada is esimaed ˆ = ρ, which, given he asympoic sandard error, is very close o 0. The value for he null hypohesis ha his coefficien equals 0 is.016, wih a prob value of Clearly his would rejec he uni roo hypohesis. 1 This can be compared o he auoregressive parameer wih no breaks of ˆ = ρ. The auoregressive parameer for he U.S. is ˆ = ρ 0.45, wih an asympoic sandard error of 0.068, compared o he auoregressive parameer wih no breaks of lowered his parameer subsanially. ˆ = ρ. Clearly, allowing for breaks has 1 There do no appear o be criical values for he uni roo hypohesis in he presences of 4 breaks.
11 I is ineresing o noe ha he poin esimaes of he break daes are remarkably similar beween he wo counries. For boh he Canadian and he U.S. inflaion raes, here is evidence of breaks in he mid 1960s, he early 1970s and he early 1980s. Of course, as noed above, he evidence is no overwhelmingly in favour of a disinc break in he 1960s for eiher counry, and here would no appear o be an economic explanaion for why here should be a break in eiher inflaion rae in his period. The 1970s break could be associaed wih he sudden increase in energy prices associaed wih he OPEC oil embargo. Clearly, he break in he early 1980s is associaed wih he sudden decrease in inflaion brough abou by he severe downurn in boh economies. There is a fourh break in he Canadian inflaion rae in 91:01, which represens a second decrease in inflaion, his ime in response o he recession of he early 1990s. Ineresingly, his break does no appear in he U.S. inflaion rae.
12 The Canadian and U.S. Nominal Ineres Raes The resuls of esimaing boh he parial srucural break model and he pure srucural break model for Canadian and U.S. ineres raes are shown on Table 3 Table 3 Srucural Break Tess Canadian and U.S. Nominal Ineres Raes Parial Break Model y c + ρy + u = j 1 Pure Break Model y = c + u j Canadian U.S. Canadian U.S. Ineres Rae Ineres Rae Ineres Rae Ineres Rae Udmax * 1.73* Wdmax * 17.95* SupF(0 1) SupF(0 ) SupF(0 3) * 1.73* SupF(0 4) * 11.06* SupF(0 5) * 10.59* SupF( 1) * 5.63 SupF(3 ) SupF(4 3) SupF(5 4) BIC * denoes significan a 95%
13 The resuls on Table 3 show ha here is very lile evidence of mean breaks in eiher he Canadian or U.S. nominal ineres raes. For insance, he UDMAX, WDMAX and he SupF(0 l) esimaed from he parial break model sugges 0 breaks for boh ineres raes. In spie of his, he BIC chooses wo breaks for he U.S. ineres rae. For he pure break model, even hough he UDMAX, WDMAX and he some of he SupF(0 l) are significan for boh ineres raes, only he SupF( 1) es is significan for Canada, suggesing breaks in he Canadian ineres rae. Once again, he BIC conradics hese resuls, choosing five breaks for each ineres rae. The esimaed breaks and heir 95% confidence inervals are summarized on Table 4.
14 Table 4 Esimaed Mean Break Daes Canada and U.S. Nominal Ineres Rae Canadian Nominal Ineres Rae U.S. Nominal Ineres Rae 5 Breaks Breaks 5 Breaks Breaks Pure Break Pure Break Pure Break Parial Break Model Model Model Model c s.e. (0.61) (1.0) (0.47) (0.1) c s.e (1.40) (1.66) (0.45) (0.41) c s.e (0.3) (0.51) (0.94) (0.19) c s.e. (.48) - (1.14) - c s.e (1.35) - (1.0) - c s.e. (0.51) - (0.3) - ρˆ s.e (0.08) T 1 65:03 74:01 63:0 76:04 95% c.i. 54:01-74:04 59:04-79:01 59:03-7:01 69:03-77:03 T 73:04 91:03 68:01 81:03 95% c.i 73:03-74:01 91:01-98:0 54:01-69:04 80:0-88:04 T 3 78:03 78:03 95% c.i 78:0-78:03 74:0-83:0 T 4 83:0 84:04 95% c.i 67:04-01:0 78:04-91:03 T 5 91:03 91:01 95% c.i. 90:03-99:01 75:01-01:0
15 The resuls on Table 4 help o explain he conradicory resuls on Table 3. The mean break model appears o be a very poor descripion of he ineres rae in eiher counry. The 95% confidence inervals surrounding he break daes for all models are ofen exremely wide, in some insances as large as years. As well, he confidence inervals ofen overlap considerably. This suggess ha his model is exremely imprecisely esimaed. For he U.S. parial break model, he auoregressive parameer, esimaed condiional on wo breaks is ˆ = ρ, which is no much differen from he value of ˆ = ρ esimaed wih no breaks. The raio for he uni roo null hypohesis ( H : ρ 1) is 3.5, for he wo break model, and i is -.14 from he model esimaed 0 = wih no breaks. Neiher of hese saisics rejecs he uni roo hypohesis. Figure illusraes he and 5 mean break models for each counry. The 95% criical value for wo breaks is 6.5, aken from Lumsdaine and Papell (1997), esimaed via Mone Carlo mehods for a series wih wo breaks in he mean.
16 Figure Esimaed Mean Breaks Canadian and U.S. Ineres Raes Canadian Ineres Rae 5 Ineres Rae (%) U.S. Ineres Rae Ineres Rae (%)
17 Perhaps he dominan feaure of Figure is he upward rend in each ineres rae, which reverses iself afer he early 1980s. The 5 break models appear o be fiing a sep funcion hrough his rend. Prior o he early 1980s, each sep is no overly large, and his likely he reason for he wide confidence inervals associaed wih he esimaed breaks. 3. Esimaing Breaks in he Mean and he Trend The Bai Perron procedure used above is designed o capure breaks in he mean of a series. If here are breaks in he rend of a series, his procedure is inappropriae. One sraighforward manner o uncover breaking rends in a series is o apply he sequenial esimaion mehodology due o Zivo Andrews (199), which was designed o es he uni roo hypohesis agains he alernaive of one endogenously deermined break in he mean and rend of a series. The break dae is deermined by he raio ha has he highes probabiliy of rejecing he null hypohesis. Lumsdaine and Pappel (1997) exended he Zivo Andrews mehodology o wo breaks. Logically, his mehodology could be exended o hree or more breaks. However, he compuaional requiremens can quickly become burdensome. In his secion I search for breaks in he mean and rend of he Canadian and U.S. inflaion and nominal ineres raes using he sequenial esimaion mehodology applied o a model of one, wo and hree endogenously deermined breaks. I use he following equaion: (3) X = c j + β jt j + ρx 1 + λi X i + ε ; j =,3, 4 k i= 1 Equaion (3) is esimaed separaely for he number of breaks, m, equal o 1, or 3. The es saisic generaed from esimaion of each model is a es of a uni roo wih m breaks, bu i is no a es of he size of m. 3 One mehod of deermining he size of 3 The model is run over he enire sample, wih he possible break daes consrained o be away from he end poins, adoping a rimming value of For each possible break dae, k is esimaed following
18 m is o consruc a saisic of he form F(m+1 m). Concepually, if he hree break model (m=3) is correc, hen he resricions imposed in he wo break model should be rejeced by he F(3 ) saisic. The same would be rue of he wo break model relaive o he one break model. However, sandard criical values for he F disribuion are no valid here, as he number of breaks is endogenous. Approximae criical values are generaed hrough Mone Carlo. Under he null hypohesis ha he wo break model is correc, we generae arificial daa using he parameers from he wo break model. This arificial series is hen esed for hree breaks. An F saisic is consruced for he null hypohesis ha he resricions are rue. This is hen repeaed 1,000 imes. 4 The F saisics are lised in descending magniude, and he 950 h is aken o be he approximae 95% criical value. This process is repeaed for he one break vs. wo break model, and a new criical value is deermined in an analogous manner. The 95% criical values are: 6.4 for he 1 vs. break model; and 6.95 for he vs. 3 break model. k max Perron (1989). We sar wih an upper bound of = 6. If he las lag included is significan, we se k = 6, oherwise reduce k by 1 unil he las lag becomes significan. If no lags are significan, se k = 0. We adop a significance level of 10%. Since he disribuion of he raio is non-sandard, he criical values for m=1 are aken from Zivo and Andrews (199), for m=, he criical values are aken from Lumsdaine and Papell (1997), and for m=3, he criical values are generaed by Mone Carlo simulaion, under he null hypohesis of whie noise. 4 This quickly becomes exremely compuaionally burdensome, so only 1,000 repeiions were aemped for each model. Thus, he criical values may be subjec o error.
19 The Canadian and U.S. Inflaion Raes A summary of he resuls of esimaing hese models for he Canadian and U.S. inflaion raes is presened on Table 5.
20 Table 5 Break in Mean and Trend Canadian Inflaion Rae U.S. Inflaion Rae 1 Break break 3 Break 1 Break Break 3 Break Model Model Model Model Model Model c (0.489) (0.471) (0.454) (0.306) (0.49) (0.34) T (0.011) (0.01) (0.011) (0.008) (0.013) (0.009) c (1.40) (.94) (.17) (1.3) (1.794) (8.998) T (0.018) (0.030) (0.08) (0.013) (0.00) (0.1) c (3.593) (3.93) (.106) (10.444) T (0.036) (0.046) (0.00) (0.1) c (3.635) - (3.859) T (0.04) - (0.040) ρˆ (0.100) (0.1) (0.19) (0.073) (0.081) (0.078) k T 1 7:04 7:04 7:04 81:03 67:01 71:03 T 8:03 8:04 81:03 74:04 T 91:03 81:03 3 F(m+1 m) Noes T is -raio for he null hypohesis of a uni roo. Criical values a 95%: break model -6.8 break model break model
21 For he Canadian inflaion rae he es does no rejec a uni roo for any of he models. However, his appears o be due o he sensiiviy of his es o he inclusion of lagged differences. When he model is re-esimaed using 1 or lags, here is no difference in he esimaed break daes, bu here is clear rejecion of he uni roo null. The F es chooses he model wih 3 breaks. The esimaed break daes are almos idenical o he las 3 break daes esimaed from he mean break model using he Bai- Perron mehodology. Furher, he sandard error on he rend variable in each of he las hree regimes is raher large, indicaing ha here may be very lile gain from including a rend break in he model. This poin is illusraed on Figure 3 which shows he relaionship beween he mean break model and he rend break model for he Canadian inflaion rae. Figure 3 Illusraion of he Mean and Trend Break Models Canadian inflaion Rae 16 1 Inflaion (%)
22 For he U.S. inflaion rae, he F saisic picks 3 breaks, and he hypohesis of a uni roo is rejeced. Alhough he break in 81:03 is idenical o ha esimaed from he mean break model, he oher wo breaks are a somewha differen daes. In a recen paper Malliaropulous (000) esimaed break dae for he U.S. inflaion rae a 81:03. However, Malliaropulous resriced his search o one break. As illusraed in Figure 4, he rend break model is aemping o make wo separae breaks ou of he inflaion behaviour in he 1970s, while he mean break model esimaes one regime over his period. When his model was re-esimaed wih he hree rend erms consrained o be 0 (no presened), he model chose idenical breaks as he Bai-Perron procedure. Figure 4 Illusraion of he Mean and Trend Break Models U.S. inflaion Rae Inflaion (%)
23 The Canadian and U.S. Ineres Raes The resuls of searching for breaks in he mean and rend of he Canadian and U.S. in ineres raes are presened on Table 6. For Canada, he F chooses hree breaks, and he daes are 80:03; 84:03 and 90:04. For he U.S., he wo break model rejecs one break, bu he daa is no consisen wih he hree break model. The break daes are 80: and 84:03, virually idenical o he firs wo breaks in Canada. 5 The esimaed breaks are illusraed on Figure 5. For he Canadian hree break model, he auoregressive parameer is ρˆ = 0.535, and he raio for he null of ρ = 1 is 8.61, which rejecs he null hypohesis of a uni roo. 6 For he U.S. wo break model, he auoregressive parameer is ρˆ = , and he raio for he null of ρ = 1 is 8.48, which also rejecs he null hypohesis of a uni roo. 7 A convenional Dickey-Fuller es of he uni roo hypohesis for each ineres rae wih no breaks yields raios of 1.85 (Canada) and.05 (U.S.), each of which each fails o rejec he null. Thus, he apparen uni roo propery of he ineres raes disappears when breaks are aken ino accoun. 5 The firs break is idenical o he break dae found by Malliaropulous (000). 6 For he 3 break model he criical value is 7.45, abulaed by Mone Carlo mehods. 7 For he break model, he criical value a 95% is 6.8 (see Lumsdaine and Pappell (1997)).
24 Table 6 Break in Mean and Trend Canadian Ineres Rae U.S. Ineres Rae 1 Break Break 3 break 1 Break break 3 break Model Model Model Model Model Model c (0.17) (0.168) (0.164) (0.136) (0.19) (0.18) T (0.004) (0.004) (0.004) -.7 (0.003) (0.003) c (0.46) (0.6) (1.455) (1.155) (.94) (.59) T (0.007) (0.104) ((0.0) (0.046) (0.03) (0.03) c (1.751) (1.846) (.769) (.868) T (0.047) - (0.034) c (0.847) - (1.39) T (0.00) - (0.016) ρˆ (0.048) (0.049) (0.054) (0.040) (0.043) (0.046) k T 1 80:03 80:03 80:03 80:0 80:0 80:0 T 88:01 84:03 84:03 84:03 T 3 9:04 91:03 F(m+1 m) 6.67* 9.71* 14.50* * denoes rejecion a 95%. Figure 5
25 Esimaed Trend Breaks Canadian and U.S. Ineres Raes 5 0 Ineres Rae (%) Ineres Rae (%)
26 4. Conclusions In his paper we have searched for breaks in he nominal ineres rae and inflaion in Canada and he U.S. over he period The daa appears o be consisen wih 4 breaks in he mean of he Canadian inflaion rae, and 3 breaks in he mean of he U.S. inflaion rae. The firs hree breaks in each counry occur a similar daes, while he Canadian inflaion rae exhibis a break in early 1991, which does no appear in he U.S. inflaion rae. There appears o be lile gain from modeling he inflaion raes wih a break in he rend funcion. In conras, he wo nominal ineres raes appear o be well modeled by a breaking mean and rend, raher han by a mean break. There appears o be 3 breaks in he Canadian nominal ineres rae and wo breaks in he U.S. ineres rae. As wih he nominal ineres raes, he firs breaks appear a similar daes in each counry. Again, here is an exra break in he Canadian ineres rae in he early 1990s. Furher, he breaks in he Canadian and U.S. inflaion raes occur a virually idenical imes, and he breaks in he wo ineres raes are a virually idenical imes. There is an exra break in boh he inflaion rae and he nominal ineres rae in Canada in he early 1990s.