PCE and CPI Inflaion Differenials: Convering Inflaion Forecass Model Specificaion By Craig S. Hakkio This specificaion describes he models ha are used o forecas he inflaion differenial. The 14 forecass generaed wih hese models provide differen ways o allow he inflaion differenial o change over ime. The Average Differenial Model Forecass 1 3 are he simples since hey are simply he average inflaion differenial esimaed over hree differen ime periods. Three sample periods of varying lengh are simple ways of allowing he average differenials o change over ime. The sample periods are: 1995q1 2007q2, 2000q1 2007q2, and 2002q3 2007q2 (used in Char 3). The hree forecass are called Average, 1995q1, Average, 2000q1, and Average, 2002q3. The esimaed differenial and RMSE for he hree sample periods for he oal and core inflaion differenial are: 1995q1 2007q2 2000q1 2007q2 2002q3 2007q2 Toal inflaion differenial RMSE Core inflaion differenial RMSE.525.541.486.526.425.651.278.553.341.620.052.366 1
2 Federal Reserve Bank of Kansas Ciy The Auoregressive Model Forecass 4 6 are based on an AR process for he inflaion differenial esimaed over he same hree sample periods. The AR model is slighly more sophisicaed han he average differenial model. The inflaion differenial is regressed agains a consan and he lagged inflaion differenial. The lag lengh is chosen by AIC. The hree forecass are called AR, 1995q1, AR, 2000q1, and AR, 2002q3. The AR model can be wrien as: K c P c P π π = a+ bk( π i π i) + ε = 1 where he lag lengh is chosen opimally (using AIC). The parameer esimaes for he hree sample periods for he oal and core inflaion differenial are: Toal inflaion differenial Core inflaion differenial (3) 1995q1 2007q2 2000q1 2007q2 2002q3 2007q2 a.385 (.143).278 (.143).326 (.135).255 (.190).454 (.150).098 (.232) b 2 -.508 (.232) RMSE.527.643.588 a.195 (.081).594 (.114) The Vecor Auoregressive Model.144 (.097).488 (.160).033 (.070).498 (.203) RMSE.427.492.330 Forecass 7 9 are derived from a VAR process for CPI and PCE inflaion esimaed over he same hree sample periods. The VAR is used o forecas fuure values for CPI and PCE inflaion and hen calculae he differenial. The lag lengh is chosen by AIC. The hree forecass are called VAR, 1995q1, VAR, 2000q1, and VAR, 2002q3.
Economic Review firs quarer 2008 3 The VAR model can be wrien as: K c c c c π a bπ i i b P i π P c = + ( + i)+ ε i= 1 K p p c c p p p π = a + ( γi π i+ γi π i)+ ε i= 1 (4) The parameer esimaes for he hree sample periods for he oal and core inflaion differenial are: Toal inflaion differenial Maximum eigenvalue (modulus) Core inflaion differenial 1995q1 2007q2 2000q1 2007q2 2002q3 2007q2 a c 2.179 (.433) c -.308 (.413) p.604 (.521) a p 1.761 (.325) γ 1 c -.600 (.310) γ 1 p.916 (.392) 2.439 (.681) -.209 (.589).402 (.785) 2.171 (.495) -.434 (.429).596 (.570) 2.196 (1.146) -1.457 (1.246) 1.905 (1.702) 2.029 (.787) -1.380 (.856) 1.751 (1.169).413.222.282 RMSE.532.654.651 γ 1 c Maximum eige value (modulus) a c 1.050 (.294) c.638 (.130) p -.123 (.142) a p 1.394 (.323) -.068 (.143) γ 1 p.316 (.155) 1.050 (.389).581 (.179) -.070 (.194) 1.680 (.429) -.065 (.197).199 (.213).748 (.511).170 (.344).468 (.442) 1.232 (.435) -.293 (.293).671 (.378).662.593.501 RMSE.415.463.312
4 Federal Reserve Bank of Kansas Ciy The Discouned Leas Squares Model While differen sample periods is one procedure ha can be used o accoun for a change in he mean, i is exreme in ha i gives equal weigh o all observaions wihin he sample period and zero weigh o observaions ouside he sample period. An alernaive would be o use discouned leas squares in which all daa are used bu older observaions ge less weigh. Specifically, he weighing facor is λ -j, where 0< λ<1. Branch and Evans (2006) sugges discouned leas squares migh work well for macroeconomic forecasing. In his paper, λ=.95, a value suggesed by Branch and Evans and by Clark and McCracken (2006). Wih his weigh, observaions from 1985q1 ge a weigh of 1 percen, observaions from 2000q1 ge a weigh of 23 percen, and observaions from 2003q3 ge a weigh of 46 percen. Discouned leas squares is used o esimae an AR model where he lag lengh is chosen by AIC (applied o a no discouned leas squares model). The resuling forecas is called DLS. The parameer esimaes for he oal and core inflaion differenial are: 1985q1 2007q2 Toal inflaion differenial a.378 (.079).162 (.115) RMSE.630 Core inflaion differenial a.142 (.054).473 (.107) b 2.066 (.106) RMSE.427
Economic Review firs quarer 2008 5 The Exponenial Smoohing Model Forecass 11 12 use exponenial smoohing. Cogley (2002) uses exponenial smoohing o esimae a ime-varying inflaion arge of he cenral bank. Le μ be he unobserved ime-varying mean inflaion diff differenial a ime and le π be he acual inflaion differenial a ime. Exponenial smoohing expresses he ime-variaion as follows: diff μ = μ + aπ μ 1 1 This procedure is simple o esimae and use o forecas. Forecas 11 is based on an esimaed smoohing parameer, while forecas 12 is derived by seing he smoohing parameer =.125 (he value used by Cogley). The wo exponenial smoohing forecass are called Exponenial, parameer esimaed and Exponenial, parameer se. 1985q1 2007q2 α esimaed α.1 Toal Inflaion Differenial α.085.1 RMSE.634.538 Core Inflaion Differenial α.175.1 RMSE.635.540 The Regime Swiching Model The regime swiching model assumes he parameers can swich beween wo differen regimes. The model can be wrien as follows: c p π π = a( S )+ ε ε N 0, σ( R ) a a ( S )= a 2 ( ) low high if S ij if S Prob S = j Si = p σ σ ( R ) = σ low high = 1 = 2 if R = 1 Prob R = j R = i = q ij 1 if R = 2
6 Federal Reserve Bank of Kansas Ciy The log-likelihood funcion is maximized numerically using he Nedler-Mean simplex algorihm over a large grid of saring values. The parameer esimaes for he oal and core inflaion differenial are as follows. Toal inflaion differenial Core inflaion differenial Sae 1 Low differenial Sae 2 High differenial α σ LR prob α σ LR prob.109 (.129).104 (.092).099 (.042).064 (.059).34.632 (.090).43.780 (.069).998 (.457).265 (.1103).66.57 The RMSE is calculaed as follows. Le e (S =s) = residual condiional on regime s = 1, 2. The RMSE is hen calculaed using he following se of residuals: e = p 1 *e (S =1) + p 2 *e (S =2). Thus, RMSE for he oal inflaion differenial is.572, and he RMSE for he core inflaion differenial is.459. The esimaed marices P = [p ij ] and Q = [q ij ] for he oal inflaion differenial are: P =. 832. 168. 085. 915 Q =. 873. 127. 349. 651 The esimaed marices P = [p ij ] and Q = [q ij ] for he core inflaion differenial are: P =. 9326. 0674. 0507. 9493. 8811. 1182 Q =. 0410. 9590
Economic Review firs quarer 2008 7 Since p 11 p 22 and large for he oal and core inflaion differenial, he long-run probabiliy of being in sae 1 is close o 50 percen. Bu, since p 11 < p 22, here is a somewha greaer long-run probabiliy of being in he high-mean sae (sae 2) han in he low-mean sae (sae 1). The long-run probabiliy for he oal inflaion differenial is 34 percen of being in he low-mean sae and 66 percen of being in he high-mean sae. The long-run probabiliy for he core inflaion differenial is 43 percen of being in he low-mean sae and 57 percen of being in he high-mean sae. The Time-Varying Parameer Model In his model, he inflaion differenial is wrien as an AR(1) process, where he parameers follow a random walk process. The AR(1) model can be wrien as follows: c p c π π = a + b( π π p 1 1)+ ε a a = a 1 + u,var u a 2 a ( )= σ u b b u b,var u b 2 σ u b = + ( )= 1 ε and u are independen The parameer esimaes for he AR(1) model are: Toal inflaion differenial Core inflaion differenial s[u a ].026.054 s[u b ].00001.062 α[2007q2].294.089 β[2007q2].242.352 RMSE.680.552 Forecas Average The Forecas average is simply he average of he 14 forecass generaed wih he previous models and sample periods.
8 federal Reserve Bank of Kansas Ciy CBO While no included in he forecas average, he CBO provides forecass for CPI inflaion (overall and core) and PCE inflaion (overall and PCE) for 2007 2017. SPF The Survey of Professional Forecasers (SPF) provides forecass for boh CPI inflaion (core and overall) and PCE inflaion (core and overall) for 2007 2009, and forecass for overall inflaion (CPI and PCE) for he nex five years and for he nex en years. Assessing he resuls This secion assesses he resuls of generaing all 14 forecass. Appendix Tables 1 and 2 show he oal and core inflaion differenial forecass for each model for several differen horizons. In addiion, o provide a sense of he long-run inflaion differenial, Appendix Chars 1 and 2 show he oal and core inflaion forecass for 2017. A few general observaions can be made. Firs, for he models esimaed over he hree differen sample periods (Average, AR, and VAR), models 1 9, he forecass of he overall and core inflaion differenials ge smaller he shorer he sample period. Second, he long-run oal inflaion differenial (2017) is broadly similar for all 14 forecass, ranging beween.3 and.5 percenage poins. However, he long-run core inflaion differenial (2017) shows greaer differences, ranging beween.05 percenage poins and.5 percen poins. Third, he forecass of he overall inflaion differenial are quie similar across models and forecas horizons once one excludes he forecass esimaed over he firs sample period (1995q1 2007q2) and he hird sample period (2002q3 2007q2). For hese forecass, he average forecas for 2008 2017 is.43 percenage poin and ranges beween.39 and.48 percenage poin. Finally, he forecass for he core inflaion differenial using he regime swiching model increases as he forecas horizon increases. For example, he forecass are.315 percenage poin (2008),.384 percenage poin (2009),.426 percenage poin (2010), and.488 percenage poin (2017). This is no oo surprising because even hough
Economic Review firs quarer 2008 9 he wo regimes are relaively persisen, he long-run probabiliy of being in he high inflaion regime is 57 percen, and he high inflaion mean differenial is.78 percenage poin, and he low inflaion mean differenial is.10 percenage poin. The regime swiching model recognizes ha he inflaion differenial is high more han half of he ime and is low less han half he ime. Therefore, while he 2007q2 core inflaion differenial is likely in he low inflaion differenial regime (88 percen probabiliy), he model expecs he core inflaion differenial o spend more han half he ime in he high differenial regime. The long-run oal inflaion differenial esimaed from he regimeswiching model is close o he resuls from he oher models. As wih he core inflaion differenial, he long-run probabiliy of being in he high inflaion regime is 66 percen, and he high inflaion differenial is.63 percenage poins and he low inflaion differenial is.11 percenage poins. In addiion, he resuls sugges ha here is an 81 percen probabiliy ha he oal inflaion differenial is currenly in he high differenial regime. The oher models, in conras o he regime swiching model, predic ha he fuure inflaion differenial will be close o he curren inflaion differenial. Since he curren inflaion differenial is generally low, he oher models generally predic a lower inflaion differenial han he regime swiching model.
10 Federal Reserve Bank of Kansas Ciy Table 1 Appendix Average oal inflaion differenial Models 2007 2008 2009 2010 2017 nex10 Average, 1995q1.525.525.525.525.525.525 Average, 2000q1.425.425.425.425.425.425 Average, 2002q3.341.341.341.341.341.341 AR, 1995q1.873.542.534.534.534.545 AR, 2000q1.822.444.438.438.438.448 AR, 2002q3.527.358.326.322.322.312 VAR, 1995q1.865.544.534.533.533.544 VAR, 2000q1.825.440.438.438.438.448 VAR, 2002q3.659.328.336.336.336.333 DLS.787.452.451.451.451.456 Exponenial, parameer esimaed Exponenial, parameer se.364.423.423.423.423.423.343.435.435.435.435.435 Regime Swich.514.477.463.458.456.462 TVP, AR1.377.394.388.388.388.399 Noes: Forecas average.589.438.433.432.432.435 Lower cenral endency Upper cenral endency.377.394.388.388.388.399.822.477.463.458.456.462 Minimum.341.328.326.322.322.312 Maximum.873.544.534.534.534.545 CBO.500.300.400.400.300.320 SPF.600.300.200.300
Economic Review firs quarer 2008 11 Table 2 Average core inflaion forecas Models 2007 2008 2009 2010 2017 nex10 Average, 1995q1.486.486.486.486.486.486 Average, 2000q1.278.278.278.278.278.278 Average, 2002q3.052.052.052.052.052.052 AR, 1995q1.333.483.481.481.481.482 AR, 2000q1.269.292.281.281.281.286 AR, 2002q3.203.091.068.066.066.077 VAR, 1995q1.281.428.460.467.468.459 VAR, 2000q1.199.244.263.266.266.261 VAR, 2002q3.116.015.047.046.046.044 DLS.268.320.309.307.307.311 Exponenial, parameer esimaed Exponenial, parameer se.172.191.191.191.191.191.170.182.182.182.182.182 Regime Swich.196.315.384.426.488.433 TVP, AR1.148.143.137.137.137.142 Noes: Forecas average.254.258.276.284.269.271 Lower cenral endency Upper cenral endency.170.143.137.137.137.142.278.320.384.426.468.433 Minimum.486.486.486.486.488.486 Maximum.052.015.047.046.046.044 CBO.6.3.5.6.3.38 SPF.3.3.3
12 Federal Reserve Bank of Kansas Ciy Char 1 Toal Inflaion Differenial, 2017 1.000 1.000.800.800.600 Average across models.600.400.400.200.200.000 Average, 1995q1 Average, 2000q1 Average, 2002q3 AR, 1995q1 AR, 2000q1 AR, 2002q3 VAR, 1995q1 VAR, 2000q1 VAR, 2002q3 DLS Exponenial, parameer esimaed Exponenial, parameer se Regime Swich TVP, AR1.000 Char 2 Core Inflaion Differenial, 2017 1.000 1.000.800.800.600.600.400 Average across models.400.200.200.000 Average, 1995q1 Average, 2000q1 Average, 2002q3 AR, 1995q1 AR, 2000q1 AR, 2002q3 VAR, 1995q1 VAR, 2000q1 VAR, 2002q3 DLS Exponenial, parameer esimaed Exponenial, parameer se Regime Swich TVP, AR1.000