Has the Business Cycle Changed? Evidence and Explanations. Appendix

Transcription

1 Has he Business Ccle Changed? Evidence and Explanaions Appendix Augus 2003 James H. Sock Deparmen of Economics, Harvard Universi and he Naional Bureau of Economic Research and Mark W. Wason* Woodrow Wilson School and Deparmen of Economics, Princeon Universi and he Naional Bureau of Economic Research

2 1. Model-Based Calculaions This secion of he appendix provides addiional deails for he model-based calculaions summarized in Tables 6 and 8 in he ex. I begins wih a descripion of he models, and hen presens expanded versions of ex Tables 6 and 8. The Rudebusch-Svensson Model The model consiss of hree equaions. The Phillips curve is specified as: π +1 = α 0 + α π1 π + α π2 π 1 +α π3 π 2 + α + ε +1, (1.1) where π is he inflaion rae and is he oupu. The IS equaion is specified as: + 1 = β 0 + β 1 + β β r ( R π ) + η +1. (1.2) where R and π are he four quarer averages of R (he Federal Funds rae) and π compued over ime periods 3 o. The model is closed using he Talor rule specificaion from Judd and Rudebusch (1998), wrien here as: R +1 = φ 0 + φ R1 R + φ R2 R 1 + φ π π φ φ 2 + ξ +1 (1.3) For our calculaions π = 400 ln(p /P 1 ), where P is he quarerl value of he U.S. GDP deflaor; = 100 ( rend ) where is he quarerl value of real GDP 1

3 and rend is he fied value from a regression of ono (1,, 2 ) over he sample period 1959:1-2002:4. The parameers in equaions (1.1) and (1.2) were esimaed over he sample period 1960:1-2002:4; he parameers of (1.3) were esimaed over 1960:1-1978:4 and 1984:1-2002:4. OLS esimaes and heeroskedasic robus sandard errors are given in he aached able. Table A.1 Parameer Esimaes for he Rudebusch-Svensson Model Parameer 1960:1-2002:4 1960:1-1978:4 1984:1-2002:4 α (0.078) α π (0.084) α π (0.073) α π (0.092) α (0.025) β (0.111) β (0.081) β (0.079) β r (0.033) φ (0.321) (0.156) φ R (0.139) (0.150) φ R (0.136) (0.107) φ π (0.050) (0.120) φ (0.086) (0.098) φ (0.099) (0.085) σ ε σ η σ ξ The Sock-Wason Srucural VAR The srucural VAR is based on a 4-variable, 4-lag VAR ha includes he logarih of oupu (), inflaion (π), ineres raes (R), and he logarihm of commodi prices (Z). The srucural model includes an IS equaion, a forward-looking New Kenesian Phillips 2

4 curve, a forward looking Talor-pe monear polic rule, and an exogenous process for commodi prices: = θ r + lags + ε, π = γ Y(δ) + lags + ε π, R = β π π + +β h/ + + lags + ε h / r, Z = lags + α ε, + α π ε π, + α r ε r, + ε z, where r = R π + k/ is he real ineres rae, π + k / is he expeced average inflaion rae i over he nex k periods, where k is he erm of he ineres rae R; Y(δ) = δ + 1/ is he discouned expeced fuure oupu, and + h / is he expeced fuure average oupu over he nex h periods measured in percenage poins. We have used generic noaion lags o denoe four unresriced lags of variables in each of hese equaions. The VAR is esimaed using = ln(gdp /GDP 1 ) where GDP is he real value of GDP; π = 400 {ln(p /P 1 ) ln(p 1 /P 2 )} where P is he GDP deflaor; R is he 1- ear Treasur Bond rae; Z = ln(pcom /Pcom -1 ) is he spo marke commodi price index. We assume ha he srucural parameers θ, γ, and δ are consan hroughou he sample period, and allow β π, β, and he coefficiens on lags o change across he wo sample periods. The values of θ, γ, and δ are specified a priori, wih θ =.002 (noe ha is he logarihm of quarerl real GDP and r is measured in percenage poins a an annual rae), γ = 0.30 and δ= (See Sock and Wason (2002) for a discussion of i= 0 3

5 hese values and for resuls using oher values of hese parameers.) Under he assumpion ha ε R is uncorrelaed wih ε and ε π, he parameers β π and β can be esimaed b IV mehods using he reduced form VAR residuals. Parameer values esimaed over he wo subsamples and heeroskesdasic robus sandard errors are given in he following able. Table A.2 Esimaed Talor Rule Coefficiens for he SW Srucural VAR Model Parameer 1960:1-1978:4 1984:1-2002:4 β π (0.194) (0.227) β (0.181) (0.146) The Smes-Wouer Models The Smes-Wouer US model (SWUS) and EuroArea model have a common srucure. The differ from anoher in wo was. Firs, he parameer values differ: he SWUS parameer values we fi o U.S. daa over and he SWEA parameer values were fi o European daa over Second, he have differen lowfrequenc characerisics: he SWEA model using uses linearl derended values of real variables, and his resuls in saionar dnamics for all of he variables in he model; he SWUS model includes common real I(1) sochasic rends shared b he models real variables and an independen I(1) sochasic rend in inflaion. The models share a common specificaion of he Talor rule 4

6 R = π 1 + ρ(r 1 π 1 ) + (1 ρ){r π (π 1 π 1 ) + r 1 } + r π (π π ) + r + η (1.4) wih π = ρ π 1 π + ε. The specificaion of he oher equaions in he models can be found in Smes and Wouers (2003a, 2003b). Our simulaions of he models used he poserior modes repored in Smes and Wouers (2003a, 2003b). The values of he Talor rule coefficiens are given in he following able. Table A.3 Full-Sample Esimaed Talor Rule Coefficiens from he Smes-Wouer Models Parameer Europe US ρ r π r r π r ρ π (consrained) σ η σ ε These values were used for he baseline versions of he models. For he pre calculaions, he Talor rule coefficiens were modified so ha he cenral bank was more accommodaive o inflaion, subjec o he consrain ha he model sill had a unique raional expecaions equilibrium. For boh models his was accomplished b seing r π =

7 Deailed resuls of he counerfacual model simulaions Table 6 in he ex repors resuls from model simulaions for each of he four models discussed above. The Base Model resuls for he Rudebusch-Svensson model were compued using equaions (1.1)-(1.2) wih parameer values shown in he column labeled 1960:1-2002:4 of Table A.1, and equaion (1.3) wih parameer values from he column labeled 1984:1-2002:4. The model was simulaed using he residuals from hese equaions over he 1984:1-2002:4 sample period. The resuls in Table 6 s column labeled Base + Pre-79 Monear Model was consruced in he same wa, excep ha he parameer values for (1.3) came from he 1960:1-1978:4 column of Table A.1. The Base Model resuls in Table 8 are he same as hose Table 6. Table 8 s Base + Pre-79 shocks resuls are compued from 1960:1-1978:4 residuals compued from (1.1)-(1.2) and he residuals from (1.3) came from he 1960:1-1978:4 column of Table A.1. The model is simulaed using he coefficien values for (1.3) from he 1984:1-2002:4 column of Table A.1. The Base Model resuls for he SVAR model were compued from he srucural VAR esimaed over he 1984:1-2002:4 sample period along wih he residuals from he period. The Base + Pre-79 Monear Model resuls were compued using he 1984:1-2002:4 residuals along wih he VAR esimaed over he 1960:1-1978:4 sample period. (Noe ha he srucural parameers θ, γ, and δ are he same in boh sample periods.) The Base Model resuls in Table 8 are he same as hose Table 6. Table 8 s Base + Pre-79 shocks resuls are compued from 1960:1-1978:4 srucural residuals ogeher wih VAR coefficien values esimaed over he 1984:1-2002:4 sample. 6

8 The Base Model resuls for he Smes-Wouers models were compued using poserior model parameer esimaes from Smes and Wouers (2003a, 2003b). The sandard deviaion are he implied populaion sandard deviaion of 4 using hese parameer values. As discussed above, he resuls for Base + Pre-79 Monear Model used he same parameers excep ha r π was reduced o r π = The following able provides addiional resuls for hese experimens. Table A.4 a. Addiional Resuls for he Rudebusch-Svensson and Srucural VAR Models Rudebusch-Svensson SW Srucual VAR Base Model Base + Pre-79 Monear Polic Base + Pre-79 shocks Base Model Base + Pre-79 Monear Polic Base + Pre-79 shocks σ( 4 ) σ(π π 4 ) σ( π ) µ( π ) π 2002: b. Addiional Resuls for he Smes-Wouer Models Smes-Wouer US Smes-Wouer EuroArea Base Model Base Model wih r π = 0.97 Base Model Base Model wih r π = 0.97 σ( 4 ) σ(π π 4 ) σ( π ) NA NA Noes: σ( 4 ) denoes he sandard deviaion of 4, and similarl for σ(π π 4 ) and σ( π ). µ( π ) denoes he mean of π. π 2002:4 is he value of π in 2002:4. 7

9 2. Nonlineariies in he Talor rule Tess for nonlineariies were carried using (1.3) esimaed over 1960:1-1978:4. The ess were conduced b adding several hreshold variables o he base specificaion. To define hese hreshold variables, le F x,0.75 denoe he 75 h percenile of he empirical disribuion of x over he sample period, and le F x, 0.25 be similarl defined. Le r = R π. The able below shows resuls wih addiional variables, he esimaed coefficiens, sandard errors and F-saisics for join significance. 8

10 Regressor Baseline Regressors consan Table A.5 Tess for Nonlineari Base Model 0.98 (0.32) 1.04 (0.35) 0.77 (0.34) 0.89 (0.25) R (0.14) 1.11 (0.14) 1.15 (0.14) 1.07 (0.14) R (0.14) 0.54 (0.13) 0.49 (0.13) 0.43 (0.13) π 0.19 (0.05) 0.28 (0.09) 0.19 (0.05) 0.22 (0.05) 0.05 (0.09) 0.05 (0.09) 0.13 (0.11) 0.04 (0,09) 0.08 (0.10) 0.07 (0.11) 0.06 (0.10) 0.06 (0.10) 1 Addiional Regressors r F > 1 r,0.75 ) 1 r < 1 Fr,0.25 ) r 1 1 r,0.75 r 0.07 (0.40) 0.27 (0.28) r > F ) 0.05 (1.00) r < F ) 0.00 (0.34) 1 r,0.25 > F ),0.75 < F ),0.25 > F ),0.75 < F ), (0.26) 0.13 (0.09) 0.02 (1.26) 0.05 (0.34) π π 8 > F π π 8,0.75 ) ( π π 8 ) 1.17 (1.58) π π 8 > F π π 8,0.75 ) π (0.13) π π 8 > F π π 8,0.75 ) 0.71 (1.65) F-saisic (p-value) for exclusion of addiional regressors 0.51 (0.73) 1.09 (0.36) 0.86 (0.46) 9

11 References Judd, John F., and Glenn D. Rudebusch. Talor s Rule and he Fed: , Federal Reserve Bank of San Francisco Economic Review 3, 1998, pp Rudebusch, Glenn, and Lars E.O. Svensson. Polic Rules for Inflaion Targeing, in John B. Talor (ed.), Monear Polic Rules, Universi of Chicago Press, Chicago, 1999, pp Smes, Frank, and Raf Wouers. An Esimaed Dnamic Sochasic General Equilibrium Model of he Euro Area, Journal of he European Economic Associaion, forhcoming 2003a. Smes, Frank, and Raf Wouers. Shocks and Fricions in U.S. Business Ccles: A Baesian DSGE Approach, manuscrip, 2003b. Sock, James H., and Mark W. Wason. Has he Business Ccle Changed and Wh?, NBER Macroeconomics Annual, 2002, pp