We Appendix for: Monetry-Fiscl Policy Interctions nd Indetermincy in Post-Wr U.S. Dt Sroj Bhttri, Je Won Lee nd Woong Yong Prk Jnury 11, 2012
Approximte model Detrend The technology process A t induces common trend in output Y t, consumption C t ; rel wge ~wt(i) P t, government purchses G t, government det B t =P t, tx revenues T t, nd trnsfers S t. In ddition, introducing non-zero stedy stte in tion lso cretes trend in nominl prices. Since we will solve the model through locl pproximtion of its dynmics round stedy stte, we rst detrend the vriles s ~Y t Y t ; Ct ~ C t ; nd ~w t(i) w t(i) : A t A t P t P t A t Note tht the scl vriles, t = B t =P t Y t, g t = G t =Y t, t = T t =Y t, nd s t = S t =Y t re lredy sttionry. We then rewrite the equilirium conditions in terms of the detrended vriles, compute the non-stochstic stedy stte, nd then tke rst-order pproximtion round the stedy stte. First order pproximtion We de ne the log devitions of vrile X t from its stedy stte X s ^X t = ln X t ln X, except for four scl vriles: ^ t = t, ^gt = g t g; ^ t = t, nd ^s t = s t s. We denote the men growth rte of the technology shock t A t =A t 1 y. The pproximted model equtions re then given y ~C t = + E ~ t Ct+1 + ~C t 1 ^Rt E t^ t+1 + + + + E t^ t+1 + ^ t + + ^d t ^ t = 1 + E t^ t+1 + 1 + ^ (1 ) (1 ) t 1 + ' + Y ~ t ~ Y t Y ~ t 1 (1 + ') (1 + ) ^R t = R ^Rt 1 + (1 R ) (^ t ^ t ) + Y Y ~ t Y ~ t + " R;t ^ t = ^ t 1 + (1 ) ^t 1 ^ t 1 + Y Y ~ t Y ~ t + g^g t + " ;t ^t = 1^t 1 + 1 ^Rt 1 ^ t Y ~ t + Y ~ t 1 ^ t + ^g t ^ t + ^s t ~Y t = ~Y t 1 + ' ( ) + [' ( ) + ] (1 g) ^g t ~Y t = Ct ~ + 1 1 g ^g t ^g t = g^g t 1 + " g;t ^d t = d ^dt 1 + " d;t ^ t = ^ t 1 + " ;t u t = u u t 1 + " u;t s t = s s t 1 + " s;t ^ t = ^ t 1 + " ;t ^ t = ^ t 1 + " ;t [' ( ) + ] (1 g) ^g t 1 ' ( ) + ^ t: ~Y t 1 + ^u t 1
where we hve de ned two scled shocks Solution nd estimtion method ^d t = (1 ) ^ t ^u t = (1 ) (1 ) 1 t : (1 + ') (1 + ) 1^ For detils of the solution method under determincy, see Sims (2002). For detils of the solution method under indetermincy, see Luik nd Schorfheide (2004). We follow the method of Luik nd Schorfheide (2004), except for one modi ction tht we descrie elow. To chrcterize the posterior distriution of the structurl prmeters, Mrkov Chin Monte Crlo simultion is used. We rst nd mode of the posterior density numericlly using csminwel y Christopher A. Sims. Then we use rndom-wlk Metropolis lgorithm to drw smple from the posterior distriution. The proposl density of the rndom-wlk Metropolis lgorithm is Norml distriution whose men is the previous successful drw nd vrince is the inverse of the negtive Hessin t the posterior mode found efore the simultion. The vrince of the proposl density is scled to chieve n cceptnce rte of round 30%. For detils of the rndom-wlk Metropolis lgorithm, see An nd Schorfheide (2007). Mrginl likelihoods re estimted using the modi ed hrmonic men estimtor y Geweke (1999). An identi ction prolem in Luik nd Schorfheide (2004) Luik nd Schorfheide (2004) use left singulr vectors corresponding to zero singulr vlues of mtrix singulr vlue decomposition (SVD) to chrcterize the full set of indetermincy solutions, or multiple equiliri, of liner rtionl expecttions (LRE) model. However, these left singulr vectors re not identi ed ecuse their singulr vlues re degenerte. This ppers to cuse numericl instility in their solution method. For exmple, smll chnges in prmeter vlues cn esily led to lrge chnge in the likelihood of LRE model under indetermincy. Becuse of this prolem, in Eq.(1) of the min text is not well identi ed. Since in our model the degree of indetermincy is t most one, is simply vector. We identify y normlizing its rst entry to its norm. With this normliztion, posterior density mximiztion nd simultion of our model is stle nd works well. The normliztion would ect the posterior distriution of the entries of the mtrix M in Eq. (1). However, those prmeters in M do not hve ehviorl interprettions. Wht mtters is the dditionl chnnel for the propgtion of the fundmentl shocks, M, whose posterior distriution is not ected y the normliztion if the prior distriution for the entries of M is t. Although our seline prior for the entries of M is not completely t, it is very di use nd the e ect of the normliztion is not signi cnt. We tried di erent speci ctions for the prior distriution for the entries of M; including uniform prior distriution over ( 5; 5) nd our results were roust to these vritions. The sme rgument pplies to those prmeters relted to the sunspot shock t. 2
Dt De nitions nd sources We use the following de nitions for our dt vriles: per cpit output = (personl consumption of nondurle+personl consumption of services+government consumption) / civilin noninstitutionl popultion; nnulized in tion = 400 log(gdp de tor); nnulized interest rtes = the qurterly verge of dily e ective federl funds rtes; tx revenues = current tx receipts + contriutions for government socil insurnce; government det = mrket vlue of privtely held gross federl det; nd government purchses = government consumption. Note tht we use sinlge price level, GDP de tor, for ll the model vriles (e.g. output, government det, tx revenues, nd government purchses). The e ective federl funds rte nd civilin noninstitutionl popultion dt were otined from the FRED dtse of Federl Reserve Bnk of St. Louis. The mrket vlue of privtely held gross federl det series ws otined from Federl Reserve Bnk of Dlls. All the other dt were tken from Ntionl Income nd Product Accounts (NIPA) tles. Mesurement equtions The mesurement equtions of our model re then given y: log(rel per cpit output) = ~ Y t ~ Y t 1 + ^ t + where the following reltionships hold Annulized in tion = 4^ t + 4 Annulized interest rtes = 4 ^R t + 4 ( + + ) Nominl tx revenue Nominl output = ^ t + Nominl government det Nominl output = ^ t + Nominl government purchses Nominl output = ^g t + g 1 = 1 + = 1 + = 1 + = = : 3
Smple mens References pre-volcker (1960:1-1979:2) post-volcker (1982:4-2008:2) 0.48 0.57 4.38/4 2.57/4 R 5.47/4 5.45/4 25 24.33 35.97 48.46 g 24.38 21.43 An, Sunge nd Frnk Schorfheide. 2007. Byesin Anlysis of DSGE Models. Econometric Reviews, 26(2-4): 113-72. Geweke, John. 1999. Using Simultion Methods for Byesin Econometric Models: Inference, Development, nd Communiction. Econometric Reviews, 18(1): 1-73. Luik, Thoms A. nd Frnk Schorfheide. 2004. Testing for Indetermincy: An Appliction to U.S. Monetry Policy. Americn Economic Review, 94(1): 190-217. Sims, Christopher, A. 2002. Solving Liner Rtionl Expecttions Models. Computtionl Economics, 20(1-2): 1-20. 4