Forecasting of a nonlinear DSGE model. Abstract. A medium-scale nonlinear dynamic stochastic general equilibrium

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1 orecasing of a nonlinear SE model y Sergey Ivashchenko Absrac A medium-scale nonlinear dynamic sochasic general equilibrium (SE) model is esimaed (54 variables 29 sae variables 7 observed variables). The model includes an observed variable for sock marke reurns. The roo-mean square error (RMSE) of he in-sample and ou-of-sample forecass is calculaed. The nonlinear SE model wih measuremen errors ouperforms AR () VAR () and he linearized SE in erms of he qualiy of he ou-of-sample forecass. The nonlinear SE model wihou measuremen errors is acually of a qualiy equal o ha of he linearized SE model. Keywords: nonlinear SE; Quadraic Kalman iler; QK; ou-ofsample forecass. JEL-codes: E32; E37; E44; E47. S. eersburg Insiue for Economics and Mahemaics (Russian Academy of Sciences); Tchaikovsky Sr. S. eersburg 987 RUSSIA glucke_ru@pisem.ne; el:

2 . Inroducion One of he mos popular approaches for analysis of he macroeconomic environmen is he use of dynamic sochasic general equilibrium (SE) models. This ype of models is he base of modern macroeconomic heory and is widely used by cenral banks and oher policy-making insiuions (Tovar 2009). SE models have a srong microeconomic foundaion. The advanage of such an approach is a descripion of models in erms of deep srucural parameers ha are no influenced by economic policy (Wickens 2008). The usage of SE models requires knowledge abou heir behavior which depends on parameer values. ifferen economeric echniques are employed for model esimaion bu he empirical lieraure has focused on he esimaion of firs-order linearized SE models (Tovar 2009). Compuaion wih linear approximaion is much faser han higher order approximaion bu is behavior can differ from ha of more accurae approximaions (see Collard and Juillard 200). Second-order approximaion can make he difference beween behavior of models and ha of approximaion much smaller. Nonlinear approximaions of SE models have several oher advanages; in paricular hey allow uncerainy o influence economic choices (Ruge-Murcia 202). The likelihood funcion is sharper for nonlinear approximaions which means a more accurae esimaion of he parameers (An and Schorfheide 2007; ernandez-villaverde e al. 200). 2

3 ecause of hese advanages of nonlinear esimaion forecasing is likely o be of high qualiy. Many sudies demonsrae he high qualiy forecasing of linear approximaions of SE models (Adolfson e al. 2007; Smes and Wouers 2004). A large porion of models forecas a small number of variables (Rubaszek and Skrzypczynski 2008; el Negro and Schorfheide 202). However in some sudies linearized SE ouperforms VAR and AR models in erms of ou-of-sample forecasing wih a large number of variables (Ivashchenko 203). In a few sudies small-scale nonlinear SE models are esimaed (ichler 2008; us e al. 202; ernandez-villaverde e al. 200). Mos of hem use only hree observed variables: oupu nominal ineres rae and inflaion (Amisano and Trisani 200; ichler 2008; alcilar e al. 203; us e al. 202). A few sudies use oher observed variable (oh 20 use addiional daa abou yield curve; Hall 202 use consumpion insead of oupu). The paricle filer is used in he sudies described above. New resuls demonsrae he grea advanage of alernaive approaches over he use of paricle filers (Andreasen 2008; Ivashchenko 204; Kollmann 204). Mos of esimaed small-scale nonlinear SE models do no provide informaion abou ou-of-sample forecass qualiy. The forecasing qualiy of nonlinear SE is nearly he same (or slighly worse) han ha of linearized SE model according o ichler (2008) (virually he only sudy ha discusses ou-of-sample forecass of esimaed SE models); bu he 3

4 corresponding model does no include observed variables which are sensiive o nonlineariies. This sudy presens an esimaed medium-scale nonlinear SE model wih seven observed variables including sock marke reurns. The SE model is described in secion 2. Secion 3 presens informaion on esimaion echniques and daa uilized. Secion 4 describes esimaion resuls and qualiy of forecass (in-sample and ou-of-sample). Secion 5 presens some conclusions. 2. Model The SE model includes four ypes of agens: householders firms governmen and he foreign secor. The srucure of he model is presened in igure. The SE model includes cenral New-Keynesian feaures (for example sicky price and adjusmen coss in invesmen). Labor Households Socks irms Transfers oods onds oreign secor Tax overnmen igure.. Srucure of SE model 4

5 TALE. The SE model variables Variable escripion Saionary variable Value of bonds bough by firms in period b H W Value of bonds bough by governmen in period Value of bonds bough by households in period Value of bonds bough by foreign secor in period b H b H W b W C Consumpion a ime c C ividends a ime d C overnmen expendiure a ime g H Habi a ime h H I Invesmens a ime i I K Capial a ime k K L Labor a ime l L M M Money sock in period m nx NX NX Ne expor in period rice of goods in period p p rice for goods of firm in period R Ineres rae in period r R S rice of socks in period S s Tax rae in period T TR Transfer from governmen in period TR W Wage in period W w X Amoun of socks bough by householders in period x X Aggregae demand in period y Oupu of firm in period y Exogenous process corresponding o elasiciy of producion funcion z Exogenous process corresponding o ineremporal z T preferences of households Exogenous process corresponding o convenional level of deb pressure z 5

6 H I L M Exogenous process corresponding o sickiness of C z households bond posiion H H z expendiure z I I efficiency of invesmens C z of labor L L C z preferences of households M M Exogenous process corresponding o governmen Exogenous process corresponding o decreasing Exogenous process corresponding o households amoun Exogenous process corresponding o liquidiy NX Exogenous process corresponding o ne expor z NX NX Exogenous process corresponding o level of price z sickiness R Exogenous process corresponding o moneary policy z R R Exogenous process corresponding o axaion policy z TR Exogenous process corresponding o ransfers policy z TR TR Exogenous process corresponding o echnological z developmen 2. Householders Households maximize he expeced sum of heir discouned uiliy funcions () wih budge resricion (2). Householders do no own capial bu hey can inves in domesic socks and bonds as a means of saving money. The uiliy funcion consiss of he propensiy o consume wih a habi effec disuiliy of labor money a he uiliy funcion and disuiliy of bond posiion deviaion from preferred level. E C M 0 C L C hc H L L C L 2 max () C L M X M H M H H X S ) H S TTR ( W L M R X (2) where C is consumpion in period L is labor supply is period M is money sock in period is he price of goods in period H is he value of 6

7 bonds bough by householders in period S is he price of socks in period X is he amoun of socks bough by householders in period is he ax rae in period T TR is he ransfer from governmen in period R is he ineres rae on bonds in period and is dividends of socks in period. 2.2 inished goods-producing firms erfecly compeiive firms produce he final good using he inermediae goods j and he CES producion echnology: /( ) ( ) / j dj (3) 0 rofi maximizaion and zero profi condiion for he finished goods producers imply he following price level and demand funcion for he inermediae good j: j. j (4) /( ) j dj (5) Inermediae goods-producing firms irms maximize heir expeced discouned uiliy funcion (6) wih resricions. The uiliy funcion consiss of dividends flow and wo rigidiies (sickiness of bond posiion and price sickiness in he Roemberg form (Lombardo and Vesin 2008)).irms are working in a marke wih monopolisic compeiion; herefore hey have a demand resricion (7). The 7

8 8 budge resricion (8) and producion funcion (9) is common. Resricion of capial evoluion (0) conains invesmen rigidiy. L I K k k p R E max (6) (7) R W L I (8) K L (9) 2 ) ( y I I I K K I (0) where is he dividends of he firm in period is he oupu of firm in period is he price for goods of firm in period I is he demand for invesmens goods in period is he aggregae demand in period is he price level for domesic goods in period is he value of bonds bough by he firm in period K is he amoun of capial used by he firm in period and L is he amoun of labor used by he firm in period. 2.4 overnmen foreign secor and balance equaions The governmen makes is decisions according o policy rules and budgeary resricions. The governmen has he following budgeary resricion:

9 9 TR M R W L M T () The moneary policy rule is as follows: R R R R R y p R R (2) The fiscal policy rules are as follows: y b (3) y b T T TR TR TR TR TR TR TR (4) T T T T T y b (5) The foreign secor is exogenous. I has a budgeary resricion (6) and is subjec o an exogenous rule (7). W W R NX (6) NX W W NX NX NX b NX NX (7) The hree balance resricions are as follows: each bond should be bough by someone (8) he amoun of socks is equal o one (9) and aggregae demand consiss of consumpion invesmens governmen

10 consumpion and ne expors (20). ormula (2) denoes how he habi is formed. 0 (8) H W X (9) C I NX (20) H h H C (2) h All he exogenous processes are AR () wih he following parameerizaion: z * 0* ( * ) * z* * (22) 3. Esimaion Of he mehods used for non-linear approximaions of SE models he perurbaion mehod is he mos widely used (Schmi-rohe and Uribe 2004) and hence i is used in his sudy. The maximum likelihood mehod is used for parameers esimaion. A few nonlinear filers can be used for calculaion of he likelihood funcion. One is he paricle filer which is used in mos sudies esimaing nonlinear SE models (ichler 2008; Hall 202; oh 20). However i is oo slow for implemenaion wih medium-scale models. Anoher is he cenral difference Kalman filer (CK) which ouperforms he paricle filer (Andreasen 2008). However he quadraic Kalman filer (QK) is used for likelihood calculaion because i produces a beer qualiy of parameers esimaions han he CK (Ivashchenko 204). The QK was slighly 0

11 slower han he CK (Ivashchenko 204) bu afer he program code was improved i became 6 imes faser and ouperformed he CK in erms of speed. The QK is based on a normal approximaion of densiy. Approximaion wih he perurbaion mehod produces equaion (23) which describes he daa generaing process for sae variables (X ). Equaion (24) describes he dependence beween observed variables ( ) and sae variables. Exogenous shocks ( ) and measuremen errors (u ) have a normal disribuion wih zero mean and covariance marices and u. X X C A A A X xx x X X X 0 (23) X S X u (24) The updaing sep is similar wih he Kalman filer owing o he lineariy of equaion (24). The predicion sep is based on an assumpion of normal disribuion of he sae variables vecor (X - ). The expeced value of vecor X is a funcion of he mean and covariance of vecors X - and. The covariance of vecor X is a funcion of he firs second hird and fourh momens of vecors X - and. However he hird and fourh momens of a vecor wih a normal disribuion are a funcion of he mean and covariance. Thus he QK compues he firs and second momens of he sae variables vecor and assumes ha i has a normal disribuion.

12 An alernaive approach for nonlinear approximaion is he pruning mehod (Kim e al. 2008). There is a nonlinear filer for he pruning approximaion (Kollmann 204). I is faser han he QK (before opimizaion of he program code) for small-scale models bu i is much slower (by abou 5 imes) for medium-scale (wih 20 sae variables) models (Kollmann 204). The SE model described above has 54 variables (29 sae variables); his is an addiional reason for he usage of he QK. The model is esimaed wih quarerly daa from he USA since 985Q unil 203Q2. The following observed variables are used: logarihm of consumpion as a fracion of (obs C ); logarihm of governmen expendiure as a fracion of (obs ); logarihm of compensaion of employees as a fracion of (obs WL ); 3-monh euro-dollar deposi rae (obs R ); growh rae (obs ); growh rae of he deflaor (obs ); and MSCI USA gross reurn (obs STR ). The SE model is esimaed 4 imes (linearized model wih he Kalman filer and second order approximaion wih he QK; wih and wihou measuremen errors for obs STR ). 4. Resuls Esimaion resuls are presened in Table 2. Some ineresing deails regarding hese are as follows. The moneary policy parameer R is less han. Many sudies have valued his parameer a greaer han (.045 ernandez-villaverde e al. 200;.66 Smes and Wouers 2004; 5.0 us e al. 202) bu in ohers i is less han (0.63 nonlinear esimaion Hall 2

13 202). Low values of R require addiional commens: he log-likelihood value of he SE model wih resricion ( R >) is less han 2900 which is much worse han wih he oher esimaions (he QK wihou measuremen errors is ; he QK wih measuremen errors is 2986.; he line esimaion wihou measuremen errors is ; and he line esimaion wih measuremen errors is ). OLS esimaion of moneary policy rule (from 990Q) produces R =0.39. TALE 2. The SE model esimaion resuls QK line wihou measur.er. wih measur. error wihou measur.er. wih measur. error aram. value sd value sd value sd value sd sd 2.60E-0 2.6E E E E-02.84E E-02.39E-02 sd 5.9E E E E E E-05 3.E E-05 sd 7.5E-02.0E-0.63E E E E E E+00 sd H 5.52E-0.6E E-0.60E E E E+00.7E+00 sd 2.32E E E E E-02.08E-02.36E E-05 sd I 3.55E E-05.92E E E E E E+03 sd L 2.25E E-05.3E E-05.00E E E+0.20E+0 sd M.77E E E-0.57E-0.00E E E E-0 sd NX 3.70E E E+00.08E E E E E-02 sd.04e E E-02.75E E E+0 4.6E-04.59E+03 sd R.93E E E E E E E E-03 sd 2.76E E E E E E E E-03 sd TR 2.28E E E E E E E E-05 sd 3.42E E E-03.82E E E E E-04 sd obs STR E E E E-03 NX 2.09E E E E-0-2.2E E E-0.22E-0 NX 5.00E E E+00.8E E E E-0.6E E-0 2.8E E-0.77E E E E E-02.62E E-0.62E E E E E E-0-2.4E E E E E E E E-05 TR 7.8E-0 6.6E E-0 4.0E E-0 9.7E E E-03 TR -3.65E-02.20E E-0 3.2E-0 6.5E E E E-0 TR -4.63E E E E E E E+00.75E E-0.26E E-0.57E E-0.88E E-0.88E E-0 8.3E E E E E E E E E E E E E E E-0 R 8.80E-0 2.E E-0.52E E-0.64E E E-02 R.5E E E-02.55E E E E E-05 R 7.54E-0.24E-0.7E E E-0 2.2E E-0.50E E-0.80E E E E E E E-05 h C 3

14 h h H H 0 0I 0L 0M 0NX 0 0R 0 0TR 0 H I L M NX R TR C L 3.50E-0.68E E-0.24E E E E-0 2.9E E E E E E-08 5.E E E E+00.0E-0 6.0E E E E-03.08E E E E E E E E E E E E E E E E E E E E E E E E E E-0.94E+0 2.2E+00.92E E+0.95E E-02.30E+00.03E E E E E E+00.02E E+00.06E-02.84E E+00.9E+0 2.3E+00.2E E E E E+00.0E E+00 9.E E E E E E E-0 -.7E E E E E E-05.8E E-03.0E-0.57E-02.78E E E E E E E E E+00.25E E+00 2.E E E-03.4E-02.25E-03.35E E E E E-0 4.2E E E E E E-0 2.2E E E E E E+00.03E E+00.5E E E E-03.37E E E E E E E E E E E E-0 8.9E E E E E E-0.28E E E E E E E E-0.25E E E E-0.56E E E E E E-0.26E E E E-0.70E E-0.30E-0 -.6E E E-0 2.2E E E-02.98E E E-03 2.E+00.08E-0 9.8E E E E E E-0.82E E-0.9E E-0.8E E E E E E E E-0 2.7E E E E E E E E-02 7.E E E E E E E E E E E E E E E E E E E-02.42E-0.3E E-0 9.7E E-0 5.8E E E E E E E-02.66E E-02.86E E E-02.03E-0.7E E-02.5E E-0.2E E-05.20E E-05.58E E-02.85E E E E E E-05.00E E-05.00E-02 2.E-05.00E E E E E E E E E E-05.3E E-05 Anoher imporan deail of he esimaion resuls is he high values of he sandard deviaion of he measuremen errors (6.6% - QK 6.7% - line esimaion 7.2% - sandard deviaion of obs STR ). This could be a resul of MSCI USA properies: i includes inernaional companies (such as ALE and JOHNSON & JOHNSON) which have a large porion of heir producion and sales in foreign counries. Idenificaion of a few sandard deviaions ( H and I ) is weak wih linear approximaion. However his problem does no 4

15 exis for he QK. The sandard deviaion of L is very sensiive o esimaion echnique (i is high for he linear esimaion and almos zero for he QK). Some auocorrelaion coefficiens ( L M and TR ) are sensiive o esimaion echnique as well (hey are close o wih one esimaion echnique and close o 0 wih anoher). TALE 3. RMSE of in-sample forecass SE QK SE QK SE line SE line VAR() AR() no meas.er. meas.er. no meas.er. meas.er. obs C (+) 4.60E E E E E E-03 obs (+) 7.90E E E E E E-03 obs (+) 5.00E E E E E E-03 obs (+).66E-03.84E E-03.74E E-03.93E-03 obs WL (+) 6.49E E E E E E-03 obs R (+).07E-03.29E-03.5E-03.5E-03.6E-03.3E-03 obs STR (+) 6.59E E E E E E-02 obs C (+2) 5.36E E E E E E-03 obs (+2).26E-02.58E-02.29E-02.29E-02.63E-02.45E-02 obs (+2) 5.22E E E E E E-03 obs (+2).88E E E-03.96E E E-03 obs WL (+2) 7.E E-03.6E E E E-03 obs R (+2).78E E-03.90E-03.90E-03.97E-03.86E-03 obs STR (+2) 6.72E E E E E E-02 obs C (+3) 5.87E E E E E E-03 obs (+3).64E E-02.72E-02.7E E-02.97E-02 obs (+3) 5.26E E E E E E-03 obs (+3) 2.0E E E E E E-03 obs WL (+3) 7.48E E-03.27E E-03.27E E-03 obs R (+3) 2.40E E E E E E-03 obs STR (+3) 6.76E E E E E E-02 obs C (+4) 6.64E E-03.08E-02.03E-02.04E E-03 obs (+4).99E E E E E E-02 obs (+4) 5.32E E E E E E-03 obs (+4) 2.E E E E E E-03 obs WL (+4) 7.72E-03.E-02.4E-02.03E-02.62E-02.08E-02 obs R (+4) 2.89E E E E E E-03 obs STR (+4) 6.87E E E E E E-02 average RMSE.48E-02.65E-02.7E-02.64E-02.76E-02.65E-02 roo mean square RMSE 2.64E E E E E E-02 forecass no worse han VAR forecass no worse han AR

16 The RMSE of he forecass are presened in Table 3 (in-sample) and Table 4 (ou-of-sample). Ou-of-sample forecass are compued for he las 22 quarers (his means he re-esimaion of parameers wih daase wihou he las quarer (from985q unil 203Q) and compuaion of forecass; he reesimaion wihou 2 quarers (from985q unil 202Q4) and so on; he las re-esimaion use daase wihou 22 quarers(from985q unil 2007Q4)). TALE 4. RMSE of ou-of-sample forecass SE QK SE QK SE line SE line VAR() AR() no meas.er. meas.er. no meas.er. meas.er. obs C (+) 5.23E E E E E E-03 obs (+).07E-02.29E E E E-03.05E-02 obs (+) 7.33E E E E E E-03 obs (+) 2.9E E E-03.86E E E-03 obs WL (+).4E-02.3E E-03.02E-02.2E-02.09E-02 obs R (+).22E-03.52E-03.64E-03.55E-03.56E-03.55E-03 obs STR (+).0E E-02.07E-0.02E E-02.0E-0 obs C (+2) 6.9E E E E E E-03 obs (+2).88E E-02.59E-02.6E-02.93E-02.95E-02 obs (+2) 8.3E E E E E E-03 obs (+2) 2.74E E E E E E-03 obs WL (+2).04E-02.8E-02.3E E-03.7E-02.24E-02 obs R (+2).72E E E E E E-03 obs STR (+2).07E-0.02E-0.03E E E E-02 obs C (+3) 8.52E E-03.3E-02.00E-02.29E-02.0E-02 obs (+3) 2.78E E E E E E-02 obs (+3) 9.04E E E E E-03.08E-02 obs (+3) 3.40E E E E E E-03 obs WL (+3).0E-02.37E-02.45E-02.00E-02.29E-02.57E-02 obs R (+3) 2.00E E E E E E-03 obs STR (+3).08E-0.03E-0.08E-0.02E-0.00E-0.02E-0 obs C (+4) 8.96E E-03.59E-02.26E-02.62E-02.27E-02 obs (+4) 3.78E E E E E E-02 obs (+4) 9.04E E-03.04E E E-03.3E-02 obs (+4) 3.77E E E E E E-03 obs WL (+4).2E-02.78E E-02.25E-02.75E-02.9E-02 obs R (+4) 2.E E E E E E-03 obs STR (+4).07E-0.02E-0.09E-0.0E-0.0E-0.0E-0 average RMSE 2.30E E E E E E-02 roo mean square RMSE 4.6E E E E E E-02 forecass no worse han VAR

17 forecass no worse han AR The VAR model produces he bes in-sample forecass; his may be explained by he larger number of parameers (VAR 84 parameers AR 2 parameers SE 64 or 65 parameers depending on he exisence of measuremen errors). The RMSEs of he ou-of-sample forecass are drasically higher han hose of he in-sample forecass because of he financial crisis of The qualiy of he in-sample forecas wih measuremen errors is beer for line and quadraic esimaions. However he siuaion wih ou-of-sample forecass is differen; line forecass wih measuremen errors are worse han wihou measuremen errors. The in-sample qualiy of line and quadraic forecass wih measuremen errors is nearly he same. Quadraic forecass wih measuremen errors ouperform all oher models in erms of ou-of-sample RMSE. I ouperforms each of he oher models for more han 2/3 of he variables. A comparison of line and quadraic esimaion wihou measuremen errors shows a small advanage of line esimaion (4 variables forecass are beer han wih quadraic esimaion of he same model) which is in line wih he resuls of ichler (2008). I should be noed ha forecass (for 2 3 and 4 quarers) of sock marke reurns by he SE model ouperform he AR and VAR models despie problems relaed o inernaional companies. 5. Conclusion In his sudy he medium-scale nonlinear SE model was esimaed. The SE model includes sock marke reurns bu observed daa (MSCI 7

18 USA gross reurn) describes inernaional companies. Thus measuremen errors (for he sock reurns variable) increase he qualiy of he model wih nonlinear esimaion (however i does no change he qualiy of he linear esimaed model). Measuremen errors have a high sandard deviaion. The qualiy of he ou-of-sample forecass of he SE models wihou measuremen errors is almos equal (slighly worse) o ha of AR () and VAR () models. The qualiy of he SE model wih linear and nonlinear esimaions is acually equal. In he case of he exisence of measuremen errors he siuaion is differen: he nonlinear SE model ouperforms all oher models (including linearized SE). Thus his sudy finds ha nonlinear SE models are more sensiive o misspecificaion (a negaive effec of sharper likelihood) and ha achieving an advanage from nonlinear approximaion requires more realisic model han in he case of a linearized model. References Adolfson M. Lindé J. and Villani M orecasing performance of an open economy SE model. Economeric Reviews 26(2-4) Amisano. and Trisani O Euro area inflaion persisence in an esimaed nonlinear SE model. Journal of Economic ynamics and Conrol 34(0)

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