Parameter Drifting in an Estimated DSGE Model on the Czech Data * Jiří Polanský Faculty of Economics and Administration, Masaryk University, Lipová 41a, 602 00 Brno and Macroeconomic Forecasting Division, Czech National Bank, Na Príkopě 28, 115 03 Praha 1. e-mail: Jiri.Polansky@cnb.cz Jařomíř Tonneř Faculty of Economics and Administration, Masaryk University, Lipováa 41a, 602 00 Brno and Macroeconomic Forecasting Division, Czech National Bank, Na Prrákopre 28, 115 03 Praha 1, corresponding author. e-mail: Jaromir.Tonner@cnb.cz Osvald VaSícek Faculty ofEconomics and Administration, Masaryk University, Lipováa 41a, 602 00 Brno. e-mail: osvald@econ.muni.cz Abstract. In the paper, we investigate a possible drifting of structural parameters in an estimated small open economy DSGE model. To do this, we first estimate the model with a Bayesian method on the Czech data and discuss results. Then, we identify trajectories of structural parameters via a non-linear filtration based on the model's second-order approximation. We identify two drifting parameters, namely the import share of export and the import share of consumption whose movements are related to the significant exchange rate movements. The rest of parameters seems to be relatively stable in time. Keywords: DSGE models, time-varying parameters, Kalman filter, Bayesian methods, Particle filter JEL classification: D58, E32, E47, C11, C13 1. Introduction The stability of an economy's structural parameters in a medium term is an important assumption for many current macro models. A majority of dynamic stochastic * Financial support of the Specific Research Fund at the Faculty of Economics and Administration and of the Research Project B2/10 at the Czech National Bank is gratefully acknowledged. The views expressed here are those of the authors, and do not necessarily reflect the position of the Czech National Bank. We are grateful to Jan Bruha, Martin Fukac and Milan Vyskrabka for many helpful comments and advices. Also, we would like to thank Michal Andrle, Jaromír Benes, David Havrlant, Tibor Hledik, Ondra Kameník, Radek Snobl, and Jan Vlcek. 1 general equilibrium (DSGE) models, based on micro foundations and exogenous processes, stands on this assumption. The possible drifting of structural parameters, caused by structural changes, might thus cause a bias of many DSGE-based analyses and forecasts with a direct consequence in frequent recalibrations. The parameter drifting in DSGE models might be influenced by the model specification. In general, parameters of DSGE models describe preferences, production structure, wage and price setting behaviour etc. However, DSGE models do not contain only equations based on agents' optimization problems. They are usually complemented with AR processes and technologies to fit a country's stylized facts.1 These exogenous processes might possibly capture some consequences of structural changes and thus allow using a "better-specified" model for a longer time. For example, high foreign direct investment inflow and the entry of the Czech Republic into the EU have affected the volumes of trade balances. Andrle at al. [2] describe the way how to cope with these issues via openness and quality technologies in a DSGE model. For a developing country, this issue might be more important because such an economy goes through frequent structural changes. Twenty years after the revolution, the Czech economy still remains on a converging path towards the more developed countries of the Western Europe. It has been hit by various shocks, some of them bringing structural changes. The European Union (EU) entry might be a good example. With this respect, naturally, there emerges a question about a projection of these changes and shocks into DSGE models' parameters and their drifting in time. There are several papers that aim to identify drifting of structural parameters. Canova [7] estimates a small New-Keynesian model with parameter drifting. He finds the stability of the policy rule parameters and varying parameters of the Phillips curve and the Euler equation. Boivin [4] estimates a Taylor rule with drifting parameters. He identifies important but gradual changes in the policy rule parameters. Fernandez-Villaverde and Rubio-Ramirez [10] estimate a DSGE model using the U.S. data and allow for parameter drifting. On a basis of 184 observations, they find out the changing parameters in the Fed's behaviour and also the drifting of pricing parameters which is correlated with changes in inflation. More recently, Fernandez-Villaverde et al. [9] build a DSGE model with both stochastic volatility and parameter drifting in the Taylor rule and estimate it non-linearly using U.S. data and Bayesian methods. They find out evidence of changes in monetary policy even after controlling for stochastic volatility. Besides, there is a literature on VAR's estimation with time-varying parameters. For example, Sims and Zha [13] do not find any change in parameters either of the policy rule or of the private sector block of their model. Instead, they identify changing variances of structural disturbances. Thus in the paper, we analyze a possible drifting of structural parameters in a relatively complex and on the Czech data estimated DSGE model.2 To do this, we let structural parameters drift and subsequently identify their trajectories via a non-linear filtration method on the model's second-order approximation. 1See [15]. 2In the paper, we do not analyze years prior to 1996 because of an incomplete data set for those years. 2 First we construct the model and check it properties. For our purpose, we need a sufficiently rich and general small open economy model, adapted to the Czech data.3 The model in the paper is based on two existing models. First, we use the model of [6] designed for the Spanish economy as our backbone framework. To cope with the Czech data, we simplify the model and extend it with several features according to [2]. To check the model's performance, we run several tests. More concretely, we carry out the Bayesian estimation of time-invariant model parameters and also check the model properties.4 These tests confirm the model usefulness for analyses based on Czech data via higher order approximations. After the initial estimation and checks, we allow several structural parameters to drift in time. We follow directly methods proposed in [10]. First, we run the Kalman filter on the first-order approximated model. This procedure is the two-step problem where the former consists of adding AR processes into the framework whereas the latter in endogenizing the deep parameters via the AR processes. Adding new exogenous processes helps us to get the model to data. Endogenizing deep parameters then show us a time-varying structure of the model. Second, we run a Particle filter on the second-order approximated model. Nonlinear filtration is necessary for model agents to anticipate future parameter movements. We identify two drifting parameters, namely the import share of export and the import share of consumption. We find the strongest relation between these parameters and significant exchange rate movements. To explain these findings, we employ a simple correlation analysis among these parameters and observables because the standard tools as decompositions to observables are not additive in the case of a non-linear world. For example, if final good producers anticipate considerable exchange rate depreciation, they try to substitute import intermediate goods for their domestic intermediate counterparts. Although the Czech economy has undergone through several structural changes, our estimation does not prove another drifting of structural parameters in the model. For example, the regime switch to the inflation targeting does not influence the Taylor rule parameters. Moreover, the entry to the EU also does not strongly influence any structural parameter. 2. Model This section provides a brief description of the model. Our objective here is to explain the motivation of the model choice, its suitability for the Czech data analysis and its basic structure. On the other hand, we do not present the full description of the model which can be found in the attached Technical Appendix. We follow 3We try to replicate significant Czech economy features. Some of them are modelled very simply, because we do not use the model for regular forecasting. In that case, the model would converge close to the CNB's g3 [2] model developed for these objectives. 4We check the model properties and its performance by the impulse response analysis, the Kalman filtration, the decomposition of endogenous variables into shocks and the decomposition of forecasts of endogenous variables from the steady-state. All of these tests are accomplished on the estimated model. 3 this structure for several reasons. First, the model itself is not the centre of the paper. Instead, it is a tool which we use for the estimations. Second, the model is based on two (Spanish and Czech) existing models, both sufficiently described in the literature and we not aim to replicate existing papers. And finally, it would be almost impossible to describe the model in full detail with a reasonable length with respect to other sections of the paper. 2.1. Motivation Most monetary DSGE models are similar to each other. They consist of several sectors with few general principles of derivation. Optimization problems imply equations describing the main behaviour characteristics of economic agents. Subsequently, the model equations is extended with many features like exogenous processes or wedges to get a final model closer to country's stylized facts.5 The number of these extensions and their variety differ with the purpose of a model, ranging from various general analyses to central banks' regular forecasts where such a core model should capture "all" the main stylized facts of an economy. For our purposes, we need a sufficiently complex and general model, extended with several features to be closer to the Czech data. First, the model should be complex to capture national accounts, wage- and price-setting behaviour and various small open economy features. Second, we need a general model for non-linear filtrations to capture some special Czech economy characteristics. On the other hand, we still aim to use as simple model as possible to be controlled. The model is based on two existing models. First, we use the model of [6], designed for the Spanish economy, as our backbone model. This model follows the current generation of DSGE models for the inflation targeting regime. It is sufficiently rich and general within its sectors structure and contains many well-known modelling features like real and nominal rigidities, technology growths, local currency pricing etc. Moreover, it is also described in literature in great detail.6 To cope with the Czech data, we extend it with several features according to [2].7 Hence, we believe that the model should provide us a sufficient rich tool for the estimations.8 The model has a relatively standard and general structure with optimizing agents and rational expectations. It contains a set of real (internal habit formation, capital adjustment costs etc.) and Calvo-type nominal rigidities with indexation parameters.9 The production structure with intermediate and final goods producing firms 5Note that adding various features into a DSGE framework should not be ad hoc. As authors in [2] note for the case of regulated prices: "In a structural model regulated prices require structural interpretation". We believe that this holds in general. 6See also [10]. 7Authors in [2] describe the new Czech National Bank's (CNB) core model and summarize the main stylized facts of the Czech economy w.r.t. their modelling principles. Also, they discuss some non-standard characteristics of the Czech economy and their corresponding ways how to structurally incorporate them into the monetary DSGE framework. 8We try to replicate all the Czech economy features, but some of them in a very simple way, because we do not use the model for regular forecasting. In that case, the model would converge close to the CNB's g3 model developed for those objectives. 9 As is noted in [6], the model does not contain the Phillips curves. Instead, the derived equations 4 enables to capture the GDP accounts while the local currency pricing mechanism enables the incorporation of a gradual exchange rate pass-through into the model mechanism. The model is closed with a debt-elastic premium according to [12]. The overall structure of the model is described in Figure 1 and more concretely presented in the next subsections. GDP Value Capital G = f(Y), Linear J, C, N Sticky in Home Prices Figure 1: Structure of the Model 2.2. Households Households consume final consumption goods, save in domestic and foreign assets, and supply differentiated labour. The individual household labour types are monop-olistically competitive which provide them a degree of power for the wage setting.10 Also, households own all firms in the model and, thus, finance them internally or receive their dividends. Households' maximize their utility function subject to budget constraint and law of motion for the capital accumulation. The utility function is separable in consumption with internal habit formation, real money balances and the labour supply. The optimization problem has the following form11 from households' and firms' optimizations are left in general forms for performing higher order approximations. 10All differentiated commodities (labour types, intermediate goods) are assumed to be packed by a bundler and then supplied to the firms as a single composite. 11 The first order condition with respect to real money balances is not necessary for the inflation targeting regime. Moreover, the first order condition with respect to Arrow securities is also not necessary because we assume complete markets and separable utility in labour. See [6] and (Erceg 5 dt{\og{cjt - hcjt-i) +vlog^ - zt. (3) where yit is the per capita production of the domestic final good, At is the neutral technology growth, a is the labour share in production of the domestic intermediate goods, 0 is the parameter associated with the fixed cost production, zt is the per capita long run growth. Assuming the Calvo price-setting with the indexation parameters, the second stage consists of the profit maximization by choosing the optimal price maxpwEt f2(pepytf^- (nn*Vi— - mc*+H i/*+r, (4) s.t. yit+r = T\^is-i— Vt+T, where 9p is the Calvo parameter for the domestic good prices, xP is the indexation parameter for the domestic good prices, mct is the real marginal cost, e is the elasticity of substitution among different types of the domestic intermediate goods. On the other hand, the intermediate importers costlessly differentiate the single foreign good which they import from the rest of the world. The packed intermediate imported good is then supplied to the final goods producers (except the government sector). The optimization problem has only one step because the import intermediate firms buys only one foreign good with the straightforward specification for the marginal cost (and hence no need for optimality conditions between two inputs). The price-setting problem has the following form oo maxp«Et ^(^m)t7^ (fl(Ks-i)XM ^ " mc^ Vu+T, (5) 0 At r=0 7 \s=l Pt+r where p|f is the price of goods of importing firms in the domestic currency, 9m is the Calvo for the import prices, nM is the imported good inflation, xM is the indexation of the imported good, mcf = ex*p is the real marginal cost in the importing sector, PW is the foreign price of the foreign homogenous final good in the foreign currency, ext'pif is its foreign price in the domestic currency, yM is the final imported good, eM is the elasticity of substitution among different types of imported goods. V M yt+r 2.4. Final Goods Producing Firms The model contains four final goods producing sectors - consumption, investment, export and government.12 Consumption, investment and export firms purchase both intermediate composite inputs. The monopolistic competition is only within the export sector. Consumption, investment and export final goods producers s = c,i,x maximize profits subject to their CES production functions maxsd sm E; oo r=0 ^7 (p*st PtSt (6) s.t. st (ns) + (l-n')*(sf(l e.s-1 where sd are the domestic consumption, investment and export, sM are the imported consumption, investment and export, p'1 are consumption, investment, and export prices, es are the elasticities of substitution among different types of consumption, investment and export goods, ns are home bias in the aggregation in consumption, investment and export and are adjustment costs in consumption, investment and exports sectors. For exporting firms, there is a second stage optimization problem associated with their market power and the Calvo price-setting max,, Et f>^r7rL^ ( f[(n?+-i)Xx#- " mct+r ) Vu+T, (7) r=0 At \s=1 Pt+r / s.t. Wt+r = ( I[(nX+s-l)XX \s=1 where pxt is the price of the exported goods in the foreign currency, 9x is the Calvo for the export prices, Iljf is the export prices inflation in the foreign currency, \x is 12 The final government spending goods are produced from domestic intermediate goods only and thus there is no optimization exercise. yt+r: 8 the indexation of the exported prices, mcf = e£*px is the real marginal cost in the exporting sector, extp* is the price of the exported goods in the domestic currency, is the demand for the products of exporting firms, ex is the elasticity of substitution among different types of exported goods. 2.5. Policy Authorities A central bank operates under the inflation targeting regime. It sets its one-period nominal interest rate through open market operations according to a Taylor-type rule of the form (/ t v? \ 7n 1-YR where r is a steady state of nominal interest rate, target is the inflation target, yr Taylor rule parameter (rates), yn Taylor rule parameter (inflation), yv Taylor rule parameter (output), Av? is the growth rate of output, At is the growth rate of population, ^m is the monetary policy shock. Hence, it targets the four period-ahead year-on-year headline inflation n4^+4. Our motivation here is to get the model closer to the official monetary policy rule of the CNB.13 For the fiscal policy, we assume a Ricardian setting of a fiscal policy treatment. Besides our effort to focus on the monetary policy and thus a simple fiscal policy, we are aware of possible ambiguities and uncertainties in supposed practical fiscal policy effects.14 Thus, we assume a simple fiscal policy according to gt = pgct + C9t, (9) where gt is the per capita level of real government consumption 2.6. The Rest of the World The rest of the world is represented by the EU and is modelled exogenously. There are many exporters in the EU and their productions enter to production function of domestic importers. Subsequently, there is a bundler. Varieties of domestic exporters are aggregated by bundler, thus exports can be also represented by one aggregate of export prices. xt = vt exty% Vt W extp" -ew W v7, (io) -I I pt log(Rf) = PRW log(Äj-1) + (1 - ) log(RW) + , (11) 13The policy rule still differs from the CNB's model because the central bank in this model also targets the output with a small weight. See [2] for the description of the CNB's monetary policy rule. 14See [3], [5], [11]. 9 log(yf) = Pyw log(yw) + (1 - pyw) log(yw) + # , (12) iog(nf) = pnw log^) + (1 - ) log(nw) + erW, (13) where vf is the export prices dispersion, ew is the elasticity of substitution among different types of world trade goods, y^v is the world demand, nw is the foreign homogenous final good prices inflation. 3. Model Estimation In this section, we present results of the time-invariant Bayesian estimation on quarterly Czech and Eurozone data and discuss the most interesting posterior values of model parameters. The posterior distributions are constructed with the Metropolis-Hastings algorithm15 of the Dynare Toolbox [8]. Finally we present the most interesting results of time-varying parameter estimation. 3.1. Data The quarterly Czech data sample covers 58 observations from 1996Q1 to 2010Q2. We use 16 time series as observables for the estimations. Seasonally adjusted national accounts data stem from the Czech Statistical Office (CZSO). Namely, we use real volumes of consumption, investment, government spending, export, import and their corresponding deflators (except the consumption and government spending deflators).16 The headline CPI inflation also comes from the CZSO. For the wages, we seasonally adjust the time series of the average nominal wage growth in the business sector which stem from the CZSO as well. The data for the labour demand are gained from the Labour Force Sample Survey's seasonally unadjusted time series for "employed in the economy". All series are seasonally adjusted to receive its trend-cyclical component. The exchange rate is the CZK/EUR while the domestic interest rate is the 3M PRIBOR. We use three foreign observables. The foreign interest rate is the 3M EURIBOR. The foreign inflation is the PPI of the effective Eurozone acquired from the Consensus Forecast. Finally, the foreign real economic activity is approximated by the foreign demand, acquired from the GDP of the effective Eurozone which stems also from Consensus Forecast.17 Because of high data uncertainty, we allow for the measurement errors in the model. Prominent examples of this uncertainty might be frequent data revisions, methodology changes, or high volatility of quarter-on-quarter dynamics of several time series, probably partly as a result of the presence of high frequency noise. Measurement errors are incorporated on levels via measurement equations where we let observables to differ from measurements. 151 million draws 16Instead of the consumption deflator, we use the CPI inflation. The government deflator is not necessary because of the simple fiscal policy treatment. 17For the definition of effective variables see Inflation Reports of the CNB. 10 3.2. Priors First of all, we set steady-state growth rates parameters. The overall growth in the model is slightly above 4.5 % a year which is approximately consistent with the previous GDP growth of the Czech economy.18 We assume that the population growth does not play any role in determining the model long-run steady-state growth, and thus, we set it to zero. We set the steady-state nominal appreciation rate to -2.4 % a year. This value corresponds approximately to the data during the relevant period until the beginning of the crisis in 2009. Adding this year to the sample shifts the rate upwards (to the less appreciation rate) because there was a considerable depreciation. In this respect, we assume that financial crises might not affect long-run steady state of an economy. Hence, with our assumption that, ceteris paribus, the Czech economy will return to the long-term appreciation, higher value would bias the calibration. On the other hand, considering the 1998-2008 period would imply stronger appreciation. The steady state inflation corresponds to the 2 % inflation target set in annual terms.19 The foreign inflation steady state is calibrated according to the inflation target of the ECB which corresponds to 2 % annually as well. The foreign demand growth for the domestic export is set at a pace of 9 % a year.20 The steady-state foreign nominal interest rate is calibrated to 4 % annually. 3.3. Posteriors In this subsection, we present point estimates of some model parameters. Our objective here is to underline and discuss the most tangible parameters which have relatively clear counterparts in the real economy. In general, we believe that the Bayesian estimation is an important tool for checking a model's calibration (that our calibrated priors are in line with data) and providing an appreciable informative message about an economy. On the other hand, we are aware of considerable data uncertainty (short time series, structural changes in the data, frequent revisions, gradual convergence of the Czech economy etc.) and possible model misspecification that might potentially bias the estimation. Point estimate of habit formation parameter is relatively high with the value slightly above 0.94. The posterior thus exceeds our prior set to 0.9. We set the prior to this value for two reasons. First, the parameter corresponds to the high level of consumption smoothing in the Czech economy. Second, the decrease of consumption expenditures during the crisis was relatively low with respect to the slump of overall real economic activity and the evolution of wages. With this respect, this fact might be probably partly explained by the social security system and support from government transfers to households. Elasticities of substitution in different sectors differ between 5.0 for the domestic goods and 9.5 for the export goods. These values correspond to the range of average markups between 25 percent and 12 percent. It might be difficult to check the 18 The overall steady-state growth is generated via the neutral technology only. 19In quarterly terms relevant for the model, this target corresponds to (2/400+1=1.005). 20The approximation of foreign demand is four times the EU GDP growth, which is assumed to be 2.5 %. See [2]. 11 Table 1: Estimated Parameters Parameter Prior Dist Posterior Lower and Upper Bound Mean Mean of a 90 % HPD interval Preferences Habits h 0.900 beta 0.9409 ( 0.9326 : , 0.9513 ) Labour supply coef. 8.832 gamma 8.8138 ( 8.7144 : , 8.9094 ) Frisch elasticity ■& 1.250 gamma 1.2556 ( 1.2330 : 1.2820 ) Wedges Euler ..euler K 1.0119 - - Forex ^forex 1.0034 - - Adjustment costs Investment K 20.000 gamma 20.1364 ( 20.0852 20.2006 ) Capital utilization Y 2 0.280 gamma 0.3229 ( 0.3054 0.3445 ) Risk premium 0.800 beta 0.8027 ( 0.7564 0.8553 ) Elasticities of substitution Domestic goods e 5.000 gamma 5.0035 ( 4.9562 5.0508 ) Import goods £M 9.000 gamma 9.0690 ( 9.0415 9.1017 ) Export goods ex 9.400 gamma 9.2646 ( 9.1852 9.3299 ) World goods ew 0.860 gamma 0.6938 ( 0.6090 0.7686 ) Consumption goods ec 7.600 gamma 7.6614 ( 7.6166 7.7022 ) Investment goods ei 7.600 gamma 7.5723 ( 7.5363 7.6029 ) Labour types 7.000 gamma 7.0235 ( 7.0055 7.0428 ) Price and wage setting Calvo dom. prices 9p 0.500 norm 0.6719 ( 0.6197 0.7600 ) Calvo exp. prices 0.080 gamma 0.2386 ( 0.1962 0.2814 ) Calvo imp. prices 6m 0.750 norm 0.7309 ( 0.7046 0.7586 ) Calvo wages 0.380 norm 0.4519 ( 0.4302 0.4744 ) Index. dom. prices 0.750 gamma 0.7111 ( 0.6509 0.7688 ) Index. imp. prices Xm 0.500 gamma 0.4519 ( 0.4187 0.4828 ) Index. exp. prices Xx 0.350 gamma 0.3600 ( 0.3157 0.4191 ) Index. wages Xw 0.920 beta 0.9529 ( 0.9176 0.9859 ) Monetary policy Taylor rule (int. rates) YR 0.960 beta 0.9544 ( 0.9389 0.9716 ) Taylor rule (output gap) Yy 0.220 gamma 0.2233 ( 0.2109 0.2363 ) Taylor rule (inflation) Yn 1.150 gamma 1.1458 ( 1.1307 1.1600 ) Fiscal policy Public consumption Pg 0.750 beta 0.7758 ( 0.7107 0.8488 ) Home bias Home bias in consump. Uc 0.280 beta 0.3453 ( 0.2878 0.4080 ) Home bias in invest. Hi 0.120 beta 0.3297 ( 0.2913 0.3608 ) Home bias in export Ux 0.350 beta 0.4178 ( 0.3873 0.4602 ) Growth rates Invest. spec. tech. Am 1.000 norm 1.0000 ( 0.9998 1.0002 ) General tech. Aa 1.009 norm 1.0090 ( 1.0088 1.0092 ) Population Al 1.000 norm 1.0000 ( 0.9998 1.0001 ) ER appreciation ex 0.994 norm 0.9942 ( 0.9940 0.9944 ) Openness tech. ao 1.0035 - - Export spec. tech. ax 1.0058 - - 12 resulting markups with the micro data because there are no corresponding official series for the Czech economy. The only series available are evolutions of prices in the food branch (agricultural prices, food production prices and food consumer prices) but these tables show only final prices without any detailed specifications of firms' cost. Besides price markups, the wage markup is 16 percent. This value might indicate a relatively significant bargaining power and labour market stickiness. The posterior values of Calvo price-setting parameters are 0.67 for domestic prices, 0.24 for export prices, and 0.73 for import prices with corresponding indexation parameters 0.71, 0.36 and 0.45. These posterior values indicate relatively flexible pricing policies of domestic firms with approximate duration of three quarters. The export sector estimation might signify a higher flexibility of exporting firms with duration slightly above one quarter. On the other hand, import sector seems to be relatively sticky with an approximate duration almost a year. The indexation of wages has the posterior 0.95 implying almost the full indexation.21 The share of domestic consumption goods in the total consumption basket (home bias in consumption) is approximately 35 percent. The similar share is for the investment sector. The home bias for the export sector is slightly higher with the posterior 0.42. The inflation parameter in the monetary policy rule has its posterior slightly above 1.14 whereas the output parameter is more than five times lower with the value of 0.22. The posterior of lagged interest rate parameter equals to 0.95, and thus corresponds to the standard smooth profile of interest rates. 3.4. Time-varying Parameter Estimation As was noted, DSGE models are usually supplemented with exogenous processes to get them closer to the data. Typical examples are sector technologies which capture important features of an economy. With this respect, these processes can be understood as time-varying parameters. We decide to incorporate four exogenous processes to the model: • First, we aim to capture some aspects of high openness of the Czech economy, especially the fact that exports are very import intensive. Thus, we assume the trade openness technology which helps to work with reexport effects in the model consistent way. log(aOt) = pao log(aOt-i) + (1 - PaO) log(ao) + tf. (14) • Second, since there is only a simple relation between government spending and consumption, we assume a government specific technology. log(aGt) = paG log(aGt_i) + £taG. (15) 21 These results might be influenced by presence of indexation parameters in the price- and wage-setting equations. In the model, prices (and wages) are changing due to the reoptimizing and the indexation. Thus, estimation of these two parameters together might be sometimes difficult to interpret. 13 • Third, since the Czech headline CPI inflation is still influenced by regulated prices, we incorporate a regulated prices technology into the model.22 This technology is only a proxy for the regulated prices goods sector. log(aRt) = paR log(aRt-i) + $R, (16) • And fourth, we added two time-varying wedges into the first order conditions of households. Namely, we insert a wedge between long-term growth of the economy and long-term real interest rate in the Euler equation and also a wedge between domestic interest rate, foreign interest rate and the exchange rate appreciation in the UIP. All these processes are according to (Andrle et al., 2009) and have forms k)g(Krkr) = Peuier logK-f) + (1 - Peuler) log^^) + , (17) log(Kf0reX) = pf0rex log(Kf-r) + (1 - p/orex) log^) + foreX. (18) Filtered trajectories of exogenous processes serve as a tool for comparison the model behaviour with our intuition, and thus can possibly show some problems and model misspecification. Figure 2 shows evolution of filtered exogenous processes. Filtration of the government specific technology tells that a ratio of government spending goods with respect to value added is high. A relation between the regulated prices technology and observed regulated prices would be beneficial. The slump of trade openness technology during the crisis shows that there was a huge decrease of reexports in the Czech economy during the first quarter 2009. It is very intuitive, because not only value added is traded. For the time-varying parameter estimation, we need to choose candidate parameters for the drifting. Our first guess comes from the Bayesian estimation. Figure 3 shows parameters whose posterior distributions are considerably bimodal. Also, we choose the parameter for the import intensity of export as a candidate because we can expect that the openness of the Czech economy was changing during the analyzed period.23 As the first exercise, we carry out a time-varying parameter estimation allowing the drifting of parameters when these movements are unanticipated by agents in the model. For parameters par = 9p, xP, nc, nx, pg, eW, we set: part = 0part-i + parss + £far • Hence, it is possible to use the first order approximation of the model and Kalman filter because applying the nonlinear filtration is not necessary (See Figure 4). First, we estimate the Calvo parameter 9p and indexation parameter \v of the domestic intermediate producers. The results indicate the relative stability of these two parameters where their movements are mutually compensating. Second, we focus on 22However, we do not incorporate a direct link to the regulated prices observable. 23In fact, we did not resist the temptation and tried to estimate all parameters as time-varying. 14 Regulated prices technology (QoQ, ann.) Trade openess tech. (QoQ, ann.) 1996:1 ,_2001 1_2006 1 Figure 2: Filtered exogenous processes the import shares of export and consumption (parameters nx and nc). The drifting of these parameters is more significant but without any trend. Another promising example is a price elasticity of exports parameter eW. Fiscal policy parameter pg seems to be stable over time. The next step is a time-varying parameter estimation allowing the drifting which is anticipated by model agents due to the higher order approximation. In such case, one needs to use a nonlinear filter because the model structure is also nonlinear. We use the Particle filter.24 For obtaining the second order approximation, we employ the Dynare Toolbox.25 The difference between first and second order approximations can be showed via following equations Vt = Vs + Ayht-i + But where ys is the steady state value of y and yht = yt — ys. The second order approximation is yt = ys + 0.5A2 + Ayht-i + But + 0.5C (yht-i 0 yht-i ) + 0.5D(u 0 ut) + E (yht-i 0 ut) where ys is the steady state value of y, yht = yt — ys, and A2 is the shift effect of the variance of future shocks. To check both models, we compare impulse responses of the first and second order approximations. The differences between the behaviour of these two approximations are relatively small when assuming one standard deviation shocks. In Figure 5, we show the comparison of five standard deviation total factor productivity shocks. The reactions are not so strong in the case of the second order approximation because risk stems into policy functions (a precautionary behaviour). 24See [1] and [14]. 25 See dynare manual [8] 15 theta_p chi_p n_c 15 10 5 0 20 10 0.4 0.68 0.675 0.67 0.665 0.6 0.8 n_x 15 10 5 0 ( 10 0.4 0.6 0.8 1 1.2 rho_g 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15 10 5 0 15 10 5 0 0.2 0.3 0.4 epsilon_W 0.6 0.8 1 1.2 Figure 3: Bimodal posterior distributions parameters theta 1996:1 2001:1 2006:1 0.716 0.714 0.712 0.71 0.708 chi 1996:1 2001:1 2006:1 0.45 0.4 0.35 0.3 0.25 1996:1 0.9 0.85 0.8 0.75 0.7 1996:1 2001:1 2006:1 rho 2001:1 2006:1 0.5 0.45 0.4 0.35 1996:1 0.74 0.72 0.7 0.68 1996:1 2001:1 2006:1 epsilonw 2001:1 2006:1 model Figure 4: Time-varying parameter estimation - Kalman filter, 1st-order approx. Another important difference between these two approximations is a shift of the steady state. Table 2 presents the shift effects when all time-varying parameters are incorporated. The neutral technological shock variance plays the most important role in explaning the shift. As the second exercise, we focus on application of the standard particle filter (see [1]). Preliminary results of nonlinear filter on the second-order approximated model can be seen in the Technical Appendix. The trajectories on Figure 6 are computed 5 0 p p n n c x ss 16 Investment deflator Export deflator Import deflator Nominal wages 2 5 0 -1 I 5 0 -5 I 4 2 0 I 0 -0.2 -0.4 10 Hours worked -2 20 0 10 Export 5 0 -5 20 0 10 10 Exchange rate -1 0 0 -10 10 20 0 Iterest rates x V / 0 10 20 0 10 Import 20 0 2 1 0 10 Consumption 20 0 10 Investment 20 20 0 10 20 0 10 20 Foreign demand Foreign inflation 10 n int. rates 20 0 10 20 0 10 20 CPI inflation Government spending 10 J0_ 20 20 0 10 20 2nd 1st Figure 5: Technology shock impulse responses (in deviation from the steady state in p.p.) 0 0 0 5 0 Table 2: A2 - shift effect (all estimated time-varying parameters) Variable Shift (%) Investment deflator 0.10 Export deflator 0.98 Import deflator 0.21 Nominal wages -0.05 Hours worked -0.17 Exchange rate -1.11 Consumption growth -0.31 Investment growth -0.44 Export growth -1.47 Import growth -1.64 Foreign demand growth 0.00 Foreign inflation growth 0.00 Interest rate -0.00 Foreign interest rate -0.00 CPI inflation 0.09 Government spending growth -0.31 averages among 50 rerunned non-linear filtrations. The nonlinear estimation confirms significant movements in imports intensity parameters. Especially during 2002 and 2003 we identify a big increase of such parameters which indicate an increase of domestic component in producing consumption and export. Other parameters seem to be stable over time. 17 theta chi 0.9 0.8 0.7 0.6 0.5 0.5 0.4 0.3 0.2 1 0.9 0.8 0.7 2003:1 n 2008:1 2003:1 rho g 2008:1 1 0.9 0.8 0.7 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 „ /"--J 2003:1 2008:1 n x 2003:1 2008:1 epsilonw 2003:1 2008:1 2003:1 2008:1 2nd -1st Figure 6: Nonlinear parameter filtration - Particle filter To complete the analysis, we need a tool for finding out, which observables are responsible for the parameter drifting. Traditional tools like endogenous variables decompositions into observables are not additive because of a nonlinear world. Thus, we employ a simple correlation analysis which shows lead and lag correlations between the drifting and observed time series. From the Figure 7, we can see a strong correlation between parameters and exchange rate movements. We find out the negative correlation between current exchange rate and current import intensity parameters. The higher domestic component of consumption and export implies lower import and thus positive net foreign assets and appreciated exchange rate. This finding is also in line with negative correlation between import and intensity parameters (mainly in the case of the second order approximation). Moreover, we find out a positive correlation between future exchange rate movements and intensity parameters. The depreciation anticipation, in this case, is a strong incentive for consumption and export goods producers to increase the domestic component. Such finding could not be revealed in the case of the first order approximation when agents do not anticipate parameter drifting. p c 18 0.5 0 -0.5 0.5 0 -0.5 0.4 0.2 0 -0.2 0.5 0 -0.5 0.5 0.4 0.2 0 -0.2 Export deflator —> nx -5 0 5 Exchange rate —> nx -5 0 5 Import —> nx -10 -5 0 5 Export deflator -- > nc -10 -5 0 5 Exchange rate -- > nc -5 0 5 Import -- > nc 10 10 \ \ \ \ ^ f 10 Y 1 10 10 0.5 0 -0.5 -1 10 0.5 0.5 -0.5 10 0.5 0 -0.5 -1 0.5 -0.5 0.5 -0.5 10 Import deflator -- > nx -5 0 5 Export -- > nx -5 0 5 Foreign demand —> nx -5 2nd 1st | Import deflator -- > nc 10 -5 0 5 Export -- > nc 10 -5 0 5 Foreign demand -- > nc -5 10 2nd 1st 10 10 > 10 \ Y 10 X / 10 10 Figure 7: Time-varying parameter in time t and observables in time t+j correlations 0 0 0 0 0 5 19 4. Conclusion In the paper, we analyze a possible drifting of structural parameters in a relatively complex and on the Czech data estimated DSGE model. The model is based on two existing models. First, we use the model designed for the Spanish economy as our backbone framework. Second, we extend the original framework by implementing several important mechanisms tailor-made for the Czech economy. Our motivation is to combine a standard approach of building DSGE models with some original ideas to obtain this type of the model in order to study essential behavioural mechanisms. To verify the model properties, we estimate the model using Bayesian technique on the quarterly Czech and Eurozone data and discuss the results. After the initial estimation and checks, we allow several parameters to drift in time. First, we impose some time-varying parameters through exogenous processes as openness technology, regulated prices technology or government specific technology into the model. Then we run the Kalman filter on the first-order approximated model with deep parameter drifting. The nonlinear filtration of the second-order approximated model is understood as the final step, when agents are aware of time-varying nature of the world. We identify two drifting parameters, namely import share of export and import share of consumption. We find out that the strongest relation is between these parameters and significant exchange rate movements. We employ a simple correlation analysis among such parameters and observables to explain these findings, because standard tools as decomposition to observables are not additive in the case of nonlinear world. If final producers anticipate significant exchange rate depreciation, they try to substitute import goods for domestic goods. Although our economy has undergone many changes during the previous fifteen years, our estimation does not confirm that structural parameters have changed in the model. For example, the switch to the inflation targeting does not influence Taylor rule parameters as well as the Czech Republic entry to the EU did not influence any structural parameter. Incorporating exogenous processes like the trade openness technology, regulated prices technology or government specific technology significantly increases model ability to replicate data and thus there is no need to add time-varying parameters. On the other hand, interpretation of exogenous processes filtration might not have a direct structural linkage. References [1] Andreasen, M. M. Non-linear DSGE Models and The Optimized Particle Filter. CREATES Research Paper, 2010-5, 2010. [2] Andrle, M., HlEdik, T., Kameník, O., and Vlček, J. Implementing the New Structural Model of the Czech National Bank. CNB Working Paper Series 2/2009, 2009. [3] Barro, R. J. Are Government Bonds Net Wealth? The Journal of Political Economy 82, 6 (1974), 1095-1117. 20 [4] BoiviN, J. Has U.S. Monetary Policy Changed? evidence from Drifting Coefficients and Real-Time Data. Journal of Money, Credit, and Banking 38, 5 (2006), 1149-1173. [5] Botman, D., Karam, P., Laxton, D., and Rose, D. DSGE Modeling at the Fund: Applications and Further Developments. IMF Working Paper 07, 2007. [6] Bijrriel, P., Fernandez-Villaverde, J., and Rubio-Ramirez, J. F. MEDEA: a DSGE model for the Spanish economy. Journal of the Spanish Economic Association, 1 (2010), 175-243. [7] Canova, F. Monetary Policy and the Evolution of US Economy. Mimeo, 2004. [8] Dynare. http://www.cepremap.cnrs.fr/dynare/. [9] Fernandez-Villaverde, J., Guerron-Quintana, P., and Rubio-Ramirez, J. F. Fortune or Virtue: Time-Variant Volatilities Versus Parameter Drifting in U.S. Data. NBER Working Paper 15928, 2010. [10] Fernandez-Villaverde, J., and Rubio-Ramirez, J. F. How Structural Are Structural Parameters. NBER Working Paper 13166, 2007. [11] Perotti, R. In Search of the Transmission Mechanism of Fiscal Policy. NBER Working Paper 13143, 2007. [12] Schmitt-GrohE, S., and Uribe, M. Closing small open economy models. Journal of International Economics 61 (2003), 163-185. [13] Sims, C. A., and Zha, T. Were There Regime Switches in U.S. Monetary Policy? The American Economic Review 96, 1 (2006), 54-81. [14] Tonner, J., VaSicek, O., Stecha, J., and Havlena, V. Estimation of Time-Variable Parameters of Macroeconomic Model with Rational Expectations. In IFAC Symposium, Computional Economics and Financial and Industrial Systems. Dogus University of Istanbul, Turkey, 2007. [15] Tovar, C. E. DSGE Models and Central Banks. BIS Working Paper 258, 2008. 21 5. Technical Appendix We recapitulate original model equations in the first section. Note that their ordering and description is the same as in the original article [6]. This is our intention, because then we can clearly specify our modifications of the original equations and also some added equations. Further we present a derivation of a steady state for the modified model. Plots of priors and posteriors are given in the fourth section. 5.1. Original Model Equations and Variables The full set of equilibrium conditions (and their derivation) can be found in [6]. We recapitulate them here. Note that all variables are stationarized, so we do not introduce a special notation to emphasize it. Steady states are denoted as variables without time index. We also omit an expectation term. (1) The FOC of households with respect to consumption, bonds, foreign bonds, capital utility, capital stock, investment, real money balances, wages and per capita hours worked where dt is an intertemporal preference shock, ct is a per capita consumption, h is the habit persistence parameter, ( is the discount parameter, zt is the per capita long run growth, yl is the growth of population, At is the Lagrangian multiplier associated with the budget constraint, tc is the tax rate of consumption, p£ is the price level of the consumption final good, pt is the price level of the domestic final good dt-\--h^+A+i---j- = At(l + to) (t^) (19) ct ~ Tt°t-i ct+1zt+1 - hct \ptj where Rt is the nominal interest rate, nt is the inflation of the domestic intermediate good Xt=f^^- (20) where RRV is the foreign nominal interest rate, extbf is an amount of foreign government bonds in the domestic currency, ext is the nominal exchange rate Xt = pxt+i RYVjextbY\$W) ext+i * zt+i nt+i ext where r(extbW, ) is the premium associated with buying foreign bonds, ThW and are the parameter and shock associated with the premium F(extbW ,tf ) = e-rbW (e-bW-exbW )+«bW rt is the real rental price of capital, ut is the intensity of use of capital, Tk is the tax rate of capital income, /zt is the investment-specific technology rt = ^'[Ut] (22) 1 - Tk where ^-1$[ut] is the physical cost of use of capital in resource terms $W=f(«t-l)+ yK-1)2 where qt is the marginal Tobin's Q, 5 is the depreciation rate of capital qsrt+1 = ^t+i^-^1— [(i - S)qt+i + n+iut+i(i - n) + sn - (23) 22 where it is the per capita investments - =* 1-5 7* "-Zt ~S 7« ~-zt lt~-zt\+fjqt+1--S lt+l — zt+1 7t+i — zt+i) (24) where S Pt it — \ rate of investment t 'j^zt is an adjustment cost function on the level of investment, A$ is the growth i - 2 i - where ^ is the per capita real money balances ^=dtv(^^l)~1 (25) Pt V zt+1 nt+i / where r is the elasticity of substitution among different types of labour, tw is the tax rate of wage income, w* is the optimal real wage in terms of the domestic final good, wt is the overall real wage index, is the per capita labour demand, 0w is the Calvo parameter for wages, \w is the indexation parameter for wages h = ^(i -rw)(w;)^xtW^ + fjew^+1 (g^)1" (^p)"1/^! (26) where tp is the labour supply coefficient, ipt is the preference shock, Ii*w = is the optimal wage inflation, •& is the inverse of Frisch labour supply elasticity. , f nxw \-v(1+#) / w* z \n( i+#) ft = Hm (n™ r"(1+*> + pewl^ j (^p) ft+i (27) (2) The intermediate domestic firms can change prices where mct is a real marginal cost, is the per capita aggregate demand of the domestic final good, 0p is the Calvo parameter for the domestic good prices, x is the indexation parameter for the domestic good prices, e is the elasticity of substitution among different types of the domestic intermediate goods g\ = Xtmctvi + peptf+1 (J^J gl+1 (28) where n* is the optimal domestic intermediate goods prices inflation g\ = XtU*yt + (39p^+1 ^ ( J£-) g\+l (29) eg] = (e - 1)g2 (30) where FOC of firms with respect to labour and capital inputs with kt is a per capita capital, a is the labour share in production of the domestic intermediate goods utkt-1 a wt . . —Jd~ = i-zt^t— (31) lr 1 — a rt t mct=(^y~a(i)>^ (32) (3) FOC of importing firms with respect to price where pp^ is the foreign price of the foreign homogenous final good in the foreign currency, extpY is its foreign price in the domestic currency, pf1 is the price of goods of importing firms in the domestic currency, yf1 is the final imported good, Tlf1 is the imported good inflation, 9M is the Calvo for the import prices, xm is 23 the indexation of the imported good, eM is the elasticity of substitution among different types of imported goods / W 9?1 = At {-fr- \yf + /SOmtZ Pt 1 UitrJ 9t+l (33) where p'f is the price of the exported goods in the foreign currency, extp'f is the price of the exported goods in the domestic currency, is the demand for the products of exporting firms, 9x is the Calvo for the export prices, II^ is the foreign homogenous final good prices inflation, II;? is the export prices inflation in the foreign currency, \x is the indexation of the export prices, ex is the elasticity of substitution among different types of exported goods 9T = Xt I -^—r I Vt + ( W~ ) 9tU (34) / i \ /fnW)x*\-Ex where Ilf1 * is the optimal import prices inflation /7TTM)xm \ 1-£M /nM*\ g?> = Atnf -y" + ßeM^+1 (^-) C\ (35) where n?* is the optimal export prices inflation in the foreign currency g? = AtE?*»f + ßexl?+1 [^-) (^j (36) £Mgf1 = (em - 1)gM2 (37) Exfff1 = (£x - 1)ffta2 (38) (4) Wages and prices evolve according to 1 = ^(^nf) +(1-^)(n*)1_e (39) 1 = °M{ nf J +(i-M(nf*)1-£M (4i) 1 = M n? ) +(i-^)(nr)1-e- (42) (5) Monetary and fiscal policy - Taylor rule, government's budget constraint and fiscal rule where R is a steady state of nominal interest rate, II is the inflation target, yr Taylor rule parameter (rates), 7n Taylor rule parameter (inflation), Yy Taylor rule parameter (output), Ayd is the growth rate of output, AL is the growth rate of population, is the monetary policy shock, Rt I I I (43) R \ R \ where bt is the level of outstanding debt w.r.t. nominal output, gt is the per capita level of real government consumption, , 9t Tt mt-i 1 yt-i , . h-i wtIf ct mt 1 bt = —d~\---r d „ +Rt-ibt-i r d ^ -(rtut-d)tK—j--tw—-j--tc---7----7 VI Vt pt-i YtVt ZtH-t Y^y^ZtUt Vt^t^t Vt pt Vt pt yf (44) 24 where Tt are the per capita lump-sum taxes, ydt To - Ti(bt - b) (45) (6) Net foreign assets evolve where ytY is the world demand, Mt is the per capita real imports, Aext is the nominal exchange rate depreciation extb W + = R^FiAextext^b^^Aext W I extPt V Pt It y?ztnt W t-1 ext_ib Yw (extP*V-ew fyj_\ _ fextPf\ f MA J V Pt J \yf J V Pt J\yfJ (46) (7) Aggregate imports and exports evolve where vf is the import prices dispersion, vf is the export prices dispersion, xt is the real per capita exports, ec is the elasticity of substitution among different types of consumption goods, is the elasticity of substitution among different types of investment goods, eW is the elasticity of substitution among different types of world trade goods, nc is a home bias in the aggregation in consumption, nl is a home bias in the aggregation in investment Mt = vM Q+i (1 - nc) pt pt Si pt ct + fit+i(1 - n*) r — €i Pt pt it it pt (47) xt = Vt Pt Pt W yt (48) where for s = c, i nc is the consumption good inflation, nt is the investment good inflation, is the imported consumption, if is the imported investment 1 - /?(! - ns) 7t+iA*+i Et_ . Pt, n s ( St+1 \ V (Ast + l) t+1 Ast+i t+1 (1 ps _ p _ 1 * 1 * I Ast where are adjustment costs T {^—) ^Mi + (i-M(nf*r- f-i + (i - 9x)(n The production of importing and exporting firms (49) (50) (51) (52) 1 yt extpt Pt Pt W ytW (53) 25 Demands for consumption, investments imports where cf is the domestic consumption, if is the domestic investment r (54) n V Pt ) ^ (55) (8) Market clearing condition - aggregate demand and supply where is the parameter associated with the fixed cost production, vp is the dispersion of the domestic intermediate goods prices, At is the neutral technology growth yd = nc M\ ° Ct + ni (El) it+gt + J-[ut]kt^ + xt (56) \pt J \ptJ zm yf = ^ Y-- (57) vt where market clearing condition - labour market where vt" is the dispersion of wages, 1t is the per capita hours worked It = vtw/d (58) v? = ep(^j v%_1 + (i-ep)(irt)-° (59) and capital accumulation ~ltktztnt = (l-5)kt-i+ztnt[l-S ^t^—zt ) it (61) Aggregate consumption and investment evolves ct = \{nc) - (cf )^ + (1 - nc)^(cf (1 - Ff))^] ^ (62) it K)- + (i -nJ)*(*f (i -H))^1](63) (9) Relative consumption and investment prices evolve nc + Q°(l -nc) I ^-J (64) 1 n' + fijtl-n*) f^-J (65) (10) Identities for inflations rates n? = T^gvn* (66) Vf - / TT irt = ^£fvn* (67) 26 Pi.) (68) ( extPt I%= ,^ P\\ (69) \ P*-1 J / W ' ~ ^-ipZi^ Aext [ ' (10) Relation among technologies and AR processes where AM is the growth rate of investment-specific technology, A a is the growth rate of neutral technology zt=Ap^ (71) & - eA» = 0 (72) log dt - Pd log dt-i - = 0 (73) log ft - pv log y>t-i - er = 0 (74) At - eAA+«A = 0 (75) Yt - eAL+«* = 0 (76) and exogenous processes log(Rf) = PRW log(Rj-i) + (1 - PRW) log(Rw) + eRW (77) log(yf) = pyw log(yj-i) + (1 - Pyw) log(yw) + £yW (78) log(irT) = pnw log(nj-i) + (1 - pnw) log(Hw) + #W (79) log(gt) = Pg log(gt-i) + (1 - Pg) log(g) + ef (80) 27 5.2. Modified and Added Model Equations (1) FOC of households with respect to consumption is modified by adding aRt regulated prices proxy dt---- - h^+A+i-1 • = At(l +tc) \ (81) ct - —ct-iatit ct+1zt+1 - hctaRt+1 \Pt J FOC of households with respect to domestic bonds incorporates K:tuler time-varying parameter Xt= p^^-^-Kluler (82) FOC of households with respect to foreign bonds incorporates premt premium, Kforex and «e"ler time-varying parameters and uip shock e"ip \ T>W forex euler * zt+i nt+i ext where premt has its own equation similar to the original log(premt) = pprem log(premt-i) + eprem - pexbWext(84) We incorporate the identity for optimal wage inflation T*W _ ^_t_ Wt (85) (2) FOC of firms is not changed (3) FOC of exporting firms has the Ttf export prices inflation in the foreign currency, qtx is the cost of exporting firm, px is the price of the exporting firm, aT = M f K + pw+i ( W- I 9tU (86) . Pt where IIx* is the optimal (star) export prices inflation in the foreign currency (tilde) g? = w*? + pexlij+1 ( I (idr-1 9%i (87) (4) Export prices dispersion is modified / (nw )xx \i-Ex 1=9x\fFj +(1-^)(nD1-e- (88) (5) Taylor rule and fiscal rule are modified, we adjust Taylor rule to capture inflation target in the Czech Republic and remove the bug26 eAL+Ayd to AL + Ayd where II4c is year-on-year CPI inflation and target is the year-on-year inflation target Rt _ (Rt-i ~R ~ \~R (Rt-iX'R, ( n4°+4 y \ R J \targett+4 J ~1T±T:— I I exp(C) (89) ytargett+4 and fiscal rule instead of (44) and (45) gt = Pg ct + eg (90) 26It depends whether parameters A are in logs or not. We express them in logs. 28 (6) Net foreign assets evolution looses premium ext_iOt_i + + ( extpl \ _ /MA \ pt j \ptj vyr (7) Aggregate imports must contain a component for exports (91) Mt = v r pt pt El ct Pt where we omit adjustment costs it + ^x+i(1 - nx) " P_l_ " Pt Xt extp* Pt (92) (93) vf = Ox /(n^x-- + (1 - Ox) (ft' Demand for imports must contain a component of exports xf1 where demand for imported exports where xf are the domestic exports (8) Market clearing condition has aggregate exports xt ' Pt \ ' ■ 1 Pt Pt H +9t + — $Mfct-i +nx I — ] xt zt^t VPt, xt K)i(xf)^1 + (1 - n*)i(xf (1 - 1?))^] ^ (9) Relative export costs evolve Pt nx + 0x(1 - nx) Pi Pt ] l (10) Identities for inflations rates ft =-ztUt wt_i fttaßt nt nf aX t (Pt-l \pt-l ('(£0 (Is) ' (94) (95) (96) (97) (98) (99) (100) (101) (102) (103) 29 pt 0 0 WtaXt = y^n, (104) nf = AextIIx (105) IW * = AextIW; W extpt it—1 xt mt xt xt—1 Mt mt_i gt—1 W W yt (106) U"*aXt =-^-r^nt (107) pt-l ydt = -4-^1, (108) yt—1 ctaRt = -^—ztfL (109) ct—1 it = -r^-ztiL; (110) ZtYLaOtaXt (111) ZtYLaOtaXt (112) ztYLaGt (113) -ZtYLaOtaXt (114) y t — w ytW—1 I4c = ncnc—inc—2nc—3 (115) (11) Relation among technologies and AR processes log(Mt) = PM log(Mt—1) + (1 - PM) log(A^) + etM (116) log(At) = pa log(At—1) + (1 - PA)log(AA) + £A (117) and time-varying parameters log(aRt) = PaR log(aRt—1) + (118) log(aOt) = Pao log(aOt—1) + (1 - paa) log(aa) + £taO (119) log(aXt) = Pax log(aXt—1) + (1 - Pax) log(ax) + £taX (120) log(aGt) = Pa'G log(aGt—1) + £taG (121) log(«e"1er ) = Peuler logfc^f ) + (1 - Peuler ) log(«e"1er ) + (122) log(Kf 0ref) = p/orex log(«t'—T) + (1 - p/ore*) log(«f°reX) + (123) 30 log(targett) = ptarget log(targett-1) + (1 - ptarget) log(n4) + ^target (124) and time-varying deep parameters Op,t = Pepep,t-1 + (1 - Pep )ep,t + & (125) Xt = PxXt-1 + (1 - PX)x + & (126) (12) Connection to observables are 100 log(nt) = mesPI - mes- obsPI = mesPI + uPI (127) 100 log(nx) = mesPX - mesff obsPX = mesPX + uPX (128) 100 log(nM) = mespM - mespM obspM = mespM + upM (129) 100log(nw) = mesf - mesf-1 obsf = mesf + uf (130) 100log = mest ~ mesf1 + 100log(7tL) obsf = mesf + ujf (131) 100log(Aext) = mesfX - mes— obsfX = mesfX + ufX (132) 100log(ct) = mesf - mesf-1 obsf = mesf + uf (133) 100log(it) = mes^ - mes^_1 obs^ = mes^ + ujf (134) 100 log ( ) = mesf - mesf-, - 100 \og{ztaXtaOt) obsf = mesf + ujf (135) xt-1 100 log\^±-) = mesf -mesfx - 100 \og(ztaXtaOt) obsf = mesf + l,>f (136) \Mt_1 / 100 log(yWt) = mesfW - mes™[ obsfW = mesfW + ufW (137) 100 log(ntW) = mesfIW - mespfW obsPtIW = mesPIW + uPIW (138) 400(Rt - 1) = mesf obsf = mesf + uf (139) 400(1?W - 1) = mesfW obsfW = mesfW + ufW (140) 100 log(n?) = mesfPI - mesf?1 obsfPI = mesfPI + ufPI (141) 100log(gt) = mesf - mesf-1 obsf = mesf + uf (142) 31 5.3. Steady State Now we are interested in finding a steady state of the model. The equilibrium is given by all model equations when we remove time index. Obtaining some steady states is more straightforward, because it is delivered directly from an individual equation. Thus steady states of technologies are immediately given as z = Az, A = AA, j = AM and yl = AL. Further we assume that u = d = p = 1. A level of domestic prices is numeraire, so p =1 and the law of one price must W hold exv = 1. We assume a nominal exchange rate appreciation for the Czech economy at pace of —2.37% annually, which implies Aex = ~420q7 +1. Export specific technology is defined in the steady state as aX = Inflation growths are derived from the steady state inflation which is inflation target. Inflation target targett is defined in annual timing, 2% annually. Due to the fact that our model works with quarterly data, the steady state of the domestic price inflation is II = + 1. Then nc = naR and IP = LP/ (eqs (101) and (102)). LI4c = (IP)4 = target (eq. (115)). The steady state of regulated prices technology is set as aR = 1. We put wedges keuler = -rV, kforex = rwra computed from equations (82) and (83). Foreign inflation steady state LW is the inflation target of the ECB, 2% annually. Together with the assumption of nominal exchange rate appreciation it delivers the steady state of the foreign inflation expressed in domestic currency, thus nW* = AexLW (eq. 106). Inflation of exports and imports prices expressed in domestic currency must be relevant to domestic inflation and nominal exchange rate appreciation, from equations (104) and (103), we get Lx = LAex and LM = LAex. Inflation of domestic exports prices in foreign currency must be the same as the foreign inflation in foreign currency, IF = (equation (105)). The steady state growths of consumption, investments, domestic product follow the overall economy growth A modified by exogenous processes, so c = = Ac (eq. (109)), i = zyl = A (eq. (110)), yd = zyl = Ay? (eq. (108)). Growht of nominal wages is given by real wage growth and inflation target Lw = zL (eq. (70)). Growth of real government spending we get from eq (113) g = zyLaG. Government specific technology growth is aG = 1. Trade openness technology growth is defined as aO = + 1 in the steady state. Exports and imports are given by overall economy growth and specific technologies in these sectors, so X = zy LaXaO, M = zYLaXaO (eqs. (112) and (111)). Foreign demand growth for the domestic exports is given ad hoc at pace of 9% a year, which implies yW = zyLaXaO (eq. (114)). Adjustment costs are zero in the steady state, so Qc = 1, Ql = 1, Qx = 1, Yc = 0, = 0, Yx = 0, Yc'der = 0, Yi,der = 0, Yx,der = 0. Also steady states of domestic and foreign nominal interest rates are given, domestic interest rate is 3% annually, it implies R = + 1 , and foreign interest rate is 4% annually, it implies Rw = + 1. Steady states of technologies and AR processes can be easily seen from equations (72) — (126), for example for (72), it is AM or if the term is missing, it is 1. Finally our observations obs are in levels (100log), thus observables (127) — (142) usually start from the level of the first observation. Measurement variables mes then ensure a proper connection to model variables. 32 Above steady states are given adhoc and can be obtained from the data, we label them big numbers. Computing other steady states can be more difficult. Begin with the easiest. Optimal domestic intermediate goods inflation and optimal wages are derived from equations (39), (40) and from parameters. i 1-e /i - o n-(1-e)(1-x)\ "•=( % ) and denoting IT"* = — uw* = (i-ewn-v-«K-x<-)z-v-«)\ Marginal costs are from equations (28), (29) and TP, n, so mc e 1 -/V^n-a-^1-*)" When we know T*, Tw* and T, we can compute from (59), (60) = (i-flp)(n*)-£ w = (i - ew){nw*)-v 1 - 9pu(1-^ ' 1 - ewii(l-^zv • Again if we know TM and parameters, we obtain from (41), (50) M(nM)-(1-£M)(i-xM)y-EM M (i _flM)(nM*)-£M M / i - 0M(nM)(1-xM)£M and with Tx, TW, (88), (94) nx - B= (i-0x)(n")-£- v 1-0X / Now we can set steady state levels of prices. Start with From TlM*, nM and equations (33) and (35) then from eqs (65) and (99) 33 From IF, LIx*, LW and equations (86) and (87), we get V ex P V n ; (M3) , adjustment costs and eq. (24) deliver _p_ \(lLz - A;)2 - k(-/lz - Ai)(7Lz) + (3\k{tilz - Ai)(~/Lz)2' and q with eq. (23) delivers *jf - (1 - S)q - 5tk r = —-. (1 — tk )u From mc and r and eqs. (32), (85) w=(l-a)(mc(-YY~a , w* = Uw*w. r y- is from ^- and eq. (64) Setting steady states of above variables is really straightforward. Much more interesting is searching for equilibrium values of real variables k,yd,i,x,c, g, M, A and /d. Solving of 9 non-linear equations (31),(57),(61),= ea;P M" nominal exports equal nominal imports, see details in [6]),(97),(90),(92),(81) and recursive (26) and (27) is necessary. 1 — a r vp zji — c=-['-) yd-n\^) l-nX{j) v^M-g , g = p9c z- hCi^ 1 d, = [1 - /3gw^1^)lF(1-^)(1-^)TL] ^(1 ~ rw>* ~ (i + rc)(z-h)(^)c ( ' ~ [i - pewz^-m-(l-^l-^^L]^((uw*))-^ M = vM q 1 k 34 To do this we set some auxiliary parameters. aw j 7L-(1-Č) 1 fpc\~€c i/W a1 = ---zfi, a2 = —, a3 =-, o4 = — — , o5 = a4n — 1 — a r vp zp nc\ p J \P/ erPW / PM \ —£x 2i2_ / x \ Ě* I —\ o6 = o4w —a7 = n I — I , o8 = v ÍIx{í - n ) I —r I , o9 = o4pfl, /^\~£c /^\~£l z-/^L \W \W (l + rc)(z-/.)(f) [1 - /^z^+^n^-^^+^V] - tw)w* ai3= [i - ^^-in-(1-'?)(1-^)7i] v(n»*)-^ ' ai4 = i?' ais = «- Substituting these parameters into above equations, we get k = aild, yd = a2kai6 (/íi)1-íllB, i = a3k, x = —M, c = a4yd - a5i - a6a7M - a4g, g = pgc, M = a\oc + ani + a$x, X = au~, (ld)ai4 = a^X. After some easy algebra ^6 , r aioC + ani a6 M = aioc + anz + as—M =^ M = —-^— where am = 1 — ag — a4 1 — ag — d . ai6a4yd — (a5ai6 — a6 azan )i c = a4y — a5« — a6a7M — a9c =>- c =- + a6a7aio + a9ai6 (ii6(Í4řÍ2(i^15 — ((Í5CÍI6 — (Í6ři7(ill)1 ří.3 ^ ai6 + a6a7 aio + a9ai6 c \ai8/ 1 where an = 0^0^(0^ + a6a7aio + a9ai6), a is = ai6a4a2a"1B — (a5ai6 — 0607011)0103 When we know ld, we can simply derive steady states of k, i, yd, c, M, A, x and g. ai6 + a6a7aio + a9 ai6 1 — Og— c o4 ' a4 From equations (58), (97), (54), (55), (96), (95) and from the above, we get 1 = ^> cd = nc (^j c, id = nl (^j 1, xd = nx (^j x, e»=^i^(£f\ i»=ni-^.(£r, ^=*r^(£.r nc \ p J n% \ p J nx \ p J yM = cM + iM + xM. 35 From equations (33), (37), (34), (38), (26), (28), (30) and from the above, we get A- g xi W exp P pM_ P -vM i-^(T)"£M al \_p_~,d P g M2 1 - POx7L (nw)xx ft* gx2 £x - 1 g1 d^tp{Yr*)-^l+i)\id)l+i) Amc yd 1 poP7Ln(x-i)(-^) g2 e - 1 g1. Net foreign assets evolution is derived from eq. (91) exbW ( expw yw ^px Y~tw (VWd^ expw M V Vd 1 - RW e'x yd YLzllyd From eq. (84) and from the fact that exbW = 0, we get prem and from eq. (48) yW x Vx W To recapitulate, first we define steady states for z, A, fi, jl, u, d, tp, , Aex, aX target, n, nc,, aR, neuler, Kforex, n*, n4c, nW, nW*, nx, nM, nx, c, i, yd, nw, g, aG, aO, X, M, yW, yc, 7*, 7x, 7c>der, yder, 7x>der, R, RW and for deep parameters 0p and x and observables (127) - (142). From simple subsequent computation we derive steady states for n*, nw*, mc, p w ttM* M fix* x P^_ pi £. Pi n r v, „.,* Pi u 1 u 1 ±± 1 u 1 ±± 11 pi pi pi pi "11 1 1 p ' Solving 9 non-linear equations delivers steady states for k, yd, i, x, c, g, M, A, Then we can compute the rest of steady states as cd, id, xd, cM, iM, xM, yM, l, gMl, gM2, gxi, gx2, f, g1, g2, exbW, prem, yW. We thus have 120 steady state values from 120 equations. f 1 1 36 5.4. Posterior distributions SE_eps_mu SE_eps_d SE_eps_A 0.05 0.1 0.15 0.2 0.25 SE_eps_mp 0.01 0.02 0.03 0.04 SE_eps_R_W 3000 2000 1000 0 2 4 6 8 10 12 14 30 20 10 0 1500 1000 500 0 30 20 10 0 SE_eps_aO 0.1 0.2 0.3 0.4 0.5 SE_eps_target 150 100 50 0 1500 1000 500 0 0 0.02 0.04 SE_omega_EX 0.2 0.4 0.6 0.8 1 1.2 SE_eps_varphi 10 -3 SE_eps_y_W SE_eps_aR 300 200 100 0 200 150 100 50 0 1500 1000 500 0 0.02 0.04 0.06 SE_eps_forex 0.02 0.04 SE_omega_R 200 150 100 50 0 400 300 200 100 0 0.02 0.04 SE_eps_prem 0.5 1 1.5 SE_eps_pi_W 0.01 0.02 0.03 0.04 0.05 SE_eps_aG 0.05 0.1 0.15 0.2 SE_eps_wedge_euler 10 -3 150 100 50 0 1500 1000 500 0 0.05 0.1 0.15 0.2 0.25 SE_omega_RW 10 -3 Figure 8: Posterior distributions 30 0 1000 500 0 0 5 2 0.5 1.5 100 40 50 20 0 0 0 0 5 10 0 5 0 5 x 10 37 SE_omega_CPI SE_omega_PIW SE_omega_PM 10 0.1 0.2 0.3 0.4 0.5 SE_omega_PX 5 10 15 SE_omega_C 5 10 15 20 25 SE_omega_M 10 20 30 40 SE_omega_G 5 10 15 20 25 gamma 1.2 1.25 1.3 " 20 1> " 15 A " 10 ft - 5 A ■ N.___ - 0 0 0.2 0.4 SE_omega_PI - 15 10 5 - 0 5 10 15 SE_omega_I 20 5 10 15 SE_omega_YW 20 20 30 h 0.88 0.9 0.92 0.94 0.96 kappa 19.5 20 20.5 10 2 4 6 8 10 SE_omega_L 10 20 30 40 50 SE_omega_X 10 20 30 SE_omega_W 40 15 10 5 0 _ 2345 psi 8.4 8.6 8.8 9 9.2 9.4 gamma2 0.25 0.3 0.35 Figure 9: Posterior distributions 10 5 0 20 5 0 0 0 10 40 10 5 0 38 Gamma_b_W alpha gamma_R 0.7 0.8 gamma_y 0.18 0.2 0.22 0.24 0.26 epsilon_M 8.8 9.2 7.5 7.6 7.7 theta_p 0.4 0.6 0.8 theta_w 0.3 0.4 0.5 0.16 0.18 0.2 0.22 0.24 gamma_pi 1.1 1.12 1.14 1.16 1.18 epsilon_x 10 epsilon_c 15 . A A ft 20 A/ 10 15 10 5 / r\ \ 0 5 0 9.2 9.4 9.6 epsilon_i 7.4 7.6 7.8 theta_M 20 0.6 0.7 0.8 0.9 chi_p 0.4 0.6 0.8 1 1.2 0.9 0.95 epsilon 4.9 5 5.1 epsilon_W 0.6 0.8 1 1.2 eta 6.966.98 7 7.027.047.06 theta_x 0.1 0.2 0.3 chi_M 0.2 0.4 0.6 0.8 1 Figure 10: Posterior distributions 0.9 0.85 40 5 0 9 39 chi_x chi_w 10 0.4 n_i 0.8 0.9 n_x 0.2 0.3 0.4 Lambda_mu 4000 3000 2000 1000 0 400 300 200 100 0 400 300 200 100 0 20 15 10 5 0 0.1 0.2 0.3 Lambda_A 0.4 0.3 0.4 Lambda_L 0.5 1.0085 1.009 1.0095 alphaO 1 1.005 wedge_uip 1.005 rho_b_W 1.01 0.4 0.5 0.6 4000 3000 2000 1000 0 200 100 0 400 300 200 100 0 30 20 10 0 0.9995 1 1.0005 alphaX 1.005 1.01 1.015 target_pi n_c 15 10 5 0 4000 3000 2000 1000 0 4000 3000 2000 1000 0 0.9935 0.994 0.9945 wedge_euler 150 f 100 50 0 0.9995 1 1.0005 dot_ex_ss 1.01 1.02 rho_g 1.005 1.01 rho_R_W 0.6 0.7 0.8 0.9 rho_y_W 0.8 0.85 0.9 0.7 0.75 0.8 Figure 11: Posterior distributions 5 0 0 0.2 0.6 0.7 20 10 0 1.01 10 5 40 Table 3: Estimated Parameters Parameter Prior Dist Posterior Lower and Upper Bound Mean Mean of a 90 % HPD interval AR coefs of shocks Intertemp. preferences. pd 0.550 beta 0.5370 ( 0.5163 ; , 0.5675 ) Hours preferences. pv 0.400 - - Public consumption pg 0.750 beta 0.7758 ( 0.7107 ; , 0.8488 ) Foreign prices pnw 0.300 beta 0.2992 ( 0.2839 ; 0.3161 ) Foreign demand pyw 0.750 beta 0.7680 ( 0.7489 ; 0.7884 ) World interest rate pRw 0.825 beta 0.8365 ( 0.8060 ; 0.8600 ) Foreign debt pbw 0.450 beta 0.4061 ( 0.3605 ; 0.4439 ) Regulated prices paR 0.300 - - General tech. pA 0.750 - - Export specific tech. paX 0.200 - - Wedge euler peuler 0.500 - - Wedge forex p forex 0.600 - - Standard devs of shocks Invest. spec. tech. 0.045 invg 0.2048 ( 0.1747 ; 0.2324 ) General tech. pA 0.010 invg 0.0078 ( 0.0045 0.0110 ) Intertemp. preferences od 0.250 invg 0.1831 ( 0.1634 0.2087 ) Hours preferences pp 0.001 invg 0.0009 ( 0.0002 0.0015 ) Monetary policy Pm 0.008 invg 0.0033 ( 0.0027 0.0038 ) Foreign prices 0.010 invg 0.0097 ( 0.0082 0.0112 ) Foreign demand yw 0.310 invg 0.1681 ( 0.1456 0.1883 ) World interest rate Rw 0.003 invg 0.0012 ( 0.0010 0.0014 ) Premium p prem 0.400 invg 0.3140 ( 0.2454 0.3652 ) Openness paO 0.095 invg 0.0609 ( 0.0448 0.0749 ) Regulated prices aR 0.012 invg 0.0111 ( 0.0089 0.0133 ) Government specific paG 0.038 invg 0.0269 ( 0.0199 0.0337 ) Population pL 0.0001 - - Government Pg 0.0001 - - Export specific aX 0.0001 - - Target ptarget 0.0100 - - UIP Puip 0.0001 - - Wedge forex forex 0.0100 - - Wedge euler euler 0.0100 - - Std of measurement errors Exchange rate pex 0.001 invg 0.0009 ( 0.0002 0.0016 ) Domestic interest rate R 0.001 invg 0.0009 ( 0.0002 0.0016 ) Foreign interest rate prw 0.001 invg 0.0008 ( 0.0002 0.0016 ) Domestic inflation pC PI 0.100 invg 0.0900 ( 0.0657 0.1128 ) Foreign inflation pPIW 0.100 invg 0.0643 ( 0.0295 0.1045 ) Import prices inflation pPM 2.000 invg 2.0983 ( 2.0442 2.1381 ) Export prices inflation pPX 3.000 invg 3.2641 ( 3.1952 3.3376 ) Investment prices inflation pPI 4.000 invg 4.0216 ( 3.9240 4.0849 ) Population L 10.000 invg 9.3616 ( 9.3189 9.4103 ) Consumption C 5.000 invg 5.3204 ( 5.2871 5.3603 ) Investment I 4.000 invg 4.1811 ( 4.1423 4.2135 ) Export X 10.000 invg 9.9154 ( 9.8412 9.9854 ) Import M 10.000 invg 9.9734 ( 9.8564 , 10.0801 ) Foreign demand pyw 10.000 invg 10.0867 ( 10.0176 10.1802 ) Nominal wages pw 1.000 invg 0.8639 ( 0.8339 0.8987 ) Government spending pg 5.000 invg 5.1637 ( 5.1101 ; , 5.2234 ) 41 5.5. Impulse responses This section presents the behaviour of the model27. All shocks are unanticipated, positive and have one standard deviation size. The model is simulated with the Dynare Toolbox [8]. The figures are in the Appendix where impulse responses of the standard model (left panels) are compared with impulse responses of the second-order version of the model (right panels). Figure 5 presents the technology shock (t4. A positive total factor productivity (TFP) shock results in positive reactions of investment, consumption and exports. Imports increase as well, partly because of a considerable import intensities of other sectors. Wages react positively to the technology shock as well. A higher productivity decreases marginal costs implying lower inflation. The reaction of inflation to the shock is not instantaneous because of the price stickiness. The nominal exchange rate appreciates. A central bank decreases its interest rate as a reaction of lower inflation and anti-inflation pressures from the appreciation. The reaction of the economy to the investment-specific technological shock (Figure 13) is similar with moderate impact on the consumption. The reaction of the model to the population shock (tL (Figure 14) implies an increase of consumption, exports and imports. The growth rate of wages falls because new workers lower wage pressures.28 The reaction of inflation and interest rate is negligible. The labour supply shock (Figure 15) decreases hours worked resulting in higher wages and lower consumption and investment. The exchange rate depreciates and thus, net exports increase. The reaction of prices and the central bank's interest rate is positive as a response to higher inflation pressures from the import prices (via a depreciation). With nominal rigidities, the one-time monetary policy shock (t17- (Figure 16) cannot spill over to the one-time decrease of the inflation. The nominal and real (because of price stickiness) interest rates rise implying a fall of consumption, investment, hours worked, and real wages. The exchange rate appreciates as a reaction to the positive inflation differential with a strong impact on net exports. The positive shock to real government spending (g (Figure 17) depreciates nominal exchange rate resulting in higher net exports. Private consumption and investment are crowded out by the positive government spending. The depreciation causes inflation pressures from import prices and the central bank reacts by increasing interest rates. The extent of interest rates increase is of limited importance for the economy since the inflation pressures are relatively small. The positive shock to foreign demand (yw (Figure 18) increases volume of exports accompanied by an increase of imports. The nominal exchange rate appreciates instantaneously implying pressures on lower inflation.29 The central banks 27Impulse responses are expressed as deviations from steady state in percentage of q-o-q growths. The shocks are unanticipated and their sizes are five standard deviations to see differences between the first and the second approximated models. 28 Note that the effects of population growth in the model are very moderate since the population growth and its economic impacts are inconsiderable with respect to some other Eurozone countries. The exception might be the last expansion of the Czech economy before the financial crisis where there was a high inflow of foreign workers. On the other hand, it should be add that the labour force time series is very volatile. 29 The nominal exchange rate appreciation is actually an increase of prices of export goods since 42 decreases interest rates to bring future inflation back to the target. The positive shock to the foreign interest rates £piw (Figure 19) causes, ceteris paribus, a negative interest rate differential with inflationary pressures from the exchange rate depreciation. The central bank raises its interest rate with negative consequences for domestic consumption and investment. The depreciation causes an increase of exports since export goods are cheaper in foreign markets. The increase of imports is very low in comparison with exports since higher imports for exports are lowered by lower imports for investment and consumption. The positive one standard error shock to foreign inflation ^jRw (Figure 20) leads to an appropriate level shift in foreign prices. Our economy protects itself against higher imported inflation via the exchange rate appreciation. The reaction of the central bank depends on the magnitude of the appreciation. In this model, the instantaneous appreciation is high enough and the central bank decreases its interest rate with the stimulus for the consumption and investment.30 The positive uncovered interest parity £up (UIP) shock (Figure 21) depreciates the nominal exchange rate. A central bank raises its interest rate as a reaction to increased inflation pressures from import prices. Net exports increase as a reaction to the depreciated exchange rate while consumption and investment decrease because of higher domestic interest rates. The similar impulse responses are after the positive forex shock £/orex (Figure 22) and a positive shock to the debt elastic premium (Figure 23) £prem. After the positive regulated prices shock (Figure 24), the headline inflation increases since regulated prices inflation is higher. A central bank increases its interest rates to decrease the net inflation below target bringing the headline inflation to the target in the future. Consumption and investment fall. The nominal exchange rate depreciates and net exports increase. The intertemporal preference shock is shown in Figure 25. Consumption, output, and wages rise whereas the investment expenditures decrease. The inflation is above target because of higher demand pressures and a central bank raises its interest rate. The nominal exchange rate depreciates and thus allows an increase of net exports. According to Andrle et al. [2], export speciic technology makes domestic intermediate goods more effective in the production of exports (a wedge between export and import deflators and the GDP deflator). Thus, the positive shock to the export specific technology (Figure 26) increases net exports. The exchange rate appreciates. The headline inflation increases as a result of inflation pressures stemming from the nontradables sector (Harrod-Ballassa-Samuelson effect). The reaction of domestic interest rate depends on the relative size of inflationary pressures form higher inflation and anti-inflationary pressures of the appreciation. In the model, the appreciation is strong enough to force a central bank to decrease interest rates. Consumption and investment react positively to the central bank's reaction. The positive kappa wedge euler shock (^uier (Figure 27) increases future inflation, decreases current interest rates, and appreciates the nominal exchange rate. As a reaction, net exports decrease. The consumption and investment decrease foreign prices are exogenous. 30In a reverse case when the appreciation is not strong enough, the central bank increases the interest rate as a reaction to the inflationary pressures from higher import prices. 43 since the shock increases the shadow value of wealth. On the whole, the shock has similar impulse responses as reverse preference shock. Impulse responses comparison between first and second order approximated models delivers practically no substantial difference when comparing one standard deviation shocks. That is why we show the comparison of five standard deviation shocks in the Appendix. We can see that reactions are not so strong in the case of the second order approximation because of precautionary behaviour (risk stems into policy functions). Another important difference embodies in the shift of steady state. Increases in steady states are mostly caused by the neutral technology shock. Table 4: A2 - shift effect Variable Shift (%) Investment deflator Export deflator Import deflator Nominal wages Hours worked Exchange rate Consumption growth Investment growth Export growth Import growth Foreign demand growth Foreign inflation growth Interest rate Foreign interest rate CPI inflation Government spending growth 0.10 0.98 0.21 -0.05 -0.17 -1.11 -0.31 -0.44 -1.47 -1.64 0.00 0.00 -0.00 -0.00 0.09 -0.31 44 Investment deflator 1 0 -1 I 5 0 -5 I 4 2 0 I 0 -0.2 -0.4 10 Hours worked Export deflator 20 0 10 Export 5 0 -5 20 0 10 10 Exchange rate Import deflator Nominal wages 0 0 0 -10 10 20 0 Iterest rates x -2 -4 -6 10 Import 20 0 2 1 0 10 n int. rates 10 Consumption 20 0 10 Investment 20 20 0 10 20 0 10 20 Foreign demand Foreign inflation 20 0 1 0 10 CPI inflation 20 0 10 20 10 20 0 10 -1 Government spending 1 0 - J0_ 20 20 0 10 20 2nd 1st Figure 12: Technology shock Investment deflator Export deflator Import deflator Nominal wages 0.2 0 -0.2 0.5 0 -0.5 I 2 0 -2 0.2 0 -0.2 0 10 20 Hours worked 0 10 Export 0.5 0 -0.5 I 2 0 -2 0 0 10 20 Exchange rate 20 0 10 20 Iterest rates 0 10 10 Import 20 10 20 n int. rates 20 0 10 0.2 0 -0.2 0.1 0.05 0 4 2 0 0.1 0 -0.1 0 10 20 Consumption 0 10 20 Foreign demand 0 10 CPI inflation 20 0.2 0.1 0 2 0 -2 1 0 -1 0.1 0.05 0 - 10 Investment 20 0 10 20 Foreign inflation 0 10 20 Government spending 0 20 0 10 20 2nd 1st Figure 13: Investment specific technology shock 5 0 5 10 20 45 1 0 -1 0.1 0 -0.1 0.05 jjnjveStment deflator 0 10 Hours worked . 1E;Xport deflator 20 0 0 10 Export 5 0 -5 20 0 0.1 10 10xS;hange rate . 10jnport deflator 20 0 0 -0.1 0 10 20 0 x 10tterest rates x 10 Import 0.1 0 -0.1 10 Consumption . 10ominal wages 20 0 10 Investment 20 0.1 0 -0.1 20 0 10 20 0 10 20 Foreign demand Foreign inflation 0.1 0 -0.1 10 20 0 10 20 0 10 20 n int. rates x 10-3P inflation Government spending 10 20 0 10 -1 20 0 0.1 0 -0.1 \ 0 J0_ 20 10 20 2nd 1st Figure 14: Population shock 0 0 0.01 0 -0.01 0.01 0 -0.01 0.02 0 -0.02 I 5 0 -5 0 10 20 Hours worked 0 10 20 Exchange rate Investment deflator Export deflator 0.02 r 0 -0.02 0.05 0 -0.05 I 5 0 -5 I 4 2 0 Import deflator Nominal wages 0 10 Export 20 0 10 x 10-3Import 20 0 10 20 x 10It-e3rest rates 10 20 n int. rates 10 20 10 20 0.01 0 -0.01 5 0 -5 I 0 -2 -4 0.01 0 -0.01 0 10 20 x 1C0-o4nsumption 10 20 n demand 0.05 0 -0.05 5 0 -5 0 10 20 CPI inflation 0 x 10I 10 20 I-n3vestment 0 10 20 Foreign inflation 0 10 20 Gxo1v0e-r4nment spending 0 10 20 2nd 1st Figure 15: Labor supply shock 0 10 20 0 0 46 Investment deflator Export deflator Import deflator Nominal wages 5-•-1 5-'-1 5-'-1 5i-1- 0 10 20 0 10 20 0 10 20 0 10 20 Hours worked Exchange rate Consumption Investment 10 Export 10 0 -10 20 0 5 0 -5 10 20 0 Iterest rates x 10 20 0 0.2 0 -0.2 10 20 0 10 20 0 10 20 Import x FF^gn demand Foreign inflation 10 n int. rates 20 0 10 CPI inflation 20 0 10 20 Government spending 10 -5 20 0 0.2 0 -0.2 0 J0_ 20 10 20 2nd 1st Figure 16: Monetary policy shock 0 0 0 0 5 0 -5 0.2 0 -0.2 0.01 0 -0.01 2 0 -2 Xny0s3ment deflator Export deflator 0.01 0 10 20 Hours worked 0 10 Export 20 0 10 20 x 10It-e3 rest rates 10 20 0 -0.01 0.02 0 -0.02 I 5 0 -5 I 4 2 0 0 10 20 Exchange rate 0 10 x 10-3 Import 20 10 20 n int. rates 10 20 5 0 -5 I 2 0 -2 I 0 -2 -4 I 5 0 -5 1Im0-p3ort deflator Nominal wages 0 x 1 10 20 nsumption 10 20 n demand 0.01 0 -0.01 1 0 -1 0 x 101 10 20 .investment 0 10 20 x 10-PI inflation 0 10 20 Foreign inflation 10 20 0.5 0 -0.5 Government spending 0 10 20 2nd 1st Figure 17: Government real consumption shock 0 10 20 0 0 0 47 Investment deflator 0.5 0 -0.5 I 5 0 -5 I 10 0 -10 0.05 0 -0.05 0 10 20 Hours worked 0 10 Export 20 0 10 20 Iterest rates 0 10 20 Export deflator 0 10 20 Exchange rate 10 Import 20 0.5 0 -0.5 0.01 0 -0.01 10 0 -10 Import deflator 0 10 20 Consumption 0 10 20 Foreign demand 10 20 0 10 20 n int. rates CPI inflation 0.5 0.2 0 -0.2 0.05 0 -0.05 1 0 -1 Nominal wages 0 10 Investment 20 0 10 20 Foreign inflation 0 10 20 Government spending 10 20 0 -0.5 0.01 0 -0.01 0 20 0 10 20 2nd 1st Figure 18: Foreign demand shock 0 0 0 Investment deflator Export deflator Import deflator Nominal wages 0.2 0 -0.2 0.5 0 -0.5 1 0 -1 0.02 0 -0.02 0 10 20 Hours worked 0 10 Export 1 0 -1 2 0 -2 20 0 0.5 0 10 20 Exchange rate 10 Import 20 0 10 20 Iterest rates 0 10 20 0 -0.5 1 0.5 0 0 10 20 Foreign int. rates 10 20 0.2 0 -0.2 I 2 0 -2 I 0 -2 -4 I 0.2 0 -0.2 0 10 20 x 10-onsumption 10 20 n demand 0.1 0 -0.1 0.01 0 -0.01 0 10 20 Investment 0 10 20 CPI inflation 0 10 20 Foreign inflation 0 10 20 Gov0-nment spending 2r 0 -2 - 0 10 20 2nd 1st Figure 19: Foreign interest rate shock 0 10 20 0 48 Investment deflator 1 0 -1 5 0 -5 20 0 -20 0.2 0 -0.2 0 10 20 Hours worked 0 0 10 Export 20 0 -20 50 0 -50 20 0 5 Export deflator 0 10 20 Exchange rate 10 20 Iterest rates 0 10 20 10 Import 20 10 20 n int. rates 10 20 1 0 -1 0.05 0 -0.05 0 -2 -4 1 0 -1 Import deflator 0 10 20 Consumption 0 10 20 n demand 0 10 20 CPI inflation 0.5 0 -0.5 0.2 0 -0.2 I 40 20 0 Nominal wages 0 10 20 Investment 0 10 20 Foreign inflation 0 10 20 Government spending 0.05 0 -0.05 0 20 10 20 2nd 1st Figure 20: Foreign prices shock 0 0 0 2 0 -2 0.02 0 -0.02 0.02 0 -0.02 I 2 0 -2 Xnv0s?ment deflator Export deflator 0.051- 0 10 20 Hours worked 0 10 Export 0 -0.05 0.05 0 -0.05 20 0 0.01 0 10 20 Exchange rate 0 10 20 x 10It-e4rest rates 10 20 0 -0.01 4 2 0 0 10 Import 20 10 20 n int. rates 10 20 5 0 -5 I 2 0 -2 I 0 -2 -4 I 2 0 -2 . 10-§ort deflator inal wages 0 x 1 10 20 nsumption 10 20 n demand 0 10 20 x 10CPI inflation 0 x 10I 10 20 I-n4vestment 0 10 20 Foreign inflation 0 10 20 Gxo1v0e-r5nment spending 2r 0 -2 - 10 20 2nd 1st Figure 21: UIP shock x 0 10 20 0 0 0 49 Investment deflator 1 0 -1 5 0 -5 I 5 0 -5 I 0.05 0 -0.05 0 10 20 Hours worked 10 Export 20 0 10 20 Iterest rates 0 10 20 5 0 -5 10 0 -10 I 2 0 -2 I 4 2 0 Export deflator 0 10 20 Exchange rate 0 10 Import 20 10 20 n int. rates 10 20 1 0 -1 0.01 0 -0.01 I 0 -2 -4 ( 0.5 0 -0.5 Import deflator 0 10 20 Consumption 10 20 n demand 0 10 20 CPI inflation 0.5 0 -0.5 0.05 0 -0.05 1 0 -1 Nominal wages 0 10 Investment 20 0 10 20 Foreign inflation 0 10 20 Government spending 0.01 0 -0.01 0 20 0 10 20 2nd 1st Figure 22: Forex shock 0 Investment deflator Export deflator Import deflator Nominal wages 0.05 0 -0.05 I 0.5 0 -0.5 1 0 -1 5 0 -5 0 10 20 Hours worked 0 10 Export 20 0 0 10 20 x 10-erest rates 10 20 1 r 0 -1 I" 2 0 -2 I 0.5 0 -0.5 I 4 2 0 0 10 20 Exchange rate 0 10 Import 20 1 0.1 0 -0.1 1 0 -1 I 0 -2 -4 0 x 1 10 20 nsumption 10 20 n demand 10 20 0 10 20 n int. rates CPI inflation 0.05 r 0.05 0 -0.05 5 0 -5 1 0 -1 10 20 0 -0.05 0 x 101 10 20 .^vestment 0 10 20 Foreign inflation 0 10 20 Gov0-nment spending 1 0 -1 0 10 20 2nd 1st Figure 23: Premium shock 0 10 20 0 0 50 Investment deflator 1 0 -1 10 0 -10 0 10 20 Hours worked 0 5 0 -5 I 1 0.5 0 10 Export Export deflator 0 10 20 Exchange rate 20 0 10 20 Iterest rates 10 2 0 -2 ( -2 -4 -6 10 Import 20 10 20 n int. rates 20 0 10 2 0 -2 0 -0.5 -1 0 -2 -4 10 0 -10 Import deflator 0 10 20 Consumption 0 10 20 x ForKgn demand 0 10 CPI inflation Nominal wages 10 Investment 20 0 10 20 Foreign inflation 20 0 10 20 Government spending 10 0 -10 0 J0_ 20 20 0 10 20 2nd 1st Figure 24: Regulated prices shock Investment deflator Export deflator Import deflator Nominal wages 0.5 0 -0.5 I 2 0 -2 I 1 0 -1 0.2 0.1 0 0 10 Hours worked 1 0 -1 20 0 10 Export 2 0 -2 20 0 0.2 10 Exchange rate 0.5 0 -0.5 20 0 10 Iterest rates 0 -0.2 20 0 x 10 Import 10 Consumption 20 0 10 Investment 20 20 0 0 -2 10 0.5 0 -0.5 20 0 1F0o-r1e6ign demand 10 n int. rates -4 20 0 10 CPI inflation 10 Foreign inflation 20 20 0 10 20 10 20 0 10 0.5 0 -0.5 Government spending 1 0 -1 - 20 0 10 20 2nd 1st Figure 25: Preference shock 0 0 0 10 20 0 0 51 Investment deflator 0.02 0 -0.02 0.05 0 -0.05 0.05 0 -0.05 I 5 0 -5 0.1 0 -0.1 Export deflator 0 10 20 0 10 Hours worked Exchange rate 0.2 0.05 0 -0.05 Import deflator 20 0 10 20 x 1C0-o4nsumption 0 10 Export 0 -0.2 20 0 0.1 0 10 20 x njt^rest rates 10 20 0 -0.1 4 2 0 10 Import 20 5 0 -5 0.1 0.05 0 0 10 20 Foreign demand 0.01 0 -0.01 5 0 -5 Nominal wages 0 10 20 0 10 20 n int. rates CPI inflation 0.02 I- 10 20 0 -0.02 0 10 20 x ^investment 0 10 20 Foreign inflation 10 20 Gxo1v0e-r4nment spending 5r 0 -5 - 20 0 10 20 2nd 1st Figure 26: Export specific technology shock 0 0 0 Investment deflator Export deflator Import deflator Nominal wages 0.5 0 -0.5 0.5 0 -0.5 1 0 -1 0.2 0 -0.2 0.5 0 -0.5 0 10 20 0 10 20 0 10 20 0 10 20 Hours worked Exchange rate Consumption Investment 0 0 10 Export 2 0 -2 20 0 0.2 10 Iterest rates 0 -0.2 20 0 0.1 0 -0.1 10 20 0 10 20 0 10 20 Import x 1F0o-r1e6ign demand Foreign inflation 10 n int. rates 20 0 10 CPI inflation 20 0 10 20 0 10 20 0 10 0.5 0 -0.5 0.1 0 -0.1 Government spending 0 20 0 10 20 2nd 1st Figure 27: Wedge euler shock x 10 20 0 52 5.6. Model Verification In Section 3, we present results of model estimation. The estimation can be understood as a tool for ensuring the model consistence with data. Because the estimation itself is not sufficient enough, we need to employ additional tools to test a model quality. This section presents some model applications for the Czech economy and other important tools to check model properties and its forecasting performance. We focus mainly on data filtering and forecasting since these are important criteria how to evaluate the model. Moreover, we use structural shock decompositions, decompositions of endogenous variables into observations or forecast decompositions into individual factors with respect to the steady state.31 By means of data filtration, we estimate and analyze past realizations of structural shocks that lie behind the evolution of observable time series.32 Analyzing the decomposition of structural shocks allows us to assess the current state of the economy and interpret observed economic data. We do not aim to explain the overall evolution of observable time series. Instead, we allow for measurement errors in the model implemented on levels (thus we have trends in the model). Such setting is able to capture middle-term and possibly long-term dynamics without information noise. ME can be understood as permanent judgments for the model filtration. The size of each error differs according its precise measurement, frequency of revisions, methodology changes etc. Thus, interest rates, exchange rate, or inflations are in fact measured without errors (or with a small sizes). National accounts data, on the contrary, with considerable errors. To sum up, a model-consistent data filtration (subtracting of noise from observables) should improve an analytic message of data since the model would be able to preserve fundamental intra-temporal as well as inter-temporal links among variables. After data filtration, we proceed to forecasting with the model. We carry out forecasting exercises via simple model simulations conditioned on exogenized foreign variables.33 The data filtration and model forecast are shown in Figures 28 - 31. To evaluate model performance, we carry out various decompositions: • A structural shock decomposition serves for comparing our intuition with model filtration. Model endogenous variables can be decomposed into individual structural shocks and thus we should be able to observe which structural shocks are responsible for a deviation of a given variable from the steady state in each period. An example of this tool is shown in Figure 32 which presents a decomposition of implied aggregate technology. The Figure indicates a dominant role of investment-specific technology over a TFP technology in the model. This result can be also seen from the model filtration since there is a downward trend of real investment from 2006Q1 to 2009 whereas the filtered consumption is stable. This analysis points out a potential shortage 3 i The detailed description and discussion of these tools can be found in Andrle [2]. 32 In line with Andrle et al. [2], we use a version of diffuse Kalman smoother since our measurement series may not be stationary. 33In other words, the values of foreign variables are fixed. Moreover, we assume that trajectories of foreign variables are anticipated. 53 CPI Inflation (QoQ, ann.) Interest Rate (%, ann.) Nominal Depreciation (QoQ, ann.) 30 20 10 0 -10 -20 -30 Nominal wages (QoQ, ann.) A /yi j \ A A II I I ................wufr V ................................1.......... Figure 28: Filtration and Forecast Foreign demand (QoQ, ann.) Foreign interest Rate (%, ann.) .--it I___i „........M___■ A. ___t ■....... Ml 1 M4ai AhfAk b . 17 VWkT^..... 1 1 V " V » h r .......................... 1 Foreign prices (QoQ, ann.) Hours Worked (QoQ, ann.) 1 i / /....... ....... 7 / 1............................. F r I/01 I/06 model ■ data — » ss~l Figure 29: Filtration and Forecast 20 15 10 I/01 I/06 I/01 I/06 model data model data ss ss I/01 I/06 I/01 I/06 model data model data ss ss 40 20 4 -20 -40 -60 I/01 I/06 I/01 I/06 model data model data ss ss 10 30 20 10 -5 -10 -10 -20 I/01 I/06 model data ss 54 Real Consumption (QoQ, ann.) 1 i lA A . it J i #w Ä//WA / ft ..Ik Art JIT if t^^Ik/ r-- .mm 1 v 11 W l. a m v» * 1 .....I...................................... Real Investment (QoQ, ann.) -40 -60 Real Export (QoQ, ann.) Real Import (QoQ, ann.) I It li . A A . . I I 1./I iAii/ - v T rf\/ - r i Figure 30: Filtration and Forecast Consumption deflator(QoQ, ann.) Investment deflator (QoQ, ann.) 25 20 15 10 5 0 -5 -10 -15 -20 Export deflator (QoQ, ann.) 1 ' ' tj ........................_........... (l J........................) 11 J, A 1 L IjL [« 1 I r W v w .......*................. V F 15 10 5 0 -5 -10 -15 -20 Import deflator (QoQ, ann.) 4 1. .....11............1..... M. . _ r 1 1 m /ii il" \ Iiv r I J .....|Af I ' ' V 1 1 1 1 Figure 31: Filtration and Forecast I/01 I/06 I/01 I/06 I/11 model data model data ss ss 60 40 20 0 -20 -40 I/01 I/06 I/11 model data model data ss ss 25 10 I/01 I/06 I/11 I/01 I/06 I/11 model data model data ss ss I/01 I/06 I/11 I/01 I/06 I/11 model data model data ss ss 55 of the model since the filtered series is below its steady state. This example greatly shows how this type of analysis is necessary. • The decomposition of an endogenous variable's deviation from its steady state into individual observables is used to evaluate which observation changes (and their size) contribute to changes of a model filtration. We can also evaluate contributions of new period observations. • The decomposition of model forecasts shows factors that are deviating the forecasted variables from their steady-states. The Figure 33 shows the domestic interest rate forecast decomposition from the steady state into individual factors. This deviation is mainly given by low foreign interest rates in the Europe. This influence is only partly compensated by setting of initial conditions. 20 15 10 5 0 -5 -10 -15 -20 -251- 2004:1 2005 1 20061 2007 1 2008 1 2009:1 2010:1 eps mu 1_ leps d 1 leps A 1 ] eps varphM 1 REST Figure 32: Decomposition of Implied Aggregate Technology Growth 56 x 10 " (R) 0.5 -0.5 -1 ] Poc. podm. ]Z_I |Z_N |Z_P -2 -2.5 -3 -3.5 -4 III/10 IV I/11 II III IV I/12 II III IV Figure 33: Decomposition of Interest Rate Forecast 0 5 57 5.7. Nonlinear filter Figure 35: Nonlinear filter 58