ESTIMATION OF THE NEW KEYNESIAN PHILLIPS CURVE IN THE CZECH ENVIRONMENT

The paper deals with the estimation of the New Keynesian Phillips curve (NKPC). First, the history of the Phillips curve and the NKPC is outlined. Next, similar research and papers regarding the NKPC are mentioned. The main goal of the paper is to estimate the parameters of the NKPC using the Bayesian techniques. These techniques are widely used for the DSGE model estimation and this paper contains links to the source foreign literature. The NKPC is estimated as part of a fully calibrated Small Open Economy (SOE) DSGE model. The SOE DSGE model consists of households, fi rms, the government and the central bank. The estimation is performed on the Czech data and the period is from 2001Q1 to 2012Q2. The fi rst output of the paper is the parameter estimates of the NKPC. The main fi nding is that the future expected infl ation plays a crucial role in setting the level of infl ation. Moreover, a shock decomposition of domestic and imported infl ation is performed and the main output is that the domestic monetary policy shock causes crucial changes in the level of both domestic and imported infl ation.


Introduction
In the last fi fty years since Phillips (1958) fi rst pointed to a possible relationship between unemployment and price and wage infl ation, the Phillips curve has become one of the most intensely debated topics in macroeconomics. The recent interest in this relationship stems partly from the fact that more and more countries have adopted infl ation targeting as their monetary policy regime. In recent research in open economy macroeconomics, New Keynesian Dynamic Stochastic General Equilibrium (DSGE) models have become increasingly popular. In fact, this school has been given its own name, New Open Economy Macroeconomics (NOEM). The New Keynesian Phillips curve (NKPC) is a key equation in these models, representing the supply side of the economy. The main feature of the NKPC is that it includes expected future infl ation. Because of rigidities in price adjustment, fi rms will base their current pricing decisions on what they expect about the future. There have been two main approaches to estimating the NKPC in the literature. The fi rst approach uses single equation methods where one estimates the curve as an isolated relationship. The second approach estimates the curve as part of a fully specifi ed model. Results from single equation methods include Galí and Gertler (1999) and Galí, Gertler and López-Salido (2001), who claim that a hybrid New Keynesian Phillips curve, including both expected future infl ation and lagged infl ation, explains well the infl ationary process in the US and the EU economy. They estimate different versions of the curve using the General Method of Moments (GMM) and fi nd that the purely forward-looking version is rejected. The backward-looking term is signifi cant, although not very important. By contrast, Fuhrer (1997), fi nds that expected future infl ation is unimportant in explaining price infl ation in the US. Smets and Wouters (2003) use Bayesian Maximum Likelihood to estimate the NKPC as part of a fully specifi ed DSGE model. They use data from the Euro area and fi nd that expected future infl ation is dominant, but also that lagged infl ation plays an important role. Adolfson et al. (2007) use the same method as Smets and Wouters (2003), but on an open economy DSGE model. They use data for the Euro area as well, and their results coincide with the ones in Smets and Wouters (2003): expected future infl ation seems to be dominant. In this paper, I will focus on the New Keynesian perspective and take the model from Galí and Monacelli (2004), which describes a simple DSGE model for a small open economy with nominal rigidities. One of the key equations in this model is the NKPC, representing the supply side of the economy. The main difference between the NKPC and the original Phillips curve is that the NKPC is forward-looking: current infl ation depends on the expectation of future infl ation. Another difference is that in the NKPC, the driving variable in the infl ation process is real marginal costs, not unemployment. The main goal of the paper is to replicate the model formulated by Galí and Monacelli (2004) and estimate the parameters of the NKPC in the Czech environment. The NKPC is estimated as part of a fully calibrated DSGE model of the Small Open Economy (SOE). Finally, the impact of all stochastic shocks on domestic and imported infl ation is discussed.

The New Keynesian Phillips curve
The key assumption underlying the NKPC is that it is either costly, or in some way diffi cult, to adjust prices every period. This could be due to some kind of menu costs of changing prices. There have been several suggestions on how to model price rigidity. Taylor (1979Taylor ( , 1980 assumed that contracts are made for several periods at the time. Then, if only a fraction of prices and wages are changed every period, both the past and the expected future will play a role in optimal price and wage setting. Calvo (1983) assumed that fi rms are not able to change their prices every period, and that the probability that a fi rm is able to change its prices in a given period is determined by an exogenous Poisson process. In this case, the duration of prices will be random, and fi rms need to form expectations about the future to set optimal prices. Rotemberg (1982) assumes quadratic costs of changing prices. In this case, it may not be optimal to change prices to what is optimal as seen from the current period only, because the optimal price in the next period might be different, and then the cost of changing the prices could exceed the gain. Therefore, one has to form expectations of future optimal prices when setting prices today. Here, I will fi rst focus on Rotemberg's assumption and assume that there exist costs of changing prices relative to both steady-state infl ation and previous period aggregate infl ation. Following Galí and Gertler (1999), I will also discuss a Calvo representation of the NKPC which assumes that some fi rms set prices according to a backward-looking rule of thumb. When we want to look at the economy of a Small Open Economy, we need to distinguish between domestic and imported infl ation. Several empirical studies have rejected the law of one price, at least in the short run, see Campa and Goldberg (2005) and Goldberg and Knetter (1997) for details. In line with Smets and Wouters (2002), I assume that there is complete pass-through to import prices at the docks, but that the importers face adjustment costs in their own price setting, so that there will be incomplete pass-through to consumer prices of imported goods.
The domestic economy has two types of fi rms, domestic producers and importers, and a continuum of each type, indexed from zero to one. Domestic producers sell their products to domestic and foreign consumers while importers only sell their products on the domestic market. The SOE model is taken from Galí and Monacelli (2004) and the NKPC is taken from Galí et al. (2001).
The consumption index is given by where is related to the degree of openness of the domestic economy.
, H t C and , where both domestic and foreign goods are defi ned as CES aggregates of a continuum of differentiated goods, indexed by i . The elasticity of substitution between domestic and foreign goods is given by 0   , and the elasticities of substitution between the different types of domestic and foreign goods are given by H  and F  , respectively.
Optimal demand for each category of goods is are the price indices of domestic and foreign goods, respectively. The aggregate price level, or the consumer price index (CPI), is In the same way, we fi nd optimal demand for each individual good within the two categories to be Domestic producers produce domestic goods by a constant return to scale Throughout the thesis, a variable without a time subscript denotes the steady-state value of that variable. Domestic goods are sold to both domestic and foreign households and also to the domestic government. We assume that the law of one price holds in the foreign economy and that foreign consumers have identical preferences for domestic goods as domestic consumers. Foreign demand for domestic goods, , where f t C is total foreign demand. Total demand for domestic goods, , where the fi rst term is domestic consumers' demand for domestic goods, the second term is foreign consumers' demand for domestic goods and the last term, G , denotes government spending.
In line with Rotemberg (1982) and Hunt and Rebucci (2005), I assume that the fi rms face quadratic costs of price adjustment. The costs arise both from changes in infl ation relative to steady-state infl ation and from changes in fi rm i's infl ation relative to previous period aggregate infl ation.
The monopolistic competition assumption is essential in New Keynesian modelling. It ensures that fi rms are willing to change output levels when demand changes, even if they do not change their prices. Importers buy the same input at a given world price.
Each importer then puts a unique brand on it and sells the fi nal product on the domestic market. The importers have monopoly power on the market for their own (branded) good.
If we log-linearise the fi rst-order conditions around the steady state which are formulated by Hunt and Rebucci (2005), we get the log-linearised versions of the two NKPC for domestic and imported infl ation assuming that all fi rms within the two sectors are equal All the variables with a hat a re percentage deviations from the steady-state level of the corresponding variable. Small characters are real variables (divided by the price index, e.g., H H p P P  ). The infl ation rates in the two prices are defi ned as and Q is the real exchange rate, defi ned as f F Q SP P  .  is the discount factor. We see that infl ation depends negatively on movements in the elasticity of demand between different types of goods. An increase in the elasticity means less market power for the fi rms and thus a lower mark-up. I therefore refer to  as a shock to market power. We also see that if real marginal costs increase, the fi rm will increase its price. Depending on  and the  parameters, the coeffi cients on expected future infl ation and lagged infl ation can vary between zero and one. For a given discount factor  , the coeffi cient on lagged infl ation must be between zero and one half. If there are no costs of adjusting infl ation relative to steady-state infl ation, that is, the 1  are zero, then the coeffi cients on lagged infl ation and expected future infl ation reduce to   respectively. This means that for  close to unity, both coeffi cients will be approximately one half. By introducing costs of deviating from steady-state infl ation, we see that we get a more fl exible Phillips curve. By setting the 2  to zero, corresponding to no costs of changing prices relative to past infl ation, we get the purely forward-looking NKPC.

The complete model
This chapter outlines the demand side of the model. This is represented by both domestic and foreign households and the domestic government. Then we specify an interest rate rule for the central bank and the equilibrium of the economy. Households are repre- sented by a continuum of infi nitely-lived individuals, indexed by j . These households consume domestic goods H C and imported goods F C . The consumers maximise the following utility function E is the conditional expectation operator,  is the discount factor, j t C is the consumption of household j in time t .
1 t C  is the aggregate consumption in the previous period and habit persistence 0 1 h   . 1 j t N  is the labour input of individual household j in the next period. Parameter  1 is the inverse elasticity of intertemporal substitution and  2 represents the Frisch elasticity of labour supply.
The utility function is maximised subject to the following budget constraint Households receive all profi ts from domestic fi rms and importers. The households also receive all revenue from price adjustment costs. It is assumed that foreigners do 1 The elasticity of intertemporal substitution measures the consumer's willingness to shift consumption between periods. When this elasticity is low, the consumers are said to be risk-averse. Thus,  also measures the relative risk aversion. 2 The Frisch elasticity of labour supply measures the response in hours of a wage change when marginal utility of consumption is kept fi xed. Thus, it measures the substitution effect of a wage change. 3 Independent and identically distributed.
(3) not hold any domestic bonds, so when aggregating the budget constraint , the net supply of domestic bonds is zero. The aggregate budget constraint then reads Substituting it for the production function, real profi ts and the market clearing condition on the market for domestic goods , , The change in net foreign bond holdings is equal to net profi ts in foreign trade. Alternatively, if the domestic country runs a current account surplus, the surplus will be put in foreign bonds.
The government spending, G , is only spent on domestic goods. It is fi nanced by a lump sum tax T and it evolves according to 1 ln ln .
The central bank follows a simple Taylor rule for interest rate setting where R is the gross interest rate defi ned as 1 R r   , r  is the degree of interest rate smoothing,   is the weight on current infl ation, y  is the weight on output growth, and r t  is an i.i.d. shock. For a detailed derivation of the SOE model, see the appendix of Alendal (2008).

Estimation
The next step is the log-linearisation of the aforementioned equations. This technique is not described in this paper. For details, see Uhlig (1995). Furthermore, the model is rewritten into Dynare in a log-linearised form.
It is an open-source add-on which works in Matlab. For more details, see Adjemian (2012). Dynare uses the Bayesian estimation techniques which are described in Koop (2003) or Hamilton (1994). The algorithm which is implemented in Dynare is described very well in Schorfheide (2000) and see Griffoli (2010) for the Dynare manual. Before the estimation, it is necessary to calibrate all the parameters present in this SOE DSGE model. The whole process of estimating DSGE models is described Villaverde (2009) or Pytlarczyk (2007. The fi nal estimated model consists of 12 log-linearised equations. ,ˆˆ, The model contains six observed time series . The time series used in this estimation are coll ected either from ARAD (Czech National Bank database) or from the Czech Statistical Offi ce. The series are: the total consumer index P , the consumer price index for domestic goods H P , the consumer index for imported goods F P , the Gross Domestic Product of the Czech Republic Y , the real exchange rate Q , the nominal wage income per hour W , and the short-term (3-month) interest rate r . The nominal wages series is used together with the total consumer price index to form a series for real wages. The data series are for the period 2001Q1 -2012 Q2. Since the model is stationary, we need to transform our data series to remove the trends by taking four lag differences of the two price indexes. Then we get the gross infl ation rates H  and F  .To relate them to the percentage deviation from the steady state, which is the variable in the estimated model, 1 is subtracted. Next, the fi rst differences of the log of the GDP and the real wages are used. The reason for these data transformations is twofold. Firstly, the trend is eliminated. Secondly, a better interpretation of the results is guaranteed. A list of the variables and their description is in Table 1.
(15)  Figure 1 shows the time series used for the estimation of the NKPC parameters.

Time series
The estimation is focused on the parameters entering the NKPC and the parameters of the shock processes. The calibrated priors of the model are listed in Table 2, as is the description of individual parameters. The DSGE models of the SOE are widely used by fi nancial institutions around the world. The vast majority of the parameters are taken from the previous foreign studies mentioned in Chapter 1. The priors of the parameters, standard errors and persistence of shocks are in Table 3. The model contains: a productivity shock, a monetary policy shock, a shock to government spending, shocks in market power for the two types of producers and, fi nally, a shock to the risk premium on holding foreign bonds. These stochastic shocks attempt to shift the model, which is in the steady state, into a different steady state. The inverse gamma distribution is used for all the standard errors. The beta distribution is in the interval 0;1 and this distribution is used for all the parameters that represent the persistence. The mean value for these parameters is 0.5, which is a very careful prior. The most common note of all the DSGE critics is regarding the sensitivity analysis of the chosen prior values. The posterior distribution is very sharp and it is obvious that the choice of the prior value plays a minor role for all the estimated parameters. The data information is much stronger than the information value of the prior. Hence, it is not possible to shift the fi nal estimate with different priors.

Results
The fi nal form of the Phillips curve is obtained by substituting the estimated parameters into and . In the estimation, the priors are set in such a way that the mode is close to fi fty-fi fty for the gross coeffi cients for expected future infl ation and lagged infl ation. Then, the higher the estimate of the parameters 1 C  and the lower the estimates of the 2 C  , the more weight is put on the expected future infl ation. It is obvious from .017 0.040 0.456 0.543 .
The model contains nine stochastic shocks and the DSGE framework allows us to quantify the contribution of each shock. Figure 2 shows the shock decomposition of the domestic infl ation. Subsequently, Figure 3 shows the shock decomposition of the foreign infl ation.

Shock decomposition of the domestic infl ation
The line shows the time series of the domestic infl ation. It is the same time series as the one shown in Figure 1. There are a few periods which indicate a plummeting of the domestic infl ation. They are the periods 2002 -2003, 2006 -2008 and 2010 -2011. The main reason for these sudden changes is the domestic monetary policy shock. The other shocks play a minor role. The fi scal policy shock tries to eliminate the domestic monetary policy shock, but the fi scal policy shock is too weak to change anything. Next, the domestic infl ation peaks in 2004, 2007, 2008 and 2012. It is mainly caused by the domestic market power shock. This shock is mainly driven by households and their demand for goods. The year 2012 differs from the other years. It is the fi rst time that the domestic monetary policy shock caused domestic infl ation to grow. Figure 3 displays the time series of the imported (foreign) infl ation. The domestic monetary policy shock causes the sudden declines and the imported market power shock causes the peaks. There is one phenomenon: the monetary policy shock caused the increase in imported infl ation in 2012.

Conclusion
The paper deals with the estimate of the NKPC, which is formulated and estimated as part of the SOEDSGE model. First, the NKPC history and the past development are described. Subsequently, the NKPC is formulated using previous foreign research. The NKPC of a SOE consists of two equations. The fi rst one represents the domestic infl ation and the latter one represents the imported infl ation. Moreover, the rest of the SOE model is briefl y described. The model contains households, which maximize their utility function using consumption and leisure time, fi rms, the government and the central bank, which controls the interest rate using the Taylor rule. The third chapter deals with the estimation of the model. In order to estimate the parameters of the NKPC, it is necessary to log-linearise the model and then calibrate all the parameters of the model. Therefore, the DSGE model is fully calibrated. Furthermore, the Bayesian techniques are used to estimate the model. The principle of the Bayesian estimation is to join information obtained from data with prior information which is delivered by the analyst. The Czech quarterly data for the period 2001Q1 -2012Q2 were used for this estimate. The main output of the paper is the estimate of the NKPC parameters. The results show that the expected future infl ation plays a dominant role in the NKPC and this fi nding can be generalised for both domestic and imported infl ation. This result seems to be similar to what was obtained by Galí and Gertler (1999), Galí et al. (2001) and Smets and Wouters (2003). Next, a shock decomposition analysis is performed. The shock decomposition quantifi es the contribution of each shock to the total level of the observed variable. The main output of this analysis is that the domestic monetary policy shock causes crucial changes in domestic and imported infl ation.