Central bank learning : comparing policy rules under E-stability

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Central Bank Learning

comparing policy rules under E-stability

Mutlu Celik (10207295) Supervisor: Gavin Goy Universiteit van Amsterdam

29-6-2016

Abstract

This paper looks at macroeconomic systems and a learning monetary authority that is determining the direction of the economy by making use of linear policy feedback rules. In this study stability under learning is a criteria for evaluating the monetary policy rules. The

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2 Verklaring eigen werk

Hierbij verklaar ik, Mutlu Celik, dat ik deze scriptie zelf geschreven heb en dat ik de volledige verantwoordelijkheid op me neem voor de inhoud ervan.

Ik bevestig dat de tekst en het werk dat in deze scriptie gepresenteerd wordt origineel is en dat ik geen gebruik heb gemaakt van andere bronnen dan die welke in de tekst en in de referenties worden genoemd.

De Faculteit Economie en Bedrijfskunde is alleen verantwoordelijk voor de begeleiding tot het inleveren van de scriptie, niet voor de inhoud.

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Central banks use monetary policy to achieve price stability. Schwartz (1995) shows that under price stability agents are able to make well-informed consumption and investment decisions, with higher efficiency in allocation. Price stability reduces inflation risk premia in nominal interest rates, this decreases real interest rates and stimulates investments. Schwartz (1995) also argues that the central bank plays an important role in forming a healthy economy and controlling the national economy to ensure stability.

In recent years researchers proposed different monetary policy rules, including the rule introduced by Taylor (1993). Taylor’s rule (1993) explains that the federal fund rates should be set depending on the current-quarter output gap and current inflation based on the output deflator. Taylor (1993) has received great attention among economists due to the accuracy in modelling the federal fund rate.

However, according to Orphanides (2001) the analysis underlying Taylor’s rule (1993) is based on unrealistic assumptions concerning the data availability and problems with the accuracy of initial data and subsequent revisions. The initial data is revised over time, however the recommendation is calculated based on the initial data. Orphanides (2001) concludes that Taylor’s rule (1993) is backward looking, which could lead to practising a wrong policy. Orphanides (2001) discovers that the recommendation with the Taylor rule based on initial data differs in comparison to revised data.

Various studies have investigated monetary policy rules (Batini and Haldane, 1999; McCallum, 1999; Taylor, 1999; Woodford, 2000). In this literature the monetary authority is controlling the economy by using a linear policy rule. The policy rule provides the central bank with feedback of how high the nominal interest rate should be. Recent literature

concerning monetary policy rules has focused on including the private sector expectations in the model (Bullard and Mitra, 2002; Aoki and Nikolov, 2006).

Woodford (2000) analyses the possibility of indeterminacy of the equilibrium, under certain policy rules. Indeterminacy of rational expectations equilibrium (REE) might be viewed undesirable. In case of indeterminacy the agent might be unable to coordinate on a particular equilibrium. When the agent is able to coordinate on an equilibrium, the risk exists that the equilibrium achieved may have undesirable properties, such as high volatility. This means that if the central banks follow the policy rule, the agent might coordinate to an undesirable equilibrium. Bullard and Mitra (2002) examine under which conditions the

coordination on a desirable equilibrium would arise. Bullard and Mitra (2002) focus on agents who do not have rational expectations and instead form forecasts by using adaptive learning.

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Under adaptive learning the agents form expectations by using the data generated in the system. The agents form their expectations by behaving as an econometrician, namely they choose a model and calculate the expectations. Expectational stability (E-stability) is used to calculate whether REE are reached under adaptive learning. Bullard and Mitra (2002) use E-stability as a criterion for good monetary policy rules. Evans and Honkapohja (2013) show that a central bank that takes action to private sector expectations can achieve a determinate and stable equilibrium. Different Taylor rules that are optimal under RE can lead to instability if private sector expectations use adaptive learning. The private sector is not capable to

converge to the REE under instability (Bullard and Mitra, 2002).

Private sector expectations under learning have been studied extensively by researchers (Bullard and Mitra, 2002; Molnár and Santoro, 2014). However, central bank learning and the performance of linear policy rules have not been investigated extensively. In this paper the central bank is learning through recursive least squares. The performance of a number of monetary policy rules is evaluated and compared whether the central bank can learn the fundamental equilibrium under different Taylor rules. Policy rules that perform well should be adopted by the central bank. The literature shows that by performing monetary policy in a systematic way, the central bank can stabilize the economy more efficiently through the interest rate (Molnár and Santoro, 2014). In this paper the analysis is done in a New Keynesian framework.

This paper is organized as follows; the second section gives an introduction into the subject and reviews related literature. The third section describes the New Keynesian model (NKM) and the policy rules followed by the fourth section which gives the results of the model under different policy rules and analyses the results. Finally, the fifth section gives the conclusion.

2 Literature review

This section starts with providing information concerning expectations and learning models. Furthermore, literature related to central bank and private sector learning will be reviewed. Subsequently, the results under the assumption of learning will be compared.

2.1 General information

Macroeconomic models are usually based on optimizing agents in a dynamic environment and this can be described in a dynamic system in equation (1)

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where 𝑦_𝑡 is a vector of economic variables at time t, such as unemployment and inflation. In the model 𝑦_𝑡+1𝑒 is a vector of current expectations of these variables for the next period t+1. Exogenous random shocks in period t are described in the vector 𝑤_𝑡, possibly following a VAR.

The question that arises in equation (1) is how are the expectations formed?

Lucas (1972) introduced rational expectations (RE), where agents form expectations based on all available information in the economy. The RE hypothesis suggests that the current expectations of the agents are equal to the actual future state, except for a small error. This hypothesis assumes that agents do not make systematic errors while forecasting, and

differences from perfect foresight are only due to shocks. Arrow (1986) discusses the strong assumptions required by the RE hypothesis, and concludes that agents are not able to gain sufficient information available in the economy in contrast to Lucas (1972).

Bullard and Mitra (2002) take a different approach for a realistic model by assuming that agents act like econometricians. In this approach, agents select a model, estimate the parameters of the perceived law of motion (PLM), which is an equation describing how agents form expectations. The agents update the model over time when information becomes

available in the system. Bullard and Mitra (2002) assume that the agents form expectations by using recursive algorithms. Under the assumption of learning, recursive least-squares method uses PLM, as described in equation (2). Bullard and Mitra (2002) follow the example of McCallum (1983). Equation (2) is the specific solution based on the NKM used in Bullard and Mitra (2002). The researchers focus on the minimal state variable solutions that are of the form:

𝑦𝑡 = 𝑎 + 𝑏𝑦𝑡−1+ 𝑐𝑤𝑡 (2)

𝑤𝑡 = 𝜑𝑤𝑡−1+ 𝑒𝑡 (3)

In equation (2) 𝑦_𝑡 is a vector of endogenous variables for period t and in equation (3) 𝑤_𝑡 is a vector of exogenous variables. In equation (3) 𝑒_𝑡 is assumed a vector of white noise terms. The shocks in the system are independent and AR(1) if 𝜑 is a diagonal matrix, otherwise VAR. The coefficients a, b and c are unknown and estimated. The coefficients of the PLM equation are updated when new data becomes available. The estimated coefficients are used to calculate the forecast for the agent in the case of Bullard and Mitra (2002). The forecast is defined in equation (4).

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where Et-1yt is the forecast for period t based on information available up to t-1. The

coefficients 𝑎_𝑡−1, 𝑏_𝑡−1 and 𝑐_𝑡−1 in equation (4) alter when new data becomes available, but not taken into account by the agents when forming expectations.

Evans and Honkapohja (2001) show that proving E-stability corresponds to stability under adaptive learning under certain conditions. Bullard and Mitra (2002) use the PLM to obtain the actual law of motion (ALM). The ALM shows how the system develops over time when the coefficients of the PLM are fixed. The ALM is calculated by substituting the forecast as defined in equation (4) into the model and rewriting this gives the ALM. E-stability is determined by mapping from the PLM to the ALM. Bullard and Mitra (2002) calculate the fixed points of the differential equation and determine E-stability if the points are locally asymptotically stable. The fixed points of the differential equation is the REE. Bullard and Mitra (2002) find that policy rules that include current expectations of inflation and output are most desirable in terms of learning and determinacy.

Alternatively, in a macroeconomic system the central bank can be the party that learns and acts as an econometrician. Both parties can be learning but for simplicity reasons in this research only the central bank is learning. The central bank has a model of the economy, which the authority estimates and updates over time. The central bank has imperfect

information concerning the coefficients in the model and is making estimates by updating the model with incoming data.

2.2 Expectations under learning

When the central bank has perfect knowledge concerning the macroeconomic

environment the bank is capable of providing correct policy recommendations (Orphanides, 2001). When the assumption of perfect knowledge is violated the central bank can learn the parameters of the model. In Aoki and Nikolov (2006) the central bank estimates the

parameters of a simple New Keynesian model. In their analysis both central bank and private agents learn the slopes of the IS and Phillips curve. Aoki and Nikolov (2006) conclude that imperfect knowledge of the economy negatively impacts the performance of the policies that perform well under the assumption of perfect knowledge.

According to Orphanides (2001), the biased estimates of the parameters lead to policy mistakes that affect the performance of the policy rules. When the information is not perfect central banks make mistakes in estimating the inputs of the policy rule, such as inflation and output gap. Aoki and Nikolov (2006) emphasize that the natural rate of interest is being

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misrepresented in particular. The biased estimation for the input of the policy rule passes on to agents’ expectations and drives the economy away from optimality.

Bullard and Mitra (2002) assume that the private sector is learning the equilibrium in the system. The baseline model is a simple forward-looking macroeconomic model analysed by Woodford (1999). The model is as following

𝑧𝑡= 𝑧𝑡+1𝑒 − 𝜎−1(𝑟𝑡− 𝑟𝑡𝑛− 𝜋𝑡+1𝑒 ) (5)

𝜋𝑡 = 𝜅𝑧𝑡+ 𝛽𝜋𝑡+1𝑒 (6)

The model is a brief description of the U.S. economy. The 𝑧_𝑡 is the output gap, 𝜋_𝑡 the inflation rate and 𝑟_𝑡 is the nominal interest rate. The variables are expressed as a percentage deviation from the long run level. The variable 𝑟_𝑡 is the deviation of the nominal interest rate from the steady state value. The superscript ‘’e’’ stands for the expectation of the variable. The parameter 𝜎 is the elasticity of intertemporal substitution of the representative household. The price stickiness is measured by the variable κ and β is the discount factor that households take into account in the dynamic equilibrium model. The 𝑟_𝑡𝑛 is the natural rate of interest and is exogenous stochastic that follows the following process

𝑟𝑡𝑛 = 𝜌𝑟𝑡−1𝑛 + 𝜀𝑡 (7)

The noise 𝜀_𝑡 is i.i.d. with mean zero and variance 𝜎_𝜀2. The equations (5)-(7) represents the model economy. The baseline specification for the policy rule in Bullard and Mitra (2002) is:

𝑟𝑡 = 𝜑𝜋𝜋𝑡+ 𝜑𝑧𝑧𝑡 (8)

Equation (8) is the contemporaneous data specification of the policy rule. Bullard and Mitra (2002) consider more policy rule specification of the central bank and analyse their

performance.

Orphanides (2001) concludes that contemporaneous data policy rules are not specified correctly. The central bank does not have the information available in the time period they require in order to make a decision. The central bank sets the interest rate in response to inflation and output gap. The policy rule in equation (8) can be changed in equation (9) because of the informational problems that arise, as in Orphanides (2001).Orphanides (2001) suggests that the central bank reacts to current expectations of inflation and output gap, called the contemporaneous expectations version in equation (9).

𝑟_𝑡 = 𝜑_𝑡𝜋_𝑡𝑒 + 𝜑_𝑧𝑧_𝑡𝑒 (9)

Bullard and Mitra (2002) suggest that including a lagged interest rate term in the policy rule captures the interest rate stabilization in central bank behaviour. Central banks want to capture the interest rate smoothing observed in their actual behaviour. Alternative

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approaches that Bullard and Mitra (2002) consider are the lagged data specification in equation (10)

𝑟𝑡 = 𝜑𝜋𝜋𝑡−1+ 𝜑𝑧 𝑧𝑡−1 (10)

In the lagged data specification the assumption is that the central bank must react to the last quarter data of inflation and output gap. The last specification that Bullard and Mitra (2002) consider is the forward expectations specification in equation (11).

𝑟𝑡 = 𝜑𝑡𝜋𝑡+1𝑒 + 𝜑𝑧𝑧𝑡+1𝑒 (11)

Bernanke and Woodford (1997) explain this specification by assuming that the central bank is reacting to the predictions of the private sector.

Bullard and Mitra (2002) evaluate the policy rules described earlier based on learnability in a forward-looking macroeconomic model. The problem Bullard and Mitra (2002) face is that under learning for the private sector there is a possibility of indeterminacy for different specified policy rules. The private sector is not able to coordinate to a particular equilibrium when indeterminacy occurs. The researchers use E-stability to conclude if the REE are stable under learning. The agents are learning the equilibrium under a specified Taylor rule. Bullard and Mitra (2002) emphasize the importance of the Taylor rule

specification in obtaining stability and determinacy. They analyse under different Taylor rules that convergence to the REE is only possible under certain parameter restrictions.

In Honkapohja and Mitra (2006) the underlying assumption is that the agents are heterogeneous. In Bullard and Mitra (2002) the agents are homogenous, where the agents are representative and so the learning rules and expectations are identical. Assuming homogeneity among agents makes the analysis easier but does not approach the reality. Honkapohja and Mitra (2006) consider stability of REE with learning, under the assumption of heterogeneous agents. Introduction of heterogeneity in learning rules affects the convergence to REE.

Honkapohja and Mitra (2006) conclude that convergence under heterogeneity does not always take place under specified learning rules.

2.3 Performance policy rules

Woodford (1994) analyses the variables chosen in monetary policy rules. He states that through the monetary policy rule the expectations become self-fulfilling. This is mainly caused by the choice of wrong parameters. The problem arises when policy rules include variables that are very sensitive to expectations. Woodford (1994) emphasizes that selecting policy rules that contain variables with forecasting power may create feedback loops where the expectations become self-fulfilling.

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Bernanke and Woodford (1997) analyse policies that are consistent with the inflation target of the central bank. The central bank can target the private sector expectations of inflation. Bernanke and Woodford (1997) determine the existence of REE when the monetary authority uses private-sector expectations as guideline to policy actions. In a dynamic model Bernanke and Woodford (1997) show that targeting private sector expectations of inflation results in not converging to REE. They state that central banks should be careful defining the policy rule and not include variables that are sensitive to the private-sector expectations.

In the next section the framework of models will be presented. The baseline model is the New Keynesian Model. The method will be defined for a learning central bank and the linear policy feedback rules will be given in the next section. The analysis is under the assumption that the private sector forms expectations by using naïve expectations.

3 Methodology

This section starts with providing the baseline model that is going to be analysed. Furthermore, the policy feedback rules are specified that are going to be analysed under learning. Subsequently, E-stability will be defined to determine if the equilibrium are stable.

3.1 The baseline model

The model is a NKM and the structure of the economy is described by a Phillips curve and an IS curve (Kerr and King, 1996 and McCallum and Nelson, 1999). The IS equation is defined as

𝑥𝑡= 𝐸𝑡𝑥𝑡+1− 𝜎−1(𝐸𝑡𝜋𝑡+1− 𝑖𝑡) + 𝑒𝑡 (12)

The economy is represented by a Phillips curve of the form

𝜋𝑡 = 𝛽𝐸𝑡𝜋𝑡+1+ 𝜅𝑥𝑡+ 𝑢𝑡 (13)

where 𝑥_𝑡, 𝜋_𝑡, 𝑖_𝑡 are, respectively at time period t, output, inflation, and the nominal interest rate. The variable 𝐸_𝑡𝜋_𝑡+1 and 𝐸_𝑡𝑥_𝑡+1 are the private sector forecasts for time period t+1 with the data available at time period t, respectively for inflation and output. The coefficient 𝜎 is the intertemporal elasticity of substitution of expenditure. The exogenous noise 𝑒_𝑡 is a AR(1) process with 𝑒_𝑡 = 𝑝_𝑒𝑒_𝑡−1+ 𝜀_𝑡 and 𝜀_𝑡 is i.d.d. with zero mean and variance 𝜎_𝜀 and |𝑝_𝑒| < 1.

The disturbance 𝑢_𝑡 is a supply shock that is a AR(1) process with 𝑢_𝑡 = 𝑝_𝑢𝑢_𝑡−1+ 𝜉_𝑡 and 𝜉_𝑡 is i.d.d. with zero mean and variance 𝜎_𝜉 and |𝑝_𝑢| < 1. The parameter 𝜅 is the degree of price stickiness and 𝛽 is the discount factor that the private sector takes into account. The central bank updates the parameters of his model and assumes these values as true, implying

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that the central bank does not take into account that it will update the coefficients in the future. For different policy rules determinacy and learnability will be determined under central bank learning. The findings will be illustrated by using the parameters β=0,99, 𝜎=0,157,𝜅=0,024 and the pe and pu are set at 0,35 based from Rotemberg and Woodford

(1999). The analysis will be done by varying as following, 0 < 𝜑_𝜋 < 10 and 0 < 𝜑_𝑥< 10. 3.2 Policy feedback rules

The equations (12) and (13) are supplemented with a policy rule to represent the behaviour of the central bank. In this research three policy rules will be analysed. The first policy rule that will be considered is the forward expectations specification in which the interest rate is defined as

𝑖𝑡 = 𝜑𝜋𝜋𝑡+1𝑒 + 𝜑𝑥𝑥𝑡+1𝑒 (14)

The second policy rule that will be analysed is the contemporaneous expectations specification in which the interest rate is defined as

𝑖_𝑡 = 𝜑_𝜋𝜋_𝑡𝑒+ 𝜑_𝑥𝑥_𝑡𝑒 (15)

The third policy rule is an expansion of the contemporaneous expectations including lagged interest rate that gives the following interest rate

𝑖𝑡= 𝜌𝑖𝑡−1+ (1 − 𝜌)(𝜑𝜋𝜋𝑡𝑒+ 𝜑𝑥𝑥𝑡𝑒) (16)

The policy rules described above will be analysed under the assumption that private sector expectations are formed by using naïve expectations defined as

𝐸_𝑡𝑧_𝑡+1= 𝑧_𝑡−1 (17)

where 𝑧_𝑡 = 𝑥_𝑡, 𝜋_𝑡. 3.3 E-stability

Evans and Honkapohja (2001) have shown that expectational stability determines stability under adaptive learning. Following Evans and Honkapohja (2001), E-stability is determined by considering a general class of models

𝑥𝑡= 𝛼 + 𝐵𝐸𝑡𝑥𝑡+1+ 𝛿𝑥𝑡−1+ 𝜓𝑤𝑡 (18)

wt = ϕwt−1+ et (19)

where 𝑥_𝑡 and w_t is a n x 1 vector of variables, 𝛼 is a n x 1 vector of constants, 𝐵, 𝛿, 𝜓 and ϕ are n x n matrices of parameters. The term e_t is a n x 1 vector of white noise terms. The MSV solution to equation (18) according to McCallum (1983) are of the form

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The matrices a, b and c are unknown and need to be estimated. The expectations of the central bank are computed by iterating equation (20) and taking the expected value of this expression.

𝐸𝑡𝑥𝑡+1 = 𝑎 + 𝑏𝑥𝑡 + 𝑐𝑐𝑤𝑡 (21)

where 𝑐𝑤_𝑡= 𝐸_𝑡𝑤_𝑡+1.

For analysing E-stability for the equilibrium the PLM must be computed, as in equation (20). The central bank uses the PLM to make forecasts of the variables of interest and the form of the PLM corresponds to the REE of interest. The ALM of 𝑥_𝑡 is obtained by substituting equation (21) in equation (18) and rewriting this expression into the form of 𝑥_𝑡 as 𝑥𝑡= (𝐼 − 𝐵𝑏)−1(𝛼 + 𝐵𝑎 + 𝛿𝑥𝑡−1+ (𝐵𝑐𝑐 + 𝜓)𝑤𝑡). The ALM describes the process

followed by the economy if forecasts are made under the assumption of the PLM. E-stability determines the stability of the REE under learning by analysing if the PLM parameters a, b and c adjust in the direction of the implied ALM parameters, which are described in equation (22)

𝑇(𝑎, 𝑏, 𝑐) = ((𝐼 − 𝐵𝑏)−1_{(𝛼 + 𝐵𝑎), (𝐼 − 𝐵𝑏)}−1_{𝛿, (𝐼 − 𝐵𝑏)}−1_{(𝐵𝑐𝑐 + 𝜓))} ₍₂₂₎

The mapping from the PLM to the ALM determines E-stability by using the following matrix equation 𝑑

𝑑𝑑(𝑎, 𝑏, 𝑐) = 𝑇(𝑎, 𝑏, 𝑐) − (𝑎, 𝑏, 𝑐) , where 𝑇(𝑎, 𝑏, 𝑐) is the mapping form.

The fixed points of the matrix equation gives the MSV solution. This solution is E-stable if the MSV point is locally asymptotically stable in that point.

In the next section the results of the model under the assumptions described earlier are given. The restrictions and conditions are given for determinacy and E-stability to hold. The analysis will be done for the different policy rules.

4 Results and analysis

In this section the analysis under different policy rules is provided. The analysis shows under the specified policy rule the conditions for determinacy and E-stability.

4.1 Forward expectations in the policy rule

4.1.1 Reduced system

In this subsection the focus is on the forward expectations specification of the policy rule. The policy rule is given by equation (14). The complete model includes equations (12), (13) and (14). It is possible to reduce the system of equations by substituting equation (14) into (12). After implementing the naïve expectations the reduced system is then given by

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𝑦𝑡 = 𝐴𝑦𝑡−1+ 𝐵𝐸𝑡𝐶𝐶𝑦𝑡+1+ 𝐶𝑤𝑡 (23)

where 𝑦_𝑡= [𝑥_𝑡, 𝜋_𝑡]′, 𝑤_𝑡= [𝑒_𝑡, 𝑢_𝑡]′, , and A is defined by A = �1 1 𝜎 𝜅 𝛽 +𝜅_𝜎� and B is defined by B = � −𝜙𝑥 𝜎 − 𝜙𝜋 𝜎 −𝜅𝜙𝑥 𝜎 −𝜅 𝜙𝜋 𝜎

� and finally C is defined by C = �1 0

𝜅 1�. 4.1.2 Determinacy

Policy rules with unique REE are associated with determinacy. The analysis of determinacy requires forward-looking variables in the system, which is the case for the forward expectations specification of the policy rule. The central banks are capable to coordinate to the equilibrium in case of determinacy. Under the forward expectations policy rule the model is rewritten to find the necessary conditions for a unique REE. The necessary conditions for determinacy to hold under this policy rule are calculated by first rewriting the system as in equation (24) �𝐼 −𝐵_𝐼 _{0 � �}_𝐸 𝑦𝑡 𝑡𝐶𝐶𝑦𝑡+1� = �𝐴 00 𝐼� � 𝑦𝑡−1 𝑦𝑡 � + �𝐶0�� 𝑤𝑡 0 � (24)

Rewriting equation (24) gives 𝑧_𝑡 = Ω𝑧_𝑡−1+ Υ𝑤�, where _𝑡 Ω = �𝐼 −𝐵_𝐼 _{0 �}−1�𝐴 0_{0 𝐼�} and Υ = �𝐼 −𝐵 𝐼 0 � −1 �𝐶0�, 𝑤� = �𝑤𝑡 _{0 �}𝑡 , 𝑧𝑡 = �_𝐸 𝑦𝑡 𝑡𝐶𝐶𝑦𝑡+1� and 𝑧𝑡−1= � 𝑦𝑡−1

𝑦𝑡 �. The problem we are facing in this particular

case is that matrix Ω and Υ cannot be calculated, due to the singular matrix B. The methods usually used analysing the conditions for determinacy, such as the Sims method and the Cristiano method, are not useful in this case. The Schur composition is used for calculating determinacy. The results are obtained numerically instead of analytically in Dynare. Table 1 summarizes the eigenvalues of the system, given the calibration chosen in this research.

In table 1 we can analyze that two eigenvalues are larger than one in modulus for two forward-looking variables. The remaining four eigenvalues are smaller than one in modulus. In this research the diagonal matrix 𝜌 is chosen with elements, 𝜌_𝑒 = 0.35 and 𝜌_𝑢 = 0.35. The two eigenvalues equal to 0.35 in table 1 are the eigenvalues of the matrix 𝜌. The eigenvalue equal to infinity appears due to the singular matrix B. The Schur decomposition gives an eigenvalue equal to infinity.

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Finally, we can conclude that determinacy holds for the chosen calibration in the forward expectations specification. The system has an unique RE solution that is dynamically stable.

Table 1: Eigenvalues of the system with forward expectations in the policy rule

Modulus Real Imaginary

0.35 0.35 0 0.35 0.35 0 0.6026 0.6026 0 0.9242 0.9242 0 1.733 -1.733 0 Inf Inf -

Table 1 gives the eigenvalues of the system for the chosen calibration described in methodology.

4.1.3 Learning

Following McCallum (1983) , the focus is on the MSV solution, which are of the form

𝑦𝑡 = 𝛼𝑦𝑡−1+ 𝑣𝑤𝑡 (25)

where 𝛼 and 𝑣 need to be calculated by the Schur decomposition. The matrix B in equation (23) is singular so the method of undetermined coefficients cannot be used. Equation (25) gives the corresponding expectation, 𝐸_𝑡𝐶𝐶𝑦_𝑡+1= 𝛼2𝑦_𝑡−1+ (𝛼𝑣 + 𝑣𝜌)𝑤_𝑡 , where

𝜌 = �𝜌_{0 𝜌}𝑒 0

𝑢� and the MSV solutions satisfy

𝐴 + 𝐵𝛼2 _{= 𝛼} ₍₂₆₎

𝐵𝛼𝑣 + 𝐵𝑣𝜌 + 𝐶 = 𝑣 (27)

The REE must satisfy the matrix equations above. The solutions to the equations (26) and (27), are 𝛼� and 𝑣̅, which is equal to the MSV solution, implying the RE solution. Equation (26) is a matrix quadratic which will have multiple solutions. For E-stability we assume equation (25) as the PLM of the central bank and using 𝐸_𝑡𝐶𝐶𝑦_𝑡+1 = 𝛼2𝑦_𝑡−1+ (𝛼𝑣 + 𝑣𝜌)𝑤_𝑡 , the ALM of 𝑦_𝑡 is obtained as

𝑦𝑡 = (A + B𝛼2)𝑦𝑡−1+ (𝐵𝛼𝑣+𝐵𝑣𝜌 + 𝐶)𝑤𝑡

This means that the mapping from PLM to the ALM takes the form

𝑇(𝛼, 𝑣) = (𝐴 + 𝐵𝛼2_{, 𝐵𝛼𝑣 + 𝐵𝑣𝜌 + 𝐶)} ₍₂₈₎

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15

𝜕𝜕

𝜕𝑑 = 𝐴 + 𝐵𝛼2− 𝛼 (29)

𝜕𝜕

𝜕𝑑 = 𝐵𝛼𝑣 + 𝐵𝑣𝜌 + 𝐶 − 𝑣 (30)

where 𝜏 is the artificial or notional time. The mapping can be computed from the PLM to the ALM. The MSV solution is the fixed point of the T-map. The results of Evans and

Honkapohja (2001) are applied for analysing E-stability, which is determined by the matrix differential equation in (31).

𝑑

𝑑𝑑(𝛼, 𝑣) = 𝑇(𝛼, 𝑣) − (𝛼, 𝑣) (31)

For the analysis of the local stability of the system in equation (31) at a RE solution, the system needs to be linearized at that RE solution. The equation for 𝛼 is independent from 𝑣. First we start with the stability of the differential equation 𝑑𝜕

𝑑𝑡 = A + B𝛼2− 𝛼 = 𝑇𝜕(𝛼) − 𝛼.

For the computation of the stability conditions, the differential equation needs to be vectorised. Using the rules for matrix differentials (Evans and Honkapohja, 2001), we get

𝑑𝑇𝜕= 𝐵(𝑑𝛼)𝛼 + 𝐵𝛼(𝑑𝛼)𝐼 (32)

The Jacobian of vec 𝑇_𝜕 is 𝐷𝑇_𝜕 = 𝜕𝜕𝑒𝜕𝑇𝛼

𝜕(𝜕𝑒𝜕 𝜕)′. Using the rules 𝑑 𝑣𝑒𝑐 𝛼 = 𝑣𝑒𝑐 (𝑑𝛼) and

𝑣𝑒𝑐 𝐴𝐵𝐶 = (𝐶′_{⊗ 𝐴)𝑣𝑒𝑐 𝐵, we obtain}

𝐷𝑇𝜕(𝛼) = 𝛼′⊗ 𝐵 + 𝐼 ⊗ (𝐵𝛼) (33)

The differential equation for 𝛼 is locally stable at 𝛼� when all the eigenvalues of 𝐷𝑇_𝜕(𝛼�) have real parts less than 1. Next we need to consider 𝑣 in equation (31). The differential equation of 𝑇_𝜕(𝛼, 𝑣) = 𝐵𝛼𝑣 + 𝐵𝑣𝜌 + 𝐶 and for the corresponding Jacobian , we have

𝐷𝑇𝜕(𝛼, 𝑣) = 𝜌′ ⊗ 𝐵 + 𝐼 ⊗ (𝐵𝛼) (34)

Given that the motion for 𝛼 converges to 𝛼�, the equation for 𝑣 is locally stable at (𝛼�, 𝑣̅) when the eigenvalues of the matrix 𝐷𝑇_𝜕(𝛼, 𝑣) have real parts less than 1. The eigenvalues of the two matrices in equation (33) and (34) need to have real parts smaller than one for E-stability to hold.

Proposition 1. Under the policy rule with forward expectations, the condition for the MSV solution (𝛼�, 𝑣̅) to be E-stable is that the equations (33) and (34) have eigenvalues with real parts smaller than one.

The matrix B is singular, therefore the Schur composition needs to be used and the switch from analytical to numerical results is implemented. The numerically results are

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16

obtained from Dynare. The output of Dynare is illustrated in table 2. Table 2 summarizes the MSV solution for the calibration chosen in this research. This results in matrix 𝛼� =

�0.562579 −1.072271_0.013502 _{0.964265 �} and matrix 𝑣̅ = �0.468067 −5.477770

0.011234 0.868534 �. Substituting 𝛼� in equation (33) and calculating the eigenvalues of the matrix 𝐷𝑇_𝜕(𝛼�) gives the vector in equation (35). The matrix 𝐷𝑇_𝜕(𝛼�) and 𝐷𝑇_𝜕(𝛼�, 𝑣̅) have a dimension of (4, 4), because of the Kronecker-product applied in equation (33) and (34).

Eigenvalues(𝐷𝑇_𝜕(𝛼�)) = [ -1.72531, -1.39549, -1.11022*10-16, 1.11022*10-16]’ (35) Substituting 𝑣̅ and 𝛼� in equation (34) and calculating the eigenvalues of the matrix 𝐷𝑇_𝜕(𝛼�, 𝑣̅) gives the vector in equation (36).

Eigenvalues(𝐷𝑇_𝜕(𝛼�, 𝑣̅)) = [-1.13645, -1.13645, 2.70237*10-17 , 2.70237*10-17] (36) Table 2: MSV solution in the forward expectations policy rule

x pi u e constant 0.000000 0.000000 0.000000 0.000000 u(-1) -1.917220 0.303987 0.350000 -0.000000 e(-1) 0.163824 0.003932 0.000000 0.350000 x(-1) 0.562579 0.013502 0.000000 -0.000000 pi(-1) -1.072271 0.964265 -0.000000 0.000000 eU -5.477770 0.868534 1.000000 -0.000000 eE 0.468067 0.011234 -0.000000 1.000000

Table 2 illustrates the MSV solution given the parameter values chosen in this research. The output gap response is set at 0.125 and the inflation response is set at 1.5.

We can conclude from equation (35) and (36) that the real parts of the eigenvalues are smaller than one for the matrix 𝐷𝑇_𝜕(𝛼�) and 𝐷𝑇_𝜕(𝛼�, 𝑣̅). According to Evans and Honkapohja (2001) the MSV solution 𝛼� and 𝑣̅ to equation (31) is E-stable .

4.2 Contemporaneous expectations in the policy rule

In this subsection we consider the contemporaneous expectation specification. The policy rule is given by equation (15). The complete model includes the equations (12), (13) and (15).

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17

After implementing the naïve expectations for the private sector the system is reduced by substituting equation (15) into (12). The reduced system is then described by

𝑦𝑡 = 𝐴𝑦𝑡−1+ 𝐵𝐸𝑡𝐶𝐶𝑦𝑡+ 𝐶𝑤𝑡 (37)

where 𝑦_𝑡= [𝑥_𝑡, 𝜋_𝑡]′, 𝑤_𝑡= [𝑒_𝑡, 𝑢_𝑡]′, , and A is defined by A = �1 1 𝜎 𝜅 𝛽 +𝜅_𝜎� and B is defined by B = � −𝜙𝑥 𝜎 − 𝜙𝜋 𝜎 −𝜅𝜙𝑥 𝜎 −𝜅 𝜙𝜋 𝜎

� and finally C is defined by C = �1 0

𝜅 1�. 4.2.2 Stability

The analysis of determinacy requires forward-looking variables in the system. In the contemporaneous expectations specification there are no forward-looking variables.

Analysing determinacy is not possible in this case, so we analyse stability.

After implementing 𝐸_𝑡𝐶𝐶𝑦_𝑡 = 𝑦_𝑡 in the system, which is described in equation (37), the equation changes into (𝐼 − 𝐵)𝑦_𝑡 = 𝐴𝑦_𝑡−1+ 𝐶𝑤_𝑡, which simplifies to equation (38).

𝑦𝑡 = (𝐼 − 𝐵)−1(𝐴𝑦𝑡−1+ 𝐶𝑤𝑡) (38)

The matrix C is stationary by the assumption that the elements |𝜌_𝑒| < 1 and |𝜌_𝑢| < 1. The necessary condition for stability under the contemporaneous expectations policy rule is that the matrix (𝐼 − 𝐵)−1𝐴 has eigenvalues smaller than 1 in modulus. The matrix (𝐼 − 𝐵)−1_{𝐴 is defined in equation (39).}

(𝐼 − 𝐵)−1_{𝐴 =} 1 𝜅𝜑𝜋+𝜎+𝜑𝑥�

𝜎 1 − 𝛽𝜑𝜋

𝜅𝜎 𝜅 + 𝛽(𝜎 + 𝜑𝑥)� (39)

The characteristic polynomial of (𝐼 − 𝐵)−1𝐴 is given by 𝑝(𝜆) = 𝜆2+ 𝑎₁𝜆 + 𝑎₀, where 𝑎₁ =

-Tr((𝐼 − 𝐵)−1𝐴) and 𝑎₀ = Det((𝐼 − 𝐵)−1𝐴). Det((𝐼 − 𝐵)−1𝐴) =𝛽𝜅𝜑𝜋𝜎+𝛽𝜎2+𝛽𝜎𝜑𝑥 (𝜅𝜑𝜋+𝜎+𝜑𝑥)2 = 𝛽𝜎 𝜅𝜑𝜋+𝜎+𝜑𝑥 (40) Tr((𝐼 − 𝐵)−1𝐴) =𝜅+𝜎+𝛽𝜎+𝛽𝜑𝑥 𝜅𝜑𝜋+𝜎+𝜑𝑥 (41)

Both eigenvalues of (𝐼 − 𝐵)−1𝐴 are inside the unit circle if and only if equation (42) and (43) are satisfied ( LaSalle, 1986, p.28).

|𝑎0|<1 (42)

|𝑎1| < 1 + 𝑎0 (43)

Equation (42) results in −(1 − 𝛽)𝜎 < 𝜅𝜑_𝜋+ 𝜑_𝑥, which holds since 0 < 𝛽 < 1 and the other parameters have positive values. Equation (43) implies the following equation

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18 Proposition 2. Stability under the policy rule with contemporaneous expectations holds if the equations (42) and (43) are satisfied. This is the case if equation (44) holds..

Stability holds for 𝜑_𝜋 > 1, even when 𝜑_𝑥 is set at zero. For values of 𝜑_𝜋 < 1 stability can hold if 𝜑_𝑥 is sufficiently high enough. Equation (44) has an economic interpretation. Equation (44) can be rewritten as 𝜑_𝜋 +(1−𝛽)

𝜅 𝜑𝑥 > 1. Equation (13) implies that a permanent

increase in the inflation leads to a permanent percentage increase of 1−𝛽

𝜅 for the output gap.

Equation (44) shows the long-run increase in the nominal interest rate for a permanent increase in the inflation rate. Equation (44) therefore implies the Taylor principle where nominal rates rise more than the increase in the inflation rate.

The instability region is illustrated in figure 1,where the parameters are set at baseline values, except for 𝜑_𝜋 and 𝜑_𝑥. The green area outside the instability region describes the stability region.

Figure 1. illustrates the instability area for the contemporaneous expectations in the policy rule. All parameters are set at baseline values given in methodology, except the inflation response and output gap response. The green area is the E-stable and stability region.

4.2.3 Learning

Following McCallum (1983) , the focus is on the MSV solution, which are of the form

𝑦𝑡 = 𝛼𝑦𝑡−1+ 𝑣𝑤𝑡= 𝛼𝑦𝑡−1+ 𝑣𝜌𝑤𝑡−1+ 𝑣𝜗𝑡 (45) 0 0.2 0.4 0.6 0.81 1.2 1.4 1.6 1.8 2 0 3 In fla tio n re sp on se

Output gap response

Instability under contemporaneous

expectations

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19

where 𝜗_𝑡= [𝜀_𝑡, 𝜉_𝑡]′ and 𝑤_𝑡 = 𝜌𝑤_𝑡−1+ 𝜗_𝑡. The matrices 𝛼 and 𝑣 need to be calculated by the method of undetermined coefficients. Equation (45) gives the corresponding expectation of the central bank, 𝐸_𝑡𝐶𝐶𝑦_𝑡 = 𝛼𝑦_𝑡−1+ 𝑣𝜌𝑤_𝑡−1 , where

𝜌 = �𝜌_{0 𝜌}𝑒 0

𝑢� and the MSV solutions satisfy

𝐴 + 𝐵𝛼 = 𝛼 (46)

𝑣 = 𝐵𝑣𝜌 + 𝐶𝜌 (47)

The REE must satisfy the matrix equations above. The matrix 𝛼 and 𝑣 are both unique matrices. The MSV solution in this case takes the form 𝑦_𝑡 = 𝛼�𝑦_𝑡−1+ 𝑣̅𝑤_𝑡 where

𝛼� = (𝐼 − 𝐵)−1_{𝐴 and 𝑣̅ = (𝐼 − 𝐵𝜌)}−1_𝐶𝜌.

For E-stability we assume equation (45) as the PLM of the central bank and using 𝐸𝑡𝐶𝐶𝑦𝑡 = 𝛼𝑦𝑡−1+ 𝑣𝜌𝑤𝑡−1 , the ALM of 𝑦𝑡 is obtained as

𝑦𝑡 = (A + Bα)𝑦𝑡−1+ (𝐵𝑣𝜌 + 𝐶𝜌)𝑤𝑡−1+ 𝐶𝜗𝑡

𝑇(𝛼, 𝑣) = (𝐴 + 𝐵𝛼, 𝐵𝑣𝜌 + 𝐶𝜌) (48)

E-stability is determined by the following differential equations :

𝜕𝜕

𝜕𝑑 = 𝐴 + 𝐵𝛼 − 𝛼 = 𝐴 + (𝐵 − 𝐼)𝛼 (49)

𝜕𝜕

𝜕𝑑 = 𝐵𝑣𝜌 + 𝐶𝜌 − 𝑣 = (𝐵𝜌 − 𝐼)𝑣 + 𝐶𝜌 (50)

The mapping can then be computed from the PLM to the ALM. The results of Evans and Honkapohja (2001) are applied. For expectational stability to hold, the requirement is that matrices 𝐵 and 𝐵𝜌 have eigenvalues with real parts less than one. The matrix B has the following eigenvalues:

𝐸𝑖𝐸𝑒𝐸𝑣𝑎𝐸𝑢𝑒𝐸(𝐵) = [0, −𝜅𝜑𝜋

𝜎 − 𝜑𝑥

𝜎]’ (51)

The first eigenvalue of matrix B is always smaller than one and the second eigenvalue is smaller than one if equation (52) holds.

𝜅𝜑𝜋+ 𝜑𝑥 > −𝜎 (52)

The eigenvalues of matrix 𝐵𝜌 are given by the product of the eigenvalues of matrix B and 𝜌. Given that matrix 𝜌 has eigenvalues smaller than one. Calculating the eigenvalues of matrix B and concluding that the real parts of the eigenvalues are smaller than one is sufficient for E-stability to hold for the contemporaneous expectations policy rule. Assuming that the

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20 Proposition 3. Under the policy rule with contemporaneous expectations the condition for the MSV solution (𝛼�, 𝑣̅) to be E-stable is that equation (52) holds.

This means that the MSV solution is E-stable. In figure 1 the stable region, the area outside the blue triangle, the green area, is also the E-stable region. This means that with the

calibration chosen in this research, E-stability and stability holds in the model as illustrated in figure 1 by the green area.

4.3 Contemporaneous expectations including lagged interest rate

In this subsection the focus is on the contemporaneous expectations including lagged interest rate of the model. Policy rules including lagged interest rate are able to capture the interest rate smoothing observed in central bank behaviour (Bullard and Mitra, 2002). The policy rule is given by equation (16). The complete model includes equations (12), (13) and (16). After implementing the naïve expectations the reduced system is then given by

𝑦𝑡 = 𝐴𝑦𝑡−1+ 𝐵𝐸𝑡𝐶𝐶𝑦𝑡+ 𝐶𝑤𝑡 (53) where 𝑦_𝑡= [𝑥_𝑡, 𝜋_𝑡, 𝑖_𝑡]′, 𝑦_𝑡𝐶𝐶 = [𝑥_𝑡𝑒, 𝜋_𝑡𝑒, 𝑖_𝑡𝑒]′, 𝑦_𝑡−1 = [𝑥_𝑡−1, 𝜋_𝑡−1, 𝑖_𝑡−1]′and 𝑤_𝑡= [𝑒_𝑡, 𝑢_𝑡, 0]′ and A is defined by A = � 1 −1_𝜎 𝜌_𝜎 𝜅 𝛽 −𝜅_𝜎 𝜅𝜌_𝜎 0 0 𝜌 � and B is defined by B = ⎣ ⎢ ⎢ ⎡ 𝜙𝑥(1−𝜌)𝜎 𝜙𝜋(1−𝜌) 𝜎 0 𝜅𝜙_𝑥(1−𝜌) 𝜎 𝜅𝜙𝜋(1−𝜌) 𝜎 0 𝜑𝑥(1 − 𝜌) 𝜑𝜋(1 − 𝜌) 0⎦ ⎥ ⎥ ⎤

and finally C is defined by C = �

1 0 0 𝜅 1 0 0 0 0� 4.3.2 Stability

Analysing determinacy for the contemporaneous expectations including lagged interest rate is not possible due to missing forward-looking variables in the system. The focus is on the stability results for this specification. After implementing 𝐸_𝑡𝐶𝐶𝑦_𝑡 = 𝑦_𝑡 in equation (53). Equation (53) changes in (𝐼 − 𝐵)𝑦_𝑡 = 𝐴𝑦_𝑡−1+ 𝐶𝑤_𝑡. After simplifying this equation we get equation (54).

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The matrix C in equation (53) is stationary by the assumption that the elements of the diagonal matrix are smaller than one.

The necessary condition for stability under the contemporaneous expectations policy rule including lagged interest rate to hold is that the matrix (𝐼 − 𝐵)−1𝐴 has eigenvalues smaller than 1 in modulus. The matrix (𝐼 − 𝐵)−1𝐴 has dimension of (3,3). The characteristic polynomial 𝑝(𝜆) of (𝐼 − 𝐵)−1𝐴 is calculated by using the formula (−𝜆)3+ 𝑡𝑟((𝐼 −

𝐵)−1_{𝐴)(−𝜆)}𝑛−1_{+ ⋯ + 𝑑𝑒𝑡(𝐼 − 𝐵)}−1_{𝐴 (Otto,1995, p. 311).}

The three eigenvalues of matrix (𝐼 − 𝐵)−1𝐴 are inside the unit circle if the equations (55) and (56) are satisfied. The equations are set up by making use of the Intermediate Value theorem and the characteristics of the characteristic polynomial of order three.

𝑝(1) > 0 (55)

𝑝(−1) < 0 (56)

Equation (55) leads to the condition in equation (57).

(1 − 𝛽)𝜑𝑥+ 𝜅(𝜑𝜋− 1) > 0 (57)

Equation (56) leads to the condition in equation (58).

(𝛽 + 1)𝜑𝑥(1 − 𝜌) + 𝜅((𝜌 − 1)𝜑𝜋 − (𝜌 + 1)) < 2𝜎(1 + 𝛽)(1 + 𝜌) (58)

Proposition 4. Stability under the policy rule with contemporaneous expectations including lagged interest rate holds if the equations (57) and (58) are satisfied.

Proposition 4 implies for stability to hold, that the policy rule cannot contain aggressive output gap response. If 𝜑_𝑥 is sufficiently low, policy rules with sufficiently high 𝜑_𝜋 can lead to stability. Comparing proposition 4 with proposition 2, we can conclude that in this case the value of 𝜑_𝑥 has priority in achieving stability. This is caused by equation (58) which restricts the value of 𝜑_𝑥 .

4.3.3 Learning

The MSV solutions are of the form

𝑦𝑡 = 𝛼𝑦𝑡−1+ 𝑣𝑤𝑡= 𝛼𝑦𝑡−1+ 𝑣𝜌𝑤𝑡−1+ 𝑣𝜗𝑡 (59)

where 𝜗_𝑡 = [𝜀_𝑡, 𝜉_𝑡, 0]′ and 𝑤_𝑡 = 𝜌𝑤_𝑡−1+ 𝜗_𝑡. The matrices 𝛼 and 𝑣 need to be calculated by the method of undetermined coefficients. Equation (59) gives the expectation of the central bank, 𝐸_𝑡𝐶𝐶𝑦_𝑡 = 𝛼𝑦_𝑡−1+ 𝑣𝜌𝑤_𝑡−1 , where

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22

𝜌 = �𝜌0 𝜌𝑒 0 0𝑢 0

0 0 0� and the MSV solutions satisfy

𝐴 + 𝐵𝛼 = 𝛼 (60)

𝑣 = 𝐵𝑣𝜌 + 𝐶𝜌 (61)

The REE must satisfy the matrix equations (60) and (61). The matrix 𝛼 and 𝑣 are both unique matrices. The MSV solution in this case takes the form 𝑦_𝑡 = 𝛼�𝑦_𝑡−1+ 𝑣̅𝑤_𝑡 where

𝛼� = (𝐼 − 𝐵)−1_{𝐴 and 𝑣̅ = (𝐼 − 𝐵𝜌)}−1_𝐶𝜌.

For E-stability we assume equation (59) as the PLM of the central bank and using 𝐸𝑡𝐶𝐶𝑦𝑡 = 𝛼𝑦𝑡−1+ 𝑣𝜌𝑤𝑡−1 , the ALM of 𝑦𝑡 is obtained as

𝑦𝑡 = (A + Bα)𝑦𝑡−1+ (𝐵𝑣𝜌 + 𝐶𝜌)𝑤𝑡−1+ 𝐶𝜗𝑡 (62)

𝑇(𝛼, 𝑣) = (𝐴 + 𝐵𝛼, 𝐵𝑣𝜌 + 𝐶𝜌) (63)

E-stability is determined by the following differential equations :

𝜕𝜕

𝜕𝑑 = 𝐴 + 𝐵𝛼 − 𝛼 = 𝐴 + (𝐵 − 𝐼)𝛼 (64)

𝜕𝜕

𝜕𝑑 = 𝐵𝑣𝜌 + 𝐶𝜌 − 𝑣 = (𝐵𝜌 − 𝐼)𝑣 + 𝐶𝜌 (65)

The mapping can then be computed from the PLM to the ALM. The results of Evans and Honkapohja (2001) are applied.

For expectational stability to hold, the requirement is that matrices 𝐵 and 𝐵𝜌 have eigenvalues with real parts smaller than one. The matrix B has the following eigenvalues:

𝐸𝑖𝐸𝑒𝐸𝑣𝑎𝐸𝑢𝑒𝐸(𝐵) = [0,0, −( −1_𝜎+𝝆_𝜎)(𝜑𝑥+ 𝜅𝜑𝜋)]’ (66)

where 𝝆 is the weighting factor in the policy rule and 𝜌 the diagonal matrix. The first two eigenvalues of B are always smaller than one and the third eigenvalue is smaller than one if equation (67) holds.

(1 − 𝝆)(𝜅𝜑𝜋+ 𝜑𝑥) < 𝜎 (67)

The eigenvalues of 𝐵𝜌 are given by the product of the eigenvalues of matrix B and 𝜌. Given that matrix 𝜌 has eigenvalues smaller than one. Calculating the eigenvalues of matrix B and concluding that the real parts of the eigenvalues are smaller than one is sufficient for E-stability to hold in the contemporaneous expectations including lagged interest rate specification.

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23 Proposition 5. Under the policy rule with contemporaneous expectations including lagged interest rate the condition for the MSV solution (𝛼�, 𝑣̅) to be E-stable is that equation (67) holds.

Equation (67) implies that the values of 𝜑_𝜋 and 𝜑_𝑥 need to be sufficiently low to satisfy the condition. The weighting factor 𝝆 decreases the value of the left-hand side of equation (67), assuming 0 < 𝝆 < 1.

5 Conclusion

The stability and determinacy for different policy rules have been analysed with a NKM model with a learning central bank. The methods explained in Evans and Honkapohja (2001) are used to calculate determinacy and E-stability. Private sector expectations under learning has been studied extensively by researchers (Bullard and Mitra, 2002; Molnár and Santoro, 2014). Central bank learning and the performance of linear policy rules have not been investigated extensively. In this study the focus was on determining E-stability and determinacy under different policy rules while the central bank is learning.

The analysis of the forward-looking specification was mainly numerically. This was because of the singular matrix B. The Schur composition was needed to calculate the E-stability and determinacy conditions. Under the calibration chosen in this research

determinacy and E-stability hold. For the other policy rules, we developed conditions under which E-stability and stability hold.

The analysis of the contemporaneous expectations specification lead to conditions that makes it possible to choose the parameters 𝜑_𝑥 and 𝜑_𝜋 under the assumption of the calibration whereby E-stability and stability hold. The analysis also leads to a condition that is in line with the Taylor principle. The condition implies that when stability holds, the nominal interest rate should increase more than the increase in the inflation rate.

Lastly, the contemporaneous expectations specification including lagged interest rate lead to conditions where the output gap response had priority for determining stability. The E-stability conditions were satisfied under sufficiently low output gap response.

In this paper it is assumed that the private agents establish their expectations through naïve expectations. This is a strong assumption, because we assume that the forecasted value is equal to the previous value, Etxt+1 = xt-1. This does not hold in the real world in most cases,

there are better assumptions to make that approach reality (Arrow and Nerlove, 1958). In this paper naïve expectations was used for simplicity reasons.

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24 References

Aoki, K., & Nikolov, K. (2006). Rule-based monetary policy under central bank learning. In NBER International Seminar on Macroeconomics 2004, 145-195. The MIT Press.

Arrow, K. J., & Nerlove, M. (1958). A note on expectations and stability.

Econometrica: Journal of the Econometric Society, 297-305.

Arrow, K. J. (1986). Rationality of self and others in an economic system. Journal of

Business, 59(4), S385-S399.

Batini, N., & Haldane, A. (1999). Forward-looking rules for monetary policy. In

Monetary policy rules, 157-202. University of Chicago Press.

Bernanke, B. S., & Woodford, M. (1997). Inflation forecasts and monetary policy (No. w6157). National Bureau of Economic Research.

Bray, M. M., & Savin, N. E. (1986). Rational expectations equilibria, learning, and model specification. Econometrica: Journal of the Econometric Society, 54(5),1129-1160.

Bullard, J., & Mitra, K. (2002). Learning about monetary policy rules. Journal of

monetary economics, 49(6), 1105-1129.

Ellison, M., & Valla, N. (2001). Learning, uncertainty and central bank activism in an economy with strategic interactions. Journal of Monetary Economics, 48(1), 153-171.

Evans, G. W., & Honkapohja, S. (2001). Learning and expectations in

macroeconomics. Princeton University Press.

Evans, G. W., & Honkapohja, S. (2013). Learning as a rational foundation for macroeconomics and finance. Rethinking Expectations: The Way Forward for

Macroeconomics, 68.

Honkapohja, S., & Mitra, K. (2006). Learning stability in economies with heterogeneous agents. Review of Economic Dynamics, 9(2), 284-309.

King, R. G., & Kerr, W. R. (1996). Limits on interest rate rules in the IS model. FRB

Richmond Economic Quarterly, 82(2), 47-75.

LaSalle, J.P., 1986, The Stability and Control of Discrete Processes, Applied Mathematical Science, 62, (Springer-Verlag: New York).

Lucas, R. E. (1972). Expectations and the Neutrality of Money. Journal of economic

theory, 4(2), 103-124.

Marcet, A., & Sargent, T. J. (1989). Convergence of least squares learning

mechanisms in self-referential linear stochastic models. Journal of Economic theory, 48(2), 337-368.

(25)

25

McCallum, B. T. (1983). On non-uniqueness in rational expectations models: An attempt at perspective. Journal of monetary Economics, 11(2), 139-168.

McCallum, B. T. (1999). Issues in the design of monetary policy rules. Handbook of

macroeconomics, 1, 1483-1530.

McCallum, B. T., & Nelson, E. (1999). Nominal income targeting in an open-economy optimizing model. Journal of Monetary economics, 43(3), 553-578.

Molnár, K., & Santoro, S. (2014). Optimal monetary policy when agents are learning.

European Economic Review, 66, 39-62.

Orphanides, A. (2001). Monetary policy rules based on real-time data. American

Economic Review, 91(4), 964-985.

Otto, B. (1995). Linear algebra with applications. Third Eddition.

Schwartz, A. J. (1995). Why Financial Stability Despends on Price Stability.

Economic Affairs, 15(4), 21-25.

Taylor, J. B. (1993, December). Discretion versus policy rules in practice.

Carnegie-Rochester conference series on public policy, 39, 195-214.

Taylor, J. B. (1999). A historical analysis of monetary policy rules. In Monetary policy

rules, 319-348. University of Chicago Press.

Rotemberg, J. J., & Woodford, M. (1999). Interest rate rules in an estimated sticky price model. In Monetary policy rules, 57-126. University of Chicago Press.

Woodford, M. (1994). Nonstandard indicators for monetary policy: can their usefulness be judged from forecasting regressions?. In Monetary Policy, 95-115. The University of Chicago Press.

Woodford, M. (2000). Pitfalls of forward-looking monetary policy. The American

Central bank learning : comparing policy rules under E-stability