Optimal monetary policy in a new Keynesian model with animal spirits and financial markets

Journal of Economic Dynamics & Control 64 (2016) 148–165

Contents lists available at ScienceDirect

Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Optimal monetary policy in a new Keynesian model with animal spirits and financial markets Matthias Lengnick n,1, Hans-Werner Wohltmann University of Kiel, Germany

a r t i c l e in f o

abstract

Article history: Received 18 December 2014 Received in revised form 25 December 2015 Accepted 10 January 2016 Available online 15 January 2016

This article presents a macro-finance-interaction model that integrates a NKM with bounded rationality and an agent-based financial market model. We derive four interactive channels between the two sectors where two channels are strictly microfounded. We analyze the impact of the different channels on economic stability and derive optimal (conventional and unconventional) monetary policy rules. We find that coefficients of optimal Taylor rules do not significantly change if financial market stabilization becomes part of the central bank's objective function. Additionally, we show that rule-based, backward-looking monetary policy creates huge instabilities if expectations are boundedly rational. Our model is externally validated by showing that it generates fat tailed output growth rates. & 2016 Elsevier B.V. All rights reserved.

JEL classification: E03 E5 G02 Keywords: Agent-based financial markets New Keynesian macroeconomics Microfoundation Optimal monetary policy Unconventional monetary policy

1. Introduction The financial crisis of 2008 has put new issues on the economics research agenda. Recently, a growing literature investigates how speculative phenomena in financial markets spill over to the real economy and whether or not real market developments feed back on financial speculation. One straightforward way to answer such questions is to integrate the standard New Keynesian Macroeconomic (NKM) model with those of the agent-based computational (ACE) finance literature. Early attempts in this area are Kontonikas and Ioannidis (2005) and Kontonikas and Montagnoli (2006) who connect a New Keynesian Macroeconomic (NKM) model with a financial market (FM) model where stock prices result from two different sources: a momentum-effect and a reversal towards the fundamental value. Those models are clearly inspired by the agent-based (chartist/fundamentalist) literature2 on financial markets. A similar approach can also be found in Bask (2011). The major drawback of these models is the lack of a consistent approach of expectation formation. The Rational Expectations (RE) hypothesis which is standard in macroeconomics is kept for the NKM part while financial markets are n

Correspondence to: Wilhelm-Seelig-Platz 1, 24118 Kiel. E-mail addresses: [email protected] (M. Lengnick), [email protected] (H.-W. Wohltmann). 1 Tel.: þ 49 431 188 1446; fax: þ 49 621 181 1774. 2 A literature overview can be found in Samanidou et al. (2007), an empirical model contest in Franke and Westerhoff (2012). For illustrative examples on exchange rate modeling see De Grauwe and Grimaldi (2005) and Bauer et al. (2009). http://dx.doi.org/10.1016/j.jedc.2016.01.003 0165-1889/& 2016 Elsevier B.V. All rights reserved.

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

149

driven by non-rational expectations that are implicitly contained in the behavior of chartists/fundamentalists (compare Brock and Hommes, 1998 for example). Some interesting work in the macro-finance-interaction literature that does not build upon NKM for the description of the real sector has been done by Westerhoff (2012) and Naimzada and Pireddu (2013). The authors use the traditional Keynesian income–expenditure model to represent the real sector. The advantage of this approach is simplicity. Models are typically of small scale so that analytical solutions are tractable. This simplicity however comes at the cost of a nonmicrofounded but ad hoc real economy. In a series of papers Paul De Grauwe3 has proposed to replace the assumption of rational expectations in standard NKMs by an evolutionary learning approach. Following the ACE-FM literature,4 agents in his model apply different forecasting heuristics and adjust their beliefs by ex post evaluation. His approach provides an adequate real sector submodel to an integrated (i.e. macro-finance-interaction) model framework because it allows to state both submodels using identical expectation hypothesis. A first approach to integrate NKM of the De Grauwe type with ACE financial markets has been proposed by Lengnick and Wohltmann (2013). The authors put the Westerhoff (2008) financial market model alongside the De Grauwe NKM and introduce two different interaction channels. In the paper at hand we will further develop this approach by providing microfoundations to the interactive channels. Additionally, we show that the integration of a financial sector could both stabilize and destabilize the macroeconomy depending on the assumptions underlying these channels. We derive optimal policy rules for different regimes and find that the central bank's response to inflation should decrease if financial market stability enters its objective function. For clarity, we want to stress that our model does not belong to the class of fully decentralized agent-based models. Such models typically simulate individual decisions and interactions without any top-down assumption5 using object-oriented computer languages. Microfoundation of macroeconomic aggregates is provided through individual autonomous (inter) actions.6 Our model does also not belong to the class of DSGE models that assume agents to form expectations by applying econometric forecasting techniques or similar learning algorithms.7 It does also not belong to the literature that incorporates parameter or state heterogeneity into a rational expectations DSGE model.8 Instead we apply (for simplicity) the representative agent (in the sense of parameter homogeneity) and introduce heterogeneous expectations in the tradition of the agent-based financial market literature. We provide external validation to this approach by showing that it is capable of generating fat tailed growth rates of output. In Section 2 we will derive an extended version of the IS-curve that gives rise to new interactive channels with the financial sector. In Section 3 we will adjust the expectations heuristics of the real sector subsystem and define the macrofinance-interaction model. The role of the different channels on (in)stability is evaluated in Section 4. In Section 5 we derive optimal simple monetary policy rules of the Taylor-type and discuss the question whether they should be forward- or backward-looking. Section 6 provides empirical validation to our approach. Section 7 concludes.

2. Microfounding an extended IS curve One important aspect on the research agenda to integrate NKM with ACE finance is the identification of the most important channels through which the different sectors influence each other. Several channels have been proposed, but all of them share two common problems: First, the interactive channels are not microfounded or empirically identified but assumed ad hoc. Second, the literature has not agreed upon which channels are most important. Typical assumptions for possible channels which affect the real sector from within the financial one are (1) the existence of wealth effects (Kontonikas and Montagnoli, 2006; Bask, 2011; Westerhoff, 2012; Naimzada and Pireddu, 2013), (2) a collateral based cost effect (Lengnick and Wohltmann, 2013) or (3) a balance-sheet based leverage targeting effect (Scheffknecht and Geiger, 2011). Typical examples for channels going in the opposite direction are (1) a misperception effect (Kontonikas and Montagnoli, 2006; Westerhoff, 2012; De Grauwe and Kaltwasser, 2012; Lengnick and Wohltmann, 2013; Naimzada and Pireddu, 2013), (2) a negative dependence on the (real) interest rate (Kontonikas and Montagnoli, 2006), or (3) a mixture of both (Bask, 2011). But even if the same type of channel is applied, its formalization is often very different. The wealth effect, for example, is formalized in Kontonikas and Montagnoli (2006) and Westerhoff (2012) by adding þ c1 sq to aggregate demand, where c1 is a positive parameter and sq the (log) stock price in period q. in contrast, Bask (2011) adds real stock price changes þc1 ðΔsq π q Þ, where πq is the inflation rate. Naimzada and Pireddu (2013) add a weighted average of the current and 3

De Grauwe (2010, 2011a,b) and De Grauwe (2011b). Beja and Goldman (1980) are a ‘classical’ source. Compare footnote 2 for more recent articles. Assumptions about aggregates (representative agents, market structure, …) are typically called ‘top-down’. In contrast, it is argued that assumptions about individuals that endogenously ‘grow’ aggregate variables (bottom-up) provide deeper microfoundation. 6 Consult Lengnick (2013) for a simple example of the mentioned literature and Dosi et al. (2010) for a more policy oriented one. For a very large-scale ACE-model consult the famous EURACE model, e.g. Dawid et al. (2014) or Dawid et al. (2016). Such models are typically modeled as object-oriented computer simulations and not as differential systems. 7 Compare Evans and Honkapohja (2001) and Adam (2005). 8 Such models are typically computationally much more involved since they feature aggregate dynamic functions that depend on integrals over fundamental parameters and/or distribution of state variables. 4 5

150

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165


 fundamental stock price þ c1 ð1 ωÞsf þ ωsq where ω is the weight and sf the fundamental value of sq.9 In the remainder of this section, we will derive channels that follow from a strict microfoundation approach to check which of the above mentioned channels and formalizations are in line with first order principles. In NKM, money is typically introduced by assuming that holding money generates utility for the household. To introduce stocks within the NKM microfoundation framework, we proceed analogously and assume that holding stocks creates utility in just the same way. Following Gali (2008, pp. 27–32) the household's period utility function is given by   Z 1q  σ N 1q þ η  U C q ; Dq ; N q ¼ 1σ 1þη

ð1Þ

with Zq being a composite index defined as: h i1=ð1  νÞ Z q ¼ α1 C 1q  ν þ α2 D1q  ν

ð2Þ

In the remainder of this paper we will call this approach stock in non-separable utility (SINU). Consumption is given by Cq, labor supply by Nq and the amount of stock demand by Dq. Utility is maximized with respect to the budget constraint: Cq þ

n  D  Bq  1 Sq Dq T q Bq W q q1 e þ þ ¼ Nq þ d þ 1 þ iq  1 q  1 þ Sq  1 Pq Pq Pq Pq Pq Pq

ð3Þ

e where Sq denotes the stock price, Tnq nominal taxes, Bq demand for bonds, Pq the goods price and d q  1 the dividend payment per stock. The costs of current (real) stock demand Sq Dq =Pq appear on the left hand side of (3) while the (real) worth of past  e stock demand plus (real) dividend receipts d q  1 þ Sq  1 Dq  1 =P q is added to the right hand side. It is assumed that divie dend payments d q  1 are earned by firms in q 1 and distributed to households at the beginning of period q. Solving the above optimization problem for an infinitely lived household yields the stock demand function10 dq ¼ xq  c3 ðsq pq Þ  c4 iq

ð4Þ

where lower case letters denote log differences, i.e. relative deviations from steady state. Interpretation of the dependencies of dq is straightforward: (1) The demand for stocks increases if an agent can afford higher consumption (which results in a higher output gap xq). (2) The higher the real price of stocks sq pq , the lesser its demand. (3) dq also depends on bond yields iq, because bonds are a substitute for stocks: If the demand for bonds becomes more profitable, stock demand would decrease. Note that stock demand does not (directly) depend on the expected stock price change between q and q þ 1 because households' behavior is not driven by a speculative motive of stock demand. The extended IS curve becomes
 1


 xq ¼ Eq xq þ 1  iq  Eq π q þ 1 þc1  Eq Δsq þ 1  π q þ 1 þc2  Eq Δiq þ 1 þ ϵxq σ

ð5Þ

with the two new (positive) constants c1 and c2. A detailed derivation can be found in the Online Appendix, Section A. The interpretation of (5) is again straightforward and closely follows Gali (2008, Chapter 2.5.2). In the case of expected (real)
 stock price increases ðEq Δsq þ 1  π q þ 1 4 0Þ, households expect future (real) stock prices to be higher than today. Hence, they expect lower stock demand for the future compared to today (dq þ 1 o dq , see Eq. (4)). Consequently, marginal utility of future consumption is lower than that of current consumption. To smooth marginal utility of consumption in q and q þ1, current consumption is increased. The same rationale holds for the expected change in government bond yields. If iq is
 expected to rise ðEq Δiq þ 1 40Þ future stock demand is expected to be lower than today ðdq þ 1 odq Þ which (as above) leads to increased current consumption and output. Given the assumptions on the wealth effect of other authors (discussed above), we can conclude from this section that Bask (2011) was closest to a channel that is in line with utility optimizing behavior although he had a slightly different
 timing ( þ c1 ðΔsq  π q Þ instead of þc1  Eq Δsq þ 1  π q þ 1 ).

3. The model One problem that has to be solved when joining a NKM model with an ACE-FM is that both are developed to run on different time scales. Financial data are available daily (sometimes even on a minute or second basis) while macroeconomic data are available quarterly (monthly at best). Hence the corresponding models are designed and estimated to fit data on very different time scales. 9

Compare Kontonikas and Montagnoli (2006, Eq. (3)); Westerhoff (2012, Eq. (2.3)); Bask (2011, Eq. (1)); and Naimzada and Pireddu (2013, Section 2.1). Compare Online Appendix, Section A, for a detailed derivation. The Online Appendix can be found on the website of the Journal of Economic Dynamics and Control. 10

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

151

Fig. 1. Time scale; indexed by days t and quarters q.

To allow for a meaningful11 integrated model, we have to make sure that both submodels still run on the time scale they are designed for. For this purpose, we assume that the financial market performs 64 increments of the time index t within one increment of the real market's time index q (Fig. 1). Quarter q consists of the days12 t ¼ 64ðq 1Þ þ 1; …; 64q.13 The justification for this assumption is straightforward: financial market agents could very well act on small time intervals, but important new information from the real sector (i.e. publication of gdp development, surveys on market expectations, forecasts of economic institutes, press conferences of CBs) is not updated daily but on a larger time scale. 3.1. Real sector The real sector of our integrated model consists of a Taylor rule (TR), an inflation equation of the Phillips-type and our extended IS curve (5). To be able to compare the model to others in the literature14 we also allow for a cost effect ð κsq Þ in the Phillips curve: Higher stock prices increase the value of collateral, hence, lower interest payments decrease marginal costs which lead to lower nominal prices e q ½π q þ 1  þ γxq  κsq þ ϵπ π q ¼ βE q

ð6Þ

e q ½-operator refers to the boundedly rational, average In the case of κ ¼ 0 Eq. (6) collapses to a standard Phillips curve. The E market expectation (defined below). The TR is depending on expected future inflation rate and output gap15:   e q ½π q þ 1  π ⋆ þ δx E e q ½xq þ 1  þ ϵi iq ¼ δπ E ð7Þ q The quarterly value of stock prices is given by the average of the corresponding daily values16: sq ¼

64q X 1 st 64 t ¼ 64ðq  1Þ þ 1

ð8Þ

Alternatively, one could assume decaying weights to account for the fact that recent information has a stronger influence on traders than older information. It has been shown that the model outcome is robust against this assumption.17 Expectations are formed in a boundedly rational way using discrete choice learning. For the output gap the set of heuristics is given by e tar ½xq þ 1  ¼ x Targeters18 : E q

ð9Þ

e sta ½xq þ 1  ¼ xq  1 Static exp:: E q
 e ext ½xq þ 1  ¼ xq þαx  xq  xq  1 Extrapolators: E q

ð10Þ ðαx 40Þ

ð11Þ

where x is the steady state of the output gap and the target level of the central bank. Compared to the original De Grauwe model, we assume a different set of expectations that is more in line with those typically assumed in ACE-FM.19 Additionally, we do not assume a hybrid form for the IS and Phillips curve, because the boundedly rational expectations 11 This means that, on the one hand, the well-known parameter estimates and calibrations that the NKM macroeconomics literature has somewhat agreed upon can be contained. On the other hand, our approach also has the ability to explain the stylized facts of financial data (fat tailed and leptokurtic returns, long lasting positive autocorrelation in absolute returns) that are found in higher frequencies. Forcing both on the same time scale would ultimately mean to give up one of the former benefits. 12 It is assumed that trading does not take place on weekends. 13 Lengnick and Wohltmann (2013, Section 2). 14 Compare literature overview in Section 1. 15 We relax this assumption in Section 5. 16 Compare Lengnick and Wohltmann (2013, Eq. (24)). 17 Compare Lengnick and Wohltmann (2013), Section 2 of the Online Appendix. 18 We call the agents following heuristic (9) targeters although they do not follow any explicitly announced CB target. Expectations of such kind are typically called stationary expectations. But for analogy with Eq. (12), however, we call both targeting. 19 Compare, for example, the model of Westerhoff (2008) or De Grauwe and Grimaldi (2006) which is based on Brock and Hommes (1998). For an alternative approach where chartism is based on a moving average rule consult Chiarella et al. (2006).

152

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

Table 1 Baseline parameterization. Real sector

Financial sector

Structure

Learning

Noise σ ϵ ¼ 0:15

σ ¼ 1:0

ζ ¼ 0:5

β ¼ 0:99

ϕ ¼ 10

γ ¼ 0:33 δx ¼ 0:5 δπ ¼ 1:5

αx ¼ απ ¼ 0:2

Structure

Learning

ℓ¼1

k ¼ 0:04

a¼1

k ¼ 0:04 e ¼ 300 m ¼ 0:975

C F

Noise σ C ¼ 0:05 σ F ¼ 0:01 σ s ¼ 0:01

Source: The Financial market parameterization is identical to Westerhoff (2008). The structural parameters of the real sector are standard in NKM (compare Gali, 2008): σ ¼ 1:0 gives rise to log utility, β ¼ 0:99 yields a steady state interest rate of about 4%. γ ¼ 0:33 follows if a unitary Frisch elasticity, a markup of 20%, constant returns to scale and price stickiness of θ ¼ 0:67 are assumed. For the NKM learning parameters we follow De Grauwe (2010, 2011a,bDe Grauwe (2010, 2011a,b) and Lengnick and Wohltmann (2013). For the new extrapolative heuristic, we assume a positive but mild trend extrapolation of 0.2. Note that the variance of noise surrounding the chartists' excess demand function is relatively high ðσ C 4σ F ; σ s Þ which reflects the fact that a large variety of such strategies exists in real markets.

approach (especially static expectations) already gives rise to persistence in line with the rule-of-thumb and habit formation idea.20 Assuming hybridity is therefore not necessary any more. A further advantage of our specification is that the special case of full price flexibility is still included in the model, while the original De Grauwe model becomes explosive for high degrees of price flexibility. This point is important because the NKM is derived by introducing real rigidities into the Real Business Cycle (RBC) model. Therefore, RBC is still incorporated in NKM as a special case. This aspect is important conceptually and should also hold for a boundedly rational version of the NKM (compare Online Appendix, Section E, for further details on this and other advantages of our specification). For inflation, heuristics are given by tar

e ½π q þ 1  ¼ π ⋆ Targeters: E q

ð12Þ

sta

e ½π q þ 1  ¼ π q  1 Static exp:: E q
 e ext ½π q þ 1  ¼ π q þ απ  π q  π q  1 Extrapolators: E q

ð13Þ ðαπ 4 0Þ

ð14Þ

with π ⋆ being the inflation target of the central bank and the steady state of π. Depending on their past performance, measured by the mean squared forecast error (MSFE), each forecasting heuristic jA ftar; sta; extg is ascribed a level of attractivity  h i 2 ej þζAy;j y A fx; πg ð15Þ Ay;j q ¼  yq  1  E q  2 yq  1 q1 j with the memory parameter ζ. The fraction of agents ωy; q applying heuristic j is given by a discrete choice model

ωy;j q ¼

expfϕAy;j q g expfϕAy;tar g þ expfϕAy;sta g þ expfϕAy;ext g q q q

and market expectations are given by the weighted average: X e j ½y e q ½y ωy;j E E q þ 1 ¼ q q þ 1 j q

ð16Þ

ð17Þ

21

De Grauwe points out that agents do not use heuristics (instead of RE) “because they are irrational, but rather because the complexity of the world is overwhelming” that ex ante calculation of mean time paths is impossible. Therefore, “heuristics [are] a rational response of agents who are aware of their limited capacity to understand the world”. In the remainder of the paper we will denote this response boundedly rational (BR) to distinguish it from strict RE. To keep the model simple, we do not add a set of heuristics and a discrete choice learning model for interest rate e q þ 1  ¼ iq . The solution of our real sector model is then expectations of Eq. (5), but only use static expectations of the form E½i 20 Introduction of hybridity into the baseline NKM is typically justified (microfounded) by assuming habit formation (Ravn et al., 2010; Smets and Wouters, 2007) or rule-of-thumb (Amato and Laubach, 2003) behavior. The BR expectations of De Grauwe clearly fall in the second category because, first, they yield the same result on the aggregate level (i.e. persistence) and, second, they follow the four criteria (compare Amato and Laubach, 2003 and Menz, 2008) for rule-of-thumb behavior: (1) They are applied if RE induce too high costs. (2) The orientation variable should be easily observable by the agents. (3) Calculating forecasts should involve virtually no computational burden. (4) Agents should learn, and learning algorithms should make sure that individual choices have converged once a steady state is reached. Additionally, Eq. (9) is also in line with habit formation since an Euler equation subject to (9) results if it is microfounded in accordance with habit formation. 21 De Grauwe (2010b,p. 415).

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

given by (compare Online Appendix, Section B) ! ! xq  1 xq 1 ¼ A q Cq þ Aq 1 Dq  sq þAq 1 Fq  sq  1 þ Aq 1 πq  1 πq

σϵxq ϵiq

tar σ  ðσ δx Þωext x;q ð1 þ αx Þ  σc1 ωs;q h

! ð18Þ

ϵπq

with the time-dependent matrices: Aq ¼

153

ð1  δπ σc1 Þωext π;q ð1 þαπ Þ

!

1  βωext π;q ð1 þ απ Þ 1 hab ext rof ðσ  δx Þðωx;q  αx ωx;q Þ ð1  δπ σc1 Þðωπ;q  απ ωext π;q Þ   A Cq ¼ @ ext 0 β ωrof π;q  απ ωπ;q  !  ! ext c1 σ ωsta ð1 þ αs Þ 1 c1 σ ωext s;q αs ωs;q s;q Dq ¼ Fq ¼ κ 0 0

γ

3.2. Financial sector We use the model of Westerhoff (2008) for the financial sector of our economy.22 In this section we will shortly describe the original Westerhoff model. Afterwards it will be adjusted to allow for interactions with the real economy. In this model agents learn from a set of two different rules: C C Chartists: eE t ½st þ 1  ¼ st þ k ½st  st  1 

h

ð19Þ

F F Fundamentalists: eE t ½st þ 1  ¼ st þ k sft  st  1

i

ð20Þ

Chartists believe in a continuation of the recently observed stock price trend while fundamentalists expect a reversal j towards the fundamental value sft. For both groups j, the excess demand for stocks Dt positively depends on the direction of the expected stock price change:  e j ½st þ 1  st þ ϵj j A fC; F g Djt ¼ ℓ E ð21Þ t t Note that the above equation denotes excess demand of institutional investors of the financial market, while the completely j microfounded equation (4) denotes households' demand. The fractions of agents Wt applying the different strategies j are determined by a discrete choice model. In addition to strategies C and F, Westerhoff (2008) also allows a ‘no trading’strategy: W jt ¼

expfeAjt g C expfeAt g þexpfeAFt g þ expfeA0t g

j A fC; F; 0g

ð22Þ

j

where At is the attractivity of strategy j that is determined as a function of past profits: Ajt ¼ ðexpfst g  expfst  1 gÞDjt  2 þ mAjt  1

ð23Þ A0t

The parameter m A ð0; 1Þ determines the memory of traders and the attractivity of no trading is normalized to ¼ 0 (i.e. no profits). Price adjustment is given by a price impact function   st þ 1 ¼ st þa W Ct DCt þW Ft DFt þ ϵst ð24Þ   C C F F that relates stock price changes positively to excess demand W t Dt þ W t Dt . The random term ϵst denotes the influence of trading strategies other than j A fC; F; 0g. Impacts from the real sector: For the first interactive channel, we follow Kontonikas and Montagnoli (2006), Westerhoff (2012), De Grauwe and Kaltwasser (2012), and Lengnick and Wohltmann (2013),23 and Naimzada and Pireddu (2013) by f assuming that the perceived fundamental value st is biased in the direction of the recent real economic development24:  t 1 ; hZ 0 ð25Þ sft ¼ h  xq q ¼ floor 64 The interpretation of (25) is that fundamental values of financial assets are not easily observable in real markets.25 Agents 22 We decided to use this model because of its straightforward assumptions and implementation. The model is also empirically validated and has successfully been used for policy analysis. For alternative models compare Dieci and Westerhoff (2010) or Tramontana et al. (2013). An interesting example on a much debated policy issue can be found in Westerhoff and Dieci (2006). 23 Compare Eq. (26) in Lengnick and Wohltmann (2013). 24 The floor function in Eq. (25) rounds a number down to the next integer. 25 See Rudebusch (2005) or Bernanke and Gertler (1999).

154

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

Fig. 2. Real and financial sector interactions (Channels).

who follow fundamental investment strategies therefore are forced to proxy the fundamental value by information they receive about the underlying real conditions of the economy (i.e. xq).26 The completely microfounded stock in non-separable utility [SINU] approach gives rise to a second channel because households' demand for stocks (Eq. (4)) has to be added to the demand of institutional investors of the financial market model (Eq. (24)). If we assume that households' (quarterly) demand for stocks is distributed evenly among the 64 days of the 1 Δdq to stock demand such that the price impact function becomes27: quarter, we have to add 64  k ð26Þ st þ 1 ¼ st þa W Ct DCt þW Ft DFt þ Δdq þϵst 64 The parameter k is introduced as a generalization that allows us to vary the intensity of the channel. The natural upperbound for k is k ¼1.28 In case of 1 4k 40 the channel is weakened, while for k¼ 0 it disappears. 3.3. Financial and real sector interaction In total, we have four channels through which the financial and real sector could impact each other (Fig. 2): Channels I and II that are in line with the literature, but assumed ad hoc, and channels IIIa and IIIb that are newly introduced by the microfounded SINU approach. The economic rationale of channel I is that the nominal value of financial assets owned by firms increases when stock prices are rising. Firms' production is largely financed by credit. If their asset side of the balance sheet increases this leads to a rise in their credit worthiness and credit rating. Consequently, they have access to cheaper credits (compare Minsky, 1986). Hence, their costs of production fall which leads to lower prices. Channel IIIa results from intertemporal utility optimization of households. If households expect increasing real stock prices, they also expect falling marginal utility of consumption for the next period. Intertemporal utility smoothing makes them increase consumption today. Channel II goes in the opposite direction and can be interpreted as follows. The true fundamental value of a given stock is f hard to identify in reality (compare Rudebusch, 2005; Bernanke et al., 1999). If the true value of st is unknown, agents have to form assumptions about it. In our model they use proxies like the recent economic development. If output is high, they assume the fundamental value to be high and adjust their demand for stocks accordingly. The fourth channel, ch. IIIb, also results from the microfounded SINU approach. According to Eq. (4), stock demand increases if (1) output increases, (2) the real stock price decreases, or (3) the nominal interest rate decreases. In all three cases, increasing demand will drive stock prices upwards. The intensity of each channel is given by the corresponding interaction parameters κ (Channel I), h (II), c1 (IIIa) and k (IIIb). In the special case of κ ¼ h ¼ c1 ¼ k ¼ 0 the two submodels operate in isolation.

4. Stability analysis In this section we are going to determine for each channel whether it is stabilizing or destabilizing the economy. For this purpose, we will vary the interaction parameters κ, h, c1 and k on an interval from zero (i.e. no interaction) upwards. All other parameters are kept constant. We report them in Table 1 and keep them as our baseline parameterization throughout the entire paper. In a first setting, the impact on (in)stability of the real sector is measured by a typical central banks' loss function which is given as a weighted sum of the unconditional variances of inflation and output29: Real Sector: Lr ¼ varðπ Þ þ 12 varðxÞ

ð27Þ

26 A more detailed explanation of Eq. (25) is found in Lengnick and Wohltmann (2013, Section 2.3). Recall that xq and st are approximately percentage deviations from the steady state. Hence one interpretation of h is that if gdp is 1% above trend, the fundamentalists belief the fundamental value to lie h% above its true counterpart. 27 Note that (net) stock demand is given by Δdq , not dq. P 28 1 1 In this case, the sum over all daily household's stock demand 64 Δdq of one quarter equals the quarterly demand from Eq. (4): 64 t ¼ 1 64Δdq ¼ Δdq . 29 Compare e.g. Svensson (2003).

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

155

3.89

3.55 0.25 2.08

2.01

0.2 0.15

1.03

0.1

κ (channel I)

κ (channel I)

0.25

0.2 1.22 0.15 0.72

0.1

0.53

0

0.42

0.05

0.05

0

0.2

0.4

0.6

0.8

1

h (channel II)

0.27 Loss: L

0

r

0.25 0

0.1

0.2

0.3

k (channel IIIb)

0.99

0.99 0.4

0.4

0.65 0.43

0.3

0.28

0.2

0.19 0.1

0.12 0

0.2

0.4

0.6

h (channel II)

0.8

1

Loss: L

r

c1 (channel IIIa)

c1 (channel IIIa)

r

0.5

0.5

0

Loss: L

0.65 0.43

0.3

0.28

0.2

0.19 0.1 0

0.12 0

0.1

0.2

k (channel IIIb)

0.3

Loss: L

r

Fig. 3. (De)stabilization of real submodel. Light gray areas denote parameterizations that yield high loss values (i.e. low welfare), while dark gray areas denote low losses. White areas, on the contrary, denote explosiveness.

The interpretation of (27) is that volatile levels of inflation and production are associated with utility losses, where output stabilization is weighted half as much as price stability. In a second setting, we calculate a similar loss function for the financial sector Financial Sector: Lf ¼ varðsÞ

ð28Þ

that associates volatile daily30 stock prices with losses. To evaluate the effect of the four different interactive channels on economic stability, we proceed as follows. First, we pick pairs of two interaction parameters with opposed direction.31 Second, we run the model for different values of the i x s interactive parameters and for different realizations of the noise terms (ϵq, ϵq, ϵπq , ϵt ) 8 q; t. Finally, we compute the average loss values (Eqs. (27) and (28)) for a given parameterization which yields (approximately) the theoretical values Lr and Lf.32 In Fig. 3 we illustrate the stabilization impact on the real sector. White areas (in the north-east) denote parameterizations for which no stable solution exists, i.e. the generated trajectories diverge/explode. Gray areas denote parameterizations that are non-explosive and the darkness indicates the corresponding loss value. The darker a region, the lower the associated loss Lr. The corresponding results for financial sector stabilization (Lf) are displayed in Fig. 4. Channel I ðκÞ: The influence of κ on welfare is clearly negative for both the real and the financial subsystem. For a given h or k, an increase of κ leads to higher loss values. For h 40:45 or k 4 0:14 it even gives rise to explosive developments. E.g. if, in the top left panel of fig. 3, we fix h ¼0.2 and let κ increase from 0 upwards, we successively reach areas of higher Lr (i.e. 30

Recall that daily stock prices are given by st, while quarterly are given by sq. For example, the parameter pair ðκ; hÞ constitutes one channel that effects the financial sector from the real sector and one channel of the opposite direction. 32 This procedure is related to the approach of Naimzada and Pireddu (2013) who also vary the interaction parameter (ω in their paper) to analyze stability. But instead of a loss function, the authors use bifurcation plots to illustrate stability impacts. 31

156

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

0.055

0.2 0.061

0.15 0.1

0.049

0.05 0

0.25

0.077 κ (channel I)

κ (channel I)

0.25

0

0.2

0.4

0.6

0.8

1

h (channel II)

0.2 0.047

0.15 0.1 0.05 0

Loss: Lf

0.041 0

0.1

0.2

0.3

k (channel IIIb)

Loss: Lf

0.5

0.5 0.084

0.2

0.056

0.1

0.047

0

0.2

0.4

0.6

h (channel II)

0.8

1

Loss: L

f

c1 (channel IIIa)

c1 (channel IIIa)

0.069

0.3

0

0.055

0.4

0.4

0.3 0.047

0.2 0.1 0

0

0.1

0.2

k (channel IIIb)

0.3

0.04 Loss: L f

Fig. 4. (De)stabilization of financial submodel. Light gray areas denote parameterizations that yield high loss values (i.e. low welfare), while dark gray areas denote low losses. White areas, on the contrary, denote explosiveness.

higher loss/volatility, lower welfare). If we fix h¼0.8, the system even becomes explosive (i.e. no stable trajectory) as soon as κ 4 0:12. The economic explanation for this explosive behavior is straightforward. If stock prices increase, inflation will fall due to (negative) cost effects (Eq. (6)). Lower inflation leads to lower inflation expectations (Eq. (14)) and therefore also to an increase in output (if the extended Taylor principle δπ 4 1 þ σc1 holds). The rising output creates a feedback mechanism that drives stock prices up further, no matter which of the opposing channels (II or IIIb) is active. If channel II is active ðh4 0Þ, a higher perceived fundamental value leads to higher demand for stocks (Eq. (25)), while if channel IIIb is active ðk 4 0Þ, households directly demand more stocks (Eq. (26)) which drives prices up. Channel II ðhÞ: For parameter h we find somewhat ambiguous results. Financial markets are always destabilized (for I and IIIa being the channel of opposite direction). For sufficiently large κ, an increase in h could even lead to explosive behavior. With respect to the real sector the results are not as clear. In combination with a significant strength of channel IIIa ðc1 4 0:1Þ, increasing values of h are neutral w.r.t. stability Lr, while for c1 o0:1 a rise in h increases stability. In combination with channel I, higher h lead to more stable developments at first. If, however, h is increased above a certain threshold, the model suddenly becomes explosive. Channel IIIa ðc1 Þ: Results for c1 are again ambiguous. When combined with channel IIIb, the impact of higher c1 is stabilizing for both the real and the financial sector. When combined with channel II, higher values of c1 are only stabilizing the real sector but are almost neutral with respect to the financial one. The economic intuition behind these results is the following. If channels IIIa and IIIb are active, an increase in output leads to higher stock demand of households (Eq. (4)) and therefore higher stock prices. Through channel IIIa (Eq. (5)), output e q ½sq þ 1  sq Þ. Higher stock prices (sq) e q ½Δsq þ 1  ¼ E depends positively on the expected future change in stock prices ðE therefore negatively affect output which dampens the original effect and stabilizes the economy. If IIIa is combined with II one would expect the same results, since channel II (just as IIIb) positively relates stock prices to output development (Eq. (25)). Channel II, however, depends on market sentiments: only if the fraction of fundamentalists

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

157

Table 2 Optimal simple rules for different channels and objective functions. Parameterization of channels: κ ¼ 0:100, h ¼ 0:500, c1 ¼ 0:200, k ¼ 0:200. Channels

Real Market

None I & II IIIa & II I & IIIb IIIa & IIIb All

δ⋆ π δ⋆ π δ⋆ π δ⋆ π δ⋆ π δ⋆ π

¼ 3:47 ¼ 3:44 ¼ 3:18 ¼ 3:45 ¼ 3:26 ¼ 3:10

Real & Fin. Market δ⋆ x δ⋆ x δ⋆ x δ⋆ x δ⋆ x δ⋆ x

¼ 1:29 ¼ 1:28 ¼ 1:25 ¼ 1:27 ¼ 1:27 ¼ 1:18

δ⋆ π ¼ 3:47 δ⋆ π ¼ 3:42 δ⋆ π ¼ 3:16 δ⋆ π ¼ 3:43 δ⋆ π ¼ 3:24 δ⋆ π ¼ 3:06

δ⋆ x ¼ 1:29 δ⋆ x ¼ 1:28 δ⋆ x ¼ 1:25 δ⋆ x ¼ 1:27 δ⋆ x ¼ 1:27 δ⋆ x ¼ 1:18

in the financial market is significantly high, we could expect the misperception effect (channel II) to have a significant impact. Obviously, this dependence on market sentiments weakens the stabilizing effect of larger values of c1: only for c1 o 0:12 we find a positive stabilization for both markets, while for c1 4 0:12 only the real sector is stabilized by further increases of c1. Channel IIIb ðkÞ: The stability impact of k also depends on the active channels: In combination with channel I, channel IIIb has a stabilizing effect on the real market as long as k is sufficiently small. At the same time, it has a destabilizing effect on the financial sector. If k is increased by too much, however, the model dynamics become explosive (compare explanation under paragraph ‘channel I (κ)’). In combination with channel IIIa, results are very different. Higher values of k have a positive impact on real sector stability but only for low values of c1. Financial market stability, in contrast, is monotonically decreasing. In a last step, we check whether the stabilizing effect of one channel could counteract the destabilizing effect of other channels by so much that a formerly explosive parameterization becomes non-explosive. As an example, we pick the parameter combinations ðh ¼ 0:8; κ ¼ 0:25Þ or ðk ¼ 0:25; κ ¼ 0:25Þ which both yield explosive dynamics (Fig. 4, top left and top right). If, in addition to these two channels, we set c1 ¼0.15 the model becomes stable again in both cases. Increasing c1 therefore shifts the unstable (white) region outwards. From this section we can conclude that there is no easy answer to the question whether interaction between financial markets and the real economy is stabilizing or destabilizing. The results depend strongly on the channels under consideration. Hence, future research has to clarify which of the proposed channels is most relevant empirically.33

5. Optimal monetary policy In this section we will derive simple optimal policy rules for the central bank. In Section 5.1 we derive optimal values for the TR parameters δπ and δx under different types of policy rules and for different objective functions. In Section 5.2 we analyze whether monetary policy should optimally be forward- or backward-looking.. In section 5.3 we derive optimal monetary policy rules if the CB is additionally allowed to use unconventional instruments 5.1. Optimal simple rule ⋆ We use the TR (7) and define the optimal simple rule [OSR] as the central banks' response ðδ⋆ π ; δx Þ that yields the ⋆ minimal loss value. We derive ðδ⋆ ; δ Þ as the minimizers of the average loss value over different realizations of the noise π x i x s vector (ϵq, ϵq, ϵπq , ϵt ) 8 q; t: ⋆ ðδ⋆ π ; δx Þ ¼ arg min Lr δπ ;δx

ð29Þ

In this context, two straightforward questions arise in our interactive model: Does the presence of a financial sector change the optimal policy rule? If it does, to what extend do the different interactive channels matter? To answer these questions, we are going to perform the optimization (29) for different cases: On the one hand, we assume different objective functions for the central bank. First, the loss function depends only on real sector stability. Real sector only: Lr ¼ varðπ Þ þ 12 varðxÞ

ð30Þ

In the second scenario we also add financial market stability varðsÞ: 1 Real & fin: sector: Lr þ f ¼ varðπ Þ þ 12 varðxÞ þ 10 varðsÞ

ð31Þ

Given this loss function the central bank tries to stabilize inflation with highest priority, followed by output and by stock prices with least priority. Note, that both Lr and Lr þ f are formulated analogously to the stability measures of the previous section. 33 Estimation of ACE models is relatively elaborate and some important questions are still open. We therefore have to leave this issue for future research. Compare Franke (2009) and Franke and Westerhoff (2012) on the estimation of ACEs.

158

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

Table 3 Percentage change in OSR policy coefficients and in volatility measures if CB switches from Lr to Lr þ f . Channels

None I & II IIIa & II I & IIIb IIIa & IIIb all

Policy coefficients

Volatility measures

δ⋆ π

δ⋆ x

varðπÞ

varðxÞ

varðsÞ

0.00%  0.56%  0.60%  0.58%  0.55%  1.39%

0.00% þ 0.05% þ 0.41%  0.04%  0.12% þ 0.40%

0.00% þ 0.46% þ 0.51% þ 0.47% þ 0.40% þ 1.14%

0.00%  0.50%  0.53%  0.51%  0.43%  1.18%

0.00%  0.08%  0.08%  0.08%  0.07%  0.37%

Table 4 Comparison of minimal loss values Lr for forward-looking, contemporaneous and backward-looking optimal simple rules. Loss values Lr þ f in parentheses. Channels

None I & II IIIa & II I & IIIb IIIa & IIIb All

Policy rule depending on e q ½π q þ 1 , E e q ½qþ 1 E

πq, xq

0.086 0.088 0.090 0.088 0.087 0.093

0.039 0.041 0.039 0.040 0.039 0.041

(0.090) (0.093) (0.094) (0.092) (0.091) (0.098)

πq-1, xq  1 (0.043) (0.046) (0.044) (0.044) (0.043) (0.046)

2.60 -1 -1 -1 -1 -1

(2.61) (-1) (-1) (-1) (-1) (-1)

On the other hand, we also vary the set of interactive channels that are operating: We start with no channels, continue with all possible pairs of two channels of opposite direction, and end by activating all channels simultaneously. The resulting ⋆ optimal values ðδ⋆ π ; δx Þ are given in Table 2. The first interesting result is that, if the central bank additionally aims to stabilize financial markets, it should less strongly respond to variations in inflation ðδ⋆ π Þ while responses to variations in output is practically unchanged. This result is ⋆ closer examined in Table 3 which shows the percentage change in both the policy coefficients ðδ⋆ π ; δx Þ and the volatility measures ðvarðπÞ; varðxÞ; varðsÞÞ that occur in the OSR if the CB minimizes Lr þ f instead of Lr. For all channel parameterizations, the CB achieves a decrease in the volatility of x and s by accepting an increase in varðπÞ. The reason is that both channels (II & IIIb) that affect the financial sector are directly related to the output gap x. Stabilizing x therefore also indirectly stabilizes the financial market. This explains the CB's higher weight for output stabilization (which also decreases varðsÞ). Put differently, the CB has to lessen its response towards inflation (decrease in δ⋆ π ; 2nd column of Table 3) if any channels are active. Hence the introduction of a new target ðvarðsÞÞ creates a conflict with the target of price stability. The change in optimal responses towards output is ambiguous: If the strongly destabilizing channel II (compare Section 4) is active, the CB's response towards x ðδ⋆ x Þ increases (Table 3, 3rd column). If channel IIIb (that weakly stabilizes the real sector; Section 4) is active, δ⋆ x is decreased instead. This change in optimal TR parameters, however, is very small and only marginally relevant for most practical considerations. Another interesting finding is that the central bank's intervention becomes weaker, the more interactive channels exist. Policy parameters are largest, if no channel is active at all. If two channels of opposite direction are added, the policy parameters fall. If all channels are active simultaneously, they are decreased further. This cannot be explained by a stabilizing effect of the interactive channels which make stabilization policy by the CB superfluous because the loss values (i.e. volatility) also increase monotonically.34 It can also not be explained by the fact that we have an increase in the number of ⋆ targets while the number of controls is unchanged35 because the reduction in δ⋆ π & δx appears ceteris paribus while the 36 number of targets is kept constant. The reduction in CB's policy intervention has to be a result of, first, an increase in the system's volatility (more noisy error terms are included in the system; right column of Table 1) and, second, an increase in the system's non-linearity (e.g. trend following chartists create momentum effects which spill over into the real sector). Both render the economy harder to control, therefore CB policy is less efficient and loss values increase (Table 4, 2nd col.). This finding is robust across both objective functions (Lr and Lr þ f ) as well as different weights37 within the objective function. 34 We show below (2nd column of Table 4) that loss values are smallest in case of no channels, increase if any two channels are activated, and increase further in case of all channels. Recall also the results of Section 4 in which we found a stabilization effect only for some channels under certain conditions. 35 Consult the literature on static and dynamic controllability theory, e.g. Tinbergen (1952) or Wohltmann and Krömer (1984). 36 In the ‘Real Market’-column the targets are var(π) and var(x), while in the ‘Real & Fin. Market’-column they are var(π), var(x) and var(s). 37 If we change, for example, the weight of output stabilization from 12 to 1 (compare Wollmershäuser, 2006), our results remain qualitatively identical.

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

159

Percentage of unstable EV

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Different Taylor−Rule arguments

Fig. 5. Stability of system matrix Aq 1 Cq for different Taylor arguments.

Loss value Lr from optimal simple rule

0.25 none ch. I & II ch. IIIa & II ch. I & IIIb ch. IIIa & IIIb all

0.2

0.15

0.1

0.05 0

0.2

0.4

0.6

0.8

Persistence in Taylor Rule (η ) Fig. 6. Effect of interest rate persistence.

5.2. History-dependent or forward-looking? It is known that an inverse relationship exists between the forward-/backward-lookingness of optimal monetary policy and that of the underlying model: The more forward-looking the model becomes, the more backward-looking the monetary policy should be and vice versa.38 Our boundedly rational model was originally composed in a forward-looking way (Eqs. (6) and (5)). However, the boundedly rational character of expectations makes the model depending on past variables (Eq. (18)) and therefore backward-looking. The question therefore arises whether monetary policy should optimally be forward- or backwardlooking, or something in between. To answer this question, we derive the (expected) loss values Lr that correspond to the optimal simple rule under three different scenarios: In scenario one, monetary policy depends on expectations only (compare Eq. (7)):   e q ½π q þ 1  π ⋆ þδx E e q ½xq þ 1  þ ϵi iq ¼ δπ E q In the second scenario it depends on contemporaneous values   iq ¼ δπ π q π ⋆ þ δx xq þ ϵiq

ð32Þ

and in a third one it depends on the most recent past:   iq ¼ δπ π q  1  π ⋆ þ δx xq  1 þ ϵiq

ð33Þ

38 This issue has been extensively discussed in macroeconomics. Consult, for example, Svensson (1997), Carlstrom and Fuerst (2000), Benhabib et al. (2003), Svensson and Woodford (2003), Eusepi (2005), and Leitemo (2008).

160

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

Fig. 7. πq-1, xq  1 Bifurcation plot of parameter απ for different monetary policy rules depending on different variables.

The resulting minimal loss values Lr are given in Table 4. The alternative loss definition Lr þ f is given in parenthesis. In analogy to the previous subsection we report values for different sets of interactive channels. Smallest loss values result if monetary policy responds to contemporaneous values of output and inflation. If it responds to agent's expectations instead, loss values increase by about 50%.39 This result is again robust across all combinations of interactive channels and different loss functions. If policy becomes backward-looking, the loss value increases dramatically in the case of no interaction between the financial and real sector. If interaction is taken into account, the model even becomes explosive (i.e. Lr -1 and Lr þ f -1). The standard results, mentioned in the beginning of this subsection, are generally confirmed in our boundedly rational NKM: The model is de facto backward-looking. Therefore, monetary policy should depend on information as recent as possible. Since computation of rational expectations is (by assumption) not possible, the most up-to-date information the CB can use is given by contemporaneous values. An interesting new aspect is that wrongly conducted (backward-looking) monetary policy could cause high volatility (large loss values) although it is strictly rule-based. In the presence of financial markets, such policy could even create explosive behavior. To verify these results, we compare the system matrices of the real market subsystem ðh ¼ c1 ¼ 0Þ for the three policy rules Eqs. (7), (32) and (33). All system matrices are time dependent. E.g. for rule (7) the system matrix is given by Aq 1 Cq 39 e q ½π q þ 1 , E e q ½xq þ 1 ) are formed from a mixture of past and contemporaneous values (πq, xq and πq-1, Note that the boundedly rational expectations (E e q ½π q þ 1   E e q ½xq þ 1 -case is in between those of the πq  xq- and π q  1  xq  1 -case. The stabilizing xq  1 ). It is therefore compelling that the loss value in the E influence of the contemporaneous π q  xq -case is strongly dominating, however.

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

161

Table 5 Conventional versus unconventional monetary policy. Parameterization of channels: κ ¼ 0:1, h ¼ 0:5, c1 ¼ 0:2, k ¼ 0:2. d

CB target

Conventional δx, δπ

Unconventional δx, δπ , δs

Lr Lr þ f

Lr ¼ 0:064 Lr þ f ¼ 0:069

Lr ¼ 0:062 Lr þ f ¼ 0:062

Table 6 Moments of financial market returns. Hill index

Kurtosis

10%

5%

2.5%

Gaussian normal

4.72

6.01

7.30

3.0

Empirical S&P 500 FTSE DAX

2.76 2.72 2.88

2.99 3.13 3.15

3.12 3.22 3.25

24.2 12.3 7.9

BR Model Westerhoff (2008) e q ½xq þ 1  e q ½π q þ 1 , E E

Channels None All

2.68 2.68

3.27 3.26

4.07 4.04

6.8 6.8

πq, xq

All

2.69

3.27

4.05

6.8

with Aq and Cq being defined in Eq. (18).40 In each time step q the fractions of agents ωy;j q using the different heuristics (i.e. the state of the learning algorithm; compare Eqs. (15) and (16)) are determined by recent economic development. To get an idea of how often the learning algorithm creates instabilities, we calculate the eigenvalues [EV] of Aq 1 Cq for 47,916 different realizations over the entire possible range of learning states. In Fig. 5 we report the percentage of learning states that result in an unstable system matrix. Obviously, the probability of becoming explosive is much lower for TRs depending on contemporaneous or expected future values of π and x. If the central bank responds to past values, we observe an increase from about 5% to 55%. As a second verification, we introduce persistence (interest rate smoothing) into the TR  n  o e q ½π q þ 1  π ⋆ þ δx E e q ½xq þ 1  þ ϵi iq ¼ ηiq  1 þ ð1  ηÞ δπ E ð34Þ q where η A ð0; 1Þ determines the degree of persistence or smoothness.41 In Fig. 6 we illustrate the (expected) loss value Lr as a function of η for different interactive channels. Smallest loss values result for η A ð0; 0:2Þ. For higher η we find exponentially increasing losses. The result, that backward-looking monetary policy destabilizes the economy, is therefore again confirmed. If a financial sector is active, losses even approach infinity. As a third verification, we follow Naimzada and Pireddu (2013) by checking if, in a bifurcation plot,42 the system loses stability earlier if the TR depends on πq  1 and xq  1 . Exemplarily,43 we show the bifurcation plots of απ (compare Eq. (14)) in Fig. 7 for the three different TRs (7), (32) and (33). For rules depending on expected and contemporaneous values (panels (a) and (b)) the system loses the unique steady state at απ  1:8. If monetary policy becomes backward-looking (panel (c)) the first bifurcation takes place already at απ  0:37. 5.3. Optimal unconventional monetary policy Another question that has been debated in the literature44 on optimal monetary policy is, whether or not the CB should also respond to over-/under-valuation of financial assets (i.e. bubbles). This is typically done by adding a stock price reaction term to the TR, e.g. þ δs  sq . Of course, we could proceed in a similar way and simply derive the optimal δ⋆ s . But the results of 40

Compare Online Appendices, Sections C and D, for system matrices of rules (32) and (33). This approach is common in the literature. See e.g. Clarida et al. (1998) and Clarida et al. (1999). 42 In the bifurcation plots we show the long run developments of the deterministic core, i.e. all stochastic terms are set to ϵiq ¼ ϵxq ¼ ϵπq ¼ ϵst ¼ 0 8 q; t. 43 A bifurcation analysis for all relevant parameters can be found in the Online Appendix, section E.2. Here, we consider απ only, because all other parameters do not give rise to bifurcations, except for αx, which produces very similar results to απ (compare Fig. 10 in Online Appendix). 44 Rudebusch (2005), Kontonikas and Ioannidis (2005), Wollmershäuser (2006), Kontonikas and Montagnoli (2006) and Castelnuovo and Nistico (2010). 41

162

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

Wollmershäuser (2006) who uses a similar45 NKM without bounded rationality already suggest that such a welfare increase takes place but is very small. In light of the policy recently performed by central banks in several advanced countries, we are going to analyze a slightly different question. Instead of applying the conventional instrument only (i.e. the interest rate), we equip the CB with another, unconventional instrument (i.e. direct purchases of financial assets) and derive the optimal mixture of both. CB If the CB's direct purchases (given by Δdq ) are added to the price impact function (26) in just the same way as the excess 46 demand of households, we get:  o k n CB  Δdq þ Δdq þ ϵst ð35Þ st þ 1 ¼ st þa W Ct DCt þW Ft DFt þ 64 The first (conventional) policy instrument is given by the TR (7) while for the second (unconventional) we assume that direct purchases are proportional to stock price misalignment:   e q ½π q þ 1   π ⋆ þ δx E e q ½xq þ 1  þ ϵi Conventional instrument: iq ¼ δπ E q   CB d f ð36Þ Unconventional instrument: Δdt ¼ δs  st s If, for example, the CB buys assets when prices are undervalued and sells when they are overvalued, we have δds o 0. The optimal (simple) mixture of conventional and unconventional instruments is then defined as (compare (29)): ⋆ d;⋆ ðδ⋆ π ; δx ; δs Þ ¼ arg min Lj δπ ;δx ;δds

j A fr; r þ f g

ð37Þ

The results of the optimization problem (37) for conventional versus unconventional policy are contrasted in Table 5. We distinguish between the case where the CB only cares for real sector stability (Lr) and the case where it also cares for financial market stability ðLr þ f Þ. As expected,47 the loss value increases in the case of conventional policy, if var(s) is added to the welfare measure. Compared to conventional monetary policy, unconventional policy leads to a welfare gain for both welfare measures. This gain is larger, if financial market stability explicitly enters the loss function. In both cases (Lr & Lr þ f ), however, the gain of additionally using unconventional instruments is very small. For simplicity we assumed in Eq. (36) that the CB knows the true fundamental stock price sf and can adjust its policy on a daily basis. In this sense, it is able to act as a fundamentalist with perfect knowledge while private agents have to form beliefs about sf (Eq. (25)). In more realistic settings, where the CB also has to form beliefs about sf, welfare gains might be lower. Otherwise, we did not consider the presence of a zero lower bound for the interest rate iq. Unconventional instruments might be more influential if the conventional measures fail (i.e. when the zero lower-bound becomes binding).

6. Empirical verification In this section we show that our model can reproduce some empirically relevant facts. Since we did not perform a fullfledged econometric estimation, we focus primarily on the question whether our model is able to reproduce some higher moments qualitatively.48 Namely, whether the distribution of endogenous variables exhibits fat tails and excess kurtosis. 6.1. Financial markets In reality, financial markets are characterized by a set of empirical stylized facts that are found throughout a wide set of different types of time series (e.g. stock prices, commodity prices, foreign exchange rates, stock indices, and interest rates). The most popular of such stylized facts are excess kurtosis and fat tails of returns. To test if our model captures these facts, we compute the Hill index49 for different tail regions (10%, 5%, 2.5%) together with the kurtosis and compare both with those of empirical time series as well as the normal distribution (Table 6). 45 The model of Wollmershäuser (2006) consists of the typical three New Keynesian equations extended by a nominal exchange rate, where the development of nominal exchange rates is modeled in several alternative ways. In one case it is given by a simplified chartist-fundamentalist model (Eq. (3.4) and footnote 6 in Wollmershäuser, 2006) similar to our stock market. 46 Implicitly we assume here that the intensity k is identical for household's and CB's purchases. This could, of course, be relaxed in future work. 47 Adding a new control while keeping the number of targets fixed must lead to a decrease of the loss value. Compare Tinbergen (1952) and Wohltmann and Krömer (1984) where the concept of static and dynamic controllability of economic systems is discussed. An alternative additional instrument to control the financial market would be the introduction of a financial transaction tax. Lengnick and Wohltmann (2013) show that such a tax could very well reduce the volatility of financial markets if it is properly conducted. 48 In the previous sections we compared model versions that were very different in nature (different policy rules, different interaction channels between the two submodels). For comparative reasons we adjusted the channels and policy parameters only while ceteris paribus keeping all other parameters unchanged. 49 The Hill estimator is a non-parametric way to measure the fatness of tails. Compare Hill (1975).

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

163

Table 7 Moments of gdp growth rates. Hill index

Kurtosis

10%

7:5%

5%

Gaussian normal

4.72

5.25

6.01

3.0

Empirical U.S., 1947–2014 U.K., 1948–2014 Germany, 1970–2014

4.59 3.09 4.00

4.16 3.80 4.44

4.81 3.46 5.41

4.4 6.3 5.3

BR Model e q ½xq þ 1  e q ½π q þ 1 , E E

Channels None

4.54

5.03

5.72

3.1

e q ½xq þ 1  e q ½π q þ 1 , E E πq, xq πq, xq

All

4.55

5.04

5.72

3.1

None All

3.98 3.96

4.29 4.24

4.61 4.57

5.4 5.0

NKM with rational Exp. e q ½xq þ 1  e q ½π q þ 1 , E E

4.72

5.25

6.01

3.0

πq, xq πq, xq (hybrid form)

4.72 4.72

5.25 5.25

6.01 6.01

3.0 3.0

Empirically, the Hill estimates are near 3 for all series and all tail regions. Compared to the normal distribution this indicates fat tails. The kurtosis is empirically not as robustly located at a certain value. It is, however, much larger than 3 in all cases (i.e. excess kurtosis or leptokurtosis). If, in our model, we set κ ¼ h ¼ c1 ¼ k ¼ 0 (no interactive channels) the financial submodel collapses to Westerhoff (2008). This model already captures the stylized facts well: The Hill estimate indicates fat tails while the kurtosis is much larger than that of the normal distribution. In the last two rows of Table 6 we report the same moments for our integrated model with all interactive channels in place.50 All moments are practically unchanged, no matter whether we apply the specification where the TR depends on expected or contemporaneous values.51 Therefore, an integrating of the Westerhoff (2008) model with a real sector does not lead to substantial improvements in empirical fit (because it is already very good). 6.2. The real economy It is known that the distribution of real economic growth is also fat tailed. Fagiolo et al. (2008), for example, report robust fat tailed gdp growth rates across different OECD countries. This fact is very interesting because it might tell something about the underlying data generating process and/or the aggregation mechanism at work.52 In Table 7 we report empirical tail indices for gdp growth rates of US, UK and Germany as well as the corresponding kurtosis. Both, tails and kurtosis, are again larger than those of the normal distribution. These results are found for all three countries. Numerically, however, they are somewhat more diverse across countries than those of financial returns. e q ½xq þ 1 ) our BR model produces gdp growth rates that e q ½π q þ 1 , E In its specification with a TR depending on expectations (E are close to the normal distribution: Hill estimates are only slightly below the corresponding Gaussian ones while the kurtosis is only slightly above three. In contrast, for the specification with contemporaneous values in the policy rule, we find much fatter tails that are reasonably close to empirical ones. The kurtosis is larger than 3 and also of empirically reasonable size. The bottom rows of Table 7 report the results of a standard New Keynesian model with rational expectations that correspond to both of our specifications. Expectedly, all moments are exactly equal to those of the normal distribution. This does not change, even if we impose a hybrid structure (i.e. backward-looking IS and Phillips curve). Our bounded rationality model thus seems to capture an important fact about empirical reality that DSGE rational expectations models cannot explain.53 50

The parameterization of the interactive channels is taken from Table 2. In its third specification (πq-1, xq  1 ) the model is too unstable (compare Fig. 5) to generate very long time series and exploit the law of large numbers. Hence it is not reported in Table 6. Similar problems arise with the (πq, xq)-specification. Here, we are also not able to generate very long time series, since with a tiny probability the simulations diverge. Instead we generate a large set of independent shorter simulations and remove those that resulted in explosive trajectories. 52 Consult Brock (1999) for an introduction (and many illustrating examples) on the relation between scaling laws and the underlying data generating (and aggregation) process. Consult Castaldi and Dosi (2009) for an application of individual firm growth rate distributions and its relation to scaling laws on the aggregate level. 53 Note that we did not perform a detailed estimation to match our model to real data, e.g. second moments of output gap, inflation and interest rates. We also did not compare our BR vs. the RE model in terms of out-of-sample forecasting. Hence we cannot claim that our model generally provides a better fit of empirical reality. We only showed that it is capable of generating leptokurtic and fat tailed distribution of output growth. 51

164

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

7. Conclusion This paper extends the work of Kontonikas and Montagnoli (2006), Bask (2011), Scheffknecht and Geiger (2011), Bask (2011), Westerhoff (2012), Naimzada and Pireddu (2013), and Lengnick and Wohltmann (2013), i.e. the macro-financeinteraction literature with non-microfounded interaction channels. It combines the macroeconomic BR-NKM of De Grauwe (2011a) with the financial ACE of Westerhoff (2008) by deriving a generalized IS curve that originates from a non-separable utility function including stocks. This approach gives rise to additional completely microfounded interaction channels with the financial sector. Once the model is set up, we perform a stability analysis with ambiguous results. The cost channel is clearly destabilizing both the real and the financial sector. For the other channels, results either differ for both sectors (i.e. stabilizing one while destabilizing the other) or they change significantly with the opposing channel. In some regimes, the interactive channels strongly feed back on each other and yield explosive dynamics. We derive optimal monetary policy rules under a set of different regimes. We find that the central bank's response to inflation decreases slightly if financial market stability enters the central bank's objective function. Another interesting result is that the optimal central bank's response to deviations of inflation from its target becomes weaker, the higher the degree of interaction between the financial and the real sector. Furthermore, we test if the standard results that monetary policy should be backward-looking if the system is forwardlooking (and vice versa) can be confirmed for our boundedly rational NKM. We have shown that because the backwardlooking nature of the expectations algorithm turns the forward-looking model into a backward-looking one, monetary policy should optimally depend on contemporaneous variables. If, instead, the policy rule becomes backward-looking, the economy is strongly destabilized. Additionally, we analyse the importance of unconventional monetary policy instruments and find that they increase welfare only marginally if applied together with conventional instruments. Finally, we perform an empirical validation of our model by showing that fat tailed gdp growth rates, an empirically important fact, are generated endogenously. While a number of research questions have been answered in this paper, others had to remain open. For example, we did only focus on simple rules when deriving optimal monetary policy. A detailed treatment of optimal unrestricted policy rules in case of boundedly rational expectations should be conducted in future research. Additionally, the assumption that agents are described by homogeneous parameterization and states could be relaxed in the future.

Acknowledgements We would like to thank Thomas Lux, Sven Offick, Sebastian Krug and Luka Scheffknecht as well as an anonymous referee for their fruitful comments and valuable suggestions.

Appendix A. Supplementary Material Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.jedc. 2016.01.003.

References Adam, K., 2005. Learning to forecast and cyclical behavior of output and inflation. Macroecon. Dyn. 9, 1–27. Amato, J.D., Laubach, T., 2003. Rule-of-thumb behaviour and monetary policy. Eur. Econ. Rev. 47, 791–831. Bask, M., 2011. Asset price misalignments and monetary policy. Int. J. Financ. Econ., 540. Bauer, C., Grauwe, P.D., Reitz, S., 2009. Exchange rate dynamics in a target zone: a heterogeneous expectations approach. J. Econ. Dyn. Control 33 (2), 329–344. Beja, A., Goldman, M.B., 1980. On the dynamic behavior of prices in disequilibrium. J. Financ. 35 (2), 235–248. Benhabib, J., Schmitt-Grohé, S., Uribe, M., 2003. Backward-looking interest-rate rules, interest-rate smoothing, and macroeconomic instability. In: Federal Reserve Bank of Cleveland Journal Proceedings, pp. 1379–1423. Bernanke, B., Gertler, M., 1999. Monetary policy and asset price volatility. Econ. Rev. 4, 17–51. Bernanke, B.S., Gertler, M., Gilchrist, S. 1999. The financial accelerator in a quantitative business cycle framework. In: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics, vol. I, pp. 1341–1393 (Chapter 21). Brock, W.A., 1999. Scaling in economics: a reader's guide. Ind. Corp. Change 8, 409–446. Brock, W.A., Hommes, C.H., 1998. Heterogeneous beliefs and routes to chaos in a simple asset pricing Model. J. Econ. Dyn. Control 22, 1235–1274. Carlstrom, C., Fuerst, T.S., 2000. Forward-Looking versus Backward-Looking Taylor Rules, Federal Reserve Bank of Cleveland Working Papers 00-09. Castaldi, C., Dosi, G., 2009. The patterns of output growth of firms and countries: scale invariances and scale specificities. Empir. Econ. 37, 475–495. Castelnuovo, E., Nistico, S., 2010. Stock market conditions and monetary policy in a DSGE model. J. Econ. Dyn. Control 34, 1700–1731. Chiarella, C., He, X.-Z., Hommes, C., 2006. A dynamic analysis of moving average rules. J. Econ. Dyn. Control 30 (9-10), 1729–1753. Clarida, R., Gali, J., Gertler, M., 1998. Monetary policy rules in practice: some international evidence. Eur. Econ. Rev. 42, 1033–1067. Clarida, R., Gali, J., Gertler, M., 1999. The science of monetary policy: a new Keynesian perspective. J. Econ. Lit. 4, 1661–1707.

M. Lengnick, H.-W. Wohltmann / Journal of Economic Dynamics & Control 64 (2016) 148–165

165

Dawid, H., Gemkow, S., Harting, P., van der Hoog, S., Neugart, M., 2014. Agentbased macroeconomic modeling and policy analysis: The Eurace@Unibi model. Bielefeld Working Papers in Economics and Management No. 01-2014. Dawid, H., Harting, P., Neugart, M., 2014. Economic convergence: policy implications from a heterogeneous agent model. J. Econ. Dyn. Control 44, 54–80. De Grauwe, P., 2011a. Animal spirits and monetary policy. Economic Theory. 47, 423–457. De Grauwe, P., 2010. The scientific foundation of dynamic stochastic general equilibrium (DSGE) models. Publ. Choice 144, 413–443. De Grauwe, P., 2011b. Top-down versus bottom-up macroconomics. CESifo Econ. Stud. 57, 499–530. De Grauwe, P., Grimaldi, M., 2005. Heterogeneity of agents, transactions costs and the exchange rate. J. Econ. Dyn. Control 29 (4), 691–719. De Grauwe, P., Grimaldi, M., 2006. Exchange rate puzzles: a tale of switching attractors. Eur. Econ. Rev. 50, 1–33. De Grauwe, P., Kaltwasser, P.R., 2012. Animal spirits in the foreign exchange market. J. Econ. Dyn. Control 36, 1176–1192. Dieci, R., Westerhoff, F., 2010. Heterogeneous speculators, endogenous fluctuations and interacting markets: a model of stock prices and exchange rates. J. Econ. Dyn. Control 34, 743–764. Dosi, G., Fagiolo, G., Roventini, A., 2010. Schumpeter meeting Keynes: a policy-friendly model of endogenous growth and business cycles. J. Econ. Dyn. Control 34 (9), 1748–1767. Eusepi, S., 2005. Comparing Forecast-Based and Backward-Looking Taylor Rules: A “Global” Analysis, Federal Reserve Bank of New York Staff Reports 198, p. 42. Evans, G., Honkapohja, S., 2001. Learning and Expectations in Macroeconomics. Princeton University Press, Princeton. Fagiolo, G., Napoletano, M., Roventini, A., 2008. Are output growth-rate distributions fattailed? Some evidence from OECD countries. J. Appl. Econom. 23, 639–669. Franke, R., 2009. Applying the method of simulated moments to estimate a small agent-based asset pricing model. J. Empir. Financ. 16, 804–815. Franke, R., Westerhoff, F., 2012. Structural stochastic volatility in asset pricing dynamics: estimation and model contest. J. Econ. Dyn. Control 36 (8), 1193–1211. Gali, J., 2008. Monetary Policy, Inflation, and the Business Cycle. Princeton University Press, Princeton. Hill, B.M., 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174. Kontonikas, A., Ioannidis, C., 2005. Should monetary policy respond to asset price misalignments? Econ. Model. 22, 1105–1121. Kontonikas, A., Montagnoli, A., 2006. Optimal monetary policy and asset price misalignments. Scott. J. Polit. Econ. 53, 636–654. Leitemo, K., 2008. Inflation-targeting rules: history-dependence or forward-looking?. Econ. Lett. 100, 267–270. Lengnick, M., 2013. Agent-based macroeconomics: a baseline model. J. Econ. Behav. Org. 86, 102–120. Lengnick, M., Wohltmann, H.-W., 2013. Agent-based financial markets and New Keynesian macroeconomics: a synthesis. J. Econ. Interact. Coord. 8 (1), 1–32. Menz, J.-O., 2008. Behavioral Macroeconomics and the New Keynesian Model, Working Paper of the Macroeconomics and Finance Series, University of Hamburg, p. 24. Minsky, H.P., 1986. Stabilizing an Unstable Economy. Yale University Press, New Haven. Naimzada, A.K., Pireddu, M., 2013. Dynamic Behavior of Real and Stock Markets with Varying Degree of Interaction. DEMS Working Paper 245, 17 pp. Ravn, M.O., Schmitt-Grohe, S., Uribe, M., Uuskula, L., 2010. Deep habits and the dynamic effects of monetary policy shocks. J. Jpn. Int. Econ. 24, 236–258. Rudebusch, G.D., 2005. Monetary policy and asset price bubbles. Fed. Reserve Bank San Franc. Econ. Lett., 5. Samanidou, E., Zschischang, E., Stauffer, D., Lux, T., 2007. Agent-based models of financial markets. Rep. Progr. Phys. 70, 409–450. Scheffknecht, L., Geiger, F., 2011. A Behavioral Macroeconomic Model with Endogenous Boom-Bust Cycles and Leverage Dynamics. Discussion Paper, University of Hohenheim. Smets, F., Wouters, R., 2007. Shocks and frictions in US business cycles: a Bayesian DSGE approach. Am. Econ. Rev. 97, 586–606. Svensson, L.E.O., 1997. Inflation forecast targeting: implementing and monitoring inflation targets. Eur. Econ. Rev. 41 (6), 1111–1146. Svensson, L.E.O., 2003. What is wrong with Taylor rules? Using judgment in monetary policy through targeting rules. J. Econ. Lit. 41 (2), 426–477. Svensson, L.E., Woodford, M. 2003. Implementing optimal policy through inflation-forecast targeting. In: The Inflation-Targeting Debate. University of Chicago Press, Chicago, pp. 19–92. Tinbergen, J. 1952. On the Theory of Economic Policy. North-Holland., Amsterdam. Tramontana, F., Westerhoff, F., Gardini, L., 2013. The bull and bear market model of Huang and Day: some extensions and new results. J. Econ. Dyn. Control 37 (11), 2351–2370. Westerhoff, F., 2008. The use of agent-based financial market models to test the effectiveness of regulatory policies. Jahrb. Natl. Stat. (J. Econ. Stat.) 228, 195–227. Westerhoff, F., 2012. Interactions between the real economy and the stock market: a simple agent-based approach. Discret. Dyn. Nat. Soc. 21 pp. Westerhoff, F.H., Dieci, R., 2006. The effectiveness of Keynes–Tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach. J. Econ. Dyn. Control 30, 293–322. Wohltmann, H.-W., Krömer, W., 1984. Sufficient conditions for dynamic path controllability of economic systems. J. Econ. Dyn. Control 7 (3), 315–330. Wollmershäuser, T., 2006. Should central banks react to exchange rate movements? An analysis of the robustness of simple rules under exchange rate uncertainty. J. Macroecon. 28, 493–519.