How costly is a misspecified credit channel DSGE model in monetary policymaking?

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How costly is a misspecified credit channel DSGE model in monetary policymaking? Takeshi Yagihashi1 Old Dominion University, Economics Department, 2021 Constant Hall, Norfolk, VA 23529, United States

A R T I C L E I N F O

A BS T RAC T

Keywords: DSGE model Financial accelerator model Credit market friction

This paper examines whether misspecification in credit market friction could be costly in the context of monetary policymaking. Using two widely known dynamic stochastic general equilibrium (DSGE) models, we simulate a hypothetical financial crisis and examine how each model performs when the misspecification occurs in the credit channel. We demonstrate that monetary policy suggested by misspecified models tends to destabilize the economy during crisis, even though one of the two models does reasonably well in estimating policy-invariant model parameters. We also show that the opportunity cost of using a misspecified model is high relative to the outcome achieved under a correctly specified model, particularly when public financial intermediation is available in the correctly specified model. Introducing labor-related variables in either the monetary policy rule or stabilization objectives has the potential of improving policy outcomes in the misspecified credit channel model.

1. Introduction Since the global financial crisis of 2008, many economists have claimed that the environment in which monetary policy operates has drastically changed (Blanchard et al., 2010; Caballero, 2010; Quadrini, 2012). In particular, it has become evident that the monetary transmission mechanism through the supply of credit (“credit channel”) matters greatly in times of financial crisis. This has led many central bankers to adopt credit channel models in their routine policy analysis.2 Around the same time, we have also witnessed the surge of a new generation of credit channel models, as seen in Adrian and Shin (2010), Cúrdia and Woodford (2010a) ,Cúrdia and Woodford (2011), He and Krishnamurthy (2013); Hollander and Liu (2016), and Meh and Moran (2010). These models commonly feature financial institutions along with detailed balance sheet structures. Such changes are mainly motivated by developments in the financial sector observed during and after the financial crisis. This new modeling practice stands in sharp contrast to the earlier generation of credit channel models, in which the role of financial institutions remained mostly passive or obscured.3 Given the numerous ways to specify the credit channel, it has become ever more important to consider what type of credit market

frictions are more relevant in the context of monetary policymaking. To answer this question, we utilize two dynamic stochastic general equilibrium (DSGE) models that are widely known in the literature. The first model is Bernanke et al. (1999, BGG) credit channel model, which assumes that lenders do not know the productivity of individual borrowers and need to pay agency costs in order to verify the financial state of the borrowers. The second model is Gertler and Karadi's (2011, GK) credit channel model, which assumes that borrowers cannot commit themselves in honoring the loan contract and hence become constrained in their ability to obtain funds. Both models generate a similar feedback loop between the real and the financial sectors (“financial accelerator effect”), with the strength of the feedback loop being dependent on the borrower's balance sheet conditions. This core mechanism of the two models matches well with the deleveraging of the financial sector and unstable overall economy observed during the recent financial crisis. However, these models take different approaches in motivating the underlying credit market friction, which makes the endogenous variables behave differently. The purpose of our paper is to examine how each model performs as a policy-guiding tool (“approximating model,” or AM). Using the other credit channel model as the “data-generating model” (DGM), we simulate a hypothetical financial crisis that increases the volatility of

E-mail address: [email protected]. The author is grateful to the editor (Dr. Sushanta Mallick) and two anonymous referees for their invaluable comments and suggestions. In the pre-crisis era, it was customary to use arbitrary dynamic equations as proxies for “credit constraints, house price effects, confidence and accelerator effects” (Harrison et al., 2005). For more details on past practice in the central bank community, see Coenen et al. (2007) and Erceg et al. (2006). 3 For a few exceptions, see Chari et al. (1995); Goodfriend and McCallum (2007). 1 2

http://dx.doi.org/10.1016/j.econmod.2017.08.023 Received 9 March 2017; Received in revised form 23 July 2017; Accepted 23 August 2017 0264-9993/ © 2017 Elsevier B.V. All rights reserved.

Please cite this article as: Yagihashi, T., Economic Modelling (2017), http://dx.doi.org/10.1016/j.econmod.2017.08.023

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long time-series data to identify model parameters, but observed data necessarily cover both crisis and non-crisis periods, making it difficult to assess model performance across different phases of the crisis.6 At a more technical level, our approach focuses on the asymptotic performance of the model based on the Kullback-Leibler Information Criterion (aka KLIC), which helps to isolate the effect of model misspecification from uncertainties that may arise from a limited sample period. Methodologically, this paper is close to two studies (Cogley and Yagihashi, 2010; Chang et al., 2013) that also use a simulation method to assess model performance when misspecification is present. However, this paper differs from them in two aspects. First, their studies calibrate policy shifts based on historical episodes in the US. In Cogley and Yagihashi (2010), policy rule parameters before the policy shift were selected to resemble the accommodative policy stance of the Federal Reserve before the Volcker disinflation started in 1980. Chang et al. (2013) prepared several combinations of their capital/labor tax rates that corresponded to the historical values in the United States that were realized during the period 1950-2003. This paper instead uses an “optimized” policy rule directly calculated from the model. In our view, this provides a cleaner laboratory for assessing how the misspecified model performs in the context of policymaking.7 Second, the aforementioned studies use models with a relatively small number of equations in their simulation, whereas the models examined in this paper match the “medium-scale” DSGE model a la Smets and Wouters (2005), Smets and Wouters (2007) and Justiniano et al. (2010), which is now the standard practice in monetary policy analysis. Thus, our results can be easily compared with more recent studies that worked with similar models. Our paper is organized as follows. The next section explains the simulation design. The third section provides the main results and the cost of misspecification. The fourth section conducts additional experiments on how to overcome the cost of misspecification. The last section summarizes this study and draws several conclusions.

key endogenous variables. In our paper, the approximating model is treated as “misspecified” if it does not fully match the data-generating model.4 The role of the approximating model is twofold. First, it is used to estimate model parameters, using a maximum-likelihood-based estimation method. Second, it is used to calculate the optimized policy rule parameters that minimize the quadratic welfare loss measure, while using the parameter estimates in the first step as input. Because the approximating model suffers from the credit channel misspecification by construct, the resulting policy outcomes will be necessarily worse than those obtained if the model used in policymaking were correctly specified. The associated cost of credit channel misspecification is assessed as follows: First, we examine how the monetary policy rule suggested by the misspecified approximating model (“AM-optimized policy rule”) affects the welfare loss measure across different phases of crisis. Second, we compare the outcomes of the AMoptimized policy rule with the outcomes of the monetary policy rule suggested by the correctly specified model (“DGM-optimized policy rule”). Our main results can be summarized as follows: First, both models generate AM-optimized policy rules that further increase the welfare loss in the DGM, contrary to the original intention by the policymaker. This occurs despite the fact that the parameters estimated using the misspecified BGG model remain relatively stable over different phases of the financial crisis. Second, when comparing the welfare losses achieved under the AM-optimized policy rule with those achieved under the DGM-optimized policy rule, we find that the former is considerably larger than the latter, suggesting that the opportunity cost of model misspecification is economically significant. We further examine whether the use of a more flexible monetary policy rule with additional variables or reassigning alternative stabilization objectives as suggested by Rogoff (1985) would improve the performance of the AM-optimized policy rule. We find that if the policymaker were equipped with the misspecified BGG approximating model, he can successfully stabilize the economy by either adding wage rate in the Taylor rule or adding labor as part of the stabilization objective. This paper makes several contributions to the monetary DSGE literature. First, the paper finds that the BGG model, widely used by many central banks, is mostly immune to the parameter invariance problem a la Lucas (1976). This finding is important because it is customary in the empirical DSGE literature to interpret the change in model parameters as structural changes rather than the consequence of model misspecification (e.g., Canova, 2009; Smets and Wouters, 2005). Second, our paper provides a detailed analysis of how the use of “unconventional” monetary policy affects model performance when credit channel misspecification is present. In particular, we show that if public financial intermediation is available to the monetary policymaker, it would help significantly in improving the policy outcome as opposed to when the policymaker is restricted to using a more conventional Taylor-type rule. Third, the type of model misspecification we examine in this paper differs from existing studies. It is customary in the literature to treat one of the two models as the more “structural” one over the other.5 We follow a more agnostic approach by altering the role of the two models in terms of generating data. Our approach is useful for policymakers who wish to refine their model by incorporating several off-the-shelf models when they do not know how robustly each model performs under potential misspecification. Finally, using a simulation method to assess the cost of a misspecified credit channel model has advantages over methods that work with observed data. Empirical studies on DSGE models usually utilize

2. Simulation design 2.1. Models We use the Bernanke et al. (1999 BGG) model and Gertler and Karadi's (2011, GK) model as the two representative credit channel models. Below we will briefly describe the rationale for choosing these models and review their key mechanisms. 2.1.1. Selection of the credit channel models The BGG model received broad attention from monetary policymakers during the recent financial crisis. In January of 2008, the Federal Reserve governor Mishkin stated in his speech that the financial accelerator mechanism featured in the BGG model describes well the nature of macroeconomic risks that the monetary policymaker faces (Mishkin, 2008). Many central banks, such as the European Central Bank, the Bundesbank, and the Riksbank, have now formally incorporated the BGG-style credit channel into their DSGE models (Gerke et al., 2013). Several empirical studies have further shown that the credit market friction in the BGG model is empirically relevant.8 The BGG model has become one of the most widely used credit channel models within and outside the academic community. 6 For example, Villa (2016) examines how well the BGG/GK models fit the data by utilizing the time-series data for the period 1983q1-2008q3. 7 It should be noted that the monetary policy we consider in this paper is restricted to a certain type of simple feedback rule that is popularized within the monetary DSGE literature. It is not the “first-best” policy that the fully-informed policymaker would freely choose to replicate the resource allocation in a decentralized, frictionless economy. 8 See for example, Christensen and Dib (2008), Christiano et al. (2014), and Merola (2015).

4 This means that we deviate from the practice of treating all models as misspecified in the sense that they inherit deviation from the reality in some way or the other. 5 See for example, An and Schorfheide (2007); Canova (2009); Canova and Sala (2009); Chang et al. (2013); Cogley et al. (2011); Cogley and Yagihashi (2010); Fernandez-Villaverde and Rubio-Ramirez (2007); Hurtado (2014); Leeper and Sims (1994); Lubik and Schorfheide (2004); Lubik and Surico (2010); Rudebusch (2005).

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The GK model features a financial accelerator mechanism similar to the BGG model. However, the model differs from the BGG model in the underlying credit market friction. More specifically, the GK model motivates the credit market friction using the moral hazard of the borrower, which is likely to be associated with the disruption of the interbank loan market and the spike of many risk-spread measures, as observed during the recent crisis. The GK model also provides a rationale for the government to act as an independent facilitator of financial intermediation, a feature not present in the BGG model. Introducing the government in this manner is often regarded as an advantage of the GK and similar models over earlier-generation credit channel models.9 In addition, a few but increasing number of empirical studies have demonstrated the empirical relevance of the GK model.10 We extend the original BGG/GK models by incorporating additional frictions (e.g., wage rigidity) and shocks (e.g., shock on marginal efficiency of investment) so that the models are similar to what is now commonly used in the macro literature as well as in actual policymaking. Below we will describe the core part of the models, leaving the details for Appendix A.

2.1.3. GK model The second credit channel model is from Gertler and Karadi (2011), GK). This model motivates the existence of the credit market friction by (1) assuming that the borrower has the option not to honor the loan contract and appropriate the funds and (2) assuming that because the lender is fully aware of such a risk, the borrower is not capable of expanding its assets beyond a certain leverage ratio. The threshold leverage ratio is determined by the condition that the benefit of continuing business for the borrower is matched by the benefit from reneging on the loan contract

ϕt =

Nt +1 = γGK [(Rtk − Rt −1)ϕt −1 + Rt −1]Nt + ωGK Qt Kt −1

(3)

(4)

where γGK is the survival rate of the borrower and ωGK is the proportional transfer from the lender to the entering borrower to be used as start-up funds. Eq. (4) shows that the borrower's net worth is the sum of the gross revenue from capital investments and the gross return on the previous period's net worth. The revenue from capital investments is increasing in both the risk spread and the leverage ratio. 2.1.4. Similarities and differences between the models The BGG and GK models share several common features. First, both models have common credit channel variables (risk spread and leverage ratio), which can be used as observables when estimating model parameters.12 Second, the two models have the same number of structural shocks.13 In particular, the marginal efficiency of investment shock and the risk premium shock are important in examining the consequences of credit channel misspecification, as researchers have pointed out that these two shocks have played a prominent role in explaining the economic fluctuation during the 2008-09 financial crisis (Christiano et al., 2014; Gali et al., 2012). Third, in both models the main role of the risk spread is to amplify the effect of structural shocks and generate a feedback loop between the real and financial variables (financial accelerator effect). Finally, the implied slow development of net worth serves as an internal propagation mechanism of the shocks. The key differences of these models are: First, the risk spread, which is at the core of both models, responds differently to the endogenous variables. In the BGG model, the risk spread (expressed as the ratio of gross returns) responds contemporaneously to the leverage ratio, and the credit market friction parameter νBGG determines the magnitude of the response. In the GK model, the risk spread is indirectly associated with the leverage ratio through the expected profitability of the borrower's business. The credit market friction parameter νGK affects how strongly the current leverage ratio responds to the expected future risk spread. Second, in the BGG model the role of government in financial intermediation is limited. If we maintain the seemingly natural assumption that the government is as uninformed as the private sector lenders about the borrower's productivity, then the government would set their contract lending rate such that the agency cost is eventually

(1)

Rtk+1

where is the gross rate of return on capital that realizes in period t + 1, Rt is the gross risk-free interest rate determined in period t, Qt and Kt are the price and quantity of the borrower's assets (i.e., installed physical capital), and Nt+1 is the net worth available at the beginning of period t + 1. The positive credit market friction parameter νBGG is derived from solving the optimal contract problem between the borrower and the lender, taking into account both the size of the (unit) agency cost and the size of the idiosyncratic productivity shock that hits the individual borrower. Eq. (1) implies that as the borrower becomes more indebted, a higher return on capital investment is needed to justify the loan contract. Any exogenous shock that affects the leverage ratio will be accompanied by an additional “financial accelerator effect” on the aggregate economy through the behavior of an external financial premium, EtRtk+1/ Rt . This financial accelerator effect is further enhanced through the accumulation process of the borrower's net worth

⎡ ⎤ ⎛ ⎞ ACt −1 Nt +1 = γBGG ⎢Rtk Qt −1Kt −1 − ⎜Rt −1 + ⎟(Qt −1Kt −1 − Nt )⎥ ⎢⎣ Qt −1Kt −1 − Nt ⎠ ⎝ ⎦⎥ + Wb, tLb,

νGK − υt

where ηt is the marginal value of the borrower's net worth, vt is the marginal value of the borrower's assets, and νGK is the fraction of assets that the borrower can possibly divert. As νGK increases, or as variable ηt or vt falls, the borrower becomes more “balance sheet constrained” (Gertler and Karadi, 2011). The constraint is assumed to be binding at all times, and the strength of the constraint depends on the expected profitability of the borrower's business captured by variables ηt and vt . The net worth of the borrower grows as follows,

2.1.2. BGG model The BGG model motivates credit market frictions through a combination of ex-ante uncertainty regarding the borrower's productivity and the agency cost (called “bankruptcy cost” in BGG) for the lender to verify the financial state of the borrower. The latter is known as the costly state verification problem, which was first introduced in Townsend (1979) and has been applied to many other credit channel models.11 To overcome the hurdle between the lender and the borrower, the two parties agree to write a loan contract such that the lending rate becomes contingent on the borrower's balance sheet condition. In aggregate, this borrower-lender relationship can be represented by the following equation

⎛ Q K ⎞νBGG EtRtk+1 = ⎜ t t ⎟ Rt , ⎝ Nt +1 ⎠

ηt

(2)

where ACt−1 is the size of the bankruptcy cost incurred in period t − 1, Wb, t is the borrower's wage rate, Lb is the (fixed) labor supply, and γBGG is the survival rate of the borrower. Eq. (2) states that the borrower's net worth is equal to the gross revenue from the capital investment net of the borrowing cost.

12 They can also serve as additional variables that enter the stabilization objective of the monetary policymaker. See Section 4 for more detail. 13 These are monetary policy shock (Smp ), technology shock ( At ), government spending shock (gt ), marginal efficiency of investment shock ( μt ), price markup shock (λp, t ), wage markup shock (λ w, t ), preference shock (bt ), and risk premium shock (Srp, t ).

9

See for example, Cúrdia and Woodford (2010b), Del Negro et al. (2017), and Gerke et al. (2013). 10 See for example, Afrin (2017) and Villa (2016). 11 See for example, Carlstrom and Fuerst (1997) and Jermann and Quadrini (2012).

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(time-invariant) preference parameters that reflect the policymaker's proclivity of cautiously adjusting the policy rate.16 The monetary policy shock Smp, t is modeled as

covered by the borrower to avoid any loss. In such an environment, there is no strong justification for the government to conduct financial intermediation, which would result in a crowding out of the private lender.14 In the GK model, the government is perceived to always honor its loan contract with the general public, which would free the government from the type of balance sheet constraint that private financial institutions must face at all times. Therefore, there is a potential efficiency gain for the government in conducting public financial intermediation, especially during financial crisis when private lenders face an unusually tight balance sheet constraint. In sum, our model choice implies that the policymaker fully acknowledges the existence of credit market frictions, but fails to understand the exact nature of them, which could lead to misjudgment of the financial accelerator effect. In addition, the role of the government differs in the context of financial market intervention, which may lead to different economic stabilization outcomes in times of financial crisis.

Smp, t = (Smp, t −1) ρmp exp(emp, t ), 2 emp, t ∼ N (0, σmp ),

where ρmp is the persistence of the monetary policy shock, emp, t is the i.i.d. component, and σmp is the size of the i.i.d. shock. We furthermore assume that the policymaker chooses its policy parameters ϕπ , ϕX , and ϕspr so as to minimize a quadratic welfare loss measure

min L 0 ≡ Γ ′t ΨΓt ,

ϕπ , ϕX , ϕspr

(7)

subject to the linearized model equations. We use inflation, output, and the interest rate as the three endogenous variables in calculating the welfare loss measure, i.e.,

lt , X lt , R lt ]′, Γt = [Π

2.2. Monetary policy

where the hat on top of the variables refers to deviation from the steady state. The 3-by-3 diagonal matrix Ψ has entries of 1, λX , λR ; these parameters represent the importance of output and interest rate stabilization relative to inflation. Approximating the social welfare loss through the weighted average of the variances of selected variables has a long tradition in the macroeconomic literature.17 This method has also been used as a means to communicate policy objectives to the general public (Evans, 2011). One unique feature of Eq. (7) is that it includes interest rate stabilization as one of the objectives. The term is expected to capture the monetary policymaker's preference for preventing the interest rate from reaching the zero-lower-bound or wasting resources in economizing money balances when the interest rate is extremely high (Woodford, 2003a). It is also in line with one of the Fed's statutory objectives to foster a “moderate long-term interest rate” (Federal Reserve Act of 1977).

In our paper, we consider two types of “unconventional” monetary policy. One is the Taylor Rule with risk spread (“Hybrid Policy”), which we regard as the monetary policy adopted specifically during the financial crisis period (defined later). The other is the monetary policy rule combined with public financial intermediation (“Dual Policy”), which is used to measure the opportunity cost of model misspecification in later analysis. We will explain each policy below. 2.2.1. Taylor rule with risk spread (the “Hybrid Policy”) The basic instrument of monetary policy in both models is the nominal interest rate, which is allowed to respond to the endogenous variables in the economy. The monetary policymaker sets the interest rate following a modified version of the Taylor rule 1− ρ ⎡ ⎛ X ⎞ ρΔX ⎤ R ρR ⎢(Πt )ϕπ (Xt )ϕX (spr )ϕspr ⎜ t ⎟ ⎥ Rt = Rt −1 exp(Smp, t ), ⎢⎣ ⎝ Xt −1 ⎠ ⎥⎦

(6)

2.2.2. Monetary policy rule combined with public financial intermediation (the “Dual Policy”) We further assume that when using the GK model, the policymaker has an additional policy option to inject credit into the economy in response to changes in the financial intermediaries’ leverage ratio. This option resembles the large-scale asset purchase implemented by the Fed during the recent financial crisis in response to the deteriorating bank balance sheet. The fraction of publicly intermediated asset is determined as

(5)

where Πt is the gross inflation observed over time t − 1 and t, Xt is the value-added output, and sprt ≡ Et (Rtk+1)/ Rt is the risk spread expressed as the ratio of future gross rate of return on capital expected at time t over the gross interest rate. The parameters ϕπ , ϕX , ρR , and ρΔX are commonly used in medium-scale DSGE estimations such as Smets and Wouters (2007) and Justiniano et al. (2010), to capture the policymaker's long-run policy stance. The rationale for including risk premium in the Taylor rule has been widely studied from both the theoretical and the empirical standpoint.15 The general consensus is that the policymaker tends to raise the policy rate when the risk spread falls and lower the rate when the risk spread rises. This type of policy is most relevant during financial crisis. For example, during the early stage of the 2008-09 financial crisis, narrative evidence suggests that the Federal Reserve had aggressively lowered the federal funds rate beyond what the standard Taylor rule with ϕspr = 0 would have suggested. We thus call the monetary policy rule in the equation (5) a “hybrid policy,” noting that it nests both the standard Taylor rule and the contingency policy adopted during the recent financial crisis. In our subsequent analysis, we treat ϕπ , ϕX , and ϕspr as policy parameters that can be freely chosen by the policymaker during the financial crisis, whereas ρR and ρΔX are

ψg, t = ψgϕtνg ,

(8)

where ϕt is the leverage ratio of the borrower and νg is the associated reaction coefficient. Eq. (8) shows that public financial intermediation is a function of the private sector's indebtedness, which also represents the degree of balance sheet constraint. Just as in the hybrid policy case, we assume that during normal times νg is restricted to zero, appealing 16 Justification of having the past interest rate and the output growth rate in the Taylor rule have been extensively studied in the literature (see Mehra, 2002; Woodford, 1999, 2003b), but there is no firm consensus in the literature as to whether ρR and ρΔX should be regarded as part of the policy parameters or not. In practice, central banks rarely communicate their policy intentions through the intended response to either past interest rate or output growth. Furthermore, in many DSGE estimations (e.g., Merola, 2015; Kamber et al., 2015) these parameters are found to remain invariant over time, which supports our treatment of these parameters as time-invariant preference. 17 In a general setting, a first-best welfare-maximizing policy is achieved when a social planner maximizes the discounted stream of utility of a representative household (“welfare”) in a frictionless economy. Rotemberg and Woodford (1997) have suggested using the second-order Taylor series approximation of the welfare as a basis for assessing alternative monetary policy rules' performances in which the central bank acts as the welfare-maximizing social planner. For related work, see also Benigno and Woodford (2005),Clarida et al. (1999), and Woodford (2002).

14 Note that it is possible that the central bank is designated to offer unsecured credit to lenders and is allowed to freely use their operating profit to cover the resulting bankruptcy cost. In practice, such a policy is rarely adopted because central banks are highly averse to incurring any loss through engaging themselves in risky financial intermediation. 15 See for example, Cúrdia and Woodford (2010a,b), Gilchrist and Zakrajsek (2011), and Yagihashi (2011).

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permanent, unanticipated change in the policy rule parameters.20 Each period is characterized as follows:

to the conventional wisdom that policymakers are generally reluctant to interfere with the private sector's business unless it is absolutely necessary to do so.18 Under this alternative policy regime, the policymaker will be able to choose four parameters ϕπ , ϕX , ϕspr and νg (instead of three in the hybrid policy case) when minimizing Eq. (7). We call this the “dual policy” because public financial intermediation (8) can be conducted independently from the monetary policy rule (5). Gertler and Kiyotaki (2010) and Gertler and Karadi (2013) show that further modification of the model allows us to analyze the effect of an equity injection, which was implemented during the recent financial crisis.19 Corresponding to the introduction of this new policy tool, the nationwide leverage ratio is redefined as

ϕc, t =

1 ϕ. 1 − ψg, t t

1. The “Pre-crisis” period. Parameters related to the credit channel are set to the benchmark values that reflect the relatively tranquil period before the crisis. Policy parameters are initialized to the optimized values, which we discuss later. 2. The “Crisis” period. The average risk spread increases and the average leverage ratio decreases, both of which are directly observable to the public. In both the DGM and AM, these changes will translate into a larger credit market friction. In addition, the standard deviation of the risk premium shock increases. This increase is not directly observable to the public and needs to be estimated by the policymaker. Policy parameters remain unchanged from the pre-crisis period. 3. The “Policy shift” period. First, the policymaker recalculates the credit market friction in the AM based on the observed changes in both the average risk spread and the average leverage ratio. Next, the policymaker estimates model parameters by matching the re-parameterized AM and the time series of eight observables in the crisis period (defined in the next subsection). Finally, the policymaker uses the newly estimated model parameters to obtain a new hybrid policy by minimizing the welfare loss in the AM. 4. The “Post-crisis” period. The average risk spread, average leverage ratio, and standard deviation of the risk premium shock all return to the pre-crisis level. Policy parameters remain unchanged from the policy shift period.

(9)

2.2.3. Comparison of outcomes across different monetary policies We use several simulation exercises to examine whether the monetary policies mentioned so far (standard Taylor rule, hybrid, dual) yield the expected stabilization outcome in our model economies (GK, BGG). We initialize the model parameters so as to replicate the financial crisis situation (described later). We also assume that the policymaker is endowed with the correctly specified model and the correct model parameters that were used in generating data. Appendix Table A.1 shows the case for the GK model. When the hybrid policy is adopted, the welfare loss reduces by 20.1% relative to the standard Taylor rule policy with ϕspr = 0 . When the dual policy is adopted, the welfare loss reduces further, which is 28.7% less relative to the standard Taylor rule policy. We also find that the reduction in welfare loss is entirely driven by the smaller standard deviation of the output. Appendix Table A.2 turns to the BGG model. Here only the standard Taylor rule policy and the hybrid policy are considered, because public financial intermediation is not regarded as a policy option. When the hybrid policy is adopted, we still see a reduction in the welfare loss relative to the standard Taylor rule policy, as in the GK model. However, the magnitude is much smaller (0.3%), as the standard deviation of the output does not fall as much as in the GK model. In summary, this experiment demonstrates that (1) the hybrid policy that targets credit market indicator outperforms a standard Taylor rule policy in both BGG and GK framework as expected, (2) in the GK model, the hybrid policy performs notably better than the standard Taylor rule policy, while in the BGG model such improvement remains quantitatively modest, and (3) in the GK model, public financial intermediation serves as an additional stabilization tool.

2.4. Estimation In preparing for the policy shift, the monetary policymaker obtains parameter estimates by fitting the AM to the observed data generated from the DGM. The process is equivalent to minimizing the distance metric known as the Kullback-Leibler Information Criterion

argminKLIC =

⎛p

∫ log⎜⎜⎝

(Y|θDGM ) ⎞ ⎟⎟p (Y|θDGM )d Y, pAM (Y|θAM ) ⎠ DGM

DGM

(10)

where pi (Y|θi ), i = GK , BGG represents the likelihood function, Y represents a vector of variables and θi represents a (subset of the) vector of parameters that appear in both models. The vector of parameters θi is further partitioned into private sector parameters and policy-related parameters

θi priv = [θ pref , θ shock ]′,

(11)

θi pol = [ϕπ , ϕX , ϕspr , ρΔX , ρR ]′,

(12)

pref

where θ is the vector of eight preference/technology parameters and θ shock is the vector of sixteen shock-related parameters (persistence, size).21 The policy-related parameters (12) are treated as known to the policymaker, and hence they will be left out of the estimation. When priv the policymaker solves the problem in Eq. (10) with respect to θAM , the priv estimates converge in probability to the “pseudo-true” estimates θl .

2.3. Timeline To simulate the recent financial crisis, we prepare three events (financial crisis, policy shift, and end of financial crisis) and four time periods that are partitioned by these events. Following the approach of Cogley and Yagihashi (2010), we assume that (1) each period has the same length of sample period of T quarters, (2) within each period, there is an immediate convergence to a new equilibrium (i.e., no learning), (3) the policymaker knows the exact date of the transition from one period to another, so that model parameters in a given period can be estimated separately from another period, and (4) policy shift is characterized as a single,

AM

Due to the presence of model misspecification, there will necessarily be priv asymptotic bias between θ priv and θl . In general, this bias can be DGM

AM

“large” in the economic sense regardless of the sample size T. Only when the AM is correctly specified is this bias expected to vanish as T approaches infinity. In actual simulation, we set T = 40, 000 quarters 20 This implies that we rule out both gradual learning by the policymaker and recurrent changes in policy in our subsequent analysis. 21 Unlike Justiniano et al. (2010) we do not include the MA parameters for the markup shocks in the set of estimated parameters. This is because these parameters are introduced to capture the high-frequency fluctuations in inflation (e.g., Smets and Wouters, 2007) and was primarily meant to improve the model fit. We exclude them from our estimation so that they do not serve as free parameters that absorb the asymptotic bias.

18 Blanchard et al. (2010) provide further discussion on why public financial intermediation is not desirable in normal times. 19 Note that our public financial intermediation does not cover policies that are aimed at solving the short-term liquidity problem (e.g., Commercial Paper Funding Facility).

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and leverage ratio as in the GK model are implied. These choices imply that the elasticity of the risk spread with respect to the leverage ratio (νBGG ) increases from 0.0126 to 0.0175, whereas the annual default rate increases from 3.42% to 5.64% from the pre-crisis to the crisis period.26 For the risk premium shock which is not available in Justiniano et al.'s (2010) model, we had to come up with our own values. We assume that the persistence parameter ρrp remains constant across periods at 0.50 , whereas the size parameter σrp increases from 0.2 in the pre-crisis period to 1 in the crisis period in a regime-switching manner (Dordal-i-Carreras et al., 2016). Before settling upon this value, we experimented with different sets of values for ρrp and σrp . Appendix Table A.3 reports the share of variances explained by the risk premium shock for selected variables based on the 10-quarter forecast error variance decomposition under the chosen parameter values. The numbers are fairly consistent with what past studies on risk premium shock have found.27 For the policy weight parameters λX and λR that appear in the welfare loss measure (7), we follow the values suggested by Woodford (2003a). Policy rule parameters ϕπ and ϕX are initialized to the values that minimize the welfare loss in the DGM during the pre-crisis period while restricting the policy responses to risk spreads and the leverage ratio to zero (i.e., ϕspr = νg = 0 ). This implies we start with a standard Taylor rule with no credit channel variables. The preference parameters of the policymaker ρR and ρΔX are set to Justiniano et al.'s (2010) estimates.

and use longer periods when the result indicates that the pseudo-true value has not been reached.22 Most of the values for θDGM are selected based on the posterior mean reported in Justiniano et al. (2010), who estimate the parameters using data from 1954Q3 to 2004Q4.23 Justiniano et al.'s (2010) estimates are in line with studies on medium-scale DSGE models. We set the prior means equal to θDGM and prior standard deviations equal to the estimates in Justiniano et al. (2010). We note that priors are introduced to facilitate numerical computation and to avoid “dilemma of absurd parameter estimates” (An and Schorfheide, 2007). In limit, the priors have no effect on pseudo-true values because of the large sample period (Gelman et al., 2000). Parameter values of the DGM, their prior distributions, and standard deviations are listed in Table 1. When estimating model parameters, the policymaker is assumed to observe eight macro variables. The number of observables is intentionally matched with the number of structural shocks in the AM to avoid indeterminacy. The variables used as observables are inflation, output, interest rate, consumption, investment, labor, wage rate, and risk spread. The first seven are fairly standard choice in the monetary DSGE literature. Risk spread is included because Christiano et al. (2014) stress the importance of this variable as part of the observables in order for credit channel models to achieve a good fit with the data, which gives the AM an advantage over the standard non-credit channel models in estimating model parameters. 2.5. Calibration

2.6. Stylized facts of the generated variables under both models

Table 2 summarizes the non-estimated parameters. Most parameter values are based on Justiniano et al. (2010), Gertler and Karadi (2011), and Bernanke et al. (1999). A few credit channel parameters take on different values in the precrisis and crisis periods, depending upon how we define the crisis. In the GK model, the fraction of assets that can be diverted (νGK ) and the proportion of assets transferred to the entering bankers (ωGK ) are chosen so that the targets for the average risk spread and leverage ratio are implied. We assume that the average risk spread increases from 100 basis points in the pre-crisis period to 150 basis points in the crisis period, reflecting the observed spike in various measures of the risk spread.24 We also assume that the leverage ratio for the pre-crisis period is five and that for the crisis period it falls to four. The level of leverage is close to the calibrated value in Gertler and Karadi (2011). The magnitude of deleveraging is matched to that in Tressel (2010), which estimated that the US leverage fell by 18% from 2007 to 2009. Parameters used in the BGG model are calibrated so that they target the same moments as in the GK model. In characterizing the borrowerlender relationship, we follow Christiano et al.'s (2014) approach that regards the borrower as financial institutions that invest in physical capital through loans to risky nonfinancial businesses.25 Thus, uncertainty about the borrower's investment project (σBGG ) and the bankruptcy cost ( μBGG ) are chosen so that the same average risk spread

Appendix Table A.4 reports how the standard deviation (both in absolute and in relative terms) of selected variables changes before and after the crisis. The standard deviations of inflation, output, and interest rate all rise during the crisis period. The standard deviation of output shows a more pronounced increase in the BGG model (+ 0.51) relative to the GK model (+ 0.09). The standard deviation of investment relative to output also rises during the crisis in both models (+ 0.31 in GK and + 0.65 in BGG), suggesting that the financial accelerator mechanism has qualitatively similar expansionary effect on investment even though the mechanism is modelled differently. In the BGG model, the relative standard deviation of labor falls slightly in the crisis period (−0.05), whereas in the GK model we do not observe any change. With respect to consumption and wage rate, the relative standard deviation remains fairly stable in both models. Appendix Table A.5 presents the changes of cross-correlation and autocorrelation of selected variables before and after the financial crisis. In the wake of the crisis, the cross-correlation between consumption and output falls in both models. Cross-correlation between wage rate and output rises in the BGG model but not in the GK model. Most autocorrelations for output, consumption, investment, labor, and wage rate rise slightly during the crisis period. Among the five variables, the autocorrelation of investment shows the largest increase in both models. These results demonstrate that both models behave similarly from the pre-crisis to the crisis period. There are subtle qualitative differ-

22 To ensure that the initial conditions are worn off, we further generated 10% more time series and used them as “burn-in.” This means in the case of T = 40, 000 , 44, 000 observations for eight variables are simulated and the first 4, 000 is discarded. 23 There are few exceptions. For the persistence of the three shocks (technology, government spending, wage markup), we use a lower value of 0.95 instead of the original values of 0.99 or 0.97. This is to avoid the possible model indeterminacy that often occurs when the persistence parameter is set too close to the boundary value of one. 24 In determining the credible magnitude of the upward spike, we considered risk spreads associated with Baa corporate bond rates, short-term commercial paper rates, and the LIBOR-OIS spread. 25 In the original work by Bernanke et al. (1999), borrowers were characterized as nonfinancial firms who received funds from the representative bank. Here, the roles of nonfinancial firms and financial institutions can be switched without loss of generality, since there is no agency problem between the bank and the entrepreneur in the BGG model (Christiano et al., 2014, footnote 13). For more related discussion on this topic, see Christiano and Ikeda (2013) and Gertler et al. (2010).

26 Note that the νBGG in our calibration is relatively smaller compared to many studies using the BGG model. This is because in these studies, the credit channel parameters tend to be either calibrated or estimated by treating nonfinancial firms as borrowers. Such treatment typically implies a lower target for the leverage ratio (e.g., two) and a higher target for the risk spread (e.g., 3–5%) than what we consider in this paper. 27 The appendix table shows that during the pre-crisis period, the contribution of the risk premium shock on inflation was 0.2% in the GK model and 0.1% in the BGG model, whereas during the crisis period these numbers jump up to 3.9% and 4.0%, respectively. A similar pattern is observed for other variables as well. Also the reported numbers for inflation, output, and interest rate are fairly close to what are found in Smets and Wouters (2007). For other related work on risk premium shock, see Gali et al. (2012) and Merola (2015).

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Table 1 Estimated parameters in the BGG/GK approximating models. Parameter value

Distri-bution

Standard deviation

Consumption habit h Inverse Frisch elasticity φ Elast.of capital utilization cost χ Investment adjustment cost S″ Calvo prices ξp

0.78 3.79 5.30 2.85 0.84

Beta Gamma Gamma Gamma Beta

0.04 0.76 1.01 0.54 0.02

Price indexation ιp

0.24

Beta

0.08

Calvo wages ξw Wage indexation ιw Persistence of monetary policy shock ρmp

0.70 0.11 0.14

Beta Beta Beta

0.05 0.03 0.06

Persistence of technology shock ρA

0.95

Beta

0.01

Persistence of government spending shock ρG

0.95

Beta

0.01

Persistence of MEI shock ρμ

0.72

Beta

0.04

Persistence of price markup shock ρp

0.94

Beta

0.02

Persistence of wage markup shock ρw

0.95

Beta

0.01

Persistence of preference shock ρb

0.67

Beta

0.04

Persistence of risk premium shock ρrp

0.50

Beta

0.04

Size of monetary policy shock 100σmp

0.22

Inv-Gamma

inf

Size of technology shock 100σA Size of government spending shock 100σG Size of MEI shock 100σμ

0.88 0.35 6.03

Inv-Gamma Inv-Gamma Inv-Gamma

inf inf inf

Size of price markup shock 100σp

0.14

Inv-Gamma

inf

0.20 0.04

Inv-Gamma Inv-Gamma

inf inf

0.20 1.00

Inv-Gamma Inv-Gamma

inf inf

*

Size of wage markup shock 100σw * Size of preference shock 100σb

*

Size of risk premium shock 100σrp Pre-crisis Crisis

Note: Parameter values in the second column are used in both the data-generating model and as the prior mean used in estimating the parameters in the approximating model. MEI shock refers to the marginal efficiency of investment shock. All model parameters are based on quarterly frequency.

the variation across periods remains well within the 95% confidence band implied by the posterior standard deviation found in Justiniano et al. (2010).29 A few shock-related parameters changed during the crisis. The most notable change occurred in the persistence of the marginal efficiency of investment shock ( ρμ ), which increased from 0.75 in the pre-crisis period to 0.92 in the crisis period. It is common to regard the marginal efficiency of investment shock as a proxy for financial market conditions; therefore, the observed change can be interpreted as correctly identifying a disturbance in the financial market.30 Other noticeable but much smaller changes in absolute terms are seen for the persistence of the risk premium shock (from 0.69 to 0.74 ), the size of the preference shock (from 0.04 to 0.07), the size of the wage markup shock (from 0.20 to 0.22 ), and the size of the price markup shock (from 0.12 to 0.13). Table 4 repeats the same exercise with the BGG model used as the DGM and the GK model as the AM (BGG-DGM/GK-AM). We observed more changes in parameter estimates across periods in this case. Within the preference/technology parameters, we found notably larger changes in the consumption habit parameter (from 0.76 to 0.92 ), the two nominal rigidity parameters (from 0.83 to 0.90 for Calvo prices; from 0.63 to 0.82 for Calvo wages), and the two indexation parameters (from 0.33 to 0.27 for price indexation; from 0.13 to 0.16 for wage indexation), compared with the case of GK-DGM/BGG-AM. We also

ences in a few moments related to labor and wage rate. However, the differences seem quantitatively minor, and therefore it is difficult to tell which single model provides a more “plausible” model dynamics in the sense of Faust (2012). From the policymaker's viewpoint, it is difficult to infer with certainty which model is more suited as a policy-guiding tool by simply studying the simulated model dynamics. 3. Results We first report the estimated model parameters in the misspecified AMs and examine the classic parameter invariance problem a la Lucas (1976). Next, we examine whether misspecification in the credit market friction leads to poor performance of the monetary policy suggested by the AM. 3.1. Preliminaries: parameter invariance problem Table 3 presents the parameter estimates when the GK model is used as the DGM and the BGG model as the AM (GK-DGM/BGGAM). Most standard errors are driven close to zero due to the large sample period. This ensures that the bias is caused by model misspecification rather than by the limited data that prevent the estimates from converging to the pseudo-true values.28 We find that most preference/technology parameters remain close to the initial pseudo-true values across different phases of the crisis. The largest change occurs for the two Calvo parameters (Calvo prices, from 0.92 to 0.88) and Calvo wages (from 0.73 to 0.70 ). However, in both cases

29 It should be noted that there is no universally accepted criterion on judging whether a change in parameter is large or small in an economic sense. In this paper we regard the change as economically significant if it is (1) larger than 10% in magnitude and/or (2) exceeds the 95% confidence band implied by our prior standard deviation, which is adopted from the posterior estimates of Justiniano et al. (2010). 30 For more discussions on the interpretation of the marginal efficiency of investment shock, see Christiano et al. (2014) and Hirose and Kurozumi (2012).

28 For the most part, the initial choice of T = 40, 000 quarters was sufficient for the estimates to converge to pseudo-true values. The only exception was detected for the post-crisis estimates of the BGG-DGM/GK-AM case, in which we increased the sample length to T = 120, 000 .

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Table 2 Non-estimated parameters in the BGG/GK approximating models. (a) Parameters common to both models Discount factor β Capital share α SS Price Markup Λpss

0.9904 0.17 0.23

SS Wage Markup Λwss MA parameter of price markup shock θp

0.15

MA parameter of wage markup shock θw SS Work Hours (in log) logLss

0.91 0.38 0.21

0.77

Government spending share of output G /X Depreciation rate δ Policymaker Preference: Output Growth ρΔX

0.025 0.24

Policymaker Preference: Interest rate smoothing ρR

0.82

Policy weight: Output λX Policy weight: Interest rate λR

0.048 0.236

(b) GK model related parameters Fraction of asset that can be diverted νGK Pre-crisis Crisis

0.3503 0.5130

Proportional transfer to the entering borrowers ωGK Pre-crisis Crisis Survival probability of lender γGK

0.0013 0.0010 0.9724

SS government share of financial intermediation ψg

0.07

Efficiency cost of public financial intermediation τGK Taylor rule (initial): Inflation ϕπ

0.001 4.51

Taylor rule (initial): Output ϕX

0.09

Taylor rule (initial): Risk Spread ϕspr

0

(c) BGG model related parameters Uncertainty about borrowers investment project σBGG Pre-crisis Crisis

0.093 0.129

Bankruptcy cost μBGG Pre-crisis Crisis Survival probability of lender γBGG

0.039 0.043 0.9724

Borrower spending share of output C e /Y Taylor rule (initial): Inflation ϕπ

0.01

Taylor rule (initial): Output ϕX

−0.03

Taylor rule (initial): Risk Spread ϕspr

0

3.11

Note: All parameters are in quarterly frequency.

observed larger changes within the shock-related parameters.31 There is no readily available explanation for many of these changes other than the use of our maximum-likelihood-based estimation method, which, through the statistical mechanism described in the equation (10), attempts to find the set of parameter estimates that minimizes the misfit of the approximating model to the generated data. Finally, we examined the change in the size of the risk premium shock, σrp . Recall that σrp is changed from 0.2 to 1 in the crisis period and back to 0.2 in the post-crisis period. We anticipated the AMs would capture these changes with some asymptotic biases. In Tables 3 and 4, we observe that the parameter estimates changed as expected under both settings. While in the GK-DGM/BGG-AM case, the size of the risk premium shock changed within the range of 0.52 − 1.14 (Table 3), the

range narrowed down to 0.48 − 0.80 (Table 4) in the BGG-DGM/GKAM case. While both AMs failed to precisely capture the rise and fall of the risk premium shock, the BGG model did slightly better than the GK model. So far, the results have suggested that the BGG model provides the policymaker with a reasonably accurate picture of what happened during and after the crisis, while successfully keeping many of the parameters policy-invariant. According to Lucas’ (1976) criterion, the BGG model would be preferred to the GK model in evaluating the consequences of alternative policies. Next, we will examine the performance of the policy prescribed based on the two models. 3.2. Stabilization of the model economy Table 5 presents the realized welfare loss in the (unknown) GK model economy when the monetary policy rule suggested by the misspecified BGG model (“BGG-optimized policy rule”) was adopted. In the crisis period, we confirmed that the welfare loss rises by 4.5% compared to the pre-crisis period (from 1.5009 to 1.5681). In the

31

For the persistence of shocks, larger changes are found for risk premium shock (from 0.49 to 0.74 ), monetary policy shock (from 0.00 to 0.13), preference shock (from 0.66 to 0.79 ), marginal efficiency of investment shock (from 0.83 to 0.91), and technology shock (from 0.96 to 0.99 ). For the size of shocks, larger changes are found for price markup shock (from 0.14 to 0.11) as well as for preference shock (from 0.05 to 0.02 ).

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Table 3 (continued)

Table 3 Pseudo-true values estimated under misspecified BGG model (GK-DGM/BGG-AM).

Subsample Subsample (1) Precrisis

(2) Crisis

(3) Policyshift

(4) Postcrisis Size: MEI shock σμ

Preference / technology parameters Consumption habit h 0.76 (0.00)

0.77 (0.00)

0.78 (0.00)

0.78 (0.00)

Inverse Frisch elasticity φ

3.79 (0.00)

3.79 (0.17)

3.79 (0.00)

3.79 (0.15)

Elast.of capital utilization cost χ

5.30

5.30

5.30

5.30

(0.10)

(0.11)

(0.10)

(0.09)

2.85

2.85

2.85

2.85

(0.05)

(0.02)

(0.01)

(0.01)

0.92

0.88

0.88

0.91

(0.00)

(0.00)

(0.00)

(0.00)

0.23

0.24

0.24

0.23

(0.01)

(0.01)

(0.00)

(0.00)

Calvo wages ξw

0.73 (0.00)

0.70 (0.00)

0.70 (0.00)

0.72 (0.00)

Wage indexation ιw

0.11 (0.01)

0.11 (0.01)

0.11 (0.01)

0.11 (0.01)

0.16

0.15

0.15

0.16

(0.00)

(0.00)

(0.00)

(0.01)

0.95

0.96

0.95

(0.00)

(0.00)

(0.00)

0.94 (0.00)

0.95

0.95

0.95

0.95

(0.00)

(0.00)

(0.00)

(0.00)

0.75

0.92

0.85

0.75

(0.01)

(0.00)

(0.00)

(0.00)

0.95

0.96

0.96

0.93

(0.00)

(0.00)

(0.00)

(0.00)

0.99

0.97

0.96

0.95

(0.00)

(0.00)

(0.00)

(0.00)

0.67

0.63

0.63

0.67

(0.00)

(0.01)

(0.01)

(0.01)

0.69

0.70

0.74

0.74

(0.00)

(0.00)

(0.00)

(0.00)

0.22

0.22

0.22

0.22

(0.00)

(0.00)

(0.00)

(0.00)

Size: Technology shock σA

0.88 (0.00)

0.88 (0.00)

0.88 (0.00)

0.88 (0.00)

Size: Government spending shock σG

0.35

0.35

0.35

0.35

(0.00)

(0.00)

(0.00)

(0.00)

Investment adjustment cost S″

Calvo prices ξp

Price indexation ιp

Shock-related parameters Persistence: mon. pol. shock ρmp

Persistence: tech. shock ρA

Persistence: gov. shock ρG

Persistence: MEI shock ρμ

Persistence: Price markup shock ρp

Persistence: Wage markup shock ρw

Persistence: Preference shock ρb

Persistence: Risk premium shock ρrp

Size: Monetary policy shock σm

Size: Price markup shock σp

(1) Precrisis

(2) Crisis

(3) Policyshift

(4) Postcrisis

6.03

6.04

6.03

6.03

(0.10)

(0.02)

(0.02)

(0.02)

0.12

0.13

0.13

0.13

(0.00)

(0.00)

(0.00)

(0.00)

0.20

0.20

0.20

0.22

(0.00)

(0.00)

(0.00)

(0.00)

0.04 (0.00)

0.05 (0.00)

0.07 (0.00)

0.04 (0.00)

0.52

1.14

1.16

0.55

(0.00)

(0.01)

(0.00)

(0.00)

40,000

40,000

40,000

40,000

*

Size: Wage markup shock σw

*

Size: Preference shock σb

*

Size: Risk premium shock σrp

Sample T

Note: The estimates in each column are obtained from the data generated by the GK model (GK-DGM) and estimated using the BGG model (BGG-AM). Numbers in parenthesis are the asymptotic standard errors. MEI shock refers to the marginal efficiency of investment shock.

policy shift period, the welfare loss further rises by 15.3% compared to the crisis period (from 1.5681 to 1.8080). This means that using a misspecified model leads to destabilization of the economy. In addition, the magnitude of the destabilization caused by the policy is larger than that caused by the crisis itself. We further found that the overall rise in the welfare loss was mainly driven by the increase in the standard deviation of output (from 5.45 to 5.87), and the fall in the standard deviation of the interest rate (from 0.57 to 0.52) did not contribute much to mitigating the increase in the welfare loss. In addition, the volatility of consumption and labor relative to output was rising, while that of investment and wage rate was falling. The economic destabilization may have been caused by two separate factors: the use of a model that badly approximated the reality in calculating the new policy rule and the use of asymptotically biased model parameters. To better understand which factor was more responsible, we repeated the same exercise with the misspecified BGG model, but this time assuming that the policymaker knew the true parameter values in all periods. Results are shown in Appendix Table A.6. We found that the welfare loss during the policy shift period became even higher compared with the previous case with biased parameters (2.1015, as opposed to 1.8080), suggesting that the destabilization was mainly due to the misspecified model rather than the changes in model parameters. Table 6 presents the realized welfare loss in the (unknown) BGG model when the “GK-optimized” policy rule is adopted. Just as in the previous case, the misspecified model suggested a policy that further increased the welfare loss by 25.1% during the policy shift period (from 1.9114 to 2.3902). Here, the interest rate also became more volatile after the policy shift, making the policy outcome slightly worse than in the previous GK-DGM/BGG-AM case. The relative volatilities also rose for consumption, investment, and wage rate. Appendix Table A.7 shows the results when the GK-optimized policy rule is adopted and the policymaker knows the true parameter values. Again, the welfare loss during the policy shift period would have been higher compared with the biased parameter case (2.4295, as opposed to 2.3902). Thus, we conclude that it was the use of misspecified models that caused the welfare loss to increase during the policy shift period, just as in the GKDGM/BGG-AM case.

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Table 4 (continued)

Table 4 Pseudo-true values estimated under misspecified GK Model (BGG-DGM/GK-AM).

Subsample Subsample (1) Precrisis

(2) Crisis

(3) Policyshift

(4) Postcrisis Size: MEI shock σμ

Preference / technology parameters Consumption habit h 0.76 (0.00)

0.78 (0.00)

0.92 (0.00)

0.87 (0.00)

Size: Price markup shock σp

(1) Precrisis

(2) Crisis

(3) Policyshift

(4) Postcrisis

6.03

6.01

6.03

6.03

(0.05)

(0.06)

(0.09)

(0.03)

0.14

0.13

0.12

0.11

(0.00)

(0.00)

(0.00)

(0.00)

0.20

0.21

0.20

0.21

(0.00)

(0.00)

(0.00)

(0.00)

0.05 (0.00)

0.03 (0.00)

0.02 (0.00)

0.03 (0.00)

0.66

0.80

0.74

0.48

(0.00)

(0.00)

(0.02)

(0.01)

40, 000

40, 000

40, 000

120, 000

*

Inverse Frisch elasticity φ

3.79 (0.13)

3.79 (0.00)

3.79 (0.16)

3.79 (0.11)

Elast.of capital utilization cost χ

5.30

5.30

5.30

5.30

(0.00)

(0.00)

(0.00)

(0.08)

2.85

2.87

2.82

2.84

(0.03)

(0.04)

(0.07)

(0.01)

0.83

0.85

0.88

0.90

(0.00)

(0.00)

(0.00)

(0.00)

0.32

0.33

0.27

0.29

(0.00)

(0.01)

(0.01)

(0.00)

Calvo wages ξw

0.63 (0.00)

0.70 (0.00)

0.82 (0.00)

0.75 (0.00)

Wage indexation ιw

0.13 (0.01)

0.14 (0.01)

0.16 (0.01)

0.13 (0.01)

0.00

0.00

0.13

0.09

Subsample

(0.00)

(0.00)

(0.00)

(0.00)

(1) Precrisis

0.98

0.96

0.99

0.99

(0.00)

(0.00)

(0.00)

(0.00)

Investment adjustment cost S″

Calvo prices ξp

Price indexation ιp

Shock-related parameters Persistence: mon. pol. shock ρmp

Persistence: tech. shock ρA

Persistence: gov. shock ρG

0.95

0.95

0.95

0.95

(0.00)

(0.00)

(0.00)

(0.00)

0.83

0.91

0.89

0.85

(0.00)

(0.00)

(0.00)

(0.00)

0.90

0.91

0.91

0.88

(0.00)

(0.00)

(0.00)

(0.00)

0.96

0.95

0.94

0.95

(0.00)

(0.00)

(0.00)

(0.00)

0.66

0.72

0.79

0.73

(0.00)

(0.00)

(0.00)

(0.00)

0.69

0.49

0.49

0.74

(0.00)

(0.00)

(0.00)

(0.03)

0.22

0.22

0.22

0.22

(0.00)

(0.00)

(0.00)

(0.00)

Size: Technology shock σA

0.88 (0.00)

0.88 (0.00)

0.88 (0.00)

0.88 (0.01)

Size: Government spending shock σG

0.35

0.35

0.35

0.35

(0.00)

(0.00)

(0.00)

(0.00)

Persistence: MEI shock ρμ

Persistence: Price markup shock ρp

Persistence: Wage markup shock ρw

Persistence: Preference shock ρb

Persistence: Risk premium shock ρrp

Size: Monetary policy shock σm

Size: Wage markup shock σw

*

Size: Preference shock σb

*

Size: Risk premium shock σrp

Sample T

Note: The estimates in each column are obtained from the data generated by the BGG model (BGG-DGM) and estimated using the GK model (GK-AM). Numbers in parenthesis are the asymptotic standard errors. MEI shock refers to the marginal efficiency of investment shock. Table 5 Realized welfare loss under misspecified BGG model (GK-DGM / BGG-AM).

(2) Crisis

(3) Policyshift

(4) Postcrisis

Policy Parameters Inflation ϕπ

4.51

4.51

4.40

4.40

Output ϕX

0.09

0.09

−0.15

−0.15

Risk spread ϕspr

0.00

0.00

−0.27

−0.27

Standard Deviation Inflation σπ Output σX Interest rate σR Consumption σC /σX Investment σI /σX Labor σL /σX Wage rate σW /σX Welfare loss L 0 (relative to previous period)

0.25 5.36 0.49 0.74 4.30 0.85 0.58 1.5009 –

0.26 5.45 0.57 0.76 4.61 0.85 0.58 1.5681 (+4.5%)

0.30 5.87 0.52 0.86 3.83 0.92 0.55 1.8080 (+15.3%)

0.29 5.67 0.43 0.86 3.48 0.93 0.56 1.6699 (−7.6%)

Note: The policy parameters in each column are obtained from the data generated by the GK model (GK-DGM) and calculated using the BGG model (BGG-AM). Standard deviations and welfare losses are obtained from the GK data-generating model. Policy parameter on the risk spread ϕspr is restricted to be zero in the pre-crisis period and crisis period.

This experiment demonstrates that both models perform poorly when a policy is sought to successfully stabilize the economy in the wake of a financial crisis. Somewhat surprisingly, the BGG model fails to provide a useful policy recommendation despite the fact that it is relatively immune to the parameter invariance problem.

3.3. The opportunity cost of using a misspecified model Finally we ask how the policymaker would have performed if he had been equipped with the correctly specified model and were able to obtain the “DGM-optimized” policy rule. Although such an outcome is 10

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Table 6 Realized welfare loss under misspecified GK model (BGG-DGM / GK-AM).

Table 8 Policy comparison under BGG-DGM.

Subsample (1) Precrisis

(1) DGM-opt: hybrid pol. (2) Crisis

(3) Policyshift

(4) Postcrisis

Policy Parameters Inflation ϕπ

3.11

3.11

6.76

6.76

Output ϕX

−0.03

−0.03

−0.09

−0.09

Risk spread ϕspr

0.00

0.00

−2.93

−2.93

Standard Deviation Inflation σπ Output σX Interest rate σR Consumption σC /σX Investment σI /σX Labor σL /σX Wage rate σW /σX Welfare loss L 0 (relative to previous period)

0.27 5.53 0.47 0.83 4.34 0.86 0.61 1.5913 –

0.29 6.04 0.56 0.84 4.99 0.81 0.61 1.9114 (+20.1%)

0.42 6.64 0.64 1.01 6.39 0.71 0.67 2.3902 (+25.1%)

0.34 6.00 0.59 0.91 5.44 0.78 0.65 1.9260 (−19.4%)

unattainable in reality, it provides a benchmark in evaluating the “AMoptimized” policy rule we have seen in the previous subsection. One can also use this benchmark to ask by how much the use of public financial intermediation would have helped to combat the financial crisis, relative to the policy suggested by the misspecified model. While the first question is relevant to both AMs, the second question is only relevant to the GK-DGM/BGG-AM case. This is because, as we explained earlier, public financial intermediation would not play an active role in improving the resource allocation in the BGG model, hence it would not be considered as a policy option. Table 7 compares the policy outcomes under GK-DGM: (1) DGMoptimized dual policy, (2) DGM-optimized hybrid policy, and (3) AM-

Table 7 Policy comparison under GK-DGM. (3) AM-opt: Hybrid pol.

Policy Parameters Inflation ϕπ

10.81

4.38

4.40

Output ϕX

0.01

−0.05

−0.15

Risk spread ϕspr

−1.33

−2.18

−0.27

Public fin.intermed.ν BGGg

− 3.33

0.00

0.00

St.Dev. (Policy shift period) Inflation σπ Output σX Interest rate σR

0.29 4.37 0.69

0.37 4.61 0.62

0.30 5.87 0.52

Welfare Losses Policy shift period (relative to AM-opt) Post-crisis period (relative to AM-opt)

1.1127 (−38.5%) 1.4106 (−15.5%)

1.2471 (−31.0%) 1.1646 (−30.3%)

1.8080 (0.0%) 1.6699 (0.0%)

3.66

6.76

Output ϕX

−0.05

−0.09

Risk spread ϕspr

0.31

−2.93

St.Dev. (Policy shift period) Inflation σπ Output σX Interest rate σR

0.32 5.96 0.59

0.42 6.64 0.64

Welfare Losses Policy shift period (relative to AM-opt) Post-crisis period (relative to AM-opt)

1.8891 (−21.0%) 1.8482 (−4.0%)

2.3902 (0.0%) 1.9260 (0.0%)

optimized hybrid policy when the BGG model is used as the AM. During the policy shift period, the welfare loss could have been further lowered by 38.5% if the policymaker had chosen the DGM-optimized dual policy (1.1127, first scenario) instead of the misspecified AMoptimized policy (1.8080, third scenario). Much of the reduction in the welfare loss would come from the lower standard deviation of output (from 5.87 to 4.37), while inflation would also contribute marginally (from 0.30 to 0.29). If the policymaker had chosen the DGM-optimized hybrid policy (second scenario) instead, he would still have been able to lower the welfare loss by 31.0% compared to the AM-optimized policy (from 1.8080 to 1.2471), although the magnitude falls short compared with the dual policy case. In the post-crisis period, both DGMoptimized dual and hybrid policies still performed notably better than the AM-optimized policy (by 15.5% and 30.3%, respectively), suggesting that the opportunity cost of model misspecification remained high even after the credit market friction returned to its normal state. We also note that the hybrid policy performed better than the dual policy in the post-crisis period, contrary to what we have seen in the policy shift period. Table 8 compares policy outcomes under BGG-DGM: (1) DGMoptimized hybrid policy, and (2) AM-optimized hybrid policy when the GK model is used as the AM. In the policy shift period, the DGMoptimized hybrid policy reduces the welfare loss by 21.0% compared with the AM-optimized hybrid policy (from 2.3902 to 1.8891). In the post-crisis period, the DGM-optimized hybrid policy reduces the welfare loss by only 4% compared with the AM-optimized hybrid policy (from 1.9260 to 1.8482). While this is still an improvement, the magnitude is much more modest compared with the earlier case of GK-DGM. In sum, these experiments demonstrate that if the policymaker were equipped with the correctly specified model, there would have been a substantial reduction in welfare loss during the policy shift period. For the GK-DGM/BGG-AM case, we find that the use of public financial intermediation (dual policy) would have allowed the policymaker to achieve an even better outcome, although such effect is limited to the policy shift period. In the post-crisis period, the welfare loss reduction from using a correctly specified model is still seen for both models, but the magnitude of the positive effect is small under the BGG-DGM. This further suggests that the opportunity cost of model misspecification is small for the GK AM in the long run.

period.

(2) DGM-opt: Hybrid pol.

Policy Parameters Inflation ϕπ

Note: DGM-opt refers to the monetary policy rule optimized based on the BGG model that was also used to generated data (BGG-DGM) and using the true model parameters. AM-opt refers to the monetary policy optimized based on the misspecified GK model, while using the estimated pseudo true values for the model parameters. Standard deviations and welfare losses are obtained from the BGG data-generating model.

Note: The policy parameters in each column are obtained from the data generated by the BGG model (BGG-DGM) and calculated using the GK model (GK-AM). Standard deviations and welfare losses are obtained from the BGG data-generating model. Policy parameter on the risk spread ϕspr is restricted to be zero in the pre-crisis period and crisis

(1) DGM-opt: Dual pol.

(2) AM-opt: hybrid pol.

Note: DGM-opt refers to the monetary policy rule optimized based on the GK model that was also used to generated data (GK-DGM) and using the true model parameters. AMopt refers to the monetary policy optimized based on the misspecified BGG model, while using the estimated pseudo true values for the model parameters. Standard deviations and welfare losses are obtained from the GK data-generating model. In columns (2) and (3), policy parameter on the public financial intermediation νg is restricted to be zero.

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4. Can policymakers be “Nudged” toward a better policy?

Table 9 Expanded taylor rule under misspecified BGG/GK models.

So far, we have focused on the policy outcome under a relatively familiar class of monetary policy rules that minimize the quadratic welfare loss of the form suggested by Woodford (2003a). In this section, we consider two policy options that are feasible in both AMs and examine whether they can be used to improve policy outcomes. 4.1. Expanding the Taylor rule The first experiment is to allow the policymaker to adjust the interest rate in response to more variables. A few studies have examined whether there is any benefit to including different asset prices in the monetary policy rule.32 In addition, the existence of labor market frictions in both models motivates us to include the wage rate in the monetary policy rule. Specifically we consider the following “expanded” Taylor rule

⎡ ⎤1− ρR ⎛ X ⎞ ρΔX ρR ⎢(Πt )ϕπ (Xt )ϕX (spr )ϕspr ⎜ t ⎟ (AddVart )ϕadd ⎥ Rt = Rt −1 exp(Smp, t ), t ⎢⎣ ⎥⎦ ⎝ Xt −1 ⎠

(13)

where AddVart is either Tobin's q (= asset price) or the wage rate, and ϕadd is the associated reaction coefficient that the policymaker can freely choose. Appendix Table A.8 shows the hypothetical welfare loss Ladd during the policy shift period when the expanded Taylor Rule (13) and the correctly specified AM are used. We confirm that each added variable contributes to further reduction in the welfare loss relative to the crisis period as well as the policy shift period when the original hybrid policy rule (5) is adopted. Table 9 shows the welfare loss using the expanded Taylor rule and the misspecified AM. For the GK-DGM/BGG-AM case, adding Tobin's q results in 68.4% larger welfare loss relative to the crisis period. Similarly, for the BGG-DGM/GK-AM case, adding Tobin's q results in 87.1% larger welfare loss relative to the crisis period. These are clearly worse outcomes than what would have been achieved under the original hybrid policy with no added variable. When the wage rate is added instead, we observe mixed results. For the GK-DGM/BGG-AM case, adding the wage rate marginally lowers the welfare loss (−0.4%) relative to the crisis period. However, for the BGG-DGM/GK-AM case, adding the wage rate results in 45.7% larger welfare loss relative to the crisis period, which is also worse compared to the outcome under the original hybrid policy. Overall, we conclude that including additional variables in the Taylor rule does not help much in improving policy outcomes under a misspecified model setting.

(1) Add Tobin's q

(2) Add Wage Rate

GK-DGM/BGG-AM Policy: Inflation ϕπ

6.71

7.51

Policy: Output ϕX

−0.14

0.01

Policy: Risk spread ϕspr

1.04

−0.26

Policy: Added variable ϕadd

0.16

−0.89

Welfare loss Ladd (relative to L 0 : crisis period) (relative to L 0 : pol.shift period)

2.6412 (+68.4%) (+46.1%)

1.5624 (−0.4%) (−13.6%)

BGG-DGM/GK-AM Policy: Inflation ϕπ

2.46

8.40

Policy: Output ϕX

−0.13

−0.03

Policy: Risk spread ϕspr

−2.32

−4.27

Policy: Added variable ϕadd

−0.14

−0.55

Welfare loss Ladd (relative to L 0 : crisis period) (relative to L 0 : pol.shift period)

3.5760 (+87.1%) (+49.6%)

2.7858 (+45.7%) (+16.5%)

Note: The policy parameters are obtained from the misspecified approximating model. Realized welfare loss L 0 uses the numbers reported in Tables 5 and 6.

type of adjustment in the stabilization objectives would be useful to prevent the destabilization of the economy caused by model misspecification. We assume that the monetary policymaker chooses policy parameters based on an “expanded” quadratic welfare loss measure, where the vector Γ in equation (7) includes a fourth variable, i.e.,

lt , X lt , R lt , n Γadd , t = [Π AddVart ]′, and the diagonal matrix Ψ has an additional parameter λadd , which represents the weight on the variance of the added variable. We then calculate the optimized hybrid policy parameters and the associated welfare loss Ladd for a range of positive values that λadd can take.35 For candidate variables, we consider ten endogenous variables (consumption, investment, capital, labor, wage rate, risk spread, asset price, leverage, rental rate of capital, and net worth), which are commonly observable in both models. Table 10 shows the welfare loss observed during the policy shift period when the GK model is used as the DGM and the BGG model is used as the AM. Here we report only cases in which the welfare loss in the policy shift period is lower compared to the original hybrid policy. When labor is added, the newly suggested policy is successful in reducing the welfare loss by 20.0% relative to the crisis period. In contrast, when risk spread is added, the welfare loss still increases relative to the crisis period (+15.3%). Furthermore the outcome is only marginally better (%−0.002 ) relative to the baseline case with lt , X lt , R lt ]′. Γt = [Π Table 11 shows the welfare loss when the BGG model is the DGM and the GK model is the AM. We report four cases that show the largest lt , X lt , R lt ]′. As reduction in losses relative to the baseline case with Γt = [Π the table shows, the realized losses are all larger than the crisis period. The best outcome is achieved when net worth is added, although this still leads to 2.8% higher welfare loss compared to the crisis period. In sum, the results generally show that reassigning the stabilization objective is mostly effective in mitigating further destabilization of the economy, rather than stabilizing the economy. This finding seems to be in conformity with Badarau and Popescu (2014), who argue that reassigning the stability objective of the central bank would have done

4.2. Assigning alternative stabilization objectives Another possible remedy is to reassign alternative stabilization objectives in the quadratic loss function. In the monetary policy literature, researchers have long discussed whether alternative welfare functions can be used to improve policy outcome. A classic example is provided by Rogoff (1985), who shows analytically that a “conservative” central banker with a strong preference towards inflation stabilization beyond the socially desired level may achieve a better outcome.33 Alternatively, researchers have argued that when the model features more structures and frictions, additional components appearing in the quadratic loss function may be justified from a welfaretheoretic ground.34 Our purpose here is to examine whether the same 32 For example, Bernanke and Gertler (2001) use Tobin's q as the proxy for asset price in the BGG model. 33 In Rogoff's case, the adjustments in the preference are motivated by the unobservable shock process that affects the time-consistent level of inflation and the trade-off between inflation and employment stabilization. Also see Canzoneri et al. (1997) and Woodford (2003a) for related work. 34 See, for example, De Paoli (2009), and Erceg et al. (2000).

35 We conducted a grid search of ten simulations per run, starting with a relatively large step size. After observing the result, we readjusted the step size so that we could eventually pin down the value of λadd that results in the minimized welfare loss criterion.

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case, meaning that the method of overcoming the model misspecification was not same for different models.

Table 10 Expanded welfare loss criterion under misspecified BGG model (GK-DGM / BGG-AM). (1) Add labor

(2) Add risk spread

Policy Parameters Relative Weight λAddVar Inflation ϕπ

0.50 5.05

0.002 4.40

Output ϕX

−0.01

−0.15

Risk spread ϕspr

−2.32

−0.27

Standard Deviation Inflation σπ Output σX Interest rate σR

0.36 5.87 0.65

0.30 4.62 0.52

Welfare loss Ladd (relative to L 0 : crisis period) (relative to L 0 : pol.shift period)

1.2544 (−20.0%) (−30.6%)

1.8079 (+15.3%) (−0.002%)

5. Conclusion This paper examines the consequence of model misspecification on monetary policymaking when the type of credit market friction is misspecified. We choose the credit channel models of Bernanke et al. (1999) and Gertler and Karadi (2011), and examine the performance of each model when the credit market friction used in the data-generating process is from the other model. We find that the use of both models equally leads to destabilization of the model economy when credit channel misspecification is present. In addition, the opportunity cost of using a misspecified model is economically significant, particularly when public financial intermediation was considered as part of an alternative policy. We further show that in some cases adding wage rate into the monetary policy rule or adding labor as part of the stabilization objectives is effective in overcoming the problem of using a misspecified credit channel model as a policy-guiding tool. Our findings provide a new perspective on what the literature has long recognized as the parameter invariance problem associated with DSGE models. We show that the relatively robust performance of Bernanke et al.'s model in estimating model parameters does not guarantee its ability to suggest the right policy. Even worse, the policymaker may presume a misspecified model is the correct one based on the fact that model parameters remain stable over different phases of the financial crisis. There is also a risk that the policymaker may prematurely discard the Gertler and Karadi's model for not being able to produce policy-invariant parameters, when in fact its performance as a policy-guiding tool is not much different from the Bernanke et al.'s model. One of the weaknesses of our experiment is that it does not explicitly consider the zero-lower-bound problem, which could significantly increase the cost of model misspecification because of the restrictions imposed on the conventional interest rate policy. Studies such as Williams (2009) report that the welfare cost of the zero lower bound was large during the recent financial crisis. Del Negro et al. (2017) further demonstrate that when the interest rate reaches the zero lower bound, a liquidity shock can have a large effect on the aggregate economy. An interesting venue to explore in future studies would be to add more structure to both the data-generating and the approximating models so that more policy options can be considered. For example, if we assume that extending credit involves additional resource cost, government subsidies could become an option to correct for the undersupply of loans (Christiano and Ikeda, 2013). Another possible extension would be the introduction of new financial shocks other than what we considered in this paper. While the discussion regarding what types of financial shock best explain the 2008-09 financial crisis is far from reaching consensus,36 it would surely be interesting to explore whether the presence of different shock(s) would alter how monetary policymakers view the cost of credit channel misspecification.

Note: The policy parameters are obtained from the misspecified BGG model. Realized welfare loss L 0 uses the numbers reported in Tables 5 and 6. Table 11 Expanded welfare loss criterion under misspecified GK Model (BGG-DGM / GK-AM). (1) Add net worth

(2) Add leve-rage

(3) Add Tobin's Q

(4) Add Rental Rate of Capital

Policy Parameters Relative Weight λAddVar Inflation ϕπ

0.008

0.06

0.04

0.15

2.61

2.58

2.56

2.46

Output ϕX

0.02

0.09

0.03

0.01

Risk spread ϕspr

0.40

1.04

−0.28

−0.56

Standard Deviation Inflation σπ Output σX Interest rate σR

0.36 6.06 0.57

0.49 6.22 0.62

0.32 6.48 0.55

0.33 6.51 0.53

1.9654 (+2.8%)

2.1849 (+14.3%)

2.1866 (+14.4%)

2.2136 (+15.8%)

(−17.8%)

(−8.6%)

(−8.5%)

(−7.4%)

Welfare loss Ladd (relative to L 0 : crisis period) (relative to L 0 : policy shift period)

Note: The policy parameters are obtained from the misspecified GK model. Realized welfare loss L 0 uses the numbers reported in Tables 5 and 6.

little in improving policy outcomes during the financial crisis. The only exception was seen when labor is added in the GK-DGM/BGG-AM case. Introducing labor into the welfare loss measure means that the policymaker places more weight on the real variables relative to nominal ones. In the context of our experiment however, adding labor effectively adjusted for the differences in the financial accelerator effects that were present in both models. It should also be noted that adding labor was not found to be effective for the BGG-DGM/GK-AM

Appendix A. Full model description We start with describing the part that is common to both credit channel models (Section A.1). The common part largely follows the specification in Justiniano et al. (2010) and Gali et al. (2012). Details about the Bernanke et al. (1999, BGG)'s model is shown in Section A.2 and details of the Gertler and Karadi (2011, GK)'s model is shown in Section A.3 (Table A.1–A.8 ).

36 See, for example, Buera and Moll (2015); Christiano et al. (2014); Goodfriend and McCallum (2007); Jermann and Quadrini (2012), and Khan and Thomas (2013).

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Table A.1 Policy comparison under correctly specified GK model. (1) Standard policy

(2) Hybrid policy

(3) Dual policy

Policy Parameters Inflation ϕπ

5.31

4.38

10.81

Output ϕX

0.12

−0.05

0.01

Risk spread ϕspr

0.00

−2.18

−1.33

Public financial intermediation νg

0.00

0.00

− 3.33

Standard Deviation Inflation σΠ Output σX Interest rate σR

0.26 5.41 0.60

0.37 4.61 0.62

0.29 4.37 0.69

Welfare loss L 0 (relative to standard policy)

1.5598 –

1.2471 (−20.1%)

1.1127 (−28.7%)

Table A.2 Policy comparison under correctly specified BGG model. (1) Standard policy

(2) Hybrid policy

Policy Parameters Inflation ϕπ

3.67

3.66

Output ϕX

−0.06

−0.05

Risk spread ϕspr

0.00

0.31

Standard Deviation Inflation σΠ Output σX Interest rate σR

0.29 6.00 0.58

0.32 5.96 0.59

Welfare loss L 0 (relative to standard policy)

1.8954 –

1.8891 (−0.3%)

Note: In above Tables A.1 and A.2, standard policy refers to the Taylor rule in which the response to the risk spread is restricted to be zero (ϕspr = 0 ) and hybrid policy refers to the Taylor rule with the possible response to the risk spread. In Table A.1 dual policy refers to the hybrid policy combined with public financial intermediation, which is available in the GK model. Model parameters excluding credit channel parameters are set to the benchmark values, while credit channel parameters are set to the crisis period values.

Table A.3 Share of variance explained by risk premium shock under correctly specified GK/BGG models. (1) Pre-crisis period

(2) Crisis period

(3) Change = (2) – (1)

GK Model Inflation Π Output X Interest rate R Risk spread spr

0.2% 0.4% 1.3% 3.3%

3.9% 7.0% 18.7% 43.7%

+3.7% +6.6% +17.4% +40.4%

BGG Model Inflation Π Output X Interest rate R Risk spread spr

0.1% 0.3% 0.7% 6.7%

4.0% 8.4% 17.3% 65.7%

+3.8% +8.0% +16.6% +59.0%

Note: The result is based on the 10-period ahead forecast error variance decomposition. Policy parameters are set to Justiniano et al.'s (2010) estimates, which are ϕpi = 2.09, ϕX = 0.07, ϕspr = 0 ).

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Table A.4 Standard deviations of the generated variables under correctly specified GK/BGG models.

GK model St.dev: Inflation σΠ St.dev: Output σX St.dev: Interest rate σR Rel.st.dev: Consumption σC /σX Rel.st.dev: Investment σI /σX Rel.st.dev: Labor σL /σX Rel.st.dev: Wage rate σW /σX BGG model St.dev: Inflation σΠ St.dev: Output σX St.dev: Interest rate σR Rel.st.dev: Consumption σC /σX Rel.st.dev: Investment σI /σX Rel.st.dev: Labor σL /σX Rel.st.dev: Wage rate σW /σX

(1) Pre-crisis period

(2) Crisis period

(3) Change = (2) – (1)

0.25 5.36 0.49 0.74

0.26 5.45 0.57 0.76

+0.01 +0.09 +0.07 +0.02

4.30

4.61

+0.31

0.85 0.58

0.85 0.58

+0.00 −0.01

0.27 5.53 0.47 0.83

0.29 6.04 0.56 0.84

+0.02 +0.51 +0.09 +0.01

4.34

4.99

+0.65

0.86 0.61

0.81 0.61

−0.05 +0.00

Table A.5 Correlations of the generated variables under correctly specified GK/BGG models. (1) Pre-crisis period

(2) Crisis period

(3) Change = (2) – (1)

GK model Cross corr: Consumption corr (C , X ) Cross corr: Investment corr (I , X ) Cross corr: Labor corr (L , X ) Cross corr: Wage rate corr (W , X ) Autocorr: Output corr (Xt , Xt−1) Autocorr: Consumption corr (Ct , Ct−1) Autocorr: Investment corr (It , It−1) Autocorr: Labor corr (Lt , Lt−1) Autocorr: Wage rate corr (Wt , Wt−1)

0.73 0.80 0.71 0.52 0.9800 0.9861 0.9691 0.9446 0.9868

0.67 0.78 0.71 0.52 0.9800 0.9861 0.9707 0.9451 0.9869

−0.06 −0.02 +0.00 +0.00 +0.0000 −0.0000 +0.0016 +0.0005 +0.0001

BGG model Cross corr: Consumption corr (C , X ) Cross corr: Investment corr (I , X ) Cross corr: Labor corr (L , X ) Cross corr: Wage rate corr (W , X ) Autocorr: Output corr (Xt , Xt−1) Autocorr: Consumption corr (Ct , Ct−1) Autocorr: Investment corr (It , It−1) Autocorr: Labor corr (Lt , Lt−1) Autocorr: Wage rate corr (Wt , Wt−1)

0.71 0.72 0.73 0.59 0.9812 0.9872 0.9786 0.9478 0.9886

0.61 0.72 0.72 0.64 0.9838 0.9881 0.9836 0.9499 0.9904

−0.10 −0.00 −0.01 +0.06 +0.0026 +0.0009 +0.0050 +0.0021 +0.0018

A.1. Model common to both credit channel models A.1.1. Household Each household k ∈ [0, 1] maximize its lifetime utility, ∞

maxEt ∑

j =0

⎡ Lt + j (k )1+ φ ⎤ ⎥, β j exp(bt + j )⎢log(Ct + j − hCt + j −1) − ΞL ⎢⎣ 1 + φ ⎥⎦

where Ct is consumption, Lt is labor supply, β is the quarterly discount factor, h is the consumption habit parameter, φ is the inverse Frisch elasticity, and ΞL is the utility weight on leisure which pins down the steady-state labor in the model. Preference shock bt is modeled as37

bt = (bt −1) ρb exp(eb, t ),

⎛ (1 − ρb )(1 − hβρb )(1 − h) ⎞ ⎟(bt − b ). Note that in actual estimation, we follow the suggestion by Justiniano et al. (2010) and normalize the (linearized) preference shock as follows: blt = ⎜ ⎝ ⎠ 1 + h + h2 Similar normalization is applied to price mark-up shock and wage mark-up shock (defined later). 37

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Table A.6 Welfare loss under misspecified BGG model and correctly estimated parameter values. Subsample (1) Precrisis

(2) Crisis

(3) Policy shift

(4) Postcrisis

Policy Parameters Inflation ϕπ

4.51

4.51

3.66

3.66

Output ϕX

0.09

0.09

−0.05

−0.05

Risk spread ϕspr

0.00

0.00

0.31

0.31

Volatility Measures Inflation σπ Output σX Interest rate σR Consumption σC /σX Investment σI /σX Labor σL /σX Wage rate σW /σX Welfare loss L 0 (relative to previous period)

0.25 5.36 0.49 0.74 4.30 0.85 0.58 1.5009 –

0.26 5.45 0.57 0.76 4.61 0.85 0.58 1.5681 (+4.5%)

0.30 6.37 0.51 0.76 4.08 0.91 0.52 2.1015 (+34.0%)

0.29 6.16 0.42 0.76 3.85 0.91 0.52 1.9454 (−7.4%)

Note: The result is based on the scenario in which GK model is used as the data-generating model and the BGG model is used as the approximating model. Model parameter values used in calculating the optimized Taylor rule are same as in the data-generating model. Policy parameter on the risk spread ϕspr is restricted to be zero in the pre-crisis period and crisis period.

eb, t ∼ N (0, σb2 ). The budget constraint for the household is

PC t t + PI t t + Tt + Dt = Rt −1Dt −1 + proft + Wt (k )Lt (k ), where Pt is price level, It is investment, Tt is lump-sum taxes, Dt is deposit, Rt is the gross interest rate on deposits, proft is the profit from owing firms, and Wt is the wage rate. A.1.2. Goods producers There are two types of goods producing firms. Final good producers aggregate the intermediate goods into a composite final good as,

⎛ Yt = ⎜ ⎝

∫0

1

1 ⎞1+ λp, t Yt (z )1+ λp, t dz⎟ , ⎠

where Yt (z ) is the total production of intermediate goods of firm z ∈ [0, 1]. Price markup shock λp, t is modeled as

1 + λp, t = (1 + λp )1− ρp (1 + λp, t −1) ρp exp(ep, t − θpep, t −1),

ep, t ∼ N (0, σp2 ). Intermediate goods producers hire labor and capital through the factor markets. These inputs are used to produce intermediate output,

Yt (z ) = max[At Ktα (z )Lt1− α (z ) − At F ; 0], where α is the capital share of income and F is the fixed cost of production. Technology shock At is modeled as

At = (At −1) ρa exp(ea, t ), ea, t ∼ N (0, σa2 ).

(14)

The overall price is determined as a geometric average of the prices for adjusters and non-adjusters 1 ⎤ λp, t ⎡ ιp ∼ 1 Pt = ⎢(1 − ξp )(Pt ) λp, t + ξp(Π1− ιpΠt −1 Pt −1) λp, t ⎥ , ⎦ ⎣ ∼ where Pt is the price chosen by the adjusters and Πt is gross inflation observed over time t − 1 and t, ξp is the Calvo price parameter, ιp is the price indexation parameter. Price markup shock λp, t is modeled as38

1 + λp, t = (1 + λp )1− ρp (1 + λp, t −1) ρp exp(ep, t − θpep, t −1),

38

⎛ (1 − ξpβ )(1 − ξp) ⎞ Note that in actual estimation, we normalize the (linearized) price markup shock as follows: λlp . t = ⎜ ⎟(log(1 + λp, t ) − log(1 + λp ) . ⎝ ξp(1 + ιpβ ) ⎠

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Table A.7 Welfare loss under misspecified GK model and correctly estimated parameter values. Subsample (1) Precrisis

(2) Crisis

(3) Policy shift

(4) Postcrisis

Policy Parameters Inflation ϕπ

3.11

3.11

4.38

4.38

Output ϕX

−0.03

−0.03

−0.05

−0.05

Risk spread ϕspr

0.00

0.00

−2.18

−2.18

Volatility Measures Inflation σπ Output σX Interest rate σR Consumption σC /σX Investment σI /σX Labor σL /σX Wage rate σW /σX

0.27 5.53 0.47 0.83 4.34 0.86 0.61

0.29 6.04 0.56 0.84 4.99 0.81 0.61

0.48 6.65 0.57 1.03 6.47 0.70 0.66

0.38 5.85 0.51 0.93 5.35 0.79 0.65

1.5913 –

1.9114 (+20.1%)

2.4295 (+27.1%)

1.8482 (−23.9%)

Welfare loss L 0 (relative to previous period)

Note: The result is based on the scenario in which BGG model is used as the data-generating model and the GK model is used as the approximating model. Model parameter values used in calculating the optimized Taylor rule are same as in the data-generating model. Policy parameter on the risk spread ϕspr is restricted to be zero in the pre-crisis period and crisis period.

ep, t ∼ N (0, σp2 ). Price adjuster z maximize the following objective, ∞

maxEt ∑ ξpjβ j j =0

⎤ ⎛ j ⎞ Λt + j ⎡ ⎢P(z )⎜∏ Π1− ιpΠ ιp ⎟Y (z ) − W L (z ) − MPK K (z )⎥ , t t+j t+j t+j t+j t + l −1⎟ t + j ⎜ ⎥⎦ Λt ⎢⎣ ⎝ l =0 ⎠

where Λt is the marginal utility of nominal income for the representative household and MPKt is the marginal product of capital. Price non-adjuster z follows the indexation rule, ι

p Pt (z ) = Π1− ιpPt −1(z )Πt −1 .

A.1.3. Capital market Demand for capital is determined through the cost minimization problem of the intermediate goods firms,

⎛ L (z ) ⎞1− α MPKt = MCt (z )At1− α ⎜ t ⎟ , ⎝ Kt (z ) ⎠ where MCt (z ) is the marginal cost. Supply for capital is determined through solving the optimal choice of investment by the capital owner. The value of physical capital can be expressed as,

Φt = βEtΛt +1MPKnet , t + (1 − δ )βEtΦt +1, where δ is the depreciation rate of capital. MPKnet , t is the marginal product of capital net of capital utilization cost,

MPKnet , t = MPKtut − Pa t (ut ), where a(ut ) is the cost of capital utilization per unit of physical capital in real terms. Capital utilization rate ut determines the amount of effective capital available for production in period t,

Kt = ut Kt−1. Physical capital accumulates according to the following law of motions

⎛ ⎛ I ⎞⎞ Kt = (1 − δ )Kt −1 + μt ⎜⎜1 − S ⎜ t ⎟⎟⎟It , ⎝ It −1 ⎠⎠ ⎝ where S represents the adjustment cost of capital that satisfies S = S′ = 0 and S″ > 0 in the steady state. Marginal efficiency of investment shock μt is modeled as,

μt = (μt −1) ρμ exp(eμ, t ),

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Table A.8 Expanded taylor rule under correctly specified GK/BGG models. (1) Add Tobin's q

(2) Add wage rate

GK Model Policy: Inflation ϕπ

4.38

4.64

Policy: Output ϕX

−0.05

0.14

Policy: Risk spread ϕspr

−2.18

−1.78

Policy: Added variable ϕadd

−0.01

−0.50

Welfare loss Ladd (relative to L 0 : crisis period) (relative to L 0 : pol.shift period)

1.2466 (−20.5%)

1.1192 (−28.6%)

(−0.04%)

(−10.3%)

BGG Model Policy: Inflation ϕπ

8.64

4.75

Policy: Output ϕX

0.14

−0.03

Policy: Risk spread ϕspr

3.69

−0.03

Policy: Added variable ϕadd

0.65

−0.75

Welfare loss Ladd (relative to L 0 : crisis period) (relative to L 0 : pol.shift period)

1.7492 (−8.5%)

1.5150 (−20.7%)

(−7.4%)

(−19.8%)

Note: The policy parameters are obtained from the correctly specified approximating model. Realized welfare loss L 0 : crisis period uses the numbers reported in Tables 5, 6. Realized welfare loss L 0 : pol.shift period uses the numbers reported in Tables A.1, A.2.

eμ, t ∼ N (0, σμ2 ). Finally, gross rate of return on holding capital is,

⎡ MPKt +1 + (1 − δ )Qt +1 ⎤ EtRtk+1 = Et ⎢ ⎥exp(Srp, t ), Qt ⎣ ⎦ where Qt = Φt / PΛ t t is the Tobin's q. Risk premium shock Srp, t is modeled as,

Srp, t = (Srp, t −1) ρrp exp(erp, t ),

erp, t ∼ N (0, σrp2 ). A.1.4. Labor market Demand for labor is determined through the cost minimization problem of the intermediate goods firms,

⎛ L (z ) ⎞−α Wt = MCt (z )At1− α ⎜ t ⎟ , ⎝ Kt (z ) ⎠ where MCt (z ) is the marginal cost. Supply for labor is determined by the employment agency, who aggregates the household labor into a homogeneous labor input as,

⎛ Lt = ⎜ ⎝

∫0

1

1 ⎞1+ λ w, t Lt (k )1+ λ w, t dk ⎟ , ⎠

where the wage markup shock λ w, t is modeled as,39

1 + λ w, t = (1 + λ w )1− ρw (1 + λ w, t −1) ρw exp(ew, t − θwew, t −1),

ew, t ∼ N (0, σw2 ). The overall wage is determined as a geometric average of wages for adjusters and non-adjusters,

39

⎛ ⎞ ⎜ ⎟ (1 − ξwβ )(1 − ξw ) l ⎜ ⎟(log(1 + λ w, t ) − log(1 + λ w ) . Note that in actual estimation, we normalize the (linearized) wage markup shock as follows: λ w . t = ⎜⎜ ξ (1 + β )⎛⎜1 + φ(1 + 1 )⎞⎟ ⎟⎟ w λ ⎝ ⎝ w ⎠⎠

18

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T. Yagihashi 1 1 ⎤ λ w, t ⎡ w W ) λ w, t Wt = ⎢(1 − ξw )(W͠ t ) λ w, t + ξw(Π1− ιwΠtι−1 ⎥ , t −1 ⎦ ⎣

where W͠ t is the wage rate chosen by the adjusters, ξw is the Calvo wage parameter, and ιw is the wage indexation parameter. Wage adjuster k maximize the following objective,

⎡ ⎤ Lt + j (k )1+ φ ξwjβ j ⎢exp( − bt + j )φ + Λt + j Wt (k )Lt + j (k )⎥ , j =0 ⎢⎣ ⎥⎦ 1+φ

∞

maxEt ∑

whereas wage non-adjuster k follows the indexation rule, w . Wt (k ) = Π1− ιwWt −1(k )Πtι−1

A.1.5. Government In both models, government spending is modeled as a fraction of the final goods produced in the economy,

⎛ 1⎞ Gt = ⎜⎜1 − ⎟⎟Yt , gt ⎠ ⎝ where gt is the government spending shock modeled as,

gt = (gt −1) ρg exp(eg, t ), eg, t ∼ N (0, σg2 ) . In the BGG model, government spending is financed as,

Gt + Dg, t = Tt + Rt −1Dg, t −1, where Dg, t is the government deposit that yields the gross interest rate Rt and Tt is the lump-sum taxes. In the GK model, government spending is financed as,

Gt + Ct fi + Dg, t = Tt + Rt −1Dg, t −1 + Rtk Qt Kg, t −1, where Ct fi is the efficiency cost of public financial intermediation and Qt Kg, t −1 is the value of public financial intermediation. In both models, the interest rate is set according to the Taylor rule, 1− ρ ⎡ ⎛ X ⎞ ρΔX ⎤ R ρR ⎢(Πt )ϕπ (Xt )ϕX (spr )ϕspr ⎜ t ⎟ ⎥ Rt = Rt −1 exp(Smp, t ), t ⎢⎣ ⎝ Xt −1 ⎠ ⎥⎦

where sprt ≡ Et (Rtk+1)/ Rt . Monetary policy shock Smp, t is modeled as

Smp, t = (Smp, t −1) ρmp exp(emp, t ), 2 emp, t ∼ N (0, σmp ).

A.1.6. Goods market equilibrium In the BGG model, the goods market equilibrium satisfies the following resource constraint,

Yt = Ct + It + Gt + Ctb + a(ut ), where Ctb is the consumption of borrowers that exit the market. In the GK model, the goods market equilibrium satisfies,

Yt = Ct + It + Gt + Ct fi + a(ut ). In both models output (= GDP) is defined as,

Xt = Yt − a(ut ). A.2. BGG model A.2.1. Financial market in the BGG model In the BGG model, the borrower m's objective is to maximize the expected profit in the next period that can be generated through capital investment. When financing the investment, the borrower is allowed to obtain funds from the (representative) lender. Thus the borrower m's balance sheet at period t is expressed as

Qt Kt −1(m ) = Nt (m ) + Bt (m ), where Qt Kt−1(m ) is the asset value at the beginning of period t and Nt (m ) is the beginning of period net worth, and Bt (m ) is the funds acquired from the lender. The objective function for the borrower can be expressed as

Vt (m ) = maxΓ bEt[Rtk+1Qt +1Kt (m )], where Rtk+1Qt +1Kt (m ) is the overall profit from capital investment and Γ b is the fraction of profit that will be kept by the borrower after sharing the 19

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profit with the lender. The lender is willing to provide the necessary funds as long as the net revenue from lending covers the opportunity cost of investing the funds into a risk-free asset. The condition is satisfied when

Γ lEt[Rtk+1Qt +1Kt (m )] = Rt Bt , where Γ l is the fraction of profit kept by the lender. The BGG model assumes that the individual borrower faces an idiosyncratic productivity shock ωBGG that makes the profit per unit of capital 2 2 and variance of σBGG such that investment potentially differ across borrowers. The shock follows the log-normal distribution with a mean of −0.5σBGG E (ωBGG ) = 1. In order for the representative lender to gain access to the realized ωBGG he is forced to pay a bankruptcy cost μBGG that is proportional to the asset value. To guarantee that the lender is guarded against the shock, the contract pre-specifies a cutoff value for the productivity shock ωBGG such that if the realized shock falls short of the cutoff value, the borrower is going to seize the residual claims net of the bankruptcy cost. Solving the optimal contract problem and aggregating across all borrowers leads to an expression that contemporaneously relates the nationwide leverage ratio of the borrower to the premium on external funds

⎛ Q K ⎞νBGG EtRtk+1 = ⎜ t t ⎟ Rt , ⎝ Nt +1 ⎠ where νBGG is the positive credit market friction parameter, which is determined through the choice of μBGG , σBGG . A.2.2. Net worth dynamics in the BGG model Aggregate net worth of the borrower is defined as the sum of the net worth for the surviving borrowers and their labor income

⎡ ⎤ ⎛ ⎞ ACt −1 Nt +1 = γBGG ⎢Rtk Qt −1Kt −1 − ⎜Rt −1 + ⎟(Qt −1Kt −1 − Nt )⎥ ⎢⎣ Qt −1Kt −1 − Nt ⎠ ⎝ ⎦⎥ + Wb, tLb, where Wb, t is the borrower's wage rate, Lb is the (fixed) labor supply, and γBGG is the survival rate of the borrower. ACt−1 is the size of bankruptcy cost incurred in period t − 1, which is determined as

ACt −1 = μBGG

∫0

ωb, t

ωBGGRtk Qt −1Kt −1dωBGG .

Borrowers who fail to survive consume the following amount before exiting the market

⎡ ⎤ ⎛ ⎞ ACt −1 Ctb = (1 − γBGG )⎢Rtk Qt −1Kt −1 − ⎜Rt −1 + ⎟(Qt −1Kt −1 − Nt )⎥ . ⎢⎣ ⎥⎦ Q K − N ⎝ t −1 t −1 t⎠ A.3. GK model A.3.1. Financial market in the GK model In GK model, the borrower m's objective is to maximize the expected lifetime net worth, ∞ j Vt (m ) = maxEt ∑ (1 − γGK )γGK (β j +1Υt , t + j +1)Nt + j (m ), j =0

where γGK is the survival rate of the borrower, β j Υt , t + j is the stochastic discount factor that applies to period t + j earnings. Beginning of period net worth Nt+1 is defined as the sum of excess return from capital investment and the (return-adjusted) net worth from the previous period,

Nt +1(m ) = (Rtk+1 − Rt )Qt Kt −1(m ) + Rt Nt (m ). The above objective function can be rewritten in recursive form as

Vt (m ) = vQ t t Kt −1(m ) + ηt Nt (m ), where

⎡ ⎤ Q K (m ) vt = Et ⎢(1 − γGK )(β Υt , t +1)(Rtk+1 − Rt ) + γGK (β Υt , t +1) t +1 t vt +1⎥ , Qt Kt −1(m ) ⎣ ⎦

⎡ ⎤ N (m ) ηt = Et ⎢(1 − γGK ) + γGK (β Υt , t +1) t +1 η ⎥. Nt (m ) t +1⎦ ⎣ As in the BGG model, the borrower is allowed to obtain funds from the (representative) lender to finance its capital investment. The lender is willing to provide funds as long as the terminal value of net worth at a given period is greater than or equal to the fund that can be appropriated by the borrower,

Vt (m ) ≥ νGK Qt Kt−1(m ), where νGK is the fraction of asset that can be diverted by the borrower. When this incentive constraint is binding, the common “threshold” leverage that applies to all borrowers can be expressed as

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ϕt =

ηt νGK − υt

.

A.3.2. Net worth dynamics in the GK model Aggregate net worth of the borrower is defined as the sum of net worth for the surviving borrowers and for the entering borrowers

Nt = Ne, t + Nn, t . The net worth for the surviving borrowers is

Ne, t +1 = γGK [(Rtk − Rt −1)ϕt −1 + Rt −1]Nt , whereas the net worth for the entering borrowers is

Nn, t +1 = ωGK Qt Kt −1, where ωGK is the proportional transfer from the lender to the entering borrowers. Thus the overall net worth of the borrower grows as follows,

Nt +1 = γGK [(Rtk − Rt −1)ϕt −1 + Rt −1]Nt + ωGK Qt Kt −1. A.3.3. Monetary policy in the GK model In the GK model, the monetary policymaker can engage in public financial intermediation in addition to the conventional interest rate policy that is also available in the BGG model. The total value of financial intermediation is defined as the sum of private financial intermediation and the public financial intermediation

Qt Kt −1 = Qt Kp, t −1 + Qt Kg, t −1. The fraction of public financial intermediation is defined as

ψg, t ≡

Qt Kg, t −1 Qt Kt −1

,

and it is endogenously determined through the following feedback rule,

ψg, t = ψgϕt

ν BGGg

,

where ν BGGg represents the degree of intervention. When public financial intermediation is implemented, an additional expenditure arises that captures the efficiency cost associated with implementing the policy

Ct fi = τGK ψg, tQt Kt −1, where τGK is the (unit) efficiency cost of public financial intermediation.

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