## Abstract

This paper investigates the sources of macrofinancial fluctuations and turbulence within the framework of an approximate linear dynamic stochastic general equilibrium model of the world economy, augmented with structural shocks exhibiting potentially asymmetric generalized autoregressive conditional heteroskedasticity. Very strong evidence of asymmetric autoregressive conditional heteroskedasticity is found, providing a basis for jointly decomposing the levels and volatilities of key macrofinancial variables into time varying contributions from sets of shocks. Risk premia shocks are estimated to contribute disproportionately to cyclical output fluctuations and turbulence during swings in financial conditions, across the fifteen largest national economies in the world.

## I. Introduction

In recent decades, the world economy has experienced extended periods of cyclical expansion and tranquility, occasionally disrupted by bouts of cyclical contraction and turbulence. Indeed, the Global Financial Crisis abruptly ended the extended period of cyclical expansion and tranquility known as the Great Moderation, generating cyclical output contractions and financial market turbulence across major advanced and emerging market economies. The Euro Area Sovereign Debt Crisis generated further cyclical output contractions and financial market turbulence in some major advanced economies, while the Taper Tantrum precipitated cyclical contractions and turbulence in some major emerging market economies.

These occasional bouts of cyclical output contraction and financial market turbulence may have a common cause. In a recent paper, Adrian, Boyarchenko and Giannone (2017) find that a tightening of financial conditions is associated with a reduction in the conditional mean and an increase in the conditional variance of output growth in the United States. They argue that these adverse effects on the conditional distribution of output growth are generated by financial amplification mechanisms.

To investigate the sources of macrofinancial fluctuations and turbulence, this paper augments an approximate linear dynamic stochastic general equilibrium (DSGE) model of the world economy with structural shocks exhibiting potentially asymmetric generalized autoregressive conditional heteroskedasticity (GARCH) effects. A refinement of the DSGE model documented in Vitek (2018), this model features a range of nominal and real rigidities, extensive macrofinancial linkages with both bank and capital market based financial intermediation, and diverse spillover transmission channels. Very strong evidence of asymmetric autoregressive conditional heteroskedasticity (ARCH) effects is found, providing a basis for jointly decomposing the levels and volatilities of output and financial conditions into time varying contributions from sets of shocks. Consistent with the finding of Adrian, Boyarchenko and Giannone (2017), risk premia shocks are estimated to contribute disproportionately to cyclical output fluctuations and turbulence during occasional abrupt swings in financial conditions, across the fifteen largest national economies in the world. This phenomenon struck all of the economies most affected by the Global Financial Crisis, the Euro Area Sovereign Debt Crisis, and the Taper Tantrum.

Accounting for the sources of macrofinancial fluctuations or turbulence within a DSGE framework is common. For example, Smets and Wouters (2007) decompose output growth fluctuations in the United States into contributions from various structural shocks using an approximate linear DSGE model. In another influential paper, Justiniano and Primiceri (2008) decompose output growth volatility in the United States into contributions from various structural shocks exhibiting symmetric stochastic volatility (SV) effects using an approximate linear DSGE model. Unlike these and related papers, this paper jointly analyzes the sources of macrofinancial fluctuations and turbulence in the world economy within a DSGE framework. To our knowledge, it is the first to add ARCH effects to a DSGE model. These are simpler to interpret than SV effects, as the conditional variances of the structural shocks are driven by the same innovations as their conditional means.

The organization of this paper is as follows. The next section develops the theoretical framework, while the following section describes the corresponding empirical framework. Estimation of this empirical framework is the subject of section four. Inference on the sources of macrofinancial fluctuations and turbulence is conducted in section five. Finally, section six offers conclusions and recommendations for further research.

## II. The Theoretical Framework

Consider a finite set of structurally isomorphic national economies indexed by *i* ∈ {1,…,*N*} which constitutes the world economy. Each of these economies consists of households, developers, firms, banks, and a government. The government in turn consists of a monetary authority, a fiscal authority, and a macroprudential authority. Households, developers, firms and banks optimize intertemporally, interacting with governments in an uncertain environment to determine equilibrium prices and quantities under rational expectations in globally integrated output and financial markets. Economy *i* * issues the quotation currency for transactions in the foreign exchange market.

### A. The Household Sector

There exists a continuum of households indexed by *h* ∈ [0,1]. Households are differentiated according to whether they are credit constrained, and according to how they save if they are credit unconstrained, but are otherwise identical. Credit unconstrained households of type *Z* = *B* and measure *ϕ*^{B} have access to domestic banks where they accumulate deposits, and to domestic property markets where they trade real estate, where 0 < *ϕ ^{B}* < 1. In contrast, credit unconstrained households of type

*Z*=

*A*and measure

*ϕ*have access to domestic and foreign capital markets where they trade financial assets, where 0 <

^{A}*ϕ*< 1. Finally, credit constrained households of type

^{A}*Z*=

*C*and measure

*ϕ*do not have access to banks or capital markets, where 0 <

^{C}*ϕ*< 1 and

^{C}*ϕ*+

^{B}*ϕ*+

^{A}*ϕ*= 1. All households are originally endowed with one share of each domestic developer, firm and bank.

^{C}In a reinterpretation of the labor market in the model of nominal wage rigidity proposed by Erceg, Henderson and Levin (2000) to incorporate involuntary unemployment along the lines of Galí (2011), each household consists of a continuum of members represented by the unit square and indexed by (*f*,*g*) ∈ [0,1] × [0,1]. There is full risk sharing among household members, who supply indivisible differentiated intermediate labor services indexed by *f* ∈ [0,1], incurring disutility from work determined by *g* ∈ [0,1] if they are employed and zero otherwise. Trade specific intermediate labor services supplied by bank intermediated, capital market intermediated, and credit constrained households are perfect substitutes.

#### Consumption and Saving

The representative infinitely lived household has preferences defined over consumption *C*_{h,i,s}, housing *H*_{h,i,s}, labor supply

where E_{t} denotes the expectations operator conditional on information available in period *t*, and 0 < *β* < 1. The intratemporal utility function is additively separable and represents external habit formation preferences in consumption and labor supply,

where 0 ≤ *α ^{C}* < 1 and 0 ≤

*α*< 1. Endogenous preference shifters

^{L}where *ι* > 0. The intratemporal utility function is strictly increasing with respect to consumption if and only if serially correlated consumption demand shock *σ* > 0, *ς* > 0, *η* > 0 and *μ* > 0. In steady state equilibrium,

The representative household enters period *s* in possession of previously accumulated property balances *ℰ*_{i,j,s} measures the price of foreign currency in terms of domestic currency. It follows that *s*, the representative household receives profit income from banks

According to this dynamic budget constraint, at the end of period *s*, the representative household holds property balances *C*_{h,i,s} at price *H*_{h,i,s} at price

##### Bank Intermediated Households

The representative bank intermediated household has a capitalist spirit motive for holding real property balances, independent of financing deferred consumption, motivated by Weber (1905). It also has a diversification motive over the allocation of real property balances across alternative assets which are imperfect substitutes, motivated by Tobin (1969). The set of assets under consideration consists of bank deposits and domestically traded real estate. Preferences over the real values of bank deposits

where serially correlated housing risk premium shock *ϕ*^{H} < 1 and *ψ*^{H} > 0. Preferences over the real values of developer specific shares

In the limit as *ψ*^{H} → ∞ there is no diversification motive over the allocation of real property balances across alternative assets which in this case are perfect substitutes.

In period *t*, the representative bank intermediated household chooses state contingent sequences for consumption *T* → ∞. In equilibrium, abstracting from the capitalist spirit motive for holding real property balances, the solutions to this utility maximization problem satisfy intertemporal optimality condition

which equates the expected present value of the gross real property return to one. In addition, these solutions satisfy intratemporal optimality condition

which equates the marginal rate of substitution between housing and consumption to the real rental price of housing. Furthermore, these solutions satisfy intratemporal optimality condition

which equates the marginal rate of substitution between leisure and consumption for the marginal trade specific labor force participant to the corresponding after tax real wage. Abstracting from risk premium shocks, the expected present value of the gross real property return satisfies intratemporal optimality condition

which relates it to the expected present values of the gross real returns on bank deposits and real estate. Finally, abstracting from the portfolio diversification motive over the allocation of real property balances these solutions satisfy intratemporal optimality condition

which equates the expected present values of the gross real risk adjusted returns on bank deposits and real estate. Provided that the intertemporal utility function is bounded and strictly concave, together with other optimality conditions, and transversality conditions derived from necessary complementary slackness conditions associated with the terminal nonnegativity constraints, these optimality conditions are sufficient for the unique utility maximizing state contingent sequence of bank intermediated household allocations.

##### Capital Market Intermediated Households

The representative capital market intermediated household has a capitalist spirit motive for holding real portfolio balances, independent of financing deferred consumption, motivated by Weber (1905). It also has a diversification motive over the allocation of real portfolio balances across alternative financial assets which are imperfect substitutes, motivated by Tobin (1969). The set of financial assets under consideration consists of internationally traded and local currency denominated short term bonds, long term bonds, and stocks. Short term bonds are discount bonds, while long term bonds are perpetual bonds with coupon payments that decay exponentially at rate *ω*^{B} where 0 < *ω*^{B} < 1, following Woodford (2001). Preferences over the real values of internationally diversified short term bond

where internationally and serially correlated duration risk premium shock satisfies *ψ*^{A} > 0. Preferences over the real values of economy specific short term bond

where serially correlated currency risk premium shocks

where *ψ*^{A} → ∞ there is no diversification motive over the allocation of real portfolio balances across alternative financial assets which in this case are perfect substitutes.

In period *t*, the representative capital market intermediated household chooses state contingent sequences for consumption *T* → ∞. In equilibrium, abstracting from the capitalist spirit motive for holding real portfolio balances, the solutions to this utility maximization problem satisfy intertemporal optimality condition

which equates the expected present value of the gross real portfolio return to one. In addition, these solutions satisfy intratemporal optimality condition

which equates the marginal rate of substitution between housing and consumption to the real rental price of housing. Furthermore, these solutions satisfy intratemporal optimality condition

which equates the marginal rate of substitution between leisure and consumption for the marginal trade specific labor force participant to the corresponding after tax real wage. Abstracting from risk premium shocks, the expected present value of the gross real portfolio return satisfies intratemporal optimality condition

which relates it to the expected present values of the gross real returns on domestic and foreign short term bonds, long term bonds, and stocks. In addition, abstracting from the portfolio diversification motive over the allocation of real portfolio balances these solutions satisfy intratemporal optimality condition

which equates the expected present values of the gross real risk adjusted returns on domestic and foreign short term bonds. Furthermore, abstracting from the portfolio diversification motive over the allocation of real portfolio balances these solutions satisfy intratemporal optimality condition

which equates the expected present values of the gross real risk adjusted returns on domestic short and long term bonds. Finally, abstracting from the portfolio diversification motive over the allocation of real portfolio balances these solutions satisfy intratemporal optimality condition

which equates the expected present values of the gross real risk adjusted returns on domestic short term bonds and stocks. Provided that the intertemporal utility function is bounded and strictly concave, together with other optimality conditions, and transversality conditions derived from necessary complementary slackness conditions associated with the terminal nonnegativity constraints, these optimality conditions are sufficient for the unique utility maximizing state contingent sequence of capital market intermediated household allocations.

##### Credit Constrained Households

In period *t*, the representative credit constrained household chooses state contingent sequences for consumption

which equates the sum of consumption and housing expenditures to the sum of profit and disposable labor income net of transfer payments, where profit income *Π*_{i,t} satisfies

which equates the marginal rate of substitution between housing and consumption to the real rental price of housing. Finally, these solutions satisfy intratemporal optimality condition

which equates the marginal rate of substitution between leisure and consumption for the marginal trade specific labor force participant to the corresponding after tax real wage. Provided that the intertemporal utility function is bounded and strictly concave, these optimality conditions are sufficient for the unique utility maximizing state contingent sequence of credit constrained household allocations.

#### Labor Supply

The unemployment rate *N*_{i,t} in unemployment *U*_{i,t}, that is *L*_{i,t}, that is *U*_{i,t} = *N*_{i,t} – *L*_{i,t}. The labor force satisfies

There exist a large number of perfectly competitive firms which combine differentiated intermediate labor services *L*_{f,i,t} supplied by trade unions of workers to produce final labor service *L*_{i,t} according to constant elasticity of substitution production function

where serially uncorrelated wage markup shock

Since the production function exhibits constant returns to scale, in equilibrium the representative final labor service firm generates zero profit, implying aggregate wage index:

As the wage elasticity of demand for intermediate labor services

In an extension of the model of nominal wage rigidity proposed by Erceg, Henderson and Levin (2000) along the lines of Smets and Wouters (2003), each period a randomly selected fraction 1 – *ω*^{L} of trade unions adjust their wage optimally, where 0 ≤ *ω*^{L} < 1. The remaining fraction *ω*^{L} of trade unions adjust their wage to account for past consumption price inflation and trend productivity growth according to partial indexation rule

where 0 ≤ *γ ^{L}* ≤ 1. Under this specification, although trade unions adjust their wage every period, they infrequently do so optimally, and the interval between optimal wage adjustments is a random variable.

If the representative trade union can adjust its wage optimally in period *t*, then it does so to maximize intertemporal utility function (1) subject to dynamic budget constraint (7), intermediate labor service demand function (32), and the assumed form of nominal wage rigidity. Since all trade unions that adjust their wage optimally in period *t* solve an identical utility maximization problem, in equilibrium they all choose a common wage

This necessary first order condition equates the expected present value of the marginal utility of consumption gained from labor supply to the expected present value of the marginal utility cost incurred from leisure foregone. Aggregate wage index (33) equals an average of the wage set by the fraction 1 – *ω ^{L}* of trade unions that adjust their wage optimally in period

*t*, and the average of the wages set by the remaining fraction

*ω*

^{L}of trade unions that adjust their wage according to partial indexation rule (34):

Since those trade unions able to adjust their wage optimally in period *t* are selected randomly from among all trade unions, the average wage set by the remaining trade unions equals the value of the aggregate wage index that prevailed during period *t* – 1, rescaled to account for past consumption price inflation and trend productivity growth.

### B. The Construction Sector

The construction sector supplies housing services to domestic households. In doing so, developers obtain mortgage loans from domestic banks and accumulate the housing stock through residential investment.

#### Housing Demand

There exist a large number of perfectly competitive developers which combine differentiated intermediate housing services *H*_{i,e,t} supplied by intermediate developers to produce final housing service *H*_{i,t} according to constant elasticity of substitution production function

where endogenous rental price of housing markup shifter

Since the production function exhibits constant returns to scale, in equilibrium the representative final developer generates zero profit, implying aggregate rental price of housing index:

As the price elasticity of demand for intermediate housing services

#### Residential Investment

There exist continuums of monopolistically competitive intermediate developers indexed by *e* ∈ [0,1]. Intermediate developers supply differentiated intermediate housing services, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive intermediate construction sector.

The representative intermediate developer sells shares to domestic bank intermediated households at price

where *s* bank intermediated household dynamic budget constraint. The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to profits

Earnings are defined as revenues from sales of differentiated intermediate housing service *H*_{i,e,s} at rental price

Motivated by the collateralized borrowing variant of the financial accelerator mechanism due to Kiyotaki and Moore (1997), the financial policy of the representative intermediate developer is to maintain debt equal to a fraction of the value of the housing stock,

given by regulatory mortgage loan to value ratio limit

The representative intermediate developer enters period *s* in possession of previously accumulated housing stock *H*_{i,e,s}, which subsequently evolves according to accumulation function

where 0 ≤ *δ*^{H} ≤ 1. Effective residential investment function

where serially correlated residential investment demand shock *χ*^{H} > 0. In steady state equilibrium, these adjustment costs equal zero, and effective residential investment equals actual residential investment.

In period *t*, the representative intermediate developer chooses state contingent sequences for residential investment *H*_{i,e,T+1} ≥ 0 for *T* → ∞. In equilibrium, demand for the final residential investment good satisfies necessary first order condition

which equates the expected present value of an additional unit of residential investment to its price, where *s* housing accumulation function. In equilibrium, this shadow price of housing satisfies necessary first order condition

which equates it to the expected present value of the sum of the future marginal revenue product of housing, and the future shadow price of housing net of depreciation, less the product of the loan to value ratio with the spread of the effective cost of bank over capital market funding. Provided that the pre-dividend stock market value is bounded and strictly concave, together with other necessary first order conditions, and a transversality condition derived from the necessary complementary slackness condition associated with the terminal nonnegativity constraint, these necessary first order conditions are sufficient for the unique value maximizing state contingent sequence of intermediate developer allocations.

#### Housing Supply

In period *t*, the representative intermediate developer adjusts its rental price of housing to maximize pre-dividend stock market value (40) subject to housing accumulation function (43) and intermediate housing service demand function (38). We consider a symmetric equilibrium under which all developer specific endogenous state variables are restricted to equal their aggregate counterparts. It follows that all intermediate developers solve an identical value maximization problem, which implies that they all choose a common rental price of housing

This necessary first order condition equates the marginal revenue gained from housing supply to the marginal cost incurred from construction. Aggregate rental price of housing index (39) satisfies

### C. The Production Sector

The production sector supplies output goods for domestic and foreign absorption. In doing so, firms demand labor services from domestic households, obtain corporate loans from domestic and foreign banks, and accumulate the private physical capital stock through business investment.

The production sector consists of a finite set of industries indexed by *k* ∈ {1, …, *M*}, of which the first *M* * produce nonrenewable commodities. In particular, the energy commodity industry labeled *k* = 1 and the nonenergy commodity industry labeled *k* = 2 produce internationally homogeneous output goods for foreign absorption, while all other industries produce internationally heterogeneous output goods for domestic and foreign absorption.

#### Output Demand

There exist a large number of perfectly competitive firms which combine industry specific final output goods *Y*_{i,t} according to fixed proportions production function

where

Since the production function exhibits constant returns to scale, in equilibrium the representative final output good firm generates zero profit, implying aggregate output price index

where *k* ≤ *M**. This aggregate output price index equals the minimum cost of producing one unit of the final output good, given the prices of industry specific final output goods.

There exist a large number of perfectly competitive firms which combine industry specific differentiated intermediate output goods *Y*_{i,k,l,t} supplied by industry specific intermediate output good firms to produce industry specific final output good *Y*_{i,k,t} according to constant elasticity of substitution production function

where serially uncorrelated output price markup shock *k* ≤ *M** and

Since the production function exhibits constant returns to scale, in equilibrium the representative industry specific final output good firm generates zero profit, implying industry specific aggregate output price index:

As the price elasticity of demand for industry specific intermediate output goods

#### Labor Demand and Business Investment

There exist continuums of monopolistically competitive industry specific intermediate output good firms indexed by *l* ∈ [0,1]. Intermediate output good firms supply industry specific differentiated intermediate output goods, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive industry specific intermediate output good sectors.

The representative industry specific intermediate output good firm sells shares to domestic and foreign capital market intermediated households at price

where *s* capital market intermediated household dynamic budget constraint. The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to net profits

where *Y*_{i,k,l,s} at price *L*_{i,k,l,s}, and other variable costs *Φ*_{i,k,l,s}. The government levies a tax on corporate earnings at rate

Motivated by the collateralized borrowing variant of the financial accelerator mechanism due to Kiyotaki and Moore (1997), the financial policy of the representative industry specific intermediate output good firm is to maintain debt equal to a fraction of the value of the private physical capital stock,

given by regulatory corporate loan to value ratio limit

The representative industry specific intermediate output good firm utilizes private physical capital *K*_{i,k,l,s} at rate *L*_{i,k,l,s} to produce industry specific differentiated intermediate output good *Y*_{i,k,l,s} according to production function:

This production function exhibits constant returns to scale, with

where internationally and serially correlated productivity shock

In utilizing private physical capital to produce output, the representative industry specific intermediate output good firm incurs a cost

where industry specific fixed cost *Φ*_{i,k,s} = 0. Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the capital utilization rate at an increasing rate,

where *η*^{K} > 0, while *μ*^{K} > 0. In steady state equilibrium, the capital utilization rate equals one, and the cost of utilizing private physical capital equals zero.

The representative industry specific intermediate output good firm enters period *s* in possession of previously accumulated private physical capital stock *K*_{i,k,l,s}, which subsequently evolves according to accumulation function

where 0 < *δ*^{K} ≤ 1. Building on Christiano, Eichenbaum and Evans (2005), effective business investment function

where serially correlated business investment demand shock *χ*^{K} > 0. In steady state equilibrium, these adjustment costs equal zero, and effective business investment equals actual business investment.

In period *t*, the representative industry specific intermediate output good firm chooses state contingent sequences for employment *K*_{i,k,l,T+1} ≥ 0 for *T* → ∞. In equilibrium, demand for the final labor service satisfies necessary first order condition

where *s* production technology constraint. This necessary first order condition equates real marginal cost *Ψ*_{i,k,l,t} to the ratio of the after tax industry specific real wage to the marginal product of labor. In equilibrium, the capital utilization rate satisfies necessary first order condition

which equates the marginal revenue product of utilized private physical capital to its marginal cost. In equilibrium, demand for the final business investment good satisfies necessary first order condition

which equates the expected present value of an additional unit of business investment to its price, where *s* private physical capital accumulation function. In equilibrium, this shadow price of private physical capital satisfies necessary first order condition

which equates it to the expected present value of the sum of the future marginal revenue product of private physical capital net of its marginal utilization cost, and the future shadow price of private physical capital net of depreciation, less the product of the loan to value ratio with the spread of the effective cost of bank over capital market funding. Provided that the pre-dividend stock market value is bounded and strictly concave, together with other necessary first order conditions, and a transversality condition derived from the necessary complementary slackness condition associated with the terminal nonnegativity constraint, these necessary first order conditions are sufficient for the unique value maximizing state contingent sequence of industry specific intermediate output good firm allocations.

#### Output Supply

In an extension of the model of nominal output price rigidity proposed by Calvo (1983) along the lines of Smets and Wouters (2003), each period a randomly selected fraction *k* > *M* *. The remaining fraction

where *k* ≤ *M** and

If the representative industry specific intermediate output good firm can adjust its price optimally in period *t*, then it does so to maximize pre-dividend stock market value (54) subject to production function (57), industry specific intermediate output good demand function (52), and the assumed form of nominal output price rigidity. We consider a symmetric equilibrium under which all industry and firm specific endogenous state variables are restricted to equal their industry specific aggregate counterparts. It follows that all intermediate output good firms that adjust their price optimally in period *t* solve an identical value maximization problem, which implies that they all choose a common price

This necessary first order condition equates the expected present value of the after tax marginal revenue gained from output supply to the expected present value of the marginal cost incurred from production. Aggregate output price index (53) equals an average of the price set by the fraction *t*, and the average of the prices set by the remaining fraction

Since those intermediate output good firms able to adjust their price optimally in period *t* are selected randomly from among all intermediate output good firms, the average price set by the remaining intermediate output good firms equals the value of the industry specific aggregate output price index that prevailed during period *t* – 1, rescaled to account for past industry specific output price inflation.

### D. The Banking Sector

The banking sector supplies global financial intermediation services subject to financial frictions and regulatory constraints. In particular, banks issue risky mortgage loans to domestic developers at infrequently adjusted predetermined mortgage loan rates, as well as risky domestic currency denominated corporate loans to domestic and foreign firms at infrequently adjusted predetermined corporate loan rates, given regulatory loan to value ratio limits. They obtain funding from domestic bank intermediated households via deposits and from the domestic interbank market via loans, accumulating bank capital out of retained earnings given credit losses to satisfy a regulatory capital requirement.

#### Credit Demand

There exist a large number of perfectly competitive banks which combine economy specific local currency denominated final corporate loans

where

Since the portfolio aggregator exhibits constant returns to scale, in equilibrium the representative global final bank generates zero profit, implying aggregate effective gross corporate loan rate index:

This aggregate effective gross corporate loan rate index equals the minimum cost of producing one unit of the domestic currency denominated final corporate loan, given the rates on economy specific local currency denominated final corporate loans.

There exist a large number of perfectly competitive banks which combine differentiated intermediate mortgage or corporate loans

where *Z* ∈ {*D*,*F*}, while serially uncorrelated mortgage or corporate loan rate markup shock

where *f*(*D*) = *M* and *f* (*F*) = *C*. Since the portfolio aggregator exhibits constant returns to scale, in equilibrium the representative domestic final bank generates zero profit, implying aggregate gross mortgage or corporate loan rate index:

As the rate elasticity of demand for intermediate mortgage or corporate loans

#### Funding Demand and Bank Capital Accumulation

There exists a continuum of monopolistically competitive intermediate banks indexed by *m* ∈ [0,1]. Intermediate banks supply differentiated intermediate mortgage and corporate loans, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive intermediate banking sector.

The representative intermediate bank sells shares to domestic bank intermediated households at price

The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments

Profits are defined as the sum of the increase in deposits

The representative intermediate bank transforms deposit and money market funding into risky differentiated intermediate mortgage and corporate loans according to balance sheet identity:

The bank credit stock *κ*_{i,s+1} equals the ratio of aggregate bank capital to assets, that is

In transforming deposit and money market funding into risky mortgage and corporate loans, the representative intermediate bank incurs a cost of satisfying the regulatory capital requirement,

where fixed cost

given regulatory capital requirement *η*^{C} > 0 and *μ*^{C} > 0. In steady state equilibrium, the bank capital ratio equals its required value, and the cost of regulation is constant.

The financial policy of the representative intermediate bank is to smooth retained earnings intertemporally, given credit losses. It enters period *s* in possession of previously accumulated bank capital stock

where bank capital destruction rate *χ ^{C}* < 0, while mortgage loan weight

where *χ*^{B} > 0. In steady state equilibrium, these adjustment costs equal zero, and effective retained earnings equals actual retained earnings.

In period *t*, the representative intermediate bank chooses state contingent sequences for deposit funding *T* → ∞. In equilibrium, the solutions to this value maximization problem satisfy necessary first order condition

which equates the deposit rate to the interbank loans rate. In equilibrium, retained earnings satisfies necessary first order condition

which equates the expected present value of an additional unit of retained earnings to its marginal cost, where *s* bank capital accumulation function. In equilibrium, this shadow price of bank capital satisfies necessary first order condition

which equates it to the expected present value of the future shadow price of bank capital net of destruction, less the sum of the marginal utilization cost of bank capital and the spread of the cost of deposit over interbank market funding. The evaluation of this result abstracts from risk premium shocks. Provided that the pre-dividend stock market value is bounded and strictly concave, together with other necessary first order conditions, and transversality conditions derived from the necessary complementary slackness conditions associated with the terminal nonnegativity constraints, these necessary first order conditions are sufficient for the unique value maximizing state contingent sequence of intermediate bank allocations.

#### Credit Supply

In an adaptation of the model of nominal output price rigidity proposed by Calvo (1983) to the banking sector along the lines of Hülsewig, Mayer and Wollmershäuser (2009), each period a randomly selected fraction 1 – *ω ^{C}* of intermediate banks adjust their gross mortgage and corporate loan rates optimally, where 0 ≤

*ω*< 1. The remaining fraction

^{C}*ω*of intermediate banks do not adjust their loan rates,

^{C}where *Z* ∈ {*D*,*F*}, while *f* (*D*) = *M* and *f* (*F*) = *C*. Under this financial friction, intermediate banks infrequently adjust their loan rates, mimicking the effect of maturity transformation on the spreads between the loan and deposit rates.

If the representative intermediate bank can adjust its gross mortgage and corporate loan rates in period *t*, then it does so to maximize pre-dividend stock market value (76) subject to balance sheet identity (78), intermediate loan demand function (74), and the assumed financial friction. We consider a symmetric equilibrium under which all bank specific endogenous state variables are restricted to equal their aggregate counterparts. It follows that all intermediate banks that adjust their loan rates in period *t* solve an identical value maximization problem, which implies that they all choose common loan rates

where *g*(*D*) = *M* and *g*(*F*) = *C*, *E*, while *h*(*D*) = 1 and *h*(*F*) = 2. These necessary first order conditions equate the expected present value of the marginal revenue gained from mortgage or corporate loan supply to the expected present value of the marginal cost incurred from intermediation. Aggregate gross mortgage or corporate loan rate index (75) equals an average of the gross mortgage or corporate loan rate set by the fraction 1 – *ω ^{C}* of intermediate banks that adjust their loan rates in period

*t*, and the average of the gross mortgage or corporate loan rates set by the remaining fraction

*ω*of intermediate banks that do not adjust their loan rates:

^{C}Since those intermediate banks able to adjust their loan rates in period *t* are selected randomly from among all intermediate banks, the average gross mortgage or corporate loan rate set by the remaining intermediate banks equals the value of the aggregate gross mortgage or corporate loan rate index that prevailed during period *t* – 1.

### E. The Trade Sector

The nominal effective exchange rate *ℰ*_{i,t} measures the trade weighted average price of foreign currency in terms of domestic currency, while the real effective exchange rate *Q*_{i, t} measures the trade weighted average price of foreign output in terms of domestic output,

where the real bilateral exchange rate *Q*_{i,j,t} satisfies

where the internal terms of trade *P*_{i,t} denotes the price of the final noncommodity output good. Finally, under the law of one price for 1 ≤ *k* ≤ *M* *,

where

#### The Export Sector

There exist a large number of perfectly competitive firms which combine industry specific final export goods *X*_{i,t} according to fixed proportions production function

where *X*_{i,k,t} = *Y*_{i,k,t} for 1 ≤ *k* ≤ *M* *, while

Since the production function exhibits constant returns to scale, in equilibrium the representative final export good firm generates zero profit, implying aggregate export price index

where *k* > *M* *. This aggregate export price index equals the minimum cost of producing one unit of the final export good, given the prices of industry specific final export goods.

##### Export Demand

There exist a large number of perfectly competitive firms which combine industry specific differentiated intermediate export goods *X*_{i,k,n,t} supplied by industry specific intermediate export good firms to produce industry specific final export good *X*_{i,k,t} according to constant elasticity of substitution production function

for 1 ≤ *k* ≤ *M* *, where serially uncorrelated export price markup shock

Since the production function exhibits constant returns to scale, in equilibrium the representative industry specific final export good firm generates zero profit, implying industry specific aggregate export price index:

As the price elasticity of demand for industry specific intermediate export goods

##### Export Supply

There exist continuums of monopolistically competitive industry specific intermediate export good firms indexed by *n* ∈ [0,1]. Intermediate export good firms supply industry specific differentiated intermediate export goods, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive industry specific intermediate export good sectors.

The representative industry specific intermediate export good firm sells shares to domestic capital market intermediated households at price

The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to profits

Earnings are defined as revenues from sales of industry specific differentiated intermediate export good *X*_{i,k,n,s} at price *X*_{i,k,n,s}. The representative industry specific intermediate export good firm purchases the energy or nonenergy commodity good and differentiates it. Fixed cost

In an adaptation of the model of nominal import price rigidity proposed by Monacelli (2005) to the export sector, each period a randomly selected fraction 1 – *ω ^{X}* of industry specific intermediate export good firms adjust their price optimally, where 0 ≤

*ω*< 1. The remaining fraction

^{X}*ω*of intermediate export good firms adjust their price to account for past industry specific export price inflation, as well as the contemporaneous change in the domestic currency denominated price of energy or nonenergy commodities, according to partial indexation rule

^{X}where 0 ≤ *γ*^{X} ≤ 1 and *μ*^{X} ≥ 0. Under this specification, the probability that an intermediate export good firm has adjusted its price optimally is time dependent but state independent.

If the representative industry specific intermediate export good firm can adjust its price optimally in period *t*, then it does so to maximize pre-dividend stock market value (113) subject to industry specific intermediate export good demand function (111), and the assumed form of nominal export price rigidity. Since all intermediate export good firms that adjust their price optimally in period *t* solve an identical value maximization problem, in equilibrium they all choose a common price

This necessary first order condition equates the expected present value of the marginal revenue gained from export supply to the expected present value of the marginal cost incurred from production. Aggregate export price index (112) equals an average of the price set by the fraction 1 – *ω*^{X} of intermediate export good firms that adjust their price optimally in period *t*, and the average of the prices set by the remaining fraction *ω*^{X} of intermediate export good firms that adjust their price according to partial indexation rule (115):

Since those intermediate export good firms able to adjust their price optimally in period *t* are selected randomly from among all intermediate export good firms, the average price set by the remaining intermediate export good firms equals the value of the industry specific aggregate export price index that prevailed during period *t* – 1, rescaled to account for past industry specific export price inflation, as well as the contemporaneous change in the domestic currency denominated price of energy or nonenergy commodities.

#### The Import Sector

There exist a large number of perfectly competitive firms which combine the final noncommodity output good

where serially correlated import demand shock *ψ*^{M} > 0. The representative final absorption good firm maximizes profits derived from production of the final private consumption, residential investment, business investment, public consumption or public investment good, with respect to inputs of the final noncommodity output and import goods, implying demand functions:

Since the production function exhibits constant returns to scale, in equilibrium the representative final absorption good firm generates zero profit, implying aggregate private consumption, residential investment, business investment, public consumption or public investment price index:

Combination of this aggregate private consumption, residential investment, business investment, public consumption or public investment price index with final noncommodity output and import good demand functions (104) yields:

These demand functions for the final noncommodity output and import goods are directly proportional to final private consumption, residential investment, business investment, public consumption or public investment good demand, with a proportionality coefficient that varies with the external terms of trade. The derivation of these results selectively abstracts from import demand shocks.

##### Import Demand

There exist a large number of perfectly competitive firms which combine economy specific final import goods *M*_{i,t} according to fixed proportions production function

where serially correlated export demand shock

Since the production function exhibits constant returns to scale, in equilibrium the representative final import good firm generates zero profit, implying aggregate import price index:

This aggregate import price index equals the minimum cost of producing one unit of the final import good, given the prices of economy specific final import goods. The derivation of these results selectively abstracts from export demand shocks.

There exist a large number of perfectly competitive firms which combine economy specific differentiated intermediate import goods *M*_{i,j,n,t} supplied by economy specific intermediate import good firms to produce economy specific final import good *M*_{i,j,t} according to constant elasticity of substitution production function

where serially uncorrelated import price markup shock

Since the production function exhibits constant returns to scale, in equilibrium the representative economy specific final import good firm generates zero profit, implying economy specific aggregate import price index:

As the price elasticity of demand for economy specific intermediate import goods

##### Import Supply

There exist continuums of monopolistically competitive economy specific intermediate import good firms indexed by *n* ∈ [0,1]. Intermediate import good firms supply economy specific differentiated intermediate import goods, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive economy specific intermediate import good sectors.

The representative economy specific intermediate import good firm sells shares to domestic capital market intermediated households at price

The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to profits

Earnings are defined as revenues from sales of economy specific differentiated intermediate import good *M*_{i,j,n,s} at price *M*_{i,j,n,s}. The representative economy specific intermediate import good firm purchases the foreign final export good and differentiates it. Fixed cost

In an extension of the model of nominal import price rigidity proposed by Monacelli (2005), each period a randomly selected fraction 1 – *ω*^{M} of economy specific intermediate import good firms adjust their price optimally, where 0 ≤ *ω*^{M} < 1. The remaining fraction *ω*^{M} of intermediate import good firms adjust their price to account for past economy specific import price inflation, as well as contemporaneous changes in the domestic currency denominated prices of energy and nonenergy commodities, according to partial indexation rule

where 0 ≤ *γ*^{M} ≤ 1, while *μ*^{M} ≥ 0. Under this specification, the probability that an intermediate import good firm has adjusted its price optimally is time dependent but state independent.

If the representative economy specific intermediate import good firm can adjust its price optimally in period *t*, then it does so to maximize pre-dividend stock market value (113) subject to economy specific intermediate import good demand function (111), and the assumed form of nominal import price rigidity. Since all intermediate import good firms that adjust their price optimally in period *t* solve an identical value maximization problem, in equilibrium they all choose a common price

This necessary first order condition equates the expected present value of the marginal revenue gained from import supply to the expected present value of the marginal cost incurred from production. Aggregate import price index (112) equals an average of the price set by the fraction 1 – *ω*^{M} of intermediate import good firms that adjust their price optimally in period *t*, and the average of the prices set by the remaining fraction *ω*^{M} of intermediate import good firms that adjust their price according to partial indexation rule (115):

Since those intermediate import good firms able to adjust their price optimally in period *t* are selected randomly from among all intermediate import good firms, the average price set by the remaining intermediate import good firms equals the value of the economy specific aggregate import price index that prevailed during period *t* – 1, rescaled to account for past economy specific import price inflation, as well as contemporaneous changes in the domestic currency denominated prices of energy and nonenergy commodities.

### F. Monetary, Fiscal, and Macroprudential Policy

The government consists of a monetary authority, a fiscal authority, and a macroprudential authority. The monetary authority conducts monetary policy, while the fiscal authority conducts fiscal policy, and the macroprudential authority conducts macroprudential policy.

#### The Monetary Authority

The monetary authority implements monetary policy through control of the nominal policy interest rate. We differentiate between flexible inflation targeting, managed exchange rate, and fixed exchange rate regimes. Under a monetary union, the leader economy follows a modified flexible inflation targeting regime, while all other union members follow fixed exchange rate regimes.

Under a flexible inflation targeting or managed exchange rate regime, the nominal policy interest rate satisfies a monetary policy rule exhibiting partial adjustment dynamics of the form

where 0 ≤ *ρ ^{i}* < 1,

*ξ*

^{π}> 1,

*ξ*

^{Y}> 0 and

*j*= 0, and this desired deviation is increasing in the expected future deviation of consumption price inflation from its target value, as well as the contemporaneous output gap. We define the output gap as the deviation of output from its potential value, which we define as that output level consistent with full utilization of private physical capital and effective labor, given the private physical capital stock and effective labor force. For the leader economy of a monetary union, the target variables entering into its monetary policy rule are expressed as output weighted averages across union members. Under a managed exchange rate regime

*j*= 1, and the desired deviation of the nominal policy interest rate from its steady state equilibrium value is also increasing in the contemporaneous deviation of the change in the nominal effective exchange rate from its steady state equilibrium value with

Under a fixed exchange rate regime, the nominal policy interest rate instead satisfies a monetary policy rule exhibiting feedback of the form

where *ξ ^{ℰk}* > 1. As specified, the deviation of the nominal policy interest rate from its steady state equilibrium value tracks the contemporaneous deviation of the nominal policy interest rate of the leader economy from its steady state equilibrium value one for one, and is increasing in the contemporaneous deviation of the change in the corresponding nominal bilateral exchange rate from its target value.

#### The Fiscal Authority

The fiscal authority implements fiscal policy through control of public consumption and investment, as well as the tax rates applicable to corporate earnings and household labor income. It also operates a budget neutral nondiscretionary lump sum transfer program that redistributes national financial wealth from capital market intermediated households to credit constrained households while equalizing steady state equilibrium consumption across households, as well as a discretionary lump sum transfer program that provides income support to credit constrained households. The fiscal authority can transfer its budgetary resources intertemporally through transactions in the domestic money and bond markets. Considered jointly, the rules prescribing the conduct of this distortionary fiscal policy are countercyclical, representing automatic fiscal stabilizers, and are consistent with achieving public and national financial wealth stabilization objectives.

Public consumption and investment satisfy countercyclical fiscal expenditure rules exhibiting partial adjustment dynamics of the form

where *Z* ∈ {*C*, *I*}, while 0 ≤ *ρ*_{G} < 1. As specified, the deviation of public consumption or investment from its steady state equilibrium value depends on a weighted average of its past deviation and its desired deviation, which in turn tracks the contemporaneous deviation of potential output from its steady state equilibrium value one for one. Deviations from these fiscal expenditure rules are captured by mean zero and serially correlated public consumption or investment shock

The tax rates applicable to corporate earnings and household labor income satisfy acyclical fiscal revenue rules of the form

where *Z* ∈ (*K*, *L*}, while 0 < *τ _{i}* < 1 and 0 ≤

*ρ*

_{τ}< 1. As specified, the deviations of these tax rates from their steady state equilibrium value depend on their past deviations. Deviations from these fiscal revenue rules are captured by mean zero and serially correlated corporate or labor income tax rate shock

The ratio of nondiscretionary lump sum transfer payments to nominal output satisfies a nondiscretionary transfer payment rule that stabilizes national financial wealth of the form

where *ζ ^{TN}* > 0. As specified, the deviation of the ratio of nondiscretionary lump sum transfer payments to nominal output from its steady state equilibrium value is increasing in the past deviation of the ratio of national financial wealth to nominal output from its target value. The ratio of discretionary lump sum transfer payments to nominal output satisfies a discretionary transfer payment rule that stabilizes public financial wealth of the form

where *ζ ^{TD}* > 0. As specified, the deviation of the ratio of discretionary lump sum transfer payments to nominal output from its steady state equilibrium value is increasing in the past deviation of the ratio of public financial wealth to nominal output from its target value. Deviations from this discretionary transfer payment rule are captured by mean zero and serially correlated transfer payment shock

The gross yield to maturity on short term bonds depends on the contemporaneous gross nominal policy interest rate according to money market relationship:

Deviations from this money market relationship are captured by internationally and serially correlated credit risk premium shock

Deviations from this interbank market relationship are captured by internationally and serially correlated liquidity risk premium shock

The fiscal authority enters period *t* in possession of previously accumulated financial wealth *t*, the fiscal authority levies taxes on corporate earnings at rate *T*_{i,t}. These sources of public wealth are summed in government dynamic budget constraint:

According to this dynamic budget constraint, at the end of period *t,* the fiscal authority holds financial wealth *δ*^{G} ≤ 1.

#### The Macroprudential Authority

The macroprudential authority implements macroprudential policy through control of a regulatory capital requirement and loan to value ratio limits. It imposes the regulatory capital requirement on lending by domestic banks, and the regulatory loan to value ratio limits on borrowing by domestic developers and firms.

The regulatory capital ratio requirement applicable to lending by domestic banks to domestic and foreign developers and firms satisfies a countercyclical capital buffer rule exhibiting partial adjustment dynamics of the form

where 0 < *κ*^{R} < 1, 0 ≤ *ρ*_{κ} < 1, *ζ*^{κ,B} > 0, *ζ*^{κ,VH} > 0 and *ζ*^{κ,VS} > 0. As specified, the deviation of the regulatory capital ratio requirement from its steady state equilibrium value depends on a weighted average of its past deviation and its desired deviation. This desired deviation is increasing in the contemporaneous deviation of bank credit growth from its steady state equilibrium value, as well as the contemporaneous deviations of the changes in the prices of housing and equity from their steady state equilibrium values. Deviations from this countercyclical capital buffer rule are captured by mean zero and serially correlated capital requirement shock

The regulatory loan to value ratio limits applicable to borrowing by domestic developers and firms from domestic and foreign banks satisfy loan to value limit rules exhibiting partial adjustment dynamics of the form

where *Z* ∈ {*D*, *F*}, while *f* (*D*) = *H* and *f* (*F*) = *S*. As specified, the deviations of the regulatory loan to value ratio limits from their steady state equilibrium values depend on a weighted average of their past deviations and their desired deviations, where 0 < *ϕ*^{Z} < 1, 0 ≤ *ρ*_{ϕZ} < 1, *ζ*^{ϕZ,B} > 0 and *ζ*^{ϕZ, V} > 0. These desired deviations are decreasing in the contemporaneous deviation of mortgage or nonfinancial corporate debt growth from its steady state equilibrium value, as well as the contemporaneous deviation of the change in the price of housing or equity from its steady state equilibrium value, respectively. Deviations from these loan to value limit rules are captured by mean zero and serially uncorrelated mortgage or corporate loan to value limit shock

The loan default rates applicable to borrowing by domestic developers and firms from domestic and foreign banks satisfy default rate relationships exhibiting partial adjustment dynamics of the form

where *Z* ∈ {*M*, *C*}, while *f* (*M*) = *H* and *f* (*C*) = *S*. As specified, the deviations of the mortgage or corporate loan default rates from their steady state equilibrium value depend on a weighted average of their past deviations and their attractor deviations, where 0 < *δ* < 1, 0 ≤ *ρ*_{δ} < 1, *ζ ^{δZ,Y}* > 0 and

*ζ*> 0. These attractor deviations are decreasing in the contemporaneous deviations of output from its potential value and the change in the price of housing or equity from its steady state equilibrium value, which affect the probability of default and loss given default, respectively. Deviations from these default rate relationships are captured by mean zero and serially uncorrelated mortgage or corporate loan default shock

^{δZ,V}### G. Market Clearing Conditions

A rational expectations equilibrium in this DSGE model of the world economy consists of state contingent sequences of allocations for the households, developers, firms and banks of all economies that solve their constrained optimization problems given prices and policies, together with state contingent sequences of allocations for the governments of all economies that satisfy their policy rules and constraints given prices, with supporting prices such that all markets clear.

Clearing of the final output good market requires that exports *X*_{i,t} equal production of the domestic final output good less the total demand of domestic households, developers, firms and the government,

where *X*_{i,j,t} = *M*_{j,i,t}. Clearing of the final import good market requires that imports *M*_{i,t} equal the total demand of domestic households, developers, firms and the government:

In equilibrium, combination of these final output and import good market clearing conditions yields output expenditure decomposition,

where the price of investment satisfies *D*_{i,t} = *C*_{i,t} + *I*_{i,t} + *G*_{i,t}.

Clearing of the final bank loan markets requires that mortgage loan supply equals the total demand of domestic developers, that is

where

The derivation of this result equates the principal and interest receipts of the banking sector to the total domestic currency denominated principal and interest payments of domestic and foreign firms.

Let *A*_{i,t+1} denote the net foreign asset position, which equals the sum of the financial wealth of households

where

where *CA*_{i,t} = *ℰ*_{i*,i,t}*A*_{i,t+1} − *ℰ*_{i*,i,t−1}*A*_{i,t} equals the sum of net international investment income and the trade balance

The derivation of this result abstracts from international financial intermediation except via the money markets and imposes restriction

## III. The Empirical Framework

Estimation and inference are based on a linear state space representation of an approximate multivariate linear rational expectations representation of this DSGE model of the world economy, expressed as a function of its potentially heteroskedastic structural shocks. This multivariate linear rational expectations representation is derived by analytically linearizing the equilibrium conditions of the DSGE model around its stationary deterministic steady state equilibrium, and consolidating them by substituting out intermediate variables assuming small capital utilization costs and abstracting from the global terms of trade shifter. The response coefficients of these consolidated approximate linear equilibrium conditions are functions of behavioral parameters that have been restricted to coincide across economies—occasionally within groups sharing a structural characteristic—and economy specific structural characteristics implied by steady state equilibrium relationships. Except where stated otherwise, this steady state equilibrium features zero inflation, productivity and labor force growth, as well as public and national financial wealth.^{2}

In what follows, *x*_{i,t} from its steady state equilibrium value *x*_{i}, while E_{t} *x*_{i,t+s} denotes the rational expectation of variable *x*_{i,t+s} conditional on information available in period *t*. Bilateral weights *x*_{i,t} across the trading partners of economy *i* are based on exports for *Z* = *X*, imports for *Z* = *M*, and their average for *Z* = *T*. In addition, bilateral weights *x*_{i,t} across the lending destinations and borrowing sources of economy *i* are based on bank lending for *Z* = *C* and nonfinancial corporate borrowing for *Z* = *F*. Furthermore, bilateral weights *x*_{i,t} across the investment destinations of economy *i* are based on debt for *Z* = *B* and equity for *Z* = *S*. Finally, world weights *x*_{i,t} across all economies are based on output for *Z* = *Y* and capital market capitalization for *Z* = *A*. Auxiliary parameters *λ*^{Z} are theoretically predicted to equal one, and satisfy *λ* = 0 and *λ*^{Z} > 0.

### A. Endogenous Variables

Core inflation depends on a linear combination of its past and expected future values driven by contemporaneous real unit labor cost according to Phillips curve

which determines the core price level

Output price inflation

Consumption price inflation

Output

Domestic demand

Investment

Public domestic demand

The response coefficients of these relationships vary across economies with the composition of their domestic demand or their trade openness.

Consumption

Reflecting the existence of credit constraints, consumption also depends on contemporaneous, past and expected future credit constrained consumption, where polynomial in the lag operator

where economy specific auxiliary parameters

Residential investment

Residential investment also depends on contemporaneous, past and expected future output and the terms of trade, where polynomial in the lag operator

which determines the shadow price of housing

which determines the rental price of housing

Business investment

Business investment also depends on contemporaneous, past and expected future output and the terms of trade. Reflecting the existence of a financial accelerator mechanism, the relative shadow price of private physical capital depends on its expected future value, as well as the contemporaneous real portfolio return and effective corporate loan rate, according to business investment Euler equation

which determines the shadow price of private physical capital

The private physical capital stock

Exports

Imports

The response coefficients of the former relationship vary across economies with their trade pattern.

The nominal property return

Reflecting the existence of a portfolio balance mechanism, the nominal property return also depends on the contemporaneous housing risk premium. The real property return

The nominal interbank loans rate

The real interbank loans rate *k* = 0 for low interbank market contagion economies, *k* = 1 for medium interbank market contagion economies, and *k* = 2 for high interbank market contagion economies, where

The price of housing

Developer profits

where economy specific auxiliary parameter

The nominal portfolio return

Reflecting the existence of a portfolio balance mechanism, the nominal portfolio return also depends on contemporaneous domestic and foreign duration and equity risk premia. The response coefficients of this relationship vary across economies with their domestic and foreign bond and stock market exposures. The real portfolio return

The nominal short term bond yield

The real short term bond yield *k* = 0 for low capital market contagion economies, *k* = 1 for medium capital market contagion economies, and *k* = 2 for high capital market contagion economies, where

The nominal long term bond yield

The real long term bond yield

The term premium

The duration risk premium *k* = 0 for low capital market contagion economies, *k* = 1 for medium capital market contagion economies, and *k* = 2 for high capital market contagion economies, where

The price of equity

Nonfinancial corporate profits

where economy specific auxiliary parameter *k* = 0 for low capital market contagion economies, *k* = 1 for medium capital market contagion economies, and *k* = 2 for high capital market contagion economies, where

Under a flexible inflation targeting or managed exchange rate regime, the nominal policy interest rate

Under a flexible inflation targeting regime *j* = 0, and the desired nominal policy interest rate responds to expected future consumption price inflation and the contemporaneous output gap. For the leader economy of a monetary union, the target variables entering into its monetary policy rule are expressed as output weighted averages across union members. Under a managed exchange rate regime *j* = 1, and the desired nominal policy interest rate also responds to the contemporaneous change in the nominal effective exchange rate. Under a fixed exchange rate regime, the nominal policy interest rate instead tracks the contemporaneous nominal policy interest rate of the economy that issues the anchor currency one for one, while responding to the contemporaneous change in the corresponding nominal bilateral exchange rate, according to monetary policy rule:

It follows that under a fixed exchange rate regime,

Bank credit depends on a weighted average of the contemporaneous money and bank capital stocks according to bank balance sheet identity

which determines the money stock

Mortgage debt

The nominal effective corporate loan rate

The corporate credit loss rate

The real effective corporate loan rate

The nominal mortgage and corporate loan rates

where *Z* ∈ {*D*,*F*}, while *f* (*D*) = *M* and *f* (*F*) = *C*. The nominal mortgage and corporate loan rates also depend on the past deviation of the bank capital ratio from its required value, as well as the past deviation of the regulatory bank capital ratio requirement from its funding cost, where *g*(*D*) = *M* and *g*(*F*) = *C*,*E*. The real mortgage and corporate loan rates

Bank retained earnings

The shadow price of bank capital

The shadow price of bank capital also depends on the contemporaneous deviation of the bank capital ratio from its required value. The bank capital stock

The regulatory bank capital ratio requirement

The desired regulatory bank capital ratio requirement responds to contemporaneous bank credit growth, as well as to contemporaneous changes in the prices of housing and equity. The regulatory mortgage and corporate loan to value ratio limits

where *Z* ∈ {*D*,*F*}, while *f* (*D*) = *H* and *f* (*F*) = *S*. The desired regulatory mortgage or corporate loan to value ratio limit responds to contemporaneous mortgage or nonfinancial corporate debt growth, as well as to the contemporaneous change in the price of housing or equity, respectively. The mortgage and corporate loan default rates

where *Z* ∈ {*M*, *C*}, while *f* (*M*) = *H* and *f* (*C*) = *S*. The attractor loan default rate depends on the contemporaneous output gap, as well as the contemporaneous change in the price of housing or equity, respectively.

The real effective wage depends on a weighted average of its past and expected future values driven by the contemporaneous and past unemployment rates according to wage Phillips curve

which determines the nominal wage

The unemployment rate depends on its past value driven by contemporaneous employment and the real effective wage according to labor supply relationship

which determines the labor force

Output depends on the contemporaneous utilized private physical capital stock and effective employment according to production function

which determines employment