A. Background

In recent years, the IMF has developed two types of macroeconomic models – dynamic stochastic general equilibrium (DSGE) models and small quarterly projection models (QPMs) – that it has used to analyze economic behavior and to forecast future developments. The DSGE models are based on theoretical underpinnings and have been found to be very useful in analyzing the effects of structural changes in the economy, as well as the effects of longer-term developments such as persistent fiscal deficits and current account deficits.⁴ And multi-country variants of these models have allowed researchers to analyze the effects of shocks in one country on economic variables in other countries. The small quarterly projection models use four or five behavioral equations to characterize the macroeconomic structure of an economy in a way that is both easy to use by modelers and comprehensible to policymakers. They focus on the key macroeconomic variables in the economy – typically output, inflation, a short-term interest rate, and sometimes the exchange rate and/or the unemployment rate. By virtue of their relatively simple and readily understandable structure, they have been used for forecasting and policy analysis purposes in central banks and by country desks in the IMF.⁵ In the past, the parameters of such models have been calibrated on the basis of the knowledge of country experts of the economic structure of the country being studied and that of similar countries.

In the series of papers that we are preparing, three important extensions will be made to the basic model. First, in all the papers in the series, Bayesian techniques are used to estimate the parameters of the model. Bayesian methods allow researchers to input their priors into the model and then to confront them with the data, in order to determine whether their priors are more or less consistent with the data. Although regime shifts in recent years (most notably, the anchoring of inflation expectations to a formal or informal target in many countries) limit the time series to relatively short periods, the approach taken in these papers will be increasingly useful over time as the lengths of usable time series are extended.

Second, in this paper, we are expanding the model to three economies – United States, the Euro area, and Japan. In a future paper, the small quarterly models of a number of countries (United States, the Euro area, Japan, emerging Asia, oil-exporting countries, Canada, Russia, other industrialized, other major oil producers, and the rest of the world) will be integrated into a small quarterly Global Projection Model (GPM), which will enable researchers to analyze the effects on a number of countries of shocks in one country and of global shocks. Moreover, the model will be programmed in such way that researchers will be able to add other countries to the model in a relatively straightforward manner. Such models will give forecasters a new tool to assist them in preparing worldwide forecasts and in carrying out alternative policy simulations in the global context. There is strong demand for such an empirically based multi-country model, both for IMF surveillance work and for helping central bank forecasters to assess the external environment in preparing their projections. Large-scale DSGE models show promise in this regard, but we are years away from developing empirically-based multi-country versions of these models. While global VARs (GVARs) have been developed for forecasting exercises, they are not very useful for policy analysis because they lack the identification restrictions necessary to obtain plausible impulse response functions.

Third, given the importance in recent years of financial-real linkages, we have been experimenting both in the previous paper and in this paper with financial variables that might be helpful in explaining economic developments and in forecasting future movements of the economy. In the earlier paper, we used a single financial variable, a US bank lending tightening measure (BLT), which contributed importantly to the explanation of movements of the US economy over the sample period and also improved out-of-sample forecasting. We continue to use the BLT variable in this paper. In future papers we will broaden the use of financial variables to include a number of financial measures of risk in the United States and other countries.

B. The Specification of The Model

We now consider a small generic open economy model that describes the joint determination of output, unemployment, inflation, a short-term interest rate and the exchange rate for the three economies. The model is fundamentally a gap model, in which the gaps of the variables from their equilibrium values play the crucial role in the functioning of the system. A number of definitions and identities are used to complete the model. We present the model specification for a single country labelled i. The specification for other countries will be very similar, although the priors for the coefficient estimates and for the standard deviations of the structural shocks will differ across economies on the basis of expert knowledge of those economies. In the next section of the paper, we will expand the model to include the US BLT variable.

A. Background

For much of the postwar period, downturns in business cycles were precipitated mainly by increases in interest rates initiated by central banks in response to periods of excess demand that gave rise to inflation pressures. Indeed, in some countries (the United Kingdom being a prominent example), actions of the fiscal and monetary authorities were considered to have brought about a stop-go economy, in which policy switched periodically back and forth between an emphasis on unemployment and economic growth, on the one hand, and an emphasis on inflation, on the other. When the economy was weak, policy eased, giving rise to expansionary pressures. When these pressures were sufficiently strong and inflation became the overriding concern, policy was tightened so that the slowing or contraction of the economy would put downward pressure on inflation and prevent it from getting out of hand.

Such policy-induced slowdowns of the economy persisted from the 1950s into the 1990s, with virtually every downturn preceded by inflationary pressures and a resulting tightening of monetary policy. However, this central bank tightening explanation cannot account for the economic slowdown of the early part of this decade, or of the current slowing of the US and other economies, since inflation pressures and interest rate increases were evidently not the main reason for these downturns. In the context of the apparent linkages between financial developments and the real economy, driven in part through asset price movements, attention has increasingly turned to the ways in which financial developments can affect the real economy. This interest has been aided by the development of theoretical models to describe and explain these linkages, in particular the financial accelerator mechanism.¹³

In our view, the more traditional types of models that allow central bank actions to play a major role in business cycle developments are still needed to explain much of the postwar period. However, the developments over the last decade or so require an extension to those models that have placed central bank actions at the center of the business cycle (and particularly in the downturns). The key factors in these most recent developments, and to a much lesser extent in earlier developments, are the financial developments that have interacted with the real side of the economy in what has come to be called financial-real linkages.

There are many different variants of financial-real linkages. Some refer to developments in financial institutions, while others focus on developments in financial markets. Within the financial institution sector, some relate to the behavior of banks and other financial institutions in dealing with perceptions of the changing risk situation facing their customers or changing attitudes to risk on their own part, while others relate to situations in which banks’ capital positions have deteriorated. In the case of financial markets, there have been cases in which liquidity has seized up and prevented potential borrowers from issuing debt, and other cases in which actual or perceived pressures on the balance sheets of lenders and/or borrowers have been the origin of the inability of the financial markets to carry out their normal intermediation functions.

These different variants can be seen in economic episodes in which the financial-real linkages made themselves felt. In normal times financial accelerators may help to explain movements in spending, but interest rate movements can typically capture most of the effects of the accelerators. It may only be in “abnormal” times that the behavior on the financial side is such as to require special treatment to pick up its effects.

What are the episodes that require special treatment? If one goes back sufficiently far into the past, one can find a number of cases. For example, the Federal Reserve Board’s MPS model included a specification of financial institution behavior relating to Regulation Q in the United States, where the ceiling on deposit rates meant that financial institutions were unable to meet the demand for mortgage loans at times of high market interest rates because they were unable to raise the needed funds, and this resulted in fluctuations in the supply of mortgages and in expenditures on housing. In the case of Canada, in the early years before banks were able to raise funds through the wholesale deposit market, the Bank of Canada operated on “credit conditions”. The ability of banks to extend loans was influenced by the amount of liquid assets on their balance sheet, which they could use to fund loans. The Bank of Canada’s RDX2 model attempted to specify the driving factors behind these credit conditions.

Over the years, with regulatory changes (such as the elimination of regulation Q in the United States) and with the increased sophistication of ways in which banks could raise funds, banks became much less constrained in their ability to extend credit, and macro models focused on the role of interest rates and other factors in determining the demand for credit. But, from time to time, there were episodes in which the normal functioning of financial institutions was disrupted for some reason and the supply of bank credit (not just its demand) became an important factor in private sector expenditures. At the same time, financial markets became increasingly important both in the direct provision of credit to business and in the development of mechanisms (such as the securitization of loans) that facilitated even further the ability of financial institutions to originate the extension of credit to households and businesses. At times, there were also unusual circumstances that affected the functioning of financial markets and influenced consumption and investment expenditures on goods and services.

A listing of some of the episodes in the relatively recent past in which behavior was affected by the supply of funds to households and businesses illustrates the nature of the issue and perhaps the difficulty of finding a simple specification that would cover all the types. In the early 1990s, in the United States, banks became reluctant to provide loans following a period in which they had suffered severe losses that impaired their capital position. Chairman Greenspan referred to this as the “financial headwinds” period and the Federal Reserve responded by reducing interest rates to levels well below what would normally be needed in the circumstances.

A second episode occurred in 1998, when the near failure of LTCM and the Russian debt default caused financial markets to seize up for a short period of time, and prevented borrowers from carrying out their planned funding. For example, there were cases of real estate developers, which had been funding their activities by short-term construction loans from banks, who were unable for a short period of time to transfer these obligations into longer-term mortgages, as the typical purchasers of such mortgages were either unwilling or unable to extend such credit at a time of market difficulties.

A third episode relates to the rise and fall of the subprime mortgage market in the United States and the associated developments in derivative markets that were associated with this market. With the spread of subprime mortgages (fueled in part by unusually low interest rates and the apparent ability of lenders to transfer their exposure to others via asset-backed securities), expenditures on housing expanded. At the same time, housing prices rose sharply, and the new financial environment in which households were able to draw on the increased value of their homes via bank loans collateralized by home equity values allowed households to increase their consumption expenditures. As interest rates returned towards neutral levels, the default of many subprime mortgage borrowers, the freezing of liquidity in the asset-backed commercial paper market, and the associated recognition of the necessity to write down the value of many structured financial instruments (well beyond those based on subprime mortgages) resulted in a situation in which both financial institutions and financial markets faced serious problems. Many financial institutions suffered significant losses (in some cases requiring recapitalization), and in such circumstances their ability and willingness to extend loans was curtailed. And financial markets were faced with considerable uncertainty about the valuation of certain kinds of financial instruments. As well, there were lingering liquidity difficulties in certain segments of the market.

The difficulties in financial markets and financial institutions showed up in the increased risk spread on issues by private sector borrowers in financial markets and in the tightening of terms and conditions on loans extended by financial institutions, as shown in the Federal Reserve’s quarterly survey of senior loan officers. As a result, investment projects had more difficulty being funded than would normally have been the case.

What does this imply for macro modeling? Consider first financial accelerators. As far as financial accelerator models are concerned, there can be an endogenous element in which the business cycle leads to increases and decreases of collateral values and hence to the ability to access funding, and an exogenous element in which exogenous shocks to asset values result in changes in the ability of borrowers to obtain financing. While the former can typically be captured to a considerable extent by interest rate movements, it will be important to try to model the latter. One issue that requires careful attention in structural DSGE models is whether financial institutions ration credit on the basis of collateral values (such as maximum loan-to-value ratios) or simply tighten terms and conditions on the loans that they are prepared to extend. A second type of financial-real linkage relates to the capital position of financial institutions (most importantly banks) and how it affects the willingness of financial institutions to extend loans. A third type of linkage relates to whether financial markets are functioning normally or are facing either liquidity difficulties or problems in evaluating risks. All the episodes that were listed above and the economic behavior patterns underlying them raise the question of whether financial-real linkages should be part of the central macro model or should be modeled via satellite models. Should they feed into the forecast in normal circumstances or only in unusual episodes? And, if the latter, can they be treated as a form of regime shift?

In this and future papers, we will attempt to integrate financial-real linkages into the type of model described earlier.¹⁴ There a number of advantages to using a small model in trying to understand and model the role of the linkages for macro economic behavior. First of all, the insights that have been developed in more complex DSGE and other models can be added to a well understood macro model to see whether they aid in the explanation of macroeconomic developments and forecasting. Second, different measures can be used to see which type of proxy is most helpful in capturing the linkages. Third, the small size of the model allows for experimentation of various types. For example, should a proxy for financial-real linkages be introduced as simply an extra variable in the model that functions continuously or should it only be allowed to affect behavior when it reaches critical threshold levels of the sort that were seen in the episodes discussed above? Fourth, by allowing for persistence in real and financial shocks and in their effects on the real economy, judgmental near-term forecasts of these shocks can play an important role in model-based, medium-term projections through the setting of initial conditions. Fifth, multi-country models with financial-real linkages will allow us to see whether cross-border financial effects have played an important role in transmitting the business cycle internationally, and to assess the relative importance of real linkages and financial linkages in transmitting shocks across countries.

In this paper, we use only one financial variable (over and above interest rates and exchange rates), the bank lending tightening variable for the United States (BLT_US). In future papers, we plan to examine the potential role of BLT variables in other countries and a variety of spread measures, such as bond spreads, swap spreads, and credit default swap spreads.

B. Model Specification Incorporating the US Bank Lending Tightening Variable

The financial variable BLT_US is an unweighted average of the responses to four questions with respect to tightening terms and conditions in the Federal Reserve Board’s quarterly Senior Loan Officer Opinion Survey on Bank Lending Practices. More precisely, for each of four questions on bank credit standards on loan applications,¹⁵ net tightening is equal to the sum of the percentage of banks responding “tightened considerably” and “tightened somewhat” less the sum of the percentage of banks responding “eased somewhat” and “eased considerably”. These net tightening variables are each weighted by one quarter to give the overall BLT variable. It is worth noting that the net tightening responses from the survey outweigh the net easing responses on average over the sample period, indicating a bias of about 5% in the variable. See figure 1 for a comparison of the base case US output gap from the US model without BLT_US (Carabenciov and others, 2008a) with the results of a regression of that output gap on BLT_US five quarters earlier.

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Comparison between Output Gap and BLT indicator

Citation: IMF Working Papers 2008, 279; 10.5089/9781451871371.001.A001

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The model with financial-real linkages makes two substantive changes to the benchmark model set out earlier. In equation 18, BLT_US is a function of ${\overline{\mathit{\text{BLT}}}}_{\mathit{\text{US}}}$, the equilibrium level of BLT_US, which itself is a random walk (equation 19), and a disturbance term, ${\epsilon }_{\mathit{\text{US}},t}^{\mathit{\text{BLT}}}$.¹⁶

As shown in equation 18, banks are assumed to tighten or ease their lending practices in part depending on their view of the expected behavior of the economy 4 quarters ahead. That is, if the output gap is assumed to be positive (a strong economy), there will be a tendency to ease lending conditions, while if it is assumed to be negative (a weak economy), there will be a tendency to tighten lending conditions.

In equation 20, the output gap is explained by the same variables as in the US version of equation 13 (a lead and lag of the output gap, the real interest rate gap, the foreign activity gap and the effective exchange rate gap), as well as by η_US, a distributed lag of ${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$. Thus, if lending conditions are easier than might have been anticipated on the basis of expectations of future economic behavior (positive ${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$), the effect will be a larger output gap and a stronger economy.

The values of the coefficients imposed in equation 21 are intended to reflect a pattern in which an increase of ${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$ (an easing of the bank lending conditions variable) is expected to positively affect spending by firms and households in a hump-shaped fashion, with an initial buildup and then a gradual rundown of the effects.

There are two ways of thinking about the way that the ${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$ variables function in the model. In the first, this proxy variable for financial tightening can be thought of as capturing the exogenous element in bank lending that has the potential to set in motion a weakening or strengthening economic situation. That is, those responsible for bank lending look forward to economic conditions about a year in the future and tighten or loosen in part on the basis of their expectations. If their actions are typical for the stage of the cycle, the interest rate variable itself may pick up the normal tightening and easing of terms and conditions on bank lending, and BLT would play little role in driving future economic developments. If, on the other hand, their actions are greater or less than is typical in light of the expected economic situation, this could have a direct effect on the ability of borrowers to access funds and to make expenditures. A second interpretation puts less emphasis on the direct effects on expenditures of the tightening or easing of bank lending conditions. Rather, from this perspective, one can consider the ${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$ variable as reflecting the views of experts on the lending side of the economy with respect to future economic and financial conditions and thereby functioning as a very useful leading indicator of economic developments.

There are a number of issues surrounding this variable. First, in the interpretation that focuses on the exogenous part of this variable, it is assumed that the part of financial-real linkages that propagates other typical shocks to the system is captured by the interest rate. This is not an unreasonable assumption, since the endogenous part of the financial accelerator mechanism intensifies the effects on the economy of other shocks and, in a macro sense, could be thought of as simply increasing the coefficient on the interest rate variable. Second, there could be an asymmetry between positive and negative shocks to BLT_US. While financial conditions that are tighter than typical will have the effect of preventing liquidity-constrained households and businesses from achieving their desired expenditures, beyond a certain point the easing of financial conditions may be less powerful in leading to increased spending. That is, once there is sufficient collateral to satisfy lenders of the safety of their loans, a further increase in the value of the collateral may not affect their behavior very much.¹⁷ Third, it is possible that small changes in financial conditions will have relatively minor effects, and only changes beyond a certain critical threshold will have the capacity to bring about economically significant changes. Fourth, given the complexity of the financial-real linkages in the economy, BLT_US may not be able to capture all of these types of linkages, and other variables (such as risk spreads) will be introduced into the output gap equations in future papers to try to pick up some of the other effects.

A. Bayesian Estimation

Bayesian estimation provides a middle ground between classical estimation and the calibration of macro models. The use of classical estimation in a situation of a relatively small sample size (which is almost always the case for time series data) often gives model results that are strange, and are inconsistent with the views of macroeconomists as to the functioning of the economy. This problem is accentuated by the simultaneity challenges to macro models, which are not handled well by simultaneous equation methods in small samples. For example, because an aggregate demand shock can lead to persistent inflationary pressures and to central bank actions to raise interest rates to offset the shock, classically estimated models using time series data will sometimes show an increase in interest rates leading to an increase in inflation. This is particularly problematic when the model is to be used for policy simulations, since it may well indicate the need for an interest rate decline to slow the rate of inflation.

Models with calibrated parameters avoid this problem, but are often criticized as representing no more than the modelers’ judgment, which may or may not be consistent with the data. While calibration is typically based on the understanding of experts of the functioning of the economy, the desire to confront the model with the data in a statistical sense has led researchers to use Bayesian estimation techniques to estimate models.

The Bayesian approach has the benefit of putting some weight on the priors of the researchers and some weight on the data over the sample period. By changing the specification of the tightness (e.g., the standard deviation) of the distribution on the priors, the researcher can change the relative weights on the priors and the data in determining the posterior distribution for the parameters. In the limit, a diffuse or noninformative distribution puts more weight on the data while a distribution with a very tight prior distribution (e.g., a small standard deviation) puts more weight on the priors.

There are a number of criteria by which researchers evaluate the success of Bayesian estimated models and decide between models with different weights placed on priors and the data. First, if an estimated model yields coefficients that are close to the priors in spite of allowing considerable weight to be placed on the data, this indicates that the priors are not inconsistent with the data. A second criterion involves seeing whether the impulse response functions (IRFs) from the model estimated with Bayesian techniques are compatible with the views of the researchers (and in the case of models built at central banks with the views of the management of the central bank) with respect to the functioning of the economy in response to shocks. Third, in comparing different variants of a given macro model (for example, one that treats shocks to output as largely demand determined and another that treats shocks as largely supply determined), researchers can use the relative magnitudes of the log data density and root mean squared errors (RMSEs) as indications of which model is more consistent with the data. And, fourth, the plausibility of the variance decomposition of the variables in the model can help to indicate whether the model is sensible.¹⁸

Bayesian estimated models are likely to have better model properties than classically estimated models, but may sometimes not fit the data as well as simple VAR models, since the sole purpose of the latter is to maximize fit. It is the combination of reasonable fit, appropriate structural results from a theoretical perspective, and the ability to give sensible results for policy simulations that gives estimated Bayesian models their strength. Also, the use of such models along with judgmental inputs for the first two quarters of the forecast period is likely to give better and more sensible forecasting results than most other models. A comparison of Bayesian-estimated Global Projection Models with competitor global models will be presented in one of the future papers.

B. Results

B.1 Estimates of coefficients

The model is estimated over the sample period 1994Q1 to 2008Q1. Tables 1 through 3 set out estimation results for the parameters for the three countries in this version of the model, showing the distribution used in the estimation, the prior mean, the prior standard deviation, the posterior mode, and the posterior standard deviation.

Table 1:

Results from posterior maximization [1]

Table 1:

Results from posterior maximization [1]

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
α eu,1	beta	0.750	0.1000	0.7170	0.0685
α ja,1	beta	0.750	0.1000	0.7589	0.1168
α us,1	beta	0.750	0.1000	0.8235	0.0514
α eu,2	gamm	0.300	0.1000	0.1401	0.0279
α ja,2	gamm	0.100	0.0500	0.0599	0.0305
α us,2	gamm	0.300	0.1000	0.1823	0.0260
α eu,3	beta	0.500	0.2000	0.1010	0.0452
α ja,3	beta	0.500	0.2000	0.2214	0.1052
α us,3	beta	0.500	0.2000	0.3649	0.2627
β eu,1	gamm	0.750	0.1000	0.9544	0.0670
β ja,1	gamm	0.750	0.1000	0.9649	0.0598
β us,1	gamm	0.750	0.1000	0.6549	0.0545
β eu,2	beta	0.100	0.0500	0.1478	0.0638
β ja,2	beta	0.100	0.0500	0.0726	0.0406
β us,2	beta	0.100	0.0500	0.0694	0.0410
βeu,3	gamm	0.200	0.0500	0.2009	0.0438
β ja,3	gamm	0.200	0.0500	0.1478	0.0399
βus,3	gamm	0.200	0.0500	0.1866	0.0455
βeu,4	gamm	0.104	0.0400	0.0516	0.0206
β ja,4	gamm	0.090	0.0400	0.0302	0.0157
β us,4	gamm	0.060	0.0200	0.0423	0.0144
β eu,5	gamm	0.052	0.0100	0.0514	0.0099
β ja,5	gamm	0.045	0.0100	0.0398	0.0090
β us,5	gamm	0.030	0.0100	0.0241	0.0085

View Table

Table 1:

Results from posterior maximization [1]

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
α eu,1	beta	0.750	0.1000	0.7170	0.0685
α ja,1	beta	0.750	0.1000	0.7589	0.1168
α us,1	beta	0.750	0.1000	0.8235	0.0514
α eu,2	gamm	0.300	0.1000	0.1401	0.0279
α ja,2	gamm	0.100	0.0500	0.0599	0.0305
α us,2	gamm	0.300	0.1000	0.1823	0.0260
α eu,3	beta	0.500	0.2000	0.1010	0.0452
α ja,3	beta	0.500	0.2000	0.2214	0.1052
α us,3	beta	0.500	0.2000	0.3649	0.2627
β eu,1	gamm	0.750	0.1000	0.9544	0.0670
β ja,1	gamm	0.750	0.1000	0.9649	0.0598
β us,1	gamm	0.750	0.1000	0.6549	0.0545
β eu,2	beta	0.100	0.0500	0.1478	0.0638
β ja,2	beta	0.100	0.0500	0.0726	0.0406
β us,2	beta	0.100	0.0500	0.0694	0.0410
βeu,3	gamm	0.200	0.0500	0.2009	0.0438
β ja,3	gamm	0.200	0.0500	0.1478	0.0399
βus,3	gamm	0.200	0.0500	0.1866	0.0455
βeu,4	gamm	0.104	0.0400	0.0516	0.0206
β ja,4	gamm	0.090	0.0400	0.0302	0.0157
β us,4	gamm	0.060	0.0200	0.0423	0.0144
β eu,5	gamm	0.052	0.0100	0.0514	0.0099
β ja,5	gamm	0.045	0.0100	0.0398	0.0090
β us,5	gamm	0.030	0.0100	0.0241	0.0085

Table 2:

Results from posterior maximization [2]

Table 2:

Results from posterior maximization [2]

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
γ eu,1	beta	0.500	0.0500	0.6859	0.0359
γ ja,1	beta	0.500	0.0500	0.7497	0.0356
γ us,1	beta	0.500	0.0500	0.7107	0.0350
γ eu,2	gamm	1.500	0.2000	1.3061	0.1658
γ ja,2	gamm	1.500	0.2000	1.0579	0.1424
γus,2	gamm	1.500	0.3000	0.9104	0.1884
γ eu,4	gamm	0.200	0.0500	0.2012	0.0511
γ ja,4	gamm	0.200	0.0500	0.1693	0.0433
γ us,4	gamm	0.200	0.0500	0.2052	0.0522
$g_{\mathit{\text{eu}}}^{\overline{Y}\mathit{\text{ss}}}$	norm	2.000	0.0500	2.2614	0.3189
$g_{\mathit{\text{ja}}}^{\overline{Y}\mathit{\text{ss}}}$	norm	1.700	0.0500	1.4444	0.3053
$g_{\mathit{\text{US}}}^{\overline{Y}\mathit{\text{ss}}}$	norm	2.200	0.0500	2.2729	0.2451
k us	gamm	20.000	0.5000	20.0773	0.4957
λeu,1	beta	0.500	0.1000	0.7867	0.0704
λ ja,1	beta	0.500	0.1000	0.8967	0.0421
λ us,1	beta	0.500	0.1000	0.8480	0.0551
λ eu,2	gamm	0.250	0.0500	0.2223	0.0407
λ ja,2	gamm	0.250	0.0500	0.1836	0.0405
λ us,2	gamm	0.200	0.0500	0.1801	0.0381
λ eu,3	gamm	0.208	0.0500	0.1061	0.0233
λ ja,3	gamm	0.180	0.0500	0.0756	0.0203
λ us,3	gamm	0.120	0.0500	0.0707	0.0264

View Table

Table 2:

Results from posterior maximization [2]

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
γ eu,1	beta	0.500	0.0500	0.6859	0.0359
γ ja,1	beta	0.500	0.0500	0.7497	0.0356
γ us,1	beta	0.500	0.0500	0.7107	0.0350
γ eu,2	gamm	1.500	0.2000	1.3061	0.1658
γ ja,2	gamm	1.500	0.2000	1.0579	0.1424
γus,2	gamm	1.500	0.3000	0.9104	0.1884
γ eu,4	gamm	0.200	0.0500	0.2012	0.0511
γ ja,4	gamm	0.200	0.0500	0.1693	0.0433
γ us,4	gamm	0.200	0.0500	0.2052	0.0522
$g_{\mathit{\text{eu}}}^{\overline{Y}\mathit{\text{ss}}}$	norm	2.000	0.0500	2.2614	0.3189
$g_{\mathit{\text{ja}}}^{\overline{Y}\mathit{\text{ss}}}$	norm	1.700	0.0500	1.4444	0.3053
$g_{\mathit{\text{US}}}^{\overline{Y}\mathit{\text{ss}}}$	norm	2.200	0.0500	2.2729	0.2451
k us	gamm	20.000	0.5000	20.0773	0.4957
λeu,1	beta	0.500	0.1000	0.7867	0.0704
λ ja,1	beta	0.500	0.1000	0.8967	0.0421
λ us,1	beta	0.500	0.1000	0.8480	0.0551
λ eu,2	gamm	0.250	0.0500	0.2223	0.0407
λ ja,2	gamm	0.250	0.0500	0.1836	0.0405
λ us,2	gamm	0.200	0.0500	0.1801	0.0381
λ eu,3	gamm	0.208	0.0500	0.1061	0.0233
λ ja,3	gamm	0.180	0.0500	0.0756	0.0203
λ us,3	gamm	0.120	0.0500	0.0707	0.0264

Table 3:

Results from posterior maximization [3]

Table 3:

Results from posterior maximization [3]

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
ϕ eu	beta	0.500	0.2000	0.8342	0.0670
ϕ ja	beta	0.500	0.2000	0.8562	0.0677
ρ eu	beta	0.500	0.1000	0.4673	0.1189
ρ ja	beta	0.500	0.1000	0.4968	0.1072
ρ us	beta	0.500	0.1000	0.2901	0.0667
${\overline{\mathit{\text{rr}}}}_{\mathit{\text{eu}}}$	norm	2.000	0.3000	1.9837	0.1904
${\overline{\mathit{\text{rr}}}}_{\mathit{\text{ja}}}$	norm	2.000	0.3000	1.3791	0.2546
${\overline{\mathit{\text{rr}}}}_{\mathit{\text{US}}}$	norm	2.000	0.3000	1.7285	0.2465
τ eu	beta	0.050	0.0300	0.0289	0.0185
τ ja	beta	0.050	0.0300	0.0375	0.0254
τ us	beta	0.050	0.0300	0.0274	0.0160
θ us	gamm	1.000	0.5000	1.0708	0.5567

View Table

Table 3:

Results from posterior maximization [3]

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
ϕ eu	beta	0.500	0.2000	0.8342	0.0670
ϕ ja	beta	0.500	0.2000	0.8562	0.0677
ρ eu	beta	0.500	0.1000	0.4673	0.1189
ρ ja	beta	0.500	0.1000	0.4968	0.1072
ρ us	beta	0.500	0.1000	0.2901	0.0667
${\overline{\mathit{\text{rr}}}}_{\mathit{\text{eu}}}$	norm	2.000	0.3000	1.9837	0.1904
${\overline{\mathit{\text{rr}}}}_{\mathit{\text{ja}}}$	norm	2.000	0.3000	1.3791	0.2546
${\overline{\mathit{\text{rr}}}}_{\mathit{\text{US}}}$	norm	2.000	0.3000	1.7285	0.2465
τ eu	beta	0.050	0.0300	0.0289	0.0185
τ ja	beta	0.050	0.0300	0.0375	0.0254
τ us	beta	0.050	0.0300	0.0274	0.0160
θ us	gamm	1.000	0.5000	1.0708	0.5567

Beginning with the output gap equations, we find that while all three economies have much more weight on the backward-looking component, β₁, than on the forward-looking component, β₂, the euro area economy has considerably more weight on the backward-looking component than do the other two economies. Also, the sum of the coefficients on the backward-looking component and the forward-looking component is appreciably lower in the US economy than in the other two economies. The Japanese economy has a somewhat weaker coefficient on the real interest rate gap, β₃, than the other two economies. The posterior estimates of the coefficient on the real exchange rate gap, β₄, are about the same for all three economies, a very different outcome than would have been expected on the basis of the prior estimates. In contrast, the coefficients on the activity variable, β₅, are roughly in line with the priors, with the US coefficient being smaller than the Japanese coefficient, which in turn is smaller than that of the euro area. This result is consistent with the relative openness of the respective economies. The response of US output gap to unexpected tightening in BLT, θ_us, is a little higher than anticipated.

Turning to the inflation equations, we see that all three economies are more forward looking than assumed by the priors, λ₁, while the effect of the output gap on inflation in each country, λ₂, is somewhat smaller than assumed by the priors. And the effect of exchange rate changes on inflation, λ₃, is much smaller than anticipated in all three economies.

In the interest rate reaction functions, the smoothing coefficient, γ₁, is larger than anticipated in all three economies. The response to deviations of inflation from target, γ₂, is lower than anticipated in all three economies.¹⁹ The coefficients on the output gap, γ₄, are about as expected for all three economies.

There are a number of other results of interest. In all three economies, the response of the unemployment gap to the output gap, or the Okun coefficient, α₂, is much smaller than anticipated. Also, in all three economies, the persistence of growth in the NAIRU to shocks, (1- α₃), is greater than anticipated. This is also the case for persistence of growth in potential output to shocks, (1- τ). Indeed, in all three economies the latter coefficient is close to unity. Similarly, the persistence of the equilibrium real interest rate to shocks, (1- ρ), is much higher than anticipated in the United States, although not in the other two economies. Finally, the expected real exchange rate is very forward looking, ϕ, for both the euro and the yen.

B.2 Estimates of standard deviation of structural shocks and cross correlations

Table 4 presents the same information for the standard deviation of the structural shocks, making use of more diffuse priors. Table 5 does the same for the cross correlations in the model. Given that the priors for the estimates of the standard deviation of the structural shocks and for the cross correlations are held with much less confidence than the priors for the coefficients, it is not surprising that the posterior results differ more from the priors than was the case for the coefficients. Particularly worth noting are the much higher posteriors than expected for the standard deviations of the BLT_US shock, the equilibrium real exchange rate shocks, the inflation shocks for all three economies, and the shock to equilibrium real interest rates in the United States. There were also a number of much lower than expected posteriors for the standard deviations of shocks, including the shocks in the two UIP equations, and the shocks to the equations for the unemployment rate gap, NAIRU, and NAIRU growth in all three economies. These results indicate that, compared to prior expectations, there is considerably less uncertainty about the shock terms pertaining to unemployment and UIP, and considerably more about the shock terms pertaining to real exchange rates and inflation.

Table 4:

Results from posterior parameters

(standard deviation of structural shocks)

Table 4:

Results from posterior parameters

(standard deviation of structural shocks)

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
${\epsilon }_{\mathit{\text{US}}}^{\overline{\mathit{\text{BLT}}}}$	invg	0.200	Inf	0.0922	0.0377
${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$	invg	0.400	Inf	0.9487	0.4592
${\epsilon }_{\mathit{\text{eu}}}^{g^{\overline{Y}}}$	invg	0.100	0.0500	0.1142	0.0351
${\epsilon }_{\mathit{\text{ja}}}^{g^{\overline{Y}}}$	invg	0.100	0.0500	0.0714	0.0217
${\epsilon }_{\mathit{\text{US}}}^{g^{\overline{Y}}}$	invg	0.100	Inf	0.2931	0.0672
${\epsilon }_{\mathit{\text{eu}}}^{\overline{Y}}$	invg	0.200	0.0500	0.1821	0.0342
${\epsilon }_{\mathit{\text{ja}}}^{\overline{Y}}$	invg	0.200	0.0500	0.6033	0.0736
${\epsilon }_{\mathit{\text{US}}}^{\overline{Y}}$	invg	0.050	0.0500	0.0291	0.0112
${\epsilon }_{\mathit{\text{eu}}}^{\overline{Z}}$	invg	1.000	Inf	4.3529	0.5711
${\epsilon }_{\mathit{\text{ja}}}^{\overline{Z}}$	invg	4.000	Inf	5.7984	0.8056
${\epsilon }_{\mathit{\text{eu}}}^{\pi }$	invg	0.500	Inf	1.0736	0.1222
${\epsilon }_{\mathit{\text{ja}}}^{\pi }$	invg	1.000	Inf	1.3525	0.1418
${\epsilon }_{\mathit{\text{US}}}^{\pi }$	invg	0.700	Inf	1.2532	0.1212
${\epsilon }_{\mathit{\text{eu}}}^{\overline{R}}$	invg	0.200	0.0400	0.1885	0.0355
${\epsilon }_{\mathit{\text{ja}}}^{\overline{R}}$	invg	0.100	0.0400	0.0795	0.0232
${\epsilon }_{\mathit{\text{US}}}^{\overline{R}}$	invg	0.200	Inf	0.9454	0.1599
${\epsilon }^{R_{\mathit{\text{eu}}}-R_{\mathit{\text{US}}}}$	invg	1.000	Inf	0.4616	0.1892
${\epsilon }^{R_{\mathit{\text{ja}}}-R_{\mathit{\text{US}}}}$	invg	0.500	Inf	0.2303	0.0939
${\epsilon }_{\mathit{\text{eu}}}^{\mathit{\text{rs}}}$	invg	0.250	Inf	0.2514	0.0325
${\epsilon }_{\mathit{\text{ja}}}^{\mathit{\text{rs}}}$	invg	0.250	Inf	0.2336	0.0341
${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{rs}}}$	invg	0.700	Inf	0.2408	0.0439
${\epsilon }_{\mathit{\text{eu}}}^{\overline{U}}$	invg	0.100	Inf	0.0304	0.0065
${\epsilon }_{\mathit{\text{ja}}}^{\overline{U}}$	invg	0.100	Inf	0.0533	0.0323
${\epsilon }_{\mathit{\text{US}}}^{\overline{U}}$	invg	0.100	Inf	0.0475	0.0203
${\epsilon }_{\mathit{\text{eu}}}^{g^{\overline{U}}}$	invg	0.100	Inf	0.0398	0.0074
${\epsilon }_{\mathit{\text{ja}}}^{g^{\overline{U}}}$	invg	0.100	Inf	0.0516	0.0148
${\epsilon }_{\mathit{\text{US}}}^{g^{\overline{U}}}$	invg	0.100	Inf	0.0472	0.0175
${\epsilon }_{\mathit{\text{eu}}}^u$	invg	0.200	Inf	0.0433	0.0061
${\epsilon }_{\mathit{\text{ja}}}^u$	invg	0.100	Inf	0.0724	0.0230
${\epsilon }_{\mathit{\text{US}}}^u$	invg	0.200	Inf	0.0967	0.0157
${\epsilon }_{\mathit{\text{eu}}}^y$	invg	0.300	0.0500	0.2591	0.0324
${\epsilon }_{\mathit{\text{ja}}}^y$	invg	0.500	0.1000	0.3923	0.0569
${\epsilon }_{\mathit{\text{US}}}^y$	invg	0.250	Inf	0.3525	0.0410

View Table

Table 4:

Results from posterior parameters

(standard deviation of structural shocks)

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
${\epsilon }_{\mathit{\text{US}}}^{\overline{\mathit{\text{BLT}}}}$	invg	0.200	Inf	0.0922	0.0377
${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$	invg	0.400	Inf	0.9487	0.4592
${\epsilon }_{\mathit{\text{eu}}}^{g^{\overline{Y}}}$	invg	0.100	0.0500	0.1142	0.0351
${\epsilon }_{\mathit{\text{ja}}}^{g^{\overline{Y}}}$	invg	0.100	0.0500	0.0714	0.0217
${\epsilon }_{\mathit{\text{US}}}^{g^{\overline{Y}}}$	invg	0.100	Inf	0.2931	0.0672
${\epsilon }_{\mathit{\text{eu}}}^{\overline{Y}}$	invg	0.200	0.0500	0.1821	0.0342
${\epsilon }_{\mathit{\text{ja}}}^{\overline{Y}}$	invg	0.200	0.0500	0.6033	0.0736
${\epsilon }_{\mathit{\text{US}}}^{\overline{Y}}$	invg	0.050	0.0500	0.0291	0.0112
${\epsilon }_{\mathit{\text{eu}}}^{\overline{Z}}$	invg	1.000	Inf	4.3529	0.5711
${\epsilon }_{\mathit{\text{ja}}}^{\overline{Z}}$	invg	4.000	Inf	5.7984	0.8056
${\epsilon }_{\mathit{\text{eu}}}^{\pi }$	invg	0.500	Inf	1.0736	0.1222
${\epsilon }_{\mathit{\text{ja}}}^{\pi }$	invg	1.000	Inf	1.3525	0.1418
${\epsilon }_{\mathit{\text{US}}}^{\pi }$	invg	0.700	Inf	1.2532	0.1212
${\epsilon }_{\mathit{\text{eu}}}^{\overline{R}}$	invg	0.200	0.0400	0.1885	0.0355
${\epsilon }_{\mathit{\text{ja}}}^{\overline{R}}$	invg	0.100	0.0400	0.0795	0.0232
${\epsilon }_{\mathit{\text{US}}}^{\overline{R}}$	invg	0.200	Inf	0.9454	0.1599
${\epsilon }^{R_{\mathit{\text{eu}}}-R_{\mathit{\text{US}}}}$	invg	1.000	Inf	0.4616	0.1892
${\epsilon }^{R_{\mathit{\text{ja}}}-R_{\mathit{\text{US}}}}$	invg	0.500	Inf	0.2303	0.0939
${\epsilon }_{\mathit{\text{eu}}}^{\mathit{\text{rs}}}$	invg	0.250	Inf	0.2514	0.0325
${\epsilon }_{\mathit{\text{ja}}}^{\mathit{\text{rs}}}$	invg	0.250	Inf	0.2336	0.0341
${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{rs}}}$	invg	0.700	Inf	0.2408	0.0439
${\epsilon }_{\mathit{\text{eu}}}^{\overline{U}}$	invg	0.100	Inf	0.0304	0.0065
${\epsilon }_{\mathit{\text{ja}}}^{\overline{U}}$	invg	0.100	Inf	0.0533	0.0323
${\epsilon }_{\mathit{\text{US}}}^{\overline{U}}$	invg	0.100	Inf	0.0475	0.0203
${\epsilon }_{\mathit{\text{eu}}}^{g^{\overline{U}}}$	invg	0.100	Inf	0.0398	0.0074
${\epsilon }_{\mathit{\text{ja}}}^{g^{\overline{U}}}$	invg	0.100	Inf	0.0516	0.0148
${\epsilon }_{\mathit{\text{US}}}^{g^{\overline{U}}}$	invg	0.100	Inf	0.0472	0.0175
${\epsilon }_{\mathit{\text{eu}}}^u$	invg	0.200	Inf	0.0433	0.0061
${\epsilon }_{\mathit{\text{ja}}}^u$	invg	0.100	Inf	0.0724	0.0230
${\epsilon }_{\mathit{\text{US}}}^u$	invg	0.200	Inf	0.0967	0.0157
${\epsilon }_{\mathit{\text{eu}}}^y$	invg	0.300	0.0500	0.2591	0.0324
${\epsilon }_{\mathit{\text{ja}}}^y$	invg	0.500	0.1000	0.3923	0.0569
${\epsilon }_{\mathit{\text{US}}}^y$	invg	0.250	Inf	0.3525	0.0410

Table 5:

Results from posterior parameters

(correlation of structural shocks)

Table 5:

Results from posterior parameters

(correlation of structural shocks)

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
${\epsilon }_{\mathit{\text{US}}}^{\overline{Y}}$,${\epsilon }_{\mathit{\text{US}}}^{\pi }$	beta	0.100	0.0300	0.0915	0.0292
${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$,${\epsilon }_{\mathit{\text{US}}}^{g^{\overline{Y}}}$	beta	0.650	0.0500	0.8759	0.0563
${\epsilon }_{\mathit{\text{eu}}}^{\overline{Y}}$,${\epsilon }_{\mathit{\text{eu}}}^{\pi }$	beta	0.100	0.0300	0.0961	0.0304
${\epsilon }_{\mathit{\text{eu}}}^y$,${\epsilon }_{\mathit{\text{eu}}}^{g^{\overline{Y}}}$	beta	0.250	0.1000	0.2476	0.1135
${\epsilon }_{\mathit{\text{ja}}}^{\overline{Y}}$,${\epsilon }_{\mathit{\text{ja}}}^{\pi }$	beta	0.100	0.0300	0.0917	0.0287
${\epsilon }_{\mathit{\text{ja}}}^y$,${\epsilon }_{\mathit{\text{ja}}}^{g^{\overline{Y}}}$	beta	0.250	0.1000	0.2156	0.1026

View Table

Table 5:

Results from posterior parameters

(correlation of structural shocks)

	Prior distribution	Prior mean	Prior s.d.	Posterior mode	s.d.
${\epsilon }_{\mathit{\text{US}}}^{\overline{Y}}$,${\epsilon }_{\mathit{\text{US}}}^{\pi }$	beta	0.100	0.0300	0.0915	0.0292
${\epsilon }_{\mathit{\text{US}}}^{\mathit{\text{BLT}}}$,${\epsilon }_{\mathit{\text{US}}}^{g^{\overline{Y}}}$	beta	0.650	0.0500	0.8759	0.0563
${\epsilon }_{\mathit{\text{eu}}}^{\overline{Y}}$,${\epsilon }_{\mathit{\text{eu}}}^{\pi }$	beta	0.100	0.0300	0.0961	0.0304
${\epsilon }_{\mathit{\text{eu}}}^y$,${\epsilon }_{\mathit{\text{eu}}}^{g^{\overline{Y}}}$	beta	0.250	0.1000	0.2476	0.1135
${\epsilon }_{\mathit{\text{ja}}}^{\overline{Y}}$,${\epsilon }_{\mathit{\text{ja}}}^{\pi }$	beta	0.100	0.0300	0.0917	0.0287
${\epsilon }_{\mathit{\text{ja}}}^y$,${\epsilon }_{\mathit{\text{ja}}}^{g^{\overline{Y}}}$	beta	0.250	0.1000	0.2156	0.1026

As shown in Table 5, the posterior results for the three types of cross correlations are mostly in line with their priors.