Financial Stress, Regime Switching and Macrodynamics: Theory and Empirics for the US, EU and Non-EU Countries

Abstract Over-borrowing and financial stress has recently become an important issue in macroeconomic and policy discussions in the US as well as in the EU. In this paper we study two regimes of financial stress. In a regime of high financial stress, stress shocks can have large and persistent impacts on the real side of the economy whereas in regimes of low stress, shocks can easily dissipate having no lasting effects. In order to study the macroeconomic dynamics, with alternative paths resulting from financial stress shocks, we introduce a macromodel with a finance-macro link which uses multiperiod decisions framework of economic agents. The agents can, in a finite horizon context, borrow and accumulate assets where however the above two scenarios may occur. The model is solved through nonlinear model predictive control (NMPC). Empirically then we use a Multi-Regime VAR (MRVAR) to study the impact of financial stress shocks on the macroeconomy in a large number of countries.


Introduction
The issues of over-borrowing and nancial stress has recently become a concern in many countries. In the US the over-borrowing of households, nancial rms and banks, and the government (States as well as Federal State) is at the forefront of the economic policy debate. In the EU it is the sovereign debt problem and the insolvency threat of the banking system in the periphery countries (Greece, Italy, Spain, Portugal) that threatens to become a cause of new nancial meltdown. Over-borrowing and unsustainable private and government debt appear to be major predictors of a nancial crises and prolonged nancial stress also in other countries, with severe impact on the macroeconomy of the country possibly with strong spillover eects to other countries.
As the empirics shows there is sometimes high leveraging, yet little nancial stress, and positive shocks to stress do not aect the macroeconomy signicantly. On the other hand, there is a nancial shock that leads to considerable economic contractions. We want to study why the economies sometimes stabilize at a prolonged boom period with little eects from nancial stress shocks, and sometimes, after nancial shocks, tend to move toward low growth or negative growth rates. In order to study the debt dynamics, with alternative paths resulting from nancial stress shocks, we introduce a nance-macro link which uses a multi-period decision framework of economic agents. The agents can, in a nite horizon context, borrow and accumulate assets where however two scenarios may occur.
In a rst scenario, there is borrowing, but there is some path with considerable growth, and a stabilization at higher or moderate growth rates can occur.
In spite of signicant borrowing, interest rates remain relatively stable, or are only slightly rising due to higher leveraging and higher risk premia. Yet, slightly rising interest rates and credit spreads do not generate macro economic feedback mechanisms to produce strong contractions and positive nancial stress shocks do not matter signicantly.
On the other hand instability can arise in the sense that shocks can be amplifying, of the sort that macro feedback mechanisms generate also real downswings, for example through rising risk premia, credit constraints or extensive eorts of deleveraging by borrowers. High leveraging of economic agents, as well as factors creating mild nonlinearities, can drive up risk premia and credit spreads. Credit helps to create booms, but rising credit spreads can be destabilizing and create busts. In fact, often credit spreads by itself do not necessarily create macroeconomic instabilities, but once there is a signicant impact on consumption and investment decisions this can result in a further decline in economic activities. This is likely to generate a vicious cycle, and the economy becomes unstable downward and may move to a low level economic activities.
In order to explore those two scenarios, we employ some model versions of the Blanchard-Fischer type (1989, ch.2) as it was designed for an open economy which allows, however, many agents in the economy to pursue debt nance.
In principle the excess of absorption over production in an economy means borrowing if not domestically then from abroad. So in our model households, there is a danger of such a self-enforcing mechanism where EU members can end up in a bad equilibrium. Those mechanisms are working for countries in the EU currency union, but may work less for stand-alone countries, for example US, UK and Japan. For countries in a lose currency union one might see such a mechanism.
Yet, instead of using a self-enforcing mechanism generated through the expectation dynamics one can also show that countries may face a vicious cycle, through non-linearities which eliminate the usual automatic stabilizers and multiplier eects. We want to show a real mechanism that also can create such a downward pushing force and can prevent recoveries from taking their path. A new perspective is taken here in the paper that allows for intertemporal behavior of agents within one regime, but simultaneously admits severe contractions and regime changes, though the agents intertemporally optimize. We want to explore the debt dynamics using two dierent model variants representing two dierent regimes. In the rst model variant we keep the interest rate on borrowing constant or it is only slightly rising reacting to leveraging. Then we relax this assumption which leads us to another regime and dynamics.
The forces we discuss here, that can bring about instability, are basically working through a positive feedback eect of nancial market and output. Rising borrowing from capital markets, bond issuing, and credit spreads, caused by previous excess leveraging, and other factors, can cause aggregate demand to fall. When aggregate demand falls utilization of capacity and thus capital utilization rates fall, and the lower income generates a lower surplus to payo future liabilities, which in turn creates greater credit spreads, lower aggregate demand and so on. In our case there are dominantly real forces not expectational forces that accelerate downturns possibly creating lock-ins into a bad equilibrium. They are basically working as positive feedback mechanism 3 between credit spread and capital utilization. This is likely to occur if there are vulnerabilities developed beforehand, that can trigger severe downturn through that mechanism.
Our model is similar to Hall (2010Hall ( , 2011, Gilchrist and Zabrajsek (2011), and Mittnik and Semmler (2011, 2012) and allows not only the credit-macro feedback mechanisms in an multi-period model, but admits also to study the contractionary eect of deleveraging as discussed in Eggertson and Krugman (2011) and Kraine (2012).
After the presentation of a model variant of low and high stress regimes, we employ a multi-regime VAR (MRVAR) to demonstrate such dynamics in the data. We use a IMF Financial Instability index (FSI) recently provided for many countries that allow us to study the empirics of the impact of nancial stress on the macrodynamics. We measure the real activities by a monthly industrial production index of the dierent countries.
As to the solution method, our model will not be solved locally through local linearization about the steady state, as used in DYNARE, or globally in an innite horizon model by Dynamic Programming, as in Ernst and Semmler (2010), but we will solve the model by Nonlinear Model Predictive Control (NMPC), which has recently been developed by Gruene

Regime of Low Financial Stress
So we presume that we are in a nite time regime with low nancial stress.
The following model might hold: In equ. (1) there is preference over log utility. The policy variables are consumption, and growth rate of capital stock, c t , g t . # But note we have a model of nite time, operating in a regime of low nancial stress. N will be the time horizon for the N −period receding horizon.
Equ. (2) represents the capital stock that increases due to investment but declines through the depreciation rate δ. There could be a stochastic shock occurring along the path, represented by the second term in equ. (2). This is the only stochastic shock we will consider. The equ (3) represents the dynamics of aggregate debt (households, rms, sovereign). $ Our debt dynamics is written here in a way which is standard if one allows for borrowing from abroad, see Blanchard and Fischer (1989, ch. 2). The interest payment on debt, rb t , increases debt but the surplus(y t − c t − i t − φ(g t k t )) negative excess absorption decreases debt. We have i t = g t k t .
Note that since consumption and investment are separate policy variables we allow here for external borrowing. Moreover, φ(g t k t ) is the adjustment cost for investment. Overall the model has two decision variables and two state variables. We presume here a quadratic adjustment cost of investment.
Assuming here r = 0 : 04, δ = 0.07 and quadratic adjustment cost of investment, we obtain the following solutions using NMPC, setting the shock equal to zero. The numerical results are shown in gure 1.
The vertical axis shows the debt to capital stock ratio and the horizontal the capital stock. Here the paths are shown for dierent initial conditions. The upper end of the two paths represents the steady state which is unique where both the trajectories end up.
Next we can also let the credit spread risk moderately rise with leveraging.
The credit spread (measured against a risk free interest rate) is made endogenous.
# Actually in the numerics we can take c = C/k, so that the two choice variables can be conned to reasonable constraints between 0 and 1.
The dierence to the model (1)-(3) above is now that we assume that the credit spread maybe a nonlinear function of the debt to capital stock ratio. Our idea here is similar to Roch and Uhlig (2012) who have an on-o scenario: With a high probability of default bond prices are low and yields are high, and the reverse holds for a low probability of default. We smooth the on-o cases, and introduce a continuum of cases where the probability of default may steadily rise starting from a low level, then rising faster, and then leveling o.
Thus we want to let the credit spread rise with the debt to capital stock ratio, rst slowly, then more rapidly, but it will nally be bounded. We use an arctan function, represented by r(b t /k t ), which gives us those properties: This is the function that has been used in Chiarella  % In de Grauwe representing there EU debt and bond yield data. 6 One would expect that with lower credit spreads, a lower steady state leveraging ratio is admissible. Again, debt is sustainable if the second term in equ.
(6), the surplus, is equal to the rst term, the interest payments on debt. As gure 2 shows the higher interest payments admits a higher steady state leveraging. Again, debt is sustainable if the second term in equ. (6), the surplus, is equal to the rst term, the interest payments on debt. Actually in this version of a low nancial stress and no macro feedback eects from slightly rising credit spreads, even a higher steady state debt is admissible and nancial stress shocks would not be destabilizing.
Thus, in a low stress regime, with little feedback eect to consumption, investment and thus on demand and output, a positive nancial stress shock would do little harm one would expect a quick mean reversion, and not much lasting eects on output.

Regime of High Financial Stress
Now let us presume that we are in a regime of high nancial stress, maybe with high leveraging but other factors also contributing to nancial stress, see below.
It is again a model in nite time so we are in a receding decision horizon of N -periods. We now not only allow for credit spreads to be endogenous, but also for a feedback eect of leveraging on demand and output.
The dierence to the rst model variant above is here now that the credit spread maybe a nonlinear function of the leveraging, as before, but there is also an endogenous eect of this on demand, output and income. Thus the major dierence to the rst model variant is that the second variant has built in an endogenous utilization of capacity and thus has endogenized both credit spread and output. This is an important macroeconomic feedback mechanism that one sometimes can observe in a regime of high nancial stress, see Hall (2010Hall ( , 2011.
We can make the actual consumption and investment demand depending on rising interest rates, triggered by rising risky yields on bonds and rising credit spreads. This would aect consumption and investment demand in the following way: with the derivatives df d(r(b/k)) < 0 and dg d(r(b/k)) < 0. Though optimal consumption and investment plans might be targeted, actual consumption and investment decline due to rising risk premia and credit spreads. The cost of loans if available at all is rising. So, overall we may have : where again du d(r(b/k)) < 0. We take (14) and can use the rising credit spread as self-enforcing mechanism reducing demand, output and capacity utilization. We can write: Now if risk premia and credit spread might rise, but is bounded, y a will decline due to higher credit cost, and thus we have lower consumption and investment demand and consequently capacity utilization falls.

'
We expect here, starting with a leveraging roughly above normal, that the feedback mechanisms of higher yields, higher credit spreads and lower output may lead to a contraction of utilization of capital stock, and possibly capital stock itself, and to a rapidly increasing debt to capital stock ratio.
The debt dynamics with endogenous credit spread and endogenous output and surplus of system (7) -(9) is shown in gure 3, using NMPC.

Financial Stress and Output Measures
Given our model variants of low and high nancial stress it is an important empirical issue to identify the high and the low nancial stress regimes. What measures can one utilize to conduct empirical estimates? One issue is to measure the nancial stress, the other is to track the interconnection of nancial stress and output. This means that there is likely to be a high stress accompanying low output and high output accompanying a low stress regime. Yet, the eects of shocks of one aecting the other may be asymmetric with respect to regimes.
Our model variants above may suggest to take leverage ratios of economic agents to measure nancial stress. So high leverage implying high nancial stress and low leverage the reverse. However, there is an issue whether the ratio of net worth to capital assets, or the reverse measure, can be used as good measure of nancial stress. This measure is greatly aected by the market valuation of assets as well as liabilities. In particular, asset valuation is heavily impacted by the condence and estimate of income streams the asset generates, as well as See Mittnik and Semmler (2012) presumed discount rates, and the liabilities such as credit instruments, short and long term loans, are strongly impacted by their corresponding risk premia.
Moreover, credit constraints, for example, as measured by the Fed index of changes in credit standards to determine the ease and tightness of obtaining credit as well as default premia and credit spreads and short term liquidity, are also important nancial stress factors for economic agents. All this will aect credit demand and supply from the nancial market and nancial intermediaries.
We thus need more extensive measures than only leverage to evaluate nancial stress.
We therefore propose to measure nancial stress empirically by taking the IMF (2011) nancial stress index, the FSI. Note that the FSI has the following components: The FSI is available for a large number of EU countries. apparent linkages between the FSI and economic activity, we would also expect a strong linkage between net worth, or leveraging, of nancial intermediaries and economic activity, since the nancial stress is aecting the balance sheets of nancial intermediaries. 10 A one-regime VAR has been used frequently to study the nancial-macro link, using the nancial accelerator. 11 Yet those one-regime VAR studies presume only local behavior of the variables, symmetry eects of shocks and mean reversion after the shocks. What we will pursue here is an MRVAR. Our MR-VAR analysis 12 takes the IMF FSI as empirical measure of nancial stress, and the growth rate of the monthly production indexthe latter is also used as a threshold variable to dene growth regimes for a selected EU countries. 10 We want to note that the nancial stress index can also be linked to some broader index of economic activity, See Hakkio and Keeton (2009) see Hakkio and Keeton (2009). 11 Estimating the nancial accelerator for the macroeconomy with a one regime VAR, see Christensen and Dib (2008). For the application of the nancial accelerator to study nancial intermediaries in a one regime VAR, see Hakkio  As measure for the performance of the macroeconomy we take a monthly production index for the dierent countries, or what is more proper in the context of our model, the growth rate of the monthly production index of the various countries we are considering. To measure output we use IP, the Industrial Production Index from the OECD (2012).
As concerning the IMF FSI, combining the three groups of variables with appropriate weight in a stress index and contrasting it with the monthly production index, one can observe clearly a counter-cyclical behavior. As an example, using Germany, this is illustrated in Figure 4, where the IP variable is shown for a threemonth moving average.
As the comparison of the smoothed growth rate of the production index and the stress index in Figure 4 shows there is less nancial stress that corresponds to good times and more nancial stress in bad times. Financial markets and nancial institutions are clearly doing better in economic booms than in recessions " . Given the apparent linkages between the FSI and economic activity, we would also expect a strong linkage between over-borrowing, nancial stress and economic activity. # A one-regime VAR has been used frequently to study the nancial-macro " This is also shown in an empirical study by Gorton (2010) who shows that there is more insolvency of nancial institutions in bad times.  The MRVAR specication applied here is as follows We estimate a standard VAR and an MRVAR model for the FSI and the industrial output growth rate with y t = (F SI t , 100∆ log(IP t )) ′ . We use AIC to discriminate between a standard VAR or an MRVAR. The AIC is given by where M = 2 is the number regimes; p j is the autoregressive order of regime j; T j is the number of observations associated with regime j;Σ j is the estimated covariance matrix of the residuals of regime j; and n denotes the number of variables in the vector y t ' .
For the case of USA, the AIC suggests a fourth-order VAR with AIC = −45.793. The threshhold for the high nancial stress state in MRVAR is estimated at 2.932. Accordingly, F SI t > 2.932 is considered as the high nancial stress state and the low nancial stress state is when F SI t < 2, 932. The AIC (M = 2, p lo = 3, p hi = 2) = -202965. Based on to the AIC values, MRVAR is a more proper specication than a one regime VAR. We have run the same model selection procedure for other 14 nations and the specication results are summarized in Table 1. The AIC statistics in Table 1 show that MRVAR is a more proper specication not only for the USA but also for all other nations under investigation. These clear statistical model selection results for all nations are also reected in dierences in dynamics of the respective two dierent regimes. We use here within − regime impulse-response functions to asses the dynamics of the two regimes. The within−regime impulse response function is a regime specic impulse response function that is calculated under an assumption that the system remains in the same regime. This is surely not a realistic assumptions, as we observe that a system frequently migrates from one regime to the other. However, a regime-specic response analysis will help us to understand the short-run dynamic behavior associated with the respective regimes.
' The AIC takes into account for possible heterogeneity in the constant terms, c j , and residual covariance, Σ j , across regimes. This AIC criterion is also applied in Mitnick and Semmler (2011). Notes: Table 1   Taking into account of the dierence in the standard deviations in the two regimes, for the same scale of nancial shocks, the responses in the high nancial stress regime is more than two times higher than that in the low nancial stress regime. This dierence provides an evidence that in a high nancial stress state an increase in nancial stress will have a much worse impact on the output than in a low nancial stress state. For a period of high nancial stress economic agents are usually income liquidity and credit constrained. So any further nancial stress shock will reduce spending more, making demand and output declining further. This is unlikely to happen in a state of low nancial stress and in a regime of higher growth where agents are less income, liquidity and credit constraints. In section 2, our theoretical model has elaborated different dynamics due to dierent nancial stress situations. For a comparable  The impacts of output shocks on nancial stress are also dierent in the two regimes (See Figure 5). In the high nancial stress regime the cumulative response of the nancial stress index to one standard deviation output growth shocks is negative and statistically signicant, settling at a level of −8.1. In other words, in the high nancial stress regime a decrease in the industrial production growth will signicantly worsen the nancial stress situation. In the low nancial stress regime, a one standard deviation output shocks have hardly any eects on the nancial stress state.

14
The dierent responses in the high nancial stress regime and in the low nancial stress regime in the MRVAR show that, depending on the nancial stress situation, the system may evolve to dierent equilibria, for which the theoretical model in the previous section has provided an economic explanation.
Since the regime-specic VARs are stationary, the regime-specic mean values of the observed variables provide a rough estimate of the regime-specic equilibria.
For USA, in the high nancial stress regime the mean growth rate of the industrial production is −0.198% and the mean FSI value is 5.384, while in the low nancial stress regime these two values are 0.235% and −1.07 respectively, implying that the system would evolve to a state with contraction in output in the high nancial stress regime and it would evolve to a state with positive output growth in the low nancial stress regime.
It is to note that these regime-specic equilibria are conducted under the assumption of no inter regime migration. This is surely not a realistic assumption, since we observe the frequent regime changes. Therefore, the quantied regime-specic equilibria should only provide a hint to gauge how the system would evolve, if no exogenous shocks leading to regime switching. months. This implies that in the high nancial stress situation an increase in nancial stress has a much worse impact on the output than in the low nancial stress situation.
In the high nancial stress regime the cumulative responses of the nancial stress index to a one standard deviation output growth shock are negatively increasing and they settle at the level of at 1.159 after two years. In other words, in the high nancial stress regime a decrease in the industrial production growth will worsen the nancial stress situation. But in the low nancial stress regime, a one standard deviation output shock has hardly any eect on the nancial stress.
The dierent responses in the high nancial stress regime and the low nancial stress regime in the MRVAR of Germany show also that, depending on the nancial stress situation, the system may evolve to dierent equilibria.
We summarize the MRVAR estimation results for all 14 nations in Table  -  Overall, the empirical analysis suggests that the stronger the position of an economy in the world in terms of output level the more autonomic is interaction between nancial stress and output and henceforth the stronger is the evidence supporting the multi-equilibria scenario predicted by the theoretical model. Generally, in the low nancial stress regime an increase in nancial stress has weaker eects on the output growth than in the high nancial stress regime. In some countries international spill-over eects may signicantly aect their own nancial stress and output growth, so that the these countries show some heterogenous response patterns.

Concluding Remarks
Often over-borrowing has led to nancial stress and nancial crisis. Historically, most severe economic crises have been preceded by a nancial crisis which has amplied the decline in real economic activity. The latter in turn has often exacerbated the nancial meltdown. On the other hand, there are many historical 20 episodes where there were moderate or even strong nancial stress shocks that, however, did not end up triggering real recessions.
In order to study the macroeconomic dynamics, with alternative paths resulting from nancial stress shocks, we rst have introduced a macromodel with a nance-macro link which uses a multi-period decision framework of economic agents. The agents can, in a nite horizon context, borrow and accumulate assets where the above two scenarios may occur. The model is solved through nonlinear model predictive control (NMPC). In contrast to studies of thenancial accelerator modelwhich is locally amplifying but globally stable and mean revertingour model can admit two basic regimes-a regime of low nancial stress and convergence toward some growth path and a scenario of greater instability. In the latter scenario large contractionary eects can be expected.
Whereas the nancial accelerator leads, in terms of econometrics, to a single regime VAR specication, the multiregime dynamics studied here requires a multiregime VAR (MRVAR) approach.
Using the IMF (2011) nancial stress index and industrial production data for the US, the EU and Non-EU countries, our method of a MRVARbased study enables us to conduct a regime specic response analysis. By using a MRVAR we could show that in a regime of high nancial stress, stress shocks can have large and persistent impacts on the real side of the economy whereas in regimes of low stress, shocks can easily dissipate having no lasting eects. The same larger eects on nancial stress and on output can arise in regimes of low output growth in contrast to periods of high output put growth. Thus empirically, we nd that nancialstress shocks and output shocks have asymmetric eects, depending on the regime the economy is in.
As we have shown, though there is heterogeneity across countries with smaller countries showing weaker channels in the the nancial-real interaction there is also much similarity in larger economies. Across countries there are common features in the sense that in larger economies (for example in the US, Germany, France and Japan), large positive nancial stress shocks in a high growth regime tend to have less of a contractionary eect than in a low growth and high stress regime. On the other hand, large reductions in nancial stress tend to induce stronger expansionary eects in low rather than in highgrowth regimes.
The latter seems to be in particular important when evaluating unconventional monetary policy. The empirical analysis presented here strongly suggests that both the timing and the intensity of policy actions matterndings that cannot be obtained by conventional, linear, singleregime analysis.   Table 2 reports the results of MRVAR. OBS hf is the number of observations in the high nancial stress regime and OBS lf the number of observations in the low nancial stress regime. T hreschhold is the the value denes the high nancial stress regime. F SI hf and ∆ log(IP ) hf are the averages the variables in the high nancial stress regime and F SI lf and ∆ log(IP ) lf are the averages of the variables in the low nancial stress regime.             One feature of our MRVAR is that the switching is based on the observable: the nancial stress index. To gauge the dierence between our approach and the often used Markov-switching approach, we also estimate a Markov-switching VAR model using the same variables, assuming that the state of the economy follows a Markov process. The results (see Fig 19) show that the high nancial stress regime dened in our MRVAR (see the fourth graph in Fig. 19) corresponds, to a large extend, to the state predicted by the estimated Markov-switching model (see the third graph in Fig. 19). Four of the ve regime switching episodes correspond to the high nancial stress regimes.  Fig. 19 is the NBER dating of economic recessions during the periods from 1980:1 to 2012:2. We have here 4 economic recessions during this period. Comparing the last two graphs in Fig. 19 we seen that four of the six high nancial stress episodes correspond to economic recessions which were dated by NBER.

Makov Switching Estimation
We also estimate a Markov-switching model for Germany(See Fig. 20). The third graph in Fig. 20 is the estimated probability of the two states and the fourth graph in Fig. 20 shows the high nancial stress regimes of the estimated MRVAR. Similar to the case of USA, one of the two unobserved states estimated by the Markov switching model corresponds quit well, though not exactly, to the high nancial stress regime.