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Yield curve inversion and the incidence of recession: a dynamic IS-LM model with term structure of interest rates.

Abstract This paper attempts to explain why yield curve inversion may serve as a leading indicator of recessions. It employs an IS-LM model with the term structure of interest rates and provides a formal phase-diagram analysis of dynamic adjustment process. It demonstrates that the occurrence of yield curve inversion is an off-equilibrium phenomenon after an adverse shock in the adjustment process of interest rates and output, and that an inverted yield curve may lead, but does not lead to, a recession.

Keywords Yield curve inversion * Recession * The IS-LM model * Term structure of interest rates * Phase diagram

JEL E00 * E30 * E40

Introduction

It has been well documented in the literature that the yield curve serves as a leading indicator of output with most being empirical studies (e.g., Estrella and Hardouvelis (1991), Estrella and Mishkin (1997), Berk (1998), among others). Theoretical research to explain such a very interesting observation seems to have lagged behind until Estrella (2005), to our knowledge. It is a subject of macroeconomics that examines the connection between interest rates and real economic activities. Hence, the IS-LM setup should be an innate candidate in modeling to investigate the issue. Since it involves the discrepancy between the short-term interest rate and the long- term interest rate, the term structure of interest rates is naturally introduced into the framework.

This paper studies how and why the yield curve inversion may serve as a leading indicator of real economic activities in a formal dynamic model. We present an IS-LM model with the term structure of interest rates introduced in Blanchard and Fisher (1989) and provide a phase-diagram analysis of the dynamic adjustment process of the output and the interest rates. The occurrence of inverted yield curve is an off-equilibrium phenomenon in the process of adjustment in the output and the interest rates after an adverse shock. While the consequent equilibrium represents the recession that follows, it is the adjustment path of the triplet--the output as well as the long-term and the short-term rates--that explains how and why the yield curve becomes inverted prior to the succeeding recession under the adverse shock. It concludes that an inverted yield curve may lead, but does not lead to, a recession, because they both are the outcomes of the adverse shock, only occurring one after the other.

Our analysis differs from the previous studies with a similar framework e.g., Blanchard and Fisher (1989), Fisher and Turnovsky (1992) in several aspects. First, we keep explicitly both the long-term interest rate and the short-term interest rate in the model, instead of eliminating the short-term rate from the term structure to look at a reduced-form model with the long- term rate left only. This is because we need to show the shape of the yield curve with both the long-term and short-term rates shown in the picture. Second, to portray the shape of a yield curve, we assume that the short-term rate is always on the LM curve, though the long-term interest rate can be away from the IS curve when it is off equilibrium. This treatment allows us to see easily the relative positions between the two interest rates that characterize the yield curve in the model. Third, the dynamic adjustment process of the long-term interest rate is formalized in light of the term structure of interest rates, assuming that the term structure may be off It warrants a globally asymptotically stable equilibrium. This result is in contrast to that of saddle-point equilibrium obtained in the previous studies, where the dynamic adjustment analysis of the long-term rate is around the LM curve after eliminating the short-term rate through the term structure. In addition, our analysis does not need to assume that the long-term bond is a consol; rather, the long-term rate can be of any kind of long-term bonds without further specification.

One of the key assumptions in our analysis is that the short-term interest rate is always on the LM curve. It is plausible and can be justified by the fact that many central banks, including the Federal Reserve of the United States, employ the short-term interest rate rather than the money supply as the primary tool in conducting monetary policy. Given the targeted short-term interest rate, the central bank adjusts the money supply to ensure the announced interest rate. This is exactly the practice of the Federal Reserve, for example. Theoretically, the LM curve in our model essentially becomes the monetary policy (MT') curve as labeled in more recent macroeconomics and monetary economics literature and textbooks. (1)

An IS-LM Model with a Term Structure of Interest Rates

Price level is assumed to be rigid as in a conventional short-run, closed-economy macroeconomic model (e.g., Carlson and Spencer 1975; Cebula 1987) when analyzing an issue on business cycle and stabilization policy. There are two markets: the loanable-funds (or equivalently, the product) market and the short-run money market. Formally, we have

Product market(IS): Y = C(Y - T) + I(R) + G, (1)

Money market(LM): (2) M/P = L(r, Y), (2)

Term structure of interest rates: R = [tau] + (1 - [alpha])r + [alpha] E (r). (3)

Y = output, R = real long-term interest rate, r = real short-term interest rate, T = tax, G = government spending, M = nominal money supply, P = price level, [tau] = term premium >0, E(r) = the expectation of r, and [alpha] [member of] (0, 1), a parameter. Also, C(*) = consumption function with 0 <C'<1, I(*) = investment function with I'<0, and L(r, Y) = demand function for the real money balance with [L.sub.r] <0 and [L.sub.Y] >0. Eqs. (1) and (2) represent the IS and LM curves, respectively, and (3) gives the term structure of interest rates that links the long-term interest rate to the short-term interest rate based on the expectations theory with a term premium. The setup given by (1)-(3) is referred to as an IS-LM model with the term structure of interest rates. (3)

The Dynamic Adjustment Process

When they are off equilibrium, variables Y, R, and r are adjusted as follows:

Y = [phi][C(Y - T) + I(R) + G - Y], [phi] > 0, (4)

r = [M - [L.sub.Y]Y]/[L.sub.r], (5)

R = [eta][[tau] + ( - [alpha])r(Y)[|.sub.LM] + [alpha]E(r) - R], [eta] > 0, [alpha] [member of] (0, 1), (6)

where X = [delta]X/[delta]t, X = Y, r, R, M. The adjustment process for the output Y follows the standard Keynesian approach that spending determines the income, as given in (4). The short-term interest rate r adjusts as featured by (5), which is derived from (2), as r is assumed to always be on the LM curve. Eq. (6) characterizes the dynamic adjustment process of the long-term interest rate R: whenever the term structure (3) does not hold, say, due to a change in r under the monetary policy, R tends to move toward r + [tau] + [alpha][E(r) - r], as guided by the term structure of interest rates. (4)

Setting Y = 0 and R = 0 in (4) and (6), we obtain the stationary loci: the Y-stationary locus is just the IS curve, whereas the R-stationary locus is given as follows:

[tau] + (1 - [alpha])r(Y)[|.sub.LM] + [alpha] E(r) - R = 0. (7)

The expectation E(r) can be general, including rational expectations. We assume it to be a function of r. For example, under the assumption of rational expectations with perfect foresight, we have E(r) = r. In this special case, the R-stationary locus becomes:

R = [tau] + r(Y)[|.sub.LM]. (7')

Note that the curve by (7') passes (Y*, R*) and is parallel to the LM curve. More generally, this R-stationary locus passes (Y*, R*) and has a slope of:

[delta]R/[delta]Y [|.sub.R=0] = [(1 - [alpha]) + [alpha[delta]E(r)/[delta] r]r'(Y)[|.sub.LM]. (8)

Since the LM curve is upward sloping (i.e., r'(Y)[|.sub.LM] > 0), it implies from (8) that the slope of R-stationary locus may be positive or negative, depending on the sign of [delta]E(r)/[delta]r - (1 - 1/[alpha]), i.e., how E(r) responds to a change in r. Figure 1 shows the case with the R- stationary locus upward sloping when [delta]E(r)/[delta]r > (1 - 1/[alpha]). That is, E(r) changes in the same direction as r, or insignificantly in magnitude if it is opposite to r.

[FIGURE 1 OMITTED]

The two stationary loci (i.e., the IS curve and R = 0) divide the Y-R space into four areas, labeled I, II, III, and IV, as shown in Fig. 1. The phase diagram is drawn accordingly based on the arrows of the field and the relative positions of the two stationary loci. From (4) and (6), we have:

[delta]Y/[delta]Y = [phi][C'(Y - T) - 1] < 0, (9)

[delta]R/[delta]R= -[eta] < 0. (10)

[delta]Y/[delta]R = [phi]I'(R) < 0, (11)

[delta]R/[delta]Y = [eta]I'[(1 -[alpha]) + [alpha] [delta]E(r)/[delta]r] r'(Y)[|.sub.LM]. (12)

The signs of [delta]Y/[delta]Y and [delta]R/[delta]R by (9) and (10) determine the directions of adjustment in Y and R, respectively, when (Y, R) is off the stationary loci; both Y and R tend to move toward their own stationary locus. The negative sign of[delta]Y/[delta]R as given in (11) implies that the IS curve is downward sloping, as usual. But the sign of

[delta]R/[delta]Y in (12) depends on if (1 - [alpha]) + [alpha][delta]E (r)/[delta]r >0, or equivalently, if [delta]E(r)/[delta]r >1 - 1/[alpha]. Conditions (9)-(12) feature the arrows of adjustment and the relative positions between the two stationary loci, which jointly determine if the equilibrium (Y*, R*) is globally asymptotically stable or a local saddle point. Formally, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (9) to (12), we can verify that the matrix NY, R) satisfies the following conditions:

(a)tr A(Y, R) = [delta]Y/[delta]Y + [delta]R/[delta]R = -[phi] [1 -C'(Y - T)] - [eta] < 0; (b)det A(Y, R) = ([delta]Y/[delta]Y)([delta]R/[delta]R) - ([delta]Y/[delta]R)([delta]R/[delta]Y) (13)

= [phi] [eta]{1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta] r]r'(Y)[|.sub.LM]I'(R)}; (14)

([delta]Y/[delta]Y)([delta]R/[delta]R) = -[phi] [eta][1 - C'(Y - T)] < 0. (15)

Note that the sign of det A(Y, R) is indicated by that of 1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r]r'(Y)[|.sub.LM]I'(R), which determines if the equilibrium is globally asymptotically stable or a saddle point, together with the other two conditions as specified in (13) and (15).

When 1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r]r'(Y) [|.sub.LM]I'(R) is positive, applying the Olech's Theorem (Olech 1963) (5) to the differential Eqs. (4) and (6), we obtain the following.

Proposition 1

Assume that the adjustment processes of (y R) are given by Eqs. (4) and (6). The equilibrium (Y*, R*) is globally asymptotically stable, if 1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r]r'(Y)[|.sub.LM]I'(R) > 0.

Proposition 1 is quite general and it can be applied to many specific cases. For example, under the assumption of rational expectation with perfect foresight, in particular, we have E(r) = r and hence [delta]E(r)/[delta]r = 1. In this case, we can verify that det A(Y, R) is positive. Then, Proposition 1 immediately leads to the following.

Corollary 1

Under the assumption of rational expectations with perfect foresight, the equilibrium (Y*, R*) is globally asymptotically stable.

More recently in the literature of monetary economics and macroeconmics, some research works replace the LM curve by the monetary policy (MP) curve. That is, r = [bar.r], a specific targeted value in the short-run interest rate, as most central banks have employed the interest rate policy in the last few decades. Technically, the short-run rate r becomes an exogenous variable and the money supply M is endogenously determined to ensure such a short rate. In this case, the LM curve is replaced by the MP curve, which is horizontal. In this case, we have r'(Y) = 0 and hence det A(Y, R) = [phi] [eta] [1 - C'(Y - T)] > 0. As a result, the three conditions as required in the Olech's theorem all hold. Hence, Proposition 1 also results in the following.

Corollary 2

If the IS-LM model is modified to the IS-MP model with a term structure of interest rates, the equilibrium (Y*, R*) is globally asymptotically stable.

The sign of 1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r] r'(Y)[|.sub.LM]I'(R) actually characterizes the relative positions between the low stationary loci. Note that 1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r]r'(Y)[|.sub.LM]I'(R) > 0 is equivalent to [1 - C'(Y - T)/I'(R) < [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r]r'(Y)[|.sub.LM], which means the slope of the IS curve is less than that of the R-stationary locus (E = 0). Since the IS curve is downward sloping, there are two subcases here: (a) [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r] > 0, i.e., the R- stationary is upward sloping. This is the case as illustrated in Fig 1, where the equilibrium (Y*, R*) is globally asymptotically stable. (b) [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r] < 0, i.e., the R-stationary locus, is also downward sloping, but flatter than the IS curve, as shown in Fig. 2. In this case, the equilibrium (Y*, R*) is still globally asymptotically stable.

What about if [1 - C'(Y - T)/I'(R) > [(1 - [alpha]) + [alpha] [delta]E(r)/ [delta]r]r'(Y)[|.sub.LM]? Then, the R-stationary locus would be downward sloping and even steeper than the IS curve. Figure 3 illustrates the corresponding phase diagram. It shows that the corresponding equilibrium is a local saddle point, rather than globally stable. Formally,

(b') det A(Y, R) = ([delta]Y/[delta]Y)([delta]R/[delta]R) - ([delta]Y/ [delta]R) - ([delta]R/[delta]Y) = [phi] [eta] {1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r]r'(Y)[|.sub.LM]I'(R) < 0; (14)

Under the condition (14') together with (13) and (15), according to Theorem 6.9.1 in Sydsaater et al. (p. 253), we have the following

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Proposition 2

Assume that the adjustment processes of (Y, R) are given by Eqs. (4) and (6). The equilibrium (Y*, R*) is local saddle point, if 1 - C'(Y - T) - [(1 - [alpha]) + [alpha] [delta]E(r)/[delta]r]r'(Y)[|.sub.LM]I'(R) < 0.

Yield Curve Inversion and Recessions

The core issue addressed in this paper is why yield curve inversion can serve as a leading indicator of a recession? Given the analysis of the equilibrium and the dynamic adjustment processes of (Y, R) developed in the previous section, we now are ready to explain it as follows.

Under an adverse shock, say, due to a financial crisis or restrictive monetary policy, the LM curve would shift leftward from [LM.sub.0] to [LM.sub.2], moving (Y, R) off the new equilibrium and below the [LM.sub.2] curve (or the MP curve if it is used alternatively), as shown in Fig. 4. Since the new equilibrium (as indicated by point [E.sub.2] in Fig. 4) is globally asymptotically stable, output Y is going to decrease with the yield curve inverted until it passes through the new [LM.sub.2] curve. This is because the short-term rate r is on curve [LM.sub.2]. After point (Y, R) passes through the new [LM.sub.2] curve, it would still take a while for it to finally reach the new equilibrium ([Y*.sub.2], [R*.sub.2]). Since the new equilibrium represents the final position of an economic contraction, the adjustment path of (Y, R) can well explain why yield curve inversion occurs before the recession occurs. Empirical evidences show that the yield curve had become inverted about 12 months or so before the economy enters a recession. By official definition of a recession, it arises after the output consecutively goes down for at least two quarters. Hence, when (Y, R) combination is below the LM curve under an adverse shock, the yield curve starts to be inverted but the recession has not yet been officially observed. It would take some time for (Y, R) to go from the initial equilibrium ([E.sub.0]) to the new equilibrium ([E.sub.2]) that characterizes the recession.

[FIGURE 4 OMITTED]

Concluding Remarks

This paper attempts to explain why yield-curve inversion can serve as a leading indicator of a recession in a formal dynamic model. We developed a version of the IS-LM model with the term structure (Blanchard and Fisher 1989). With such a framework, we provide a formal phase-diagram analysis to illuminate how an adverse shock may cause yield curve inversion as well as a subsequent recession. The phase-diagram analysis of off-equilibrium (Y, R) here is very general without restriction on the type of long-term bond and the dynamic adjustment process in R is naturally based on the term structure of interest rates. It shows that the equilibrium (Y*, R*) is globally asymptotically stable, rather than a saddle-point equilibrium under very general conditions, including the rational expectations. A key technical assumption made in this paper is that the short-term interest rate is always on the LM curve, which helps us easily see the shape of the yield curve along the path of an off-equilibrium (Y, R). Finally, it is worth noting that the occurrence of inverted yield curve is an off-equilibrium phenomenon in the dynamic adjustment process after an adverse shock. That is, yield curve inversion itself is a by-product of an adverse shock, but occurs ahead of the recession. Hence, it only leads, but does not lead to, the succeeding recession.

References

Berk, J. M. (1998). The information content of the yield curve for monetary policy: a survey. De Economist, 146(2), 303-320.

Blanchard, 0. J., & Fisher, S. (1989). Lectures on macroeconomics. Cambridge: MIT Press.

Carlson, K. M., & Spencer, R. W. (1975). Crowding out and its critics. Federal Resserve Bank of St Louis Review 57(2), 2-17.

Cebula, R. J. (1987). The deficit problem in perspective. Lexington: Lexington Books.

Estrella, A. (2005). Why does the yield curve predict output and inflation? The Economic Journal, 115 (July), 722-744.

Estrella, A., & Hardouvelis, G. A. (1991). The term structure as a predictor of real economic activity. Journal of Finance, 46(2), 555-576.

Estrella, A., & Mishkin, F. S. (1997). The term structure of interest rates and its role in monetary policy for the European central bank. European Economic Review, 41, 1375-1401.

Fisher, W. H., & Turnovsky, S. J. (1992). Fiscal policy and the term structure of interest rates: an intertemporal optimizing analysis. Journal of Money, Credit, and Banking, 24(1), 1-26.

Olech, C. (1963). On the global stability of an autonomous system on the plane. Contributions to Differential Equations, Vol. 1. Interscience.

Romer, D. (2008). Macroeconomic theory (3rd ed.). New York: McGraw-Hill Irwin.

Sydsxter, K., Hammond, P., Seierstad, A., & Strthm, A. (2005). Further mathematics for economic analysis. London: Financial Time/Prentice Hall.

(1.) Romer (2008) has shown the equivalence between the two treatments in this regard.

(2.) In theory, the interest rate in the money demand function should be the nominal rate. Here, it is converted by Fisher equation, as usual.

(3.) Blanchard and Fisher (1989) developed such a model to study the effects of a change in fiscal policy such as the 1981-1983 tax cuts. But they did not explicitly label both the long-term and the short-term interest rates in their graphic analysis; instead, the short-term rate was eliminated and indirectly represented by the long-term rate from the term structure and the Fisher equation. Hence, their treatment did not explicitly present yield curve in the picture.

(4.) Note that the driving force behind the term structure of interest rates is arbitrage. Hence, when it is off the structure, the same driving force would apply to pull the long-term rate back to the position as determined by the term structure.

(5.) See also, for example, Sydsaeter et al. (2005, p. 251).

The authors would like to thank Richard Cebula, Chris Coombs and the audiences in the 72nd International Atlantic Economic Society (IAES) Conference at Washington DC for their comments on earlier versions. Bill Yang is grateful to the Department of Finance and Economics, Georgia Southern University for the research support (Summer 2011). The standard disclaimer applies.

X. H. Wang

Department of Economics, University of Missouri-Columbia, Columbia, MO, USA

B. Z. Yang *

Department of Finance and Economics, Georgia Southern University, Statesboro, GA, USA e-mail: billyang@georgiasouthern.edu

Published online: 16 February 2012

[C] International Atlantic Economic Society 2012

DOI 10. 1007/s11294-012-9350-7
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Comment:Yield curve inversion and the incidence of recession: a dynamic IS-LM model with term structure of interest rates.
Author:Wang, X. Henry; Yang, Bill Z.
Publication:International Advances in Economic Research
Article Type:Report
Geographic Code:1USA
Date:May 1, 2012
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