On the Need for a New Approach to Analyzing Monetary Policy

Abstract

Previous articleNext article FreeOn the Need for a New Approach to Analyzing Monetary PolicyAndrew Atkeson, Patrick J. Kehoe, Andrew AtkesonUniversity of California, Los Angeles, Federal Reserve Bank of Minneapolis, and NBER Search for more articles by this author , Patrick J. KehoeFederal Reserve Bank of Minneapolis, University of Minnesota, and NBER Search for more articles by this author , University of California, Los Angeles, Federal Reserve Bank of Minneapolis, and NBERFederal Reserve Bank of Minneapolis, University of Minnesota, and NBERPDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinked InRedditEmailQR Code SectionsMoreModern models of monetary policy start from the assumption that the central bank controls an asset price, namely, the short rate, as its policy instrument. In these models, this policy instrument is then linked to the economy through the agents’ Euler equation for nominal bonds. More abstractly, the Euler equation links the policy instrument to the economy through the model’s pricing kernel. To be useful, a model of how monetary policy affects the economy should account for how the pricing kernel has moved with the short rate in postwar U.S. data.1In this paper, we use data on the dynamics of interest rates and risk to uncover how the pricing kernel has moved with the short rate in postwar U.S. data. Our two main findings are as follows:• Most (over 90%) of the movements in the short rate correspond to random walk movements in the conditional mean of the pricing kernel. We refer to these movements as the secular movements in the short rate.• The remaining movements, which we refer to as the business cycle movements, correspond to movements in the conditional variance of the pricing kernel associated with changes in risk.Standard models used for monetary policy analysis are inconsistent, by construction, with these regularities and, hence, do not capture how the pricing kernel moves with the short rate. We argue that this inconsistency is a serious problem if we want to use these models to understand monetary policy and the macroeconomy. We argue that a new approach to analyzing monetary policy is needed.Here we sketch a new approach to analyzing monetary policy. To do so, we build an economic model consistent with the comovements of interest rates and risk found in U.S. data. Using this model, we interpret postwar monetary policy as follows:• Secular movements of the short rate arise as a result of random walk movements in the Fed’s inflation target.• Business cycle movements of the short rate arise as a result of the Fed’s endogenous policy response to exogenous business cycle fluctuations in risk. The Fed chooses this policy response to maintain inflation close to its target.In our economic model, the Fed is simply responding to exogenous changes in real risk over the business cycle—specifically, to exogenous changes in the conditional variance of the real pricing kernel—with the aim of maintaining inflation close to a target level. Clearly, this view differs substantially from the standard view of what the Fed does over the business cycle. In the standard view, risk plays no role. Instead, the Fed’s policy is a function of its forecasts of economic variables that enter the mean of the pricing kernel, such as expected real growth and expected inflation. This policy is often summarized by a Taylor rule. Our interpretation of the historical record is that, over the business cycle, what the Fed actually did has little to do with these forecasts about changes in conditional means of growth and inflation. Instead, policy mainly responded to exogenous changes in real risk.While we find our model helpful in interpreting the data, it represents, at best, a start to a new approach. Going beyond this specific model, our empirical findings lead us to raise two broader questions to be answered in future research in monetary policy analysis.The first question regards the secular movements in the Fed’s policy instrument: Why did the Fed choose such large secular movements in its policy instrument, namely, the short rate? In our economic model, we mechanically describe the secular movements in Fed policy as arising from a random walk inflation target. Our approach here is similar to that followed in many recent monetary models. The main problem we see with this approach is that it attributes the vast bulk of the movements in the Fed’s policy instrument to a purely mechanical factor. Thus, while this approach may be adequate as a statistical description of Fed policy, it seems useless for answering fundamental questions beyond a superficial level: Why did the great inflation of the 1970s occur? Why did it end? Is it likely to occur again? How can we change institutions to reduce that likelihood?We argue that, to answer such questions, a deeper model of the forces driving the secular component of policy is needed. We briefly discuss some ambitious attempts by Orphanides (2002), Primiceri (2006), and Sargent, Williams, and Zha (2006) at modeling these forces, but we find them wanting. We are led to call for a new approach to modeling the economic forces underlying the secular movements in Fed policy.The second question regards the business cycle comovements between the Fed’s policy instrument and the macroeconomy as captured in the Euler equation: How do we fix our models so that they capture this link? The Euler equation in standard monetary models links the short rate to expectations of growth in the log of the marginal utility of consumption and inflation. Canzoneri, Cumby, and Diba (2007) document that this Euler equation in these models does a poor job of capturing this link between policy and the economy at business cycle frequencies.We offer a potential explanation for the failure of the Euler equation. Existing research nearly universally imposes that the conditional variances of these variables that enter the Euler equation are constant. Thus, all the movements in the pricing kernel in these models arise from movements in conditional means. With our model of the pricing kernel, we find precisely the opposite, at least for the business cycle. That is, over the business cycle, nearly all of the movements in the Euler equation come from movements in conditional variances and not from conditional means.Given this finding, we argue that recent attempts to fix this Euler equation by making the conditional means of the pricing kernel more volatile while continuing to assume that the conditional variances are constant are misguided. We argue that instead researchers should be looking for a framework that delivers smooth conditional means and volatile conditional variances of the pricing kernel at business cycle frequencies. That is, researchers should come to terms with the fact that, at business cycle frequencies, interest rates move one for one with risk.In terms of antecedents for this work, our pricing kernel builds on the work of Backus et al. (2001) and Backus, Foresi, and Telmer (2001). Our economic model is a pure exchange economy with exogenous time‐varying real risk. Since the early contribution by McCallum (1994), a large literature has studied interest rates in such economies. Examples include Wachter (2006), Bansal and Shaliastovich (2007), Gallmeyer et al. (2007), and Piazzesi and Schneider (2007). Our model draws most heavily from the work of Gallmeyer, Hollifield, and Zin (2005).Our paper proceeds as follows. Section I documents four key regularities regarding the dynamics of interest rates and risk that we use to guide our construction of the pricing kernel. Section II documents that standard monetary models are inconsistent with these regularities and lays out our pricing kernel. Section III presents our two main findings regarding the comovements of the short rate and the pricing kernel in postwar U.S. data. Section IV presents the economic model we use to interpret these findings. Section V discusses the two broader questions for monetary policy research that follow from our findings. Section VI concludes.1. Throughout this paper, we consider models in which all variables are conditionally lognormal, and we use the term pricing kernel as shorthand for the log of the pricing kernel.I. The Behavior of Interest Rates and Risk: EvidenceEmpirical work in finance over the past several decades has established some regularities regarding the dynamics of interest rates and risk that any useful analysis of monetary policy must address. In this paper, we focus on the implications of four of these regularities for the analysis of monetary policy. We will argue that standard monetary models are not consistent with these regularities and that a new approach is needed if we are to build models for monetary policy analysis that are consistent with these regularities. We document these four regularities here. Two of the regularities regard the dynamics of interest rates and two regard the comovements of interest rates and risk.A. Dynamics of Interest RatesTo document the first two regularities, we use a traditional principal components analysis to summarize the dynamics of the yield curve. This analysis reveals the following two regularities.1. The first principal component accounts for a large majority of the movements in the yield curve. Because it is associated with similar movements in the yields on all maturities (essentially parallel shifts in the term structure), this component is commonly referred to as the level factor in interest rates. It also has the property that it is (nearly) permanent and is well modeled by a random walk. Here we will refer to the first principal component as the secular component of interest rates in order to emphasize that permanence. In the data, this secular component corresponds closely to the long rate.2. The second principal component accounts for most of the remaining movements in the yield curve. Because it is associated with changes in the difference between the short rate and the long rate—with changes in the slope of the yield curve—it is commonly referred to as the slope factor in interest rates. This component also captures most of the movements in interest rates at business cycle frequencies. Here we will refer to this component as the business cycle component of interest rates in order to emphasize that property. In the data, this business cycle component is essentially the yield spread between the long rate and the short rate.We document these two regularities here. We use monthly data on the rates of U.S. Treasury bills of maturities of 3 months and imputed zero coupon yields for maturities of 1–13 years over the postwar period from 1946:12 to 2007:12. For 1946:12–1991:2, we use data from McCulloch and Kwon (1993) for these series; for 1991:3–2007:12, we use CRSP (Center for Research in Security Prices) data for the 3‐month T‐bill rate and data from Gurkaynak, Sack, and Wright (2006) for the other zero coupon rates. (In the rest of our analysis, we use the 3‐month T‐bill rate as our measure of the short rate and the 13‐year zero coupon rate as our measure of the long rate.)Our principal components analysis of the yield curve uses the traditional procedure (closely following that of Piazzesi [forthcoming, sec. 7.2]). We focus on the first two principal components, which together account for over 99% of the variance of the short rate and over 99.8% of the total variance of all yields. In figure 1, we plot the short rate and the first two principal components of the yield curve that result from our analysis.2Fig. 1. Short rate and the secular and business cycle components. The short rate is the 3‐month T‐bill rate. The secular and business cycle components are the first two principal components derived from a decomposition of the covariance matrix of a vector of 14 yields: the 3‐month rate and the imputed zero coupon yields for maturities $$k=1,\ldots ,13$$ years over 1946:12–2007:12. For the period 1946:12–1991:2, we use data from McCulloch and Kwon (1993), and for the period 1991:3–2007:12, we use data from Gurkaynak et al. (2006).View Large ImageDownload PowerPointTo document our first regularity, we note that the first principal component accounts for over 90% of the variance of the short rate. (It also accounts for over 97% of the total variance of all yields.) This component’s monthly autocorrelation is over .993. Figure 1 demonstrates visually that this component captures the long secular swings in the short rate. Figure 2 demonstrates that it also corresponds closely to the long rate.Fig. 2. Long rate and the secular component. The long rate is the imputed zero coupon yield for 13‐year bonds over 1946:12–2007:12.View Large ImageDownload PowerPointTo document our second regularity, we show in figure 3 that the second principal component is very similar to the yield spread between the short rate and the long rate. This component’s monthly autocorrelation is .957. Figure 1 demonstrates that, barring one exception in the early 1980s, this component captures well the business cycle movements in the short rates.Fig. 3. Yield spread and the business cycle component. The yield spread $$y^{L}_{t}-i_{t}$$ is defined as the difference between the imputed zero coupon yield for 13‐year bonds and the 3‐month T‐bill rate. For the business cycle component, see caption to fig. 1.View Large ImageDownload PowerPoint2. We have scaled these principal components so that the short rate’s loadings on each of these components are equal to one.B. Interest Rates and RiskWith regard to the dynamics of interest rates and risk, decades of empirical work have revealed that movements in the business cycle component of interest rates are associated with substantial movements in risk. Specifically, this work has found two regularities regarding the comovements of interest rates and expected excess returns.3. Movements in the difference between the short rate and the long rate—that is, the yield spread—are associated with movements in risk, defined as the expected excess returns to holding long‐term bonds of a similar magnitude.4. Movements in the short rate relative to foreign currency short rates are associated with movements in risk, defined as the expected excess returns to holding foreign currency bonds of a similar magnitude. We follow much of the literature in interpreting movements in expected excess returns as movements in the compensation for risk.3 Before we cite some of the work documenting these regularities, let us describe them more precisely. We begin with the regularity on the yield spread and the expected excess returns to holding long bonds. We use the following notation to describe these empirical results. Let $$P^{k}_{t}$$ denote the price in time period t of a zero coupon bond that pays off 1 dollar in period $$t+k$$ and let $$p^{k}_{t}=\mathrm{log}\,P^{k}_{t}$$. Then the (log) holding period return, that is, the return to holding this $$k$$‐period bond for one period, is $$r^{k}_{t+1}=p^{k-1}_{t+1}-p^{k}_{t}$$. The (log) excess return to holding this bond over the short rate $$i_{t}$$ is $$r^{k}_{xt+1}=r^{k}_{t+1}-i_{t}.$$ The risk premium on long bonds is the expected excess return $$E_{t}r^{k}_{xt+1}$$. Many researchers have run return forecasting regressions of excess returns against the yield spread similar to the regression (1)rxt+1k=αk+βk(ytk−it)+εt+1k, where $$y^{k}_{t}\equiv -p^{k}_{t}/k$$ is the yield to maturity on this bond. Regressions of this form have been run for 20 years, starting with the work of Fama and Bliss (1987). (See also the work of Campbell and Shiller [1991] and Cochrane and Piazzesi [2005].)Note that, under the hypothesis that the risk premia on long bonds are constant over time, the slope coefficient $$\beta ^{k}$$ of this regression should be zero. In the data, however, these regressions yield estimates of $$\beta ^{k}$$ that are significantly different from zero, with point estimates typically greater than one for moderate to large $$k$$.We emphasize the magnitude of this slope coefficient here because these regression results thus imply that the risk premium on long bonds moves more than one for one with the yield spread. More precisely, note that a finding that the slope coefficient $$\beta ^{k}\geq 1$$ implies that (2)Cov(Etrxt+1k, ytk−it)≥Var(ytk−it), which, by the use of simple algebra, implies that the variance in the risk premium on long bonds is greater than that of the yield spread: (3)Var(Etrxt+1k)≥Var(ytk−it).The fourth regularity regarding movements in the spread between the short rate and foreign currency–denominated short rates and the expected excess returns to holding foreign currency–denominated bonds is simply a consequence of the empirical finding that exchange rates are well approximated by random walks, as documented by Meese and Rogoff (1983) and much subsequent work.To see this, let (4)rxt+1∗=it∗+et+1−et−it denote the (log) excess return on a foreign short bond with rate $$i^{*}_{t}$$, where $$e_{t}$$ is the log of the exchange rate. If exchange rates are a random walk, then $$E_{t}e_{t+1}=e_{t}$$, so that (5)Etrxt+1∗=it∗−it. That is, the expected excess return on a foreign bond is simply the interest differential across currencies.3. The bulk of the asset‐pricing literature interprets measured returns as capturing the total payoffs to owning an asset and accounts for differences in returns as arising from differences in risk. In doing so, this literature assumes that measured returns do not leave out some portion of total returns, such as taxes, transactions costs, or liquidity services that both differ across assets and vary over the business cycle.II. Toward an Economic ModelIn this section, we present the result that standard models, by assumption, cannot match the dynamics of interest and risks that we have discussed. We then present a simple model of the pricing kernel that is consistent with these dynamics.A. The Standard Euler EquationConsider, first, the link between the short rate and macroeconomic aggregates built into standard monetary models. We begin with representative agent models. The short‐term nominal interest rate enters standard representative consumer models through an Euler equation of the form (6)11+it≡exp(−it)=βEtUct+1Uct1πt+1, where $$i_{t}$$ is the logarithm of the short‐term nominal interest rate $$1+i_{t}$$; $$\beta $$ and $$U_{ct}$$ are the discount factor and the marginal utility of the representative consumer, respectively; and $$\pi _{t+1}$$ is the inflation rate. Analysts then commonly assume that the data are well approximated by a conditionally lognormal model, so that this Euler equation can be written as (7)it=−EtlogUct+1Uct1πt+1−12VartlogUct+1Uct1πt+1.A critical question in monetary policy analysis is, What terms on the right‐hand side of (7) change when the monetary authority changes the interest rate $$i_{t}$$? The traditional assumption is that conditional variances are constant, so that the second term on the right‐hand side of (7) is constant. This leaves the familiar version of the Euler equation: (8)it=−EtlogUct+1Uct+Etlogπt+1+constant. Thus, by assumption, standard monetary models imply that movements in the short rate are associated one for one with the sum of movements in the expected growth of the log of the marginal utility of the representative consumer and expected inflation. The debate in the literature on the effects of monetary policy might thus be summarized roughly as a debate over how much of the movement in the short rate is reflected in the expected growth of the log of marginal utility of consumption (representing a real effect of monetary policy) and how much of the movement is reflected in expected log inflation (representing a nominal effect of monetary policy). A resolution of this debate in the context of a specific model depends on the specification of its other equations. However, virtually universally, the possibility that movements in the short rate might be associated with changes in the conditional variances of these variables is ruled out by assumption.We have described the standard Euler equation in the context of a model with a representative consumer. Our discussion also applies to more general models that do not assume a representative consumer. To see this, note that we can write equations (6)–(8) more abstractly in terms of a nominal pricing kernel (or stochastic discount factor) $$m_{t+1}$$ as (9)exp(−it)=Etexpmt+1. In a model with a representative agent, this pricing kernel is given by $$\mathrm{exp}\,( m_{t+1}) =\beta U_{ct+1}/( U_{ct}\pi _{t+1}) $$ and (9) is the representative agent’s first‐order condition for optimal bond holdings. In some segmented market models, (9) is the first‐order condition for the subset of agents who actually participate in the bond market; in others, (9) is no single agent’s first‐order condition. In general, (9) is implied by lack of arbitrage possibilities in the financial market.Using conditional lognormality, we see that (9) implies that (10)it=−Etmt+1−12Vartmt+1, and with constant conditional variances, we have that (11)it=−Etmt+1+constant. Thus, the more general assumption made in the literature is that movements in the short‐term interest rate are associated with movements in the conditional mean of the log of the pricing kernel and not with movements in its conditional variance.Standard monetary models with constant conditional variances are clearly inconsistent with the evidence on the comovements of interest rates and risk. We can see this by considering the following proposition:Proposition 1. In any model with a pricing kernel in which variables are conditionally lognormal and conditional second movements are constant, risk is constant.Proof. Let $$m_{t+1}$$ be (the log of) the pricing kernel and let $$r_{t+1}$$ be any log asset return. Lack of arbitrage implies the standard asset‐pricing formula: (12)1=Etexp (mt+1+rt+1). Taking logs of (12) and using conditional lognormality gives $$0=E_{t}m_{t+1}+E_{t}r_{t+1}+\frac{1}{2}\mathrm{Var}_{t}( m_{t+1}+r_{t+1}) \mathrm{.}\,$$Using (10) implies that the expected excess return on this asset is (13)Etrt+1−it=−12Vart(rt+1)−Covt(mt+1, rt+1). If conditional second moments are constant, then expected excess returns are constant. Hence, risk is constant. QEDProposition 1 implies that, when we log‐linearize our models and impose that the primitive shocks have constant conditional variances, risk is constant. Our reading of the literature on monetary policy is that these assumptions are nearly universal. Yet, as we have seen, the evidence is clear that risk is not constant. This seems a serious problem if we want to use these models to understand what in the macroeconomy moves when the short rate moves.B. A Simple Model of the Pricing KernelHere we present a simple model of the pricing kernel that is consistent with the evidence on interest rates and risk that we have discussed. This model serves as a statistical summary of the joint dynamics of interest rates and risk. In the next section, we use this model to decompose movements in the short rate observed in postwar U.S. data into movements in the conditional mean of the pricing kernel and its conditional variance. This model is similar to the “negative” Cox‐Ingersoll‐Ross model analyzed by Backus et al. (2001) augmented with a random walk process and an independently and identically distributed (i.i.d.) shock to the pricing kernel. To analyze the expected excess returns on foreign bonds, we extend the model to having two countries and two currencies in a manner similar to that in the 2001 work of Backus, Foresi, and Telmer.1. The Home Country Pricing KernelThe model has two state variables, $$z_{1t}$$ and $$z_{2t}$$, that govern the dynamics of the pricing kernel, interest rates, and risk. One state variable follows a random walk with (14)z1t+1=z1t+σ1ε1t+1, and the other follows an AR1 process with heteroskedastic innovations given by (15)z2t+1=(1−φ)θ+φz2t+z2t1/2σ2ε2t+1. The innovations $$\varepsilon _{1t+1},\varepsilon _{2t+1}$$ are independent, standard, normal random variables. Because these state variables are independent and all yields will be linear combinations of these variables, they correspond to the principal components of the yield curve implied by this pricing kernel. We will show below that $$z_{1t}$$ is a level factor and $$z_{2t}$$ is a slope factor. To emphasize its persistence, we refer to $$z_{1t}$$ in the model as the secular component of interest rates. Because it is stationary, we refer to $$z_{2t}$$ in the model as the business cycle component of interest rates. (We calibrate our model so that the secular and business cycle components in the model correspond closely to the secular and business cycle components that we have identified in the data.)We use these two state variables to parameterize the dynamics of the pricing kernel. The (log of the) pricing kernel $$m_{t+1}$$ is given by (16)−mt+1=δ+z1tσ1ε1t+1−(1−λ2/2)z2t+z2t1/2λε2t+1+σ3ε3t+1, where $$\varepsilon _{3t+1}$$ is a third independent, standard, normal random variable.2. The Short RateGiven this stochastic process for the pricing kernel, we use the standard asset‐pricing formula $$i_{t}=-\mathrm{log}\,E_{t}\mathrm{exp}\,( m_{t+1}) $$ to solve for the dynamics of the short rate. Because the pricing kernel is conditionally lognormal, we have that (17)it=−Etmt+1−12Vart(mt+1), so that movements in the short rate correspond to a combination of movements in the conditional mean of the log of the pricing kernel and movements in the conditional variance of the log of the pricing kernel. Observe that the conditional mean of the log of the pricing kernel is given by (18)Etmt+1=−δ−z1t+(1−λ2/2)z2t and that the conditional variance of the log of the pricing kernel is given by (19)12Vart(mt+1)=12(σ12+σ32)+λ22z2t. We thus have that (20)it=δ−12(σ12+σ32)+z1t−z2t. Note that the structure of this model implies that the state variable $$z_{1t}$$ is the secular component of the short rate and the state variable $$z_{2t}$$ is the business cycle component of the short rate.In contrast to standard monetary models, this model allows for variation over time in the conditional variance of the pricing kernel. As (19) makes clear, that variation corresponds to business cycle movements in the short rate, with the extent of that variation governed by the parameter $$\lambda $$. In particular, $$\lambda $$ governs how movements in the business cycle component of the short rate are divided between movements in the conditional mean of the (log of the) pricing kernel and the conditional variance of the (log of the) pricing kernel. Specifically, the response of the conditional mean of the pricing kernel to $$z_{2t}$$ is $$1-\lambda ^{2}/2$$, and the response of $$1/2$$ of the conditional variance is $$\lambda ^{2}/2$$. Thus, if $$\lambda =0$$, then here, as in the standard model, the conditional variance of the pricing kernel is constant, and all movements in $$z_{2t}$$ correspond to movements in the conditional mean of the log of the pricing kernel. In contrast, if $$\lambda =\sqrt{2}$$, then the conditional mean of the pricing kernel does not respond to movements in $$z_{2t}$$, while $$1/2$$ of the conditional variance of the pricing kernel responds one for one with $$z_{2t}$$. If $$\lambda > \sqrt{2}$$, then the conditional mean and the conditional variance of the pricing kernel move in opposite directions when the business cycle component of the short rate moves.3. Longer‐Term Interest RatesTo solve for longer‐term interest rates, we use the standard asset‐pricing formula (21)ptk=logEtexp(mt+1+pt+1k−1) to set up a recursive formula for bond prices. These prices are linear functions of the states $$z_{1t}$$ and $$z_{2t}$$ of the form (22)ptk=−Ak−Bkz1t−Ckz2t, where $$A_{k}$$, $$B_{k}$$, and $$C_{k}$$ are constants. Then we can use standard undetermined coefficients to derive this proposition:Proposition 2. The coefficients of the bond prices are given recursively by Ak=δ+Ak−1+Ck−11−φθ−12Bk−1+12σ12−σ32,Bk=Bk−1+1,Ck=−1−λ2/2+Ck−1φ−12λ+Ck−1σ22, with $$A_{1}=\delta -( \sigma ^{2}_{1}+\sigma ^{2}_{3}) /2$$, $$B_{1}=1$$, and $$C_{1}=-1$$.Proof. To find these prices, we start with $$k=1$$ to find the price of the short‐term bond, using the asset‐pricing formula (21) with $$p^{0}_{t+1}=0,$$ so that pt1=logEtexp(mt+1)=Etmt+1+12Vart(mt+1), so plugging into both sides gives −A1−B1z1t−C1z2t=−δ−z1t+12σ12+σ32+z2t.For $$k> 1,$$ we write the coefficients at $$k$$ as functions of the coefficients at $$k-1$$ as follows. Given our form in (22), we know that pt+1k−1=−Ak−1−Bk−1z1t+1−Ck−1z2t+1. Using the form of the dynamics of the state variables (14) and (15), we have pt+1k−1=−Ak−1−Bk−1z1t−Bk−1σ1ε1t+1−Ck−11−φθ−Ck−1φz2t−Ck−1σ2z2t1/2ε2t+1. Note, then, that this bond price is conditionally lognormal. Combining this bond price with our form for $$m_{t+1}$$ gives logEtexpmt+1+pt+1k−1=Etmt+1+pt+1k−1+12Vartmt+1+pt+1k−1=−δ−Ak−1−Ck−11−φθ+12Bk−1+12σ12−Bk−1+1z1t−−1−λ2/2+Ck−1φz2t+12λ+Ck−1σ22z2t+σ32. Using ptk=−Ak−Bkz1t−Ckz2t then gives recursive formulas for the coefficients of bond prices and yields. QED4. Level and Slope Factors $$\approx $$ Secular and Business Cycle ComponentsWe now show that, in our model, the secular component of interest rates $$z_{1t}$$ corresponds to a level factor that leads to parallel shifts in the yield curve and that the business cycle component $$z_{2t}$$ corresponds to a slope factor that leads to changes in the spread between the long and short rates.Since yields are related to prices by $$y^{k}_{t}=-p^{k}_{t}/k,$$ this implies that yields can be written as ytk=z1t+1kAk+Bkz1t+Ckz2t. Thus, the implications of this model for the yield curve and its movements depend on the behavior of the coefficients $$