The effect of a shock in (expected) inflation on the term structure : analysis using an ane term structure model with macro factors.

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The effect of a shock in (expected) inflation on the

term structure

Analysis using an affine term structure model with macro factors

Thesis for the degree of MSc. Econometrics, track Financial Econometrics University of Amsterdam, Faculty of Economics and Business

Author: Guus Smal1 Supervisors: Prof. Dr. H.P. Boswijk2 Drs. G. De Lange3 March 12, 2014

A Gaussian multi factor model as in Ang and Piazzesi [2003] is used to determine the effect of a shock in (expected) inflation on the term structure. The reduced form of the model is estimated to overcome estimation difficulties, after which the structural form is derived from it. Impulse response functions (IRs) are subsequently used for analysis. The counterintuitive result of a negative effect of an inflation shock on yields is obtained. For a shock in expected inflation, positive responses of the yield curve are observed. Responses to the shocks diminish with maturity. These obervations are robust over IRs of the structural model, the VAR model from the reduced form and an unrestricted VAR.

1_{University of Amsterdam, student nr. 5745632, guussmal@gmail.com} 2

University of Amsterdam, Department of Quantitative Economics

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1 Introduction

In recent years, pension funds in the Netherlands have been confronted with very low interest rates. Central banks’ have moved short term interest rates to historically low levels. In addition, longer terms bonds were used to stimulate the economy.

“Accommodative policy is appropriate, in my view, because the economy is operating well below its potential and inflation is undesirably low. If it were positive to take interest rates into negative territory I would be voting for that” [San Francisco Federal Reserve Bank President Janet Yellen – February 2010]

Pension funds’ projected liabilities, of which a large part is due in well over thirty years, are usually discounted with market interest rates. The funding ratio, which is the most important indicative figure for a pension fund’s health, is calculated as

F R = Discounted Bonds + Other Assets Discounted Liabilities

The low interest rates posed a challenge to politicians and pension fund boards, as reduc-tions in pension rights were thought to be necessary to increase the funding ratios to healthy levels.

”We cannot exclude the possibility that pension payments will be cut next year. It depends on the development of the interest rates and returns.”[Chairman of General Civilian Pension fund (ABP) Henk Brouwer - October 2013]

It is common practice to partially hedge interest rate risk. The fixed income portfolio consists of cashflows in the future, securing a part of the liabilities, no matter what happens to the interest rate. The protection that the bond portfolio gives can be extended by derivative contracts, such as interest rate swaps. The hedging is usually done on the basis of duration, because it is very costly to exactly replicate a fraction of the projected liable payments.

Another challenge pension funds face is inflation. They typically want to correct their payments to retirees for changes in purchasing power, but will only grant this indexation if the funding ratio is high enough. Assets offering protection against inflation are not readily available. Although inflation linked bonds trade in the market, these are not available for every maturity. Other assets have been argued to be a protection from inflation risk. For example, commodities and stocks have often been suggested (See Bodie [1976],Bodie [1983],Schotman and Schweitzer [2000]). In Figure 2, the risks are visualized. It is clearly visible that an increase in inflation and a decrease in interest rates have a similar effect on an unprotected portfolio: the projected payments become higher.

It is a generally accepted view that interest rates move together with inflation. If this is true, in the sense that if a percentage point change in inflation would imply a one percent point change in interest rates, the two risks would partially offset each other. This has been a reason for pension funds not to hedge their interest rate risk entirely, as the inflation risk is difficult to hedge on its own. The idea that inflation and interest rates move together is very

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Figure 2: The graphs indicate the liabilities for a representative pension fund. The effect of a level shock of minus a percentage point in interest rates and a plus 10 percent point shock in inflation is visualized.

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intuitive, in the sense that everybody knows that a lender should be compensated for the loss of purchasing power. Fisher [1930] formalized this idea in the Fisher equation, which equates the nominal interest rate to the real interest rate and the expected inflation. It remains unclear however, whether the classic Fisher equation holds under shocks in the economy. A great number of papers has examined the Fisher equation in a VAR environment (for example Mishkin [1992]), but has not delivered unambiguous results. Expected inflation series seem to have different time series properties than realized inflation series, which indicates that the two measures are different in nature.

Another line of research examines central bank policy, which is focused on inflation tar-geting by setting the short term interest rate. In this research field, rules that say how much the central bank should change the interest rate if inflation or the output gap changes are estimated. It appears that expected inflation and realized inflation are judged differently (Clarida et al. [1998]).

Affine term structure models have been used to describe the dynamics of the yield curve. Duffie and Kan [1996] develop a framework for term structure models with latent factors. Ang and Piazzesi [2003] take a discrete Gaussian version of their model and show that it is equivalent to a VAR model with no arbitrage restrictions. They include macroeconomic variables to explain movements in the yield curve better. From this and other papers that use a variant of their model, inflation is shown not to have a one-to-one effect on yields and the effect is different along the yield curve. From this research, it has become clear that expected inflation, in the form of surveys, exhibits a hidden factor that is not correlated with current and past values of inflation and is not well explained by other macroeconomic variables. If the correlation of yields and realized inflation proves not to hold as in the Fisher equation, the hedging strategy of leaving interest rate risk open to allow for simultaneous movements of inflation and yields is questionable.

In this thesis the effect of both realized and expected inflation on the term structure is examined. A multi factor model as in Ang and Piazzesi [2003] is estimated with both inflation and expected inflation. The impulse response functions are derived and used to analyze the changes in yields following a shock. The reduced form is estimated with realized inflation and expected inflation separately and with both, for different lag lengths. First we are interested in the sign and magnitude of the IRs of yields to a shock in (expected) inflation. It is hypothesized that the responses of yields to the shocks are positive and diminish over time. Another relevant question is: How does this differ along the yield curve? We expect smaller responses for yields with a longer maturity. In addition, we explicitly address the difference between expected and realized inflation. If expected inflation is driven by inflation and a hidden factor, the model with expected inflation should do better. Therefore, we expect the responses from yields to a shock in realized inflation to be less when the reduced form is estimated with both expected and realized inflation. The structural form of the model with one lag is derived from the reduced form. We expect the effects to be similar to the effects found in the reduced form specification.

This thesis is organized as follows: Section 2 gives a short introduction to affine term structure models. It starts with the Vasicek model, quickly goes through the Duffee and Kan framework and ends with a discussion of multi factor models that incorporate macroeconomic factors. Section 3 examines results of studies on the dynamics of inflation, expected inflation

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and yields. One line of literature examines the Fisher equation by cointegration models. Another considers the ‘Taylor rule’, which describes central bank policy. Last, results of multi factor models with inflation or expected inflation are examined. Section 4 gives the specifications we will employ, derives the impulse response functions and shows an analytical result. Section 5 considers the choice of data and estimation results. Section 6 concludes.

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2 Affine term structure models

In this section a short introduction is given of ATSMs, showing the reader the development towards the ATSM with macro factors. For a more complete discussion of interest rate models, textbooks such as Brigo and Mercurio [2006], Piazzesi [2010] and Filipovi´c [2009] are very instructive.

2.1 One factor models

In the seminal paper by Vasicek [1977] a no arbitrage argument is used, similar to the famous Black and Scholes [1973] argument, to derive a partial differential equation for bond prices. Denote the price at t of a zero coupon bond which pays $1 at maturity T ≥ t by P (t, T ). Then the compounded interest rate is

R(t, T ) = − log P (t, T ) T − t The instantaneous short rate rt is then

rt= lim T →tR(t, T ) = lim T →t− log P (t, T ) − log P (t, t) T − t = −∂ log P (t, t) ∂t

where the partial derivative is with respect to the second argument. The short rate rtis assumed to follow an Itˆo process

drt= f (t, rt)dt + ρ(t, rt)dWt (2.1) where f (t, rt) and ρ2(t, rt) are the drift and variance of the process. Wt denotes a Wiener process under the objective measure P . Vasicek [1977] assumes that the price of a zero coupon bond that pays 1 unit of money at maturity depends on the maturity and the instantaneous interest rate only, i.e. P (t, T ) = P (t, T, rt). Applying Itˆo’s lemma, we find a stochastic differential equation for the bond price:

dP (t, T, rt) = P (t, T, rt)(µ(t, T, rt)dt + σ(t, T, rt)dWt) (2.2) µ(t, T, rt) = 1 P (t, T, rt) (∂P ∂t + f ∂P ∂rt +1 2ρ 2∂2P ∂r2 t ) σ(t, T, rt) = 1 P (t, T, rt) ρ∂P ∂rt

The no arbitrage argument is as follows: Suppose an investor chooses to sell the amount V1 of one bond and buy V2 of another. Then his total wealth V = V2− V1 satisfies (from

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now on we will omit arguments for readabilty) dV = dV2− dV1

= V2(µ2dt + σ2dWt) − V1(µ1dt + σ1dWt) = (V2µ2− V1µ1)dt + (V2σ2− V1σ1)dWt

The investor can choose V1 and V2 such that the stochastic term becomes zero by setting V1 = V σ2 σ1− σ2 V2 = V σ1 σ1− σ2

This implies that the portfolio should earn the instantaneous riskless rate. dV = rtV dt

V2µ2− V1µ1 = rtV Substituting the values for V1 and V2 delivers

µ1− rt σ1

= µ2− rt) σ2

Because the equality holds for any maturity, this is called the market price of risk, as it denotes the excess return for an extra unit of risk. Vasicek [1977] sets q = µ−rt

σ , but it can also be time dependent. Once q is specified, writing rt= µ − σq and substituting for µ and σ results in the partial differential equation

∂P ∂t + (µ − σq) ∂P ∂r + 1 2 ∂2P ∂r2 − rP = 0 (2.3)

Once the functional form of the drift and variance of the spot rate process in ( 2.1) have been specified, bond prices can be derived from solving ( 2.3) subject to the boundary condition P (s, s) = 1. Yields R(t, T ) of bonds at time t with maturity T are then computed as R(t, T ) = −_{T −t}1 log P (t, T, rt).

Vasicek [1977] specifies the short rate process to be an Uhlenbeck-Ornstein process drt= κ(θ − rt)dt + σdWt,

with κ, θ and σ constants. The specification of the drift leads to a mean reverting process, as long as κ > 0. This process for the short rate is able to produce negative values, as neither µ(r, t) or σ(r, t) is restricted. Cox et al. [1985] specify the short rate to satisfy the stochastic process

drt= κ(θ − rt)dt + σ √

rtdWt

with the parameters constants. This specification has the benefit that under the condition that 2κθ ≥ σ2, the short rate will become negative with an infinitely small probability, when its starting value is positive. In addition, the variance of the short rate will increase with its level, which seems plausible as well. However, the distribution of the short rate is not normal, but non central chi-square. This means that the model is less tractable. The two specifications for the stochastic differential equation of the short rate above are the most well known. Many other authors have contributed different one factor model specifications, usually to address a certain problem or empirical phenomenon.

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2.2 Multi factor models

The one factor specification is problematic, because it does poorly in approximating observed term structures. In addition, the very nature of one stochastic process determining all bond prices simultaneously implies that bond price changes are perfectly correlated, which is clearly rejected by empirical evidence. Cox et al. [1985] already suggest using more factors in the short rate equation. Hull and White [1990] empirically test two factor Vasicek and CIR models and find that these do a far better job than the single factor models. Notably, the correlation between the yield movements does not have to be one and a hump shaped form of the yield curve, which is often observed, can be reproduced.

Duffie and Kan [1996] develop a general framework for multi factor ATSMs. They specify the short rate rtto be dependent on factors xtthat follow a stochastic process with mean µt and diffusion σtσt0: r(xt) = −log P (t,T )_{T −t} , with T maturity. Bond prices are usually calculated using risk neutral pricing. Under no arbitrage, an equivalent probability measure is assumed to exist, under which bond prices, discounted with the risk free rate, become martingales. A compatible price function then satisfies

P (t, T ) = f (xt, T − t) = E[exp{− Z T

t

r(xs)ds}|xt] (2.4) where E denotes the expectation under the risk neutral measure. To connect with the previous section, Vasicek [1977] used the short rate itself as the factor and set the function r as the identity.

The authors model f (t, T −t) directly by f (t, T −t) = eA(T −t)+B(T −t) xt_{. This specification}

and the boundary condition f (t, 0) = 1 deliver differential equations for A(T − t) and B(T − t). An important result is that f (t, T − t) is exponential affine in the factors if and only if the drift and diffusion are affine combinations of the factors. This model is very tractable and is therefore used extensively.

The factors xtcan be arbitrary variables, but Duffie and Kan [1996] suggest taking yields, directly or indirectly. The direct inclusion of yields is straightforward. The yields can also be said to depend on unobservable factors xt, that can be derived from the equations

Yit= R(t, t + τi) = A∗(τi) + B∗(τi) xt.

This approach is backed by the finding that a lot of the variation in the term structure can be explained by just a couple of principal components of the yields (Litterman and Scheinkman [1991]). This change of variables leads to a pricing function f∗(t, τ ) = eA∗∗(τ )+B∗∗(τ ) Yt_,

where Yt is a cross section of yields at time t. Some practical problems can arise if the covariance matrix of the factors is factor dependent. The Gaussian case, with a time-homogeneous covariance matrix, does not pose such problems.

Dai and Singleton [2000] give a canonical representation of multi factor models and check the properties of models with three level factors and m(0 ≤ m ≤ 3) factors driving the volatility. They show that many extant models are unnecessarily restrictive. They conclude that there is a trade off between restricting the correlation between factors and the complexity of the volatility structure. Among models with three factors, m ∈ {1, 2} gives the best results.

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2.3 A model with macro factors

The framework of Duffie and Kan [1996] initially used yields only to describe the yield curve. However, as Duffee [2011] describes, only half of the variation in bond risk premiums can be explained by variations in bond yields. There is a hidden factor, that has a small effect on the yield curve, but a large effect on the dynamics. Macroeconomic fundamentals could be responsible for this hidden factor, but Duffee finds that it is not correlated by current or past inflation or other macroeconomic variables.

Ang and Piazzesi [2003] design a model that incorporates macroeconomic variables next to unobservable factors. They adopt the Gaussian version of the Duffie and Kan [1996] framework, to connect with the literature that has researched the topic in a VAR environment. The Gaussian specification turns the affine term structure model in a VAR model, which makes analysis convenient. The short rate is an affine function of both observable and unobservable factors, rt = δ0 + δ110 Xto+ δ120 Xtu. The observable factors are assumed to be orthogonal to the unobserved factors. This is because the authors wanted to interpret the dynamics in the light of the Taylor rule (see next section). This orthogonality is clearly violating empirical evidence, as it is assumed that yields do not affect macro variables, which would make central bank policy redundant. It makes the parameter space a lot smaller, which is also convenient. The unobservable factors X_tu follow a VAR(1) process and the observables X_to follow a VAR(p) process.

X_tu = µu+ φu₁Xt−1+ σuut

X_to = µo+ φo₁X_t−1o + . . . + X_t−po + σoo_t. Then the process for both factors can be written as

Xt= µ + Φ1Xt−1+ . . . + σt. and the short rate equation becomes simply

rt= δ0+ δ10Xt.

In continuous time, dXt= µ(Xt)dt + σWt under the risk neutral measure Q and dXt= µ∗(Xt)dt + σWt∗with µ∗(Xt) = µ(Xt) + σλtunder the objective measure P . There are thus three parameter sets: one under the risk neutral measure, one under the objective measure and one that determines the change of measures (λ). Only two need to be estimated to know all three.

Under no arbitrage, the price of a non dividend paying asset Vt satisfies Vt= EtQ[exp(−rt)Vt+1]

= E_tP[exp(−rt) dQ dPVt+1].

In which the short rate rt is the one period rate. The Radon-Nikodym derivative dQ_dP, that converts the risk neutral measure to the data generating measure, has a density process ξt= Et(dQ_dP). Duffie [2010] shows that

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ξt+1 is assumed to follow the process ξt+1= ξt exp(− 1 2λ 0 tλt− λ0tt+1),

where λt is the market price of risk associated with the corresponding shocks t and is assumed to satisfy λt= λ0+ λ1Xt, allowing time dependence.

In order to price assets in the economy, a pricing kernel is used. It is defined by mt+1= exp(−rt)ξt+1/ξt. Intuitively, the discount factor is amplified by the change of measures. The gross return of any asset in the economy Rt+1satisfies Et(mt+1Rt+1) = 1. Now bond prices can be calculated recursively in this discrete setting by P (t, n + 1) = pn+1_t = Et(mt+1pnt+1). The dynamics of the factors, the short rate and the pricing kernel constitute a Gaussian factor model. Bond prices are exponentially affine in the state variables as in pn

t = exp(An+Bn0Xt). Difference equations for An and Bn are derived in the appendix of Ang and Piazzesi [2003]. An and Bn are functions of the risk neutral parameters.

The model that Ang and Piazzesi developed was used extensively. Maybe the most exten-sive model is used in Ang et al. [2008], in which regime switching is also present.

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3 Yields, inflation and expected inflation

3.1 The Fisher equation.

A discussion of the difficulties in studying the Fisher equation can be found in the introduction of Sun and Phillips [2004]. The Fisher equation is

it,T = rt,T + πt,T ,

equating the nominal interest rate it,T to the real rate rt,T and inflation πt,T.They discrimi-nate between the ex ante Fisher equation and the ex post Fisher equation. The ex post Fisher equation is a mere identity and determines the real rate as an unobservable variable, after inflation has been observed. However, nominal interest rates are forward looking measures and inflation and real rates are backward looking. An investor determines the interest rate he demands before he knows what the inflation will be. The forces that determine the nominal rate are the expected inflation and the real rate. Therefore, inflation and real rates do not have to exhibit the same statistical qualities as the nominal rates. This is problematic and many studies do circumvent this nuisance by simply taking the ex post rates.

Evans and Lewis [1995] show for U.K. data that when they allow the inflation process to change over time, there is a one to one correspondence between inflation and interest rates. Mishkin [1992] only finds a strong Fisher effect in post war USA during periods in which interest rates and inflation exhibit a common trend. Koustas and Serletis [1999] reject the Fisher effect for a score of developed countries. Sun and Phillips [2004] account for the bias and reject the Fisher equation.

With the introduction of inflation linked securities, it became possible to gauge expected inflation and the real interest rate from the market. Intuitively, an investor who has a nominal bond wants to be compensated for inflation risk, so a nominal bond will exhibit an inflation risk premium. Evans [2003] uses a model with different regimes and finds that the inflation risk premium is too high for short maturities and too low for long maturities. He finds that the Fisher equation is a rather poor approximation for the dynamics over the term structure. Chen et al. [2005] find a steep upward sloping risk premium in the US and reject the Fisher Hypothesis. It remains vague what the dynamics are in this case, even when inflation and its risk are priced by the market.

3.2 The Taylor rule

Taylor [1993] comments on technical central bank policy rules that dictate the change of interest rates. Central banks generally consider measures of inflation, expected inflation and the output gap in order to determine their policy. However, a large part of the final decision is based on discretion. A policy rule can be helpful though to clarify policy and estimations of the rule have shown to be quite accurate in following the Fed policy. A basic form of the Taylor rule is as follows:

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It states that the target short term interest rate r∗ is a function of the long term interest rate ¯

r and expectations of inflation πt,T and the output gap yt. π∗t and yt∗ are optimal points. Clarida et al. [1998] fit mechanic policy rules like the above to the actions of central banks of two groups of three countries each. They find that central banks that had an inflation target looked at expected inflation, rather than realized inflation. Interestingly, estimations of the policy rule of the central banks of Germany, Japan and the U.S.A. resulted in non-significance for lagged inflation when it was added to the model already containing expected inflation. The other coefficients did not change much. The point estimate of the coefficient for inflation was even negative for the policy rule of the Fed.

3.3 Results from Affine Term Structure Models

Ang and Piazessi include inflation and the output gap to account for the latent factors that are used in the framework of Duffie and Kan [1996]. Three versions of their VAR model are estimated: They estimate a model without no arbitrage restrictions, one with the no arbitrage restriction, and one lag of the macro variables and an extended version with twelve lags of the macro variables. They derive impulse response functions for their models and find positive responses of short term yields (1 month, 1 year and 5 years) to a shock in inflation. The response decreases with maturity in all their models. However, there are significant differences in the impulse responses among different model specifications. Notably, they find less hump-shaped impulse response functions for the model with one lag than for the model with twelve lags.

H¨ordahl et al. [2006] estimate a term structure model while using a structural macro model, rather than unobservable factors. First they estimate the joint dynamics of inflation, the inflation target and the output gap. In equilibrium, the short rate is determined by these dynamics. Subsequently, with a stochastic discount factor and no arbitrage, the term structure is constructed. They find that an inflation shock has primarily an effect on the curvature of the yield curve. The responses of yields are statistically significant for maturities up to seven years, but very small.

Ang et al. [2008] estimate an affine term structure model under regime switching. These authors succeed in determining the inflation risk premium from nominal bonds and inflation time series. They specify that expected inflation is a function of current inflation, regime and unobservable factor. The inflation risk premium increases with maturity, fully accounting for the usually upward sloping nominal bond curve. Nominal bond rate variation is found to be caused by changes in expected inflation and expected inflation has a greater influence on the term spread for longer maturities. In the above studies, expected inflation is derived from inflation linked bond prices or is determined endogenously.

Ang et al. [2007] compare four methods to forecast inflation. They conclude that surveys prove to be the best. Term structure models do a poor job at forecasting inflation. Combi-nations of the methods do not seem to improve the performance of survey based forecasts. Chernov and Mueller [2012] use surveys to implement expected inflation into their model. They find that expected inflation is driven by inflation, the output gap and a hidden factor. This factor has a great impact on expected inflation, but is not related to past or present values of inflation or other macroeconomic variables. Neither does it directly affect yields.

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The model with surveys outperforms the model without the survey data. Furthermore, the authors find that an additional factor is needed when information from TIPS is included in the model. Hence, there is different information in both measures. This difference might reflect other concerns. A trader will be more concerned about the risk premium and making profits, whereas an economist looks at economic activity.

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4 Specifications

4.1 Structural and reduced form

In our analysis we use a Gaussian multi factor model as in Ang and Piazzesi [2003].The major advantage of this model is that it allows calculation of classical impulse response functions. From the discussion in the previous section it is clear that it might not be the best model from a ATSM point of view. Notably, the variances are assumed to be constant over time, which is in contradiction with empirical data. However, it may serve as a starting point and it may approximate more complex models.

We specify fm as the Nm macro factors and fl as the Nl latent factors. The factors are assumed to follow the following process, in which p lags of the macroeconomic factors and one lag of the latent factors are present:

Ft= (fm 0 t . . . fm 0 t−p+1fl 0 t ) 0 (4.1) = (F_tm0)F_tl0) (4.2) = c + ρFt−1+ ξt (4.3) ξt∼ IID N (0, ΣξΣ 0 ξ) (4.4)

We know from Ang and Piazzesi [2003] that the yields are affine combinations of the factors. It is assumed that Nl yields are measured without measurement error, in order to back out the latent factors. Ne yields are measured with measurement error. We can then write the yields as Y_t1= A1+ B1Ft (4.5) Y_t2= A2+ B2Ft+ ut (4.6) ut∼ IID N (0, ΣuΣ 0 u) (4.7)

with ut the measurement error.

Hamilton and Wu [2012] pointed out that the specification used by Ang and Piazzesi [2003] is unidentified, as the structural form has more parameters than the reduced form. The restrictions that were intended to optimize the estimation procedure, were in fact identifying restrictions. Furthermore, they showed that the parameters found by Ang and Piazzesi [2003] correspond to a local maximum of the likelihood. The difficulties with likelihood estimation are due to a large and very flat solution space. The reduced form of the above model is (the derivation is in the appendix):

f_tm= am+ φmlYt−11 + φmmFt−1m + ξtm (4.8) Y_t1= a1+ φ11Yt−11 + φ1mFt−1m + ψftm+ B1lξtl (4.9) Y_t2= a2+ φ21Yt1+ φ2mFtm+ ut (4.10)

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Note first that the equation for Y_t1 has f_tm on the RHS, so that the error terms of the upper two equations are orthogonal. The same is the case for the second and third equation. Because the three blocks can separately be seen as SUR equations, we can estimate each block equation by equation by OLS, delivering the same results as FGLS or full information maximum likelihood. This is a result due to Kruskal [1960]. The reduced form can be transformed into a VAR-model (see appendix):

  f_tm Y_t1 Y_t2  = Φ0+ Φ1   f_t−1m Y_t−11 Y_t−12  + · · · + Φp   f_t−pm Y1 t−p Y_t−p2  + A   ξ_tm ξ_tl ut   , (4.11) where Φ0=   am a∗₁ a∗₂,  Φ₁ =   φmm1 φml 0 φ∗_1m1 φ∗₁₁ 0 φ∗_2m1 φ∗₂₁ 0  , Φ_j =   φmmj 0 0 φ∗_1mj 0 0 φ∗_2mj 0 0  

for 2 ≤ j ≤ p, where j indicates the jth column of the coefficient matrix. The matrix

A =   INm 0 0 ψ Bl₁ 0 φ21ψ + φ2m1 φ21B1l INe  

transforms the original errors.

This specification allows direct calculations of IRs. We do not need to know all the parameters that determine the model. The risk neutral dynamics and the parameters for the short rate are not necessary in order to obtain the IRs of the original model with one lag. Actually, when the reduced form is estimated, the only thing we have to do is infer B1 and B2 and the coefficient matrix for the factors ρ under the objective measure. This poses no restrictions on the matrix B if there are no restrictions on the risk neutral parameters (see Hamilton and Wu [2012]

4.2 Mapping between structural and reduced form parameters

The benefit of estimating the reduced form is seen when the structural parameters are inferred from the reduced form parameters for the model with one lag. From the derivation of the reduced form we have that

φmm = ρmm− ρmlB1l−1B m 1 (4.12) φml = ρmlB1l−1 (4.13) φ1m= B1lρlm− B1lρllB1l−1B1m (4.14) φ1l = B1lρllB1l−1 (4.15) ψ = B₁m (4.16) φ2m= B2m− Bl2B1l−1B m 1 (4.17) φ2l = B2lB1l−1 (4.18)

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Struc. form # parameters Red. form # parameters ρmm NmNm φmm NmNm ρml NmNl φml NmNl ρlm NlNm φ1m NlNm ρll NlNl φ1l NlNl B₁m NlNm φ2m NeNm B₁l NlNl φ2l NeNl B₂l NeNm ψ NlNm B₂m NeNl

where ( 4.14) and ( 4.16) nicely hold because we consider the case for one lag only. If the parameters are listed, we see that there are N_l2 parameters that we cannot infer from the reduced form parameters. The variance of the residuals in the reduced form equation for Y_t1 is V (B₁lξ_tl) = B₁lV (ξ_tl)Bl₁0. It is commonly assumed that the latent factors are independent and therefore V (ξt) = INl. Then it is straightforward to estimate B

l

1 by numerically solving B₁lB₁l0 = ˆV (B₁lξ_tl) or equivalently vech(B₁lB₁l0) = vech( ˆV (B₁lξ_tl)), where ˆV (Bl₁ξ_tl) is just the average outer product of the residuals dBl

1ξtl and vech stacks the elements on and below the diagonal into a vector. Once B₁l is calculated, the rest of the structural parameters can be calculated analytically by ˆ ρml= ˆφmlB[l−11 (4.19) d B₁m= ˆψ (4.20) c Bl₂= ˆφ2lBcl 1 (4.21) d B₂m= ˆφ2m+ cB₂lBcl 1 −1 d B₁m (4.22) ˆ ρll= cB1l −1 ˆ φ1lBcl 1 (4.23) ˆ ρlm= ˆφ1m+ ˆφ1l (4.24) ˆ ρmm= φmm+ ˆρlmBcl 1 −1 (4.25) As our interest lies in the effect of a shock in the macro economic factors on yields and we can compare the specifications, the parameters of the VAR specification ( 4.11) are expressed in terms of the structural parameters. First consider the structural specification ( 4.26). When the process for the factors is included, it can be written as:

Y_t1 = A1+ B1ρFt−1+ B1ξt (4.26)

Y_t2 = A2+ B2ρFt−1+ B2ξt+ ut ut∼ N (0, ΣuΣ

0

u) , which leads to the coefficients of the macro factors

[B₁mρmm+ Bl1ρlm] for yields Yt1 and (4.27) [B₂mρmm+ Bl2ρlm] for yields Yt2. (4.28)

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In the specification ( 4.11), the coefficients of the macro factors for the yields are

φ∗_1m= φm₁ + ψφmm and (4.29)

φ∗_2m= φ21φ∗1m+ φ2mφmm. (4.30) When the applicable equations from ( 4.12)-( 4.18) are substituted in ( 4.29), we get

φ∗_1m= B₁mρmm+ B1lρlm− (B1lρll+ Bm1 ρml)Bl−11 Bm1 (4.31) φ∗_2m= B₂mρmm+ B2lρlm− (B2lρll+ Bm2 ρml)Bl−11 Bm1 . (4.32) The apparent difference between the coefficients is determined by the last cross terms in the equations in ( 4.31).

When the impulse response functions are compared, both contain a matrix that is taken to the n-th power to obtain the coefficients for the effects of a shock n periods ago. These matrices are Φ1 =   φmm1 φml 0 φ∗_1m1 φ∗₁₁ 0 φ∗_2m1 φ∗₂₁ 0  ρ = ρmm ρml ρlm ρll

for the reduced form VAR and for the structural model respectively. The former is post multiplied by a matrix D and the latter premultiplied by a matrix B and postmultiplied by a matrix Σξ in order to get the complete impulse responses. A way to explore similarities as we found above is to diagonalize the matrices of which powers are taken and subsequently compare the two matrix series. This route is not pursued here, as it is out of the scope of this thesis.

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5 Data and estimation

5.1 Choice of variables

From Section 3, it is clear that there is a multitude of options when it comes to choosing which variables to include in the model. We restrict ourselves to including inflation and expected inflation. Other variables, such as the output gap or unemployment could also be included. We assume that these variables will be controlled for by the latent factors.

For estimation we used the euro swap rates to construct the zero coupon yield curves. These are available from January 1999. We assumed that the 1, 5 and 30 year yield were measured without error and the 3, 10 and 20 year yield were measured with error.

For inflation, we used the HICP All Items for the Eurozone. This measure is available from January 1996. Initially HICP ex. Tobacco was considered, but this measure goes back to January 2001 only. The two indices show a correlation larger than 0.99.

In general there are two data sources that can be understood to measure expected inflation: Market data of inflation linked securities and survey data. The first principal component of the survey of the European Commission and the inflation linked euro swap was taken as our expected inflation. The inflation linked derivative has been quoted since June 2004. The survey is held since 1999. As the survey is published quarterly, interpolation was used in order to get monthly data. The realized and expected inflation data are monthly year on year data, to account for seasonality. The data of 2013 is left out with the intent of testing the model.

A note is in order here about the timing of the variables. In the model, a variable has a time indication t. However, realized inflation at this point is the inflation in the period (t − 12, t). Expected inflation and yields at t respectively cover the period (t, t + 12) and (t, t + 12 · M ), with M maturity in years. The two inflation measures and the short and long yield are shown in Figure 3. First the inflation variables are studied in their own right. An augmented Dickey Fuller test is performed on both. It rejects the null of a unit root for the expected inflation series when three differenced lags are included in the auxiliary regression. A unit root in the inflation series is not rejected. The two series exhibit a correlation of 0.77. Next the ACF and PACF are shown in Figure 4. The ACFs of both series are similar and show significant autocorrelation for a long horizon. The PACF of inflation shows a lot of lags being significant, when controlled for all the previous lags. The PACF of expected inflation cuts off after one lag. The ACFs indicate that expected inflation is more persistent, but the PACF and unit root of the inflation series contradict this conclusion.

The apparent differences between the time series’ properties are in line with findings of Sun and Phillips [2004] and Mishkin [1992]. As the series behave differently, they might play a different role in the dynamics we’re about to study.

As a preliminary study, the 1-year yield and the 30-year yield are regressed on the con-temporaneous values and one lag of realized and expected inflation as well as on one lag of the 1-year yield and the 30-year yield. The unit root and the high correlation of the series should make us very careful in giving any interpretation to the estimated parameters. The results are in Table 1a and 1b.

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(a) HICP EU AI, (1999-2012) (b) Principal component of the EC survey and the BE expected inflation,(2004-2012)

(c) The 1 year zero coupon yield, (1999-2012). (d) The 30 year zero coupon yield, (1999-2012).

Figure 3: Source: Bloomberg, EuroStat

coefficients t-values c 0.0019 1.5062 inflation 0.0923 1.5995 exp. inflation 0.0026 6.4259 inflation(-1) -0.1881 -3.2827 exp. inflation(-1) -0.0012 -2.8002 1 year yield(-1) 0.9406 46.5590 30 year yield(-1) 0.0370 1.2662

(a) regression of 1-year yield

coefficients t-values c 0.0923 3.1996 inflation 0.0228 0.2979 exp. inflation 0.0019 3.5343 inflation(-1) -0.1249 -1.6418 exp. inflation(-1) -0.0015 -2.5353 1 year yield(-1) 0.0692 2.5819 30 year yield(-1) 0.8673 22.3730

(b) regression of 30-year yield

Table 1: Results of a simple regression of the yields on their lags and contemporaneous and lagged values of the inflation measures.

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(a) ACF of inflation (b) ACF of expected inflation

(c) PACF of inflation (d) PACF of expected inflation Figure 4: time series properties of the realized and expected inflation series

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as the yields are highly autocorrelated. Furthermore, we see positive coefficients for the contemporaneous values of inflation and expected inflation, but negative values for their lagged values. The t-value of the 1 year yield for the 30 year yield is twice the t-value of the 30 year yield for the 1 year yield. This feeds the idea that there is much information in the 1 year yield.

The t-values in the regression of the 30 year yield are a lot smaller than the ones in the regression of the 1 year yield, except for the constant, which is more significant and for the coefficient for the lagged expected inflation, which is only slightly smaller. As the inflation measures are autocorrelated and highly correlated with each other, it is interesting to see that the t-values are still large, certainly for the regression of the 1 year yield. This finding adds to the idea that expected inflation and realized inflation should be treated differently.

5.2 Parameter estimation for the one lag models

In the previous section we compared the coefficients for the lagged values of the macroeco-nomic factors in the reduced form and the structural model with one lag. The reduced form was estimated and the structural parameters were derived. The vector of macroeconomic factors is f_tm = (πt−1,t πt,t+1e )0. Two tables in appendix C list the parameters in matrix form and in parentheses the standard deviations are given. The standard deviations were cal-culated using a block bootstrap technique. The standard deviations for the reduced form can also be derived using standard OLS techniques, because the reduced form was estimated with OLS. However, the structural parameters are derived from the reduced parameters and hence the inference of variance becomes difficult. To compare the variances, a single method to obtain both is desirable. These arguments lead to the use of a bootstrap method. Blocks of five observations are resampled into 999 vectors of original length. Hall et al. [1995] indicate that blocks of size n13 should be used when estimating the variance, which comes down to

10313 _{u 4.7 This technique preserves time series characteristics such as the autocorrelation,}

which is a dominant feature.

First we look at the estimates for the reduced form. We see that the lagged values of the macrovariables determine the current values for the bigger part. The coefficient of expected inflation for realized inflation is not significant1 and vice versa. The coefficients in φml are all insignificant.

The same observation holds for the yields in Y_t1. The lagged values are highly signifi-cant, but the macro factors do not influence these yields significantly in the reduced form estimation.

For the yields in Y_t2, things are different, because there are no direct lagged values in the VAR equation. For the 3-year yield, the coefficient for realized inflation is significant. For all three yields, expected inflation is significant. The coefficient is positive for the 3-year yield and negative for the 10- and 20-year yield. In addition, most coefficients for the lagged

1

Insignificant means here that the t-ratio, based on the bootstrap standard error, is smaller than 2 in absolute value. We will use this definition in the rest of the thesis. This is a rough measure, because of the use of the bootstrap technique and the small number of observations. However the goal here is not to investigate validity and properties of these uncertainty measures.

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values of the yields in Y_t1 are significant. The 30-year yield however seems not to influence the 3-year yield.

The estimates of ψ are puzzling. ψ denotes the effect of contemporaneous values of the macro variables on the yields Y_t1. There is no significant effect observed here, which indicates that the contemporaneous values of the macro factors add little to the information that is in the lagged values of the yields.

The matrix B₁l determines a lot of the parameters. This matrix was estimated from the residuals of the SUR regression. The standard errors indicate that it differs a lot over different bootstrap samples, which might be problematic.

For the structural model, we first consider the process for the factors. The macroeconomic factors are largely determined by their lagged values. Realized inflation has a significant negative effect on expected inflation, which is strange. In addition, expected inflation has a significant positive effect on realized inflation. The latent factors have very little influence on the macro economic factors. The latent factors themselves are determined by their lagged values and the macroeconomic variables. Realized inflation has a negative effect on the latent factors, whereas expected inflation has a positive effect.

When we look at the coefficient matrix B, it appears that the coefficients for the macroe-conomic factors are significant and the coefficients for the latent factors are not. This would imply that yields are almost completely determined by the macroeconomic factors. The co-efficients for both realized inflation and expected inflation are positive, except for the 3-year yield, on which only expected inflation has a significant, positive, effect.

It is interesting to see that in the reduced form estimation, the yields are determined largely by their lagged values, but in the structural specification, it is the macro factors that have a significant effect on the yields and not the latent factors. Hence, there is some information in the yields that is not in the latent factors.

For the models to be stationary, we can check on the eigenvalues of the matrices Φ1 and ρ. All eigenvalues are within the unit circle. Eigenvalues close to unity show that the processes are quite persistent.

5.3 Factor analysis

There are some stylized results in interest rate models with latent factors. The variation in the yields is well explained by the three principal components of the yield time series. It is questionable whether this is due to some underlying macro economic forces or a mere artefact of highly correlated series. The factor loadings2 are plotted in Figure 5a. They nicely follow the patterns of level, slope and curvature, that are found in many studies (e.g. Litterman and Scheinkman [1991]). The loadings of the curvature factor are quite high for the 30-year yield, which could reflect the unusual movement of the curve in recent turbulent times.

The principal components are plotted in Figure 5b. They account for 99.8% of the variation in the yield data. For comparison, the 1- and 30-year yield time series are plotted underneath.

2

The loadings of a principal component are the elements of a normalized eigenvector of the covariance matrix of the yields. These elements are multiplied by the respective yields to obtain the principal component. In this case, the first loading of the n-th factor corresponds to the contribution of the 1-year yield to the n-th factor.

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(a) The loadings of the three principal compo-nents. The x-axis shows the position of the yields, ordered by maturity.

(b) The first three principal components. The first shows a lot of variation and this diminishes for the second and third. The x-axis shows time in months.

(c) The latent factors plus a constant as in Ftl+

Bl−1₁ A1 = B1l−1(Y 1

t − B1mFtm) for the model

with one lag. The x-axis shows time in months.

(d) Time series of 1- and 30-year yield for com-parison. The last 103 data points should be compared with the principal components. The x-axis shows time in months.

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1 · 10−3 1 · 10−1 1 · 10−1 1 · 10−1 variance inflation exp. infl inflation exp. infl explained

pc1 2.2875 0.7024 -0.0022 0.5597 93.20% pc2 4.9818 0.5607 0.0013 0.2554 99.28% pc3 2.7467 0.4846 -0.0004 0.3131 99.80% infl - - - 0.5620 exp.infl - - 0.0078 -se1 0.6595 0.0651 0.0007 0.0502 se2 0.6595 0.0651 0.0007 0.0534 se3 0.6595 0.0651 0.0007 0.0626 se4 - - 0.0009 0.546 R2 0.9174 0.7068 0.9504 0.8239

Table 2: Results of a simple regression of realized and expected inflation on the principal components. On the right the cumulative portion of the variance of the yields by the factors is given.

The first principal component has peaks in common with the 1-year yield. The other two principal components are not easily linked to movements in the yields. When the principal components are compared with realized and expected inflation, there is less apparent common movement if any at all. The sharp increase-drop-increase pattern that is in both the inflation series and not so much in the yields cannot be found in the principal components either.

The estimated latent factors from the structural model with both inflation and expected inflation and one lag are in Figure 5c. They are calculated as F_tl+ Bl−1₁ A1 = B1l−1(Yt1− B₁mF_tm) so a small constant is added. The first latent factor clearly has some features, notably the middle peak, that are both present in the 1-year yield and the first principal component. The second latent factor seems highly correlated with the first one. The third latent factor is different from the first two and follows the 30-year yield.

It is in our interest to know whether the macro factors correspond to (a linear combination of) the principal components of the yields. The latent factors are assumed to account for these components, but as we saw the macro factors do a better job in explaining the yields in the structural model. A simple regression of the macro factors is performed to examine this a bit further.

The results are in Table 2. The coefficients for the variables and their OLS standard deviations are given. A constant is included in each regression but not in the table. Variations of realized inflation are for 92% explained by the three principal components versus only 71% of variation in expected inflation. This is in line with the observation of Chernov and Mueller [2012] that expected inflation exhibits a hidden factor that is not present in the principal components or realized inflation.

When expected inflation is added to the regressors, the results in the third column tell that the coefficients of the second and third principal component have become insignificant. The other way around this is not the case. All three coefficients in the fourth column become lower, but stay significant. This too is in line with a hidden factor present in the expected inflation series.

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5.4 IRs for the reduced form

5.4.1 Realized inflation only

Model selection is not a trivial matter in this case. First, the model that we use for analysis, the VAR-model as specified in ( 4.11), is not directly estimated. The reduced form in ( 4.8), ( 4.9) and ( 4.10) is estimated, after which the VAR is constructed. It is possible that the resulting VAR model is not stationary. Stationarity is needed in order to perform our analysis, so only stationary models can be selected. The number of lags to include can be decided upon by looking at the AIC, corrected for the small sample size. The model was estimated initially with inflation only. This means the vector in ( 4.11) is f_tm Y_t10 Y_t200=

πt−12,t Y1 0 t Y2 0 t 0

We estimated the reduced form for the complete time span of the inflation series, for the time span of expected inflation and for the time span that takes in the data before the big drop in 2009 (see Figure 3). The results are given in Figure 6, which makes analysis of the models with different lag lengths easy. Per lag inclusion, the AIC and stationarity are determined. Furthermore, we found autocorrelation in the residuals. The Figure shows the number of lags in the residual series for each variable in ( 4.11) for which the null of zero autocorrelation was rejected. As autocorrelation points in the direction of misspecification, the models with the lowest AIC and the lowest autocorrelation should be compared.

When the entire time span is considered (Figure 6a), the model is stationary for one lag, becomes non stationary for six lags and then is stationary again for ten and eleven lags. The AIC steadily increases with lag length and therefore selects the model with only one lag. There is heavy autocorrelation in the residuals. Notably, the measurement errors are profoundly autocorrelated, as all twelve lags of the measurement error of the 10 year and the 20 year yield show significant autocorrelation, no matter how many lags are included.

For the later period (Figure 6b) a different picture emerges. Stationarity is observed for the model with lag lengths of five and higher. The AIC exponentially increases, and suddenly becomes negative when the eighth lag is included. This anomaly might be due to problems with computing the inverse of the covariance matrix. However, when the identity is added to the covariance matrix, the same pattern appears. The AIC selects the model with five lags if the large negative numbers are the complication of calculation difficulties and eight lags if they are not. The autocorrelation is less than for the entire time span but still substantial.

For the first 110 periods (Figure 6c), before the big drop in inflation, the model is always stable, the AIC selects the model with one lag or the model with nine lags. The autocor-relation is a lot less for the errors in the first two blocks in ( 4.8) and ( 4.9), but still very much present in the measurement errors. In Figures 7a- 9b the responses of inflation and the yields on a one standard deviation increase in inflation are given, along with confidence intervals, again constructed with the block bootstrap technique, for models with different lag lengths over different time spans. A note must be made here on the confidence intervals. Usually, these are interpreted to indicate the uncertainty of predicted values. In this case they must be interpreted differently. They indicate what the variation is in the IRs over different bootstrap samples. A small confidence interval for the IR therefore indicates that most IRs retrieved from the bootstrap samples adhere to the IR found for the original sample.

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(a) 1999-2012

(b) 2004-2012

Figure 6: Descriptive statistics of the constructed model with inflation only. Per extra lag included, AIC, stationarity and the number of lags in the residual series of ( 4.11) are shown.

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(a) (1999 - 2012), 1 lag

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(a) (2004-2012), 5 lags

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(a) (1999-2008), 1 lag

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A counterintuitive result is the reason for the inclusion of so many figures. For inflation, we see an initial positive response, which quickly declines. For the yields however, hardly a significant positive response is found and we see a significant negative response after some time. This is not in line with theory, that says higher inflation means higher yields. This observation is not due to some unlucky lag length selection or the big drop in inflation, but found for all estimations.

However, it is in line with the observations made in Section 3. The Fisher equation was rejected in several studies. In addition, central bank policy focuses on expected inflation rather than on realized inflation. Another feature that is present in all six figures 7a- 9b is that the response seems to be stronger for yields on the short end of the curve than the yields on the long end.

The dynamics differ substantially when different lag lengths are considered for the same period. For the entire time span, the model with one lag (Figure 7a) delivers smooth IRs. For inflation, it starts positive, diminishes quickly and goes to zero. For the yields, the initial response is positive, but quickly turns negative. With maturity, the IR becomes less steep. For longer maturities, the IR goes up at the end, whereas for the yields on the short end of the curve, it decreases. It should be noted that for the yields that were assumed to be measured with error, the initial response at t = 1 is very much different from the response of the other yields. For yields with maturity 10 and 20, it is negative and for the yield with maturity 3 very low.

When five lags are included, a different picture emerges (Figure 7b). The IRs of both inflation and yields show a positive response first, that increases in the initial period. Sub-sequently, the responses diminish quickly and stabilize. This shows that lag length is very important, as it heavily influences functional forms and values of the IRs.

For the period 2004-2012, the model with 5 lags (Figure 8a) shows an initial increase in the IRs of both inflation and yields. Then after a big drop, the responses are all negative around 20 months after the shock. A recovery of the responses leads to positive responses from inflation after 35 months, after which the responses die out. The responses from yields go up a little bit but stay negative.

For the model with eight lags (Figure 8b), the behaviour of the IRs is less smooth than before. Moreover, in the initial period, the IR becomes more complex. Remember that this was a model with very low AIC. This could have something to do with the strange initial period. The IRs are similar to the ones from the model with 5 lags, but more complex and volatile in the initial period.

The estimation of the model over the period 1999-2008 was purely run to check for robustness. It might be that the drop in inflation in 2009 could account for the observed counterintuitive results.

The model with one lag gives the IRs in Figure 9a. The response of inflation starts at 1 and declines quickly to zero. The responses of yields are near zero initially and show a sharp decrease, after which the IRs increases but stays negative. The IRs for the 10, 20 and 30 year yield are quite comparable to the IRs in Figure 7a.

For the model with nine lags, the IR of inflation (Figure 9b) first steeply drops to zero, then becomes negative but stays close to zero. The yields show an initial response of around zero. Then a drop is observed after which the responses go to zero. This model again

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Figure 10: Descriptive statistics of the model with expected inflation only, 2004-2012.

showed a very low AIC and the complicated non smooth development of the responses is again observed.

Summarizing, a positive response from yields, which was hypothesized, is only seen in the first few months. The confidence intervals indicate that this positive initial response is sample dependent. After that, the responses from the yields to a shock in inflation are negative for most of the estimations.

5.4.2 Expected inflation only

In Section 3 it became clear that earlier work with inflation in affine term structure models delivered interesting impulse responses for inflation and expected inflation. Notably, the responses to inflation were not that profound and expected inflation had a hidden factor in it that improves the model. Now, the model is estimated with expected inflation as macro factor, leaving out realized inflation.

The vector in ( 4.11) is now f_tm Y_t10 Y 20t 0 = πe_t,t+12 Y_t10 Y_t20 0 . Estimation results are in Figure 10.

The model is stationary for lag lengths one, two, eight and nine. The AIC selects the model with one lag and eight lags. Autocorrelation is again present, but far less than in the model with inflation. The IRs are given for the models with one and eight lags in Figure 11a and 11b, respectively.

Recall that expected inflation is the principal component of break even inflation and survey data. This explains the scale of the responses of yields in comparison to the original shock. First consider the IRs of the model with one lag in Figure 11a. For expected inflation itself, we see that the response reduces to negative values after around 40 periods and then stays close to zero.

For the yields on the short end of the curve, the IR increases steeply in the first 10 months, after which it steadily declines. For the 5 year yield, we see a positive response that becomes smaller at first, but then it becomes larger again to a maximum at around 25 months. We see this same pattern for the longer maturity yields, with the maximum shifting to the right, to 40 months for the 30 year yield.

The model with 8 lags shows less smooth IRs (Figure 11b). The response from expected inflation first goes up and then steeply decreases, to become slightly negative after around 40 months. The yields show a positive, upward sloping IR initially. After a few months, the

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(a) (2004-2012), 1 lag

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(a) 2004-2008

Figure 12: Descriptive statistics of the model with both inflation and expected inflation.

IRs decrease steeply to more or less stable values for the 1 and 3-year yield. The IRs for the other yields are clearly upward sloping after the decrease.

The two IRs from the models with expected inflation only have two important common-alities: 1) the responses from yields to shocks in expected inflation are significantly positive and 2) the yields on the short end of the curve react stronger to shocks in expected inflation. 5.4.3 Both inflation and expected inflation

It remains to see what happens if both the realized and expected inflation are included in the model. Now the vector of the VAR in ( 4.11) is

f_tm0 Y_t10 Y_t200 =πt−12,t πet,t+12 Y1 0 t Y2 0 t 0 .

We should see some effects that were present in the IRs when the series were included separately. Some characteristics could disappear as well, because only one of the two variables is responsible for a certain characteristic and they are highly correlated.

From Figure 12a, it becomes clear that only the model with one lag proves to be stationary. This is strange, as the other estimations gave multiple lag lengths for which the model was stationary. Apparently the inclusion of more lags of both realized and expected inflation complicates the dynamics.

In Figure 13a and 13b the responses of the yields and (expected) inflation to shocks in respectively inflation and expected inflation are shown. The response to a shock in inflation of inflation itself is positive and declines quickly to zero. The same is practically the case for the response of expected inflation. At first, there is a strong response, but it decreases to low values quickly.

The yields show an initial small response to a shock in inflation, but become negative after a few months. The responses then increase again towards zero. Taking into account the confidence intervals, no significant positive response from any yield is observed. A few months after the shock, the responses from yields are significantly lower than zero.

The responses to a shock in expected inflation show different graphs. Expected inflation itself responds positive and decreases to negative values after 30 months, after which the response goes back to zero. The response from inflation is positive at first, grows to a maximum at around 15 months and decreases to a negative minimum after 40 months.

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(a) responses to a one standard deviation shock in inflation shock, (2004-2012), 1 lag.

(b) responses to a one standard deviation shock in expected inflation.

Figure 13: Responses of inflation, expected inflation and yields to a shock in expected and realized inflation, (2004-2012), 1 lag.

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The yields show an initial positive response, which increases first to a maximum that occurs after around 15 months for the 1-year yield. This maximum shifts right with longer maturities and is observed after around 25 months for the 20-year yield. The 30-year yield shows a IR that is a bit different from the IRs of the other yields. It is less concave and has a maximum that is less to the right. However, the response stays relatively high for a longer time. The responses from yields to a shock in expected inflation are significantly larger than zero after the first couple of months, in which the variance is apparently quite large. The response from inflation looks like the responses from yields but is very different from the response of expected inflation.

5.4.4 Comparison of results of the reduced form IR’s

Summarizing the observations from the IRs analysis, it can be said that an inflation shock has a non significant, positive effect on yields in the short run and a significant negative effect on the long run. Yields on the short end of the curve react more strongly to shocks in inflation, which is indicated by the larger IRs.

Shocks in expected inflation lead to significant positive responses of yields over a horizon of 60 months. Yields on the short end of the curve show larger responses than the yields on the long end.

In two of the IRs, 11a and 13b, the maximum of the IR shifted right as yields with longer maturity are considered. Thus there is some evidence that the yields with a longer maturity do not only react less but also delayed to shocks in expected inflation.

The IRs show that lag length is very important. With the inclusion of more lags, the IRs become more complicated. Especially in the first few months after a shock, the differences are large. This impairs the accuracy of estimates for policy purposes or other practical use.

Compare Figure 7a and 13a. The responses to a shock in inflation seem to be less in the model with both expected and realized inflation. Notably, the yields on the long end show a minimum of -0.30%-point in 7a and this mininum is -0.25%-point in 13a. In addition, the responses from yields to an inflation shock become more alike. At first, the responses from the 1,3 and 5-year yield were strictly decreasing on the entire 60-month horizon. When expected inflation is added to the model, all the yields show a ’boomerang-like’ response curve, that starts positive, has a negative minimum and stays negative.

Figures 11a and 13b can be compared as well. In Figure 11a, the response is declining initially for the 5,20 and 30-year yield. The movement of the maximum to the right with maturity is large. In Figure 13b the maxima of the responses are considerably higher, but towards the end of the 60-month horizon the impulse responses are of the same level in both figures. The responses from the yields are all steeply increasing in the first period. The initial decrease in the response from the 5, 10 and 20 year yield is gone. The movement of the maximum response to the right as maturity goes up is less profound.

5.5 IRs for the structural form

The mapping from the reduced form parameters to the structural parameters as in section 4 is only straightforward for models with one lag included. When more lags are in the models,

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estimation is still possible but rather involved. Therefore we only analyze the IRs of the structural model with one lag of both realized and expected inflation.

They are in Figure 14a and 14b. There is very little difference from the IRs of the reduced form. Actually it’s hard to distinguish between the Figures with the bare eye. The transformation from the reduced form to the structural form implies some restrictions and does affect the parameters and their variances, as was shown earlier. The IRs are virtually the same and the block bootstrap technique delivers confidence intervals that are much alike. As it is uncommon to find a negative effect of inflation shocks on yields, we check for robustness by estimating an unrestricted VAR with one lag, in which we included both macroeconomic variables and the yields that were assumed to be measured without error. The IRs are in Figure 15, which indicates that these are not very different from the IRs from the model with no arbitrage restrictions. The AICs for the unrestricted VAR that we got as output did not show the strange pattern we observed for our restricted VAR. In addition, the AICs were lowest for the specifications with the maximum possible lag length. This is an indication that the restrictions we imposed by estimating the reduced form do have implications. We did not examine this in more detail and it is left for further research.

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(a) Responses of yields to a one standard deviation shock in realized inflation.

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Figure 15: Responses to a shock in realized and expected inflation. The two upper left graphs show the responses of realized inflation, the upper right the responses of expected inflation and the other graphs show the responses of the yields in Y1

t in increasing order.

6 Conclusion

We started out to examine whether a shock in inflation or expected inflation had an effect on yields. This matters for investors, and especially investors with long term liabilities that grow with inflation.

Without doubt, our most surprising finding is that a shock in realized inflation triggered a negative response of all yields after some months. This finding is robust over different time spans and lag lengths. It answers our first question and rejects the hypothesis that the response of yields to an inflation shock is positive. The impulse response functions differed a lot among models with different lag lengths, but in general the responses showed big movements in the first 1 5 to 25 months and diminish after that. As the impulse responses do not generally decrease monotonically and big movements are seen a couple of months after the shock, the hypothesis that the IRs monotonically diminish over time is not supported. After the big movement the IRs die out, but IRs of a stationary model always go to zero in the end.

For shocks in expected inflation, the responses of yields are positive. Next, we did see that the maximal response occurred several months after the shock. Therefore, the hypothesis that the response is diminishing with time is rejected. For the responses along the term struc-ture it is found that they generally become less with maturity. Thus our second hypothesis stands.

When both realized and expected inflation are included in the model, we find that the responses of yields to an inflation shock become less and the responses to a shock in expected

The effect of a shock in (expected) inflation on the term structure : analysis using an ane term structure model with macro factors.