The expected real yield and inflation components of the nominal yield curve

North American Journal of Economics and Finance 39 (2017) 1–18

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North American Journal of Economics and Finance j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e c o fi n

The expected real yield and inflation components of the nominal yield curve Ronald H. Lange Economics Department, Laurentian University, Sudbury, Ontario P3E 2C6, Canada

a r t i c l e

i n f o

Article history: Received 1 June 2016 Received in revised form 25 October 2016 Accepted 26 October 2016

JEL Classification: E43 E44 E52 Keywords: Yield curve Real interest rates Expected inflation, factor model State-space model Regime-switching estimation

a b s t r a c t The term structure of real yields and expected inflation are two unobserved components of the nominal yield curve. The primary objectives of this study are to decompose nominal yields into their expected real yield and inflation components and to examine their behaviour using state-space and regime-switching frameworks. The dynamic yield-curve models capture three well-known latent factors – level, slope, and curvature – that accurately aggregate the information for the nominal yields and the expected real and inflation components for all maturities. The nominal yield curve is found to increase slightly with a slope of about 120 basis points, while the real yield curve slopes upward by about 20 basis points, and the expected inflation curve is virtually flat at slightly above 2 per cent. The regime-switching estimations reveal that the nominal yield, real yield and expected inflation curves have shifted down significantly since 1999. Crown Copyright Ó 2016 Published by Elsevier Inc. All rights reserved.

1. Introduction Generally, real yields are expected to be a more important determinant for the short-end of the nominal yield curve because of the influence of monetary policy shocks, while inflation expectations are expected to be relatively more important for the longer end because of the influence of shocks to inflation expectations. Although the nominal yield curve is widely used for pricing bonds, as well as for forecasting output and inflation, there has been little research on these two key determinants of the yield curve largely because they are unobservable. The real yield curve is particularly important for monetary policy since real yields affect all intertemporal savings and investment decisions in the economy. Real yields are also important because they are often used to gauge the stance of monetary policy relative to the neutral or natural real rate of interest in a Taylor-type rule. In addition, real yields are of interest to investors that focus on the cost of capital. The term structure of inflation expectations is very important for monetary policy in a country like Canada that has adopted explicit targets for inflation. Inflation expectations also play a crucial role in hedging strategies of portfolio investors. Consequently, both policy makers and market participants have an interest in the central roles that expected real rates and inflation play in the dynamics of asset prices over time. This study attempts to fill this gap in the literature by capturing some stylized facts about these two important unobservable components of the nominal yield curve in Canada. A previous study by Ang, Bekaert, and Wei (2008) uses an affine

E-mail address: [email protected] http://dx.doi.org/10.1016/j.najef.2016.10.016 1062-9408/Crown Copyright Ó 2016 Published by Elsevier Inc. All rights reserved.

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R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

term-structure model to identify the structure of real rates and inflation risk premia to fill this gap for the U.S. In general, the affine term-structure models tend to be restricted in the range of term structure yields that they can handle in order to obtain closed form solutions. Some previous studies also rely on survey data for inflation expectations to derive expected real yields and risk premia. For example, Joyce, Lildholdt, and Sorensen (2010) use survey data on long-term inflation expectations for the U.K. Similarly, Chernov and Mueller (2012) use information in the term structure of survey-based forecasts of inflation to estimate a factor hidden in the nominal term structure of yields. However, survey data on inflation expectations differ considerably in their breadth of coverage, the frequency and time span over which they are accessible, and the extent to which they may be distorted by certain biases due to the heterogeneity of agents surveyed. This study on Canada contributes to the literature by overcoming some of the limitations of these previous studies by first decomposing the nominal yield curve into its expected real rate and inflation components using structural VAR techniques instead of relying on survey data. Secondly, the study uses a state-space methodology that estimates dynamic latent factors for the level, slope and curvature of the three term-structure curves. These yield curve factors are able to quite accurately aggregate information from a relatively larger set of term structure data than that restricted by the affine term-structure models in the finance literature. The empirical methodology in this study proceeds in four steps. First, the nominal yield curve is decomposed into the term structure of theoretical yields that are consistent with the expectations hypothesis plus a rolling term premia using the stationary vector-stochastic process in Campbell and Shiller (1987, 1991). Second, the longer term theoretical yields are then decomposed into ex ante real yields and expected inflation using the structural VAR methodology developed by Blanchard and Quah (1989), where structural shocks are identified by the long-run restriction that inflation expectation shocks have a permanent effect on longer term yields, while ex ante real rate shocks have only a temporary effect. Third, the nominal yields and the real and inflation components are estimated using the dynamic Nelson and Siegel (1987) model in a state-space framework to capture three latent factors – level, slope and curvature – that can accurately capture the three term-structure curves. Finally, a Markov-switching VAR framework is used to capture regime shifts in the variances and in the contemporaneous responses of the level, slope and curvature factors to innovations in inflation and the monetary policy rate, the primary nominal and real sources of the regime switches. To preview the conclusions, the yield curve decompositions capture credible stylized facts about ex ante real yields, expected inflation, and the term premium. The dynamic yield-curve models identify a nominal yield curve that increase from about 5 per cent to slightly over 6 per cent with a slope of over 120 basis points, an ex ante real yield curve that slopes upward by about 20 basis points to plateau at slightly less than 3 per cent, and an expected inflation curve that is virtually flat at slightly above 2 per cent. The regime-switching estimations stochastically divide the sample period into high- and low-variance regimes in about 1999 for all three curves. The dynamic yield-curve models for the two regimes indicate that the nominal yield curve has shifted down by almost 4.5 percentage points to about 3.50 per cent in the current regime, while the real yield curve has shifted down by almost 3.5 percentage points to about 1.25 per cent and the expected inflation curve has shifted down by over 1.0 percentage points to about 1.60 per cent. The following section briefly discusses some recent research on the state-space representation of the dynamic factor model for the nominal term-structure of interest rates and on the regime-switching behaviour of interest rates, including attempts to extract real yields from nominal interest rates. Section 3 outlines the methodology for the decomposition of the nominal yields and presents the results for nominal yields and the ex ante real yield and expected inflation components. Section 4 outlines the dynamic yield-curve methodology and discusses the results for the nominal yield curve and its components. The regime-switching approach used in this study and the results are presented in Section 5. The final section briefly discusses the implications of the empirical results for monetary policy and market practitioners and outlines some directions for further research. 2. Some previous research 2.1. Dynamic yield curve model he dynamic factor or yield-curve model can traced to Diebold and Li (2006), who provide essentially a time-series extension of the exponential components framework by Nelson and Seigel (1997) that assumes the term structure of interest rates to be a function of three unobservable components. They extend the framework by computing the values of the exponential factor loadings and using ordinary least squares to obtain three time-varying parameters that are interpreted as factors corresponding to the level, slope, and curvature of the yield curve. Diebold, Rudebusch, and Aruoba (2006) extend the model in a state-space form with the Kalman filter and a VAR transition equation. The extension allows for a simultaneous fit of the yield curve at each maturity and maximum-likelihood estimates, as well as optimal filtered and smoothed estimates of the underlying factors. They find that the three time-varying parameters in the state-space form of the Nelson-Seigel model can be estimated efficiently for the United States. However, they do not find evidence of strong own-dynamics of the term structure factors. Diebold et al. (2006) also extend the dynamic latent-factor framework further by complementing the empirical NelsonSeigel model with a nonstructural VAR model for real activity, inflation, and the monetary policy instrument. Overall, they find that causality from macroeconomic variables to the yield curve is much stronger than from the yield curve to the

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

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macroeconomy for the U.S. However, Lange (2013) found relatively strong bidirectional causality between latent yield-curve factors and the macroeconomic fundamentals for Canada using the dynamic latent-factor approach. More recently, Geyer, Hanke, and Weissensteiner (2016) evaluate the performance of inflation forecasts backed out from the nominal and real yield curves in the United Kingdom, using the Nelson–Siegel framework to model the break-even inflation term structure.1 They base their analysis on the 1-day break-even inflation derived from Nelson–Siegel factors, which avoids the need for a direct estimation of the inflation risk premium. 2.2. Regime switching in interest rates Many studies document regime-switching behaviour in nominal interest rates. For example, Hamilton (1988), Gray (1996), Sola and Driffill (1994), Bekaert, Hodrick, and Marshall (2001), Ang and Bekaert (2002), and Bansal and Zhou (2002) allow for regime-switching in mean reversion parameters. The use of regime-switching estimation to analyse the expectations theory of the term structure of interest rates can be traced to a important article by Hamilton (1988). He modelled the short rate as a univariate AR(4) process with the mean and variance driven by a 2-state, first-order Markov process. His model captured a shift in the short rate that coincided with the change in the Fed’s operating procedure in 1979–82 and seemed to generate expected future short rates that were consistent with the long yield being governed by the expectations hypothesis. However, Driffill (1992) re-estimated Hamilton’s model using different data for the bond yield to find that the model could no longer reconcile the data with the hypothesis. Sola and Driffill (1994) and Kugler (1996) extended Hamilton’s switching model to a bivariate VAR model in the spirit of Campbell and Shiller (1987, 1991). The specification in the first difference of the short rate and the term spread remedied the possible unit root problems that were inherent in Hamilton’s univariate model. Sola and Driffill broadly confirmed Hamilton’s timing of a regime switch in US short-term rates and were able to generate data with stochastic shifts of regime that are consistent with the expectations hypothesis. Bekaert et al. (2001) explored whether the regime shift in the US data on the term structure was due to a generalized ‘peso’ problem, in which a high interest-rate regime occurred less frequently in the data sample than was rationally anticipated by market participants. They estimated a 3-regime univariate model using data from seven countries in order to capture different draws from the same regime-switching process. However, they were unable to reject the expectations hypothesis only for shorter term maturities when regime switching induced by peso effects was combined with a small time-varying term premium. One strand of the research has also focused on the properties of regime switching in interest rates and the econometric performance of the models. Ang and Bekaert (2002a) used both univariate and term spread models, and a cross-sectional approach to show that 2-state models could replicate the nonparametric drift and volatility functions found in the literature. In a related study (2002b), they incorporated extra information from the term structure and from short rates in other countries to assess the performance of regime-switching models. They found that incorporating the extra information improves the overall fit of the models only for the USA and that adding the US interest rate improved regime classification for Germany and the UK. Overall, the research on regime switching of interest rates found similar results in univariate, multicountry, and term spread models. Typically, the models produced one regime with a unit root (or near-unit root) and lower conditional volatility, and a second regime with considerable mean reversion and higher conditional volatility. The first regime was usually associated with ‘normal’ periods when monetary policy smoothing makes interest rates behave like a random walk. The second regime was associated with periods when extraordinary shocks drive up interest rates, triggering more volatility and mean reversion. In general, there was very little improvement in the fit of the regime-switching models by allowing for state-dependent transition probabilities. 2.3. Expected real yields and inflation The anomalies in the nominal term structure data have also been examined by focusing on regime switching in the processes for inflation and real interest rates. Garcia and Perron (1996) found evidence for the constancy of the ex ante real rate over long time periods, but that a random walk component in the real rate subjects the mean to occasional shifts. Their model simultaneously identifies inflation and real factor sources behind the regime switches and analyses how they contribute to nominal interest rate variation. Evans and Wachtel (1993) documented the existence of inflation regimes. Evans and Lewis (1995) established that long-term movements in nominal rates reflect one-for-one movements in expected inflation generated by a Markov-switching model of inflation, which is consistent with the long-run Fisher relationship. Chen (2001) found that persistent shifts in the implied term premium still existed even after allowing for regime switching in inflation and real rates, which does not support the expectations hypothesis. Evans (2003) formulates a model with regime switches for U.K. real and nominal yields and inflation.

1 The break-even inflation rate is the gap between the yields on a nominal and an indexed-linked bond, which should be a good measure of inflation expectation according to the Fisher hypothesis. . .

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Most relevant for this study is the research by Ang et al. (2008) that specify a no-arbitrage term structure model with both nominal bond yields and inflation data to identify the term structure of real rates, expected inflation and inflation risk premia. They employ a three-factor representation of yields that consist of an observed inflation factor, which switches regimes, a latent regime-switching factor for the nominal term structure, and a latent factor that represents a time-varying but regime-invariant price of risk factor. Only the conditional mean and volatility of inflation and the term structure factor are allowed to switch regimes in their model. Their benchmark and best-fitting model allows for four regimes, that allows for different regimes for the real factors and inflation. Consequently, two different regime variables affect the drift and variance of the process of the latent term-structure factor and two different regime variables affect the drift and variance of the inflation process. Ang et al. (2008) find that a regime with low short real rates and high inflation occurs about 72 per cent of the time and switches between a regime with low and volatile real rates and inflation that occurs about 20 per cent of the time. They find that the term structure of real rates assumes a fairly flat shape around 1.3 per cent and that the real rate curve is downward sloping in some regimes. Expected inflation always approaches the unconditional mean of inflation of 3.94 per cent in their specification because inflation is assumed to be a stationary process. Their model matches an upward-sloping nominal yield curve by generating an inflation risk premium that increases in maturity on average to 1.14 per cent. The decompositions of nominal yields into real yields and inflation components at various horizons indicate that variation in inflation compensation (expected inflation and inflation risk premia) explains about 80 per cent of the variation in nominal rates at both short and long maturities. Some previous research has relied on survey data of inflation expectations to derive real yields and various risk premia. Joyce et al. (2010), for example, exploit information from UK real and nominal bonds using consensus forecasters’ expectations of average inflation from five to ten years ahead from 1992 to 2007 to construct inflation forecasts and inflation risk premia. Their affine term-structure model imposes no-arbitrage restrictions across nominal and real yields so that interest rates can be decomposed into expected real policy rates, expected inflation, real term premia and inflation risk premia. They find that expected real short rates and expected inflation account for about 60% of the variance of nominal 1-year forward rates, but that the contribution of expected inflation tails off at medium to long horizons where term and risk premia start to account for a larger portion of .of the variance. Chernov and Mueller (2012) also present an affine term-structure model where a ‘‘hidden factor” is extracted from inflation surveys. They show that this hidden inflation-survey factor is a significant price of the risk factor and that it improves the model’s forecasting performance of inflation and yields. Some recent research has also relied on inflation-indexed bonds to extract inflation expectations and risk premia. Abrahams, Adria, Crump, and Moench (2015), for example, estimate an affine term-structure model for the joint pricing of real and nominal bond yields for both the U.S. and the U.K. They explicitly accommodate liquidity risk premia and show that variations in nominal term premia are primarily driven by variations in real term premia rather than inflation and liquidity risk premia. Similarly, they find that their model substantially improves forecasts of inflation by including liquidity risk. 3. The nominal yield curve decomposition 3.1. Decomposition of longer term yields The yield to maturity of a nominal bond that matures in n periods, Rntþi , can be decomposed into two components: a weighted average of expected future yields to maturity of a 1-period debt instrument that matures in one period, Et rtþi , and a risk premium, Et hn;t , the so-called rolling premium:

Rnt 

n1 X wi Et rtþi þ Et hn;t ;

ð1Þ

i¼0

P where wi  g i ð1  gð1  g n ), g  1=ð1 þ R Þ, Et hn;t  n1 i¼0 wi Et hn1;tþi , and E denotes expectations conditional on information available at time t; R⁄ is the sample mean of Rnt , and Et hn;t , the rolling premium is a weighted average of the expected 1-period holding premia, Et hn1;tþi . Eq. (1) is an identity and does not impose any restrictions on the data. The modern version of the expectations hypothesis imposes the restriction that the rolling premium Et hn;t or the sequence of holding premia Et hn1;tþi are constant. In the pure version, the term premium is 0, so that Et hn;t ¼ 0. The weights wi sum to unity and decline monotonically with the horizon i, so expected short rates or holding premia of the near future carry a larger weight than expected short rates or holding premia of the distant future. The variables for the nominal long-term yield Rnt and the 1-period short rate rtþi in (1) are allowed to follow a linear stochastic process so that they are stationary in first differences rather than in levels. Consequently, the short rate can be subtracted from both sides of the term structure equation (1) and rearranged so that the term spread can be related to expected changes in the short rate: n1 X wi Et r tþi  r t  Rnt  r t  Et ht : i¼0

ð2Þ

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

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Eq. (2) can be written in terms of the stationary expected successive changes in the short-term interest rate Et r tþi as follows: n1 X fðg i  g n Þ=ð1  g n gEt Dr tþi  Rnt  r t  Et ht

ð3Þ

i¼1

In this specification, Drtþi is a stationary stochastic process and the term spread Rnt  rt is a stationary process. Following Campbell and Shiller (1987, 1991), the process of generating long- and short-term interest rates is modelled as a vector autoregression with two variables: the change in the short rate, Drt , and the spread between the long and short rate, St ¼ Rt  rt . The VAR system can be written as the following matrix equation:



Dr t St





¼

aðLÞ bðLÞ cðLÞ

dðLÞ



   l1t Drt1 ¼ l2t St1

ð4Þ

where the polynomials in the lag operators a(L), b(l), c(L) and d(L) are of order p. From the stationary specification in Eq. (3), the matrix equation (4) becomes a stationary vector-stochastic process. The VAR can be used for multiperiod forecasting of Drt and includes the variable St, which according to (3) is the optimal forecast of future Drt . In the modern version of the expectations hypothesis, the actual spread St equals the theoretical spread, St plus a constant c, or St ¼ Et St þ c. The intuitive explanation is that if the term premia are constant, all relevant information of market participants is embodied in the yield spread, which is included in the bivariate system in (4). Campbell and Shiller (1991) use a number of informal tests on the correlation coefficient between the actual and theoretical spreads, the ratio of their standard deviations, as well as the regression coefficient for the theoretical spread on the actual spread, to confirm that the theoretical spread of the expectations hypothesis tracts the actual spread very closely.2 In this VAR approach, the long-run behaviour of interest rates is inferred from their short-run behaviour in the sample period, rather than being estimated directly. This approach avoids the need to estimate multiperiod regressions with overlapping errors. It also avoids approaches that estimates the term premium by subtracting model-generated forecasts of inflation and real interest rates from observed long-term nominal interest rates. Hardouvelis (1994) uses the vector autoregressive methodology of Campbell and Shiller (1987, 1991) and confirms using the informal test measures mentioned above that, with the exception of the United States, the actual term spread tracks the theoretical spread of the expectations hypothesis very closely in the G7 countries that includes Canada. 3.2. The ex ante real and expected inflation components In this section, the theoretical longer term yields derived from the stationary vector-stochastic process from the Campbell and Shiller (1987, 1991) VAR methodology are decomposed into ex ante real yields and expected inflation using the structural VAR methodology developed by Blanchard and Quah (1989).3 The methodology identifies structural shocks using the long-run restriction that inflation expectation shocks have a permanent effect on the longer term yields, while ex ante real rate shocks have only a temporary effect. In this framework, inflation expectations are characterized as a stochastic process corresponding to the permanent component of nominal yields and ex ante real yield expectations correspond to the stationary component. This suggests that inflation expectations and nominal yields move one for one in the long run; that is, they are cointegrated (1,1), while real yields are stationary. The assumptions may be somewhat controversial because if both nominal yield and expected inflation are affected by a common set of permanent shocks, then the ex ante real rate will also be affected by the same shocks (Evans, 1998). However, the interpretation in this study is that nominal yields and expected inflation respond one-for-one to the permanent set of shocks only in the long run, which is consistent with what Mishkin (1992) calls a long-term Fisher effect. A short-term Fisher effect, on the other hand, is a stronger assumption in that it implies a constant real interest rate. Evans and Lewis established in a cointegrating relationship that long-term movements in nominal rates reflect one-for-one movements in expected inflation generated by a Markov-switching model of inflation. Consequently, the ex ante real rate of interest is stationary (0) and the relationship is consistent with a long-run Fisher relationship in their study. A number of other earlier studies also find that nominal interest rates and inflation contain unit roots. For example, Mishkin (1992) finds that both nominal yields and inflation over different maturities contain unit roots. Evans and Wachtel (1993) and Ball and Cecchetti (1990) also report that inflation contains a unit root. Also, more recently, Ang et al. (2008) present a generalize ARMA model of expected inflation (Appendix D) that allows for an additional factor for inflation that is composed of a stochastic expected inflation term plus a random shock. The additional factor yields results similar to their benchmark model. In addition, in all of the models assessed by Ang et al. (2008), the inflation risk premium is stochastic. The decompositions of their nominal yields into real yields and inflation components at various horizons indicate that variation in inflation compensation (both expected inflation and inflation risk premia) explains about 80 per cent of the variation in nominal rates at both short and long maturities. This large variation is consistent their measure of inflation compensation following a stochastic process. In addition, their main finding that the inflation components are the main driver of term spreads is not dependent on their term structure model having regimes. 2 3

Under the expectations hypothesis, all three test measures should equal unity. St. Amant (1996) also uses the methodology to decompose U.S. long-term interest rates.

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Similarly, the assumption of stationary real yields may also be controversial. However, Garcia and Perron (1996) show that the average value of the ex-post real rate for the U.S. is subject to occasional jumps caused by important structural events, such as the sudden rise in oil prices in 1973 and the huge increase in the federal budget deficit in the middle of 1981. These events took place before the sample period in this study and can explain the systematic non-rejection of the random walk hypothesis in their study and, by implication, the nonstationarity of the ex ante real interest rate found in other studies. In addition, measures of ex post real interest rates for the 1986–2013 sample period in this study, such as the real overnight rate and theoretical yield for the 10-year-and-over maturity, easily reject the null hypothesis of a unit root using Dickey-Fuller tests with two lags and a constant at conventional levels. The Blanchard and Quah (1989) structural VAR methodology is applied to an autoregressive system composed of two variables: the (theoretical) nominal longer term yield (tRt ) from the Campell and Shiller (1987, 1991) decomposition and the real rate that is equal to the theoretical yield minus the rate of inflation (rrt ¼ tRt  pt ). It is assumed that the theoretical interest rate fluctuations are a function of two non-autocorrelated and orthogonal types of shocks: inflation expectations (ept ) and ex ante real interest rate shocks (errt ) The orthogonality assumption only implies that disturbances to the real interest rate and inflation expectations are not systematically correlated, but that the cumulative effect of the structural shocks can be correlated. It is also assumed that inflation expectations are best characterized as a stochastic process corresponding to the permanent component of nominal interest rates and ex ante real rate expectations as the temporary stationary component. The structural bivariate VAR model can be given the following moving-average representation using the Wold decomposition theorem: 1 X

xt ¼ A0 et þ A1 et1 þ    ¼

Ai eti ¼ AðLÞet ;

ð5Þ

i¼0

where et ¼









ept DtRt 0 errt and xt ¼ rrt . The variance of the structural shocks is normalized so that Eðet et Þ ¼ I. The structural VAR

is identified by first estimating the following VAR:

Dxt ¼

Y

Dxt1 þ    þ

Y

t

Dxtq þ et

ð6Þ

q

where et is a vector of estimated residuals, q is the number of lags, and Eðet e0t Þ ¼ R is the covariance matrix of the residuals. The estimated VAR is then inverted to obtain the following moving-average representation:

xt ¼ et þ C 1 et1 þ    ¼

1 X C i eti ¼ CðLÞet :

ð7Þ

i¼0

The residuals of the models in the reduced form (7) are related to the structural residuals in the following way:

et ¼ A0 et ;

ð8Þ

which implies that ¼ A0 Eðe e and thus ¼ ¼ R. The main identifying restriction is that the matrix of long-run effects of the reduced-form shocks (e) and their variance R, that is C(1) in (7), are related to the equivalent matrix of structural shocks e, that is A(L) in (5), through the following relation: Eðet e0t Þ

0 0 t t ÞA0

Að1Þ ¼ Cð1ÞA0 ;

A0 A00

Eðet e0t Þ

ð9Þ

where the matrix C (1) is calculated from the estimated VAR in (7). The restriction in (9) means that ex ante real rate shocks do not affect the nominal longer term yields in the long run. The following structural decomposition is obtained from the restrictions:

DtRt ¼ Ap ð1Þept þ Ap ðLÞept þ Arr ðLÞerrt ;

ð10Þ

Eq. (10) presents the moving-average components of the different types of shocks to the nominal (theoretical) longer term interest rates. The A ðLÞ terms represent the temporary components of the shocks. The first two terms on the righthand side of (10) represent the measure of inflation expectations and the last term represents the measure of ex ante real interest rates. Once the structural shocks have been identified with (10), the effects of the shocks are cumulated to provide estimates of the ex ante real rate of interest and expected inflation over the sample period. 3.3. The empirical decomposition The yields for the estimations were obtained from a database of constant-maturity, zero-coupon yields for the Government of Canada bond market.4 The sample period for the estimations is from January 1986 to February 2013, a total of 326 monthly observations for each nominal yield. The estimations are based on 12 maturities: 3, 6, 12, 24, 36, 48, 60, 72, 84, 96,

4 The database provides best-fit, zero-coupon yields based on an exponential spline-based model for historical bond closes as described in Bolder, Johnson, and Metzer (2004).

7

25

25

20

20

15

15

10

10

5

5

0

0

-5

1960

1965

1970

1975

1980

Actual

1985

1990

1995

Theoretical

2000

2005

2010

Percent

Percent

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

-5

Term

25

25

20

20

15

15

10

10

5

5

0

0

-5

percent

percent

Fig. 1. Decomposition of the long-term nominal yield into its theoretical yield and a term premium for sample 1957:01–2013:01.

-5 1960

1965

1970

1975 TRL

1980

1985 EXRRL

1990

1995

2000

2005

2010

EXPINFL

Fig. 2. The theoretical long-term yield from Fig. 1 and the estimates of the long-term ex ante real rate and expected inflation components for sample 1957:01–2013:01.

108, and 120 months. The yields are measured as average of the month to facilitate the estimations with monthly macroeconomic data The choice of lag length for the VAR was guided by the Akaike Information, Hannan-Quinn and Schwarz Bayesian Criteria, and sequential likelihood-ratio tests from 1 to 4 lags. The Akaike and Hannan-Quinn criteria choose 3 lags and the Schwarz Bayesian Criteria chooses 2 lags, and the likelihood-ratio test chooses 3 lags. Consequently, three lags were chosen for each stage of the estimations, so the sample period for all estimations starts in 1986:12. Fig. 1 presents the decomposition from the stationary vector-stochastic process from the Campbell and Shiller (1987, 1991) bivariate matrix equation (4) for the 10+-year government of Canada bond yield, chosen because of its availability over a long sample period. The figure indicates that the theoretical yield based on the expectations hypothesis tracks the actual yield very closely with a term premium that is stationary around a mean of 58 basis points. The informal tests confirm that the theoretical yield from the expectations hypothesis tracts the actual yield very closely, with a correlation coefficient equal to 0.95, a ratio of their standard deviations equal to 1.17, and the coefficient for a regression of the theoretical yield on the actual yield equal to 1.11, which can be constrained to equal one. Fig. 2 presents estimates of the ex ante real rate and expected inflation from the theoretical yield in Fig. 1 for the cumulated shocks from the Blanchard and Quah (1989) structural decomposition in equation (10). The ex ante real rate component is equal to 2.97 per cent and the expected inflation component is 3.73 per cent, for a total that is equal to the average theoretical yield of 6.70 percent in Fig. 1. Table 1 presents means and standard deviations for the nominal yields for the 12 constant-maturity, zero-coupon yields for the full sample period (1986:12–2013:02) and two stochastic regimes (discussed later). The mean of the nominal yields at 5.52 per cent decomposes into the mean of the theoretical yields at 4.95 per cent and the mean premia for liquidity and risk of 61 basis points. The mean theoretical yields decompose into the mean ex ante real yield components of 2.81 per cent and expected inflation components of 2.14 per cent. The standard deviations indicate that the theoretical longer term yields are slightly more volatile than the actual yields.

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Table 1 Yield curve components. Nominal yields

Theoretical yields

Ex ante real yields

Expected inflation

Term premiums

2.14 1.99

0.61 0.80

Mean St.Dev.

5.52 2.69

4.95 3.14

Full sample:1986:12–2013:02 2.81 2.25

Mean St.Dev.

7.83 1.95

7.35 2.74

Regime 2: 1986:12–1998:12 4.66 1.46

2.69 1.50

0.48 1.02

Mean St.Dev.

3.55 1.32

2.83 1.54

Regime 1:1999:01–2013:02 1.22 1.44

1.61 1.15

0.72 0.54

Notes: Means and standard deviations of the nominal and theoretical yields, the ex ante real and expected inflation components of the theoretical yields, and the term premium for the full sample and the two regimes.

4. Yield curve methodology and estimation results 4.1. Dynamic latent factor methodology The foundation for the dynamic latent factor model used in this study is the Nelson and Siegel (1987) functional form, which is a convenient and parsimonious 3-component exponential algorithm. Diebold and Li (2006) reformulate the original Nelson-Seigel exponential expression for the yield curve as

yt ðsÞ ¼ b1t þ b2t

    1  eks 1  eks s þ b3t  eks ; k ks

ð11Þ

where yt (s) is the continuously compounded yield at maturitys. The Nelson-Seigel yield curve corresponds to a discount curve that begins at zero maturity and approaches zero at infinite maturity. The b1t, b2t, and b3t are time-varying parameters, and the parameter k controls the exponential decay rate.5 Small values of the decay parameter k produce slow decay that better fit the longer term maturities, while large values produce fast decay that better fit the shorter maturities. The decay parameter also governs where the loading on b3t reaches a maximum. The three time-varying parameters can be construed as dynamic latent factors. The loading on b1t is 1, a constant that does not decay to zero in the limit. An increase in b1t increases on all yields equally since the loading is identical on all maturities. Consequently, it is viewed as a long-term factor and is typically interpreted as the level (Lt) factor. The loading on b2t is ð1  eks Þ=ðksÞ, a function that starts at 1 and decays monotonically to zero. An increase in b2t increases short yields more than long yields since the short rates load more heavily on b2t, resulting in a change in the slope of the yield curve, defined as short-minus long-term yield. It is viewed as a short-term factor and is interpreted as the slope (St) factor. The loading on b3t is ðð1  eks Þ=ksÞ  eks , which starts at zero, increases, and then decays to zero. An increase in b3t will have little effect on the very short and long yields, but will increase medium-term yields since they load more heavily on it. It may be viewed as a medium-term factor and interpreted as the curvature (Ct) factor. The time-varying parameters b1t, b2t, and b3t, or the state variables Lt, St, and Ct, and the decay parameter k are able to capture a variety of shapes of the yield curve through time, such as upward sloping, downward sloping, and inversely humped. The factor representation has the advantage of aggregating information from a large set of yields so that the representation does not depend on the set of yields chosen. The important contribution of Diebold et al. (2006) is extending the Nelson-Siegel form into a state-space form that is useful because the Kalman filter delivers maximum-likelihood estimates and optimal filtered and smoothed estimates of the underlying factors. The dynamic movements or evolution of the yield curve factors, Lt, St, and Ct, are assumed to follow a vector autoregressive process of order one, VAR(1), to achieve a state-space representation of the model. The state-space system consists of two equations: transition and measurement equations. The transition equation that specifies the dynamics of the state vector in matrix notation is

2

3 2 Lt  lL a11 6 7 6 4 St  lS 5 ¼ 4 a21 Ct  lC a31

a12 a22 a32

3 2 3 gt ðLÞ Lt1  lL 76 7 6 7 a23 54 St1  lS 5 þ 4 gt ðSÞ 5; Ct1  lC gt ðCÞ a33 a13

32

ð12Þ

where t = 1,. . .,T. The measurement equation is the specification of the term structure itself that relates the set of N yields to the three unobserved factors (Lt, St, Ct)

5

The estimation approach assumes that there is a single constant decay parameter k for all maturities s.

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

2

3

2







 32

3 2 3 Lt e1t 76 7 6 6 7 6 7 7 ... 4 ... 5 ¼ 6 4...  ...    54 St 5 þ 4 . . . 5: ksN ksN 1e 1e yðsN Þ 1  eksN Ct eNt ksN ksN yðs1 Þ

1

1eks1 ks1

1eks1 ks1

 eks1

9

ð13Þ

The transition equation (12) may be written in vector notation as

ðf t  lÞ ¼ Aðf t1  lÞ þ gt

ð14Þ

where ft is a 3  1 vector of yield-curve factors (Lt, St, Ct), l is a 3  1 vector of means, A is a 3  3 matrix of VAR coefficients, and gt is a 3  1 vector of disturbance errors. Similarly, the measurement equation (13) that relates the set of N yields to the three unobserved factor may be written as

yt ¼ Kf t þ et ;

ð15Þ

where yt is a N  1 vector of observed yields (s) at time t, K is a N  3 matrix of factor loadings, ft is the 3  1 vector of factors, and et is a N  1 vector of disturbance errors. The white-noise (WN) transition and measurement disturbances (gt, et) are assumed to be orthogonal to each other and to the initial state to obtain the linear-least squares optimality of the Kalman filter:



gt et



   0 Q ; 0 0

 WN

0 H

 ;

ð16Þ

and

Eðf 0 g0t Þ ¼ 0;

ð17Þ

Eðf 0 e0t Þ ¼ 0

ð18Þ

where Q is the variance-covariance for the transition equation (14), and H is the variance-covariance matrix for the measurement equation (15). The Q matrix is assumed to be non-diagonal, so that shocks to the three yield-curve factors are correlated, while the H matrix is assumed to be diagonal, so that deviations of yields of various maturities from the yield curve are uncorrelated. The estimation of the decay factor k is the key to estimating the state-space model. An optimization algorithm maximizes the likelihood function estimated by the Kalman filter. The filter computes the optimal yield predictions and the corresponding prediction errors, after which the Gaussian likelihood function is evaluated using the prediction-error decomposition of the likelihood function for the predictions and the states. It sequentially updates the measurement and transition equations until an optimal yield prediction is obtained. In the first step for a given decay parameter k, the measurement coefficient is determined, then an OLS regression is run for each month t to obtain the latent factors ft and the measurement errors et. The resulting matrix of measurement errors is used to calculate the measurement disturbance covariance matrix H. In the second step, the latent variables are treated as dependent variables in a VAR(1), which is used to estimate the transition matrix A, the mean state vector l, and the transition errors gt. The transition errors are then used to compute the transition disturbance covariance matrix Q. The (3  3) transition matrix A contains 9 parameters, the (3  1) mean state vector l contains 3 parameters, the measurement matrix K contains the free parameter k, the transition disturbance covariance Q contains 3 variances and 3 covariances, and the measurement disturbance covariance matrix H contains freely estimated disturbance variances for each of 12 yields. In Diebold and Li (2006), the decay parameter k was fix to maximize the loading on the medium-term factor at exactly 30 months in order to use ordinary least squares to estimate the measurement matrix K . In this study, the parameter k is freely estimated along with the betas (factors) in the loading matrix by linear least squares as in Diebold et al. (2006). In total, there are 31 parameters to be estimated by numerical optimization. 4.2. Dynamic factor estimation results Table 2 presents the mean and standard deviations of the three latent factors for the full sample period and two stochastic regimes (discussed later) for all three models that are estimated with the vector transition equation (14).6 The means and standard deviations of the estimated level factors (Lt) for the ex ante real yield and expected inflation models are very close to those reported in Table 1 from the yield decompositions. However, the estimated mean level of the nominal factor at 6.47 per cent is relatively larger than the mean nominal level, although the standard deviation is about the same. The estimated slope factor (St) for the nominal model indicates an average slope (short minus long) of about 1.29 per cent, which seems plausible. The relatively small slopes for the real yield and expected inflation models suggest that the slope of the nominal yield curve is due to the premia for liquidity and risk, which is consistent with the findings by Ang et al. (2008) for the inflation premia. The first panel of Table 3 presents the estimation results for the VAR(1) transition equation (14) for the latent level, slope, and curve factors (Lt ; St ; C t ) for the nominal, ex ante real yield and expected inflation models. The estimates of the A matrix 6

The Kalman smoother based on the full sample estimates was used to obtain the optimal extractions of the estimated factors.

10

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

Table 2 Characteristics of the estimated dynamic factors. Nominal yield

Ex ante real yield

Mean

St. dev.

Mean

Expected inflation St. dev.

Full Sample: 1986:12–2013:02 2.73

Mean

St. dev. 1.93

^ Lt ^t S

6.47

2.33

2.30

2.10

1.29

3.68

0.20

0.26

0.02

0.17

^t C

3.30

4.00

0.75

0.86

0.03

0.25

^ Lt ^t S

8.56

1.46

1.59

2.69

2.47

2.36

3.40

0.14

0.29

0.02

0.21

^t C

0.81

2.60

0.51

1.01

0.04

0.32

^ Lt ^t S

4.41

0.90

1.46

1.61

1.10

3.23

2.30

0.19

0.21

0.03

0.13

^t S

1.99

2.72

0.93

0.66

0.08

0.16

Regime 2: 1986:12–1998:12 4.60

Regime 1: 1999:01–2013:02 1.12

Notes: Means and standard deviations of the estimated yield-curve factors – level, slope, curvature – for the nominal, ex ante real and inflation curves for the full sample and the two regimes.

indicate highly persistent own dynamics, with estimated own-lag coefficients of 0.99, 0.98, and 0.88, respectively, for the nominal yield curve factors. The level and curvature factors for the real yield curve and the level factor for the expected inflation factor also indicate quite persistent own dynamics with coefficient above 0.91. The cross-factors dynamics for the nominal yield curve are also quite important with a small significant effects of St-1 on Lt and of Ct-1 on Lt and St. These results for the nominal yield curve are consistent with Lange (2013) that finds the cross-factor dynamics to be relatively more important in Canada than those for the U.S. in Diebold et al. (2006). In fact, the cross-factor dynamics are much stronger in the real yield and expected inflation models, with St-1 having quite large effects on Lt and Ct in both models. Ct-1 also has relatively large effects on Lt and St in the inflation model. The much larger cross-dynamic effects on the real yield and expected inflation models indicates that dynamics of the nominal yield curve are more complex than suggested by the nominal factors themselves. The estimates of the mean levels of the factors are 5.94 per cent for the nominal curve, 2.49 per cent for the real curve and 2.25 per cent for the expected inflation curve. All estimates of mean levels of the state vector l are highly significant and appear plausible since they are relatively close to the mean values of the decomposed yields in Table 1.

Table 3 Nominal yield, ex ante real yield and expected inflation factor models (1986:12–2013:02). State

VAR transition A matrices Nominal Yields

Ex ante Real Yields

Expected Inflation

Lt1

St1

Ct-1

l

Lt1

St1

Ct1

l

Lt1

St1

Ct1

l

0.99 (0.00) 0.02 (0.02) 0.03 (0.03)

0.01 (0.00) 0.98 (0.02 0.01 (0.03)

0.01 (0.00) 0.08 (0.02) 0.88 (0.03)

5.94 (1.73) 0.33 (1.16) 2.93 (1.01)

0.97 (0.01) 0.01 (0.00) 0.02 (0.01)

0.22 (0.10) 0.66 (0.04) 0.37 (0.07)

0.01 (0.03) 0.02 (0.01) 0.92 (0.02)

2.49 (0.93) 0.21 (0.04) 0.76 (0.26)

0.91 (0.01) 0.00 (0.00) 0.00 (0.00)

0.60 (0.13) 0.71 (0.04) 0.24 (0.02)

0.63 (0.60) 0.09 (0.01) 0.68 (0.03)

2.25 (0.60) 0.01 (0.33) 0.01 (0.04)

St

Lt 0.05 (0.00) – –

Ct 0.07 (0.02) 1.83 (0.12) 3.36 (0.21)

Lt 0.18 (0.01) –

Ct

St 0.10 (0.01) 1.36 (0.09) –

St 0.01 (0.00) 0.01 (0.00) –

Ct 0.01 (0.00) 0.00 (0.00) 0.03 (0.00)

Lt Lt St Ct

Lt

Transition covariance Q Lt St 0.25 0.00 (0.02) (0.01) – 0.04 (0.00) – –

matrices Ct -0.03 (0.01) 0.04 (0.00) 0.10 (0.01)

LR/Wald Tests v2ð3Þ ¼ 390:9

P-value 0.00

Tests for diagonality of Q matrices LR/Wald Tests P-value 0.00 v2ð3Þ ¼ 140:1

v2ð3Þ ¼ 274:3

0.00

v2ð3Þ ¼ 99:0

0.00

–

LR/Wald Tests

v2ð3Þ ¼ 26:8 v2ð3Þ ¼ 93:1

P-value 0.00 0.00

Notes: Estimates of the VAR transition A matrices and the transition covariance Q matrices for the nominal yield, ex ante real yield and expected inflation factor models. The likelihood ratio and Wald tests are for the diagonality of the Q matrices. Bold estimates denote significance at the 5 per cent level and standard errors appear in brackets.

11

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18 7.00

7.00

Nominal Nominal Real

6.00

6.00

Real Inflation Inflation

5.00

4.00

4.00

3.00

3.00

2.00

2.00

1.00

3

6

12

24

36

48

60

72

84

96

108

120

percent

percent

5.00

1.00

Months

Fig. 3. Average observed (blue) and estimated (black) yields for the nominal yield curve, and average empirical proxy (blue) and estimated (black) yields for the ex ante real yield and expected inflation curves for the sample 1986:12–2013:02. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In the second panel, all but one of the covariance terms on the off-diagonals of the Q matrices for all three models (slope factor for the level in the real yield model) are significant. The cross-dynamics from the transition A matrices in the first panel and the covariances in the second panel suggest overall quite strong dynamics among the factors for the nominaland real-yield curves and the expected inflation curve. The two tests of diagonality in the bottom panel strongly reject the restriction that the Q matrix is diagonal for all three models. The three factor representation from each model has the advantage of aggregating information from a large set of yields from the vector measurement equation (15), so that the representation does not depend on the set of yields chosen. The plots of the implied average fitted-nominal, -real and inflation curves against the average data-based yield curves in Fig. 3 indicate an overall good fit and that the three latent factors provide flexible approximations to the nominal and real yield and expected inflation data. There are only small discrepancies at the 3-month horizon. The residuals between fitted and observed yields are often interpreted as pricing errors. The average nominal yield curve slopes upwards, increasing from about 5 per cent for the 3-month yield to over 6 per cent for the yield for 10-year maturity, which is consistent with the estimated slope factor St of 1.29 for the nominal model in Table 2. However, the ex ante real yield curve increases only slightly from about 2.65 per cent to about 2.85 per cent, so that the real term spread is only about 20 basis points, which is also consistent with the estimate for St of 0.20 in Table 2. The curve for expected inflation is flat at about 2.12 per cent, which is also consistent with the estimate of 0.02 for St in Table 2 and quite close to the mid-point of the inflation targets that were adopted in 1991. Overall, the upward slope of the nominal term structure of about 120 basis points is due to the premia for liquidity and risk that increases with maturity, which is consistent with the findings by Ang et al. (2008). 5. Regime-switching yield curves In this section, a VAR model with regime-switching variances and contemporaneous response coefficients are used to detect any potential breaks in the sample and to identify whether the sources of any structural instability or volatility are due to monetary policy and inflation. The estimations are also used to determine whether the responses of the latent variables to innovations in the monetary policy rate and inflation are asymmetric or symmetric. 5.1. Regime-switching methodology The general reduced-form VAR system for the regime-switching estimation may be written as

X t ¼ AðLÞX t1 þ lt ;

ð19Þ

where Xt is a vector of macroeconomic variables, A(L) is a matrix of polynomials in the lag operator L, l is the vector of VAR residuals, and t = 1, . . ., T. The data vector for VAR in this section is given by

^tÞ X t ¼ ðyt ; pt ; it ; ^Lt ; ^St ; C where yt is a measure of output growth,

ð20Þ

pt is inflation,it is the monetary policy rate, ^Lt is the latent level factor for the esti-

mated nominal yield, ex ante real yield or expected inflation models from the previous section, ^ St is the estimated slope factor ^ t is the estimated curvature factor for the models. The data vector implies a relatively weak notion for the three models, and C of an ordering of contemporaneous variables. Nonfinancial variables are ordered first so that economic variables do not react contemporaneously to financial variables, with inflation reacting to output growth in the previous period. This is consistent

12

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

with the view of the transmission mechanism in Canada where it is believed that output and prices take some time to respond to financial conditions. The financial variables are ordered according to maturity, with the monetary policy rate occurring before the level, slope and curvature factors for the longer term yields. The innovation model for the three latent factors for each model – level (l), slope (s), curvature (c) – are assumed to be affected contemporaneously by innovations in inflation (lp ) and the monetary policy rate (li ), as well as by own shocks (t).7

level factor : ll ¼ alp lp þ ali li þ tl

ð21Þ

slope factor : ls ¼ asp lp þ asi li þ asl ll þ ts

ð22Þ

curvature factor : lc ¼ acp lp þ aci li þ acl ll þ acs ls þ tc

ð23Þ

The final feature of the VAR model is the identification of ‘‘fundamental” or ‘‘structural” shocks. A general form of the innovation model relating the (observable) reduced-form VAR residuals and the (unobserved) structural shocks can be written as Alt ¼ Bmt , where A and B are square matrices of structural parameters on the estimated reduced-form residuals lt and the underlying structural disturbances mt, respectively. If B is an identity matrix, then the general form reduces to the standard innovation model lt ¼ A1 0 mt , which maps the estimated reduced-form residuals on to the contemporaneous structural disturbances. The contemporaneous relationship between the VAR innovations in the residuals in Eqs. (21)–(23) can be expressed in the following matrix equation:

2

1

6 6 aip 6 6 a1p 6 6 4 asp acp

0

0

0

1

0

0

ali

1

0

asi

asl

1

aci

acl

acs

32

3

2

lp 1 76 7 6 0 76 lr 7 6 0 76 7 6 6 7 6 07 76 ll 7 ¼ 6 0 76 7 6 0 54 ls 5 4 0 lc 0 1 0

1 0 0 0 1 0 0

0 1

0

0 0

32

3

mp 76 7 0 76 ms 7 76 7 6 7 07 76 ml 7 76 7 0 54 ms 5 1 mc

0 0 0 0

ð24Þ

where a-coefficients are used to indicate responses to (observable) VAR l-innovations in the residuals.8 The matrix equation (24) has 15 unknown parameters (including 5 shock variances) to be estimated from 15 residual variances and covariances for the 1-state model. Since the innovation model in Eq. (24) assumes a Choleski lower triangular ordering of the variables in (20), it is a just-identified system of equations since it has n(n  1)/2 zero restrictions. The regime-switching estimation allows all parameters in the innovation model in matrix equation (24) to switch regimes. The estimation relies on the maximum-likelihood technique of Hamilton (1989, 1990) that uses the ExpectationMaximization (EM) algorithm for the estimation of the regime-switching parameters.9 The estimation assumes an ergodic Markov chain so that the probability of being in a state given last period’s state is constant over time

pðst ¼ ijst1 ¼ iÞ ¼ /ðui Þ;

ð25Þ

where p (st) is the probability of being in a regime i (=1, 2), st is a binary state variable subject to Markov switching, and / is the (1-sided) normal cumulative density function. The us are the coefficients for the transition probability of remaining in regime i. They characterize the conditional density function and measure the degree of regime persistence. The innovation responses are estimated by maximizing the likelihood function over two unobserved states. The probability distribution from the maximum likelihood value for the observed values for each state at each period of time and the probability of remaining in a regime given past information are used to obtain a joint conditional density function for each state. The sum of the joint density functions for two states yield the following log-likelihood function (LLF) for T periods

LLF ¼

T X X pðst Þdðljst Þ;

ð26Þ

t1 St

where l is the innovation matrix equation (24). The log-likelihood function (26) is maximized with respect to the parameter vector {as; us; rs}, where as are the contemporaneous response coefficients for the latent factors, us are the coefficients for the transition probability of remaining in regime i, and rs are the variances of the innovation model (24). 5.2. Regime-switching estimation results The macroeconomic variables for the regime-switching estimations include the 12-month percentage change in the consumer price index (p), the measure used as the inflation target in Canada, the overnight financing rate as the instrument of 7 Monetary policy also reacts contemporaneously to innovations in output growth and inflation since they are ordered first in the general specification of the VAR system. 8 For illustration purposes, it is assumed that the signs of all reaction coefficients are positive. However, this will not be the case for the factors for expected inflation, which are expected to react negatively to innovations in the monetary policy rate. 9 Under the procedure for applying the EM algorithm, the hidden Markov chain is inferred in the expectation step for a given set of parameters, then the parameters for the hidden Markov chain are re-estimated in the maximization step. These two steps are repeated until convergence is achieved.

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

13

monetary policy (i), and real GDP in expenditure-based 2007 chain-linked dollars. The data for real GDP are interpolated to a monthly frequency by computing a distribution for each series that maintains the sum of the measure across each quarter and assumes that the interpolation error is described by a first-order random walk. The monthly levels for real GDP are expressed as log-differences and scaled by 100 so that changes can be interpreted as month-to-month percentages. Since Fig. 3 indicates that the latent level, slope and curvature factors for the three yield curves fit the observed data for the 12 maturities quite well, the yield curve factors for each curve were examined separately for regime shifts. The Markovswitching VAR framework outlined in Eqs. (25) and (26) for the innovation model in Eqs. (21)–(23) or the matrix equation (24) is used to capture regime shifts in the variances and the responses of the level, slope and curvature factors for each of the yield curves to innovations in inflation and the monetary policy rate, the primary sources of nominal and real regime switches. A triangular Choleski ordering is assumed for the measure of the monthly growth in real GDP (interpolated), the 12-month growth of the CPI, the monetary policy rate and the three latent factors.10 Fig. 4 presents the probabilities of being in the high-variance regime and Table 4 presents the response coefficients and standard deviations for both the 1- and 2-regime estimations. Fig. 4 indicates that the curves for the nominal yield and ex ante real yield switched predominately from the high-variance regime to the low-variance regime in about 1998, while the curve for expected inflation switched regimes about a year later. The factors for all three curves experienced the highvariance regime again during the monthly downturns in economic activity in Canada in the early-2000s and during the global recession that began in 2008. The estimates for the conditional variances or r-standard deviations of the innovations in bottom half of Table 4 reveal different levels of within-regime variance, with the volatility of the innovations in the monetary policy rate being about three times larger in regime 2 than in the current regime 1 and the volatility of the factors for all three curves being at least twice as large in regime 2 as in regime 1. The standard deviations of the innovations in inflation are virtually the same in both regimes, but about three times larger than in the 1-regime estimation. Overall, the regime-switching estimation captures conditional heteroskedasticity in the form of Markov switching in the scale r2 ðst Þ of the variance, where the error terms of the equations for the policy rate and the latent factors for the three curves switch stochastically and discretely between high- and low-variance regimes. The estimates for the a-response coefficients for the factors to one-standard deviation shocks to inflation and the policy rate are presented in the top half of Table 4.11 The insignificant contemporaneous responses of the level of the nominal yield curve to an innovation in the inflation rate for both the 1- and 2-regime estimations are quite interesting because the response coefficients for the real yield curve are equal (and can be constrained) to -1.0, while the contemporaneous coefficients for the expected inflation are equal to 1.0. These coefficients indicate that the responses of both curves are exactly offsetting, which can account for the insignificant contemporaneous response of the nominal level factor to an innovation in inflation. However in the long-run, the nominal yield and expected inflation are expected to move one-for-one since expected inflation is constructed as a stochastic permanent component of nominal yields. The responses of the curvature factor for the real yield curve are slightly smaller for both estimations at about 0.8. The responses of the factor for the real slope (short minus long) are not significant, suggesting that the inflation innovations have about the same effect on short- and long-term inflation expectations. The responses of the level of the real yield curve to an innovation in the monetary policy rate are about 0.5 for both the 1and 2-regime estimations, suggesting that about half of the policy innovation is immediately incorporated into the overall real yield curve. The policy innovation also triggers a contemporaneous small reduction in the level of the expected inflation curve. The positive coefficients for the slope of the nominal yield curve (short minus long) policy at about 1.25 indicates that an innovation in the monetary policy rate increases the short end of the yield curve and decreases the long end, suggesting that an innovation in the policy rate increases the short-term rate and decreases long-term inflation expectations, so that the overall slope decreases (narrows) by more than one. The responses of the coefficients for the slope factor of the expected inflation curve indicate a small increase in the slope (widening), which suggests that the increase in the policy rate decreases inflation at the short end by more than inflation expectations at the long end, consistent with an expectations interpretation of the yield curve. An innovation in the monetary policy rate has also slightly larger effects on the curvature factors for both the nominal and expected inflation curves. The relatively larger effects on the nominal slope and curvature factors suggests that the liquidity and risk premia may also have declined in response to an increase in the policy rate. Overall, the flexibility of the (unrestricted) variance switching does not appear to have induced many regime-dependent asymmetric responses of the yield curve factors to innovations in inflation and the monetary policy rate. Only the slope and curvature factors for the expected inflation curve are significant at about 0.5 in regime 1, suggesting that the inflation expectations may have become more responsive to an inflation innovation in the current regime due to inflation targeting. The curvature factor for the real yield curve is also significant in the low-variance regime 1. The transition probabilities u11 and u22 are equal to about 0.95 for all curves, indicating that both regimes are very persistent. The regime-switching estimates in Fig. 4 stochastically divide the sample from 1986 to 2013 into high- and low-variance regimes in about 1998–99. The division is consistent with some recent studies that find bond yields, both in the U.S. and abroad, fell below levels that were consistent with standard macroeconomic fundamentals, such as inflation, GDP growth, and fiscal balances since the late-1990s (Gruber & Kamin, 2009; Rudebusch, Swanson, & Wu, 2006). Kulish and Rees 10

The measure for the monthly growth of GDP was included in the VAR as a regime-independent variable. The response coefficients for the slope and curvature factors to innovations in the level and slope factors in matrix equation (24) are not presented to conserve space in the table. 11

14

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18 Nominal Yield Curve

1.00

1.00

0.50 0.00

0.50

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

2010

2012

Real Yield Curve

1.00

1.00

0.50 0.00

0.50

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

2010

2012

Expected Inflation Curve

1.00

0.00

1.00

0.50 0.00

0.00

0.50

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

2010

2012

0.00

Fig. 4. The probabilities of being in the high-variance regime for the nominal yield, ex ante real yield and expected inflation curves for the sample period 1986:12–2013:02.

Table 4 Estimates of the regime-switching parameters from the yield curve models (1986:12–2013:2). Nominal yield

Ex ante real yield

Expected inflation

One regime

Low-variance

High-variance

One regime

Low-variance

High-variance

One regime

Low-variance

High-variance

rp

0.02 (0.73) 0.12 (0.82) 0.02 (0.18) 0.18 (1.37) 1.29 (6.69) 1.43 (10.54) 0.16

rl

0.05

rs

1.33

rc

3.26

uii

–

1.03 (61.74) 0.00 (0.01) 0.82 (10.52) 0.59 (10.58) 0.01 (0.12) 0.16 (2.66) 0.43 (20.85) 0.12 (18.23) 0.09 (17.42) 0.13 (18.83) 0.08 (17.22) 0.95 (2.70)

1.01 (21.41) 0.00 (0.02) 0.81 (15.02) 0.47 (11.33) 0.06 (1.21 0.01 (0.36) 0.37 (17.66) 0.43 (18.19) 0.26 (17.80) 0.21 (17.58) 0.14 (20.80) 0.95 (2.39)

1.01 (154.4) 0.23 (2.32) 0.28 (1.71) 0.29 (3.65) 0.11 (7.70) -0.18 (7.29) 0.16

0.09

0.05 (0.84) 0.15 (1.01) 0.02 (0.11) 0.08 (1.38) 1.27 (5.60) 1.34 (7.54) 0.38 (21.19) 0.42 (16.72) 0.27 (14.95) 1.24 (19.51) 0.90 (18.02) 0.95 (2.70)

1.02 (38.17) 0.01 (0.17) 0.77 (18.06) 0.48 (13.74) 0.06 (1.24) 0.02 (0.61) 0.16

ri

0.01 (0.19) 0.15 (1.01) 0.03 (0.48) 0.18 (1.37) 1.29 (6.69) 2.04 (9.51) 0.41 (20.78) 0.13 (16.58) 0.17 (18.56) 0.70 (17.14) 0.35 (16.16) 0.94 (40.45)

1.02 (141.00) 0.47 (3.69) 0.59 (12.61) 0.09 (3.91) 0.14 (3.46) 0.29 (5.83) 0.41 (19.82) 0.13 (16.58) 0.03 (15.76) 0.05 (18.89) 0.07 (14.96) 0.96 (53.96)

1.00 (77.70) 0.15 (1.05) 0.23 (1.00) 0.03 (2.70) 0.10 (6.42 0.16 (4.69) 0.39 (17.65) 0.45 (16.89) 0.06 (18.40) 0.10 (15.18) 0.17 (15.28) 0.95 (2.38)

alp asp acp ali asi aci

0.10 0.21 0.03 0.07 –

0.11 0.17 0.01 0.02 –

Notes: Subscripts for a indicate response coefficients for the level, slope and curvature to innovations in inflation and the monetary policy interest rate denoted as p and i, respectively. The standard deviations r are denoted with subscripts for inflation, the policy rate, and the level, slope and curvature factors. Figures in parentheses are t-statistics based on maximum-likelihood standard errors. Bold values are for discussion in the text.

(2011) argue that this increasing pattern of cross-correlations along the yield curve between inflation-targeting, small-open countries, such as Australia, Canada, Norway, and Sweden, and the large U.S. economy since about 1997 can be accounted for by the relatively greater persistence of foreign than domestic shocks on longer-term interest rates. Lange (2014) finds overall very strong links between the yield curves in Canada and the U.S, with the U.S. yield curve accounting for as much as 45 per cent of the variation of the movement in the level factor and about 30 per cent of the movements in the slope and the curvature factors of the Canadian curve. Also, Lange (2015) captures a ‘financial globalization’ regime that begins in late 1996, where the responses of longer term spreads, an important measure of the stance of monetary policy in Canada, are found to have increased by about one third to changes in U.S. longer-term yields. Recently, there has also been some focus on the potential sources of the low long-term yields in the U.S. and the increased correlation of international long-term interest rates. Byrne, Fazio, and Fiess (2010) focus on a global saving glut, Ciccarelli and Mojon (2010) on global inflation, and Diebold, Li, and Yue (2008) on global yield-curve factors as potential sources of the increased correlation. In light of the virtually simultaneous regime shifts in all three yield curves, the term structure data were re-estimated using the dynamic yield-curve model in the vector equations (14) and (15) for the high-variance regime

15

9.0

9.0

8.0

8.0

7.0

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R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

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2.0 1.0

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5.00

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3.00

3.00

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2.00

2.00 Regime2 Regime2 Regime1 Regime1

1.50 1.00

Percent

Percent

Fig. 5. Average observed (blue) and estimated (black) yields for the nominal yield curves in high-variance regime 2 (1986:12–1998:12) and low-variance regime 1 (1999:01–2013:02). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3

1.50

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36

48

60

72

84

96

108

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1.00

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Fig. 6. Average empirical proxy (blue) and estimated (black) yields for the ex ante real yield curves in the high-variance regime 2 (1986:12–1998:12) and low-variance regime 1 (1999:01–2013:02). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(1986:12–1998:12) and the low-variance regime (1999:01–2013:02). The yield curves for both the estimated and empiricalbased values for the 12 yields for both the high- and low-variance regimes are presented in Figs. 5–7. Fig. 5 indicates that the nominal yield curve has shifted down by about 430 basis points from an average of about 7.85 per cent in regime 2 to about 3.55 per cent in the current regime 1, equal to the declines in estimates for the mean levels for the nominal yields in the lower panels of Table 1 and close to declines in the estimated level factors in Table 2. Fig. 5 indicates that, with the exception of the estimated the 3-month yield, the nominal yields are estimated with very small errors relative to the observed yields in both regimes. Fig. 6 indicates that the real yield curve has shifted down about 345 basis points from about 4.65 per cent in regime 2 to about 1.20 per cent, accounting for about 85 per cent of the downward shift in the nominal yield curve. The decline is also close to those reported in Tables 1 and 2. The expected inflation curve in Fig. 7 has shifted down about 110 basis points from about 2.70 per cent in regime 2 to about 1.60 per cent in the current regime 1, almost exactly equal to the declines reported for the mean level of expected inflation in Table 1 and the estimated level factors for inflation in Table 2.12 The low level of inflation expectations in regime 1 largely reflects the decrease in average inflation since the global recession in 2008 to about 1.6 per cent in the lower bound of Canada’s inflation target band. The very large drop in the average real component of the yield curve in the current regime is consistent with recent research that finds a declining trend in estimates of the equilibrium interest rate. Laubach and Williams (2003, 2015), for example, use a state-space framework to show that the decline in the natural or neutral rate of interest is related to a decrease in economic activity and potential output. The original Laubach and Williams (2003) framework has been extended by Johannsen and Mertens (2016) and Kiley (2015) to find that the declining trend has become more firmly established since the start of the Great Recession. Several trends have been cited as possible factors contributing to a decline in the long-run equilibrium real rate. Bernanke (2005) has advanced the ‘‘global savings-glut hypothesis” that has resulted from high saving rates in many emerging market countries, coupled with a lack of suitable domestic investment opportunities in those countries, as putting downward pres12 The slight difference in the decrease in sum of the average nominal yields and the real and expected inflation components suggests an increase in the risk premium component of the nominal yield curve of about 25 basis points in regime 1.

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18 3.00

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2.75

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16

1.50 Regime2 Regime2

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Regime1 Regime1

3

6

12

24

36

48

60

72

84

96

108

120

1.25 1.00

Months

Fig. 7. Average empirical proxy (blue) and estimated (black) yields for the expected inflation curves in the high-variance regime 2 (1986:12–1998:12) and low-variance regime 1 (1999:01–2013:02). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

sure on rates in advanced economies. The shift to the right of the global savings schedule as a cause of the low global real rates has also been supported by Caballero, Farhi, and Gourinchas (2008). Summers (2014), on the other hand, argues that the lower global real rates are due to a shift to the left of the global investment schedule due to ‘‘secular stagnation.” He attributes the decline in the marginal propensity to invest to the small amount of physical capital that the revolutionary information technology firms with high stock market valuations have needed. Gordon (2014, 2016) argues that a slowdown of productivity growth due to technological change has been an important factor that has contributed to low global real rates. He has also points to overall demographic trends resulting in a larger share of the population in age cohorts with high saving rates. 6. Concluding remarks The methodology in this study first uses the expectations hypothesis to decompose nominal term-structure yields into longer term theoretical yields that are consistent with the sum of 1-period changes in the short-term interest rate over the longer horizons and a rolling term premium that reflects liquidity and risk concerns using the stationary vectorstochastic VAR methodology in Campbell and Shiller (1991). The theoretical longer term yields are then decomposed into ex ante real yields and expected inflation using the structural VAR methodology developed by Blanchard and Quah (1989), where structural shocks are identified by the long-run restriction that inflation expectation shocks have a permanent effect on longer term yields, while ex ante real rate shocks have a only temporary effect. This suggests that inflation expectations and nominal yields move one for one in the long run; that is, they are cointegrated (1,1), while real yields are stationary. In this framework, inflation expectations are characterized as a stochastic process corresponding to the permanent component of nominal yields and ex ante real yield expectations correspond to the stationary component. The nominal yields and the real and inflation components were then estimated using the dynamic Nelson-Seigel model by applying the Kalman filter in a state-space framework to capture the three well-known latent factors that represent the level, slope and curvature of the yield curves. The cross-dynamics from the VAR parameters in the transition matrices and the covariances suggest overall quite strong and complex dynamic interactions among the factors for all three curves. The three factor representation from each model is able to accurately aggregate the information for the set of 12 nominal yields and their ex ante real and expected inflation components to characterize the yield curves in terms of the statistical properties of the average observed or empirical-proxy yields and the corresponding components for each maturity. The nominal yield curve increase from about 5 per cent to slightly over 6 per cent with a slope of over 120 basis points. On the other hand, the ex ante real-yield curve sloped upward only by about 20 basis points to plateau at about 2.85 per cent and the expected inflation curve was virtually flat at about 2.12 per cent. Regime-switching estimations stochastically divide the sample period into high- and low-variance regimes in about 1999. The estimates for the innovation response coefficients for the level factor to an inflation innovation for the real yield curve are equal to 1.0 in both regimes, while the contemporaneous coefficients for the expected inflation are equal to 1.0 in both regimes, so that the responses of both the real yield and expected inflation curves are exactly offsetting. These asymmetric responses can account for the insignificant response of the nominal level factor to a contemporaneous innovation in inflation. However, in the long-run, the nominal yield and expected inflation would move one-for-one since expected inflation is constructed as the stochastic permanent component of nominal yields. The responses of the level of the real yield curve to an innovation in the monetary policy rate suggest that about half of the policy innovation is incorporated into the overall real yield curve in the current period, while the policy innovation also triggers a contemporaneous small reduction in the level of the expected inflation curve. The dynamic yield-curve models for the two regimes indicate that the nominal yield curve has shifted down by about 430 basis points to over 3.50 per cent in the current regime, while the real yield

R.H. Lange / North American Journal of Economics and Finance 39 (2017) 1–18

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curve has shifted down by almost 350 basis points to about 1.20 per cent and the expected inflation has shifted down by over 100 basis points to about 1.60 per cent in the current regime. These results have a few implications for monetary policy. Most notably, inflation expectations are well anchored at about 2 per cent in Canada, which is an inflation-targeting country. Also, the relatively large drop in the real yield curve to about 1.20 per cent indicates the importance of the effects financial globalization on a small open economy like Canada. The drop is also consistent with recent search that suggests a marked decline in the neutral or natural rate of interest (Laubach & Williams, 2015), which is quite important for gauging the overall stance of monetary policy in a Taylor-type policy rule. An important limitation of the research in this study is the derivation of term (liquidity and risk) premia where it is essentially assumed to be residual that is backed out of the nominal yield curve using the Campbell and Shiller (1991) methodology for the expectations hypothesis. The derivation of the risk premia is particularly important for investment in the economy because it is a component of the cost of capital. It is also the important for total inflation compensation for portfolio investors. In addition, the market price of risk can be expected to depend on regime shifts. Consequently, the derivation of the risk premia for the nominal yield curve requires a more rigorous modelling, such as that used in pricing models of bond yields. Although the finance literature on bond pricing in the affine term-structure models is extensive, in recent years, there has been a move to use model forecasts to examine the ability of term structure models to describe the deviations from the expectations hypothesis. Duffee (2002), for example, is a useful starting point for modelling the market price of risk and forecasting future yields. Another limitation of the decomposition used in this study is the assumption based on the Blanchard and Quah (1989) methodology that inflation expectations are a permanent and stochastic component of nominal longer term yields. Some attempts to generate inflation expectations using forecasts from an estimated autoregressive process may provide a useful alternative methodology for comparative purposes. The recent research by Furceri and Pescatori (2014) suggests such an approach for generating inflation expectations at various term structure horizons. 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