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Term structure estimation in the presence of autocorrelation
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North American Journal of Economics and Finance
Term structure estimation in the presence of autocorrelation Januj Juneja ∗ College of Business, San Diego State University, San Diego, CA, United States
a r t i c l e
i n f o
Article history: Received 15 September 2013 Received in revised form 22 February 2014 Accepted 24 February 2014 Keywords: Affine term structure model Principal components analysis Autocorrelation misspecification Monte Carlo simulation Maximum likelihood estimation
a b s t r a c t This paper assesses the effects of autocorrelation on parameter estimates of affine term structure models (ATSM) when principal components analysis is used to extract factors. In contrast to recent studies, we design and run a Monte Carlo experiment that relies on the construction of a simulation design that is consistent with the data, rather than theory or observation, and find that parameter estimation from ATSM is precise in the presence of serial correlation in the measurement error term. Our findings show that parameter estimation of ATSM with principal component based factors is robust to autocorrelation misspecification. © 2014 Published by Elsevier Inc.
1. Introduction In the widely used class of affine term structure models (ATSM) of Duffie and Kan (1996), bond yields are assumed to rely on a low dimensional state space (Piazzesi, 2003, chap. 12). Several authors assume that this state space is comprised of certain yields or linear combinations of yields which are chosen to fit the model exactly (e.g., Chen & Scott, 1993; Duffee, 2002; Fisher & Gilles, 1996; Pearson & Sun, 1994).1 Other yields are taken to be measured with independent and identically distributed (IID) error relative to the model. Since principal components analysis (PCA) is a widely used and easy to implement data reduction tool, it would seem that principal component based factors would provide both a convenient and reasonable specification for affine models. However, while reasonable and
∗ Tel.: +1 619 594 8397. E-mail address: [email protected] 1 We use ATSM to refer to dynamic term structure models in which the yield is an affine function of the state variable. http://dx.doi.org/10.1016/j.najef.2014.02.007 1062-9408/© 2014 Published by Elsevier Inc.
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convenient, a specification that uses PCA is reported to produce implied errors which are highly autocorrelated (Adrian et al., 2013; Duffee, 2011; Hamilton & Wu, 2012, 2014). A model which produces implied errors, computed as the difference between the actual yield and model-implied yield, that are highly autocorrelated contradicts the often assumed IID measurement error structure typically specified for the estimation of affine models.2 In this paper, we design and run a Monte Carlo experiment to assess the impact of autocorrelation in measurement errors on parameter estimates obtained from the estimation of ATSM when PCA is used to extract factors. We decide to limit our study to the class of ATSM of Duffie and Kan (1996) because it has been widely studied in the literature (Andersen & Benzoni, 2010; Cassola & Luis, 2003; Christensen, Diebold, & Rudebusch, 2009, 2011; Collin-Dufresne, Goldstein, & Jones, 2008; Cox, Ingersoll, & Ross, 1985; Dejong, 2000; Duarte, 2004; Duffee, 2002; Jegadeesh & Pennacchi, 1996; Langetieg, 1980; Peroni, 2012; Vasicek, 1977). Additionally, we recognize that serial correlation can be a potential issue for any ATSM (Dejong, 2000; Dempster & Tang, 2011; Juneja, 2013) regardless of whether pricing factors are latent, extracted using individual yields, or estimated from PCA. However, the recent increase in the popularity of application of PCA to extract factors for use in modeling ATSM motivates us to focus on PCA (Ang & Piazzesi, 2003; Collin-Dufresne et al., 2008; Graveline & Joslin, 2011; Joslin, Priebsch, & Singleton, 2010; Joslin, Singleton, & Zhu, 2011a; Joslin, Le, & Singleton, 2013a).3 We study the Joslin, Singleton, and Zhu (2011) (hereafter, JSZ) normalization of the class of ATSM. The JSZ normalization retains all the basic properties of the affine term structure model and its usage enables us to conduct repeated optimizations over a large number of simulation trials using predominantly ordinary least squares estimations, which significantly reduces the computational burden of the estimation relative to other normalizations of the affine term structure models contained in the class of Duffie and Kan (1996). Our decision to focus on parameter estimation follows from recent work done by Dempster and Tang (2011) and Juneja (2013). Dempster and Tang (2011) design Monte Carlo experiments to assess the impact of serial and cross sectional correlation in measurement errors on parameter estimates obtained from Kalman filtering estimation of alternative affine term structure model specifications. The authors conclude that serial correlation in measurement errors do affect parameter estimates obtained from alternative affine term structure model specifications and propose a new specification to solve the problem. Our Monte Carlo experiment is constructed to be completely consistent with the data and not observation or econometric theory (Bauer, Rudebusch, & Wu, 2012; Dempster & Tang, 2011). Approaches that are completely consistent with data and experimentation (e.g., data driven simulation methodologies) are advantageous relative to alternative approaches in cases where econometric modeling, economic analysis, and economic decisions are engaged separately (Yang & Cheng, 2010). In such cases, experimentation (e.g., data driven methods) yields more insight than an approach that is justified by theory or observation. This is because experimentation enables us to appropriately separate econometric modeling, economic analysis, and economic decisions. In the current study, the construction of an experiment enables us to first, build a simulated model for the term structure and generate serial correlation in measurement errors, which comprises econometric modeling, and second, provide estimates and analyze (e.g., run statistical tests) an ATSM in the presence of varying amounts of serial correlation. The latter is what comprises economic analysis. Thirdly, and finally, constructing the experiment enables us to draw conclusions regarding the precision of parameter estimates in the presence of varying amounts of serial correlation. The last one comprises economic decisions. All three of these steps are constructed separately and independently, but sequentially. This means that for a data driven approach, the parameter estimates for the model update following the update in the data and measurement error structure enabling us to assess the effects of serial
2 In affine specifications, the measurement error is the difference between the actual yield and the yield predicted by the model (Fisher & Gilles, 1996). 3 Raisman and Zohar (2004) and Adrian, Crump, and Moench (2013) also use PCA in the formal development of a term structure model. However, Raisman and Zohar (2004) model their factors using an ARIMA process and not an ATSM. As an alternative to pricing the term structure using a model contained within the class of affine term structure models of Duffie and Kan (1996); Adrian et al. (2013) propose a regression based approach to the pricing of interest rates.
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correlation in the measurement error term on parameter estimates properly. The reason for this is that the parameter estimates for the model reflect changes in the data and measurement error structure which differ across simulation trials. The implication here is that we do not know exactly how serial correlation will affect the data across simulation trials before running the simulation trial. On the other hand, an approach which is justified by theory or observation is characterized by parameter estimates that are constructed and updated based upon theory or fixed values that have been observed and not on the actual data or actual measurement error structure.4 Since data and measurement error updates are made based upon fixed values (e.g., treating the serial correlation coefficients as fixed before running the simulation trial), we know exactly how the serial correlation in the measurement errors will impact the data from simulation trial to simulation trial. The former approach leads to more appropriate analysis because we are unaware of the exact impact of serial correlation on the data before conducting economic analysis. Put differently, generating serial correlation in measurement errors and economic analysis are independent, separate, and sequential processes. Thus, a data-driven simulation methodology, like the one used in this study, is more advantageous to investigate the effects of serial correlation in the measurement error term on parameter estimates implied by the JSZ normalization than one which is justified by econometric theory or observation. Still other approaches address the issue of autocorrelation using models specified with lag operators. Monfort and Pegoraro (2007) propose a discrete-time term structure model of interest rates that, among other things, captures the historical dynamics of the factor driving alternative shapes in the term structure of interest rates using lagged values. Joslin, Le, and Singleton (2013b) develop a family of Gaussian macro-dynamic term structure model of interest rates in which bond yields are a function of risk factors which follow a vector autoregressive process of order ∞ > q ≥ 1. Models such as these account for serial correlation in the term structure of interest rates because lagged values contain information about the degree to which data are auto correlated. Moreover, there may also be implications for residuals of using approaches that employ macro factors to extract term structure estimates. Information contained in macro factors, such as inflation, can be useful for thinking about future returns in bond markets (e.g., Chernov and Mueller, 2012) or equity markets (e.g., Konchitchki, 2011 or Konchitchki, 2013). In conjunction with this literature and such approaches is the existence of omitted variables. Such variables may also capture output (e.g., Ludvigson & Ng, 2009) or volatility (e.g., Joslin, 2010) which can drive future returns. If residuals are reflecting macro factors, then autocorrelation in model estimates could be reflecting autocorrelation in macro variables and although they are omitted, these variables could also be contributing to serial correlation in model estimates. Omission of macro variables from the model does not preclude them from contributing to serial correlation in the parameter estimates if any of the omitted variables (e.g., inflation, output, or volatility) is correlated. In contrast to prior studies, we explore the impact of serial correlation on maximum likelihood estimates by designing a Monte Carlo experiment that relies upon work done by Collin-Dufresne et al. (2008) (hereafter, CDGJ). Our experiment relies on the construction of a simulation study based upon a measurement error assumption. This assumption consists of constructing two sets of independent and identically distributed errors; a low error case in which we add a half basis point standard deviation shock to the yield curve and a high error case in which we add a two basis point standard deviation shock to the yield curve. Our choice of magnitudes for the error shocks follows from CDGJ who note on p. 771 of their article, “the latter (two basis points) represents an extremely large value, as IID errors of that magnitude generate negative serial correlation in daily yield changes that are between −0.10 and −0.25, which are clearly at odds with the near-zero autocorrelations observed in the data.” Our results imply that despite noted model misspecification due to serial correlation in affine term structure models, parameter estimation associated with the JSZ normalization is reasonably precise. Our results are based upon an experiment in which data is simulated from a three factor Vasicek model with parameters estimated using Kalman filtering. Then, we construct a serial correlation error structure using the simulated data and add the errors to the simulated yields (which are generated to be
4
Within each simulation trial, the simulated yield is treated as the actual or observed yield.
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Table 1 Descriptive statistics. Maturity
Mean
Stand. Dev.
Maximum
Minimum
1-Month 3-Month 6-Month 9-Month 12-Month 2-Year 3-Year 5-Year 6-Year 7-Year 9-Year 10-Year
0.0415 0.0417 0.0424 0.0431 0.0439 0.0462 0.0469 0.0506 0.0510 0.0525 0.0548 0.0547
0.0180 0.0182 0.0181 0.0180 0.0178 0.0156 0.0150 0.0125 0.0127 0.0113 0.0092 0.0103
0.0682 0.0687 0.0711 0.0733 0.0750 0.0760 0.0761 0.0771 0.0771 0.0775 0.0777 0.0780
0.0102 0.0100 0.0098 0.0098 0.0099 0.0127 0.0140 0.0210 0.0177 0.0256 0.0258 0.0138
This table reports descriptive statistics for LIBOR rates and extrapolated zero coupon yields from June 21, 1996 until July 2, 2008.
free of error). This construction enables us to generate yield curve data which is completely consistent with observed serial correlations. Within the context of each simulation trial, the simulated yield is treated as the observed yield. Finally, we estimate the RPC specification of the JSZ normalization using each simulated dataset in the presence of a measurement error assumption and find that model estimation is quite accurate in the presence of model misspecification caused by serial correlation. We maintain that our simulation study enables us to properly examine the effects of serial correlation on parameter estimates implied by the JSZ normalization of the Gaussian ATSM should autocorrelation be bounded by the measurement error assumption. The remainder of this paper is outlined as follows. We begin with a data description in Section 2. In Section 3, we provide an overview of the two Gaussian normalizations of affine term structure models studied in this paper. Then, we provide all the details of the Monte Carlo experiment in Section 4 and the results in Section 5. Finally, we conclude in Section 6. 2. Data The dataset we used in our study comprises daily observations of Libor rates with maturities of 1-month, 3-month, 6-month, 9-month, and 12-month and swap rate quotes for 2-year, 3-year, 4year, 5-year, 7-year, and 10-year from June 21, 1996 until July 02, 2008. Collected from Bloomberg, the dataset resulted in 3034 observations after eliminating special days like Christmas or New Year Day, when no trading took place. Our choice of Libor and swap rate quotes follows from CDGJ. Some descriptive statistics for the LIBOR rates and extrapolated zero coupon yields are shown in Table 1. 3. Gaussian normalizations used in this study 3.1. Dai and Singleton (2000) normalization Mostly following the structure of Dai and Singleton (2000), the specification of an n-factor continuous-time affine term structure model begins with the risk neutral dynamics (Q) for unobservable Markov state vector G, ˜ − Gt )dt + ˜ dGt = K(
St dWtQ ;
(1)
where WtQ is an n-dimensional independent standard Brownian motion under Q, Gt is an n × 1 vector, ˜ is an n × 1 vector, and St is an n × n matrix with the ith diagonal element: K˜ and are n × n matrices, Sii,t =
ai + bi Gt ;
(2)
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where ai is a scalar and bi is an n × 1 vector. The short term interest rate is given by rt + o + 1 Gt ;
(3)
where o is a scalar quantity and 1 is an n × 1 vector. Within this framework, Duffie and Kan (1996) demonstrated that prices of zero-coupon bonds are
Pt,T
⎡ ⎤ T − rv dv t ⎦ = e(Bt,T Gt +At,T ) . = EtQ ⎣e
(4)
Zero coupon yields implied by Eq. (4) are yt,T =
G −Bt,T t At,T −1 Log Pt,T = − . T −t T −t T −t
(5)
In Eqs. (4) and (5) above, Bt,T and At,T are solutions to the following system of ordinary differential equations (i.e., Riccati equations), 2 dAt,T 1 n ˜ KB ˜ t,T + ˙ Bt,T ] ai − 0 , = − 2 dt i=1 i
(6)
2 dBt,T 1 n = −K˜ Bt,T + ˙ Bt,T ] bi − 1 . 2 dt i=1 i
(7)
The ODEs given in Eqs. (6) and (7) can be solved via numerical integration beginning from the initial conditions AT,T = 0 and BT,T = 0n×1 . To obtain zero-coupon bond prices, we also need to know the dynamics of the state vector in the actual probability measure (P) and this requires knowledge of the market prices of risk. The market prices of risk are assumed to be given by (t) =
St ;
where is an n × 1 vector of constants. The assumed market prices of risk lead to the following dynamics of the state vector under P: dGt = K( − Gt )dt +
St dWt ;
(8)
where Wt is an n-dimensional vector of standard independent Brownian motions under P, and K = K˜ −
˚
˜ + = K −1 (K˜
(9)
).
In Eqs. (9) and (10), the jth row of ˚ (an n × n matrix) is j bj and the jth element of
(10) (an n × 1 vector)
is j aj . See Dai and Singleton (2000) for more information. 3.2. Joslin, Singleton, and Zhu (2011) Normalization JSZ (2011) develop a Gaussian term structure model representation in which pricing factors are taken to be portfolios of yields that are observationally equivalent (e.g., Joslin et al., 2010; JSZ; or, Joslin et al., 2013a) to the Dai and Singleton (2000) normalization described above (e.g., JSZ, 2011a, 2011c). Q Given the factors, they show that the entire yield curve can be obtained from r∞ , the long-run mean T of the short-term interest rate under Q; , the speed of mean reversion of the factors under Q; P P and ˙ p , the conditional covariance matrix of factors. To develop their representation, they start with the following discrete-time characterization of the factors (state vector) Gt ∈ RN : P P Gt = OG + 1G Gt−1 + ˙G ∈ Pt ,
Gt =
Q OG
Q + 1G Gt−1
+ ˙G ∈ Q t ,
(11) (12)
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rt = OG + 1G Gt .
(13)
Here rt is the one-period spot rate, ˙G ˙GT is the conditional covariance matrix of Gt , and ∈t ∼ N(0, IN ). Given the system in Eqs. (11)–(13), prices for a zero coupon yield of maturity m are given by
m−1
Dt,m = EtQ e
−
i=0
rt+i
= eAm +Bm Gt
(14)
where (Am , Bm ) satisfy the well-known Riccati difference equations
Am+1 − Am = 0Q Bm +
1 B Ho Bm − 0 , 2 m
(15)
Bm+1 − Bm = 1Q Bm − 1 ,
(16)
subject to the initial conditions A0 = 0 and B0 = 0. The loadings for the corresponding bond yield are am = − Am /m and bm = − Bm /m. The parameters used for pricing are Q = (k0Q , k1Q , G , 0 , 1 ). G Applying invariant transformations to the system given by Eqs. (11)–(13) (e.g., Dai & Singleton, 2000; Joslin, Singleton, & Zhu, 2011), the risk factors can be replaced by portfolios of yields, Pt , and the Q Q distribution (risk-neutral) of Pt can be fully characterized by Q = (∞ , Q , ˙P ). The parameters of P Q Q , 1P ) and ˙ P . The system is given by the P distribution (historical) are (OP P P Pt = OG + 1P Pt−1 + ˙P ∈ Pt ,
Pt =
Q OP
Q + 1P Pt−1
(17)
+ ˙P ∈ Q t
(18)
rt = OP + 1P Pt ; where
Q Q (OP , 1P , OP , 1P )
(19) are
invariant transformations. The yields are given by
Q analytical (i.e. closed-form) functions of (k∞ , Q , ˙P ) owing to the Q P P P parameters are given by = (OP , 1P , k∞ , Q , ˙P ).5 Model implied
yt,m = AP + Bp Pt ,
(20)
where (Ap , Bp ) are explicit functions of (Am , Bm ). Under Case P of the RPC specification of the JSZ normalization, ordinary least squares (OLS) P , P ) from which we can also compute a fairly gives the maximum likelihood estimates of (OP 1P accurate initial value for the covariance matrix ˙P ˙PT . While, JSZ investigate several specifications (see Table 1, p. 946 in JSZ), we choose to estimate Case P of the RPC specification because several parameters can be estimated via OLS which reduces the computational burden of the estimation and this will be useful for our simulation study discussed in the next section. In the RPC specification, the pricing factors, which are assumed to fit the model exactly, are derived from the application of PCA to the data (for more details on the background of the JSZ normalization, see JSZ (2011a, 2011b, 2011c)). 4. Monte Carlo simulation experiment In this section, we describe the Monte Carlo experiment used in this study to assess the impact of serial correlation on maximum likelihood parameter estimates obtained from the JSZ normalization of the Gaussian ATSM. In Section 4.1, we provide a description of the data generating process in detail. We refer the reader to Section 3.1 for the relevant background. In Section 4.2, we describe estimation of the JSZ normalization and refer the reader to Section 3.2 for more information on the JSZ normalization. In Section 5, we present the results of the simulation experiment.
5
Q Q Q Q If G is stationary under Q, then r∞ and k∞ are connected through the following relation: r∞ = k∞
m1 i=1
(−Q1 )
−i
. If m1 = 1,
Q Q Q and r∞ is simply −k∞ /1Q for any stationary model. In our applications, then Q1 is not a repeated root of the Jordan form of 1G Q this is the case, and so we follow JSZ (2011) and estimate their normalization in terms of r∞ because of its intuitive economic interpretation.
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4.1. Data generating process In our analysis, we simulate data from a three factor Gaussian model through the following steps. We assume that both K and ˙ are diagonal matrices and so the state variables are independent which means that our data are being generated from a three-factor Vasicek model. We begin by simulating the latent factor process based upon the application of an Euler discretization scheme to Eq. (8) for each day in the sample. The latent factor process is simulated using an Euler discretization scheme based upon Eq. (8) and computed for each day covering the entire sample period. To use an Euler discretization scheme to simulate G, we write the process for G as Gt = Gt−1 + K( − Gt )t + ˙z.6 Eqs. (9) and (10) are then used to convert the parameters under P to the parameters under Q. We then use Eqs. (4), (6) and (7) to generate zero-coupon prices for each day in the sample and compute yields based upon Eq. (5) in which A and B satisfy Eqs. (6) and (7). Although closed form expressions for coefficients A and B do exist, in their absence, MATLAB’s ODE45 provides fast and efficient estimates of these coefficients (Chacko & Das, 2002; Piazzesi, 2003). We construct yields with time to maturity
= 0.083, 0.25, 0.5, 0.75, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 years because these maturities correspond to the actual data. Parameter values used to simulate the yield curve are obtained from the estimation of a three-factor Vasicek model via a Kalman filtering algorithm.7 Estimation of the three factor Vasicek model using the daily dataset described in Section 3.1 led to the following parameters; 1 = 0.01825, 2 = 0.6563, 3 = 0.000002024, 1 = 0.0000019, 2 = 0.5961, 3 = −0.0000438, 1 = 0.0084, 2 = 0.0099, 3 = 0.0079, 1 = 0.0058, 2 = 0.00025, and 3 = 0.0021. To construct the error, we follow Dempster and Tang (2011) and assume that errors evolve according to the following equation: εt = Pεt−1 + t ,
(21)
where P is a B × B matrix and t is a B × 1 random vector distributed jointly normal with mean 0 and covariance matrix . Here, B is the number of bond maturities. While this setup is identical to Dempster and Tang (2011), in our design, we assume that serial correlation in yields is attributed to serial correlation in the factors and so we construct factors using PCA applied to the simulated yields and use the factors in our construction of the error term. The intuition for this design relies on the fact that within the class of affine term structure models, one first constructs the factor and then uses the factor to construct the yield and so any serial correlation in yields arises from serial correlation in the factor. To compute serial correlation in the factors, we run a regression with yields as the dependent variable and factors as the independent variable and store the residual for each OLS regression and exactly estimate the specification given in Eq. (21) for each residual using OLS to obtain serial correlation coefficients and use those serial correlation coefficients to construct the error term as the fitted value from the OLS regression. Finally, we add the aforementioned simulated error term to the simulated yields. Performing the steps described above enables us to construct yield curve data which has a serial correlation structure that is consistent with the data. Next, as motivated by CDGJ, we estimate Case P of the RPC specification of the JSZ normalization under using a measurement error assumption. 4.2. Estimation of the JSZ normalization As mentioned in the introduction, the focus of our simulation study is on the JSZ normalization and so for each simulated dataset, we re-estimate Case P of the RPC specification of the JSZ normalization using the measurement error assumption. Implementation of Case P requires the Q P , and 9 for K P ). Estimaestimation of 22 parameters (1 for r∝ , 3 for Q , 6 for ˙P ˙P , 3 for KOP 1P tion of Case P using the daily dataset described in Section 3.1 led to the following parameters; Q r∞ = 0.0644, Q = −0.0017, Q = −0.0399, Q = −0.0399, 1 = 0.00080, 2 = 0.00079, 3 = 0.00069, 1 2 3
6
t is the time between observations and z is a standard normal shock term. Estimation via the Kalman filter is covered well in the literature, and so we refer the reader to Chen and Scott (1993), Fisher and Gilles (1996), Dejong (2000), Bolder (2001), Christensen et al. (2009, 2011), or Dempster and Tang (2011) for more details. 7
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Table 2 Panel A: Differences in means. Parameter
G1:Mean
G2:Mean
Q r∞ 1Q 2Q 3Q 0 kP,1 0 P,2 0 P,3 1P,11 1 P,12 1 P,13 1 P,21 1P,22 1 P,23 1 P,31 1 P,32 1P,33 1 2 3 12 13 23
−0.01386 0.00544 0.00373 0.00468 −0.0000052 −0.0000026 0.0000025 −0.52676 −0.00702 −0.00933 −0.00681 −0.50476 −0.00829 −0.00908 −0.00880 −0.49861 0.000075 0.000078 0.000078 −0.00559 −0.01595 −0.00237
−0.01040 0.00559 0.00472 0.00569 −0.0000072 −0.0000081 −0.0000010 −0.69095 −0.00915 −0.01573 −0.00960 −0.68252 −0.01494 −0.01670 −0.01475 −0.66152 0.000282 0.000287 0.000295 −0.00936 −0.00949 −0.02692
t-Test statistic
P-value
−2.03095 −0.65867 −1.38624 −1.42184 0.36353 0.94617 0.61879 13.62774 0.21607 0.64717 0.28166 14.60534 0.66048 0.76783 0.59184 13.1823 −50.1471 −50.1365 −47.3128 0.34516 −0.58413 2.18210
0.04239 0.51018 0.16583 0.15523 0.71625 0.34418 0.53613 0 0.82896 0.51759 0.77824 0 0.50903 0.44268 0.55403 0 0 0 0 0.73001 0.55920 0.02922
Panel A reports means for parameters obtained from the maximum likelihood estimation of Case P of the JSZ normalization. The first sample consists of yields simulated based upon the low-error case. The second sample consists of yields simulated based upon the high-error case. The column titled G1: Mean reports means for parameter estimates from the first sample. G2: Mean reports means for the parameter estimates from the second sample. Rows which were highlighted in bold contain parameters whose means were significantly different across samples.
p,1
p,2
p,3
p,11
=
p,31
=
12 = 0.4688, 13 = −0.4270, 23 = 0.4305, KOP = 0.0015, KOP = −0.000028, KOP = −0.0011, K1P p,12 p,13 −0.2521, K1P = 0.0976, K1P p,32 p,33 1738, K1P = −0.0679, k1P =
= −0.0192,
p,21 K1P
p,22 = 0.0300, K1P = Q parameters for r∝ and
−0.0171,
p,23 K1P
= 0.0081, K1P
0.0129. The Q are used as initial conditions P for the maximum likelihood estimation, but under Case P, the maximum likelihood estimates of KOP P and K1P are given by the OLS estimates corresponding to a VAR model of portfolios of yields which also provides good starting conditions for the maximum likelihood estimation of ˙P ˙P . Hence, the P and K P are estimated through OLS regressions applied to VAR 12 parameters corresponding to KOP 1P models of portfolios of simulated yields which also yield initial conditions for the maximum likelihood estimation of the 6 parameters of ˙P ˙P (see Joslin et al. (2011a) for more information). 5. Results We repeat each simulation based optimization of the RPC specification of the JSZ normalization N = 1000 times and carry out a paired sample t-test (i.e., difference in mean test) and a rank-sum test (i.e., difference in median test) to test for statistical differences between the parameters estimated in the high error case and those estimated in the low error case. Table 2 reports means and medians for each parameter and group as well as results from the statistical tests for their differences based upon the simulation trials. We assess the effect of a large amount of serial correlation on estimation of the parameters. In the high error case (i.e., 2 basis points), most of the parameter estimates are not statistically distinct from those parameter estimates obtained from the low-error case (i.e., 0.5 basis points). For 14 out of 22 parameters, we fail to reject the null hypothesis that the mean is statistically different across both groups. Furthermore, for 15 out of 22 parameters, we fail to reject the null hypothesis that the median is statistically different across both groups. Both tests were conducted at the 5% level. The Please cite this article in press as: Juneja, J. Term structure estimation in the presence of autocorrelation. North American Journal of Economics and Finance (2014), http://dx.doi.org/10.1016/j.najef.2014.02.007
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Table 3 Panel B: Differences in medians. Parameter
G1:Median
G2:Median
Rank sum test statistic
P-value
Q r∞ 1Q 2Q 3Q 0 kP,1 0 P,2 0 P,3 1P,11 1 P,12 1 P,13 1 P,21 1P,22 1 P,23 1 P,31 1 P,32 1P,33 1 2 3 12 13 23
−0.00164 0.00523 0.00392 0.00528 −0.0000043 −0.0000017 0.0000010 −0.51317 −0.00472 −0.01340 −0.00797 −0.49993 0.00067 −0.01255 −0.00078 −0.48338 0.0000654 0.0000668 0.0000666 −0.00126 −0.00489 0.00187
−0.00165 0.00517 0.00420 0.00533 −0.0000073 −0.0000048 0.0000028 −0.75852 −0.01390 −0.00846 −0.01543 −0.75291 −0.02205 −0.01330 −0.01865 −0.71951 0.000249 0.000250 0.000253 −0.00714 −0.00168 −0.02474
994,038 996,155 987,937 997,300 1,009,545 1,012,277 999,913 1,169,075 1,003,033 1,008,399 1,004,028 1,178,477 1,010,479 1,010,017 1,010,012 1,163,500 507,245 1,008,250 510,178 996,117 1,039,108 509,810
0.6168 0.7365 0.3306 0.8046 0.4837 0.3618 0.9638 0 0.8445 0.5408 0.7847 0 0.4397 0.4611 0.4614 0 0 0.5345 0 0.7343 0.0028 0
Panel B reports medians for parameters obtained from the maximum likelihood estimation of Case P of the JSZ normalization. The first sample consists of yields simulated based upon the low-error case. The second sample consists of yields simulated based upon the high-error case. The column titled G1: Median reports medians for parameter estimates from the first sample. G2: Median reports medians for the parameter estimates from the second sample. Rows which were highlighted in bold contain parameters whose means were significantly different across samples.
implication of Table 3 is that should the magnitudes of the errors be bounded by the measurement error assumption, parameter estimation of the RPC specification of the JSZ normalization is robust to autocorrelation misspecification.8 6. Conclusion A common assumption in the term structure literature is that a certain number of yields or a linear combination of yields fits the model exactly. Other yields are taken to be measured with IID error relative to the model. One drawback of this approach is that the errors are not IID. In fact, they are highly auto-correlated. In this work, we investigated the extent to which serial correlation in measurement errors impacts maximum likelihood estimates obtained from ATSM, when principal components are used as risk factors. We acknowledge that serial correlation is potentially an issue for any dynamic ATSM; however, we limited our focus to principal components analysis because of its recent dramatic increase in popularity as a tool to extract risk factors (e.g., Beckmann, Belke, & Dobnik, 2012; Vasishtha & Maier, 2013). Our experiment focused on repeated estimations of Case P of the RPC specification of the JSZ normalization using the measurement error assumption. We chose to focus on Case P of the RPC specification of the JSZ normalization because it retains all the basic properties of
8 We also carried out our simulation experiment using monthly data. In particular, we simulated data on monthly zero coupon yields using a three factor Vasicek (1977) model with parameters estimated via the Kalman Filter. The zero coupon yield data used to estimate the Vasicek model was extrapolated from coupon bearing U.S. Treasury yields obtained from the Federal Reserve Bank website. Then, for each simulation trial, we repeated each step necessary to construct the serially correlated errors. We followed the steps for estimation of the JSZ normalization as outlined in the paper assuming the low error and high error cases and then ran statistical tests to examine the differences in the parameter estimates and found out that the results were similar. The main difference was computational cost, as the entire simulation experiment finished much faster with monthly data.
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the affine term structure model and enabled us to conduct repeated maximum likelihood estimations over 1000 simulation trials in a reasonable amount of time and using predominantly OLS, which significantly reduced the computational burden of the estimation relative to other normalizations of the affine term structure models contained in the class of Duffie and Kan (1996). The data generating process for the zero-coupon yields used in this study was obtained from the estimation of a Vasicek model. Statistical tests across re-estimations led us to conclude that should autocorrelation be bounded by the measurement error assumption, then serial correlation does not materially affect model estimation. Our results depend upon the magnitude of IID Gaussian measurement errors. We maintain that should serial correlation be bounded by the measurement error assumption, which is supported by CDGJ, then, our experiment yields a reasonable framework for which to study the impact of serial correlation in measurement errors on maximum likelihood parameter estimates obtained from Gaussian ATSM. The main implication of our findings is that parameter estimation from the JSZ normalization of the class of ATSM is robust to autocorrelation misspecification, provided it is within the context of the measurement error assumption. These are important findings regarding the relationship between serial correlation and the maximum likelihood estimates of ATSM given that Dempster and Tang (2011) find that assumptions regarding the serial correlation in the measurement error term do impact parameter estimates associated with affine term structure models. Our approaches differ in that the simulation design in the current study was constructed completely from the data and not observation or theory, which is more advantageous for investigating this problem. While the focus of this paper has been on the effects of serial correlation in the measurement error term on parameter estimation of the JSZ normalization, there are some important issues left unresolved. These areas pertain to the economic implications of serial correlation in the measurement error term on affine term structure model estimation. Such implications would include the examination of differences in risk premia, term premia, in-sample fitting error, out-of-sample forecasting accuracy arising due to serial correlation. We leave these issues to future work. Acknowledgements The author is especially thankful to the editor Hamid Beladi and an anonymous referee for very helpful comments that greatly improved the paper. The author also thanks Professor Chris Lamoureux and Professor A.D. Amar for helpful comments. References Adrian, T., Crump, R. K., & Moench, E. (2013). Pricing the term structure with linear regressions. Journal of Financial Economics, 110, 110–138. Andersen, T., & Benzoni, L. (2010). Do bonds span volatility risk in the US Treasury market? A specification test for affine term structure models. Journal of Finance, 65, 603–652. Ang, A., & Piazzesi, M. (2003). A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables. Journal of Monetary Economics, 50, 745–787. Bauer, M. D., Rudebusch, G. D., & Wu, J. C. (2012). Correcting estimation bias in dynamic term structure models. Journal of Business and Economic Statistics, 30, 454–467. Beckmann, J., Belke, A., & Dobnik, F. (2012). Cross-section dependence and the monetary exchange rate model—A panel analysis. North American Journal of Economics and Finance, 23, 38–53. Bolder, D. (2001). Affine term structure models: Theory and implementation. Working paper. Bank of Canada. Cassola, N., & Luis, J. B. (2003). A two factor model of the German term structure of interest rates. Applied Financial Economics, 13, 783–806. Chacko, G., & Das, S. (2002). Pricing interest rate derivatives: A general approach. Review of Financial Studies, 15, 195–241. Chen, R. R., & Scott, L. (1993). ML estimation for a multifactor equilibrium model of the term structure. Journal of Fixed Income, 3, 14–31. Chernov, M., & Mueller, P. (2012). The term structure of inflation expectations. Journal of Financial Economics, 106, 367–394. Christensen, J., Diebold, F. G., & Rudebusch, G. D. (2009). An arbitrage-free generalized Nelson–Seigel term structure model. The Econometrics Journal, 12, 33–64. Christensen, J., Diebold, F. G., & Rudebusch, G. D. (2011). The affine arbitrage-free class of Nelson–Seigel term structure models. Journal of Econometrics, 164, 4–20. Collin-Dufresne, P., Goldstein, R., & Jones, C. (2008). Identification of maximal affine term structure models. Journal of Finance, 63, 743–795. Cox, J., Ingersoll, J., & Ross, S. (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–408.
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Dai, Q., & Singleton, K. J. (2000). Specification analysis of affine term structure models. Journal of Finance, 55, 1943–1978. Dejong, F. (2000). Time series and cross-section information in affine term structure models. Journal of Business and Economic Statistics, 18, 300–314. Dempster, M. A. H., & Tang, K. (2011). Estimating exponential affine models with correlated measurement errors: Applications to fixed income and commodities. Journal of Banking and Finance, 35, 639–652. Duarte, J. (2004). Evaluating an alternative risk preference in affine term structure models. Review of Financial Studies, 17, 379–404. Duffie, D., & Kan, R. (1996). A yield factor model of interest rates. Mathematical Finance, 6, 379–406. Duffee, G. (2002). Term premia and interest rate forecasts in affine models. Journal of Finance, 57, 405–433. Duffee, G. (2011). Forecasting with the term structure: The role of no-arbitrage restrictions. Working paper. Johns Hopkins University. Fisher, M., & Gilles, C. (1996). Estimating exponential-affine models of the term structure. Working paper. Federal Reserve Board. Graveline, J., & Joslin, S. (2011). G10 Swap rates and exchange rates. Working Paper. University of Minnesota and Massachusetts Institute of Technology. Hamilton, J., & Wu, J. C. (2012). Identification and estimation of Gaussian affine term structure models. Journal of Econometrics, 168, 315–331. Hamilton, J., & Wu, J. C. (2014). Testable implications of affine term structure models. Journal of Econometrics, 178, 231–242. Jegadeesh, N., & Pennacchi, G. (1996). The behavior of interest rates implied by the term structure of Eurodollar futures. Journal of Money, Credit, and Banking, 28, 426–446. Joslin, S. (2010). Pricing and hedging volatility risk in Fixed Income markets. Working paper. University of Southern California. Joslin, S., Le, A., & Singleton, K. J. (2013a). Why Gaussian macro-finance term structure models are (nearly) unconstrained factor VAR’s. Journal of Financial Economics, 109, 604–622. Joslin, S., Le, A., & Singleton, K. J. (2013b). Gaussian macro-finance term structure models with lags. Journal of Financial Econometrics, 11, 581–609. Joslin, S., Priebsch, M., & Singleton, K. J. (2010). Risk premiums in dynamic term structure models with unspanned macro risks. Journal of Finance (forthcoming). Joslin, S., Singleton, K. J., & Zhu, H. (2011a). A new perspective on Gaussian DTSM’s. Review of Financial Studies, 24, 926–970. Joslin, S., Singleton, K. J., & Zhu, H. (2011b). Supplement to ‘A new perspective on Gaussian DTSM’s’. http://www.stanford.edu/ ∼kenneths/ Joslin, S., Singleton, K. J., & Zhu, H. (2011c). Matlab code for estimating Gaussian term structure models using the normalization scheme in Joslin, Singleton, and Zhu. http://www.stanford.edu/∼kenneths/ Juneja, J. (2013). An evaluation of alternative methods used in the estimation of Gaussian term structure models. Review of Quantitative Finance and Accounting (forthcoming). Konchitchki, Y. (2011). Inflation and nominal financial reporting: Implications for performance and stock prices. The Accounting Review, 86, 1045–1085. Konchitchki, Y. (2013). Accounting and the Macroeconomy: The case of aggregate price-level effects on individual stocks. Financial Analysts Journal (forthcoming). Langetieg, T. (1980). A multivariate model of the term structure. Journal of Finance, 35, 71–97. Ludvigson, S., & Ng, S. (2009). Macro factors and bond risk premia. Review of Financial Studies, 22, 5027–5067. Monfort, A., & Pegoraro, F. (2007). Switching VARMA term structure models. Journal of Financial Econometrics, 5, 105–153. Pearson, N., & Sun, T. S. (1994). Exploiting the conditional density in estimating the term structure: An application to the Cox, Ingersoll, and Ross model. Journal of Finance, 4, 1279–1304. Peroni, C. (2012). Testing linearity in term structures. Applied Financial Economics, 22, 651–666. Piazzesi, M. (2003). Affine term structure models. In L. P. Hansen, & Y. Ait-Sahalia (Eds.), Handbook of financial econometrics. Amsterdam: North-Holland. Raisman, H., & Zohar, G. (2004). Short-term predictability of the term structure. Journal of Fixed Income, 14, 7–14. Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188. Vasishtha, G., & Maier, P. (2013). The impact of the global business cycle on small open economies: A FAVAR approach for Canada. North American Journal of Economics and Finance, 24, 191–207. Yang, X., & Cheng, J. (2010). A data-driven approach for building macroeconomic decision support system. In 2010 Institute of Electrical and Electronics Engineers International Conference on Systems, Man, and Cybernetics October 13, (pp. 1054–1061). Dr. Januj A. Juneja is an Assistant Professor of Finance in the College of Business Administration at San Diego State University. The focus of his recent research lies in the assessment of mathematical and statistical methods used to estimate interest rate models. Dr. Juneja obtained his Ph.D. in Finance from the University of Arizona.
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North American Journal of Economics and Finance
Term structure estimation in the presence of autocorrelation Januj Juneja ∗ College of Business, San Diego State University, San Diego, CA, United States
a r t i c l e
i n f o
Article history: Received 15 September 2013 Received in revised form 22 February 2014 Accepted 24 February 2014 Keywords: Affine term structure model Principal components analysis Autocorrelation misspecification Monte Carlo simulation Maximum likelihood estimation
a b s t r a c t This paper assesses the effects of autocorrelation on parameter estimates of affine term structure models (ATSM) when principal components analysis is used to extract factors. In contrast to recent studies, we design and run a Monte Carlo experiment that relies on the construction of a simulation design that is consistent with the data, rather than theory or observation, and find that parameter estimation from ATSM is precise in the presence of serial correlation in the measurement error term. Our findings show that parameter estimation of ATSM with principal component based factors is robust to autocorrelation misspecification. © 2014 Published by Elsevier Inc.
1. Introduction In the widely used class of affine term structure models (ATSM) of Duffie and Kan (1996), bond yields are assumed to rely on a low dimensional state space (Piazzesi, 2003, chap. 12). Several authors assume that this state space is comprised of certain yields or linear combinations of yields which are chosen to fit the model exactly (e.g., Chen & Scott, 1993; Duffee, 2002; Fisher & Gilles, 1996; Pearson & Sun, 1994).1 Other yields are taken to be measured with independent and identically distributed (IID) error relative to the model. Since principal components analysis (PCA) is a widely used and easy to implement data reduction tool, it would seem that principal component based factors would provide both a convenient and reasonable specification for affine models. However, while reasonable and
∗ Tel.: +1 619 594 8397. E-mail address: [email protected] 1 We use ATSM to refer to dynamic term structure models in which the yield is an affine function of the state variable. http://dx.doi.org/10.1016/j.najef.2014.02.007 1062-9408/© 2014 Published by Elsevier Inc.
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convenient, a specification that uses PCA is reported to produce implied errors which are highly autocorrelated (Adrian et al., 2013; Duffee, 2011; Hamilton & Wu, 2012, 2014). A model which produces implied errors, computed as the difference between the actual yield and model-implied yield, that are highly autocorrelated contradicts the often assumed IID measurement error structure typically specified for the estimation of affine models.2 In this paper, we design and run a Monte Carlo experiment to assess the impact of autocorrelation in measurement errors on parameter estimates obtained from the estimation of ATSM when PCA is used to extract factors. We decide to limit our study to the class of ATSM of Duffie and Kan (1996) because it has been widely studied in the literature (Andersen & Benzoni, 2010; Cassola & Luis, 2003; Christensen, Diebold, & Rudebusch, 2009, 2011; Collin-Dufresne, Goldstein, & Jones, 2008; Cox, Ingersoll, & Ross, 1985; Dejong, 2000; Duarte, 2004; Duffee, 2002; Jegadeesh & Pennacchi, 1996; Langetieg, 1980; Peroni, 2012; Vasicek, 1977). Additionally, we recognize that serial correlation can be a potential issue for any ATSM (Dejong, 2000; Dempster & Tang, 2011; Juneja, 2013) regardless of whether pricing factors are latent, extracted using individual yields, or estimated from PCA. However, the recent increase in the popularity of application of PCA to extract factors for use in modeling ATSM motivates us to focus on PCA (Ang & Piazzesi, 2003; Collin-Dufresne et al., 2008; Graveline & Joslin, 2011; Joslin, Priebsch, & Singleton, 2010; Joslin, Singleton, & Zhu, 2011a; Joslin, Le, & Singleton, 2013a).3 We study the Joslin, Singleton, and Zhu (2011) (hereafter, JSZ) normalization of the class of ATSM. The JSZ normalization retains all the basic properties of the affine term structure model and its usage enables us to conduct repeated optimizations over a large number of simulation trials using predominantly ordinary least squares estimations, which significantly reduces the computational burden of the estimation relative to other normalizations of the affine term structure models contained in the class of Duffie and Kan (1996). Our decision to focus on parameter estimation follows from recent work done by Dempster and Tang (2011) and Juneja (2013). Dempster and Tang (2011) design Monte Carlo experiments to assess the impact of serial and cross sectional correlation in measurement errors on parameter estimates obtained from Kalman filtering estimation of alternative affine term structure model specifications. The authors conclude that serial correlation in measurement errors do affect parameter estimates obtained from alternative affine term structure model specifications and propose a new specification to solve the problem. Our Monte Carlo experiment is constructed to be completely consistent with the data and not observation or econometric theory (Bauer, Rudebusch, & Wu, 2012; Dempster & Tang, 2011). Approaches that are completely consistent with data and experimentation (e.g., data driven simulation methodologies) are advantageous relative to alternative approaches in cases where econometric modeling, economic analysis, and economic decisions are engaged separately (Yang & Cheng, 2010). In such cases, experimentation (e.g., data driven methods) yields more insight than an approach that is justified by theory or observation. This is because experimentation enables us to appropriately separate econometric modeling, economic analysis, and economic decisions. In the current study, the construction of an experiment enables us to first, build a simulated model for the term structure and generate serial correlation in measurement errors, which comprises econometric modeling, and second, provide estimates and analyze (e.g., run statistical tests) an ATSM in the presence of varying amounts of serial correlation. The latter is what comprises economic analysis. Thirdly, and finally, constructing the experiment enables us to draw conclusions regarding the precision of parameter estimates in the presence of varying amounts of serial correlation. The last one comprises economic decisions. All three of these steps are constructed separately and independently, but sequentially. This means that for a data driven approach, the parameter estimates for the model update following the update in the data and measurement error structure enabling us to assess the effects of serial
2 In affine specifications, the measurement error is the difference between the actual yield and the yield predicted by the model (Fisher & Gilles, 1996). 3 Raisman and Zohar (2004) and Adrian, Crump, and Moench (2013) also use PCA in the formal development of a term structure model. However, Raisman and Zohar (2004) model their factors using an ARIMA process and not an ATSM. As an alternative to pricing the term structure using a model contained within the class of affine term structure models of Duffie and Kan (1996); Adrian et al. (2013) propose a regression based approach to the pricing of interest rates.
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correlation in the measurement error term on parameter estimates properly. The reason for this is that the parameter estimates for the model reflect changes in the data and measurement error structure which differ across simulation trials. The implication here is that we do not know exactly how serial correlation will affect the data across simulation trials before running the simulation trial. On the other hand, an approach which is justified by theory or observation is characterized by parameter estimates that are constructed and updated based upon theory or fixed values that have been observed and not on the actual data or actual measurement error structure.4 Since data and measurement error updates are made based upon fixed values (e.g., treating the serial correlation coefficients as fixed before running the simulation trial), we know exactly how the serial correlation in the measurement errors will impact the data from simulation trial to simulation trial. The former approach leads to more appropriate analysis because we are unaware of the exact impact of serial correlation on the data before conducting economic analysis. Put differently, generating serial correlation in measurement errors and economic analysis are independent, separate, and sequential processes. Thus, a data-driven simulation methodology, like the one used in this study, is more advantageous to investigate the effects of serial correlation in the measurement error term on parameter estimates implied by the JSZ normalization than one which is justified by econometric theory or observation. Still other approaches address the issue of autocorrelation using models specified with lag operators. Monfort and Pegoraro (2007) propose a discrete-time term structure model of interest rates that, among other things, captures the historical dynamics of the factor driving alternative shapes in the term structure of interest rates using lagged values. Joslin, Le, and Singleton (2013b) develop a family of Gaussian macro-dynamic term structure model of interest rates in which bond yields are a function of risk factors which follow a vector autoregressive process of order ∞ > q ≥ 1. Models such as these account for serial correlation in the term structure of interest rates because lagged values contain information about the degree to which data are auto correlated. Moreover, there may also be implications for residuals of using approaches that employ macro factors to extract term structure estimates. Information contained in macro factors, such as inflation, can be useful for thinking about future returns in bond markets (e.g., Chernov and Mueller, 2012) or equity markets (e.g., Konchitchki, 2011 or Konchitchki, 2013). In conjunction with this literature and such approaches is the existence of omitted variables. Such variables may also capture output (e.g., Ludvigson & Ng, 2009) or volatility (e.g., Joslin, 2010) which can drive future returns. If residuals are reflecting macro factors, then autocorrelation in model estimates could be reflecting autocorrelation in macro variables and although they are omitted, these variables could also be contributing to serial correlation in model estimates. Omission of macro variables from the model does not preclude them from contributing to serial correlation in the parameter estimates if any of the omitted variables (e.g., inflation, output, or volatility) is correlated. In contrast to prior studies, we explore the impact of serial correlation on maximum likelihood estimates by designing a Monte Carlo experiment that relies upon work done by Collin-Dufresne et al. (2008) (hereafter, CDGJ). Our experiment relies on the construction of a simulation study based upon a measurement error assumption. This assumption consists of constructing two sets of independent and identically distributed errors; a low error case in which we add a half basis point standard deviation shock to the yield curve and a high error case in which we add a two basis point standard deviation shock to the yield curve. Our choice of magnitudes for the error shocks follows from CDGJ who note on p. 771 of their article, “the latter (two basis points) represents an extremely large value, as IID errors of that magnitude generate negative serial correlation in daily yield changes that are between −0.10 and −0.25, which are clearly at odds with the near-zero autocorrelations observed in the data.” Our results imply that despite noted model misspecification due to serial correlation in affine term structure models, parameter estimation associated with the JSZ normalization is reasonably precise. Our results are based upon an experiment in which data is simulated from a three factor Vasicek model with parameters estimated using Kalman filtering. Then, we construct a serial correlation error structure using the simulated data and add the errors to the simulated yields (which are generated to be
4
Within each simulation trial, the simulated yield is treated as the actual or observed yield.
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Table 1 Descriptive statistics. Maturity
Mean
Stand. Dev.
Maximum
Minimum
1-Month 3-Month 6-Month 9-Month 12-Month 2-Year 3-Year 5-Year 6-Year 7-Year 9-Year 10-Year
0.0415 0.0417 0.0424 0.0431 0.0439 0.0462 0.0469 0.0506 0.0510 0.0525 0.0548 0.0547
0.0180 0.0182 0.0181 0.0180 0.0178 0.0156 0.0150 0.0125 0.0127 0.0113 0.0092 0.0103
0.0682 0.0687 0.0711 0.0733 0.0750 0.0760 0.0761 0.0771 0.0771 0.0775 0.0777 0.0780
0.0102 0.0100 0.0098 0.0098 0.0099 0.0127 0.0140 0.0210 0.0177 0.0256 0.0258 0.0138
This table reports descriptive statistics for LIBOR rates and extrapolated zero coupon yields from June 21, 1996 until July 2, 2008.
free of error). This construction enables us to generate yield curve data which is completely consistent with observed serial correlations. Within the context of each simulation trial, the simulated yield is treated as the observed yield. Finally, we estimate the RPC specification of the JSZ normalization using each simulated dataset in the presence of a measurement error assumption and find that model estimation is quite accurate in the presence of model misspecification caused by serial correlation. We maintain that our simulation study enables us to properly examine the effects of serial correlation on parameter estimates implied by the JSZ normalization of the Gaussian ATSM should autocorrelation be bounded by the measurement error assumption. The remainder of this paper is outlined as follows. We begin with a data description in Section 2. In Section 3, we provide an overview of the two Gaussian normalizations of affine term structure models studied in this paper. Then, we provide all the details of the Monte Carlo experiment in Section 4 and the results in Section 5. Finally, we conclude in Section 6. 2. Data The dataset we used in our study comprises daily observations of Libor rates with maturities of 1-month, 3-month, 6-month, 9-month, and 12-month and swap rate quotes for 2-year, 3-year, 4year, 5-year, 7-year, and 10-year from June 21, 1996 until July 02, 2008. Collected from Bloomberg, the dataset resulted in 3034 observations after eliminating special days like Christmas or New Year Day, when no trading took place. Our choice of Libor and swap rate quotes follows from CDGJ. Some descriptive statistics for the LIBOR rates and extrapolated zero coupon yields are shown in Table 1. 3. Gaussian normalizations used in this study 3.1. Dai and Singleton (2000) normalization Mostly following the structure of Dai and Singleton (2000), the specification of an n-factor continuous-time affine term structure model begins with the risk neutral dynamics (Q) for unobservable Markov state vector G, ˜ − Gt )dt + ˜ dGt = K(
St dWtQ ;
(1)
where WtQ is an n-dimensional independent standard Brownian motion under Q, Gt is an n × 1 vector, ˜ is an n × 1 vector, and St is an n × n matrix with the ith diagonal element: K˜ and are n × n matrices, Sii,t =
ai + bi Gt ;
(2)
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where ai is a scalar and bi is an n × 1 vector. The short term interest rate is given by rt + o + 1 Gt ;
(3)
where o is a scalar quantity and 1 is an n × 1 vector. Within this framework, Duffie and Kan (1996) demonstrated that prices of zero-coupon bonds are
Pt,T
⎡ ⎤ T − rv dv t ⎦ = e(Bt,T Gt +At,T ) . = EtQ ⎣e
(4)
Zero coupon yields implied by Eq. (4) are yt,T =
G −Bt,T t At,T −1 Log Pt,T = − . T −t T −t T −t
(5)
In Eqs. (4) and (5) above, Bt,T and At,T are solutions to the following system of ordinary differential equations (i.e., Riccati equations), 2 dAt,T 1 n ˜ KB ˜ t,T + ˙ Bt,T ] ai − 0 , = − 2 dt i=1 i
(6)
2 dBt,T 1 n = −K˜ Bt,T + ˙ Bt,T ] bi − 1 . 2 dt i=1 i
(7)
The ODEs given in Eqs. (6) and (7) can be solved via numerical integration beginning from the initial conditions AT,T = 0 and BT,T = 0n×1 . To obtain zero-coupon bond prices, we also need to know the dynamics of the state vector in the actual probability measure (P) and this requires knowledge of the market prices of risk. The market prices of risk are assumed to be given by (t) =
St ;
where is an n × 1 vector of constants. The assumed market prices of risk lead to the following dynamics of the state vector under P: dGt = K( − Gt )dt +
St dWt ;
(8)
where Wt is an n-dimensional vector of standard independent Brownian motions under P, and K = K˜ −
˚
˜ + = K −1 (K˜
(9)
).
In Eqs. (9) and (10), the jth row of ˚ (an n × n matrix) is j bj and the jth element of
(10) (an n × 1 vector)
is j aj . See Dai and Singleton (2000) for more information. 3.2. Joslin, Singleton, and Zhu (2011) Normalization JSZ (2011) develop a Gaussian term structure model representation in which pricing factors are taken to be portfolios of yields that are observationally equivalent (e.g., Joslin et al., 2010; JSZ; or, Joslin et al., 2013a) to the Dai and Singleton (2000) normalization described above (e.g., JSZ, 2011a, 2011c). Q Given the factors, they show that the entire yield curve can be obtained from r∞ , the long-run mean T of the short-term interest rate under Q; , the speed of mean reversion of the factors under Q; P P and ˙ p , the conditional covariance matrix of factors. To develop their representation, they start with the following discrete-time characterization of the factors (state vector) Gt ∈ RN : P P Gt = OG + 1G Gt−1 + ˙G ∈ Pt ,
Gt =
Q OG
Q + 1G Gt−1
+ ˙G ∈ Q t ,
(11) (12)
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rt = OG + 1G Gt .
(13)
Here rt is the one-period spot rate, ˙G ˙GT is the conditional covariance matrix of Gt , and ∈t ∼ N(0, IN ). Given the system in Eqs. (11)–(13), prices for a zero coupon yield of maturity m are given by
m−1
Dt,m = EtQ e
−
i=0
rt+i
= eAm +Bm Gt
(14)
where (Am , Bm ) satisfy the well-known Riccati difference equations
Am+1 − Am = 0Q Bm +
1 B Ho Bm − 0 , 2 m
(15)
Bm+1 − Bm = 1Q Bm − 1 ,
(16)
subject to the initial conditions A0 = 0 and B0 = 0. The loadings for the corresponding bond yield are am = − Am /m and bm = − Bm /m. The parameters used for pricing are Q = (k0Q , k1Q , G , 0 , 1 ). G Applying invariant transformations to the system given by Eqs. (11)–(13) (e.g., Dai & Singleton, 2000; Joslin, Singleton, & Zhu, 2011), the risk factors can be replaced by portfolios of yields, Pt , and the Q Q distribution (risk-neutral) of Pt can be fully characterized by Q = (∞ , Q , ˙P ). The parameters of P Q Q , 1P ) and ˙ P . The system is given by the P distribution (historical) are (OP P P Pt = OG + 1P Pt−1 + ˙P ∈ Pt ,
Pt =
Q OP
Q + 1P Pt−1
(17)
+ ˙P ∈ Q t
(18)
rt = OP + 1P Pt ; where
Q Q (OP , 1P , OP , 1P )
(19) are
invariant transformations. The yields are given by
Q analytical (i.e. closed-form) functions of (k∞ , Q , ˙P ) owing to the Q P P P parameters are given by = (OP , 1P , k∞ , Q , ˙P ).5 Model implied
yt,m = AP + Bp Pt ,
(20)
where (Ap , Bp ) are explicit functions of (Am , Bm ). Under Case P of the RPC specification of the JSZ normalization, ordinary least squares (OLS) P , P ) from which we can also compute a fairly gives the maximum likelihood estimates of (OP 1P accurate initial value for the covariance matrix ˙P ˙PT . While, JSZ investigate several specifications (see Table 1, p. 946 in JSZ), we choose to estimate Case P of the RPC specification because several parameters can be estimated via OLS which reduces the computational burden of the estimation and this will be useful for our simulation study discussed in the next section. In the RPC specification, the pricing factors, which are assumed to fit the model exactly, are derived from the application of PCA to the data (for more details on the background of the JSZ normalization, see JSZ (2011a, 2011b, 2011c)). 4. Monte Carlo simulation experiment In this section, we describe the Monte Carlo experiment used in this study to assess the impact of serial correlation on maximum likelihood parameter estimates obtained from the JSZ normalization of the Gaussian ATSM. In Section 4.1, we provide a description of the data generating process in detail. We refer the reader to Section 3.1 for the relevant background. In Section 4.2, we describe estimation of the JSZ normalization and refer the reader to Section 3.2 for more information on the JSZ normalization. In Section 5, we present the results of the simulation experiment.
5
Q Q Q Q If G is stationary under Q, then r∞ and k∞ are connected through the following relation: r∞ = k∞
m1 i=1
(−Q1 )
−i
. If m1 = 1,
Q Q Q and r∞ is simply −k∞ /1Q for any stationary model. In our applications, then Q1 is not a repeated root of the Jordan form of 1G Q this is the case, and so we follow JSZ (2011) and estimate their normalization in terms of r∞ because of its intuitive economic interpretation.
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4.1. Data generating process In our analysis, we simulate data from a three factor Gaussian model through the following steps. We assume that both K and ˙ are diagonal matrices and so the state variables are independent which means that our data are being generated from a three-factor Vasicek model. We begin by simulating the latent factor process based upon the application of an Euler discretization scheme to Eq. (8) for each day in the sample. The latent factor process is simulated using an Euler discretization scheme based upon Eq. (8) and computed for each day covering the entire sample period. To use an Euler discretization scheme to simulate G, we write the process for G as Gt = Gt−1 + K( − Gt )t + ˙z.6 Eqs. (9) and (10) are then used to convert the parameters under P to the parameters under Q. We then use Eqs. (4), (6) and (7) to generate zero-coupon prices for each day in the sample and compute yields based upon Eq. (5) in which A and B satisfy Eqs. (6) and (7). Although closed form expressions for coefficients A and B do exist, in their absence, MATLAB’s ODE45 provides fast and efficient estimates of these coefficients (Chacko & Das, 2002; Piazzesi, 2003). We construct yields with time to maturity
= 0.083, 0.25, 0.5, 0.75, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 years because these maturities correspond to the actual data. Parameter values used to simulate the yield curve are obtained from the estimation of a three-factor Vasicek model via a Kalman filtering algorithm.7 Estimation of the three factor Vasicek model using the daily dataset described in Section 3.1 led to the following parameters; 1 = 0.01825, 2 = 0.6563, 3 = 0.000002024, 1 = 0.0000019, 2 = 0.5961, 3 = −0.0000438, 1 = 0.0084, 2 = 0.0099, 3 = 0.0079, 1 = 0.0058, 2 = 0.00025, and 3 = 0.0021. To construct the error, we follow Dempster and Tang (2011) and assume that errors evolve according to the following equation: εt = Pεt−1 + t ,
(21)
where P is a B × B matrix and t is a B × 1 random vector distributed jointly normal with mean 0 and covariance matrix . Here, B is the number of bond maturities. While this setup is identical to Dempster and Tang (2011), in our design, we assume that serial correlation in yields is attributed to serial correlation in the factors and so we construct factors using PCA applied to the simulated yields and use the factors in our construction of the error term. The intuition for this design relies on the fact that within the class of affine term structure models, one first constructs the factor and then uses the factor to construct the yield and so any serial correlation in yields arises from serial correlation in the factor. To compute serial correlation in the factors, we run a regression with yields as the dependent variable and factors as the independent variable and store the residual for each OLS regression and exactly estimate the specification given in Eq. (21) for each residual using OLS to obtain serial correlation coefficients and use those serial correlation coefficients to construct the error term as the fitted value from the OLS regression. Finally, we add the aforementioned simulated error term to the simulated yields. Performing the steps described above enables us to construct yield curve data which has a serial correlation structure that is consistent with the data. Next, as motivated by CDGJ, we estimate Case P of the RPC specification of the JSZ normalization under using a measurement error assumption. 4.2. Estimation of the JSZ normalization As mentioned in the introduction, the focus of our simulation study is on the JSZ normalization and so for each simulated dataset, we re-estimate Case P of the RPC specification of the JSZ normalization using the measurement error assumption. Implementation of Case P requires the Q P , and 9 for K P ). Estimaestimation of 22 parameters (1 for r∝ , 3 for Q , 6 for ˙P ˙P , 3 for KOP 1P tion of Case P using the daily dataset described in Section 3.1 led to the following parameters; Q r∞ = 0.0644, Q = −0.0017, Q = −0.0399, Q = −0.0399, 1 = 0.00080, 2 = 0.00079, 3 = 0.00069, 1 2 3
6
t is the time between observations and z is a standard normal shock term. Estimation via the Kalman filter is covered well in the literature, and so we refer the reader to Chen and Scott (1993), Fisher and Gilles (1996), Dejong (2000), Bolder (2001), Christensen et al. (2009, 2011), or Dempster and Tang (2011) for more details. 7
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Table 2 Panel A: Differences in means. Parameter
G1:Mean
G2:Mean
Q r∞ 1Q 2Q 3Q 0 kP,1 0 P,2 0 P,3 1P,11 1 P,12 1 P,13 1 P,21 1P,22 1 P,23 1 P,31 1 P,32 1P,33 1 2 3 12 13 23
−0.01386 0.00544 0.00373 0.00468 −0.0000052 −0.0000026 0.0000025 −0.52676 −0.00702 −0.00933 −0.00681 −0.50476 −0.00829 −0.00908 −0.00880 −0.49861 0.000075 0.000078 0.000078 −0.00559 −0.01595 −0.00237
−0.01040 0.00559 0.00472 0.00569 −0.0000072 −0.0000081 −0.0000010 −0.69095 −0.00915 −0.01573 −0.00960 −0.68252 −0.01494 −0.01670 −0.01475 −0.66152 0.000282 0.000287 0.000295 −0.00936 −0.00949 −0.02692
t-Test statistic
P-value
−2.03095 −0.65867 −1.38624 −1.42184 0.36353 0.94617 0.61879 13.62774 0.21607 0.64717 0.28166 14.60534 0.66048 0.76783 0.59184 13.1823 −50.1471 −50.1365 −47.3128 0.34516 −0.58413 2.18210
0.04239 0.51018 0.16583 0.15523 0.71625 0.34418 0.53613 0 0.82896 0.51759 0.77824 0 0.50903 0.44268 0.55403 0 0 0 0 0.73001 0.55920 0.02922
Panel A reports means for parameters obtained from the maximum likelihood estimation of Case P of the JSZ normalization. The first sample consists of yields simulated based upon the low-error case. The second sample consists of yields simulated based upon the high-error case. The column titled G1: Mean reports means for parameter estimates from the first sample. G2: Mean reports means for the parameter estimates from the second sample. Rows which were highlighted in bold contain parameters whose means were significantly different across samples.
p,1
p,2
p,3
p,11
=
p,31
=
12 = 0.4688, 13 = −0.4270, 23 = 0.4305, KOP = 0.0015, KOP = −0.000028, KOP = −0.0011, K1P p,12 p,13 −0.2521, K1P = 0.0976, K1P p,32 p,33 1738, K1P = −0.0679, k1P =
= −0.0192,
p,21 K1P
p,22 = 0.0300, K1P = Q parameters for r∝ and
−0.0171,
p,23 K1P
= 0.0081, K1P
0.0129. The Q are used as initial conditions P for the maximum likelihood estimation, but under Case P, the maximum likelihood estimates of KOP P and K1P are given by the OLS estimates corresponding to a VAR model of portfolios of yields which also provides good starting conditions for the maximum likelihood estimation of ˙P ˙P . Hence, the P and K P are estimated through OLS regressions applied to VAR 12 parameters corresponding to KOP 1P models of portfolios of simulated yields which also yield initial conditions for the maximum likelihood estimation of the 6 parameters of ˙P ˙P (see Joslin et al. (2011a) for more information). 5. Results We repeat each simulation based optimization of the RPC specification of the JSZ normalization N = 1000 times and carry out a paired sample t-test (i.e., difference in mean test) and a rank-sum test (i.e., difference in median test) to test for statistical differences between the parameters estimated in the high error case and those estimated in the low error case. Table 2 reports means and medians for each parameter and group as well as results from the statistical tests for their differences based upon the simulation trials. We assess the effect of a large amount of serial correlation on estimation of the parameters. In the high error case (i.e., 2 basis points), most of the parameter estimates are not statistically distinct from those parameter estimates obtained from the low-error case (i.e., 0.5 basis points). For 14 out of 22 parameters, we fail to reject the null hypothesis that the mean is statistically different across both groups. Furthermore, for 15 out of 22 parameters, we fail to reject the null hypothesis that the median is statistically different across both groups. Both tests were conducted at the 5% level. The Please cite this article in press as: Juneja, J. Term structure estimation in the presence of autocorrelation. North American Journal of Economics and Finance (2014), http://dx.doi.org/10.1016/j.najef.2014.02.007
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Table 3 Panel B: Differences in medians. Parameter
G1:Median
G2:Median
Rank sum test statistic
P-value
Q r∞ 1Q 2Q 3Q 0 kP,1 0 P,2 0 P,3 1P,11 1 P,12 1 P,13 1 P,21 1P,22 1 P,23 1 P,31 1 P,32 1P,33 1 2 3 12 13 23
−0.00164 0.00523 0.00392 0.00528 −0.0000043 −0.0000017 0.0000010 −0.51317 −0.00472 −0.01340 −0.00797 −0.49993 0.00067 −0.01255 −0.00078 −0.48338 0.0000654 0.0000668 0.0000666 −0.00126 −0.00489 0.00187
−0.00165 0.00517 0.00420 0.00533 −0.0000073 −0.0000048 0.0000028 −0.75852 −0.01390 −0.00846 −0.01543 −0.75291 −0.02205 −0.01330 −0.01865 −0.71951 0.000249 0.000250 0.000253 −0.00714 −0.00168 −0.02474
994,038 996,155 987,937 997,300 1,009,545 1,012,277 999,913 1,169,075 1,003,033 1,008,399 1,004,028 1,178,477 1,010,479 1,010,017 1,010,012 1,163,500 507,245 1,008,250 510,178 996,117 1,039,108 509,810
0.6168 0.7365 0.3306 0.8046 0.4837 0.3618 0.9638 0 0.8445 0.5408 0.7847 0 0.4397 0.4611 0.4614 0 0 0.5345 0 0.7343 0.0028 0
Panel B reports medians for parameters obtained from the maximum likelihood estimation of Case P of the JSZ normalization. The first sample consists of yields simulated based upon the low-error case. The second sample consists of yields simulated based upon the high-error case. The column titled G1: Median reports medians for parameter estimates from the first sample. G2: Median reports medians for the parameter estimates from the second sample. Rows which were highlighted in bold contain parameters whose means were significantly different across samples.
implication of Table 3 is that should the magnitudes of the errors be bounded by the measurement error assumption, parameter estimation of the RPC specification of the JSZ normalization is robust to autocorrelation misspecification.8 6. Conclusion A common assumption in the term structure literature is that a certain number of yields or a linear combination of yields fits the model exactly. Other yields are taken to be measured with IID error relative to the model. One drawback of this approach is that the errors are not IID. In fact, they are highly auto-correlated. In this work, we investigated the extent to which serial correlation in measurement errors impacts maximum likelihood estimates obtained from ATSM, when principal components are used as risk factors. We acknowledge that serial correlation is potentially an issue for any dynamic ATSM; however, we limited our focus to principal components analysis because of its recent dramatic increase in popularity as a tool to extract risk factors (e.g., Beckmann, Belke, & Dobnik, 2012; Vasishtha & Maier, 2013). Our experiment focused on repeated estimations of Case P of the RPC specification of the JSZ normalization using the measurement error assumption. We chose to focus on Case P of the RPC specification of the JSZ normalization because it retains all the basic properties of
8 We also carried out our simulation experiment using monthly data. In particular, we simulated data on monthly zero coupon yields using a three factor Vasicek (1977) model with parameters estimated via the Kalman Filter. The zero coupon yield data used to estimate the Vasicek model was extrapolated from coupon bearing U.S. Treasury yields obtained from the Federal Reserve Bank website. Then, for each simulation trial, we repeated each step necessary to construct the serially correlated errors. We followed the steps for estimation of the JSZ normalization as outlined in the paper assuming the low error and high error cases and then ran statistical tests to examine the differences in the parameter estimates and found out that the results were similar. The main difference was computational cost, as the entire simulation experiment finished much faster with monthly data.
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the affine term structure model and enabled us to conduct repeated maximum likelihood estimations over 1000 simulation trials in a reasonable amount of time and using predominantly OLS, which significantly reduced the computational burden of the estimation relative to other normalizations of the affine term structure models contained in the class of Duffie and Kan (1996). The data generating process for the zero-coupon yields used in this study was obtained from the estimation of a Vasicek model. Statistical tests across re-estimations led us to conclude that should autocorrelation be bounded by the measurement error assumption, then serial correlation does not materially affect model estimation. Our results depend upon the magnitude of IID Gaussian measurement errors. We maintain that should serial correlation be bounded by the measurement error assumption, which is supported by CDGJ, then, our experiment yields a reasonable framework for which to study the impact of serial correlation in measurement errors on maximum likelihood parameter estimates obtained from Gaussian ATSM. The main implication of our findings is that parameter estimation from the JSZ normalization of the class of ATSM is robust to autocorrelation misspecification, provided it is within the context of the measurement error assumption. These are important findings regarding the relationship between serial correlation and the maximum likelihood estimates of ATSM given that Dempster and Tang (2011) find that assumptions regarding the serial correlation in the measurement error term do impact parameter estimates associated with affine term structure models. Our approaches differ in that the simulation design in the current study was constructed completely from the data and not observation or theory, which is more advantageous for investigating this problem. While the focus of this paper has been on the effects of serial correlation in the measurement error term on parameter estimation of the JSZ normalization, there are some important issues left unresolved. These areas pertain to the economic implications of serial correlation in the measurement error term on affine term structure model estimation. Such implications would include the examination of differences in risk premia, term premia, in-sample fitting error, out-of-sample forecasting accuracy arising due to serial correlation. We leave these issues to future work. Acknowledgements The author is especially thankful to the editor Hamid Beladi and an anonymous referee for very helpful comments that greatly improved the paper. The author also thanks Professor Chris Lamoureux and Professor A.D. Amar for helpful comments. References Adrian, T., Crump, R. K., & Moench, E. (2013). Pricing the term structure with linear regressions. Journal of Financial Economics, 110, 110–138. Andersen, T., & Benzoni, L. (2010). Do bonds span volatility risk in the US Treasury market? A specification test for affine term structure models. Journal of Finance, 65, 603–652. Ang, A., & Piazzesi, M. (2003). A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables. Journal of Monetary Economics, 50, 745–787. Bauer, M. D., Rudebusch, G. D., & Wu, J. C. (2012). Correcting estimation bias in dynamic term structure models. Journal of Business and Economic Statistics, 30, 454–467. Beckmann, J., Belke, A., & Dobnik, F. (2012). Cross-section dependence and the monetary exchange rate model—A panel analysis. North American Journal of Economics and Finance, 23, 38–53. Bolder, D. (2001). Affine term structure models: Theory and implementation. Working paper. Bank of Canada. Cassola, N., & Luis, J. B. (2003). A two factor model of the German term structure of interest rates. Applied Financial Economics, 13, 783–806. Chacko, G., & Das, S. (2002). Pricing interest rate derivatives: A general approach. Review of Financial Studies, 15, 195–241. Chen, R. R., & Scott, L. (1993). ML estimation for a multifactor equilibrium model of the term structure. Journal of Fixed Income, 3, 14–31. Chernov, M., & Mueller, P. (2012). The term structure of inflation expectations. Journal of Financial Economics, 106, 367–394. Christensen, J., Diebold, F. G., & Rudebusch, G. D. (2009). An arbitrage-free generalized Nelson–Seigel term structure model. The Econometrics Journal, 12, 33–64. Christensen, J., Diebold, F. G., & Rudebusch, G. D. (2011). The affine arbitrage-free class of Nelson–Seigel term structure models. Journal of Econometrics, 164, 4–20. Collin-Dufresne, P., Goldstein, R., & Jones, C. (2008). Identification of maximal affine term structure models. Journal of Finance, 63, 743–795. Cox, J., Ingersoll, J., & Ross, S. (1985). A theory of the term structure of interest rates. Econometrica, 53, 385–408.
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The focus of his recent research lies in the assessment of mathematical and statistical methods used to estimate interest rate models. Dr. Juneja obtained his Ph.D. in Finance from the University of Arizona.
Please cite this article in press as: Juneja, J. Term structure estimation in the presence of autocorrelation. North American Journal of Economics and Finance (2014), http://dx.doi.org/10.1016/j.najef.2014.02.007