Abstract
This paper proposes a cumulated sum (CUSUM) test for the null hypothesis of quantile cointegration. A fully modified quantile estimator is adopted for serial correlation and endogeneity corrections. The CUSUM statistic is composed of the partial sums of the residuals from the fully modified quantile regression. Under the null, the test statistic converges to a functional of Brownian motions. In the application to US interest rates of different maturities, evidence in favor of the expectations hypothesis for the term structure is found in the central part of the distributions of the Treasury bill rate and financial commercial paper rate, but in the tails of the constant maturity rate distribution.
1 Introduction
The cointegration methodology developed by Engle and Granger (1987) has given rise to numerous studies of long run equilibrium relationships among nonstationary economic variables, such as stock price and market fundamentals (Campbell and Shiller 1987, 1988a,b), short- and long-term interest rates (Engle and Granger 1987; Stock and Watson 1988; Hansen 1992), and aggregate consumption and income (Campbell 1987; Engle and Granger 1987). There is a substantial literature on cointegration tests for time series models. Traditional testing procedures include Engle and Granger (1987) who propose a unit root test on the residuals from the cointegrating regression allowing for heteroskedasticity. Phillips and Ouliaris (1990) analyze the nonstandard asymptotic properties of the Engle-Granger tests. In contrast, Johansen (1988, 1991, 1995) proposes maximum likelihood test statistics that follow multivariate unit root distributions. Among various approaches the residual based Engle-Granger type tests have been popular due to computational convenience. For example, in a bivariate setting, the residuals
In the Engle-Granger test, if the (augmented) Dickey-Fuller test rejects that the residuals contain a unit root, then we reject the null hypothesis of no cointegration against the alternative of cointegration. However, since long run equilibrium relationship is of particular interest to economists, some authors focus on the null of cointegration using residual based procedures (Park, Ouliaris, and Choi 1988; Park 1990; Shin 1994). More recently, Xiao and Phillips (2002) apply the conventional cumulated sum (CUSUM) test for structural change to cointegrating regression residuals and develop a consistent residual based test for the null of cointegration. Particularly, the fully modified approach proposed by Phillips and Hansen (1990) is used for serial correlation and endogeneity corrections. To examine the fluctuation in the equilibrium error from the cointegrating regression, Xiao and Phillips (2002) calculate the cumulative sum of the residuals from the fully modified OLS regression. Under the null of cointegration, the residuals are expected to follow a stable process and the CUSUM test statistic converges to a functional of Brownian motions free of nuisance parameters. Under the alternative of no cointegration, the fluctuations in the residuals have larger order of magnitude and the test statistic diverges to infinity asymptotically. This robust test is easy to implement and has good finite sample performance. Furthermore, Xiao (2009) extends the cointegration methodology to quantile regressions. In the quantile cointegrating regression, leads and lags of the integrated regressors are included to account for endogeneity. The cumulative sum of the resulting residuals has the same asymptotic behavior as that from the fully modified regression considered by Xiao and Phillips (2002). However, in the dynamic model, selecting the lengths of leads and lags can be an issue.
Tests based on the CUSUM statistics were originally introduced to investigate whether a regression relationship is stable over time (Brown and Durbin 1968). Brown, Durbin, and Evans (1975) test constancy of the regression coefficients based on the cumulated sum of squared recursive residuals. Ploberger and Kramer (1992) apply the CUSUM test of parameter stability to OLS residuals from a stationary model. Hao and Inder (1996) propose a diagnostic OLS based CUSUM test for structural change in cointegrated regression models. Although used for different purposes, it should be emphasized that the cointegration models studied by Hao and Inder (1996) and Xiao and Phillips (2002) have the same behavior under the null hypotheses, but have different behaviors under the alternatives.
This paper extends the analysis of Xiao and Phillips (2002) to the case of conditional quantiles, since the long run relationship among nonstationary time series may not be uniform. In practice, locations in the distribution other than the mean may matter for cointegration analysis. To examine the equilibrium relationships across different quantiles of the distribution of the response variable, the CUSUM test is employed to test the null hypothesis of quantile cointegration. Similar to the OLS regression, in the quantile regression with I(1) regressors, due to serial correlation between the regression disturbance and the innovation of the integrated regressors and long run endogeneity in the data, the quantile estimator for the cointegrating coefficients is second-order biased and depends on nuisance parameters. In this case, it is difficult to make inference. To solve this problem, this paper uses a Phillips-Hansen type fully modified quantile estimator. The resulting limit distribution is mixed normal so that it provides a standard inference procedure. The CUSUM test statistic is composed of partial sums of the residuals from the fully modified quantile regression. Under the null of quantile cointegration, the test statistic has the same limit distribution as that from Xiao and Phillips (2002).
A great number of empirical papers find that the interest rate process is I(1) and test for cointegration among interest rates of different terms to maturity. The seminal paper of Campbell and Shiller (1987) demonstrates that present value models of the term structure imply cointegration of short- and long-term interest rates. Engle and Granger (1987), Stock and Watson (1988), Boothe (1991), Hansen (1992), Hall, Anderson, and Granger (1992), Mandeno and Giles (1995), and Downing and Oliner (2007) among others also discuss the expectations theory of the term structure of interest rates. A more complete review of work on the expectations hypothesis of the term structure is provided in Iacone (2009). These papers consider various US interest rate series, including different yield series on the federal funds rate, Treasury bill rate, and commercial paper rate. The results are mixed. This paper applies the residual based quantile cointegration test to several sets of US interest rate data. The quantile version of the CUSUM test rejects the expectations hypothesis of the term structure in certain quantiles of the interest rate distributions.
The remainder of this paper is organized as follows: In Section 2, the model is set up. The asymptotic theory for the fully modified quantile estimator and test statistic is developed. In Section 3, the empirical application is discussed. Section 4 concludes the paper. All proofs are provided in the Appendix.
2 Theory
2.1 The model
Let {wt} be an m-vector time series generated by
and
where “⇒” denotes weak convergence of the associated probability measures in the space D[0, 1] under the Skorohod metric, [Tr] signifies the integer part of Tr, and Bw(r), 0≤r≤1, is a vector of Brownian motions with covariance matrix Ω. In general, the long run covariance matrix is defined as
Consider yt as a scalar, xt as a k-dimensional vector, and m=k+1. We have
Thus, the long run covariance Ω is of the form:
Assuming that the elements of
However, least squares methods are confined to estimating conditional mean functions for linear models. In order to account for the nonlinearity in the long run relationship, the analysis is generalized to the entire conditional distribution of yt, which is represented by quantile τ with τ∈(0, 1). Using the quantile regression method introduced by Koenker and Bassett (1978), any percentage point in the distribution of the response variable yt conditional on observed covariates xt can be estimated. This method provides a comprehensive picture of the conditional distribution (Koenker and Bassett 1978; Koenker and Hallock 2001). Denote ℱt as the information set up to time t, so xt∈ℱt. Let F(·) and Ft(·)=Pr(ut<·|ℱt) be the unconditional and conditional cumulative distribution functions of ut and define the τth unconditional and conditional quantiles of ut by
where θ(τ)=(α′(τ), β′(τ))′. The quantile dependent regression coefficient vector θ(τ) can characterize the possibly nonlinear long run relationship between yt and xt. Nonetheless, for each τ the conditional quantile of yt is formulated as a linear function of θ(τ) so that the unknown coefficients can be estimated by linear programming (Koenker and Hallock 2001).
2.2 Estimation method
Consider the following objective function with asymmetric weights on positive and negative residuals:
Here, function ρτ(·) is the asymmetric absolute deviation loss function defined by ρτ(u)=u(τ–I(u<0)), also known as the check function (Koenker and Bassett 1978). Define ψτ(u)=τ–I(u<0). For the τth quantile, the unknown regression coefficients are estimated by minimizing the sum of asymmetrically weighted absolute residuals such that
where
2.3 Assumptions
The following assumptions are imposed in order to derive the asymptotic distribution of the test statistic.
Assumption 1Define
where
where
Assumption 2The conditional distribution function, Ft(u)=Pr(ut<u|ℱt), is absolutely continuous and has a continuous density function ft(u) such that
Assumption 3The density function ft(υT) is uniformly integrable for any sequence
Assumptions 2 and 3 are two standard technical assumptions in the quantile regression literature such as Xiao (2009), Qu (2008), and Oka and Qu (2011)[2]. These two assumptions ensure that the conditional density function is uniformly continuous, bounded, and integrable in some neighborhood of the τth quantile.
2.4 Theoretical results
If the above assumptions hold, the estimates of the regression coefficients will converge to their true values. Let DT be a diagonal matrix. The presence of DT is due to the different convergence rates of the coefficients associated with the deterministic trend and integrated regressors. When the regression model contains an intercept such that dt=1, we have
Theorem 1Under H0and Assumptions 1–3,
where
The asymptotic representation is composed of integrals of Brownian motions and a bias term. In particular, the asymptotic distribution of the cointegrating coefficient estimator
where
Following Phillips and Hansen (1990), define
where
The long run variance
where
where
and
The regression coefficient estimator after the modification,
where
where
Consequently, the fully modified estimator
where
Theorem 2Under H0and Assumptions 1–3,
where
In the linear regression case, the cumulated sum of the residuals, which is
Under H0, the cumulated sum of the fully modified residuals converges to a functional of Brownian motions. For the quantile regression model, the residuals from the fully modified quantile regression is calculated as
In particular,
where
Define
Then, for a certain quantile τ, the asymptotic representation of the CUSUM test statistic is as follows:
Theorem 3Under H0and Assumptions 1–3,
For each quantile level τ, the asymptotic distribution is the same as that from the linear case. Simulated critical values are tabulated in Tables 1 and 2 from Hao and Inder (1996) and Table 1 from Xiao and Phillips (2002). Hao and Inder (1996) also consider the CUSUM statistic, however, for testing the null hypothesis of parameter constancy. Under the null of no structural change, the test statistic from their model has the same asymptotic distribution as the cumulative sum statistic here. The limit distributions are different under the alternatives. The empirical size and power properties of the CUSUM test for cointegration are evaluated via simulation by Xiao and Phillips (2002). In the case of quantile regression, the finite sample properties of the test are similar to those from the fully modified OLS regression.
y−x | τ=0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
---|---|---|---|---|---|---|---|---|---|---|
Bank discount | ||||||||||
4−13 | 0.9277 | 0.9519 | 0.9657 | 0.9743 | 0.9823 | 0.9917 | 1.0041 | 1.0185 | 1.0306 | |
CST(τ) | 1.3405* | 1.4009* | 1.4586** | 1.1296 | 0.5869 | 1.1693* | 2.5175** | 4.8285** | 6.8210** | |
4−26 | 0.8977 | 0.9112 | 0.9283 | 0.9578 | 0.9699 | 0.9783 | 0.9972 | 1.0198 | 1.0446 | |
CST(τ) | 1.4817** | 1.4740** | 1.4375** | 1.5938** | 1.2855* | 0.9978 | 1.1742* | 1.1906* | 1.6425** | |
13−4 | 0.9702 | 0.9798 | 0.9949 | 1.0044 | 1.0161 | 1.0258 | 1.0314 | 1.0433 | 1.0702 | |
CST(τ) | 8.5412** | 4.9614** | 3.3197** | 1.3636* | 0.6999 | 1.0054 | 1.3845* | 1.2737* | 1.1013 | |
13−26 | 0.9725 | 0.9673 | 0.9662 | 0.9780 | 0.9891 | 0.9928 | 1.0019 | 1.0081 | 1.0124 | |
CST(τ) | 2.0737** | 1.5570** | 1.3513* | 1.5347** | 1.0337 | 0.9155 | 1.9900** | 3.0591** | 4.8101** | |
26−4 | 0.9564 | 0.9726 | 0.9982 | 1.0113 | 1.0242 | 1.0339 | 1.0543 | 1.0618 | 1.0496 | |
CST(τ) | 3.7237** | 1.8984** | 1.2797* | 0.9947 | 1.1424 | 1.4907** | 1.3302* | 1.3294* | 1.0870 | |
26−13 | 0.9878 | 0.9921 | 0.9963 | 1.0029 | 1.0088 | 1.0145 | 1.0288 | 1.0234 | 1.0129 | |
CST(τ) | 4.8220** | 2.9392** | 1.8578** | 0.8833 | 0.9543 | 1.4324** | 1.1671 | 1.3898* | 1.8279** | |
Coupon equivalent | ||||||||||
4−13 | 0.9210 | 0.9445 | 0.9587 | 0.9665 | 0.9748 | 0.9851 | 0.9998 | 1.0101 | 1.0238 | |
CST(τ) | 1.3901* | 1.4046* | 1.2724* | 0.9406 | 0.7558 | 1.0460 | 3.4397** | 4.3309** | 6.8739** | |
4−26 | 0.8845 | 0.8960 | 0.9117 | 0.9390 | 0.9503 | 0.9604 | 0.9752 | 1.0020 | 1.0222 | |
CST(τ) | 1.4312** | 1.5441** | 1.4277** | 1.3336* | 1.1337 | 1.2139* | 1.2262* | 1.2755* | 2.1178** | |
13−4 | 0.9777 | 0.9861 | 1.0015 | 1.0133 | 1.0231 | 1.0343 | 1.0406 | 1.0512 | 1.0768 | |
CST(τ) | 8.6429** | 5.1868** | 3.4836** | 1.3518* | 0.8191 | 1.0880 | 1.2547* | 1.3962* | 1.1125 | |
13−26 | 0.9609 | 0.9555 | 0.9553 | 0.9673 | 0.9755 | 0.9798 | 0.9880 | 0.9927 | 0.9979 | |
CST(τ) | 2.3203** | 1.5408** | 1.2494* | 1.4563** | 1.0802 | 0.8867 | 1.7646** | 3.1131** | 4.6295** | |
26−4 | 0.9758 | 0.9926 | 1.0156 | 1.0320 | 1.0473 | 1.0552 | 1.0755 | 1.0807 | 1.0732 | |
CST(τ) | 3.8391** | 2.2514** | 1.5080** | 1.3153* | 1.3036* | 1.3447* | 1.3707* | 1.3316* | 0.9762 | |
26−13 | 1.0004 | 1.0045 | 1.0087 | 1.0165 | 1.0218 | 1.0286 | 1.0412 | 1.0360 | 1.0269 | |
CST(τ) | 4.1117** | 2.9041** | 1.7448** | 0.8042 | 1.0676 | 1.3622* | 1.2202* | 1.3708* | 1.8768** |
US daily Treasury bill data is from 01/02/2002 to 04/30/2012. The number of observations is 2585. Let * denote rejection of the null of cointegration at 5% level and let ** denote rejection at 1% level.
y−x | τ=0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
---|---|---|---|---|---|---|---|---|---|---|
30−60 | 1.0056 | 1.0036 | 1.0026 | 1.0039 | 1.0041 | 1.0041 | 1.0041 | 1.0052 | 1.0156 | |
CST(τ) | 1.5898** | 2.4501** | 2.0793** | 1.1205 | 1.1426 | 1.7477** | 3.7407** | 6.6798** | 4.6846** | |
30−90 | 1.0069 | 1.0062 | 1.0047 | 1.0061 | 1.0082 | 1.0096 | 1.0090 | 1.0101 | 1.0326 | |
CST(τ) | 1.5954** | 2.3190** | 2.5919** | 2.3279** | 1.7541** | 1.8461** | 2.5133** | 4.4216** | 2.9653** | |
60−30 | 0.9828 | 0.9947 | 0.9961 | 0.9962 | 0.9962 | 0.9963 | 0.9964 | 0.9950 | 0.9926 | |
CST(τ) | 6.0914** | 6.9691** | 4.4223** | 2.2500** | 1.2470* | 1.1233 | 1.6134** | 2.0012** | 1.7048** | |
60−90 | 1.0095 | 1.0063 | 1.0043 | 1.0041 | 1.0043 | 1.0042 | 1.0047 | 1.0049 | 1.0102 | |
CST(τ) | 1.4585** | 2.3730** | 1.7650** | 1.0464 | 0.9105 | 1.5000** | 3.8005** | 6.4077** | 6.6321** | |
90−30 | 0.9656 | 0.9885 | 0.9893 | 0.9903 | 0.9922 | 0.9936 | 0.9928 | 0.9906 | 0.9898 | |
CST(τ) | 3.3449** | 4.2920** | 2.7547** | 1.9796** | 1.4799** | 2.1144** | 2.3797** | 2.1914** | 1.6938** | |
90−60 | 0.9877 | 0.9950 | 0.9954 | 0.9960 | 0.9960 | 0.9943 | 0.9940 | 0.9938 | 0.9896 | |
CST(τ) | 7.7150** | 7.0606** | 4.1914** | 1.8809** | 0.8963 | 1.5375** | 1.4186* | 2.0230** | 1.9401** |
US daily financial commercial paper rate data is from 01/02/1997 to 04/30/2012. The number of observations is 3748.
Under H1, the cumulated sum of the fully modified residuals diverges to infinity. Note that the fully modified residuals of the quantile regression model can be rewritten as:
As shown in Lemma 1 of Xiao and Phillips (2002), under H1, as T→∞, M→∞, and M/T→0,
Furthermore, we have
The following theorem presents consistency of the test:
Theorem 4Under H1and Assumptions 1–3, as T→∞, Pr(CST(τ)>AT)→1 for any nonstochastic sequence
Therefore, under the alternative of no cointegration the test statistic diverges and the divergence rate depends on the bandwidth.
3 Empirical study
The theoretical model can be applied to examining the term structure of interest rates, and, more specifically, to testing the hypothesis of cointegration of short- and long-term interest rates. The expectations theory of the term structure of interest rates suggests that interest rates should be cointegrated if can be characterized as I(1) processes.
According to the generalized-present-value (GPV) model, any long-term yield can be expressed as a function of current and expected short-term yields (Boothe 1991). Engle and Granger (1987) and Campbell and Shiller (1987) test the model of the term structure and find that the yields of domestic bonds that differ only by term to maturity are intimately linked. Since interest rates in the US are usually characterized as nonstationary processes, the spread between these interest rates of different terms to maturity should be stationary if the expectations theory of the term structure holds. This means that short- and long-term rates are cointegrated.
Many papers have tested the expectations hypothesis for the term structure using US yield series on the federal funds rate (Hansen 1992), Treasury bill rate (Stock and Watson 1988; Boothe 1991; Hall, Anderson, and Granger 1992; Hansen 1992; Mandeno and Giles 1995), and commercial paper rate (Downing and Oliner 2007). In this paper, three sets of data are considered. The data comprises daily observations of interest rates of different yields from the U.S. Department of the Treasury and the Board of Governors of the Federal Reserve System. The interest rate statistics include daily Treasury bill rates,[6] U.S. government securities/Treasury constant maturities (nominal) interest rates,[7] and financial commercial paper rates.[8] The frequency is one business day. Unavailable or missing observations of holidays are omitted. The longest series, which is the Treasury bill constant maturity rate, is from January 4, 1982 to April 30, 2012 and contains 7584 observations. The shortest series, which is the Treasury bill rate, is from January 2, 2002 to April 30, 2012 and contains 2585 observations.
The regression model contains an intercept term but no time trend (dt=1), as a result of which yt=α(τ)+β(τ)xt+ut(τ). Nine representative quantile levels, τ=0.1, 0.2, …, 0.9, are considered. The bandwidth for the fully modified kernel estimator is
As a pretest to investigate the properties of the interest rate series, the augmented Dickey-Fuller (ADF) test does not reject that the logarithm of each rate has a unit root, but strongly rejects a unit root in the first difference of the series. Thus, the interest rates are I(1).
The estimation and test results are summarized in Tables 1–3. For each quantile τ, the fully modified coefficient estimate
y−x | τ=0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
---|---|---|---|---|---|---|---|---|---|---|
3−6 | 0.9465 | 0.9605 | 0.9660 | 0.9687 | 0.9730 | 0.9780 | 0.9828 | 0.9880 | 0.9924 | |
CST(τ) | 0.8859 | 1.1161 | 1.3232* | 1.2241* | 1.2130* | 1.2300* | 1.1889* | 1.3899* | 1.4494** | |
3−12 | 0.9071 | 0.9338 | 0.9462 | 0.9499 | 0.9560 | 0.9642 | 0.9742 | 0.9879 | 1.0071 | |
CST(τ) | 0.8836 | 1.2439* | 1.4801** | 1.4236* | 1.3526* | 1.4633** | 1.6802** | 1.7879** | 1.7508** | |
6−3 | 1.0056 | 1.0104 | 1.0161 | 1.0210 | 1.0265 | 1.0305 | 1.0334 | 1.0373 | 1.0517 | |
CST(τ) | 1.9344** | 1.6523** | 1.3564* | 1.2062* | 1.2235* | 1.2491* | 1.2896* | 1.1003 | 1.0039 | |
6−12 | 0.9594 | 0.9730 | 0.9772 | 0.9807 | 0.9845 | 0.9899 | 0.9956 | 1.0028 | 1.0133 | |
CST(τ) | 1.0650 | 1.0816 | 1.3913* | 1.6695** | 1.6807** | 1.9304** | 2.0034** | 2.0739** | 2.0344** | |
12−3 | 0.9846 | 1.0064 | 1.0211 | 1.0293 | 1.0363 | 1.0432 | 1.0507 | 1.0605 | 1.0839 | |
CST(τ) | 1.9336** | 1.8389** | 1.6954** | 1.7005** | 1.5964** | 1.6644** | 1.7136** | 1.6424** | 1.2180* | |
12−6 | 0.9808 | 0.9952 | 1.0037 | 1.0079 | 1.0111 | 1.0173 | 1.0209 | 1.0253 | 1.0377 | |
CST(τ) | 2.2030** | 2.1713** | 2.0231** | 2.0425** | 2.0922** | 1.7259** | 1.5582** | 1.2095* | 0.9516 |
US daily Treasury bill constant maturity data is from 01/04/1982 to 04/30/2012. The number of observations is 7584.
As Tables 1 and 2 show, short- and long-term bank discount and coupon equivalent Treasury bill rates and financial commercial rates are generally cointegrated in the central part of the distribution, but not in the tails. In the first panel of Table 1, 4-, 13-, and 26-week Treasury bill bank discount rates are cointegrated with each other for the inner quanitles. Also, the 90% quantile of the 13-week bank discount rate is cointegrated with the 4-week rate. Similarly, the 90% quantile of the 26-week rate and the 4-week rate are cointegrated. From the second panel of Table 1, the corresponding coupon equivalent Treasury bill rates follow a similar pattern. There is evidence of cointegration when the quantile level is between 0.4 and 0.6. The upper quantiles of the 13- and 26-week coupon equivalent rates are cointegrated with the 4-week rate.
In Table 2, for the 30-, 60-, and 90-day financial commercial rates, the null of cointegration is retained around the median in most cases. However, the distributions of the 30- and 90-day rates are not cointegrated in either direction. In addition, the fully modified coefficient estimates
The results from the model with the Treasury constant maturities interest rates or the Treasury yield curve rates are different. In many cases, the null of cointegration is retained in either the lower or upper tail. In Table 3, the Treasury constant maturities interest rates of shorter terms, such as the 3- and 6-month rates, are cointegrated with rates of longer terms in the lower tail of the distribution. The 6-month constant maturities rate is cointegrated with the 3-month rate in the upper tail. Similarly, there is cointegration relationship between the 90% quantile of the 12-month rate and the 6-month rate. However, the distribution of the 12-month rate is not cointegrated with the 3-month rate. Also, when cointegration is not rejected, the fully modified coefficient estimates
In general, the expectations hypothesis for the term structure is supported in the inner quantiles for Treasury bill rates and financial commercial rates. For the constant maturity Treasury rates, cointegration relationships are found in the tails of the interest rate distributions. One explanation may be that each interest rate itself exhibits asymmetric adjustment dynamics over the business cycle[9] (Koenker and Xiao 2004). Hence, results from the multifactor (term structure) model also display some asymmetry in the median, lower, or upper quartiles of the interest rate distributions.
4 Conclusion
This paper provides a cumulated sum test for the null hypothesis of quantile cointegration. In order to correct serial correlation and long run endogeneity, a Phillips-Hansen type fully modified quantile estimator for the cointegrating coefficients is employed to remove the second-order bias and nuisance parameters. For this semiparametric correction, the long run covariance between the regression disturbance and the innovation of the I(1) regressors is estimated using the Bartlett kernel with the plug-in bandwidth, as suggested by Andrews (1991) and Xiao and Phillips (2002). The CUSUM test statistic is composed of the partial sums of the residuals from the fully modified quantile regression. For each quantile, under the null of cointegration, the test statistic converges to a functional of Brownian motions.
The model is applied to several sets of daily US interest rate data. The US interest rates are found to be nonstationary and integrated of order one. Over various quantiles, evidence of cointegration is mixed. In the conditional quantile context, the expectations hypothesis for the term structure is retained only in part of the interest rate distribution for certain data sets.
In addition, several extensions can be made to the model. For example, it is relevant to incorporate structural changes in the quantile regression, since the long run relationship among the nonstationary variables may not be time-invariant. Also, ignoring the possibility of structural break can affect the power of cointegration tests (Gregory 1994; Gregory and Hansen 1996). In empirical studies, for example, Hansen (1992), Hall, Anderson, and Granger (1992), and Mandeno and Giles (1995) all consider a possible regime shift in the term structure data due to the Federal Reserve’s policy change. By allowing for structural changes, one can estimate and test the quantile cointegration relationship, which is possibly unstable over time.
Funding: Supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China 14XNF018.
Funding source: Renmin University of China
Award Identifier / Grant number: 14XNF018
Funding statement: Supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China 14XNF018
Appendix
A Proofs
A.1 Proof of Theorem 1
Theorem 1 can be proved following the procedure in Section A.1 from Xiao (2009), since under the null hypothesis of cointegration the model in this paper has the same asymptotic behavior as the quantile cointegrating regression considered by Xiao (2009). Let “Σ” denote summation over t from t=1 to t=T unless otherwise specified. For a single quantile τ, the unknown quantile regression coefficients
If
where
and
To derive the asymptotic distribution of the consistent estimator
From the invariance principle stated in Assumption 1 and Assumption 2, for the first term,
where
Similar to the derivation on page 257–258 in Appendix A.1 from Xiao (2009), under Assumption 3 the second term converges to
Consequently,
where “:=” signifies definitional equality. Since G(ϕ) is convex and differentiable, there is only one solution to the minimization problem of G(ϕ). From the first order condition that
the unique minimizer of G(ϕ) is
By Lemma A of Knight (1989) and the convexity lemma of Pollard (1991),[12] since GT(ϕ) is minimized at
A.2 Proof of Theorem 2
Similar to the case of fully modified OLS regression, the fully modified quantile estimator is consistent, which is also shown in the following derivation. The regression coefficient estimator after the modification
where
Then, we have
where
A.3 Proof of Theorem 3
For the fully modified quantile regression, we have
where in general
The CUSUM test statistic is given by
where
A.4 Proof of Theorem 4
Under H1 we have
where
Analogues to Lemma 1 from Xiao and Phillips (2002), under H1 as T→∞, M→∞, and M/T→0,
where Ω*=Ωxwη, Δ*=Δxwη,
Since
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