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FACULTY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF MATHEMATICS AND STATISTICS A threshold cointegration analysis of Norwegian interest rates Berner Larsen Sta-3900 Master's Thesis in Statistics April, 2012

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FACULTY OF SCIENCE AND TECHNOLOGY DEPARTMENT OF MATHEMATICS AND STATISTICS A threshold cointegration analysis of Norwegian interest rates Berner Larsen Sta-3900 Master's Thesis in Statistics April, 2012 Acknowledgements First and foremost I would like to thank my supervisors Associate Professor Edvin L. Bredrup and Associate Professor Sigrunn Holbek Sørbye at Department of Mathematics and Statistics, University of Tromsø. Second, I would like to thank my colleague Associate Professor Svein Oskar Lauvsnes at Bodø Graduate School of Business, University of Nordland, for reading through some versions of this thesis and giving me valuable advice. Also, I am grateful to University of Tromsø and Bodø Graduate School of Business, University of Nordland for adapting the study progression and the workload, respectively, such that it has been posssible to complete this Master s Degree program in Statistics on a part-time basis. Finally, I would like to express my deepest gratitude to my wife Lillian for being patient all those weekends and evenings when I have been married to my books. i ii Abstract In this thesis we generalize the Hansen and Seo test in the R package tsdyn, which tests a linear cointegration model against a two-regime threshold cointegration model, to the case of three regimes in the alternative hypothesis. As the Lagrange Multiplier (LM) test statistic used in the Hansen and Seo test in tsdyn is different from the LM statistic described in Hansen and Seo (2002), we generalize both these LM statistics, and show that they are equal under certain conditions. The Hansen and Seo test uses the SupLM statistic which is the maximum of this LM statistic when the two thresholds vary over the set of all possible threshold values. The grid search algorithm, which is necessary when maximizing this LM statistic, is also extended to the case of three regimes, and it is rewritten such that if the cointegration value β is given, it really maximizes the LM statistic under the constraints specified by the user. In our empirical studies we have examined thoroughly the bivariate time series consisting of the monthly NIBOR rates of the maturities tomorrow next and 12 months. When modeling this bivariate time series, we find strong evidence for a two-regime TVECM being superior to a linear VECM, and in our out-of-sample forecasting the two-regime SETAR model gives much better prediction of the cointegration relation than a linear AR model. When testing a two-regime SETAR model for the cointegration relation against a three-regime model, the two-regime model cannot be rejected at any reasonable significance level. In addition, we show how influential a few outliers may be by removing them from the time series and rerunning some of the statistical tests. Also, we have tested all the 66 possible pairs of Norwegian interest rates for cointegration, and we have tested the term spread of each pair for threshold effects, i.e., testing a linear model against a two-regime model, as well as testing a two-regime model against a three-regime model. We find a lot of cointegrated pairs, and we find evidence for a two-regime model in approximately 50 % of the cases, and evidence for a three-regime model in some cases in this univariate time series analysis. At last, we simulate a bivariate time series with a three-regime threshold cointegration model as data generation process, and estimate a three-regime threshold cointegration model from this simulated time series. Thus, we illustrate that the thresholds which our version of the Hansen and Seo test detects as optimal, are close to the original thresholds used in the simulation. As expected, a linear model for this bivariate time series is strongly rejected, and there is strong evidence for a three-regime threshold model for the cointegration relation being superior to both a linear model and a two-regime threshold model. iii iv Contents List of tables List of figures vii ix 1 Introduction 1 2 The time series models and the statistical tests used in this thesis White noise processes Autoregressive moving average models Model checking Unit root models The ADF test The KPSS test SETAR(p) models Testing for the number of regimes in SETAR models Vector autoregressive models The asymptotic Portmanteau test The ARCH-LM test Normality tests Vector error correction models The Johansen cointegration rank test Threshold vector error correction models Checking the residuals of an estimated TVECM The Hansen and Seo test Testing for three-regime threshold cointegration The derivation of the SupLM statistic in the case of three regimes Some matrix formulas The three-regime TVECM written in matrix form The least squares estimators of the parameters The LM statistics The SupLM statistic The implementation of the SupLM test in the case of three regimes Summary and concluding remarks Analysis of the NIBOR rates of maturities tomorrow next and 12 months The data set Estimation of an unrestricted VAR model v Contents 4.3 Testing for unit roots Estimation of a VECM Estimation of a TVECM Estimation of a TAR model for the cointegration relation Out-of-sample forecasting of the term spread Summary Analysis of the NIBOR rates of the maturities tomorrow next and 12 months Testing of Norwegian interest rates Analysis of a simulated TVECM with three regimes Simulation Estimation of a TVECM Testing a two-regime threshold model against a three-regime threshold model Summary Conclusion and discussion 79 Bibliography 85 Appendices 87 A The source code of TVECM.HSTest 89 B The source code of TVECM.XHSTest 95 C The R code chunks used in Chapter 4 and vi List of Tables 4.1 The coefficients of the estimated VAR model The ARCH and normality tests of the estimated VAR model of NIB12MTN The Portmanteau test of the estimated VAR model of NIB12MTN ADF test of NIBTN: τ 3, φ 2 and φ 3 tests ADF test of NIBTN: τ 2 and φ 1 tests ADF test of diff(nibtn): τ 3, φ 2 and φ 3 tests KPSS test of NIBTN: ˆη µ and ˆη τ tests ADF test of NIB12M: τ 3, φ 2 and φ 3 tests ADF test of NIB12M: τ 2 and φ 1 tests ADF test of diff(nib12m): τ 3, φ 2 and φ 3 tests KPSS test of NIB12M: ˆη µ and ˆη τ tests The values and the critical values of the trace statistic The values and the critical values of the λ max statistic The coefficients of the estimated VECM AIC, BIC and SSR for different TVECMs The coefficients of the TVECM with β = 1.071, lag=2 and threshold value γ = The results of the ARCH-LM test for the residuals The results of the Doornik and Hansen test for nonnormality in the residuals The results of the Hansen and Seo test The minimum SSR for NIB12MTN and TestNIB12MTN The results of the setartest of snib12mtn The results of the setartest of stestnib12mtn The P-values of the Ljung-Box tests of the ARMA(p, q) models The coefficients of the SETAR(3) model for the data set snib12mtn The results of the setartest with thdelay=0 on the term spread of pairs of Norwegian interest rates The results of the setartest with optimal thdelay on the term spread of pairs of Norwegian interest rates The results of the setartest of snib12mtn without the outliers in autumn The number of interest rate pairs with P-value 0.05 in the setartest The coefficients of the simulated TVECM The coefficients of the estimated TVECM from the simulated data The results of the Hansen and Seo test for the simulated time series y t The results of the setartest of ssim vii List of Tables viii List of Figures 4.1 Plots of the time series NIB12M and NIBTN Plots of the time series TestNIB12M and TestNIBTN Diagnostic plots for the ADF test of the time series NIBTN Diagnostic plots for the ADF test of the time series NIB12M Plots of the cointegration relations Plots of R t, r t and s t against s t Plot of the grid search for β and the threshold γ The responses of r t, R t and s t to s t 1 in the TVECM when nthresh= The responses of r t, R t and s t to s t 1 in the TVECM when nthresh= The residuals in the estimated VECM and TVECM Plots of the SETAR tests of snib12mtn Plots of the SETAR tests of stestnib12mtn Plots of the estimated ACF and PACF functions from stestnib12mtn Plots of the estimated ACF and PACF functions from snib12mtn Predictions of snib12mtn for the last ten months of Predictions of stestnib12mtn for the last ten months of Plot of the simulated time series y t = [ y 1t y 2t ] Plot of the simulated w t Plots of y 1t, y 2t and w t against w t Plot of the grid search for the threshold γ The responses of y 1t, y 2t and w t to w t 1 in the estimated TVECM Plots of the SETAR tests of w t ix List of Figures x Chapter 1 Introduction Cointegration has since it was introduced in Granger (1981), attached much attention among economists because it is a tool for testing the existence of and finding stable longrun relationships between nonstationary variables. For example, time series of interest rates are often nonstationary, but the Expectations Hypothesis which states that a longterm interest rate is an average of expected future short-term rates plus a risk premium, implies that there exists a stable linear long-run relationship between the short-term and the long-term interest rate (Hall, Anderson, and Granger 1992). A lot of papers have investigated the relationship between short-term and long-term interest rates, see e.g., Modigliani and Shiller (1973), Engsted (1996), Campbell and Shiller (1991), Musti and D Ecclesia (2008), Arize, Malindretos, and Obi (2002), and Buigut and Rao (2010). Some of the results support the Expectations Hypothesis, and some do not, so other theories than the Expectations Hypothesis have been presented to explain the term structure of interest rates, but it is generally accepted that interest rates of different maturities should not deviate too much from each other (Siklos and Wohar 1996). The first tests for cointegration were proposed in Engle and Granger (1987), while Johansen (1988) and Johansen and Juselius (1990) have developed a procedure to test for the number of cointegration relations, i.e., long-run relationships, between the variables, and to find these cointegration relations. In this thesis we consider only bivariate time series, so the number of cointegration relations is either 0 (i.e., no cointegration) or 1. When modeling a bivariate time series consisting of two interest rates of different maturities, the long-run relationship is typically the term spread, i.e., the difference between the interest rates, or more generally, a linear combination of the interest rates with one coefficient normalized to 1, and the other coefficient nearby 1. If we in our model include an error correction term containing this long-run relationship, we achieve that at each time point adjustments are performed due to deviations from the long-run equilibrium, the larger deviations the larger adjustments. However, in economic applications it is often unrealistic that the adjustments should be done at each time point. For example, there may be transaction costs, so that arbitrage opportunities between two markets only arise when the price difference is large enough to imply net gains to traders (Clements and Galvão 2004). To take into account such nonlinear behavior, Balke and Fomby (1997) introduced the threshold cointegration model, which allows the adjustment to be made only when the deviation from the long-run equilibrium is larger than an upper threshold and/or smaller than a lower threshold. Stigler (2011) gives both an overview of the field threshold coin- 1 Chapter 1. Introduction tegration and a description of how such a data analysis may be conducted by using the R package tsdyn. With this paper as a starting point, we will analyse NIBOR rates, downloaded from by using threshold cointegration. We will also analyse the term spread by using nonlinear autoregressive time series models described in Di Narzo, Aznarte, and Stigler (2011). When a threshold cointegration model is estimated, it is of crucial interest to test whether this nonlinear model is superior to a linear cointegration model. Hansen and Seo (2002) proposed a test which tests a linear cointegration model against a two-regime threshold cointegration model, and this test is implemented in the R package tsdyn. In Hansen and Seo (2002) and Seo (2003) monthly U.S. Treasury bond rates are modeled by using tworegime and three-regime threshold cointegration models, respectively. We downloaded these U.S. interest rates from St. Louis Federal Reserve Bank at stlouisfed.org/fred2/, but we were not able to reproduce the results in Seo (2003). Therefore, we have examined the algorithm of the Hansen and Seo test in the package tsdyn thoroughly. Our main contribution is the generalization of the Hansen and Seo test in the R package tsdyn to the case of three regimes in the alternative hypothesis. As the Lagrange Multiplier (LM) test statistic used in the Hansen and Seo test in tsdyn is different from the LM statistic described in Hansen and Seo (2002), we generalize both these LM statistics, and show that they are equal under certain conditions. The Hansen and Seo test uses the SupLM statistic which is the maximum of this LM statistic when the two thresholds γ 1 and γ 2 vary over the set of all possible threshold values. However, the function LM(γ 1, γ 2 ) is a highly irregular function such that we have to perform a grid search when maximizing this function. The global maximum of a function under explicitly given constraints is unique, i.e., the maximum value is unique, but there may be more than one point which give this maximum value. However, neither the implementation of the Hansen and Seo test used in Seo (2003) nor the implementation in the package tsdyn gives the user full control over the constraints used when maximizing LM(γ 1, γ 2 ), which may explain why we did not succeed in reproducing the results in Seo (2003). Therefore, we have made a new algorithm for the grid search in Chapter 3, which covers both the case of two regimes and the case of three regimes in the alternative hypothesis. In the case of three regimes, the algorithm is quadratic in the number of possible threshold values, and hence very time consuming as the P-value of the test statistic is estimated by using bootstrapping. Though, it is preferable with an algorithm which maximizes correctly under the given constraints. In our empirical studies we have examined thoroughly the bivariate time series consisting of the monthly NIBOR rates of the maturities tomorrow next and 12 months, which was the first pair of Norwegian interest rates we found where a two-regime threshold model is significantly better than a linear model. We analyse both this bivariate time series by using functions for multivariate time series analysis, and the cointegration relation by using functions for univariate time series analysis. Our out-of-sample forecasting shows that a threshold model gives much better prediction of the cointegration relation than a linear model. In addition, we analyze the effect of removing 6 outliers from the tomorrow next rates and 2 outliers from the 12 months rates by using interpolation. Thus, we show how influential a few outliers may be. 2 Chapter 1. Introduction As there exist NIBOR rates of 9 different maturities and interest rates on Norwegian government bonds of 3 different maturities, we may make 66 pairs of Norwegian interest rates. Due to the fact that our version of the Hansen and Seo test is very time-consuming, we are not able to test all these pairs of interest rates for threshold cointegration. Rather, we have tested all these pairs for cointegration, and we have tested the term spread of each pair for threshold effects, i.e., testing a linear model against a two-regime model, as well as testing a two-regime model against a three-regime model. We find a lot of cointegrated pairs, and we find evidence for a two-regime model in approximately 50 % of the cases, and evidence for a three-regime model in some cases in this univariate time series analysis. However, a threshold cointegration analysis of the corresponding bivariate time series is a topic for further research. At last, we simulate a bivariate time series with a three-regime threshold cointegration model as data generation process, and estimate a three-regime threshold cointegration model from this simulated time series. Thus, we illustrate that the thresholds which our version of the Hansen and Seo test detects as optimal, are close to the original thresholds used in the simulation. As expected, a linear model for this bivariate time series is strongly rejected, and there is strong evidence for a three-regime threshold model for the cointegration relation being superior to both a linear model and a two-regime threshold model. The rest of this thesis is organized as follows: In Chapter 2 we give an overview of all time series models and statistical tests used in this thesis. Chapter 3 describes our generalization of the Hansen and Seo test in the R package tsdyn to the case of three regimes in the alternative hypothesis. Chapter 4 contains our analysis of Norwegian interest rates. In Chapter 5 we simulate a time series which follows a three-regime vector error correction model, and we analyse this simulated time series by using the same tools as in Chapter 4. In Chapter 6 we summarize the results of this thesis. Appendix A and B contain the R code of the original version in tsdyn and our revised version of the Hansen and Seo test, respectively. Appendix C contains all the R code chunks which were run in Chapter 4 and 5 to perform the data analysis. 3 4 Chapter 2 The time series models and the statistical tests used in this thesis This chapter contains all the time series models and the statistical tests used in later chapters of this thesis. All the materials in this chapter are found in the R documentation of the R functions mentioned in the text, and in Shumway and Stoffer (2006), Tsay (2010), Lütkepohl (2007), Pfaff (2008a) and Juselius (2006). 2.1 White noise processes A univariate time series x t is an ordered set of random variables x 1, x 2, x 3,... where x i is the value at time point i. Also, we will use the same notation x t for a realization of this stochastic process. In applications, we usually know only one single realization of finite length T of the data-generating process, such that the task is to find a model for the data-generating process which fits well to the known data x 1,..., x T. A white noise process is a simple data-generating process, but it is important as more complicated time series models usually are defined by using a white noise process as error term. It is defined as follows: Definition 2.1. A white noise process is a data-generating process ɛ t such that E(ɛ t ) = 0, E(ɛ 2 t ) = σ 2 for all t and E(ɛ t ɛ τ ) = 0 for all t τ. If in addition ɛ t N(0, σ 2 ), we say that ɛ t is Gaussian white noise. If we replace the condition of uncorrelatedness with the stronger assumption of independence, the process is said to be independent white noise. 2.2 Autoregressive moving average models When modeling the term spread in Chapter 4, we will try an ARMA(p,q) model which is defined as follows: 5 2.3. Unit root models Definition 2.2. The time series x t is an ARMA(p,q) process if x t = c + φ 1 x t φ p x t p + ɛ t + θ 1 ɛ t θ q ɛ t q (2.1) where c is a constant, p and q are nonnegative integers, φ p 0, θ q 0 and ɛ t is a white noise process. The integer p is called the AR order, while the number q is the MA order. Using the lag operator L defined by Lx t = x t 1, we may write the model as: (1 φ 1 L φ p L p )x t = c + (1 + θ 1 L + + θ q L q )ɛ t. The polynomial φ(l) = 1 φ 1 L φ p L p is the AR polynomial, while the polynomial θ(l) = 1 + θ 1 L + θ 2 L 2 + θ q L q is the MA polynomial. We assume that the AR and MA polynomials have no common factors; otherwise

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