The dynamics between the ECB's policy rate and the swap curve : an analysis based on linear and nonlinear Granger causality

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The dynamics between the ECB’s policy

rate and the swap curve:

an analysis based on linear and nonlinear Granger causality

A Master’s Thesis to obtain the degree in

Financial Econometrics

University of Amsterdam Faculty of Economics and Business

Amsterdam School of Economics Carried out at Sprenkels & Verschuren

Author: Rachel Ho Student number: 10079742

Email: rpsy.ho@gmail.com

Date: December 30, 2014 Supervisor: Prof. Dr. C.G.H. Diks Second reader: Prof. Dr. H.P. Boswijk In-company supervisor: E. Schotanus MSc

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Abstract

This study aims to analyze causal relationships between the ECB’s policy interest rate and the swap curve. The policy rate is the price that banks pay to borrow funds from the ECB. Because the ECB has a monopoly power, it has full control in determining this interest rate. Consequently, this directly influences the short term money market interest rates, and subse-quently the swap curve. In turn, the changes in the swap curve may affect investment decisions and the funding ratio of pension funds. Therefore, it is of importance to study the dynamic relationship between the ECB’s policy rate and the swap curve. The approach is to conduct lin-ear and nonlinlin-ear Granger causality tests, which are the Toda & Yamamoto (1995) augmented Granger causality test and the nonparametric Diks & Panchenko (2006) test, respectively. The nonlinear causalities are visualized by introducing Generalized Additive Models (GAM). This methodology framework is followed using both bivariate and five-variate VAR models. Based on economic theory, one would expect to find Granger causality effects from the ECB’s policy interest rate on the swap curve. However, statistical results show that the reverse is true. It seems as though the one-year swap rate tends to have the strongest relationship with the ECB rate. These findings imply that, even though the ECB uses its policy rate to influence interest rates on the money markets, the decision to alter its policy rate may be caused by the swap curve in the past. Through subsampling it has become clear that the observed causalities only hold in the period before the financial crisis. It appears that all relationships failed to exist in an unstable economy after the crisis.

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Acknowledgements

I would like to take this opportunity to show my gratitude to a number of people who have made it possible to write my master’s thesis successfully.

First, I would like to express my special appreciation to my supervisor Professor Dr. Cees Diks for his great support and guidance during the past six months. During our meetings he would always succeed in providing fundamental insights and clarifications at times when I was strug-gling with my investigation. It had stimulated me to grow and move forward. And for that, I am very grateful.

Furthermore, I would like to thank my in-company supervisor Ewout Schotanus at Sprenkels and Verschuren for the useful comments and for providing the necessary data. I would also like to thank Marieke Klein and Bertjan Kobus for their effort during the process. The brainstorm-ing sessions have helped broaden my scope on the project.

And lastly, I would like to take this moment to thank my family and friends who have supported me throughout the entire process. They have never stopped believing in me and encouraged me to accomplish my goals.

Hereby, I proudly present my master thesis in conclusion to the Master’s in Econometrics at the University of Amsterdam.

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Introduction

September 4th, 2014 was the day that the European Central Bank (ECB) did an announcement, that surprised the financial markets. The ECB decided to cut its policy interest rates to new record lows (The Wall Street Journal, 2014). Ever since the global financial crisis in 2008, which led to the virtual breakdown of the money market, the central bank has responded with several monetary policy tools. The most important measure was to lower its key interest rates in order for banks to build up large liquidity reserves (European Central Bank, 2010). What followed after, was a stagnant growth in the euro zone and weakening inflation. Since the main objective of the ECB is to maintain price stability in the euro zone, it has led to a worrisome statement from the ECB president Mario Draghi:

”In August, we see a worsening of the medium-term inflation outlook, a downward move-ment in all indicators of inflation expectations. Most, if not all, the data we got in August on GDP and inflation showed that the recovery was losing momentum.” (The Wall Street Journal, 2014)

In an effort to combat the low inflation, the policy interest rate is used as a monetary policy tool. It is defined as the rate at which the ECB lends money to banks. By changing this interest rate, the ECB influences the interest rates on the money markets. This will in turn affect the swap curve, which consists of swap rates for different maturities. Subsequently, the changes in the swap curve are of great interest to pension funds, because the level of both assets and liabilities of pension funds are influenced by the swap curve. In addition, throughout the last decade the demand of interest rate swaps by pension funds and insurance companies has increased largely to cover the interest rate risk (Syntrus Achmea, 2012). This leads to the implication that a change in the policy interest rate of the European Central Bank may be evaluated to manage

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Rachel Ho CHAPTER 1. INTRODUCTION

certain risks by pension funds. This study will therefore address the following central research question:

”What is the dynamic relationship between the policy interest rate of the ECB and the swap curve?”

The expectation is that the policy interest rate of the central bank influences the swap curve, and in particular the short term swap rates. This is mainly because the ECB uses its interest rate as a tool to control the short term money market. This suggests a strong connection between the policy rate and the one-year swap rate. Longer term swap rates may be affected in a relatively smaller manner.

The objective of the thesis is to explore causal linkages between the policy interest rate and various swap rates. Because the main interest lies in predicting the dynamics between the key variables based on their own past, a time series analysis is conducted in this study. The causal relationships among the time series are investigated based on the concept of Granger causality. The time series X is said to Granger cause Y if the values of X help predict future values of Y (Hamilton, 1994). Granger causality has become a valuable and popular method for causality analysis in time series, since it was introduced by Granger (1969). The approach is to conduct linear and nonlinear Granger causality tests in this thesis. Beforehand, a preliminary analysis is carried out to give an indication of the order of integration and cointegration. The analysis consists of the Augmented Dickey-Fuller (ADF) test and the Johansen cointegration test.

Often, such pretests are conducted to obtain the right model specification. However, this may lead to severe pretest biases (Toda & Yamamoto, 1995). Hence, instead of the conventional linear Granger causality test an alternative testing procedure is used. This is widely known as the Toda & Yamamoto (1995) augmented Granger causality test, which is robust to the order of integration and cointegration. This test is conducted using Vector Auto Regressive (VAR) models in a bivariate and five-variate setting. In addition, a nonparametric Granger causality test is introduced. It is referred to as the nonparametric Diks-Panchenko test, since it was developed by Diks & Panchenko (2006). This test does not make use of the assumption of a linear model. Hence, it can be used to examine nonlinear causal relationships. This test is not only applied to the logarithmic returns, but also on the residuals of the estimated VAR models. By performing the latter, one can determine the nature of the causality. The nonlinear causalities may be visualized by introducing Generalized Additive Models (GAM). A GAM is a linear model, in which the predictor consists of a sum of smooth functions of explanatory variables. Estimating and plotting the smooth functions will give an idea of the interdependence between the swap rates and the policy interest rate of the ECB. The GAM models are specified using

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CHAPTER 1. INTRODUCTION Rachel Ho

logarithmic returns and VAR residuals as well, depending on the causalities found through the Diks-Panchenko test. The results obtained through this methodology framework are ultimately investigated by analyzing subsamples with respect to the periods before and after the global financial crisis.

Empirical data has been obtained to analyze the structure of the data. The dataset consists of time series of the ECB’s policy rate and swap rates with maturities of one, two, five and ten years. The level of the ECB’s policy rate is set by the ECB Governing Council every month. The decision could either be an interest rate cut or rise, or no change at all. The level of the swap rates is determined on a daily basis. It is considered as the rate a certain party either pays or receives until the time of maturity, assuming the contract is issued on that specific date. The full sample of 5326 observations covers a period between October 1, 1999 and April 30, 2014.

The remainder of the thesis is organized as follows. The economic background of the rela-tionship between the ECB’s policy rate and the swap curve is discussed in Chapter 2. Next, Chapter 3 provides a detailed description of the research methodology used to examine causal linkages between the key variables. Then the data is presented in Chapter 4, along with the results of the preliminary analysis. Chapter 5 contains the results obtained using the linear and nonlinear Granger causality tests, followed by the results of estimating the GAM models in Chapter 6. After that, the same research methodology framework is applied on the subsam-ples before and after the event of the crisis. The results of which are presented in Chapter 7. Ultimately, a summary and conclusions are given in the last chapter of this thesis.

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Chapter 2

Theory and previous studies

The policy interest rate of the European Central Bank (ECB) is the main instrument to maintain price stability in the Eurozone. This policy rate is called the refinancing rate. It is defined as the rate at which the ECB lends money to commercial financial institutions, such as banks. By changing this interest rate, the ECB can influence the interest rates on the money markets, and in particular the swap curve. Subsequently, the changes in the swap curve affect investment decisions and the funding ratio of pension funds. This suggests that a change in the ECB’s policy rate can be evaluated to manage certain risks by pension funds. Therefore, it is of importance to study the dynamic relationship between the ECB’s policy rate and the swap curve. The first step is to discuss the economic background of this relationship, which will be done in the first section of this chapter. Section 2.2 describes the importance of the relationship with respect to pension funds. At last, previous studies regarding the ECB’s policy interest rate and the swap curve are discussed in Section 2.3.

2.1 Economic background

In order to understand the connection between the European Central Bank’s policy rate and the swap curve, first brief descriptions of both key concepts are given in the subsections below. Subsection 2.1.1 describes the economic background of the European Central Bank regarding its monetary policy and how the economy as a whole is affected. Secondly, the swap curve is explained in Subsection 2.1.2, along with its link to the policy interest rate.

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CHAPTER 2. THEORY AND PREVIOUS STUDIES Rachel Ho

2.1.1 European Central Bank

The European Central Bank is the central bank for Europe’s currency, the euro.1 Its main task is to define and implement monetary policy for the euro area. This means maintaining price stability, in other words keeping the inflation rates below but close to two percent over the medium term. High inflation would mean that prices rise rapidly. Over time, one would not receive the same amount of goods and services for the same amount of money. To prevent this from happening, the ECB uses a number of monetary policy tools. One of the most important tools is the decision making with regards to the level of key interest rates. These interest rates influence the economy and price levels in the euro area. A visualization is given in Figure 2.1 (ECB, 2000). This chart illustrates the various channels through which monetary policy actions affect the entire economy.

Figure 2.1: Transmission mechanism of monetary policy (ECB, 2000)

It can be roughly divided into two stages. First, a change in the policy interest rate leads to changes in financial market conditions. This is observable as changes in market interest rates, the exchange rate, asset prices, credit conditions and overall liquidity in the economy. Subsequently,

1

All information provided about the European Central Bank (n.d.) is retrieved from its website https://www. ecb.europa.eu/mopo/intro/html/index.en.html.

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Rachel Ho CHAPTER 2. THEORY AND PREVIOUS STUDIES

in the second stage of the transmission mechanism, these changes in financial market conditions influence the level of spending by households and firms and price levels. Alternatively, a more direct way of influencing the economy is by generating expectations about inflation, which in turn may affect pricing decisions directly. There are three key interest rates, which are used as monetary policy tools. The main interest rate is the refinancing rate used for main refinancing operations, which controls the liquidity with respect to the banking system. The transactions are generally conducted with a frequency and maturity of one week. The other two interest rates are the deposit rate, at which banks make overnight deposits with the ECB, and the marginal lending rate, which facilitates overnight credit to banks from the ECB.

In this study, the focus is on the refinancing rate, which will be referred to as the ECB’s policy interest rate in the remainder of the thesis. The level of the ECB’s policy interest rate is the price that banks pay to borrow funds from the ECB. Because the ECB has a monopoly power, it has full control in determining this interest rate. Consequently, this directly influences the short term money market interest rates. The money market is characterized by its high liquidity and short term maturities. It involves borrowing and lending in the short term, with maturities of one year or less. A few examples of financial instruments used in the money market are Treasury bills, federal funds and repurchase agreements. Next to changes in the short term interest rates, medium and long term interest rates are affected by expectations of future policy interest rates of the central bank. In addition, the market expectations about the future development of short term rates may particularly influence longer term interest rates.

2.1.2 The swap curve

It is clear now that the European Central Bank may have a certain control over the economy by using a powerful monetary policy tool, the ECB’s policy interest rate. By changing this interest rate, the ECB contributes to changes in the short term money market. However, the question that remains, is how this mechanism may affect the swap market. Therefore, an interest rate swap is explained first to obtain clearer understandings about the swap market.

An interest rate swap is a derivative, the value of which is derived from the value of the interest rate. It is an agreement between two parties in which interest payments are being exchanged during a certain amount of time based on a specified principal amount. One party has agreed to pay a floating rate, while the other party has agreed to pay a fixed rate. A commonly used floating rate is a short term interbank rate, such as the 6 month Euribor rate (European Commission, 2014). Euribor is short for Euro Interbank Offered Rate. The Euribor rates are derived from the interest rates at which a large number of European banks borrow funds from one another (European Money Markets Institute, 2014). The Euribor rates are considered to

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form the most important benchmark for a great number of financial products such as mortgages and interest rate swaps. Hence, in a swap contract one has agreed to exchange the 6 month Euribor rate for some fixed rate. So, if two parties engage in a 10-year swap agreement, it will involve exchanging 20 (floating) Euribor payments against 10 yearly fixed payments. These fixed payments are based on some fixed rate, also known as the swap rate. It represents the rate at which the market is willing to exchange the 6 month floating rate.

However, the swap rate does not only contain a single interest rate, it represents a whole curve, consisting of swap rates for maturities of one to fifty years. It is better known as the swap curve. The swap rates differ from one another as they all correspond with different maturities. This is comparable with a savings deposit in a bank. Freezing a certain amount of money on a savings deposit for one year may generate 2% interest, while doing the same for ten years may generate 3% interest.

So, by engaging in a swap agreement, one must pay the 6 month Euribor rate. The other is obliged to pay the swap rate. Because the Euribor rates are derived from the rates at which banks lend money to one another, the short term Euribor rate, in particular, is indirectly influenced by the ECB’s policy interest rate. Subsequently, the swap rates will be affected through the change of the 6 month Euribor rate. If at one point, the 10-year swap rate equals 3%, this implies that the expected floating rate for the next 10 years will be 3%. Hence, a decrease in the short term Euribor rate may lead to a decrease in the swap rates. This leads to the overall expectation that the short term swap rates are affected in a more visible way by the fluctuations in the policy interest rate than the long term rates. As the 6 month Euribor rate changes, the 1-year swap rate is more likely to adjust its level than the 10-year swap rate. Thus, through various channels, the European Central Bank has a certain influence on the swap curve. A schematic illustration is given in Figure 2.2.

Figure 2.2: Relationship between ECB’s policy interest rate and the swap curve

2.2 The effect on pension funds

Throughout the last decade, more parties have become evolved in the swap market. One of the main reasons is the increasing demand of interest rate swaps by pension funds and insurer companies to cover the interest rate risk. The focus in this thesis is merely on pension funds.

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Rachel Ho CHAPTER 2. THEORY AND PREVIOUS STUDIES

By analyzing the dynamic relationship between the ECB’s policy rate and the swap curve more insights may be provided with respect to pension funds’ risk management. First, the relationship between the swap market and pension funds is discussed in this section.

The balance sheet of a pension fund comprises of investments on the asset side and pension provision on the liabilities side. The technical reserves (in Dutch: Technische Voorziening) are defined as the present value of the unconditional pension liabilities. The financial situation of a pension fund is generally measured by the funding ratio. The funding ratio (in Dutch: dekkingsgraad ) equals the technical reserves divided by the value of the investments. A funding ratio of 100% means that a pension fund has the right amount of resources to fulfill its nominal pension obligations when investments are assumed to have a risk-free return. Both sides of the balance sheet are sensitive to interest rate fluctuations. The difference between the effect on the investments and the effect on the technical reserves is called the interest rate risk (in Dutch: renterisico).

In the New Financial Assessment Framework (in Dutch: Nieuw Financieel Toetsingskader (nFTK)) is stated that a sufficient amount of capital is required to be able to pay out the pension benefits if that specific amount of capital is invested risk-free (De Nederlandsche Bank, 2008). Because the swap rate is assumed to be the risk-free rate, pension funds make use of a interest rate term structure derived from the swap rate in calculating the present value of the pension liabilities. This term structure is provided by the Dutch Bank (in Dutch: De Nederlandsche Bank (DNB)). Hence, the value of the technical reserves is strongly dependent on the swap rate and its fluctuations. A decrease of the swap rate means that more capital is required to pay out the pension benefits. This leads to an increase of the technical reserves. The rate at which the technical reserves increase if the swap rate decreases by one percent point is called the duration of the pension fund. On average, the duration is equal to 20 for a Dutch pension fund (Syntrus Achmea, 2012). This means that a 1% point decrease of the swap rate leads to a 20% increase of the technical reserves. Young pension funds have relatively high durations compared to old pension funds. This is because a pension fund with older participants is less sensitive to interest rate fluctuations.

A large proportion of the investments comprises of fixed-income securities such as bonds and mortgages. The average duration of the total investment portfolio is around 2 to 3, which is much lower than the duration of the technical reserves (Syntrus Achmea, 2012). A decrease of the swap rate will lead to a relatively higher increase of the technical reserves than the increase of the value of the investment portfolio. This causes the funding ratio to decrease. On the other hand, if the swap rate increases, the technical reserves will decrease relatively more than the investment portfolio. This results in an increasing funding ratio. Because of these fluctuations, additional measures must be taken to decrease the interest rate risk. This is why interest rate

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swaps play an important role in balancing the interest rate risk.

There are two types of swaps: receiver swaps and payer swaps. A receiver swap is a swap for which the buyer receives the fixed rate and pays the other party the floating rate (Pacific Invest-ment ManageInvest-ment Company (PIMCO), 2008). For a payer swap, it is the other way around. Swap agreements are traded over-the-counter at which both parties engage in an individual con-tract. Banks are usually the other party whom pension funds engage with in a swap agreement. The use of interest rate swaps enables investors to pay fixed rates on a loan instead of floating rates or vice versa. The main reason for pension funds to do this is to balance its interest rate risk (Syntrus Achmea, 2012). This can be achieved by using receiver swaps. If the swap rate increases, it will lead to a loss on the swap. However, the value of the technical reserves will decrease by a greater proportion. And if the swap rate decreases, the swap will generate profits. Because the technical reserves increase relatively more, it must be compensated by engaging in a sufficient number of swap agreements.

2.3 Previous studies

Previous studies have investigated the cointegration relationship between the swap spread and its determinants in the US. Toyoshima (2012) found an existing relationship between the interest rate swap spread and its four determinants: the corporate bond spread, the slope of the yield curve, the T-bill and Eurodollar (TED) spread and yield volatility. Another study concluded that a rise in the treasury rates is associated with a rise in the swap spread (Toyoshima & Hamori, 2012). Furthermore, several studies have attempted to explain the spread between the Euro Overnight rate (EONIA) and the ECB’s policy rate. The increase in the EONIA spread is mostly explained by the current liquidity deficit (Linzert and Schmidt, 2008).

However, there are no studies involved in the investigation of the direct relationship between the central bank’s policy interest rate and the swap curve. What differentiates this thesis from prior studies is the study of the co-movement between the ECB’s policy rate and the swap curve, instead of attempting to explain the swap spread. The approach is to conduct a time series analysis in order to make pronouncements regarding the future dynamics between the key variables. Furthermore, the focus of this thesis is the testing of the existence of a (bidirectional) relationship of the ECB’s policy rate and the swap curve using (non)-linear Granger causality tests.

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Chapter 3

Research methodology

In order to study the co-movement between the ECB’s policy rate and the swap curve, a number of methods and models are used in this thesis. The research methodology will be discussed in this chapter. First, unit root tests are presented in Section 3.1. Next, cointegration tests are discussed in Section 3.2. Regardless of the outcome of these tests, a Vector Auto Regressive (VAR) model is specified to perform the parametric linear Granger causality test. This will be discussed in more detail in Section 3.3. In addition to the analysis of linear causalities, the nonparametric Diks-Panchenko test for nonlinear Granger causality is presented in Section 3.4. These Granger causality tests give an indication of the existence of causal relationships. In order to provide visualizations of these relationships, Generalized Additive Models are introduced in the final section.

3.1 Unit root testing

Unit root testing is of importance to determine whether time series variables are stationary or not. Standard regression techniques, such as Ordinary Least Squares (OLS), require that variables are covariance stationary. Without stationary variables, the standard assumptions for asymptotic analysis will not be valid. Hence, invalid estimates are produced, which are known as spurious regression results (Granger & Newbold, 1974). T-statistics are highly significant and R-squared is high, although no relation exists between variables. To avoid spurious regressions, a well known unit root test is introduced in this section. That is the Augmented Dickey-Fuller (ADF) test, which uses the existence of a unit root as the null hypothesis.

Consider a time series variable denoted by xt. The testing procedure for the Augmented

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CHAPTER 3. RESEARCH METHODOLOGY Rachel Ho xt= α + γt + βxt−1+ p−1 X i=1 φi∆xt−i+ et, (3.1)

where α is a constant, γ is the coefficient on a time trend and ∆xj = xj− xj−1is the differenced

series of xt. In order to determine the existence of a unit root, the testing problem is formulated

as follows: H0: β = 1 versus Ha: β < 1. The t-ratio of ˆβ − 1, ADF-test = ˆ β − 1 std( ˆβ), (3.2)

where ˆβ denotes the least squares estimate of β, is called the Augmented Dickey-Fuller unit root test (Tsay, 2010).

3.2 Cointegration testing

Two or more time series are said to be cointegrated if they share a common stochastic drift. To obtain a better understanding of cointegration testing, first consider a k-dimensional VAR(p) time series yt with the following representation

yt= α + Φ1yt−1+ . . . + Φpyt−p+ t, (3.3)

where α is a k-dimensional constant vector and the innovation t is assumed to be Gaussian.

Multivariate Granger causality analysis is usually performed by fitting a VAR model to the time series. In case of cointegrated time series, a VECM model must be specified as the cointegrating relations are not explicitly apparent in a VAR representation. A vector error-correction model for the VAR(p) process ytis given by

∆yt= α + Πyt−1+ Φ∗1∆yt−1+ . . . + Φ∗p−1∆yt−p+1+ t, (3.4)

where Φ∗_j = − p X i=j+1 Φi, j = 1, . . . , p − 1, Π = αβ0 = Φp+ Φp−1+ . . . + Φ1− I = −Φ(1).

The term Πy_t−1 represents the error-correction term. It is the only term in Eq. (3.4) which in-cludes potential I (1) variables. In order for ∆ytto be I (0) it must be the case that Πyt−1is also

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Rachel Ho CHAPTER 3. RESEARCH METHODOLOGY

I (0). Hence, Πy_t−1 must contain all the possible cointegrating relations. These cointegration relations are examined, by applying the Johansen cointegration test.

As mentioned earlier and visible in Eq. (3.5), if the variables in ytare all integrated of order

one, implying that the differenced terms are stationary, this leaves us only the error-correction term Πy_t−1to introduce long-term stochastic trends. The intuition is that I (1) time series with a long-run equilibrium relationship cannot drift too far apart from the equilibrium because economic forces will act to restore the equilibrium relationship (Zivot & Wang, 2003). The long-term dynamics are delong-termined by the rank of matrix Π. The goal of the Johansen cointegration test is to test the rank of this matrix (Tsay, 2010).

Consider the hypotheses

H0: Rank(Π) = m versus Ha: Rank(Π) > m.

To perform the Johansen test, the following likelihood ratio (LR) statistic is introduced

LRtr(m) = −(T − p) k

X

i=m+1

ln(1 − ˆλi), (3.5)

where T corresponds to the sample size, p is the chosen lag length and ˆλi is the i -th eigenvalue

of the matrix Π. This test is referred to as the trace cointegration test (Tsay, 2010). The Johansen testing procedure is conducted sequentially for m = 0, 1, 2, . . . until no rejection of H0 occurs for the first time. Then one may assume that there are m cointegration relations. In

the case of rank zero, which means no cointegrating relations exist, the system is stationary in first differences. If Π has full rank, the system yt is said to be stationary in levels. However,

if Π is restricted to reduced rank r, then one may conclude there are r existing cointegrating relations among the variables in yt.

3.3 Linear Granger causality

Relations between variables may be analyzed by performing the Granger causality test on VAR specifications. Consider two time series Xt and Yt in the bivariate VAR model

Yt= c1+ A(p)Xt−1+ B(p)Yt−1+ Y,t

Xt= c2+ C(p)Xt−1+ D(p)Yt−1+ X,t,

(3.6)

where A(p), B(p), C(p) and D(p) are lag polynomials of order p and the disturbances are identically independent distributed with zero mean and constant variance. The time series X

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CHAPTER 3. RESEARCH METHODOLOGY Rachel Ho

is said to Granger cause Y if the values of X help predict future values of Y (Hamilton, 1994). So in the example above, the test of the joint restriction that the coefficients of the polynomial A(p) are equal to zero, is called the linear Granger causality test.

Hypothesis testing in levels VAR’s will cause no problems if the variables are stationary. However, if the variables are assumed to be integrated or cointegrated, the conventional asymp-totic theory no longer holds according to Sims et al. (1990). A VAR may be estimated in first-order differences, if the variables were known to be integrated and not cointegrated. On the other hand, if the variables were known to be cointegrated, hypothesis testing can gener-ally be done using a VECM specification. Nevertheless, it is often unknown a priori whether variables are stationary, integrated or cointegrated. Therefore, pretests such as the ADF test and the Johansen cointegration test are usually conducted to obtain the ’right’ specification. Toda & Yamamoto (1995) state that this method of hypothesis testing may suffer from severe pretest biases. Hence, an alternative testing procedure is created, which is robust to the order of integration and cointegration. This test is widely known as the Toda & Yamamoto (1995) augmented Granger causality test. The procedure is based on a five-step testing framework.

First, the maximum order of integration m must be determined. This can be done by con-ducting the Augmented Dickey-Fuller test for each time series. For instance, for the bivariate case, if Xt is integrated of order 1 and Yt is integrated of order 0, the maximum order of

in-tegration m is equal to 1. The second step is to specify a VAR model in levels. Determining the lag length of the model is based on the Bayes Information Criterion (BIC). Let the lag length be equal to p. Next, specify the augmented VAR model with lag length equal to p + m. So, the maximum order of integration is added to the number of lags to correct for the use of the asymptotic theorems. Now, a Wald test may be carried out for the first p variables with p degrees of freedom. If the null hypothesis of no Granger causality is rejected, one may conclude that Xt Granger causes Yt, or vice versa.

These Granger causality results may be compared with the results derived from the Johansen cointegration test. In the case of cointegrated series, there must be causality present. Hence, this provides a cross-check on the Granger causality results.

3.4 Non-linear Granger causality

In the previous section, a parametric linear time series model is assumed to perform the well known Granger causality test. This way, linear causal linkages may be found through empirical data. In addition, nonlinear causal relationships are investigated using a nonparametric non-linear Granger causality test, developed by Diks & Panchenko (2006). The background of the Diks and Panchenko test will be discussed in this section.

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Suppose that FX,t and FY,t denote the information sets consisting of past observations of

Xtand Yt respectively up to and including time t. One may assume that YtGranger causes Xt

if, for k ≥ 1

(Yt+1, . . . , Yt+k)|(FX,t, FY,t) ∼ (Yt+1, . . . , Yt+k)|FX,t, (3.7)

where 0 ∼0 denotes equivalence in distribution. In practice, k = 1 is often used. In that case, Granger causality is tested by comparing the conditional distribution of Yt+1 with and without

past and current observed values of Xt. Now, suppose that XtlX = (Xt−lX+1, . . . , Xt) and

YtlY = (Yt−lY+1, . . . , Yt) are the delay vectors, with lX, lY ≥ 1. The null hypothesis that past

observations of XtlX contain no additional information about Yt+1 (beyond that in YtlY) can

be formulated as

H0 : Yt+1|(XtlX; YtlY) ∼ Yt+1|YtlY. (3.8)

Let Yt+1be denoted by Ztto avoid notational confusions. Diks & Panchenko (2006) then propose

to drop the time index and also lx = ly = 1 is assumed. Hence, under the null hypothesis

presented in Eq. (3.8), the conditional distribution of Z given (X, Y ) = (x, y) is equivalent to the distribution of Z given Y = y. Therefore, the null hypothesis can be rewritten as

fZ|X,Y(z|x, y) = fZ|Y(z|y). (3.9)

Moreover, the null hypothesis can be reformulated in terms of ratios of joint distributions. fX,Y,Z(x, y, z) fY(y) = fX,Y(x, y) fY(y) ·fY,Z(y, z) fY(y) . (3.10)

In other words, the conditional independence of the distributions of Z and X given Y is tested using the null hypothesis in Eq. (3.10). Diks & Panchenko (2006) show that it implies that

q ≡ E[fX,Y,Z(X, Y, Z)fY(Y ) − fX,Y(X, Y )fY,Z(Y, Z)] = 0. (3.11)

Let W = (X, Y, Z) denote a random vector with the invariant distribution of (XtlX, YtlY, Yt+1).

Suppose that ˆfW(Wi) is a local density estimator of the random vector W at Wi defined by

ˆ

fW(Wi) = (2)dW(n − 1)−1Pj,j6=iIijW where dW is the dimension of W and IijW = I(||Wi−

Wj|| < nwith I(·) the indicator function and n the bandwidth, which depends on the sample

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CHAPTER 3. RESEARCH METHODOLOGY Rachel Ho Tn() = (n − 1) n(n − 2) X i ( ˆfXi,Yi,Zi(X, Y, Z) ˆfY(Yi) − ˆfX,Y(Xi, Yi) ˆfY,Z(Yi, Zi)). (3.12)

If the parameters are chosen as lX = lY = 1 and n = Cn−β(C > 0,1₄ < β < 1₃) , then Diks &

Panchenko (2006) prove that the DP test statistic in Eq. (3.12) satisfies √

n(Tn(n) − q) Sn

d

−→ N (0, 1), (3.13) where−→ denotes convergence in distribution and Sd _nis an estimator of the asymptotic variance of Tn(·) (Diks & Panchenko, 2006). Following Diks and Panchenko’s suggestion of implementing

a one-tailed version of the test, will lead to a rejection of the null hypothesis if the observed value of the left-hand-side of Eq. (3.13) is too large.

3.5 Generalized Additive Models

The methodology described so far is used to give an indication of the existence of causal relation-ships. However, the structure of the causal linkages remains unknown, in the case of nonlinear Granger causality. To overcome this problem, generalized additive models are introduced. While these models were originally developed by Hastie & Tibshirani (1986), it was Wood (2006) who provided more insights into the implementation of such a model. Hence, his notation is followed in this thesis.

A generalized additive model (GAM) is a generalized linear model in which the linear pre-dictor consists of a sum of smooth functions of covariates. Generally, it takes on the following structure (Wood, 2006):

g(µi) = X∗iθ + f1(x1i) + f2(x2i) + f3(x3i, x4i) + . . . , (3.14)

where µi ≡ E(Yi), Yi is a response variable of some exponential family distribution, X∗i contains

strictly the parametric model components with parameter vector θ and the fj are smooth

func-tions of the covariates xk. According to Wood (2006), a GAM model allows for more flexibility

and convenience by specifying the model only in terms of smooth functions, instead of defining parametric relationships. However, the smooth functions still need to be represented in some way and the degree of smoothness must be considered.

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function fj. If bij(x) is the i-th basis function, then fj can be represented as

fj(x) = q

X

j=1

bij(x)βij, (3.15)

for some values of the unknown parameters, βij. Hence, by estimating βij a linear model in Eq.

(3.14) is obtained. The simplest example of a basis function is the polynomial basis such that the smooth function fj can be written as

fj(x) = β1+ xβ2+ x2β3+ x3β4+ . . . . (3.16)

While the polynomial basis tend to have some issues regarding properties of fj over its whole

domain, the spline basis is more optimal in that sense. The latter is shown to have good ap-proximation theoretic properties. One example is the cubic regression spline, which is a curve consisting of sections of cubic polynomial joined together such that they show continuity, even for their first and second derivatives. Therefore, one must define knots at which the sections emerge. However, the cubic regression spline can only smooth with respect to one covariate. This disadvantage leads to the introduction of the thin plate regression spline. It may be visu-alized as the two-dimensional analog of the cubic regression spline. By choosing this basis, no knots have to be defined. Thin plate regression splines arise from the consideration of truncating the space of the wiggly components of the thin plate spline (Wood, 2006). In other words, the degree of smoothness is taken into account by minimizing the integral of the squares of the second derivatives.

By determining the degree of smoothness for each smooth function, one can control the shape of the function. It also affects the curve fitting to the data. In order to control the model’s smoothness, a ’wiggliness’ penalty is added to the least squares fitting objective. This is carried out by minimizing

||y − Xβ||2_{+ λ}

Z 1

0

[f00(x)]2dx, (3.17) where the integral of the squares of second derivatives penalizes models that are too ’wiggly’ (Wood, 2006). And because f is linear in its parameters, the penalty can be written as

Z 1

0

[f00(x)]2dx = βTSβ, (3.18) where S is a matrix of known coefficients, determined by the choice in basis functions. Hence,

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CHAPTER 3. RESEARCH METHODOLOGY Rachel Ho

the penalized regression spline fitting problem becomes the minimization of

||y − Xβ||2+ λβTSβ (3.19) with respect to β. Then, it can be shown that the penalized least squares estimator of β is given by

ˆ

β = (XTX + λS)−1XTy, (3.20) and the influence matrix A by

A = X(XTX + λS)−1XT. (3.21) The trade off between model fit and model smoothness is controlled by the smoothing parameter λ. For large λ, a large penalty is given to the estimated smooth function leading to a straight line estimate for fj. On the other hand, λ = 0 leads to an un-penalized regression spline estimate.

These findings imply that if λ is too high, the data is over smoothed. And if the value of λ is too low, the estimated function will be too wiggly. Either way, the spline estimate ˆf will differ from the true function f . Therefore, a suitable criterion might be to choose the optimal λ by minimizing M = 1 n n X i=1 ( ˆfi− fi)2, (3.22)

where ˆfi and fi are respectively defined as ˆf (xi) and f (xi). However, M may not be used

directly, since the true function f remains unknown. The alternative is to choose the optimal λ by minimizing the generalized cross validation score (GCV) (Wood, 2006)

Vg =

nPn

i=1(yi− ˆfi)2

[tr(I − A)]2 . (3.23)

Now that the smooth functions fj are determined using the thin plate regression splines and the

degree of smoothness parameter λ is optimized by minimizing the GCV score, one may estimate the GAM models by penalized maximum likelihood. The response variable Yi is assumed to be

of the Gaussian distribution. Therefore, the GAM models will be estimated using penalized least squares.

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Chapter 4

Data and preliminary analysis

The dataset used for this thesis consists of time series of the ECB’s policy rate and swap rates with maturities of one, two, five and ten years. The data are obtained from Bloomberg. The full sample of 5326 observations covers a period between October 1, 1999 and April 30, 2014. The levels of the swap rates are determined by the market on a daily basis. Each swap rate corresponding to a specific date is considered as the rate a certain party either pays or receives until the time of maturity, assuming the contract is issued on that specific date. So, entering a swap agreement today could give a different swap rate than entering the agreement tomorrow. These fluctuations are caused by fluctuations in the market rate, which in turn is influenced by the ECB’s policy rate.

The level of the ECB’s policy rate is set by the ECB Governing Council. The Governing Council meets every month. The economic situation is assessed and decisions on the key interest rates are normally taken during that meeting. After this meeting, a press conference is held to announce the decision on the key interest rates. The decision could either be an interest rate cut or rise, or no change at all. Because the swap rates represent daily rates, the policy interest rate is assumed to remain constant in between monthly changes. However, as these daily rates are rather unknown, one might argue that the assumption of constant rates may affect the results in a significant matter. An alternative method would be to perform data interpolation. Nevertheless, the downside is that the estimated daily ECB data through interpolation may contain future values of the ECB rates. Therefore, the approach of keeping rates constant is preferred in this study.

The time series of the ECB’s policy rate and the swap rates and their returns are displayed in Figure 4.1 and Figure 4.2. The following notation is used: ”ECB” is the ECB’s policy rate and ”Swap1”, ”Swap2”, ”Swap5” and ”Swap10” are the swap rates for maturities of one, two, five and ten years respectively. All variables are in natural logarithms. The returns are defined

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CHAPTER 4. DATA AND PRELIMINARY ANALYSIS Rachel Ho 2000 2002 2004 2006 2008 2010 2012 2014 −6.0 −5.5 −5.0 −4.5 −4.0 −3.5 −3.0 ECB

(a) ECB levels

2000 2002 2004 2006 2008 2010 2012 2014 −0.6 −0.4 −0.2 0.0 0.2 ECB returns (b) ECB returns 2000 2002 2004 2006 2008 2010 2012 2014 −6.0 −5.5 −5.0 −4.5 −4.0 −3.5 −3.0 Swap1 (c) Swap1 levels 2000 2002 2004 2006 2008 2010 2012 2014 −0.10 −0.05 0.00 0.05 0.10 0.15 Swap1 returns (d) Swap1 returns 2000 2002 2004 2006 2008 2010 2012 2014 −5.5 −5.0 −4.5 −4.0 −3.5 −3.0 Swap2

(e) Swap2 levels

2000 2002 2004 2006 2008 2010 2012 2014 −0.2 −0.1 0.0 0.1 0.2 Swap2 returns (f) Swap2 returns

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Rachel Ho CHAPTER 4. DATA AND PRELIMINARY ANALYSIS 2000 2002 2004 2006 2008 2010 2012 2014 −5.0 −4.5 −4.0 −3.5 −3.0 Swap5 (g) Swap5 levels 2000 2002 2004 2006 2008 2010 2012 2014 −0.10 −0.05 0.00 0.05 0.10 0.15 Swap5 returns (h) Swap5 returns 2000 2002 2004 2006 2008 2010 2012 2014 −4.2 −4.0 −3.8 −3.6 −3.4 −3.2 −3.0 −2.8 Swap10

(i) Swap10 levels

2000 2002 2004 2006 2008 2010 2012 2014 −0.05 0.00 0.05 0.10 Swap10 returns (j) Swap10 returns

Figure 4.1: Time series (part II)

as rt = ln(Pt) − ln(Pt−1), where Pt represents the closing interest rate on day t. The time

series of the ECB’s policy rate and the swap rates are all very similar, implying a high level of cointegration between the variables. Though, Swap10 does show more volatility through time. Based on the development of the time series, one may assume non-stationarity. However, there is no reason to assume the existence of a trend in the time series on the basis of the graphs. Therefore, the Augmented Dickey-Fuller (ADF) test is carried out under the null hypothesis of a unit root against the alternative of stationarity. The trend-stationary alternative is not considered, since no trend is assumed.

The ADF test is conducted for both the logarithmic levels (log-levels) and the logarithmic returns (log-returns). The lag lengths are selected using the Bayes Information Criterion (BIC), which are consistently zero except for the log-returns of Swap2. The optimal lag order is in this case equal to 2. This is due to the fact that the BIC-value corresponding with lag zero (= −7.9036) is slightly higher than the BIC-value of lag 2 (= −7.9038). Therefore, the number

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CHAPTER 4. DATA AND PRELIMINARY ANALYSIS Rachel Ho

Table 4.1: Results ADF test

Variables ADF-statistic ECB (0) 0.999 rECB (0) 0.000** Swap1 (0) 0.992 rSwap1 (0) 0.000** Swap2 (0) 0.973 rSwap2 (2) 0.000** Swap5 (0) 0.952 rSwap5 (0) 0.000** Swap10 (0) 0.900 rSwap10 (0) 0.000**

The variables are in log-levels and log-returns. The number of lags in parenthesis are determined based on the BIC. The reported numbers for the ADF test are p-values, for which (**) corresponds to statistical significance at 1% level.

of lags varies in this case. The results are presented in Table 4.1. The p-values of the ADF test are reported in the second column. The null hypothesis of the existence of unit roots is not rejected in the case of log-levels and is rejected for the log-returns. Because the variables turn out to be non-stationary in levels and stationary in returns, one may conclude the variables are most likely to be integrated of order one.

Another important part of the preliminary analysis is the cointegration testing, since the variables turn out to be integrated of order one. To investigate if a common stochastic drift is shared, the Johansen cointegration test is conducted for the different pairs of variables. The results are shown in Table 4.2. All variables are in log-levels. The number of lags are, once again, selected using the BIC. The test results imply that the variables are indeed cointegrated. This is in line with our expectations based on the economic background of their relationships. While all pairs contain one single cointegrating vector, in the five-variate case two cointegrating vectors

Table 4.2: Results Johansen cointegration test

Variables lags Cointegration-vectors

ECB Swap1 1 1**

ECB Swap2 1 1**

ECB Swap5 1 1**

ECB Swap10 1 1*

All five variables 4 2**

The variables are in log-levels. The number of lags are determined based on the BIC. The number of reported cointegration vectors are based on the trace statistic of the Johansen cointegration test. (*)/(**) corresponds to statistical significance at 5% and 1% level respectively.

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Rachel Ho CHAPTER 4. DATA AND PRELIMINARY ANALYSIS

are found based on the Johansen cointegration test. The last conclusion is rather unexpected. Since all pairs of variables each contain one cointegration relation, this would imply that the five-variate model must have at least four cointegration relations. Instead, the Johansen test shows a number of two relations in the five-variate case. This may be caused by insufficient power of the test, resulting in insufficient evidence to reject the null hypothesis.

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Chapter 5

Results Granger causality

The results of the linear and nonlinear Granger causality tests are presented in this chapter. To analyze the dynamic relationship between the ECB’s policy rate and the swap curve, a four-step methodology framework is used. First, the Toda and Yamamoto (1995) testing procedure is followed to obtain the linear Granger causality results. This is done for both the bivariate VAR models and the five-variate VAR model. By doing the latter, one can analyze the effects on one another while controlling for the effects of the other variables. The second step is to extract the residuals of the VAR models and test if any linear Granger causality is left. This is a check to make sure that all linear Granger causality is covered. So, no significant linear effects between residuals must be present. Next, the nonparametric Diks & Panchenko (2006) test is applied to the log-returns to investigate the presence of nonlinear Granger causality. This step is repeated, but this time the VAR residuals are used in the Diks & Panchenko (2006) test to determine the nature of the causality. This is done in a bivariate and five-variate VAR representation.

5.1 Results linear Granger causality test

The results obtained via the Toda and Yamamoto augmented Granger causality test are pre-sented in Table 5.1. The reported numbers are p-values. The notations ”*” and ”**” denote the statistical significance of the corresponding p-value at the 5% level and 1% level, respectively. Directional causalities are presented by the symbol →. Since all variables were tested to be integrated of order one, the maximum order of integration m equals one. The lag length for each pair of variables in the bivariate system is also set to one, based on the Bayes Information Criterion. The number of lags for the five-variate system was found to be four. All variables used are in logarithmic levels.

From the results, one can instantly observe the absence of linear Granger causality in the bivariate models. This may be caused by the presence of omitted variable bias, since important

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Rachel Ho CHAPTER 5. RESULTS GRANGER CAUSALITY

Table 5.1: Results linear Granger causality test

Variables Bivariate Five-variate

X Y X → Y Y → X X → Y Y → X

ECB Swap1 0.6028 0.5381 0.5812 0.1703 ECB Swap2 0.2319 0.2312 0.4901 0.0003** ECB Swap5 0.3920 0.6204 0.0566 0.0002** ECB Swap10 0.4508 0.5337 0.0475* 0.0083**

The reported numbers are p-values. The number of lags is equal to one in the bivariate case, whereas a number of four lags are identified in the five-variate case. The lags are determined based on the BIC. (*)/(**) correspond with statistical significance at 5% and 1% level respectively.

factors are excluded from the models. The dependent variable may be less explained by its regressor, which leads to a greater residual variance. Hence, no significant effect is visible in the bivariate results of Table 5.1. On the other hand, the five-variate model shows a positive number of Granger causalities, for which one pair is bidirectional. That is, ECB and Swap10 Granger cause each other. Other significant Granger causalities found are Swap2 → ECB and Swap5 → ECB. It is quite remarkable that three out of the four Granger causality effects is caused by the swap rates, instead of the ECB Granger causing the swap rates. The latter is what is expected initially, since the European Central Bank uses its policy rate as a tool to influence interest rates on the money markets. Evidently, this does not necessarily hold on an empirical basis.

Besides applying the testing procedure on the logarithmic variables, the same procedure is done for the residuals to check if all linear Granger causality is covered. By extracting the residuals of each VAR model (bivariate and five-variate as well), it can easily be tested if any linearity is still present in the residuals. One may assume that this is not the case, since the VAR models are assumed to be linear and hence contain all linear causalities. The null hypothesis of no Granger causality could not be rejected in all cases. This holds for both the bivariate and the five-variate models. Therefore, one may conclude that the residuals do not contain any linear causality.

5.2 Results nonlinear Granger causality test

The results of the nonparametric Diks & Panchenko (2006) test, applied to the bivariate and the five-variate model, are displayed in Table 5.2. The test is applied on the logarithmic returns and the VAR residuals. By performing the latter, one can determine the nature of the causality. This can be explained by the fact that the residuals of the VAR models do not contain any linear causalities, since all linear effects are covered in the linear VAR model. The first set of

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CHAPTER 5. RESULTS GRANGER CAUSALITY Rachel Ho

Table 5.2: Results non-linear Granger causality test

Variables Log-returns VAR res (bivariate) VAR res (five-variate) X Y X → Y Y → X X → Y Y → X X → Y Y → X ECB Swap1 0.9430 0.0216* 0.8731 0.0243* 0.9791 0.0024** ECB Swap2 0.9798 0.0556 1.0000 0.0518 0.9463 0.0024** ECB Swap5 0.9874 0.0921 0.9981 0.1031 0.9637 0.0276* ECB Swap10 0.9997 0.0306* 0.9995 0.0249* 0.9988 0.0116*

The reported numbers are p-values. The first set of results is based on the log-returns variables. The second and third set are based on the VAR residuals of the bivariate and five-variate model respectively. (*)/(**) correspond with statistical significance at 5% and 1% level respectively.

results are obtained using the log-returns. The second and third set of results are obtained using the VAR residuals of the bivariate and five-variate model, respectively. The lags are set to lx = ly = 1. The constant C is set to 8.0 suggested by Diks and Panchenko (2006). Using the

optimal rate β = 2₇ that Diks and Panchenko (2006) provided, leads to the implication that the bandwidth value n is chosen to be n = Cn−β = 8.0 · 5326−

2

7 = 0.68929, with the number of

observations n equal to 5326.

The implementation of these values leads to quite unilateral results. This time, the ECB’s policy rate does not have a single significant nonlinear effect on the swap rates. On the other hand, different significant unidirectional relations were found, in which the swap rate Granger causes the ECB rate. In contrast to the linear results, an additional significant relationship Swap1 → ECB is found through the nonparametric Diks-Panchenko test, in the log-returns and both residual cases as well. In the same way, the unidirectional relationship Swap10 → ECB is found. The presence of a significant causal relationship between Swap1, Swap10 and ECB in the VAR residuals implies that these relationships are strictly nonlinear. This also holds for the relationships Swap2 → ECB and Swap5 → ECB in the five-variate case. Moreover, it seems some nonlinear effects are present in the bivariate case, whereas no linear Granger causality is observed in the previous section.

Once again, the implied results are not in line with what was expected a priori. The expected relationship between the ECB’s policy interest rate and the several swap rates is characterized by the policy rate having a supposable influence on the swap rates. The nonparametric Diks & Panchenko (2006) test indicates that this is not necessarily the case, since unidirectional relationships in opposite directions are found. This suggests that the level of the swap rates in the past may play a significant part in taking decisions regarding the future ECB’s policy interest rate. In addition, it could also be implied that past values of the swap curve represent an important part of the economy, which could lead to more active changes in monetary policy by the European Central Bank.

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Rachel Ho CHAPTER 5. RESULTS GRANGER CAUSALITY

Concluding remarks

Two Granger causality tests are conducted to examine the existence of causal relationships between the policy interest rate of the ECB and the swap rates with maturities m = 1, 2, 5, 10. The Toda & Yamamoto (1995) linear Granger causality test is performed using VAR models in a bivariate and five-variate representation.1 The results show that no linear Granger causality is present in the bivariate models, while a positive number of Granger causalities are found in the five-variate model. These are Swap2 → ECB, Swap5 → ECB and Swap10 ↔ ECB. The last relationship seems to be bidirectional.

The nonlinear causal relationships are investigated through the nonparametric Diks & Panchenko (2006) test. The test is applied on the logarithmic returns and the residuals of the estimated VAR models. The results show that the ECB’s policy rate does not have an effect on the swap rates. However, multiple effects were found from the swap rates onto the ECB rate. It appears that all the swap rates have a nonlinear effect using the five-variate VAR residuals. The log-returns and the bivariate VAR residuals show significant (non)linear effects from Swap1 and Swap10 onto ECB.

These results are quite remarkable and not in line with our expectations. The central bank uses its policy rate as a tool to influence interest rates on the money markets, but the results show that the influence is in the opposite direction. The implication is that the swap curve may play a significant part when the ECB Governing Council takes decisions on its policy rate. Also, the fact that the policy rate is assumed to be constant in between monthly changes may have a possible effect on the outcome.

1_{The Toda & Yamamoto (1995) test is applied on VAR models in levels, regardless of the possible existence}

of integration and cointegration. Because of its robustness against the order of integration and cointegration, no VECM or VAR model in first differences need to be specified.

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Chapter 6

Results Generalized Additive Models

The nonlinear causal relationships found in the previous chapter are investigated through Gen-eralized Additive Modeling (GAM). By specifying and plotting such a model, it is possible to visualize the interdependences between the policy interest rate of the European Central Bank and the swap curve. There are two general structures to be considered, which are given by

yt= f1(xt−1) + f2(yt−1) + i, (6.1)

yt= f (xt−1, yt−1) + i, (6.2)

where yt represents the response variable, fj is the smooth function to be estimated and i

is the error term. The main difference between the two GAM models lies in specifying an interaction term in the second model in Eq. (6.2), rather than implementing one covariate in each smooth function such as in Eq. (6.1). The second model implies that the response variable may be explained by a certain interaction of past variables, instead of considering the individual effect of one variable on one another. Let the model structure in Eq. (6.1) be denoted by the Individual model structure and the one in Eq. (6.2) by the Interaction model structure for convenience reasons. The most optimal model structure will be determined based on the Bayes Information Criterion. This is a suitable way of model selection, since it amounts to penalizing the complexity of a model. Once the optimal GAM model structure is chosen, the results obtained by estimating that specific GAM model will be presented in the next section. The second section visualizes the estimated GAM models by introducing plots.

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Rachel Ho CHAPTER 6. RESULTS GENERALIZED ADDITIVE MODELS

6.1 Results optimal GAM model

In order to determine which model structure is preferred, both models need to be estimated first. This means that a basis function and the degree of smoothness for each smooth function must be specified. So, all smooth functions are determined using thin plate regression splines. The minimization of the generalized cross validation score leads to maintaining control over the degree of smoothness. And lastly, all GAM models are estimated using penalized least squares. Implementing this estimation method leads to the Individual model structure being more optimal, in the sense that lower BIC-values were found compared to the Interaction model structure. This suggests that an interaction term between the key variables does not lead to a better model. Thus, this section will continue describing the results of estimating GAM models of the type as in Eq. (6.1).

Recall that multiple significant nonlinear causal relationships were found in the direction Swap → ECB in the previous chapter. In order to provide more insights into the structure of these relationships, the following representation is investigated:

ECBt= f1(SwapNt−1) + f2(ECBt−1) + i, (6.3)

where ECBt is the response variable, f1(SwapNt−1) is a smooth function of the lagged swap

rates for maturities of N = 1, 2, 5, 10 years and f2(ECBt−1) is a smooth function of the lagged

response variable ECB. By estimating these models, the function ˆf1(SwapNt−1) allows us to

perceive the interdependence structure of the relationship Swap → ECB in a more comprehen-sive manner.

Even though only a select number of significant nonlinear relationships were found through the Diks & Panchenko (2006) test, it is still preferable to investigate all possible relationships regardless of their insignificance. This is mainly due to the possibility that the Diks-Panchenko test, and tests in general, may suffer from sensitivity to deviations from the null hypothesis. Therefore, all possible relationships are analyzed through GAM modeling. The results are pre-sented in Table 6.1. The first set of results are obtained using the log-returns time series as variables. The other two sets are based on the filtered residuals of the bivariate and five-variate VAR models. These last results can be considered as strictly nonlinear relations, since all lin-earity is covered by the VAR models.

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CHAPTER 6. RESULTS GENERALIZED ADDITIVE MODELS Rachel Ho

Table 6.1: Results GAM estimation

Variables Log-returns VAR res (bivariate) VAR res (five-variate) yt xt f1(xt−1) f2(yt−1) f1(xt−1) f2(yt−1) f1(xt−1) f2(yt−1)

ECB Swap1 0.0000** 0.8171 0.0066** 0.9844 0.0012** 0.9230 ECB Swap2 0.3019 0.9822 0.9152 0.9673 0.3840 0.0938 ECB Swap5 0.1683 0.9490 0.4087 0.9811 0.1365 0.8341 ECB Swap10 0.4936 0.9842 0.9273 0.8490 0.5281 0.2796

These results are obtained through estimation of the smooth functions in the model yt = f1(xt−1) + f2(yt−1).

The corresponding p-values are reported in the table. The first set of results is based on the log-returns time series. The second and third set are based on the VAR residuals of the bivariate and five-variate case, respectively. (*)/(**) correspond with statistical significance at 5% and 1% level respectively.

From this table, it is obvious that only one significant relation is found, that is Swap1 → ECB. This holds for the log-returns time series and the filtered residuals of both the bivariate and five-variate model as well. This suggests that the one-year swap rate has a certain effect on the policy interest rate of the ECB. Because the residual time series show significance, it is safe to say that at least a part of the observed effect appears to be nonlinear. The overall effect present in the log-returns time series could also contain some linearity. However, it is more plausible to assume this is not the case, since no linear effect caused by Swap1 was found through the Toda and Yamamoto linear Granger causality test. Nevertheless, it is still good to keep in mind that the results do not necessarily have to coincide, as the investigated models and methods differ from one another. By plotting the estimated GAM model, it is possible to gain more insights into this matter, which is investigated in the next section.

Regardless of the possible nature of the causality found in the one-year swap rate, it is worth noticing that all other swap rates share no significant relationship with the central bank’s policy rate. This coincides with our anticipation, in some way. The initial expectation was that particularly the short term swap rates would be influenced by the policy rate. This suggests a strong connection between the variables ECB and Swap1. Although it turns out to be a reversed directional relationship, it remains the most likely to be expected relation. However, this anticipation does not rule out the fact that the nonlinear relationships between the ECB rate and the other swap rates, found through the Diks & Panchenko (2006) test, have disappeared now. The most plausible reason could be that the models used in both methods do not coincide, leading to different results.

Furthermore, one may notice that the estimations of f2(ECBt−1) all turned out to be

insignificant. Because of this insignificance, it would make sense to eliminate the smooth function f2 from the model in Eq. (6.3), as it is not serving any purpose. This can also be viewed in

the insignificant results in Table 6.1. By simplifying the model to ECBt = f1(SwapNt−1) + i

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significant relation Swap1 → ECB is found through estimation of the simplied model.

6.2 GAM visualizations

The next step is to visualize the interdependence between the one-year swap rate and the ECB’s policy rate by plotting the estimated GAM model. Figure 6.1 depicts a plot of the estimated smooth function ˆf1(Swap1t−1) using the log-returns time series. The x-axis contains all the data

points in the black area and the y-axis represents the response. The mere focus will be on the center area above the black area, since most of the data points are located at that area. Hence, moving away to the borders will lead to more unreliable results. This can also be observed from the dashed lines around the graph, which represent the two standard error bounds. While approaching the end points of the plot, the standard error bounds diverge substantially.

−0.10 −0.05 0.00 0.05 0.10 0.15 −0.01 0.00 0.01 0.02 dSwap1.l s(dSw ap1.l,4.15)

Figure 6.1: GAM plot of ˆf1(Swap1t−1) using log-returns

The first thing to point out by analyzing the plot, is that the relationship between the variables Swap1 and ECB is not strictly linear. It rather has the shape of a parabola at the center. By zooming further in on the center area of the plot, the following is observed: if the one-year swap rate either increases or decreases by a one percent point, then this leads to a decrease of the expected ECB rate in the future. Diverging with greater distance from the center may also result in a greater decrease in the ECB rate, although with high uncertainty.

Next to the plot in Figure 6.1, two additional GAM plots are obtained using the residuals from the bivariate and five-variate VAR model. These plots are depicted in Figure 6.2. The use of the residuals, instead of the usual log-returns time series, separates the nonlinear from the linear effects. The first thing to notice about the plots is that they show a lot of similarities

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CHAPTER 6. RESULTS GENERALIZED ADDITIVE MODELS Rachel Ho −0.10 −0.05 0.00 0.05 0.10 0.15 −0.02 −0.01 0.00 0.01 0.02 res.Swap1.l s(res .Sw ap1.l,7.09) (a) bivariate −0.10 −0.05 0.00 0.05 0.10 0.15 −0.01 0.00 0.01 0.02 res.Swap1.l s(res .Sw ap1.l,5.94) (b) five-variate

Figure 6.2: GAM plots of ˆf1(Swap1t−1) using VAR residuals

in the center area. However, this time, the nonlinear effect implies no change in the expected policy rate, if the one-year swap rate slightly increases or decreases. And if a greater change in the swap rate were to occur, then the ECB rate is expected to decrease. But because of the greater error bounds, this result must be interpreted with caution.

Concluding remarks

The nonlinear causal relationships found in Chapter 5 are further examined by specifying and estimating GAM models, such as in Eq. (6.3), in which the ECB’s policy rate depends on smooth functions of each swap rate and its own past. The results show that only one significant relation was found, that is Swap1 → ECB. Initially, it was expected that the policy rate would influence the short term swap rate, hence implying a strong connection between the variables ECB and Swap1. Even though the results show the opposite relationship, it remains the most likely expected relationship. However, all other swap rates share no significant relationship with the ECB rate in the GAM model estimation unlike before in the Diks & Panchenko (2006) test. The most plausible reason is that the models and methods do not coincide, which led to different outcomes.

By plotting the estimated GAM models it is possible to visualize the interdependence be-tween the policy rate and the one-year swap rate. From the log-returns plot, the following may be observed: a one percent point change in the one-year swap rate causes the expected ECB’s policy rate to decrease in the future. A greater change in the one-year swap rate may lead to

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a greater decrease in the ECB rate. However, it must be noted that these findings come with high uncertainty, as the standard error bounds diverge substantially. The VAR residuals plot shows a slightly different outcome: the nonlinear effect implies no change in the ECB rate at a slight change of Swap1. For the overall effect we must focus on the log-returns plot.

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Chapter 7

Before and after the crisis

In order to determine whether the results in the previous chapters may assume to hold in general, a sensitivity analysis is conducted with respect to the selected sample period. This type of analysis is to test and check how robust the models in fact are. Therefore, subsamples are considered, before and after the global financial crisis in 2007-2008. A brief description of the subsamples is given in the first section. The same methodology framework, presented in Chapter 3, is used. Hence, the Toda & Yamamoto (1995) linear Granger causality test and the Diks & Panchenko (2006) nonlinear Granger causality test are conducted to determine the existence of causalities, both in a bivariate and five-variate setting. These results are presented in Section 7.2 and 7.3 respectively. Then, these causalities are investigated and visualized by the use of GAM models in the last section of this chapter.

7.1 Subsamples

Ever since the global financial crisis in 2007-2008, central banks all over the world had to take additional nonstandard measures to get their economies back on track. This also meant high cuts in the policy rates of the central banks to provide a liquidity boost into the economy. This policy rate development is also visible in Figure 4.1.a, in which the European Central Bank decided to lower its policy interest rate enormously in the year of 2008. In the period after, it could be the case that steering monetary policy into another direction (that means focusing on an objective other than maintaining price stability) might result in yet to be observed causalities. Therefore, two subsamples are considered to investigate whether the previous results over the full sample coincide with the results through subsampling. In the case of mixed results, one might argue that our original results presented in Chapter 5 are not robust, in the sense that the results show sensitivity towards the period(s) of investigation.

The dynamics between the ECB's policy rate and the swap curve : an analysis based on linear and nonlinear Granger causality