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Empirical Analysis of the Relationship
between the Yield Curve and
Macroeconomic Variables using German
Data
Diplomarbeit
bei Prof. Dr. Svetlozar T. Rachev
Institut fur Statistik und Okonometrie
Universitat Karlsruhe (TH)
vorgelegt von Thomas Bernhard Reckers
Betreuer: Dipl.-Wi.-Ing. Markus Hochstotter
20. April 2006
Ehrenwortliche Erklarung
Ich versichere hiermit wahrheitsgemaß, die Arbeit selbstandig angefertigt, alle benutzten
Hilfsmittel vollstandig und genau angegeben und alles kenntlich gemacht zu haben, was
aus Arbeiten anderer unverandert oder mit Abanderung entnommen wurde.
Karlsruhe, den 20. April 2006
(Thomas Reckers)
I
Contents
1 Introduction 5
2 Theories about the term structure of interest rates 6
2.1 Traditional economic theories of the term Structure . . . . . . . . . . . . 6
2.1.1 expectations Theory . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Liquidity preference theory . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Market segmentation theory . . . . . . . . . . . . . . . . . . . . . 7
2.1.4 Preferred habitat theory . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Quantitative term structure models . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Short-rate equilibrium models . . . . . . . . . . . . . . . . . . . . 8
2.2.1.1 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . 9
2.2.1.2 The Cox-Ingersoll-Ross model . . . . . . . . . . . . . . . 10
2.2.2 No-arbitrage models . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2.1 The Ho and Lee model . . . . . . . . . . . . . . . . . . . 10
2.2.2.2 The Hull and White model . . . . . . . . . . . . . . . . 11
3 Literature review 12
3.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Exchange rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Budget deficit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Oil price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 External interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Data 24
4.1 Interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Consumer price index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 GDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Ifo business climate index . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Unemployment rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Exchange rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.7 Budget deficit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.8 Oil price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.9 External interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
5 Methodology 35
5.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Unit root test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4 Granger causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Empirical analysis 42
6.1 Autoregressive Moving Average Processes . . . . . . . . . . . . . . . . . . 42
6.2 The Box-Jenkins Methodology . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 OLS estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.4 VARMA modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.5 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.6 Time frame estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.7 (G)ARCH modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.7.1 ARCH LM Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.7.2 (G)ARCH model estimation . . . . . . . . . . . . . . . . . . . . . 63
7 Conclusions 66
Bibliography 67
A Schwarz criterion 71
B Akaike criterion 71
C Adjusted R2 71
D Log-likelihood function 71
E Skewness 71
F Kurtosis 72
G Jarque-Bera (JB) test of normality 72
H Ito process 72
2
Glossary
adf Augmented Dickey-Full test
cpi Consumer price index in %
dcpi 1st difference of the consumer price index in %
dfiscal 1st difference of the new indebtness of the fiscal deficit
dfx 1st difference of the nominal DEM/EUR US$ exchange rate
dgdp 1st difference of the quarterly gross domestic product in %
difo 1st difference of the Ifo business climate index
djapanlong 1st difference of the Japanese government bond yields
djapanshort 1st difference of the Japanese call money rates
djapanslope 1st difference of japanslope
dmths3 1st difference of the 3-months money market rate reported by Frankfurt
doil 1st difference oil price per barrel in Euro
drealfx 1st difference of the real effective exchange rate for DEM/EUR
dslope1y10y 1st difference of the 10-years 1-year slope
dslope1y10y 1st difference of the 10-years 3-months slope
due 1st difference of the quarterly change in % of the unemployment rate
duslong 1st difference of the 10-years yield for US government bonds
dusshort 1st difference of the treasury bill rates
dusslope 1st difference of usslope
dyr1 1st difference of the 1-year yield on listed German Federal securities
dyrs10 1st difference of the 10-years yield on listed German Federal securities
dyrs5 1st difference of the 5-years yield on listed German Federal securities
e.g. for example
fiscal New indebtness of the fiscal deficit
fx Nominal DEM/EUR US$ exchange rate
3
gdp Quarterly gross domestic product in %
i.e. id est, that means
ifo Ifo business climate index
japanlong Japanese government bond yields
japanshort Japanese call money rates
japanslope Difference of japanlong and japanshort
LB Ljung-Box
LM Lagrange Multiplier
LR Likelihood ratio
mths3 3-months money market rate reported by Frankfurt
oil Oil price per barrel in Euro
realfx Real effective exchange rate for DEM/EUR
SIC Schwarz information criterion
slope1y10y 10-years 1-year slope
slope1y10y 10-years 3-months slope
ue Quarterly change in % of the unemployment rate
uslong 10-years yield for US government bonds
usshort Treasury bill rates
usslope Difference of uslong and usshort
V ARMA Vector autoregressive moving-average
yr1 1-year yield on listed German Federal securities
yrs10 10-years yield on listed German Federal securities
yrs5 5-years yield on listed German Federal securities
4
1 Introduction
Theoretical and empirical models for the analysis of interest rates represent one of
the most discussed areas in economic research. The level of interest rates plays an
important role in an economy because it strongly influences the extent of investments.
This thesis attempts to conduct an empirical analysis for the German term structure
of interest rates. The term structure of interest plots the relationship between the
interest rate and the maturity of debt and is influenced by several latent and observable
parameters. In this analysis, the focus is on interpreting the impact of macroeconomic
variables on the yield curve but in the empirical section, univariate models are also
estimated and compared with multivariate models.
In the second chapter, a brief overview about traditional and quantitative term structure
models is given. A review of the literature, which deals with the relationship of interest
rates and macroeconomic variable, follows. The data that will be used for the latter
analysis is introduced in the fourth chapter and the subsequent chapter presents basic
econometric tests with these time series. The main part of this thesis is represented
by the empirical analysis. Different uni- and multivariate models are estimated. Based
on these estimation, forecasting and (G)ARCH models are also assessed. Finally, a
summary of the result concludes the paper.
5
2 Theories about the term structure of interest rates
This chapter introduces traditional and quantitative theories about the term structure
of interest rates. Traditional concepts attempt to explain the yield curve and its shape in
an intuitive way, whereas quantitative theories are precise mathematical models (Cairns,
2004).
2.1 Traditional economic theories of the term Structure
The following term structure theories are all based on traditional economic theory. Each
of these theories emphasises in a different direction. Although these models might be
slightly out of date, they provide some useful hints about how the term structure evolves.
2.1.1 expectations Theory
Among the traditional models, the expectations theory is the oldest approach (Deppner,
1992).Under the assumption that investors view all assets as perfect substitutes, re-
gardless of their maturities, Fisher (1932) basically argues that long-term interest rates
reflect the expectations about future short-term rates.
Let i be the interest rate of a bond. For an n-period bond, the expectations theory
states that the yield of this bond is an average of the expected short-term yields over
the maturity of the bond (Hubbard, 2002):
in,t =i1,t + ie1,t+1 + ... + ie1,t+n−1
n(1)
In line with the expectations theory, the slope of the yield curve reflects what the expecta-
tion of the market about future interest rates are. A downward-sloping (upward-sloping)
curve means that investors presume that future short-term rates will fall (rise) and in
a flat-curve environment, market participants predict short-term will move sideways in
the next periods.
However, this would imply that short-term rates are mostly expected to rise (since the
yield is usually upward-sloping) which is inconsistent with the actual empirical observa-
tions.
6
2.1.2 Liquidity preference theory
Investors tend to prefer short-term securities, according to the liquidity preference theory,
because long-term assets are likely to be more volatile and hence have a higher price risk
than short-term paper. However, the majority of borrowers favours issuing long-term
bonds because they can hedge their future cash flows for a longer period of time. This
controversial preference for maturities leads to a constitutional weakness on the long
end of the yield curve. In order to raise the desirability of long-term bonds, the issuers
will offer a risk-premium to the creditor and aims to convince the investors who are
indifferent between short- and long-term assets (Hicks, 1950).
2.1.3 Market segmentation theory
The market segmentation theory behaves in an opposite way to the expectations theory.
It assumes that the expectation about future interest rates are uncertain and the par-
ticipants of the capital market are acting risk-averse which means that every individual
will invest in maturities which he prefers (Culbertson, 1957).
Consequently, debtors and creditors will only buy and sell asset with certain maturi-
ties so that the capital market is broken down into delimited and non-substitutable
sub-markets. Between these sub-markets, there is no interaction due to the supposed
risk-aversion and the equilibrium interest rates in every segment is a result of the exist-
ing demand and supply of fixed income assets.
Although it is true that certain investor groups will prefer different maturities (e.g. a
pension fund will most probably prefer long-term assets whereas a bank might prefer
to invest in the short end), in the real world short and long maturities are competing
against each other.
Hence, the market segmentation theory is not a very reasonable description for the term
structure of interest rates.
2.1.4 Preferred habitat theory
Finally, the preferred habitat theory attempts to merge the previously mentioned mod-
els. Investors have a view that different maturities can be substitutes but not perfect
substitutes and they tend to prefer certain maturities but they can be induced to buy
other securities (Modigliani and Sutch, 1966). Hence, they will only switch the maturity
if they get offered a term premium which compensates them for buying a less-preferred
7
maturity. Under this model, the spread between short and long rates will mainly depend
on the expected change of the long rate but it may also be influenced by the demand and
supply of securities of different maturities, risk aversion, transaction costs and arbitrage
opportunities. The yield for an n-period bond using the preferred habitat theory equals
the average of expected future one-period yields plus the hn,t (habitat) term premium.
in,t =i1,t + ie1,t+1 + ... + ie1,t+n−1
n+ hn,t (2)
The term premium is the crucial difference between the preferred habitat and the previ-
ous theories. While it is always zero under the expectations theory and always positive
under the liquidity preference theory, the preferred habitat theory assumes that the pre-
mium can take on positive as well as negative values. It is obvious that the shape of
the yield curve depends on the expectations about future short-term yields and the size
of the term premium. Since investors typically prefer shorter instead of longer maturi-
ties, hn,t increases with higher maturities. Thus, the preferred habitat theory perfectly
explains why the term structure is mostly upward-sloping even if short-term rates are
not expected to rise soon. A flat (downward-sloping) curve suggests that the market
expects a slight (more significant) fall of short rates.
The preferred habitat theory is therefore the most logical way of explaining the usual
upward-sloping curve and the movement together of short- and long-term rates.
2.2 Quantitative term structure models
There are two main classes of interest rate models: Equilibrium (short-rate) and no-
arbitrage models. Equilibrium models have the disadvantage that their calculated bond
price does not exactly match the observed market price. No-arbitrage models use today’s
term structure as an input and hence, they are consistent with the current yield curve.
Generally, these models are widely used for the pricing of interest rates derivatives (Hull,
1999).
2.2.1 Short-rate equilibrium models
Originally, interest rate models were based on the assumption of specific one-dimensional
dynamics for the instantaneous spot rate process r (Brigo and Mercurio, 2001). Short-
8
rate equilibrium models follow an Ito process of the form
dr = m(r)dt + s(r)dz
The instantaneous drift, m, and instantaneous standard deviation, s, are assumed to be
functions of r but are independent of time. In this model, all rates move in the same
direction but not necessarily by the same amount. It does not imply that the term
structure always has the same shape (Hull, 1999).
According to Cairns (2004), three basic characteristics are desirable for short-rate mod-
els.
• Interest rates should be positive.
• r(t) should be autoregressive so that it will reach some long-term target and will
not drift to plus or minus infinity or to zero.
• If possible, the model should include a simple formula for bond and derivative
pricing.
The main disadvantage of short-rate models is their endogeneity and the small number
of parameters makes it difficult to properly reproduce some typical shapes of the yield
curve (Brigo and Mercurio, 2001).
2.2.1.1 The Vasicek Model
The model of Vasicek (1977) is probably the best known model.
The risk neutral process for r in the Vasicek Model is
dr = a(b− r)dt + σdz (3)
where a, b and σ are constants. The key feature of the model is its mean-reverting
structure. The stochastic term σdz is normally distributed. In the model:
• b represents the risk-neutral long-term mean risk-free rate
• a represents the rate at which r reverts back to this long-term mean
• σ represents the local volatility of short-term interest rates
The shape produced by the this model can be upward sloping, downward sloping, or
slightly ”humped”(Hull, 1999; Cairns, 2004).
9
2.2.1.2 The Cox-Ingersoll-Ross model
The main disadvantage of the Vasicek model is that the risk-free rate of interest can
become negative. Even if the probabilities of r(t) becoming negative are small (either
because the timescale is short or because the volatility of r(t) is small), there are many
circumstances and parameter sets where the probability of negative rates can be signif-
icant.
Thus, the model proposed by Cox, Ingersoll and Ross (Cox et al., 1985) keeps rates of
interest positive.
dr = a(b− r)dt + σ√
rdz (4)
The mean-reverting drift is the same as in Vasicek’s model, but the standard deviation is
proportional to√
r, i.e. as the short-term interest rate increases , its standard deviation
increases(Hull, 1999; Cairns, 2004).
2.2.2 No-arbitrage models
No-arbitrage models attempt to solve the described problems which occur when short-
rate models are used. Whereas equilibrium models deliver today’s term structure as an
output, no-arbitrage models treat it as an input. Furthermore, their drift usually de-
pends on time which means that the average path taken by the short rate is influenced
through the initial zero-curve (Hull, 1999).
Cairns (2004) divides no-arbitrage models into two classes. Multifactor, time-
homogeneous models are calibrated to historical data and a close fit between theoretical
and observed prices is achieved by regular calibration of stochastic variables. Secondly,
time-inhomogeneneous, no-arbitrage models use observed prices as a direct input and
therefore the theoretical price exactly matches the observed price. These models are
preferred by market practitioners, because they make sure that the prices of derivatives
from different market makers match and there are no arbitrage opportunities.
2.2.2.1 The Ho and Lee model
Ho and Lee (1986) presented their model in the form of a binomial tree of bond prices
with two parameters: the short-rate standard deviation and the market price of risk of
the short rates. The continuous time limit of the model is:
dr = θdt + σdz (5)
10
where σ, the instantaneous standard deviation of the short rate, is constant and θ(t) is
function of time to ensure that the model fits the initial term structure (for the analytical
calculation of θ(t) see equation (6))
θ(t) = Ft(0, t) + σ2t (6)
Here, F(0,t) is the instantaneous forward rate for a maturity t as seen at time zero
and the subscript t denotes a partial derivative with respect to t. As θ(t) approximately
equals Ft(0, t), the average direction of the short rate in the future will be moving roughly
equal to the slope of the instantaneous forward curve.
This model exactly fits to the current term structure of interest rates, but it is very
inflexible when it comes to choosing the volatility structure (Hull, 1999; Cairns, 2004).
2.2.2.2 The Hull and White model An extension of the Vasicek model is the frame-
work designed by Hull and White (1990).
dr = a
(θ(t)
a− r
)dt + σdz (7)
where a and σ are constants and. The model can be characterised as the Vasicek model
with a time-dependent variable. At time t the short rate reverts to θ(t)a
at rate a. The
θ(t) function can be derived analytically:
θ(t) = Ft(0, 1) + aF (0, t) +σ2
2a(1− e−2at) (8)
The last of the three terms is usually very small that is why, on average, r approximately
follows the slope of the initial instantaneous forward rate curve. One advantage over the
Ho and Lee model is that this model can represent a wider range of volatility structures.
11
3 Literature review
The yield curve is a very useful indicator for the transmission of macroeconomic policy
(Turnovsky, 1989). On the one hand, the short end of the curve is determined by
monetary policy markers and depends essentially on the current level of inflation, the
inflation target of a central bank and the level of output activity. However, the long
end of the curve can not directly be influenced by the central bank and therefore is also
driven by other factors such as the exchange rate, the fiscal deficit, the expectations
about future short-term interest rates and external long-term interest rates (see Butter
and Jansen, 2004; Dua et al., 2004). Consequently, there should exist a relationship
between macroeconomic variables and the term structure of interest rates. The term
structure can be used to forecast GDP growth and future inflation. Movements in
macroeconomic fundamentals will have an impact on the shape of the curve and the
level of interest rates. Although, economic theory would suggest that the impact of the
macroeconomy on yields is stronger than vice versa.
For that reason, a model which aims to describe the finance-macro relationship has to
investigate the impact of the financial market on the macroeconomy and the opposite
effect.
3.1 Inflation
The empirical and theoretical relationships between inflation and interest rates are one
of the most discussed subjects in the economic and econometric field.
In his seminal work, Irving Fisher (1932) stated that the nominal rate of interest is the
equilibrium real return plus the fully anticipated rate of price changes in a market which
is well functioning (Fama, 1975). 1 According to Fisher’s theory, the nominal rate has
a one-to-one relationship with the expected inflation since the real rate is independent
of price changes. Therefore, the term structure could be a predictor for future inflation.
However, the outcome of empirical studies is very diverse and the theory is only
validated by some studies.
Inflation targeting is one important element of monetary policy. The European Central
1 The Fisher equation leads to the following conclusion:
1 + i = (1 + r)(1 + π)
π is the anticipated inflation rate, r is the real rate of interest and i is the nominal rate of interest.
12
Bank (ECB) aims to keep the CPI of the Euro Area of below, but close to, 2% and the
Bundesbank was also widely known for its monetary policy on inflation targeting. A
strong increase in the long-term interest rates will be interpreted as a rise in inflation
expectation. A central bank who has an implicit inflation target will consequently raise
short-term interest rates in order to lower inflation expectations of the market and to
indirectly influence the long end of the yield curve (Schich, 1999).
In the next section, some of the empirical papers which deal with the relationship
between the yield curve, current and expected inflation rates will be presented. Firstly,
there is a larger amount of research which uses the term structure to forecast future
inflation.
As already mentioned, Fisher’s ”The theory of interest”which attempts to link current
nominal interest rate and expected inflation (1932) was the starting point for an
enormous amount of economic research. In his book, Fisher uses annual data from
the US and the UK for the period between 1820 to 1924 to verify his hypothesis. The
empirical analysis proves that there is a correlation of 0.8 and higher between long-term
interest rates and price inflation when the inflation is lagged over 20 or more years.
The majority of studies about the US economy dealing with the Fisher effect suggest
that there is indeed a positive relationship between interest rates and inflation but they
do not confirm Fisher’s claim of a one-to-one relationship (Cooray, 2002). In an earlier
study by Yohe and Karnosky (1969), prices level changes are confirmed to be the main
determinant of changes in the nominal interest rates. Their results suggest that the
time horizon of forming inflation expectation increases as the term to maturity of the
fixed-income security increases.
Mishkin (1992), analysing monthly US Data from January 1953 to December 1990,
argues that the level of inflation and interest rates may be the result of a stochastic
trend and the apparent short-term forecasting relationship between both is spurious.
However, cointegration test for a common trend of both series suggest that a long-run
Fisher effect might exist. Looking at different subperiods, Fisher observes that the
long-run Fisher effect is particularly strong when both series seem to exhibit a stochastic
trend (January 1953 to October 1979). For periods when it seems as inflation does not
have a trend, the Fisher effect is not measurable (November 1979 to December 1990).
There are a few pieces of research which attempt to investigate whether the Fisher
effect is valid for German data. In a multicountry study, Mishkin (1984) only finds very
small evidence for the existence of the effect in Germany. Unlike the previous result,
Yuhn (1996), using quarterly data from 1973:II to 1993:II for short- and long-term
13
nominal interest rates and inflation rates for five industrialised countries (US, UK,
Japan, Germany and Canada), points out that German inflation rate and interest rate
have the strongest relationship among the investigated countries. His empirical study
uses Johansen’s cointegration tests in order to verify the link.
With the implementation of a threshold cointegration model which takes into account
that inflation and interest rates in developed countries fluctuate in a narrow band,
Weidmann (1997) shows that there is indeed a one-to-one relationship between the two
variables.
Although the Bundesbank did not introduce their inflation target of 2% per year before
1986, it interpreted its task to safeguard the internal value of the DM. The ECB, which
conduct the monetary policy for Germany, has adopted this inflation target. Higher
inflation will thus increase the probability of a rise in nominal short-term rates by the
central bank. Since actual and expected inflation do not change significantly in the
short-run, one should actually expect how the real interest rate and inflation will evolve
due to the known policy of the inflation targeting bank (Leidermann and Svensson,
1995).
Bernanke and Mihov (1997) use a (VAR)-based approach in order to analyse how the
Bundesbank has responded to changes in money growth and which impacts the German
central bank’s policy had on prices and other key variables. As already mentioned, the
Bundesbank did not have an official inflation target until 1986, it still had an implicit
target (the so-called ”‘unavoidable inflation”’). Comparing the deviation between the
actual and the targeted money growth as well as the deviation of the actual inflation
and the ”‘unavoidable inflation”’ in the period from 1975 to 1995, the authors notice
that it seems as if the money growth target is subservient because the volatility of the
inflation targets is lower and it declines steadily (except in the oil-shock-year, 1980).
Taking also other computations of the paper into account, the Bundesbank turns out
to be rather an inflation targeter than a money targeter.
3.2 GDP
Current inflation and the expectations about the future level of inflation can explain
much of the nominal interest rate but the shape of the term structure can also explain
future real economic activity.
Forecasting real economic growth is fundamental for policymakers and businesses. How-
14
ever, macroeconomic models are often very complex and inaccurate. Using financial
data as an input to predict the real GDP growth makes sense because the figures are
immediately available and are precise.
The term spread which is the difference between short-term and long-term interest rates
has been a useful predictor for real economic growth. While an upward sloping suggests
that the economy will expand, a negative yield spread is associated with future economic
tightening.
There are two explanations for this empirical relationship which are discussed among
researchers.
The first reason for the correlation may be that the term spread reflects the stance of
monetary policy. If the policymakers raise short-term interest rates, long-term rates are
usually not increasing one-to-one with them but slightly less. Hence, the spread tightens
and even might become negative. Higher interest rates slow down overall spending and
economic growth will stagnate. Therefore, a small or negative slope of the yield curve
will be an indication for slower growing economy in the future.
Secondly, the yield spread might reflect the market expectations about future economic
growth. If businesses and market participants anticipate that the real income will in-
crease in the future. The increase in expected future real income will cause a rise in
borrowing by businesses because they expect profitable investment opportunities. As
they typically issue long-term debt, the supply for these bonds will increase and prices
will go down. The yield of the longer paper will increase relative to the short-term rates
and the curve will consequently become steeper (Bonser-Neal and Morley, 1997).
In an analytical rational expectations models, Estrella (2005) attempts to explain the
rationale for the significance of the yield slope. According to this model, monetary policy
is the main reason for the forecasting power of the term structure. Essentially, monetary
policy reacts if there are deviations from targeted inflation or potential output growth.
If the monetary policymakers systematically aim to achieve these goals, the predictive
power of the term structure is more directly dependant on macroeconomic variables.
The quality of the prediction is therefore the result of the form of the monetary policy
reaction function.
In a multicountry empirical analysis for 11 industrial countries (1972:1 - 1996:4), Bonser-
Neal and Morley (1997) aim to measure the forecasting power of the term spread. In the
in-sample model, the explanatory power of Germany’s term spread (on the 10-years gov-
ernment bond and the 3-month interbank rate) is among the highest, only exceeded by
Canada and the United States. On average, the slope of curve accounts for about 30% of
15
the future real GDP growth in Germany, if one looks at a 1- to 3-years forecast horizon.
Regardless of the investigated country, the coefficient confirms the positive relationship
between the term spread and real GDP growth. In order to testify that the yield spread
is the best predictor for future growth, the authors compare the out-of-sample forecast
power of the yield spread model with two alternative forecasting models, a model which
uses the lagged real GDP growth and the second model combines both variables in one
equation. The root mean squared error (RMSE) statistic which measure the accuracy
of the different forecasts suggests that the pure yield spread model outperforms the al-
ternative models.
In one of the first multicountry studies, Harvey (1991) analyses quarterly data from the
G-7 countries (period 1970:1 - 1989:4). He uses the term spread in quarter t between a
bond that has five quarters to maturity and a bond that has one quarter to maturity
to make a forecast for the GNP growth from quarter t+1 to quarter t+5. The forecast
for Germany has an adjusted R2 of almost 30% and is only exceeded by the US and
Canada.
Plosser and Rouwenhorst (1994) conduct an empirical study for the US, Germany,
France, Canada and the UK. When they investigate quarterly data, their main find-
ing is that the term spread is a better forecaster of real output and consumption than
of nominal output or consumption growth in Germany (period 1960:1 - 1991:3). In a
second exercise, they regress the annualised growth rate of industrial production on the
short-term rate (1 year) or on both the short term rate and the term spread between
year k and year 1 where k is the forecast horizon. Interestingly, the coefficient of the
term spread for Germany and the US does not change significantly if the short-term rate
is added to the equation. Consequently, the predictive power of the spread is not caused
by the fact that the yield spread is correlated with the level of the short-term rate which
might point out that the information content of the curve slope is not due to the level
of the short-term rate but rather reflects market expectations about future growth.
Although it seems as the empirical relationship between the slope and growth variable
is evident, there are also studies which compute a much lesser correlation coefficients.
Bange (1996) evaluates the influence of the yield spread on future growth for Japan,
Germany and the US. Among the three countries, Germany exhibits the smallest cor-
relation if the dependent variable is the quarterly growth rate of industrial production
while the 1-period lagged variable of the dependent variable is the first explanatory
variable and the second explanatory variable is the spread (between the rate on three
month loans and the yield of a long-term government bond with a maximum maturity
16
of four years). Even though the period chosen by Bange (1973:1 - 1988:1) is very similar
to the Harvey’s selection and the regression equation are comparable, Bange computes
a R2 which is 20% lower for Germany and 15% higher for Japan compared to Harvey’s
results. This might refer to the fact that the achieved correlation depends very much on
the formulation of the regression equation and the chosen data period.
3.3 Exchange rate
Germany, as the so-called export world champion who exports the largest amount of
goods in US$, relies heavily on trade relationship with foreign countries. Although the
largest trading partner are mainly in Europe, many exports go into the United States,
Asia and the Middle East. Thus, exchange rates especially affect the performance of
domestic businesses who generate a large revenue in US$ or in currencies which are
correlated/pegged with the US$. A depreciating US$ vs. the local currency harms
domestic corporations because their spending are usually in the local currency and hence,
prices are fixed in this currency. If the foreign currency decreases in value, domestic
companies have to offer their products at a higher price and the export demand lessens.
In the opposite case (an appreciation of the domestic currency), export demand increases.
In other words, the value of a currency has an impact on the economic growth, if a
country has large trading activities. So far, this example has not taken price changes
into consideration that means this model, which reflects the value of one currency in
terms of another, looks at the movements of the nominal exchange rate (Hubbard, 2002).
The real exchange rate takes also the purchasing power of the currency into account. It
can be expressed by the following equation where EXr is the real exchange rate, EX the
nominal exchange rate, P the domestic price index and Pf the foreign price index.
EXr =EX ∗ P
Pf
(9)
The percentage change of the real exchange rate is computed as follows:
∆EXr
EXr
=∆EX
EX+
∆P
P− ∆Pf
Pf
(10)
An international term structure model is used by Inci and Lu (2004) to explain exchange
rates. As we have already discussed in the previous sections, the term structure includes
a significant amount of information about the market’s expectation of future inflation
17
and economic growth. According to the authors, the uncovered interest rate parity
relationship is too simplistic in order to describe the effects between short- and long-
term interest rates. Their model which combines the yield for bonds of eight different
maturities for up to 5 years denominated in US$ /DEM and exchange rate data for
the US$/DEM pair is used to conduct an in-sample forecast (from January 1974 to
December 1998). Their best-performing contemporaneous model has a mean of 0.5053
$/DEM compared to the actual value of 0.5055 $/DEM and a volatility of 0.1072%
compared to the actual volatility of 0.1095% and one can therefore conclude that the
empirical model very well explains currency movements.
Byeon and Ogaki (1999) examine the empirical link between exchange rates and the
yield curve. They regress the log of real exchange rates for a number of developed
countries (April 1973 - April 1995) on the short-term real interest rate differential and
the long-term real interest rate differential. If the equation includes both differentials,
the coefficient for the long-term interest rate differential is negative for all the examined
exchange rates. However, the coefficient of the short-term real interest rate differential is
typically positive for the majority of the investigated exchange rates. The model suggest
that the relationship between short- and long-term interest rates as well as exchanger
rates is of a very complex nature. Another empirical result of the study is that the effect
of monetary policy actions on the exchange rate mainly depends on how the long-term
rate reacts to this change. If the central bank raises interest rates and the long end
shifts upwards as well, the domestic currency appreciates. In case that the long-term
rate moves sideways, the higher short-term rate will cause the domestic currency to
depreciate. An interesting exercise is conducted by Clostermann and Schnatz (2000)
who are using a ”synthetic”euro exchange rate to explore the determinants of the real
effective euro exchange rate. Using quarterly data from 1975:1 to 1998:4, they find out
that real interest rate differential is significant when they estimate a single equation
error correction model (SEECM). According to their regression equation, a rise of 1% in
the short-term in the real interest rate differential results in a 1.3% increase in the real
effective exchange rate.
3.4 Budget deficit
The impact of fiscal policy on interest rates is a widely debated issue. A number of
theoretical approaches attempt to explain why the fiscal deficit might have an impact
on interest rates (Correia-Nunes and Stemitsiotis, 1995). According to an often uttered
18
thesis, an increase of the government deficit might lead to a rise of real interest rates.
Higher interest rates will cause a weaker consumer demand and faltering businesses. A
high level of public debt will also trigger inflationary pressure and hence, the central
bank will raise the short-term rate. As market participants would expect a continuous
tightening of the monetary policy, long-term rates will go upwards in line with the short
end of the curve.
The loanable fund model assumes the existence of a market where consumer demand
(household’s savings S) and demand for funds meet (D fiscal deficit, Z private invest-
ment) meet, hence an increase of the deficit leads to lower investments and higher savings.
Public debt issues could also influence the risk premium which investors are willing to
pay for long-term bonds instead of short-term paper (Decoudre, 2005).
Correia-Nunes and Stemitsiotis (1995) empirically analyse the relationship between nom-
inal and real long-term interest rates and budget deficits. They concentrate on long-term
rates because these are a key determinant of capital accumulation and play a central
role in the transmission of macroeconomic policies to the economy. In their study, they
investigate annual data from 1970 to 1993 for 10 OECD countries including Germany.
In order to estimate the nominal long-term rates iL, the real short-term rate rs, the
budget to GDP ratio d and the expected inflation rate πe is taken.
iL = α0 + α1rs + α2πe + α3d + u (11)
The employed equation leads to a R2 which is above 80% in every country. The coefficient
for the budget to GDP ratio is significant in all countries with the theoretically predicted
positive sign.
Secondly, they test whether the long-term interest rates might not be only influenced
by the current fiscal deficit but also by the stock of accumulated public debt which
reflects the supply of sovereign bonds in the market. In this model, investors will ask for
an additional risk premium, if the country’s debt relative to income is high. However,
adding this ratio to the equation does not improve the performance of the estimation.
Decoudre (2005) rebuilds Correia-Nunes and Stemitsiotis’ model with the same data.
He criticises that the fiscal surplus data is not cyclically-adjusted and includes interest
payments on debt. Repeating the estimation with the identical data but the cyclically
adjusted primary balance, Germany’s R2 goes down from 0.84 to 0.55 and the t-statistic
is not significant anymore. Due to the result of the estimation, Decoudre concludes that
Correia-Nunes and Stemitsiotis’ analysis can not be regarded as statistical and robust.
19
This brief overview shows how debatable it is whether the public deficit influence interest
rates or not but it seems reasonable to include the variable in the subsequent following
model.
3.5 Oil price
At first glance, the likely relationship between the yield curve and the oil price may
not be obvious but there are indications that high energy prices, as a major input to
production, are causing recessions and lower economic growth. Consequently, oil price
shocks which lead to economic depression and excessive inflation would also have an
impact on monetary policy. If one looks closer at the statistical relationship between
political events in the Middle East and recession in the US which were caused by an oil
price shock, the belief of a connection is supported. However, the lag between crisis in
the oil-producing countries and recession in the US is quite stretched. There also seems
to be some empirical evidence for a link between the oil price, productivity and inflation
(Barsky and Kilian, 2004).
It has to be pointed out that the linkages between energy and economic aggregates are
of a complex nature. Firstly, a rise of energy prices can have an effect on the aggregate
production function. Demand may also alter due to an oil price increase. In addition,
an upward shift of energy prices will usually cause a transfer of income from the oil
importing countries to the oil exporting countries and the income decrease will trigger a
lessening of consumer spending which will reduce aggregate demand and output. Lastly,
the energy price increase might also lead to other actions, for instance, monetary policy
could be forced to tighten and would lower real economic activity (Bjørnland, 2000).
Although the majority of the research in this area deals with the US economy, there are
a few studies which undertake multi-country studies. Carrying out multivariate vector
autoregressions, Jimenez-Rodrıguez and Sanchez (2004) explore the effects of oil price
shocks on the real economic activity in the main industrialised OECD countries (period
1972:3 - 2000:4) and estimate linear/non-linear models. The Granger-causality test show
that the different oil prices variables are not statistically significant at a 5% critical level
for most of the countries if the real GDP of period is estimated with lagged values of
real GDP, real effective exchange rate (REER), real oil price, real wage, inflation, short
and long-term interest rates. Furthermore, they observe the response to oil price shocks
in different countries and find out that, in the case of positive movements in oil prices,
there is a similar pattern reaction among the net oil importing countries and the largest
20
negative short-run effect occurs within a year. As one would expect, the strongest re-
sponse takes place in the US (the accumulated loss GDP growth is 3.2%) , whereas
Germany’s economy only reacts half as strong as the US with 1.6%.
In a recent study of Schmidt and Zimmermann (2005), the effects of oil price shocks
are especially analysed for the German economy. Since 1970, the openness of the Ger-
man economy (an indication is the ratio of the im- and export divided by GDP) has
more than doubled. During the same period, the usage of energy in percent of GDP
has halved. In order to take these developments into account, the year 1986, where
the oil price dropped from 40$ to 20$ due to a production increase, divides the sample
into two subperiods. Comparing the two sample periods, their main finding is that oil
price shocks only contribute a declining and limited extent to business cycle fluctuations
because the importance of the energy use has been decreasing.
The previously presented research dealt with the impact of the oil price on GDP growth
but energy prices also influence inflation. LeBlanc and Chinn (2004) conduct an empir-
ical investigation about the effects of oil price changes on inflation in the G-5 countries.
There are two main reason why it is the common belief that European countries might
respond more to oil price increases: The European countries tend to have stronger labour
unions which are more likely to enforce wage increases due to rising inflation and since
product market competition is not as concentrated as in the US, European producers
will possibly pass along the higher wages to consumers. This wage-price spiral does
usually not occur in the US where workers are more likely to absorb through higher fuel
prices and increases in other energy-related products.
Using a short-run Phillips curve approach, the authors estimate an equation which has
inflation on the left hand side and lags of inflation, unemployment rate, interest rates
and the percentage change in nominal oil prices on the explanatory side of the equation.
Like Schmidt and Zimmermann (2005), they also find out that the one-year inflationary
impact of higher oil price has declined when they estimate the coefficients for the period
from 1980:1 to 1990:4 and extend the sample period by one observation each time up to
2001:4. In the following step, the whole sample is compared to the other G-5 countries.
In Germany, the oil price has only a very small impact. A 10% increase in the oil price
leads only to a maximum inflationary effect of 0.5% depending on the estimation model,
whereas the effect on US inflation is only slightly stronger with 0.8%. The US, the UK
and Japan show generally stronger responses to oil price increases than Germany and
France. As a conclusion, one can say that inflation in the G-5 countries is not very
sensitive to oil prices and that there are no significant differences in the relationship
21
between the US and other G-5 countries. The previously mentioned effects for Europe
might be offset by the higher energy sensitivity of the US economy.
Having looked at different empirical studies, it seems as if the impact of the oil price
on the GDP growth and inflation has diminished in the past decades, but was more
important in the 70’s.
3.6 External interest rates
In a world, where financial markets are highly integrated and capital can be moved within
seconds, it is sensible to also include external interest rates. In this thesis, the American
and Japanese interest rates will be included because they are the two other major bond
markets. The following equation (see Chinn and Frankel, 2005) shows the theoretical
linkage between the difference in nominal interest rates between two economies on assets
of equal maturity and default risk, when expressed in common currency terms.
(ikt − ik∗t) = [ikt − ik∗t − (ft,t+k − EXt)] + (ft,t+k − EXet+k) + (EXe
t+k − EXt) (12)
The term in square brackets is called covered interest rate differential (ft,t+k is the
forward exchange rate from period t to t + k and EXt is the exchange in period t),
(ft,t+k −EXet+k) is the exchange rate risk premium and (EXe
t+k −EXt) is the expected
appreciation/depreciation of the exchange rate. When both of these terms turn zero,
the interest rate differential equals the expected depreciation.
Chinn and Frankel (2005) analyse how US and European interest rates have influenced
each other in the past decades. For money market rates, they find evidence that the
US real rates have a greater impact on the German real rate than the reverse (period
1973-2004) and this pattern does not change with the inception of the EMU. In terms of
real long-term rates, a vector error correction equation over the identical sample period
is formulated to investigate the causality relations. Up to 1995, US rates did not re-
spond to Europeans rates at all but the opposite was frequently the case. In the second
subperiod from 1996 on, this relationship changes and it is observable that sides react
to the movement of the counterpart.
Engsted and Tanggard (2005) use monthly US and German long-term government bond
yields (from 1975 to 2003) to implement a VAR-based variance decomposition of bond
returns. Their empirical study confirms the results of Chinn and Frankel: US real inter-
est rates have a stronger predictive power for German real rates and excess bond return
than vice-versa.
22
A vector-autoregressive framework is used by Wang et al. (2004) to examine linkages
between government bond markets (indices) of five industrialized countries (US, Japan,
Germany, UK and Canada) during the period of January 1986 to December 2000. Firstly,
a cointegration test shows no evidence of a long-run relationship during most of the
sample period. However, e.g. Smith (2002) supplies evidence for the existence of coin-
tegration among bond markets for a similar period and countries. We can therefore
conclude that the outcome of these test highly depends on the specification of the data
set. Multivariate Granger causality tests are used to study the short-term dynamics
among the markets. The results at the 10% significance level show a clear dominance of
the US, whereby German rates may be influenced by the US and Japan. However, the
5% significance level does merely indicate an influence of the US on Germany, therefore
it appears that the linkages are generally quite weak. After composing a directed graph
based on likelihood ratio tests, variance decompositions are applied in order to quantify
the economic significance of economic variables. According to the results, 74% of the
movements in the German bond market are caused by domestic factors. The remaining
share is originated by Japan (about 21%) and the US (about 5%)
In an earlier work, Clare and Lekkos (2000) undertake a decomposition of the covari-
ation between long-term bonds. In their sample period (August 1991 - October 1999,
weekly data), on average 70% of the German yield curve slope variation is determined
by German factors. The covariance between the German and US yield slope seems to
be mainly driven by macroeconomic fundamentals rather than by risk premia.
Having examined different research on the correlation of bond markets, one cannot deny
that there are linkages among them. Thus, it is appropriate to estimate models with
external rates as well as models which only consist of domestic variables.
23
4 Data
This chapter introduces the time series data used. Time series are taken from different
sources (Deutsche Bundesbank, Federal Statistical Office, Ifo and International Financial
Statistics). If not stated otherwise, data from 1992 on is for the whole federal territory
of Germany and earlier data only covers the former West Germany.
The study uses quarterly data because the object of the study is to look at possible long-
run relationships between the yield curve and macroeconomic fundamentals. Certain
data sets are published in a monthly interval, therefore the average of three months is
taken in order to obtain quarterly data. In case that only yearly data is available, the
weighted average is used for the first, second and third quarter, while the fourth quarter
of the year contains the actual annual figure.
As the official time series for the yield data of the Deutsche Bundesbank begins at the
end of 1972 and the collapse of the Bretton-Woods system of fixed exchange rates also
falls in the same period, the analysed period starts in the first quarter of 1973 (including
131 observations until 2005Q3).
4.1 Interest rates
In order to investigate the impact of macroeconomic data on the term structure, one
needs the financial data derived from listed German Federal securities. Figure 1 plots the
monthly average of 3-months (mths3), money market rate reported by Frankfurt (Time
series key: su0107), 1-year (yr1), 5-years (yrs5) and 10-years (yrs10) yield of the term
structure on listed Federal securities (Time series key: wz9808, wz9816 and wz9826).
Figure 2 shows the slope of the yield curve, i.e. the difference between the 10-years
yield and 1-year yield (slope1y10y)as well as the difference between the 10-years and the
3-months yield (slope3m10y). All this data is extracted from the time series database
of the Deutsche Bundesbank. The key descriptional statistics are presented in Table 1.
24
Figure 1: 3-months, 1-year, 5-years and 10-years yield
Figure 2: Slope 10 years - 1 year and Slope 10 years - 3months
25
slope3m10y slope1y10y mths3 yr1 yrs5 yrs10
Mean 1.140 1.198 5.762 5.705 6.560 6.997Maximum 4.610 3.680 14.370 12.660 11.200 11.210Minimum -4.750 -2.280 2.040 2.050 2.710 3.240Std. Dev. 1.722 1.180 2.855 2.448 1.997 1.743Skewness -0.806 -.0517 0.911 0.581 0.076 -0.015Kurtosis 3.726 2.738 3.084 2.519 2.187 2.391
Table 1: Descriptive statistics of the yield data
mths3 yr1 yrs5
mths3 1 0.936 0.770yr1 0.936 1 0.894yrs5 0.770 0.894 1yrs10 0.675 0.795 0.973
slope3m10y 0.657 0.469 0.190slope1y10y 0.595 0.538 0.228
yrs10 slope3m10y slope1y10y
mths3 0.675 0.657 0.595yr1 0.795 0.469 0.538yrs5 0.973 0.190 0.228yrs10 1 0.107 0.117
slope3m10y 0.107 1 0.846slope1y10y 0.117 0.846 1
Table 2: Adjusted R-squared of the yield variables
4.2 Consumer price index
The consumer price index (cpi) is computed monthly by the Federal Statistical Office of
Germany and was published in November 2005 (see figure 3).
26
Figure 3: CPI
4.3 GDP
Quarterly real GDP growth (price-adjusted to 1995) is also provided by the Federal
Statistical Office (see figure 4).
27
Figure 4: GDP
4.4 Ifo business climate index
As a monthly indicator for the economic situation, the Ifo (Institute for Economic Re-
search) in Munich publishes the business climate indices. As the business climate index
is only available from 1980Q1 on, it will be approximated by the mean of the business
situation and expectations index between 1973Q1 and 1979Q4. From 1991Q1 on, the
index covers the reunified Germany.
28
Figure 5: Ifo business climate index
4.5 Unemployment rate
As a proxy for the current economic situation, the unemployment rate can be useful.
The seasonally-adjusted time series is obtained from the Federal Statistical Office (Time
series code: UACC02).
29
Figure 6: Unemployment rate
4.6 Exchange rate
In order to examine the influence of the German exchange rate, nominal and real ex-
change rates are going into the model (see Chapter 3.3). As the US$ is the lead-
ing world currency, the DEM/EUR US$ exchange rate is used in the data set (Code
DEM:134..RF.ZF... and EUR: 134..RF.ZF...). The real effective exchange rate includes
the price-adjusted exchange rate of other major currencies and the weighted average of
the different real exchange is calculated by using the value of trade with the respective
countries (Code: 134..REUZF..., Start: 1975:1). Both time series (see figure 7 and table
3) are provided by the International Financial Statistics.
30
Nominal exchange rate Real effective exchange rate
Mean 1.031855 92.27707Maximum 1.665 122.21Minimum 0.714 68.45Std. Dev. 0.218682 14.76788Skewness 0.69082 0.069216Kurtosis 2.612219 1.89151
Table 3: Descriptive statistics of exchange rate data
Figure 7: Nominal exchange rate and real effective exchange rate
4.7 Budget deficit
Published by the Federal Statistical Office, the new indebtness covers the increase of
public debt per year including subsovereign debt. As the budget deficit is only available
on a year basis, quarterly data is obtained by dividing the annual movement by four.
31
Figure 8: Budget Deficit
4.8 Oil price
The UK brent crude oil price in Euro is extracted from the ZIS database of the Deutsche
Bundesbank. One can clearly recognise the two major oil crises starting at the end of
1973 and at 1979.
32
Figure 9: Oil price per Barrel in Euro
4.9 External interest rates
Lastly, short and long-term rates from Japan and the US are used to estimate the
German yields. These time series are also taken from the IFS database. For the US,
treasury bill rates (IFS code: 11160C..ZF...) and 10 years government bond yields
(IFS code: 11161...ZF...) are taken. Call money rates (IFS code: 15860B..ZF...) and
government bond yields (IFS code: 15861...ZF...) provide the basis for the Japanese
data.
33
uslong usshort japanlong japanshort
Mean 7.786 6.137 5.033 4.359Maximum 14.847 15.087 9.533 13.037Minimum 3.620 0.916 0.653 0.001Std. Dev. 2.554 2.992 2.838 3.594Skewness 0.753 0.700 -0.024 0.418Kurtosis 3.145 3.741 1.581 2.443
Table 4: Descriptive statistics of the external interest rates data
Figure 10: Short- and long-term interest rates for Japan and the US
34
5 Methodology
Having introduced the data series in the previous chapter, the next step is a deeper
analysis of the time series properties. With these results, the subsequent section can
undertake the actual empirical analysis. For all the empirical test and calculations,
EViews 5.1 is used.
5.1 Stationarity
The concept of stationarity is vital in time series analysis because it can strongly
influence its properties. In order to be weakly or covariance stationary, a time series
must satisfy the following conditions (Verbeek, 2004).
E(yt) = µ < ∞
V ar(yt) = σ2 < ∞
cov(yt, yt−k) = E{(yt − µ)(yt − k)} = γk, k = 1, 2, 3, ...
Weak stationarity implies that a process has a constant mean, a constant variance and
a constant autocovariance structure. Strong stationarity would require that the joint
distribution of a time series is invariant under time shift which is almost impossible to
prove empirically (Tsay, 2002).
The use of non-stationary data leads to several problems. Time series with a unit root
might trend over time. Therefore, the regression of two unrelated time series might have
a high R2. This phenomenon is called spurious regression. In addition, the ’t-ratios’ do
not follow a t-distribution if the input variables of the regression are not stationary. An
unexpected ’shock’ might also cause difficulties when dealing with non-stationary series
(Brooks, 2002).
5.2 Unit root test
Most economic time series exhibit strong trends. A random walk with drift is the sum
of an infinite number of random variables which are being generated by the same zero-
mean, constant-variance distribution. If this is given, the variance of yt would obviously
35
be infinite and the random walk is a non-stationary process (also in case µ equals zero).
yt =∞∑i=0
(µ + εt−i) (13)
yt is said to be integrated of order one, I(1). Taking the first difference produces a
stationary process (Greene, 1997).Generally a non-stationary series is integrated of order
d, I(d), if it becomes stationary after being first differenced d times.
In this master thesis, the augmented Dickey-Fuller (adf) Test is employed which can
accommodate higher-order autoregressive processes in εt.
∆yt = γ∗yt−1 +
p−1∑j=1
φj∆yt−j + εt (14)
where
φj = −p∑
k=j+1
γk
and
γ∗ = (
p∑i=1
γi)− 1
The adf test has the null hypothesis γ∗ = 0 and is rejected if γ∗ < 0. In Equation (14),
the regressions estimates a pure random walk model. By adding µ or βt to the equation,
one can include an intercept or a linear time trend (Enders, 1995).
In table 5, we have computed the adf test statistics for the examined time series. The
second, third and fourth column show the chosen number of lagged difference terms, the
t-statistics and the critical value level to which H0 (Yt has a unit root) can be rejected.
The appropriate lag length is determined by the Schwarz criterion whereby a maximum
of 12 lags is considered. The test were conducted for level and first difference data.
Most of the level data indeed has a unit root. However, the slope and 3-months money
market time series as well as the GDP and the Ifo index data seem to be stationary
according to the adf test. The first difference data is clearly stationary for all data sets
(the high t-Statistic for the budget deficit is due to the artificial generation of quarterly
data out of yearly data).
36
Level DataLag (SIC) t-Statistic Reject H0
yrs10 1 -1.57 noyrs5 1 -1.74 noyr1 1 -2.47 no
mths3 1 -3.74 1%slope3m10y 1 -4.59 1%slope1y10y 1 -3.31 5%
cpi 0 -1.99 nogdp 0 -4.67 1%ifo 1 -4.37 1%ue 2 -1.90 no
fxrate 1 -2.11 norealfx 0 -0.84 nofiscal 9 -2.32 nooil 1 -2.05 no
1st DifferenceLag (SIC) t-Statistic Reject H0
yrs10 0 -8.02 1%yrs5 0 -8.21 1%yr1 0 -8.16 1%
mths3 0 -7.78 1%slope3m10y 0 -8.12 1%slope1y10y 0 -8.33 1%
cpi 0 -10.76 1%gdp 0 -13.60 1%ifo 0 -6.13 1%ue 1 -4.53 1%
fxrate 0 -8.70 1%realfx 0 -9.80 1%fiscal 8 -2.68 10%oil 0 -9.90 1%
Table 5: Adf test statistics
37
5.3 Multicollinearity
In case of multicollinearity, the explanatory variables are intercorrelated which means
that they have an either exact or approximately exact linear relationship. Since this
phenomenon refers to the condition of the explanatory variables, it is not possible to
test for multicollenearity but only to measure the degree of it. The following rules of
the thumb can help detecting the strength (Gujarati, 1995).
• High R2 but few significant t ratios
• High pair-wise correlation among regressors
• Examination of partial correlations
• Auxiliary regressions
Table 6 shows the pair-wise correlation of all regressors. According to Gujarati, the
correlation coefficient should be in excess of 0.8 in order to have serious problems with
multicollinearity which does not apply in this sample.
38
dcpi dgdp difo due dfiscal dfxrate drealfx
dcpi 1.000 -0.086 -0.121 -0.137 -0.024 0.050 0.026dgdp -0.086 1.000 0.355 -0.051 0.013 0.012 0.030difo -0.121 0.355 1.000 -0.148 -0.098 0.025 -0.088due -0.137 -0.051 -0.148 1.000 0.037 0.078 -0.109
dfiscal -0.024 0.013 -0.098 0.037 1.000 -0.134 0.220dfxrate 0.050 0.012 0.025 0.078 -0.134 1.000 -0.496drealfx 0.026 0.030 -0.088 -0.109 0.220 -0.496 1.000
doil 0.051 -0.009 0.032 -0.030 -0.013 0.396 -0.283djapanlong 0.069 0.101 0.030 -0.128 -0.071 0.167 -0.098djapanshort 0.148 -0.225 -0.169 -0.154 -0.087 -0.073 0.056djapanslope -0.121 0.317 0.204 0.084 0.047 0.193 -0.124
duslong 0.109 0.037 0.099 -0.044 -0.068 0.208 -0.304dusshort -0.019 0.054 0.134 -0.136 -0.022 0.055 -0.199dusslope 0.122 -0.041 -0.094 0.145 -0.030 0.109 -0.008
doil djapanl. djapansh. djapansl. dusl. dussh. dussl.
dcpi 0.051 0.069 0.148 -0.121 0.109 -0.019 0.122dgdp -0.009 0.101 -0.225 0.317 0.037 0.054 -0.041difo 0.032 0.030 -0.169 0.204 0.099 0.134 -0.094due -0.030 -0.128 -0.154 0.084 -0.044 -0.136 0.145
dfiscal -0.013 -0.071 -0.087 0.047 -0.068 -0.022 -0.030dfxrate 0.396 0.167 -0.073 0.193 0.208 0.055 0.109drealfx -0.283 -0.098 0.056 -0.124 -0.304 -0.199 -0.008
doil 1.000 0.416 0.219 0.043 0.299 0.241 -0.062djapanlong 0.416 1.000 0.446 0.191 0.377 0.216 0.040djapanshort 0.219 0.446 1.000 -0.793 -0.029 -0.086 0.091djapanslope 0.043 0.191 -0.793 1.000 0.289 0.241 -0.070
duslong 0.299 0.377 -0.029 0.289 1.000 0.670 -0.025dusshort 0.241 0.216 -0.086 0.241 0.670 1.000 -0.759dusslope -0.062 0.040 0.091 -0.070 -0.025 -0.759 1.000
Table 6: Correlation table of the explanatory variables
5.4 Granger causality
To determine the direction of causality between two related stationary variables, Granger
(1969) suggests a testable definition of causality. The test basically assumes that the
information relevant to the forecast of the pair of variables is solely contained in the time
series data on these variables (Gujarati, 1995). We can say that y does not Granger-
39
cause x if for all s > 0 the mean squared error of a forecast of xs+t based on (xt, xt−1, ...)
is the same as the mean squared error of a forecast of xt+s that uses both (xt, xt−1, ...)
and (yt, yt−1, ...). However, the existence of ”Granger causality”does not imply that y
is the result or the result of x. In case of non-stationary and cointegrated variables, a
error correction model can be used to determine the Granger causality (see Koop, 2000).
Gujarati (1995) distinguishes three cases of causality:
1. Unidirectional causality exists if the set of estimated coefficients indicates that
just one of the variables has lagged values which include statistically significant
information when the other variable is dependent in the regression equation.
2. Feedback or bilateral causality occurs if both variables are statistically significantly
different from zero in both regression equations.
3. Independence describes the case when the other variable can not help improving
the forecast of the dependent series.
Table 7 shows all the results of the pairwise granger causality tests with lags 2 and 4. For
each macroeconomic variable, there are two rows of results. The first of these two rows
shows the probability according to the F- statistic for the null hypothesis ”macroeco-
nomic variable does not (Granger-)cause the yield variable”. The second row includes
the probability for the null ”yield variable does not (Granger-)cause the macroeconomic
variable”.
The output of the tests provides some evidence about possible causality relationships.
There is strong statistical evidence that all yield variables are (Granger-)causing dcpi.
This would suggest that there is indeed some kind of fisher effect (see section in Germany
3.1). Additionally, the relationship between the yield variables and dcpi is unidirectional
from the yield variables to dcpi. The tests indicate a feedback relationship from the yield
variables to the gdp and ifo time series. Interestingly, only the short end of the curve
and the slope variables seems to have a bilateral relationship with the due series. As one
would expect, the lagged duslong time series includes statistical significant information
for all the yield variables if they are dependent in the regression equation. dfiscal, dfxrate,
drealfx, doil and djapanlong do not contain any statistical significant information.
To summarize, duslong is the only macroeconomic variable which clearly
(Granger-)causes the term structure. In the other direction, yield variables seem to
have an impact on dcpi, ifo and gdp.
40
dyrs
10dyrs
5dyr1
mth
s3sl
ope1
0yrs
1yr
slop
e10y
rs3m
ths
Lag
s2
42
42
42
42
42
4
yie
ld9
dcp
i0.
099
0.19
30.
003
0.02
50.
000
0.00
00.
016
0.00
00.
032
0.00
20.
016
0.01
2dcp
i9
yie
ld0.
629
0.78
60.
588
0.55
80.
865
0.81
60.
538
0.57
80.
624
0.29
50.
175
0.36
4yie
ld9
gdp
0.09
60.
022
0.01
60.
004
0.00
50.
001
0.00
40.
034
0.00
50.
168
0.00
20.
084
gdp
9yie
ld0.
484
0.74
90.
448
0.77
90.
019
0.05
20.
010
0.00
50.
003
0.00
30.
007
0.00
9yie
ld9
ifo
0.11
00.
066
0.33
30.
017
0.15
90.
043
0.00
20.
263
0.06
20.
410
0.00
80.
449
ifo
9yie
ld0.
479
0.08
90.
151
0.06
30.
007
0.01
70.
000
0.00
30.
007
0.03
50.
002
0.00
5yie
ld9
due
0.29
60.
435
0.24
00.
438
0.53
20.
590
0.00
30.
025
0.03
70.
181
0.01
30.
060
due
9yie
ld0.
865
0.37
80.
638
0.23
20.
123
0.08
00.
159
0.20
20.
110
0.09
90.
083
0.14
7yie
ld9
dfisc
al0.
700
0.11
70.
633
0.90
50.
913
0.90
70.
539
0.54
30.
355
0.14
30.
535
0.26
1dfisc
al9
yie
ld0.
793
0.97
10.
633
0.90
50.
340
0.67
00.
915
0.98
00.
641
0.92
40.
982
0.99
9yie
ld9
dfx
rate
0.61
00.
659
0.85
80.
755
0.33
30.
682
0.40
40.
864
0.60
10.
807
0.34
80.
939
dfxra
te9
yie
ld0.
779
0.48
70.
858
0.75
50.
381
0.88
20.
670
0.93
40.
611
0.72
60.
451
0.67
9yie
ld9
dre
alfx
0.51
40.
505
0.35
50.
515
0.56
00.
700
0.28
10.
255
0.22
30.
465
0.42
70.
452
dre
alfx
9yie
ld0.
939
0.95
30.
355
0.51
50.
313
0.50
50.
060
0.18
90.
134
0.34
00.
041
0.09
6yie
ld9
doi
l0.
485
0.10
40.
942
0.41
20.
613
0.29
30.
799
0.39
70.
805
0.87
90.
963
0.26
7doi
l9
yie
ld0.
910
0.74
80.
942
0.41
20.
945
0.66
60.
264
0.69
40.
529
0.78
50.
074
0.22
1yie
ld9
dja
pan
long
0.32
30.
034
0.97
80.
526
0.63
10.
778
0.48
40.
686
0.98
30.
994
0.50
40.
998
dja
pan
long
9yie
ld0.
925
0.64
20.
978
0.52
60.
632
0.31
70.
709
0.30
90.
618
0.27
50.
533
0.21
9yie
ld9
dusl
ong
0.23
00.
307
0.30
80.
609
0.52
00.
610
0.99
70.
875
0.98
70.
958
0.91
00.
956
dusl
ong
9yie
ld0.
045
0.12
50.
103
0.32
00.
031
0.20
20.
010
0.06
70.
070
0.31
90.
035
0.08
4
Tab
le7:
Gra
nge
rpai
rwis
eca
usa
lity
test
s
41
6 Empirical analysis
This chapter is the main part of this thesis and covers the empirical analysis of the re-
lationship between the yield curve and macroeconomic variables. Firstly, autoregressive
moving-average models are estimated. In the second step, a conventional OLS regression
is applied in order to find out which variables drive the yield curves. The fusion of the
uni- and multivariate model is implemented with the VARMA model. Furthermore, the
forecasting quality of the autoregressive models is backtested. In order to investigate
whether the explanatory power of different macroeconomic variables changes over time,
a ’moving’ estimation is used. The ’moving’ estimation computes the coefficients for a
10 year period and will move from the oldest available time frame to the most recent
one. In the last section, (G)ARCH models are employed.
6.1 Autoregressive Moving Average Processes
Univariate models are basically attempting to model and to predict data using only
information contained in their own past values and possibly current and past values of
an error term (Koop, 2000). Autoregressive models depend only upon the values that
the variable took in previous periods plus an error term. An AR(p) model with p lags
can be formulated as
yt = µ + β1yt−1 + β + ... + βpyt−p + εt
where εt is white noise which is independently and identically distributed (iid) with
E(ut) = 0 and V ar(ut) = σ2 (see Brooks, 2002). The moving average process MA(q) is
a qth order moving average mode.
yt = µ + εt + θ1εt−1 + ... + θqεt−q
The term is essentially a linear combination of white noise processes, so that yt depends
on the current and previous values of ε.
Extending the model with further explanatory variables, leads to a vector autoregressive
moving average (VARMA) process (see Lutkepohl, 1993). Thus, the model for yt with p
autoregressive terms, q moving average terms and r explanatory variables (in addition
to the autoregressive term) has the following shape (this equation is just for the first
42
explanatory variable).
yt = µ+β1yt−1+ ...+βpyt−p+β1,1x1,t−1+ ...+β1,px1,t−p+ ...+βr,1xr,t−1+ ...+βr,pxr,t−p+εt
and
εt = ut + θ1ut−1 + ... + θqut−q
Although univariate ARMA with small values of p or q seems to be quite simple at first
glance, they can sometimes produce better out-of-sample forecasts than large macro-
econometric models (see Hamilton, 1994).
6.2 The Box-Jenkins Methodology
The Box-Jenkins methodology is a procedure to estimate an ARMA in a systematic
manner and it basically includes three or four steps:
1. Identification
2. Estimation
3. Diagnostic checking
4. Forecasting (see section 6.5)
Identification involves determining the order of the model required which means deter-
mining the appropriate values for p, q and d. This is done by using graphical procedures
which are called correlogram. The correlograms are the result of the autocorrelation
function (acf) and the partial autocorrelation function (pcf). The autocorrelation func-
tion is computed out of the sample covariance at lag k, γk, and the sample variance,
γ0
γk =
∑(yt − y)(yt+k − y)
nand γ0 =
∑(yt − y)2
n(15)
where n is the sample size and y is the sample mean. With γk and γ0, we can define the
sample autocorrelation function at lag k which is the ratio of the sample covariance to
sample variance.
ρk =γk
γ0
(16)
43
Similarly, the partial autocorrelation funcion ρkk measures the correlation between an
observation k lags ago and the current observation, after removing the effects of observa-
tions at intermediate periods (i.e. from yt+1 to yt−k+1). However, it is mostly difficult to
use graphical plots in order to identify the appropriate model, because the correlograms
of real data do barely exhibit the typical patterns. Therefore, information criteria are
used to identify the correct lag selection. In figure 11, the correlograms for dyrs10, dyr1
and slope10y1y are plotted . The first difference of the data has to be taken for the model
because stationarity is desirable in ARMA models. In chapter 5.2, the null hypothesis
of the adf test is rejected for all time series i.e. the data is stationary. Non-stationary
coefficients have the property that previous values of the error term keeps on effecting
the current value of yt as time progresses. As expected, there is no distinct pattern in
the diagrams and it seems as only the 1st lag is statistically significant. Interpreting
the correlogram would thus lead to an AR(1) model without a moving average term.
However, the explanatory power of the correlogram seems to be limited.
Hence in the second step estimation, different models are estimated and computed so
that the correlation and information criterion can be compared. Every combination
with up to AR(5) and MA(5) processes is estimated which leads to 35 different options.
In table 8, the results of the estimation are displayed with the adjusted R2 and the
Schwarz information criterion (SIC) for every model with up to AR(2) and MA(2)
terms (the Akaike criterion is also computed but not included in the table, it will also
not be discussed further due to its preference for models with a higher number of
parameters). All the models are estimated without a constant because the t-statistic of
the constant parameter does not indicate statistical significance and the coefficients are
close to zero. A R2 with an asterisk indicates that the model has the highest correlation
for the chosen time series. The other asterisk marks the model with the lowest figure
for the information criteria (see Gujarati, 1995).
Only in 2 out of 6 cases, the model with the highest R2 has also the lowest information
criterion but the values of the SIC are very close to each other for every time series
wherefore it is reasonable to use the adjusted R2 as an information criterion. In 5
out of the 6 models, the highest R2 is obtained if either only AR terms or only MA
terms are employed in the equation. If the estimations with up to five autoregressive
and five moving-average terms are also taken into account, the results slightly change.
For dyrs10, the ARMA(0,1) still has the lowest SIC, but the highest R2 (0.232) can be
achieved with (4,3) combination. This is similar in the case of dyrs5. The ARMA(0,1)
44
has the lowest SIC, whereas the highest R2(0.193) is an ARMA(4,5)-model. An increased
number of parameters improves significantly the adjusted R2 of dyr1. With (5,5) terms,
the correlation goes up to 0.315 compared to 0.203 in the (0,2)-model. The R2 improves
only marginally by 1.5%, but, as opposed to the previous models where the lowest
SIC only included a low number of ARMA terms, the SIC dmths3 is the lowest for an
ARMA(4,3). Lastly, the slope time series dslope1y10y and dslope3m10y are tested. In
both cases, higher order models outperform the results of table 8 in terms of correlation
and the SIC. For dslope1y10y, the R2 doubles if an ARMA(4,4)-model is employed and
the lowest SIC is obtained with three AR and two MA terms. The highest R2(0.329)
for dslope3m10y is attained with a (3,3)ARMA-model and the SIC is minimised with
an ARMA(4,3) (for the higher lag order results see also table 13). In conclusion, we can
say that best models in terms of the information criterion can have a high as well as
a low number of parameters depending on the series although the SIC is the criterion
which has a stiff penalty term Brooks (2002).
The results indicate that the lower the maturity of the term structure, the higher is
the fit of the model in terms of R2 (merely dyrs5 does not confirm this trend). This
observation contrasts with the estimation in chapter 6.3, where the outcome was exactly
in the opposite if one looks at the correlation. It leads to the possible conclusion that
models for higher maturity paper should preferably be multivariate, whereas models for
dyr1 and dmths3 better perform if they are univariate.
After the estimation of the parameters, the model has to be validated and checked with
diagnostic checking. Diagonistic checking essentially means testing for autocorrelation.
If the residuals are white noise, the particular fit can be accepted. The Ljung-Box (LB)
statistic is a simple test in order to test the joint hypothesis that all m of the correlation
coefficients ρk are simultaneously equal to zero. Basically, it validates the linear inde-
pendence in time series. The LB statistic is approximately chi-square distributed (see
Hamilton, 1994).
LB = T (T + 2)m∑
k=1
ρ2k
T − k∼ χ2
m (17)
The computation of the LB statistics for all the models which are investigated in table
8 are displayed in table 9, additionally the Q-statistics for an ARMA(3,3),(4,4) and
(5,5) are computed. As the series represents the residuals from ARIMA estimation, the
appropriate degrees of freedom are adjusted to represent the number of autocorrelations
less the number of AR and MA terms previously estimated. For each model, we have
45
chosen a set of two lags, one shorter lag and one longer lag. According to the Q-
statistics, the joint null hypothesis of the autocorrelation coefficients being jointly zero
can be rejected for all the analysed residuals, as the p-value of the Q-statistic is not
close to one in any case. However, there is also barely clear evidence for strong serial
correlation, i.e. p-values which are zero, and it seems as there is no possibility to avoid
autocorrelation in these ARIMA estimations. Adding more ARMA parameters to the
equation does not lower the LB statistics. Taking the 2nd difference of the time series
does lower the information content and autocorrelation is still present.
dyrs10 dyrs5 dyr1AR MA Adj. R2 SIC Adj. R2 SIC Adj. R2 SIC
1 0 0.112 0.878 0.100 1.258 0.154 1.754∗0 1 0.156∗ 0.847∗ 0.129∗ 1.245∗ 0.200 1.8112 0 0.118 0.906 0.097 1.292 0.163 1.7790 2 0.151 0.883 0.124 1.281 0.203∗ 1.8451 1 0.140 0.875 0.112 1.274 0.176 1.7572 1 0.137 0.914 0.113 1.304 0.181 1.7871 2 0.137 0.909 0.108 1.308 0.170 1.7942 2 0.131 0.951 0.113 1.333 0.175 1.824
dmths3 dslope1y10y dslope3m10yAR MA Adj. R2 SIC Adj. R2 SIC Adj. R2 SIC
1 0 0.276 1.905 0.136 1.021∗ 0.178 1.9000 1 0.280 2.112 0.163 1.099 0.196 2.0342 0 0.247 1.906 0.154 1.038 0.159 1.8940 2 0.410∗ 1.941 0.164 1.127 0.255∗ 1.9871 1 0.282 1.927 0.160 1.023 0.176 1.9322 1 0.306 1.854∗ 0.175∗ 1.042 0.251 1.809∗
1 2 0.277 1.963 0.157 1.055 0.223 1.9032 2 0.280 1.920 0.172 1.076 0.233 1.862
Table 8: ARMA Models
46
Figure 11: ACF and PACF (10 lags) for dyrs10, dyr1 and slope10y1y
47
dyrs10 dyrs 5 dyr1AR MA Lag Q-stat. Prob. Q-stat. Prob. Q-stat. Prob.
1 0 2 4.877 0.027 2.833 0.092 2.909 0.08810 18.617 0.029 15.686 0.074 7.798 0.555
0 1 2 0.406 0.524 0.251 0.617 2.355 0.12510 10.270 0.329 10.364 0.322 6.064 0.733
2 0 3 4.249 0.039 3.162 0.075 2.684 0.10110 15.052 0.058 13.319 0.101 6.972 0.540
0 2 3 1.019 0.313 1.141 0.285 2.980 0.08410 8.995 0.343 9.004 0.342 6.463 0.596
1 1 3 0.599 0.439 0.742 0.389 0.910 0.34010 9.561 0.297 8.965 0.345 6.149 0.631
2 1 4 0.360 0.549 0.404 0.525 1.372 0.24110 9.633 0.210 9.234 0.236 4.260 0.749
1 2 4 1.094 0.296 1.191 0.275 1.256 0.26210 9.323 0.230 9.874 0.196 5.985 0.542
2 2 5 6.138 0.013 3.538 0.060 1.679 0.19510 9.171 0.164 6.530 0.367 4.212 0.648
3 3 10 3.698 0.448 7.559 0.109 5.926 0.20520 14.671 0.401 20.616 0.112 17.694 0.221
4 4 10 4.296 0.117 3.450 0.178 5.477 0.06520 14.122 0.293 16.614 0.165 20.467 0.059
5 5 11 7.273 0.007 3.079 0.079 5.211 0.02220 13.859 0.180 12.514 0.252 11.020 0.356
dmths3 dslope1y10y dslope3m10yAR MA Lag Q-stat. Prob. Q-stat. Prob. Q-stat. Prob.
1 0 2 1.241 0.265 2.119 0.146 0.935 0.33410 11.437 0.247 11.611 0.236 8.009 0.533
0 1 2 3.497 0.061 4.278 0.039 0.613 0.43410 17.325 0.044 13.835 0.128 14.530 0.105
2 0 3 3.000 0.083 0.433 0.510 3.153 0.07610 11.579 0.171 12.819 0.118 11.221 0.190
0 2 3 4.765 0.029 2.103 0.147 4.584 0.03210 7.879 0.445 9.810 0.279 9.276 0.320
1 1 3 1.310 0.252 0.177 0.674 1.169 0.28010 10.039 0.262 11.482 0.176 8.015 0.432
2 1 4 7.526 0.006 2.834 0.092 7.163 0.00710 9.056 0.249 10.081 0.184 8.688 0.276
1 2 4 8.768 0.003 0.660 0.416 1.359 0.24410 10.059 0.185 13.687 0.057 3.120 0.874
2 2 5 12.092 0.001 2.183 0.140 12.453 0.00010 12.752 0.047 9.643 0.140 18.113 0.006
3 3 10 7.143 0.129 5.791 0.215 13.098 0.01120 19.093 0.161 15.076 0.373 23.312 0.055
4 4 10 6.2505 0.044 6.896 0.032 7.565 0.02320 17.184 0.143 18.991 0.089 16.795 0.157
5 5 11 3.603 0.058 4.113 0.043 3.4793 0.06220 19.487 0.034 16.816 0.079 9.3607 0.498
Table 9: Q-statistics (Ljung-Box)
6.3 OLS estimation
To continue the analysis, it is useful to compute an ordinary least square estimation for
every yield variable using the macroeconomic variables which we discussed in chapter
3 as explanatory variables. Table 10 provides the coefficients, the t-statistics and the
48
adjusted R2 of the regression. For the external interest rates from Japan and the US,
the appropriate variables are chosen so that they match the regressand. The long-term
yields (dyrs10 and dyrs5 ) are explained with duslong and djapanlong, the short end
(dyr1 and dmths3 ) matches with dusshort and djapanshort and the slope variables are
regressed on the difference of the foreign long and short time series.
Dep. Var. dyrs10 dyrs5 dyr1Expl. Var β t-Stat. β t-Stat. β t-Stat.dcpi 0.002 0.031 0.002 0.031 0.095 1.024dgdp -0.015 -0.794 -0.017 -0.762 0.002 0.053difo 0.028 2.447 0.039 2.824 0.014 0.627due -0.158 -1.767 -0.249 -2.331 -0.491 -2.801drealfx 0.013 1.014 -0.005 -0.360 -0.034 -1.483dfiscal 0.004 1.213 0.002 0.588 0.000 -0.033doil 0.011 1.235 0.014 1.329 0.048 3.122djapan 0.269 4.232 0.356 4.700 0.113 1.485dus 0.311 6.464 0.322 5.612 0.229 3.923Adj. R2 0.520 0.540 0.336SIC 0.460 0.816 1.738
Dep. Var. dmths3 dslope1y10y dslope3m10yExpl. Var β t-Stat. β t-Stat. β t-Stat.dcpi 0.134 1.318 -0.064 -0.873 -0.104 -1.047dgdp -0.012 -0.311 0.004 0.140 0.001 0.034difo -0.024 -0.984 0.015 0.866 0.055 2.381due -0.602 -3.116 0.296 2.230 0.343 1.906drealfx -0.005 -0.184 0.034 1.883 0.003 0.140dfiscal -0.002 -0.296 0.003 0.646 0.005 0.736doil 0.044 2.582 -0.023 -1.986 -0.019 -1.220djapan 0.140 1.677 0.044 0.719 0.189 2.281dus 0.157 2.438 0.147 2.501 0.165 2.073Adj. R2 0.242 0.137 0.137SIC 1.932 1.253 1.860
Table 10: OLS estimations for the yield variables (Period: 1975Q1-2005Q3)
The overview of these results gives a first indication in which direction macroeconomic
data tends to influence yields of different maturities.
As the theory suggest, the inflation rate has a positive influence on yields. The shorter
the maturity, the statistical significance of the price level rises and the βs also increase.
Long-term nominal yields are strongly influenced by expectations about future inflation,
49
however it seems as the current inflation level only has a very small impact on the long
end and is not statistically significant in terms of the t-statistics. Consequently, the
slope of the curve gets tighter when inflation is going up.
There are three variables which characterise economic growth: dgdp, difo and due.
dgpd and difo are going in the same direction, i.e. an increase of the variable implies
economic growth and a diminishing Ifo or gdp points at a recession, whereas the
indication mechanism of the unemployment is working in the opposite way. According
to the t-statistics, the statistical significance of the Ifo index compared to the GDP
growth is higher for every equation and the βs of the GDP are not always indicating
the theorictically assumed sign. The β of the Ifo for the 3-months is negative which is
not the expected sign but for all the other yield parameter it is positive. Surprisingly,
the unemployment rate is a very strong explanatory when it is used for the yield level.
For the curve slope, this variable would be expected to move in the opposite direction
(lower unemployment rate indicates higher growth and therefore a steepening curve,
see chapter 3.2), but it moves in line with the spread between short and long rates.
The effect of drealfx and dfiscal on the term structure is not consistent and not
statistically significant, whereas doil has a positive impact on yield levels and a negative
on the term spread.
The foreign interest rates from Japan and the US have by far the largest coefficients,
which means that their influence on the German yields is extremely strong. If the
Japanese and US long-term rates rise by one percent and all the other explanatory
variables are fixed, dyrs10 gains 0.58%. The βs become smaller as the maturity
decreases. This might hint at the fact that the long end of the cure is more integrated
globally and the short end, which is driven by the independent monetary of each
country or monetary system, is therefore less correlated with foreign interest rates.
The adjusted R2 is decreasing as the maturity gets shorter. Especially for the slope
variable, the correlation is very low.
Butter and Jansen (2004) conduct a similar exercise for the German long-term interest
rate (10-years Bund) over the period from 1982Q1 to 2001Q4. Their results are quite
similar and they also find out that the Japanese and US interest rate have the strongest
impact on long-term rates in Germany. Additionally, their equation contains a long-run
element which is the regression output of the level data. As the additional long-run
coefficient does not improve the adjusted R2 in this estimation, it is not necessary to
include the term.
50
Finally, the variables with the five lowest t-statistics are dropped (see table 11) and
compared to the results of table 10. Table 12 documents which variables are chosen
for the different regressand. The adjusted R2 only decreases significantly for dmths3
by 6%. Therefore, one can conclude that the calculated correlation from the first OLS
estimation is only due to the input of the most significant regressors.
Dep. Var. dyrs10 dyrs5 dyr1Expl. Var β t-Stat. β t-Stat. β t-Stat.dvar1 0.021 2.048 0.032 2.757 -0.528 -3.284dvar2 -0.197 -2.388 -0.271 -2.809 0.043 2.917dvar3 0.306 5.179 0.396 5.703 0.131 1.897dvar4 0.320 6.851 0.359 6.543 0.287 4.991Adj. R2 0.512 0.536 0.336SIC 0.351 0.708 1.721
Dep. Var. dmths3 dslope1y10y dslope3m10yExpl. Var β t-Stat. β t-Stat. β t-Stat.dvar1 -0.702 -3.214 0.291 2.429 0.072 2.875dvar2 0.010 0.490 0.033 1.939 0.516 2.535dvar3 0.158 1.683 -0.023 -1.991 0.107 1.134dvar4 0.262 3.355 0.128 2.275 0.254 2.687Adj. R2 0.183 0.145 0.142SIC 2.335 1.071 2.196
Table 11: OLS estimations with four explanatory variables
dyrs10 dyrs5 dyr1 dmths3 dslope1y10y dslope 3m10ydvar1 difo difo due due due difodvar2 due due doil doil drealfx duedvar3 djapanlong djapanlong djapanshort djapanshort doil djapanslopedvar4 duslong duslong dusshort dusshort dusslope dusslope
Table 12: Explanatory variables
This section should have given an idea how intense the macroeconomy is steering the
term structure and which are the main drivers for different maturities. The following
sections will deepen the analysis.
51
6.4 VARMA modelling
After the estimation of a pure univariate models in section 6.2 and a multivariate model
in section 6.3, we will merge both models for a so called vector autoregressive moving-
average (VARMA) model (see section 6.1). Vector autoregressive models were introduced
by Sims (1980) and have been very popular since then due to their flexibility and the
”a theoritical”approach. In this section, the estimation is not truly a VARMA models
because only the yield and slope variables are dependent in the OLS but it combines
univariate and multivariate parameters such as VARMA models do.
In section 6.2, we selected the best-fitting ARMA model (with up to two and five, respec-
tively, ARMA terms) according to the Schwarz information criterion and the adjusted
R2. For each time series, the ARMA model with lowest SIC and the highest R2 for up
to two and five ARMA terms is taken. If the additional ARMA terms do not result in
a higher R2 or lower SIC, only the models with up to two ARMA terms are considered.
Since in some cases, additional parameters produce higher R2 or lower SIC, these models
are accordingly estimated. There is also the possibility (see dyrs10 and dyrs5 ) that the
highest R2 and lowest SIC match in one model. In table 12, the four macroeconomic
variables with the highest statistical significance for each yield series in the OLS esti-
mation are specified. The autoregressive terms of these four variables are included with
no, one or two lags in the uni-/multivariate models. This procedure leads to between
6 and 12 models per series: The best-fitting (according to SIC and the adjusted R2)
models out of table 8 with no, one or two lags of macroeconomic variables combined
with the best ARMA models with up to five AR and MA terms with no, one or two lags
of macroeconomic variables.
The goal of this exercise is to determine whether the macroeconomic data delivers addi-
tional information to the autoregressive estimation. Table 13 summarises all the results.
The examination of the results shows that the additional macroeconomic variables can
have ambiguous effects. For the long end of the curve (dyrs10 and dyrs5 ), autoregres-
sive macroeconomic variables do neither improve the correlation nor the information
criterion. Therefore, you can conclude that if the estimation is done with lagged terms,
it is optimal to choose an univariate model. The outcome for the further variable is
not so distinct. Adding the first lag of macroeconomic variables to the univariate dyr1
models yields for the (1,0)- and (0,2)-model to a better models in terms of correlation
and the selection criterion. However, two lags of macroeconomic variables do not make
sense as the SIC increases and correlation only slightly rises. For dmths3, an ARMA(2,1)
52
and (3,2)-model enhances the R2 and the SIC if the multivariate equation is employed
(with the first lag of macroeconomic variables). Additional variables do not help in the
other investigated models for dmths3, as either the R2 does not significantly pick up
or the information criteria drops. Still, univariate short-rate models partially improve
when multivariate parameters are added. Finally, the slope time series are analysed.
The multivariate case only improves ARMA(1,0) for dslope1y10y and ARMA(2,1) for
dslope3m10y.
Altogether, we can conclude that multivariate models can be useful if we have only a
few ARMA terms and are especially well working at the short end of the curve, whereas
they are less helpful for the long-term interest rates and the yield spread.
Expl. Variables dyrs10 Expl. Variables dyrs5AR MA Macro Adj. R2 SIC AR MA Macro Adj. R2 SIC
0 1 0.156 0.847 0 1 0.129 1.2450 1 1 Lag 0.135 0.970 0 1 1 Lag 0.129 1.3430 1 2 Lags 0.123 1.109 0 1 2 Lags 0.114 1.4814 3 0.232 0.756 4 5 0.193 1.3944 3 1 Lag 0.204 1.057 4 5 1 Lag 0.220 1.4784 3 2 Lags 0.197 1.184 4 5 2 Lags 0.214 1.614
Expl. Variables dyr1 Expl. Variables dmths3AR MA Macro Adj. R2 SIC AR MA Macro Adj. R2 SIC
1 0 0.154 1.754 0 2 0.410 1.9411 0 1 Lag 0.256 1.750 0 2 1 Lag 0.339 1.9631 0 2 Lags 0.255 1.877 0 2 2 Lags 0.396 1.9230 2 0.197 1.845 2 1 0.306 1.8540 2 1 Lag 0.261 1.767 2 1 1 Lag 0.428 1.7850 2 2 Lags 0.268 1.882 2 1 2 Lags 0.417 1.8385 5 0.315 1.840 3 2 0.417 1.7445 5 1 Lag 0.313 1.973 3 2 1 Lag 0.455 1.7115 5 2 Lags 0.409 1.943 3 2 2 Lags 0.470 1.759
n.a. n.a. 3 3 0.426 1.759n.a. n.a. 3 3 1 Lag 0.437 1.773n.a. n.a. 3 3 2 Lags 0.485 1.759
Expl. Variables dslope1y10y Expl. Variables dslope3m10yAR MA Macro Adj. R2 SIC AR MA Macro Adj. R2 SIC
1 0 0.136 1.021 0 2 0.255 1.9871 0 1 Lag 0.228 0.997 0 2 1 Lag 0.219 1.9971 0 2 Lags 0.264 1.072 0 2 2 Lags 0.304 1.9432 1 0.175 1.042 2 1 0.251 1.8092 1 1 Lag 0.219 1.069 2 1 1 Lag 0.335 1.8142 1 2 Lags 0.288 1.086 2 1 2 Lags 0.305 1.8113 2 0.320 0.914 3 3 0.329 1.7933 2 1 Lag 0.320 0.978 3 3 1 Lag 0.270 1.8303 2 2 Lags 0.360 1.043 3 3 2 Lags 0.265 1.8454 4 0.348 0.963 4 3 0.273 1.7374 4 1 Lag 0.310 1.087 4 3 1 Lag 0.181 1.8654 4 2 Lags 0.371 1.085 4 3 2 Lags 0.290 1.850
Table 13: VARMA models
53
6.5 Forecasting
One of the advantages of the models which are discussed in section 6.2 and 6.4 is that
they provide an easy framework for forecasting models. Therefore, the best performing
models of the mentioned section are used to produce in-sample and out-of-sample fore-
casts. In-sample forecasts estimate the parameters over the whole sample period. This
method measures the average relationship over the entire period and produces a forecast
with information that was not available at the time when the forecast was computed
(e.g. calculate a forecast for 1995Q1 using the estimation for the period from 1975Q1
to 2005Q3). On the other hand, out-of-sample models only use observations which we
were actually available when the forecast was computed (e.g. forecasting a variable for
1995Q1 with a sample from 1975Q1 to 1994Q4). In this section, two kinds of out-of-
sample models are employed. Firstly, a time frame forecast, which always uses the same
number of observations, i.e. keeps on increasing the start/end date by one observation
after each estimation (rolling time frame). Secondly, the sample period is extended
by one observation each time and the start date keeps identical (recursive time frame).
Usually, one would expect that in-sample outperform out-of-sample models because they
include more information (Brooks, 2002).
As only 40 observations (10 years) are used for the rolling time frame forecast, the
number of parameters should be kept small in order to avoid overfitting due to a high
degrees of freedom. Therefore, only the maximum of two lags of autoregressive and
moving-average terms as well as only one lag of macroeconomic variables (chosen ac-
cording to table 12) are used to estimate the coefficients which makes it possible to
compare the performance of the three forecasting methods. The ARMA lag order is
selected according to table 13 (lag order with the highest R2 and lowest SIC). For the
rolling forecast, the first time frame is 1975Q1 to 1984Q4 so that the first forecast is
1985Q1 and the last is 2005Q3. The recursive forecast works out in the same manner
and starts with the sample period from 1975Q1 to 1984Q4, then subsequently adds one
observation. The in-sample forecast estimates the coefficients for the whole sample and
also computes the predictions from 1985Q1 to 2005Q3.
Only one period forecasting models are tested in this thesis because the quality of the
prediction significantly decreases if multi-period forecasts are conducted.
Table 14 summarises the results of the estimated forecasting models. For each ARMA
(one pure ARMA model and an ARMA model with macroeconomic variables) order,
three different forecasting models (rolling time frame, rol. ; recursive time frame, rec.
54
and out-of-sample, out) are tested. In order to compare the models, two performance
indicators are computed. The first indicator simply measures whether the forecast went
in the correct direction. The absolute error is the sum of the absolute value of the dif-
ference between the observed and the forecasted variables. Therefore, it is desirable to
minimise the absolute error.
For every estimated model, more than the half of the predicted values went in the correct
direction which shows that these forecasting models are generally working. As already
mentioned in the first paragraph of this chapter, the time frame (in-sample) model
should perform worst, whereas the out-of-sample estimation should be the best model
according to the theory. The absolute error, which can be regarded as the more exact
gauge compared to the direction indicator, verifies this behaviour in the vast majority of
the tested ARMA orders, i.e. the absolute error is the highest for the rolling time frame
models and the lowest for the out-of-sample models. However, the amount of forecasts
in the correct direction does not necessarily originate from the out-of-sample model, in
approximately half of the cases an in-sample model outperforms the out-of-sample model
in terms of the direction prediction.
Adding the macroeconomic variables to the forecasting equation does not enhance the
quality of the prediction (with the exception of dyrs5 where the addition of the macro-
economic term significantly improves the forecast). The absolute error even increases
slightly in almost every ARMA model if the macroeconomic are attached to the equa-
tion.
The comparison of the results of the forecasts for the different time series shows that the
models for the slope variables are the best working ones with lowest absolute error. The
models for the long end (dyrs10 and dyrs5 ) are in the middle and the shorter maturities
(dyr1 and dmths3 ) are overall the worst working models in terms of the absolute error.
In table 14, we have used the highest adjusted R2 or the lowest SIC as the selection
criteria (for dyrs10 and dyrs5 the criteria chose the same ARMA order), but there is
no perceivable relationship between the quality of the forecast and the chosen selection
criterion.
55
Expl. Variables dyrs10 Expl. Variables dyrs5Model AR MA Macro Dir. Abs. err. AR MA Macro Dir. Abs. err.
rol. 0 1 57.83% 21.901 0 1 50.60% 27.683rec. 0 1 60.24% 21.378 0 1 53.01% 26.416out 0 1 59.04% 21.170 0 1 54.22% 26.264rol. 0 1 1 Lag 59.04% 23.036 0 1 1 Lag 67.47% 23.973rec. 0 1 1 Lag 60.24% 21.538 0 1 1 Lag 80.72% 19.201out 0 1 1 Lag 62.65% 20.751 0 1 1 Lag 89.16% 17.566
Expl. Variables dyr1 Expl. Variables dmths3Model AR MA Macro Dir. Abs. err. AR MA Macro Dir. Abs. err.
rol. 1 0 53.01% 27.401 0 2 63.86% 25.167rec. 1 0 53.01% 26.745 0 2 57.83% 27.271out 1 0 53.01% 26.700 0 2 57.83% 24.509rol. 1 0 1 Lag 56.63% 30.298 0 2 1 Lag 59.04% 30.429rec. 1 0 1 Lag 54.22% 29.382 0 2 1 Lag 65.06% 27.841out 1 0 1 Lag 57.83% 27.638 0 2 1 Lag 65.06% 25.053rol. 0 2 56.63% 28.038 2 1 62.65% 22.854rec. 0 2 54.22% 28.141 2 1 60.24% 22.918out 0 2 55.42% 27.361 2 1 63.86% 22.356rol. 0 2 1 Lag 51.81% 32.687 2 1 1 Lag 53.01% 31.676rec. 0 2 1 Lag 54.22% 29.782 2 1 1 Lag 65.06% 26.980out 0 2 1 Lag 55.42% 27.851 2 1 1 Lag 65.06% 26.201
Expl. Variables dslope1y10y Expl. Variables dslope3m10yModel AR MA Macro Dir. Abs. err. AR MA Macro Dir. Abs. err.
rol. 1 0 73.49% 17.930 0 2 66.27% 21.441rec. 1 0 73.49% 18.098 0 2 63.86% 19.612out 1 0 73.49% 17.933 0 2 63.86% 19.618rol. 1 0 1 Lag 72.29% 20.028 0 2 1 Lag 61.45% 26.914rec. 1 0 1 Lag 67.47% 19.335 0 2 1 Lag 63.86% 22.385out 1 0 1 Lag 69.88% 16.762 0 2 1 Lag 72.29% 19.482rol. 2 1 77.11% 17.843 2 1 71.08% 20.467rec. 2 1 75.90% 18.239 2 1 63.86% 19.461out 2 1 71.08% 17.623 2 1 65.06% 19.334rol. 2 1 1 Lag 68.67% 23.671 2 1 1 Lag 60.24% 27.746rec. 2 1 1 Lag 66.27% 21.705 2 1 1 Lag 63.86% 25.126out 2 1 1 Lag 77.11% 17.039 2 1 1 Lag 67.47% 19.951
Table 14: Forecasting models
56
6.6 Time frame estimation
This sections builds open chapter 6.3. The OLS estimation with the four most significant
variables is being repeated but a time frame estimation over 40 observations (10 years)
is applied. This exercise can demonstrate how the coefficient, t-statistics and correlation
changes over time, i.e. it is possible to gauge the influence of the different explanatory
variables.
The first estimation covers the time period from 1975Q1 to 1984Q4, then the frame
moves to 1975Q2-1985Q1 and so forth up to the last observation. A program which is
implemented in EViews automatically estimates each time frame and reads out the co-
efficients, the t-Statistics and the adjusted R2. The result of this process are the plots in
figure 12 to figure 17. Each of these series has 84 observations from 1984Q4 to 2005Q3.
Observing the trend of the correlation throughout the investigated period, one can recog-
nise that there are common trends among the different time series. In 1991 (right afer
the German reunification), the R2 reaches the peak for every dependent variable and the
t-Statistics for the majority of the explanatory variable also rise significantly during this
period. This period was characterised by an inverted yield curve with short-term rates
almost touching the 10% level. The inversion lasted from about 1991Q1 to 1994Q4. The
second powerful increase of the goodness-of-fit occurs from about 2001 on, when the ECB
started cutting rates and the low-yield era in Europe commenced. Another interesting
feature of the results is that the R2 of longer paper is much less volatile compared to the
shorter paper and the term spread. The range of the R2 of dyrs10 and dyrs5 is about
0.2, whereas the range of the R2 of dyr1, dmths3, slope1y10y and slope3m10y is more
than three times higher. The stability of the long-term rates estimation gives evidence
that the relationship between the long end and macroeconomic variables is more robust
in the long-run than the link between the short end as well as the slope which seems to
have a quite volatile connection with the macroeconomy.
It is also worth looking at the individual results. For both long yield time series (dyrs10
and dyrs5 ), the statistical significance and the coefficient of djapanlong decreases during
the observed period (see figure 12(a), 12(b), 13(a) and 12(b)). The opposite movement
is taken by duslong. The t-Statistics of due and difo for dyrs10 are trending similarly.
The next group is the short end of the curve (see figure 14 and 15). The two time
frame estimations exhibit very analogue outcomes. The doil coefficients drops and the
t-Statistic is rather volatile. In terms of the external interest rates, the effect of the
Japanese rates are overall slightly stronger than the US rates rate. The unemployment
57
rate (due) is the only explanatory variable with a t-Statistic which never goes below
one. However, the influece of this variable approximately halves in the post reunifica-
tion period.
(a) Coefficients (b) t-Statistics (c) Adjusted R2
Figure 12: Time frame estimation for dyrs10
(a) Coefficients (b) t-Statistics (c) Adjusted R2
Figure 13: Time frame estimation for dyrs5
(a) Coefficients (b) t-Statistics (c) Adjusted R2
Figure 14: Time frame estimation for dyr1
58
(a) Coefficients (b) t-Statistics (c) Adjusted R2
Figure 15: Time frame estimation for dmths3
(a) Coefficients (b) t-Statistics (c) Adjusted R2
Figure 16: Time frame estimation for dslope1y10y
(a) Coefficients (b) t-Statistics (c) Adjusted R2
Figure 17: Time frame estimation for dslope3m10y
Finally, the estimation is applied to the slope series dslope1y10y and dslope3m10y. The
phenomenon of the weakened effect of due after the reunification. Recently, the statis-
tical significane of dusslope has increased enormously which resulted in an increase of
correlation. Overall the link between the slope and the macroeconomy seems to be the
59
weakest that is why the time frame estimation is difficult to interpret.
In conclusion, the time frame regression is an useful experiment but as each estimation
has only 40 observations, a few outliers might distort the results.
6.7 (G)ARCH modelling
One of the major problems when analysing financial data is volatility clustering. In
financial time series, volatility tends to appear in bunches which means that periods
with very high (low) volatility tend to follow periods with higher (low) volatility. The
result of this behaviour are volatility cluster, one could think of that as ’autocorrelated
volatility’ (Brooks, 2002). Another relevant stylized fact of financial data is the tendency
of financial asset returns to have distributions that exhibit heavy tails. In figure 18, the
graphs for the second order difference are plotted and the volatility cluster are readily
identifiable.
(a) ddyrs10 (b) ddmths3 (c) ddslope1y10y
Figure 18: The second difference time series of yrs10, mths3 and slope1y10y
The classical linear regression model (CLRM) essentially assumes that the error term εt
is normally distributed, i.e. the error have zero mean and the variance is constant and
finite over all values of xt.
yt = β1 + β2xt + εt where εt ∼ (0, σ2)
The assumption of the constant variance of the error terms is called homoscedasticity. If
the variance changes changes over time, this occurrence is known as hetroscedasticity. If
the errors are assumed to be homoscedastic, but are hetroscedastic, an implication would
be that standard error estimates might be wrong.
ARCH (Autoregressive conditionally heteroscedastic) modelling attempts to parame-
60
terise the volatility clustering and the hetroscedastic behaviour of financial data. If the
errors have a zero mean, the below-mentioned can be assumed.
σ2t = var(εt|εt−1, εt−2, ...) = E[ε2
t |εt−1, εt−2, ...] (18)
Equation (18) states that the conditional variance of a zero mean normally distributed
random variable εt is equal to the conditional expected value of the square of εt. The
ARCH approach models the conditional variance of the error term, σ2, to depend on the
immediately pervious value of the squared error.
σ2t = α0 + α1ε
2t−1 (19)
This equation is also known as ARCH(1) model (the conditional variance depends on
one lagged squared error) and originates from Engle (1982). If the model is extended to
the general case, a ARCH(q) model is obtained where the error variance depends on q
lags of squared errors.
σ2t = α0 + α1ε
2t−1 + α2ε
2t−2 + ... + αqε
2t−q (20)
Despite offering a framework for the analysis of time series models of volatility, ARCH
models have barely been used due to a number of problems. There is no clear approach
to determine the value of q (number of lags of the squared residuals in the model).
The value of q might be very large in order to capture all of the dependence in the
conditional variance which might lead to a large conditional variance model that is not
parsimonious.
To solve these issues, Bollerslev (1986) introduced GARCH (Generalised ARCH) models.
The GARCH model extends Engle’s model with a term which allows the conditional
variance to be dependent upon previous lags (a moving-average component).
σ2t = α0 + α1ε
2t−1 + β1σ
2t−1 (21)
Equation (21) is known as a GARCH(1,1) model. If the model can be generalised with
q lags of squared errors and p lags pf the conditional variance.
σ2t = α0 +
q∑i=1
αiε2t−i +
q∑j=1
βjσ2t−j (22)
61
The three terms specified in the equation can be interpreted as a weighted function of a
long-term average value (α0), information about volatility during the previous periods
(αiε2t−i) and the fitted variance from the model during the previous periods (βjσ
2t−j). The
advantage of GARCH over ARCH is that it is more parsimonious, and avoids overfitting.
Since σ2t is the conditional variance, its value must always be strictly positive, as the
squares of lagged errors are not negative by definition. Therefore, all the coefficients
are required to be non-negative. In case one or more coefficients would be negative,
the conditional might turn negative as well if there is a large lagged squared innovation
term. Thus, the sufficient condition for the coefficients is αi ≥ 0 ∀ i = 0, 1, 2, ..., q
The basic intention of ARCH/GARCH models is to correct the deficiencies of the ordi-
nary least square method by computing a prediction for the variance of each error term.
This prediction turns out often to be of interest for financial data. Since these meth-
ods are treating hetroscedasticity as a variance to be modelled and deal with the issues
of volatility clustering, they have become a commonly used tool for financial decision
concerning risk analysis, portfolio selection and derivative pricing (Engle, 2001).
6.7.1 ARCH LM Test
Engle (1982) suggests a Lagrange Multiplier test for ARCH effects in the residuals.
The specification of heteroskedasticity was motivated by the observation that in many
financial time series, the magnitude of residuals appeared to be related to the magnitude
of recent residuals.
The ARCH LM test statistic is computed from an auxiliary test regression. To test
the null hypothesis that there is no ARCH up to order q in the residuals, the following
regression is estimated:
ε2t = β0 + (
q∑i=1
βsε2t−i) + vt (23)
where ε is the residual. The regression is computed with the squared residuals on a
constant and lagged squared residuals up to order q. In EViews, two test statistics
are reported. The F-statistic is an omitted variable test for joint significance of all
lagged squared residuals. The Obs*R-squarded statistic is Engle’s LM test statistic,
computed as the number of observations times the R2 from the test regression. The
exact finite sample distribution of the F-statistic under H0 is not known but the LM
statistic is asymptotically distributed χ2(q) under quite general conditions. Table 15
presents the results of the ARCH LM test for the first differenced data. Overall, the
62
dyrs10 dyrs5Lags F-stat Prob. Obs. R2 Prob. F-stat Prob. Obs. R2 Prob.
1 3.737 0.055 3.687 0.055 1.214 0.273 1.221 0.2692 2.516 0.085 4.954 0.084 0.669 0.514 1.356 0.5083 2.205 0.091 6.482 0.090 2.286 0.082 6.708 0.0824 2.119 0.083 8.247 0.083 3.726 0.007 13.817 0.0085 1.742 0.130 8.527 0.130 3.185 0.010 14.752 0.011
dyr1 dmths3Lags F-stat Prob. Obs. R2 Prob. F-stat Prob. Obs. R2 Prob.
1 2.665 0.105 2.651 0.103 29.992 0.000 24.644 0.0002 3.159 0.046 6.158 0.046 6.216 0.003 11.578 0.0033 2.866 0.039 8.297 0.040 17.560 0.000 38.082 0.0004 2.421 0.052 9.338 0.053 7.482 0.000 24.984 0.0005 1.915 0.097 9.308 0.097 5.561 0.000 23.676 0.000
dslope1y10y dslope3m10yLags F-stat Prob. Obs. R2 Prob. F-stat Prob. Obs. R2 Prob.
1 0.664 0.417 0.671 0.413 34.950 0.000 27.839 0.0002 1.419 0.246 2.841 0.242 11.939 0.000 20.530 0.0003 1.274 0.286 3.828 0.281 42.253 0.000 64.456 0.0004 1.292 0.277 5.160 0.271 32.075 0.000 64.845 0.0005 0.821 0.537 4.167 0.526 8.573 0.000 33.103 0.000
Table 15: ARCH LM test statistics
null hypothesis can be cleary rejected in some cases, but there are also time series where
heteroskedasticity is not definitely evident. The presence of ARCH effects in dmths3 and
dslope3y10y is obvious. For all lags, the test statistic are highly significant. For dyr1
and dyrs10, the evidence is not as obvious as in the previously mentioned time series,
but the null is predominantly rejected at a confidence level of 90%. The results for dyrs5
are not clear. For one and two lagged squared error terms, the time series seems to
be homoskedastic, whereas for more than two lagged terms, autocorrelation probably
occurs. Finally, the test statistics of dslope1y10y do not indicate ARCH effects.
6.7.2 (G)ARCH model estimation
After the introduction to the concept of GARCH modelling and the confirmation of
the existence of ARCH effects in most models, the models are finally estimated and
63
compared to the pure ARMA models.
In order to compare the restricted ARMA models with the unrestricted ARMA-GARCH
model, the previously measures (adjusted R2 and SIC) are used but likelihood ratio (LR)
tests are also performed. The likelihood ratio test is a hypothesis test which compares the
maximised values of the log-likelihood function (LLF) (assuming normally distributed
errors). The LR test statistic asymptotically follows a Chi-squared distribution
LR = −2(Lr − Lu) ∼ χ2m (24)
where m are the number of restrictions, Lu the LLF of the unrestricted model and Lr
the LLF of the restricted model. As the restricted residual sum of the square is always
at least as big as the unrestricted sum of squares, Lr ≤ Lu is always given.
In the (G)ARCH estimation, models with no or one lagged squared error (ARCH) terms
and no or up to five conditional variance (GARCH) terms are estimated combined with
a maximum of five ARMA lags. Like, in section 6.2, the ARMA parameters do not
include a constant.
In comparison to the pure linear ARMA estimation, the goal of the GARCH estimation
is to improve the SIC and to reject the null of the LR test (i.e. the unrestricted GARCH
model is superior to the pure ARMA model). It is important to note that the difference
of the Schwarz Criteria between the ARMA and the corresponding ARMA-GARCH
model with the equivalent ARMA lag order strongly correlates (above 99%) with LR
test for both models. The adjusted R2 is not used as a selection criterion because it
usually decreases when GARCH models are estimated and its information content is
limited (see Brooks, 2002). Table 16 presents the results for dyr1, dmths3, dslope1y10y
and dslope3m10y, as dyrs10 and dyrs5 do not improve when GARCH terms are added
to the equation, they are not documented in the table.
Although, there are a large number of LR test statistics for dyrs10 where the null can be
rejected, the SIC only marginally decreases for a GARCH(0,1) and GARCH(1,1) with ei-
ther ARMA(3,2) or ARMA(3,3) parameters. For these estimations, the p-value of the JB
statistic also indicates normally distributed residuals. In case of dyrs5, a GARCH(0,1)
model with an ARMA(2,3) or (2,4) is the only combination which fulfills both require-
ments and the residuals are normally distributed. In section 6.7.1, the ARCH LM test
already showed that there is no clear evidence for ARCH effects in dyrs10 and dyrs5,
which is confirmed by the just presented results.
Turning from the long end of the curve to the short end, GARCH models achieve sig-
64
nificant drops in the SIC and the LR test heavily increases. For dyr1, a GARCH(1,1)
model extended with an ARMA(4,2) term performs best and if the ARMA order is lim-
ited to two lags, a GARCH(1,0) with ARMA(0,1) is chosen. A simple ARMA(0,1) with
a GARCH(1,2) process for dmths3 outperforms even models with higher lag orders. The
residuals of the dyr1 and dmths3 are clearly not gaussian distributed which questions
the validity of the LR tests.
Finally, the time series for the slope of the curve are studied. Although, the ARCH
LM test did not show any ARCH effects for dslope1y10y, the LR test strongly rejects
the null and the SIC improves by 0.5 for the best ARMA-GARCH (0,1)-(1,2). The
p-value of the JB statistic is pretty low but not as low as for the models for the shorter
maturities. Despite the presents of distinct ARCH effects, the GARCH estimation for
dslope3m10y does not perform better than for dslope1y10y, whereby the best model is
an ARMA-GARCH (0,1)-(1,1).
GARCH estimations are distinctly enhancing for some time series, as table 16 docu-
ments, even if the residuals are not normally distributed and the LR test therefore can
not be regarded as theoretical valid. However, the strong correlation of the LR test and
the difference of the SIC between the ARMA and ARMA-GARCH models gives evidence
that the outcome of the LR test is still relevant. A further analysis, which would go
beyond the scope of this master thesis, should therefore incorporate econometric models
with stable non-Gaussian variables (see Rachev and Mittnik, 2000, chapter 14).
Time series AR MA ARCH GARCH SIC R2 JB (p-value) LR stat.dyr1 4 2 1.854 0.212 0.000 41.941
4 2 1 1 1.636 0.270 0.0000 1 1.811 0.200 0.000 35.0430 1 1 0 1.616 0.186 0.000
dmths3 0 1 2.112 0.280 0.000 84.5180 1 1 2 1.611 0.256 0.000
dslope1y10y 0 1 2.112 0.280 0.000 84.4550 1 1 3 1.649 0.251 0.015
dslope3m10y 0 1 2.034 0.196 0.000 67.2940 1 1 2 1.629 0.176 0.716
Table 16: GARCH models
65
7 Conclusions
This master thesis covered a very broad discussed area in economic and econometric
research. Certainly, the theoretical and empirical part of this essay could therefore only
analyse a small extract of possible research directions in the area.
In the literature review, a selection of some recent research about the relationship of yield
curve and macroeconomic variables was presented. In chapter 6, where the empirical
estimations and tests were summarised, the outcome provided some interesting results.
With the employed estimations, it is possible to quantify and compare the influence of
macroeconomic variables on different maturities of interest rates as well as the shape
(slope) of the yield curve. In the second step, univariate models were estimated and
compared versus multivariate models. In this context, a main conclusion is that univari-
ate models are far better for estimating short maturities and multivariate models should
be preferred for longer maturities (compare section 6.2 and 6.3). Forecasting models
performed satisfactory, but adding macroeconomic variables in the forecasting equation
did not notably improve the quality of the forecast. Lastly, an univariate GARCH es-
timation was computed and led to better fitting models for the short end of the curve
and the term spread.
Further research could focus on e.g. multivariate GARCH models or the reaction of the
term structure of interest rates after the occurrence of economic shocks.
66
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A Schwarz criterion
SIC = ln (σ) +2k
T(25)
where σ is the residual variance, k=p+q+1 is the total number of parameters estimatedand T is the sample size (Brooks, 2002).
B Akaike criterion
AIC = ln (σ) +k
Tln T (26)
where σ is the residual variance, k=p+q+1 is the total number of parameters estimatedand T is the sample size (Brooks, 2002).
C Adjusted R2
As the R2 never falls when additional regressors are added to the equation, the ad-justed R2 takes into account the loss of degrees of freedom associated with adding extravariables.
R2 = 1−(
T − 1
T − k(1−R2)
)(27)
where T is the number of observations and k the number of parameters in the model(Brooks, 2002).
D Log-likelihood function
The log-likelihood function is computed as follows and assumes normally distributederrors (Brooks, 2002).
l = −T
2(1 + log(2π) + log
(ε′ε
T
)) (28)
E Skewness
Skewness is a measure of asymmetry of the distribution of the series around its mean.Skewness is computed as:
S =1
N
N∑i=1
(yi − y
σ
)3
(29)
71
The skewness of a symmetric distribution, such as the normal distribution, is zero.Positive skewness means that the distribution has a long right tail and negative skewnessimplies that the distribution has a long left tail (Brooks, 2002).
F Kurtosis
Kurtosis measures the peakedness or flatness of the distribution of the series. Kurtosisis computed as
K =1
N
N∑i=1
(yi − y
σ
)4
(30)
The kurtosis of the normal distribution is 3. If the kurtosis exceeds 3, the distribution ispeaked (leptokurtic) relative to the normal; if the kurtosis is less than 3, the distributionis flat (platykurtic) relative to the normal (Brooks, 2002).
G Jarque-Bera (JB) test of normality
The JB test of normality is an asymtotic, or large-sample, test. It is based on the OLSresiduals and uses the following test statistic:
JB = n
(S2
6+
(K − 3)2
24
)(31)
Under the null hypothesis, Jarque and Bera showed that asymptotically (i.e., in largesamples) the JB statistic in equation (31) follows the chi-square distribution with 2 df.
H Ito process
If a variable x follows an Ito process (Hull, 1999):
dx = a(x, t)dt + b(x, t)dz (32)
where dz is a Wiener process and a and b are functions of x and t. The variable x has adrift rate of a and a variance rate of b2. Ito’s lemma shows that a function, G, of x andt follows the process
dG =
(∂G
∂xa +
∂G
∂t+
1
2
∂2G
∂x2b2
)dt +
∂G
∂xbdz (33)
where the dz is the same Wiener process as in equation (33). Thus, G, also follows anIto process. It has drift rate of
72
∂G
∂xa +
∂G
∂t+
1
2
∂2G
∂x2b2 (34)
and a variance rate of (∂G
∂x
)2
b2 (35)
73
