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1
Reexamining the Expectations
Hypothesis of the term structure of
interest rates: An Out-of-Sample
Forecasting Perspective
Julia Draeb
Dr. Jing Li
Miami University
March 5, 2021
Abstract
The expectations hypothesis of the term structure of interest rates implies that the
long term and short-term interest rates are linked. This paper determines whether that
link can help improve the forecast of future interest rates and how that forecasting
power differs across term differences. With rolling windows, the out-of-sample fore-
casting errors based on the autoregressive distributive lag models are compared to the
autoregressive model. From the forecasting perspective, I find evidence that the link-
age becomes weaker as the term difference increases. This finding complements the
previous study of Li and Davis (2017).
2
Introduction and Literature
Review
Interest rates drive financial activity and
have serious implications for the over-
all health of the economy. At a macro
level, interest rates affect the money
multiplier, and, by extension, the rate
of inflation in the economy. Further-
more, interest rates reflect the cost of
borrowing money, as a result, they af-
fect investment decisions for nearly
every large-scale purchase from individ-
uals and firms.
For example, mortgage rates are tied to
the bond yield and have enormous ef-
fects on the housing market including
housing starts and refinance decisions.
Lower interest rates will lower the
overall mortgage cost of a house which
will incentivize people to purchase new
homes. Furthermore, homeowners will
refinance their home mortgage to take
advantage of low interest rates and low
mortgage costs. On the other hand,
when interest rates are high, home
builders will increase housing starts to
sell homes at a premium. The behavior
associated with changes in interest rates
in the housing market have larger im-
plications on the entire United States’
economy. In fact, in 2018, the housing
market accounted for nearly 15% of
the United States GDP (BEA). The
Great Recession exemplifies the effects
of the housing market on our economy.
There were extremely low interest rates
on subprime mortgage loans that cre-
ated a bubble in the housing market. In
2008, homeowners were unable to pay
their mortgage loans and crashed the
mortgage market. These defaults also
flowed into the secondary mortgage
market and eventually led to the bank
crisis and stock market collapse. The
Dow Jones finally recovered from pre-
Great Recession in March 2013, illus-
trating how devastating this housing
market crash was on the economic ac-
tivity of the United States.
Moreover, interest rates also influence
student loans and, in turn, the decision
of whether a student attends college.
On one hand, if interest rates are low,
then the cost of borrowing money is
low, and a student may be able to af-
ford a higher education when they ordi-
narily would not. However, if interest
rates on student loans is too high, an
applicant may decide to abstain or de-
fer from college because the cost of
borrowing is too high. Student loans
compiled to over 1.5 trillion in 2019
which is about 7.5% of GDP, making it
another huge market where interest
rates dictate behavior (BEA).
Interest rates show up in a plethora of
other industries that have grandiose ef-
fects on each market including returns
on savings, credit card debt, car loans,
and other loans. Therefore, accurately
forecasting interest rates is an incredi-
bly important and highly researched
topic.
3
A starting point to understand the in-
teraction between long-term and short-
term interest rate is the expectations
hypothesis. In the absence of liquidity
premium and other market frictions
like regulation or taxation, the expecta-
tions hypothesis states that investing in
multiple short-term bonds should re-
turn the same as a long-term bond. In
essence, this is an application of the no
arbitrage condition to the bond market.
The no arbitrage condition states that
portfolios with the same present dis-
counted payoff must cost the same. If
there is arbitrage opportunity, people
would quickly take advantage of the
profit by selling the overvalued asset
and buying the undervalued asset. This
process of market correction does not
stop until the condition of no arbitrage
is satisfied.
For example, this hypothesis states that
investing in two 1-year bonds consecu-
tively should yield equivalent returns to
investing in one 2-year bond. This
holds intuitively 1. If investing into 1-
year bonds for two years yields more
than a 2-year bond, then then investors
will choose to buy the 1-year bonds
and, in turn, increase the price and de-
crease the yield. The converse also
holds: if a 2-year bond is expected to
return more than two 1-year bonds,
then investors will invest in the 2-year
bond. These corrections will occur until
1 The five-year bond is typically higher than the one-year bond because of liquidity premium.
the markets return to long run equilib-
rium where each investment decision
yields equivalent expected returns.
These strategies are referred to as the
buy-and-hold strategy and the roll-over
strategy. The first strategy is to buy and
hold onto a 2-year bond with an annual
interest rate, i2,t. The latter strategy is to
invest into a 1-year bond with an inter-
est rate of i1,t and then roll over those
returns into another 1-year bond at an
unknown rate, i1,t+1. The no arbitrage
condition implies that these returns
should be equal in equilibrium:
(1 + 𝑖2,𝑡)2
=
(1 + 𝑖1,𝑡)(1 + 𝑖1,𝑡+1) (1)
I assume the future short-term rate,
i1,t+1 , can be predicted perfectly. This
equation can be simplified because the
interest rates are so small that multiply-
ing them together makes it negligible.
Therefore, equation (1) leads to the fol-
lowing approximation:
𝑖2,𝑡 ≈1
2(𝑖1,𝑡 + 𝑖1,𝑡+1) (2)
This equation implies that the 2-year
interest rate is only greater than the 1-
year interest rate when the market ex-
pects the 1-year rate to increase in the
next period. Therefore, the shape of
the yield curve indicates the market’s
expectation about future interest rate:
4
𝑖2,𝑡 > 𝑖1,𝑡 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑖1,𝑡+1 > 𝑖1,𝑡 (3)
This equation indicates that the yield
curve is upward sloping: as term length
increases, the yield increases.
The Fisher Equation states that the
nominal interest rate, i, is equal to the
real interest rate, r, plus expected infla-
tion π.
𝑖 = 𝑟 + π (4)
Therefore, with a fixed real interest
rate, the yield curve is upward sloping if
the market expects the inflation rate
will rise in the future. Conversely, the
yield curve is inverted or downward
sloping if the market expects deflation
or economic downturn.
To compare the interest rates of an s-
year long-term bond to a 1-year bond,
we can generalize equation (2) into the
following:
𝑖𝑠,𝑡 ≈1
𝑠(𝑖1,𝑡 + 𝑖1,𝑡+1 + ⋯
+ 𝑖1,𝑡+𝑠−1) (4)
Clearly, the link between long term and
short-term rates is measured by 1/s,
which will become weaker as s in-
creases. Therefore, when s gets suffi-
ciently large, the short run and long run
rates are unrelated or independent from
each other. This paper will attempt to
verify this implication of the expecta-
tions hypothesis from the forecasting
perspective.
The expectations hypothesis is so im-
portant because it implies that we can
forecast future spot interest rates based
off previous long-term and short-term
interest rates. Imposing the assumption
that the markets are in long run equilib-
rium without market frictions, the hy-
pothesis expects the future spot interest
rate to yield the present discounted dif-
ference between the long-term and
short-term rates. This hypothesis im-
plies we can forecast future spot inter-
est rates; however, this relationship is
difficult to test.
The primary challenge for a statistical analysis of interest rates is nonstation-arity. Nonstationary variables are varia-bles that have means and variances that change over time which can make tra-ditional regression theory invalid. To conduct a causal study based on a re-gression, nonstationary variables must be transformed to a stationary process through detrending and differencing unless they are cointegrated. Nonsta-tionary variables typically are trending and highly persistent. In the literature, many studies find that interest rates have unit roots, and therefore, can be modelled as a generalized random walk or a nonstationary autoregressive pro-cess. Even though this is a simplistic model, complex research and analysis rarely forecasts interest rates more ac-curately than the random walk model (Fuqua School of Business). It is gener-ally accepted that interest rates need to be differenced one time to become a
5
stationary process, in other words, it is an I(1) process.
One way of modelling the relationship between nonstationary variables is by using the cointegration method. A data set is cointegrated if there is some sta-tionary linear combination of the non-stationary time series. In other words, there is a long run relationship between the movement of these nonstationary variables. Graphically, we see interest rates trending together over time with-out getting significantly far apart (Fig-ure 1-A, Figure 1-B). Examining graphs is informal. Thus, formal cointegration tests such as those in Engle and Granger (1987) can be applied to the time series2.
In previous literature, Hall & Anderson
(1992) determines that cointegration of
term structures is an appropriate model
for testing the relationship between
long-term and short-term interest rates.
In fact, Hall finds evidence of cointe-
gration for nearly all the interest rates
tested aside from when the spread in-
corporates very long-term or very
short-term rates. Hall then used an er-
ror correction model to confirm the ex-
istence of cointegration. This model
also provided evidence that long-term
rates drive the term structure and
short-term rates fluctuate to return to
long run equilibrium, which is evidence
that the expectations hypothesis exists.
2 Cointegration will be the focal point of my master’s thesis
Aside from cointegration of the term
structures, there has also been an abun-
dance of literature regarding the best
method for forecasting interest rates.
Sarno and Thorton (2003) found that
forecasting improves when accounting
for nonlinearities and asymmetries of
the interest rates. Later, in an analysis
to determine the best Federal Funds
Rate forecasting, Sarno, Thorton, and
Valente (2006) found the gap between
the current and target rate performs the
best when using conventional
measures. On the other hand, the sim-
ple univariate reaction function per-
forms the best when using hit ratios
and market timing tests. Furthermore,
Hoffman and Rasche (1996) found that
cointegration is only beneficial to the
model at longer forecast horizons. Fi-
nally, Bidarkota (1998) finds the one-
period ahead models perform substan-
tially better than the multi-step ahead
forecasts.
In a newer literature, Li and Davis
(2017) finds the cointegration test be-
comes an inappropriate test of the ex-
pectations hypothesis as the term dif-
ference increases because the relation-
ship between short-term and long-term
rates gets weaker. This is very intuitive:
the 3-Month Rate should have more
predictive power for forecasting the 1-
Year Rate as opposed to the 10-Year
Rate. Therefore, the 3-Month Rate
6
would be more strongly cointegrated
with the 1-Year Rate than with the 10-
Year Rate. The Li and Davis paper can
explain why Hall found cointegration
for shorter term differences but not in
longer term differences.
This paper will be an extension of Li
and Davis (2017) where I will use out-
of-sample forecasting to determine if
the relationship between interest rates
gets weaker as the term difference in-
creases. More specifically, I will look at
whether the forecasting power de-
creases as the term difference rises. My
conjecture is that, for instance, the 3-
month interest rate can help forecast 1-
year interest rate better than 10-year in-
terest rate.
Data and In-Sample Fitting
The data set consists of 467 monthly
observations from September 1981 to
August 2020. The beginning and end-
ing dates are chosen given the data
availability. These data are gathered
from the St. Louis Federal Reserve
Economic Data database and incorpo-
rate four different length interest rates.
To downplay the impact of factors
such as default risks, this paper focuses
on the interest rate of government
bonds. The longest-term interest rate is
the 10-year treasury constant maturity
rate, followed by the 1-year treasury
constant maturity rate and the 3-month
treasury constant maturity rate. The
rates are returns on treasury securities;
a relatively risk-free bond issued by the
United State Treasury to sell govern-
ment debt. Lastly, the shortest-term
rate is the federal funds rate, an over-
night rate set by the Federal Open Mar-
ket Committee (FOMC) for interbank
lending. This is the interest rate banks
will charge other banks for overnight
loans on excess liquidity. The federal
funds rate is targeted by the FOMC
and is used to control the supply of
money, in turn controlling the econ-
omy. When the economy needs stimu-
lus, the FOMC will purchase govern-
ment bonds from the depository insti-
tutions which will give these institu-
tions more liquidity and decrease the
federal funds rate. The opposite occurs
when the economy is growing too
quickly. The summary statistics for
these four interest rates are shown in
Table 1.
The summary statistics show similar
means for the federal funds rate, 3-
month rate, and 1-year rate, at 4.166,
3.933, and 4.286 respectively (Table 1).
However, the 10-year rate has a mean
of almost double at 7.505. This is be-
cause the 10-year rate is a longer-term
asset than the other rates. The holding
period makes these assets less liquid
than the other interest rates. Therefore,
the depository institution needs to offer
a higher yield to sell this illiquid asset.
This trade-off is referred to as the li-
quidity premium. Moreover, the 10-
7
year rate has the lowest standard devia-
tion which is also expected. The longer
length of the 10-year bonds will buffer
them from short run market fluctua-
tions, making them more stable. On the
other hand, the federal funds rate and
the 1-year interest rate vary dramatically
because they are directly affected by
monetary policy (Table 1).
Li and Davis (2017) formalized the idea
that the closer the term difference, the
more strongly cointegrated the data
would be. This is very intuitive and can
be visualized in Figure 1-A and Figure
1-B. Clearly, the federal funds rate is
more closely related to the 3-month
rate and has a weaker relationship with
the 10-year rate. We see graphically that
the 10-year rate is significantly
smoother and has less fluctuations
overall than the other rates.
Figures 1-A and 1-B both show down-
ward trends in overall interest rates,
and, in 2008-2012, the federal funds
rate was nearing the Zero Lower
Bound. Interest rates are unstable when
they are negative, implying that money
will lose value in depository institu-
tions. Nearing the Zero Lower Bound
incentivizes people to consume because
money is earning minimal interest in
the banks. Therefore, the Federal Re-
serve will lower interest rates to just
above the Zero Lower Bound during
recessions to stimulate the economy,
such as in 2008-2012.
Another notable factor in these graphs
is the consistent downward trend start-
ing in the 1980s. Bernanke examines
the increased savings in 2005 and at-
tributes the downward trend to a
“Global Saving Glut”. He determined
that the higher savings in the United
States is due to increased foreign in-
vestment and the aging population.
First, emerging market nations are in-
creasing their reserves by investing into
the United States. The influx of foreign
investment increased the supply of
bonds and therefore decreased the in-
terest rate, otherwise thought of as the
price of the bond. Continuing, the ag-
ing population of the United States has
caused people to demand safer assets
such as treasury securities. This has a
compounding effect on the market and
pushes the interest rate even lower.
The first empirical methodology I used
was a first order autoregressive model
which uses the history of the variable as
a key regressor to estimate that variable
in the next period. When the auto-
regressive coefficient is restricted to be
one, the autoregressive model is re-
duced to a random walk. This type of
modeling is very intuitive for interest
rates because previous month’s interest
rate will provide a reasonable guess for
this month’s interest rate. The predic-
tion error is captured in the error term.
The equation is as followed:
8
𝑦𝑡 = ∑ 𝛽𝑗𝑦𝑡−𝑗
𝑗
+ 𝜇𝑡 (5)
These results of the in-sample fitting of
the autoregressive models using the
whole sample are below in Table 2.
These results confirm that the first
lagged term of an interest rate is a good
regressor to predict itself for the whole
sample. Specifically, the lagged 10-year
rate is particularly effective in predict-
ing the next interest rate in the next pe-
riod, having a coefficient of 0.9886
which implies the interest rate today
follows the interest rate from yesterday
very closely. Also, the federal funds rate
autoregressive model has the highest R-
squared statistic of 0.9930 which indi-
cates this model accounts for 99.3% of
the variability in the data. These beta
values are close to one which are sug-
gestive of a unit root. This finding is
common in literature and is consistent
with the persistence or smoothness
shown by data. Unit roots make tradi-
tional regression theory invalid (e.g., t
statistics do not follow t distribution).
Nevertheless, this paper is mainly con-
cerned with out-of-sample forecasting.
In fact, I expect good forecasts because
a regression with unit root time series
can produce super-consistent coeffi-
cient estimates.
The next modeling methodology is the
autoregressive distributive lag model
which is an extension of the autoregres-
sive model that will incorporate the
lagged term of related, different-length
interest rates. Literature by Bentzen
(2001) claims that the autoregressive
distributive lag model remains valid
when the variables are nonstationary
and cointegrated. In this paper, this
model can be useful to determine if the
interest rates are related and how
strongly they are related to other inter-
est rates. Simplistically, this model will
give insight into if these interest rates
are dependent on the lagged value of a
different interest rate. The equation for
this model is given as followed:
𝑦𝑡 =
∑ 𝛽𝑗𝑦𝑗−1
𝑗
+ ∑ 𝛼𝑘𝑥𝑡−𝑘
𝑘
+ 𝜇𝑡 (6)
To best forecast interest rates, this pa-
per will employ the principle of parsi-
mony. Simply put, the principle of par-
simony is a guiding scientific principle
that states the most simplistic explana-
tion for the observation is preferred. By
application of this principle, I will use
the first-order lagged models with only
one lagged term instead of incorporat-
ing the second or third lagged terms.
Following the principle of parsimony in
this paper we let j=1 in the AR model
(1), and j=1, k=1 in the ADL model
(2).
Let me emphasize that because my goal
is forecasting, I can ignore the endoge-
neity issue such as omitted variable
bias. Consequently, no other control
9
variables are needed in models (1) and
(2).
Below are Tables 3-A, B, and C sum-
marizing the in-sample autoregressive
distributive lagged models for the entire
sample of data. Overwhelmingly, these
additional lagged interest rates were not
statistically significant, however, we can
still derive economic significance from
the models. Clearly, adding the lagged
value of shorter-term interest rates to
the 10-year rate did not improve the
model (Table 3-A). When predicting a
shorter-term difference, such as the 1-
year to the 3-month and federal funds
rate, the coefficients become more sig-
nificant and t-values increases. In fact,
the lagged federal funds rate has a t-
value of -2.52 for the 1-year rate which
makes the value significant at the 95%
level (Table 3-B). The 3-month rate
nearly doubled in significance when
predicting the 1-year rate, t-value of -
0.638, compared to the 10-year rate, t-
value of 0.341. Lastly, the federal funds
rate holds similar significance when
added to the 3-month rate, being signif-
icant at the 90% level (Table 3-C).
Utilizing the entire data set does have
advantages, such as finding overall pat-
terns and getting baseline statistics,
however, breaking the data into sub-
samples is able to account for potential
structural changes and illustrate the dy-
namic evolvement of the relationship.
Therefore, I will be using a rolling win-
dow with a size of 60 to section the
data. A rolling window will section the
data into 60 observations and run a re-
gression on those 60 data. The size of
the window is 5 years which is suffi-
cient time to determine a trend and
predict the next period. The regression
will first regress observations 1 to 60,
then 2 to 61, and so on and so forth
until the last window from 407 to 467,
making 407 regressions total. The roll-
ing window will log those statistics on
each subset of data. This is useful for a
few reasons. First, we can directly test
the predictive power of each subset of
data. Using the window for the first 60
observations, we can predict the 61st
observation and compare to the actual
observation. The rolling window makes
it so we can test the accuracy of the
prediction with the sample we have.
Next, a rolling window also helps visu-
alize additional statistics throughout the
sample size. Below are figures plotting
the t-statistic for the first lagged of the
x-variables for the 10-year, 1-year, and
3-month rates used in the autoregres-
sive distributive model above. In these
figures, the t-statistics on the lagged x-
variables for the 10-year rate fail to be
higher than 1.96 in absolute value and
therefore are often not statistically sig-
nificant (Figure 2-A). However, the t-
statistics are larger when using the
lagged variables of the 3-month and
federal funds rate to predict the 1-year
rate, consistently getting a magnitude of
over 1.96 (Figure 2-B). Lastly, for the
10
3-month rate, the t-statistics rise further
and more consistently have a large
enough value to be statistically signifi-
cant (Figure 2-C).
Rigorously speaking, using 1.96 as the critical value is problematic since the variables have unit roots. Here I use it just as a benchmark. What is important here is the pattern shown in the magni-tude of t-values.
Overall, the findings in Figure 2 agree with Li and Davis (2017). The linkage between interest rates gets weaker as the difference in term increases.
Out-of-Sample Forecasting
In-sample fitting has the drawback that
the same data are used twice. First, they
are used for the estimation of the
model, and then, the computing of the
forecasting error. A more realistic and
relevant question is, how does the
model perform in terms of predicting a
value not used in the estimation pro-
cess. So, in this section I will examine
the out-of-sample forecasting perfor-
mance.
Using a rolling window, I model a sub-
set of data and predict the next value in
the dataset. With that prediction, I can
obtain a forecasting error term by com-
paring the actual value to the predicted
value from the rolling window data.
This is different from the in-sample fit-
ting because I am not using the actual
value in the dataset to help make the
prediction. Therefore, this out-of-sam-
ple method is a more credible way of
evaluating the performance of models.
I will continue to use the 60 observa-
tions for the rolling window and create
forecasting errors for the 407 rolling
window predictions. I used the differ-
ent interest rates to create the Out-of-
Sample Forecasting Error for the auto-
regressive lag model and compare to
the benchmark autoregressive model.
The figures below create an easy visual-
ization for how accurate the different
regression methods were for each inter-
est rate.
Figure 3-A shows the different fore-
casting errors for the 10-year interest
rates. Here, the autoregressive model
performs better than the autoregressive
distributive models with the 1-year, 3-
month, and federal funds rates. In Fig-
ure 3-A, the autoregressive model has
an error term that is more consistently
around zero than the other models.
This indicates that the autoregressive
model predicts the actual value of the
interest rate more accurately and there-
fore is a better performing forecasting
model. Furthermore, the autoregressive
model has an average square error of
5.4140 e-02 whereas the other models
are slightly higher at around 5.6 e-02
(Table 4). The lower average square er-
ror term again signals that the auto-
regressive model forecasts the 10-year
11
interest rate better than all the auto-
regressive distributive models.
However, the autoregressive model is
not always the best predictive model. In
fact, as the term difference between the
interest rates decrease, the autoregres-
sive distributive models have more pre-
dictive power. For example, when us-
ing the rolling window to forecast the
1-year interest rate, the error term is
more concentrated near zero, and only
gets larger than -1 once (Figure 3-B).
The mean square error term for these
autoregressive distributive models is
also below the autoregressive model. In
fact, incorporating the lagged 3-month
interest rate reduces the mean squared
error term from 5.1092 e-02 to 5.0644
e-02 and incorporating the federal
funds rate reduces it to 4.7886 e-02
(Table 4). The lower error terms and
the lower mean squared error term
both indicate that incorporating other
lagged interest rates improves the fore-
casting for the 1-year interest rate.
Lastly, the autoregressive distributive
model performs even better for the 3-
month interest rate using the lagged
federal funds rate. In Figure 3-C, the
autoregressive distributive model con-
sistently has a lower error term than the
autoregressive model. Including the
lagged federal funds rate lowers the
mean square error term from 4.4366 e-
02 to 4.0832 e-02 (Table 4). This is the
most positive change in Table 4.
To demonstrate the differences be-
tween the mean squared error terms of
the autoregressive models and the auto-
regressive distributive model, a two-
sample t-test was conducted. The null
hypothesis states that the mean of the
squared errors for the autoregressive
model is equal to that of the auto-
regressive distributive. The higher the
magnitude of the t-statistic, the more
evidence that this null hypothesis is re-
jected and that the mean square error
for these models are not equal. For
these regressions, the autoregressive
process will be used as a baseline that
each autoregressive distributive process
will be compared to.
The 10-year interest rate has the lowest
t-statistics for each autoregressive dis-
tributive process at -0.3397, -0.2949,
and -0.2680. These t-statistics demon-
strate that adding additional lagged
term rates does not drastically change
the mean squared error from the auto-
regressive model. Also, the negative
value indicates that the autoregressive
model has the lowest mean squared er-
ror and has the most predictive power
out of all the 10-year interest rate mod-
els.
On the contrary, the t-statistic for the
3-month interest rate is 0.4419 which
indicates that adding the lagged federal
funds rate decreases the mean squared
error by a larger amount compared to
the 10-year interest rate. Although this
12
value is not statistically significant be-
cause the magnitude is not greater than
1.96, I can still derive economic value
from comparing it to the other models
t-statistics. Clearly, this t-statistic is pos-
itive and larger than the t-statistics for
all the 10-year interest rates. The posi-
tive magnitude indicates that adding the
lagged federal funds rate to the 3-
month forecasting model decreases the
mean squared error and therefore is a
better predictive model. The larger t-
statistic shows that adding a lagged var-
iable into the 3-month rate improves
the forecasting more than adding a
lagged variable to the 10-year rate.
This is evidence that the term differ-
ence matters when employing the ex-
pectations hypothesis and using lagged
rates to predict future spot rates. These
out-of-sample forecasts demonstrate
that the 10-year out of sample forecast-
ing is not improved by adding any of
the lagged interest rates, however, both
the 1-year and 3-month rates are im-
proved by adding lagged rates. The 10-
year interest rate has the highest mean
squared error for the autoregressive
models and the 3-month interest rate
has the lowest (Table 4). The t-statistics
solidify this evidence because the mag-
nitude of the t-statistics increase from
the 10-year rate being the smallest to
the 3-month being the largest. Overall,
the autoregressive models improve as
the term difference decreases.
Conclusion
In all, this paper centers on out-of-sam-
ple forecasting and extends Li and Da-
vis (2017) by showing that the linkage
between the interest rates becomes
weaker as the difference in terms rises.
At larger term differences, the lagged
other interest rates do not improve in-
sample fitting or out-of-sample fore-
casting in the ADL model relative to
the AR model. Adding these lagged
terms also increases the mean squared
error for out-of-sample 10-year interest
rate regressions which implies the best
model out of those in this paper is the
first order autoregressive model. How-
ever, the story changes when the term
difference decreases, as shown by the
1-year and 3-month regressions. The
regressions for the in-sample fitting
show that the lagged interest rates are
more statistically significant. Also, the
out-of-sample forecasting was im-
proved for both the 1-year and 3-
month interest rate compared to the
baseline autoregressive model.
The results in this paper have long
reaching implications for economic
theory and policy. First, this paper sup-
ports the Federal Open Market Com-
mittee targeting short term interest
rates to effect long-term interest rates.
This paper provides evidence that simi-
lar length interest rates are cointe-
grated. Therefore, the monetary policy
actions to target short-termed interest
13
rates will flow to longer-termed interest
rates. An extension of these findings
may investigate how monetary policy
ripples through the other interest rates.
Also, this paper lends evidence to the
expectations hypothesis when the term
difference is relatively short. Therefore,
we can predict interest rates more accu-
rately by incorporating a lagged term of
similar length interest rates. Accurately
predicting interest rates influences mar-
kets across the nation and can impact
investment decisions everywhere.
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& Valente, Giorgio (2006) Federal
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(federalreserve.gov)
[8] Bentzen, Jan & Engsted, Tom. A re-
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14
Table 1: Summary Statistics
Table 2: First-Order Autoregressive Model
10-Year 1-Year 3-Month Federal Funds
Beta 0.9886*** 0.9833*** 0.9824*** 0.9854*** Standard Error 0.0041 0.0043 0.0045 0.0038 R-Squared 0.9921 0.9914 0.9902 0.9930
Note: Stars indicate significance of coefficient. * if p<0.1, ** if p<0.05, *** if p<0.01
Table 3-A: Autoregressive Distributive Lagged Model for 10-Year Rate
Independent Lagged Variable 1-Year Rate 3-Month Rate Federal Funds Rate
Alpha -0.0038 0.0039 -0.0019 Standard Error 0.0128 0.0114 0.0104 R-Squared 0.9921 0.9921 0.9921
Note: Stars indicate significance of coefficient. * if p<0.1, ** if p<0.05, *** if p<0.01
Table 3-B: Autoregressive Distributive Lagged Model for 1-Year Rate
Independent Lagged Variable 3-Month Rate Federal Funds Rate
Alpha -0.0305 -0.0824* Standard Error 0.0478 0.0327 R-Squared 0.9914 0.9915
Note: Stars indicate significance of coefficient. * if p<0.1, ** if p<0.05, *** if p<0.01
Table 3-C: Autoregressive Distributive Lagged Model for 3-Month Rate
Independent Lagged Variable Federal Funds Rate
Alpha -0.0811 Standard Error 0.0490 R-Squared 0.9902
N Mean Standard Deviation Maximum Minimum
10 Year 467 7.505 3.121 15.61 0.01 1 Year 467 4.286 3.405 16.52 0.10 3 Month 467 3.933 3.229 15.32 0.62 Federal Funds Rate 467 4.166 3.430 15.87 0.05
15
Table 4: Mean Squared Prediction Error Across Different Models
1-Year Au-toregressive Distributive Model
3-Month Autoregres-sive Distrib-utive Model
Federal Funds Rate Auto-regressive Dis-tributive Model
Autoregres-sive Model
10-Year Interest Rate 5.6664 e-02 5.6299 e-02 5.6091 e-02 5.4140 e-02 1-Year Interest Rate 5.0644 e-02 4.7886 e-02 5.1092 e-02 3-Month Interest Rate 4.0832 e-02 4.4366 e-02
Table 5: t-Statistics for Equal Mean of Squared Prediction Errors of AR and ADL
Models
1-Year Auto-regressive Distrib-utive Model
3-Month Auto-regressive Dis-tributive Model
Federal Funds Au-toregressive Dis-tributive Model
10-Year Interest Rate -0.3397 -0.2949 -0.2680 1-Year Interest Rate 0.0614 0.4519 3-Month Interest Rate 0.4419
16
Figure 1-A: Federal Funds Rate and 3 Month Rate
Time
Pe
rce
nt
1990 2000 2010 2020
05
10
15 Federal Funds Rate
3 Month Rate
Figure 1-B: Federal Funds Rate and 10 year Rate
Time
Pe
rce
nt
1990 2000 2010 2020
05
10
15 Federal Funds Rate
10 year Rate
17
18