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

draebjn@miamioh.edu

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.

References

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tion hypothesis of the term structure.

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Heather M. & Granger, Clive W.J.

(1992) A Cointegration Analysis of

Treasury Bill Yields. The Review of

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126.

[3] Hoffman, Dennis L.& Rasche, Rob-

ert H. Assessing Forecast Performance

in a Cointegrated System (1996) Journal

of Applied Econometrics 11, 495-517.

[4] Bidarkota, Prasda V. The Compara-

tive Forecast Performance of Univari-

ate and Multivariate Models: An Appli-

cation to Real Interest Rate Forecasting

(1998) International Journal of Fore-

casting 14, 457-468.

[5] Sarno, Lucio & Thornton, Daniel L.

The Dynamic Relationship Between

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(2003) Journal of Banking & Finance

27, 1079-1110.

[6] Sarno, Lucio & Thornton, Daniel L

& Valente, Giorgio (2006) Federal

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from FRB: Speech, Bernanke — The

Global Saving Glut and the U.S. Cur-

rent Account Deficit –March 10, 2005

(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