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 [email protected]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).
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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
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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 masterβs 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
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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-
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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
βGlobal Saving Glutβ. 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 monthβs interest
rate will provide a reasonable guess for
this monthβs interest rate. The predic-
tion error is captured in the error term.
The equation is as followed:
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π¦π‘ = β π½ππ¦π‘βπ
π
+ ππ‘ (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:
π¦π‘ =
β π½ππ¦πβ1
π
+ β πΌππ₯π‘βπ
π
+ ππ‘ (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
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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.