40
and the bottom left panels, is less pronounced and more gradual. With the flexibility brought
by the additional liquidity premium factor, these two models are also able to fit TIPS yields
almost perfectly.
The problem with Model NL can be further seen by looking at the model-implied 10-year
breakeven inflation, as shown in the upper right panel of Figures 5. The 10-year breakeven
rate implied by Model NL, which by construction should equal the 10-year TIPS breakeven
inflation, appearstoo smooth compared to its data counterpartand missesmostof the short-run
variations in theactual series. The poor fitting of the TIPS breakeven inflation rates highlights
the difficulty that the 3-factor model has in fitting nominal and TIPS yields simultaneously.
29
In contrast, the 10-year breakeven inflation rates implied by Models L-I and L-IId, shown in
the middle and bottom left panels, show substantial variations that roughly match those of
the actual TIPS breakeven inflation rate. In particular, the model-implied and the TIPS-based
breakeven inflation rates peak locally at the beginning of 2000, in the middle of 2001 and
2002, and so on, and the magnitude of their variation are also similar. In these two models,
the gap between the model-implied and the TIPS-based breakeven inflation rates is thesum of
TIPS liquidity premium and TIPS measurement errors.
To quantify the improvement in terms of the model fit, Panels B and C of Table 4 provide
three goodness-of-fit statistics for TIPS yields at the 5-, 7- and 10-year maturities and TIPS
breakeven inflation at the 7- and 10-year maturities, respectively. The first statistic, CORR,
is simply the sample correlation between the fitted series and its data counterpart. Consistent
with the visual impression from Figure 5, allowing a TIPS liquidity premium component
improves the model fit for raw TIPS yields and even more so for TIPS breakeven inflation,
with the correlation between model-implied 10-year TIPS breakeven and the data counterpart
rising from 32% to over 90% once we move from Model NL to the other models. The next
two statistics are based on one-step-ahead model prediction errors from the Kalman Filter, v
t
,
defined in Equation (C-15) in Appendix C, and are designed to capture how well each model
can explain thedatawithoutresorting to largeexogenous shocks or measurement errors. More
specifically, the second statistic is the root mean squared prediction errors (RMSE), and the
third statistic is the coefficient of determination (R
2
), defined as the percentage of in-sample
29
Given the flexible nature of latent-factor model used in this paper,it ispossible that there may exist another
local maximum of the likelihood function underwhich the TIPS yieldsare fitted better,producing a closer match
between the model-implied and the TIPS-based breakeven inflation rates. However, such a fit would certainly
come at the expense of otherundesirable featuresof the model.
23
60
variations of each data series explained by the model:
R
2
=1 ¡
P
T
t=2
v
2
t
P
T
t=2
(y
t
¡
y)
2
;
(52)
where y
t
is the observed series and
ydenotes its sample mean.
30
As we can see from Panels
Band C of Table 4, based on these two metrics, the improvement moving from Model NL to
models with a liquidity factor is notable even for TIPS yields. In other words, the seemingly
reasonable fit of Model NL for raw TIPS yields is only achieved by assuming large exogenous
shocks to the state variables. The fit of Model NL for TIPS breakevens is even worse, with a
R
2
of ¡18:12% at the 10-year maturity. In comparison, all other models with aTIPS liquidity
factor explain more than 88% of the time variations of TIPS breakevens at both maturities.
[Insert Figure 5 about here.]
5.3 Matching Survey Inflation Forecasts
It is conceivable that a model with more parameters like Model L-IId could generate smaller
in-samplefitting errors for variables whosefit is explicitly optimized, but produce undesirable
implications for variables not used in the estimation. To check this possibility, we examine
the model fit for a variable that is not used in our estimation but is of enormous economic
interest, the expected inflation. In particular, we examine how closely the model-implied
inflation expectations mimic survey-based inflation forecasts. Ang, Bekaert, and Wei (2007)
recently provide evidence that survey inflation forecasts outperforms various other measures
of inflation expectations in predicting future inflation. In addition, survey inflation forecast
has the benefit of being a real-time, model-free measure, and hence not subject to model
estimation errors or look-ahead biases that could affect measures based on in-sample fitting of
realized inflation.
31
30
Unlike in a regression setting, a negative value of R
2
could arise because the model expectation and the
prediction errorsare not guaranteed to be orthogonal in a small sample.
31
Alternatively, we could compare the out-of-sample forecasting performance of various models. However,
we doubt the usefulness of such an exercise for two reasons. First, the sample available for carrying out such
an exercise is extremely limited due to the relatively short sample of TIPS. In addition, the large idiosyncratic
fluctuations associated with commonly used price indices would lead to substantial sampling variability in any
metric of forecast performance we use and further complicate the inference problem.
24
39
Panel D ofTable4 reportsthethreegoodness-of-fit statistics, CORR, RMSE and R
2
,for 1-
and 10-year ahead inflation forecasts from the SPF. Among the models, Model NL generates
inflation expectations that agree least well with survey inflation forecasts, producing large
RMSEs and small R
2
statistics at both horizons. This poor fit is especially prominent at the
1-year horizon: the RMSE is large, the correlation between the model and survey forecast is
essentially 0, and the R
2
ishighly negativeat ¡52%. In contrast, all other models, which have
aliquidity factor, generate a reasonable fit with the survey forecasts at both horizons. The
best fit is achieved by Models L-II and L-IId, both of which generate correlations above 80%
and small RMSEs at both horizons and explain a large amount of sample variations in survey
inflation forecasts. Models L-II and L-IId also improve notably upon Model L-I, suggesting
that some cyclical variations in TIPS yields might not be due to movements in the real yields.
Overall, Model L-IId doesnot seemto sufferfrom an overfitting problem. Aswewillseefrom
later sections, this model also generate sensible implications for TIPS liquidity premiums and
inflation risk premiums, further supporting our conclusion.
Avisual comparison of the model-implied inflation expectations and survey forecasts can
help us understand the results in Table 4. The left panels of Figure 6 plot the 1-year infla-
tion expectation based on Models NL, L-I and L-IId, together with the survey forecast. It can
be seen that the Model-NL-implied 1-year inflation expectation contains a large amount of
short-run fluctuations that are not shared by its survey counterpart. It also fails to capture the
downward trend in survey inflation forecasts during much of the sample period. In compari-
son, implied 1-year inflation expectation based on the other models show a visible downward
trend, consistent with the survey evidence. It is interesting that although Models L-I and L-IId
exhibit similar fit to TIPS yields and TIPS breakevens, as can be seen from Figure 5, they are
more differentiable based on their implications for inflation expectations. In particular, while
the 1-year inflation expectation implied by ModelL-IId bearsa high resemblance to the1-year
survey forecast, the same series implied by Model L-I appears to be much more variable than
the survey counterpart.
It is also not surprising that Model NL produces a larger RMSEs for 10-year inflation
expectations than the L-I and L-II models: the upper middle panel of Figure 6 shows that
Model NL completely misses the downward trend in the 10-year survey inflation forecast
since the early 1990s and implies a 10-year inflation expectations that moved little over the
sample period. This is the flip side of the discussions in Section 5.2, where we see a Model-
25
30
NL-implied 10-year real rate that is too variable and is used by the model to explain the entire
decline in nominal yields in the 1990s. Overall, the near-constancy of the long-term inflation
expectation is the most problematic feature of Model NL. Models L-I and L-IId, on the other
hand, produce 10-year inflation expectations that are clearly downward trending, though the
model-implied values are a bit lower than the survey forecast in the early 1990s, as shown in
the two lower panels in the middle column of Figure 6. As can be recalled from Figure 5, the
long-term real yields in these models also display a downward trend, but a much weaker one
compared to that in Model NL.
Finally, the three right panels of Figure 6 plot the model-implied inflation risk premiums
at the 1- and 10-year horizons for the three models under consideration. One immediately
notable feature is that Model-NL implies an inflation risk premium, shown in the upper right
panel, that is negative and increasing over time in the 1990-2007 period. In contrast, as men-
tioned in Section 2, most of the existing studies not using TIPS find that average inflation risk
premium has been positive historically. Furthermore, studies such as Clarida and Friedman
(1984) indicate that the inflation risk premium likely was positive and substantial in the early
1980s and probably has come down since then. As can be seen from Figure 7, which plots
the 10-year inflation risk premium estimates together with the 95% confidence bands for the
three models, even after we take into account sampling uncertainties, long-term inflation risk
premiums implied by Model NL remain negative over most of the sample period. In compar-
ison, the two models that allow for a liquidity premium, Models L-I and L-IId, both generate
10-year inflation risk premiums that are positive and fluctuate in the 0 to 1% range over the
same sample period. The short-term inflation risk premiums implied by these two models, on
the other hand, are fairly small, consistent with our intuition.
[Insert Figure 6 about here.]
[Insert Figure 7 about here.]
5.4 Summary
In summary, we find that Model NL, which equates TIPS yields with true underlying real
yields, fares poorly along a number of dimensions, including generating a poor fit with the
TIPS data as well as unreasonable implications for inflation expectations and inflation risk
26
35
premiums. This underscores the need to take into account a liquidity premium in modeling
TIPS yields. In contrast, models that allows for a TIPS liquidity premium, Models L-I and L-
II, improves upon Model NL in all three aspects and in particular produce long-term inflation
expectations that agree quite well with survey forecasts.
Among models allowing aliquidity premium in TIPS yields, Models L-II and ModelL-IId
generate short-term inflation expectations that matches survey counterparts better than Model
L-I, suggesting itisimportantto allow fora systematiccomponent in TIPS liquidity premiums.
Finally, alikelihood ratio testrejects Models L-IIin favor ofour preferred model, Model L-
IId, which features a deterministic trend in TIPS liquidity premium that isdesigned to capture
the “newness effect” during the early years of TIPS. In the remainder of our analysis we’ll be
mainly focusing on this model.
6 TIPS Liquidity Premium
6.1 Model Estimates of TIPS Liquidity Premiums
Once we estimate the model parameters and the state variables, we can calculate the TIPS
liquidity premiums at various maturities based on Equation (35). The top and the middle
panels of Figure 8 plot the 5-, 7- and 10-year liquidity premiums implied by Models L-I
and L-II, respectively, while the bottom panel shows the the deterministic and the stochastic
components of the liquidity premiums based on Model L-IId.
Three things are worth noting from this graph: First, all three panels show that liquidity
premiums exhibit substantial time variations at all maturities. The substantial variabilities at
maturities as long as 10 years are in part due to thefact that the independent liquidity factor is
estimated to be very persistent under the risk-neutral measure. As can be seen from Table 3,
the risk-neutral persistence of the liquidity factor, ~•
⁄
=~• + ~¾
~
‚
1
,is estimated to be very small
at around 0.1 in all models and is tightly estimated, with a standard error of about 0.006. In
contrast, thepersistenceparameterunderthephysicalmeasure, ~•, isnot asprecisely estimated,
with typical values of around 0.20 and typical standard errors of around 0.27.
Second, the term structure of TIPS liquidity premiums is relatively flat at all times under
Model L-I, while under Model L-II, the term structure has a mild downward-sloping behavior
27
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