model-implied risk-free real yields and BEI, for Models NL-noIE, NL, and LII, respectively.
By construction, the model-implied TIPS yields (TIPS BEI) and the model-implied risk-free real
yields (BEI) coincide under Model NL(-noIE), after adjusting for the indexation lag.
The top and middle left panels of Figure 4 show that both versions of Model NL interpret the
decline in 10-year TIPS yields from 1999 to 2004 as part of a broad downward shift in real yields
from 7% in early 1990s to about 2% around 2003. The less than 5% decline in the 10-year
nominal yield over the same period is therefore attributed almost entirely to a lower real yield,
leaving little room for lower inﬂation expectations or risk premiums. However, the literature
generally ﬁnds that long-term inﬂation expectations likely have edged down over this period,
and it is hard to imagine economic mechanisms that would generate such a large decline in
long-term real yields. Furthermore, although the two NL models match the general trend of TIPS
yields, they both have problems ﬁtting the time variations, frequently resulting in large ﬁtting
errors, especially in the early part of the sample and again during the recent ﬁnancial crisis. In
contrast, the bottom left panel of Figure 4 shows a less pronounced and more gradual decline in
real yields based on Model LII, which is able to ﬁt TIPS yields almost perfectly, as shown by the
juxtaposition of the red and black lines.
In addition, the top and middle right panels of Figure 4 show that the 10-year BEI implied by
the two NL models, which by construction should equal the 10-year TIPS BEI after adjusting for
the indexation lag, appears too smooth compared to the actual data and misses most of its
short-run variations. The poor ﬁtting of the TIPS BEI highlights the difﬁculty that the 3-factor
model has in ﬁtting nominal and TIPS yields simultaneously. In contrast, the 10-year BEI implied
by Model LII, shown in the bottom right panel of Figure 4, exhibits substantial variations that
closely match those of the actual TIPS BEI. In particular, the model-implied and the TIPS-based
Model-implied true breakevensare calculated as the difference between model-implied nominal yieldsand
model-implied real yields of comparable maturities. Model-implied valuesare calculated using smoothed estimates
ofthe state variables. Resultsfor Model LI are broadly similarto those forModel LII and are reported in the
See Kozicki and Tinsley (2006), forexample.
BEI plunge toward the end of 2008 following the Lehman collapse, consistent with reports of
substantial liquidation of TIPS holdings over this period.
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 BEI at
the 7- and 10-year maturities, respectively. The ﬁrst statistic, CORR, is the simple sample
correlation between the ﬁtted series and its data counterpart. The next two statistics—the root
mean squared prediction errors (RMSE) and the coefﬁcient of determination (R
)—are based on
one-step-ahead model prediction errors from the Kalman Filter, and are designed to capture how
well each model can explain the data without resorting to large exogenous shocks or measurement
All three statistics suggest that including a TIPS-speciﬁc factor improves the model ﬁt
for raw TIPS yields and even more so for TIPS BEI.
[Insert Figure 4 about here.]
C. Matching Survey Inﬂation Forecasts
Next, we brieﬂy examine how closely the model-implied inﬂation expectations mimic
survey-based counterparts. Ang et al. (2007) 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.
Panel D of Table 4 reports the three goodness-of-ﬁt statistics, CORR, RMSE and R
and 10-year ahead SPF inﬂation forecasts. When survey inﬂation forecasts are not used in the
estimation, Model NL-noIE generates inﬂation expectations that departs signiﬁcantly from survey
inﬂation forecasts, as can be seen from the large RMSEs and small and even negative R
For a briefaccount ofthe episode, see Hu and Worah (2009).
is deﬁned as the percentage ofin-sample variationsof each data seriesexplained by the model. Unlike
in a regression setting, a negative value of R
could arise because the model expectation and the prediction errors are
not guaranteed to be orthogonal in a small sample.
Avisual comparison between the model-implied inﬂation expectations and survey forecasts,
plotted in the ﬁrst two top panels of Figure 5, show that Model NL-noIE fails to capture the
downward trend in survey inﬂation forecasts during much of the sample period, and implies that
the 10-year inﬂation expectation barely moved. This is the ﬂip side of the discussions in Section
B, where Model NL-noIE implied a 10-year real rate that was too variable and explained the
entire decline in nominal yields during the 1990s. Adding information from survey inﬂation
forecasts brought Model NL-implied inﬂation expectations more in line with survey forecasts, as
can be seen from the ﬁrst two middle panels of Figure 5. Model NL then explains the smaller
decline in nominal yields by generating a sustained increase in inﬂation risk premiums from
around -1.5% in the early 1990s to the current level of slight above zero, in contrast to previous
ﬁndings of an overall positive and generally declining inﬂation risk premiums over the past three
decades (see Section VII). By comparison, Model LII, which allows for a TIPS-speciﬁc factor
and is shown in the bottom right panel of Figure 5, generates 10-year inﬂation risk premiums that
are mostly positive and ﬂuctuate in the 0 to 0.5% range, and short-term inﬂation risk premiums
that are fairly small and became persistently negative during the recent ﬁnancial crisis.
D. Out-of-Sample Forecasting
It is conceivable that a model with more parameters like Model LII could generate smaller
in-sample ﬁtting errors for variables whose ﬁt is explicitly optimized, but preforms worse out of
sample. To check this possibility, we run an out-of-sample forecasting horse race between the
four term structure models estimated above, a random walk model, the monthly Michigan survey,
the monthly Blue Chip Financial Forecasts (BCFF) survey, and the quarterly SPF, and report the
root mean squared errors (RMSEs) in Table 5. Three conclusions emerge: First, the term structure
models perform as well as the two professional surveys—SPF and BCFF—over a common
sample period, and much better than the Michigan survey and the random walk model. Second,
within the term structure models, adding survey inﬂation forecasts help improve the forecasting
performance of Model NL. Finally, Model LII outperforms Model NL, with the improvement
more pronounced at the longer 2-year horizon.
[Insert Table 5 about here.]
In addition, the robustness checks reported in the Supplementary Appendix show that the
parameter estimates and good performance of Model LII remain largely intact when re-estimated
over a pre-crisis sample. Therefore, we will mainly focus on this model in the remainder of our
E. The Effect of Indexation Lags
Graph A of Figure 6 shows the Model LII-implied differences between the indexed bond
,speciﬁed in equation (28), and the real yields y
,speciﬁed in equation (15), for
maturities of 5, 7, and 10 years. Those estimates are generally small but rose to 30 basis points at
the 10-year maturity and 70 basis points at the 5-year maturity in December 2008, when 3-month
CPI inﬂation was running at an annual rate of about 13%. The differences between indexed
bond yields and fully-indexed real yields are estimated to be almost entirely due to the expected
divergence between inﬂation over the 2.5 months prior to time t and inﬂation over the 2.5 months
before the maturity of the bond, the second term on the left hand side of equation (27). The last
term, the indexation lag premium, is estimated to be generally small, varying between -5 and 3
basis points at the 10-year maturity, consistent with Risa (2001), and slightly larger at shorter
maturities. Similar patterns are observed in all models we estimate.
VI. What Drives the TIPS Spread
In this section, we examine estimates of the TIPS spread from Model LII and show them to be
mostly explained by proxies of TIPS liquidity. Other factors, such as CPI seasonality, TIPS
deﬂation ﬂoors and special demand for nominal Treasuries, also account for a small portion of
their variations, though the effects of those factors appear to be much less signiﬁcant.
A. Model Estimates of the TIPS Spread
The TIPS spread implied by Model LII is plotted in Graph B of Figure 6 for maturities of 5, 7
and 10 years. Three main features emerge: First, this spread exhibits substantial time variations at
all maturities. Such variability at maturities as long as 10 years is in part due to the estimated high
persistence of the TIPS-speciﬁc factor under the risk-neutral measure.
Second, the term
structure of the estimated TIPS spread is downward sloping in 2001-2004 and 2008-2011. A
market price of risk on the independent TIPS-speciﬁc factor that is on average positive, as is the
case here, would contribute to a downward-sloping term structure of the TIPS spread.
Finally, the TIPS spreads were fairly high (1.5-2% range) when TIPS were ﬁrst introduced but
had been steadily declining until around 2004, likely reﬂecting the maturing process of a
relatively new ﬁnancial instrument. The spread surged to record-high levels (3%) after the
Lehman bankruptcy, reﬂecting a sharp increase in transaction and funding costs for TIPS as well
as heightened risk aversion.
The heavy ﬂight-to-safety ﬂows into the nominal Treasury market
likely also contributed to larger demand imbalances between nominal Treasuries and TIPS, which
in turn should lead to a larger liquidity premium in TIPS yields relative to their nominal
counterparts. Similarly sharp rises in liquidity premiums and/or illiquidity measures were seen in
other key markets during the recent ﬁnancial crisis, including equity and corporate bond
B. Link to Observable TIPS Liquidity Measures
Given the unobserved nature of the TIPS-speciﬁc factor in our model, one may question
whether it is indeed capturing TIPS liquidity rather than other components of TIPS yields that are
Ascan be seen from Table 3, its risk-neutral persistence, ~
=~ + ~
,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.
These estimates are somewhat largerthan those in Pﬂueger and Viceira (2013) but broadly in line with those in
Abrahams et al. (2013).
See, for example,Bao et al. (2011), Dick-Nielsen et al.(2012),and Brennan, Huh, and Subrahmanyam (2012).
orthogonal to nominal Treasury yields. In this section, we provide strong evidence in favor of a
liquidity premium interpretation of the TIPS spreads by linking them to various observable
measures of TIPS liquidity, while controlling for other factors that might have contributed to a
wedge between TIPS yields and indexed bond yields.
One immediate difﬁculty we face is the lack of real-time, forward-looking measures of
liquidity conditions in the TIPS market that are available over a reasonably long sample period.
For example, as shown in the Graph A of Figure 7, one widely used measure of illiquidity, the
bid-ask spread, only became available for TIPS in 2003 from TradeWeb.
Ameasure that is
available over a longer sample is the relative trading volumes of TIPS versus nominal Treasury
coupon securities, plotted in Graph B.
This measure remained low up to mid 2004 and then rose
substantially, suggesting steady improvement in TIPS liquidity during the pre-crisis sample
period. The rise in relative trading volumes coincides roughly with the decline in our estimated
TIPS spread over the same time window, with the two series showing a highly negative
correlation of about 80% over the period of 1999 to 2007.
[Insert Figure 7 about here]
Another observable measure of TIPS liquidity used in the literature is the average absolute
ﬁtting errors from the Svensson TIPS yield curve, plotted in Graph C of Figure 7. This measure
captures funding constraints and limits to arbitrage that prevent investors from eliminating
deviations of yields from their fundamental values as measured from a ﬁtted yield curve.
plausible that during a ﬁnancial crisis, capital becomes more scarce and risk aversion run higher,
leaving signiﬁcant arbitrage opportunities unexploited. According to this measure, liquidity
TradeWeb data, ThomsonReuters.
We construct the measure as13-week averages of weekly dealer transaction volumesin TIPS divided by those
in nominal Treasury coupon securities using data reported by primary dealers and collected by the Federal Reserve
Bank of New York under Government Securities DealersReports(FR-2004).
Similar measureshave been used by Fontaine and Garcia (2012)and Hu, Pan,and Wang (2012)to measure the
liquidity of nominal Treasury securities,and by Grishchenko and Huang (2013) forTIPS.
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