Keywords: Treasury inflationprotected securities (TIPS), expected inflation, inflation risk premium, liquidity, SPF, term structure
Abstract:
This paper presents a joint study of yields on nominal Treasury securities and those on Treasury InflationProtected Securities (TIPS) in a noarbitrage asset pricing framework. Since its inception in 1997, the market for TIPS has grown substantially and now comprises about 8% of the outstanding Treasury debt market. More than a decade's TIPS data thus accumulated is a rich source of information to academic researchers and market participants alike. Because TIPS are securities whose coupon and principal payments are indexed to the price level, information about yields on these "real bonds" has direct implications for asset pricing models, many of which are written in terms of real consumption. Meanwhile, realtime TIPS data has attracted much attention from policy makers and market participants as a source of information about the state of the economy. In particular, the differential between yields on nominal Treasury securities and on TIPS of comparable maturities, often called the "breakeven inflation rate" (BEI) or "inflation compensation", has been used by policy makers and market participants as a proxy for market's inflation expectation. For example, the minutes of FOMC meetings often take note of changes in TIPS yields since the previous meeting,^{1} and explicit references to TIPS breakeven rates in speeches by Fed officials are common.^{2} Similarly, financial press frequently cite TIPS breakeven rates when discussing inflation expectations.
However, two difficulties arise in such interpretation of the TIPS breakeven inflation. First, TIPS breakeven inflation contains the inflation risk premium, which is the extra compensation investors in nominal bonds demand for bearing inflation risks. Second, TIPS has only been introduced recently and during its existence has been a less liquid instrument compared to nominal Treasury securities. The additional "liquidity premium" TIPS investors require for holding such instruments will drive up TIPS yields and depress the TIPS breakeven inflation. While the inflation risk premium has been studied by many researchers,^{3} the liquidity premium embedded in TIPS breakeven inflation has to our knowledge never been examined in the asset pricing literature.
The goal of this paper is to document the existence of liquidity premium in TIPS yields and to quantitatively characterize its behavior. To this end, we estimate and contrast several models for real and nominal yields, where the liquidity premium is either ignored or modeled as following different processes. All models we use are from the affineGaussian noarbitrage term structure family and allow a rich dynamics in both the inflation risk premium and in nominal and real term premia.
Our main findings can be summarized as follows. First, a general 3factor affine term structure model that ignores the liquidity premium generates large pricing errors for TIPS as well as counterfactual implications for inflation expectations and inflation risk premiums. In comparison, after incorporating an additional liquidity factor, the three 4factor models that we estimate lead to notably smaller TIPS pricing errors, generate reasonable inflation risk premiums, and produce modelimplied inflation expectations that agree well with survey inflation forecasts. Second, the liquidity premium estimates from all 4factor models share the feature that it was large (12%) in the early years but declined in recent years, consistent with the notion that TIPS market liquidity conditions have improved over time and that TIPS pricing has possibly become more "efficient". In particular, around 80% of the variations in our estimates of TIPS liquidity premiums can be explained by variables related to the liquidity conditions in the TIPS market. Third, our best model for TIPS liquidity premiums has three parts, including a deterministic trend that captures the gradual but steady decline in TIPS liquidity premiums from the early years, a TIPSspecific factor that is independent of the nominal bond factors, as well as a component that is correlated with the rest of the economy. Finally, a variance decomposition shows that TIPS liquidity premiums explain more than 40% of variations in TIPS breakeven inflation, while that percentage declines to about one quarter in the long run.
The results in this paper shows that one needs to be careful in using the TIPS breakeven inflation rate as a proxy for inflation expectation, since an economically significant TIPS liquidity premium, on top of the inflation risk premium, could drive a large wedge between the TIPS breakeven inflation and inflation expectation. This problem seems to be especially severe in the early years of the TIPS market.
The remainder of this paper is organized as follows. In Section 2, we provide evidence that TIPS yields and TIPS breakeven inflation contain an additional factor, likely reflecting the illiquidity of TIPS, beyond those driving the nominal interest rates. Section 3 spells out the details of the noarbitrage models we use, including the specification of the additional liquidity factor, and Section 4 explains the empirical methodology. Section 5 presents the main empirical results based on one model assuming zero TIPS liquidity premium and three models in which the TIPS liquidity premium is assumed to follow different specifications. Section 6 provides further discussions on the model estimates of the TIPS liquidity premiums and shows that they are indeed linked to the liquidity conditions in the TIPS markets. Section 7 provides some variance decomposition results for TIPS yields, TIPS breakeven inflation and nominal yields. Finally, Section 8 concludes.
In this section we present some simple analysis suggesting there exists a factor that is important for explaining the variations in TIPS yields but not as crucial for modeling nominal interest rates. We further argue that this factor is related to the illiquidity of the TIPS market. This serves as the motivation for introducing a TIPSspecific factor when we model nominal and TIPS yields jointly in later sections.
Simple Regression Analysis
In our first analysis, we regress the 10year TIPS breakeven inflation rate, defined as the spread between the 10year nominal yield and the 10year TIPS yield, on 3month, 2year and 10year nominal yields plus a constant:^{4}
Principal Components Analysis
Next, we conduct a principal component analysis (PCA) of the cross section of the nominal and TIPS yields. It is well known that, in the case of nominal yields, three factors explain most of the nominal yield curve movements. This is confirmed in Panel B of Table 1, which shows that more than 97% of the variations in weekly changes of 3 and 6month and 1, 2, 4, 7 and 10year nominal yields can be explained by the first three principal components. However, once we add the 5, 7 and 10year TIPS yields, at least four factors are needed to explain the same amount of variance. Panel C of Table 1 reports the correlations between the first four PCA factors extracted from nominal yields alone and the first four PCA factors extracted from nominal and TIPS yields combined. It is interesting to note that, once we add TIPS yields to the analysis, the first, the second and the fourth factors largely retain their interpretations as the level, slope and curvature of the nominal yields curve, as can be seen from their high correlation with the first, the second and the third nominal factors, respectively. However, the third PCA factor extracted from nominal and TIPS yields combined is not highly correlated with any of the nominal PCA factors. The results shown here and the simple regression analysis shown above have an interesting parallel with the literature on the unspanned stochastic volatility (USV) effect: using a simple regression analysis, CollinDufresne and Goldstein (2002) argued that bond derivatives contain a factor that is not spanned by the yield curve factors, and Heidari and Wu (2003) reported evidence for unspanned stochastic volatility using a PCA analysis.^{5}
A Case for TIPS Liquidity Premium
A promising interpretation of the TIPSspecific factor we found above is that it reflects a "liquidity premium": investors would demand a compensation for holding a relative new and illiquid instrument like TIPS, especially in the early years.
Indeed, several measures related to TIPS market liquidity conditions, as well as anecdotal reports, indicate that the liquidity in TIPS market was much poorer than that of nominal securities, and that TIPS market liquidity improved over time, although this improvement was not a smooth, steady process. The top panel of Figure 1 shows the gross TIPS issuance over the period 19972007. The TIPS issuance dipped slightly in 20002001 before rising substantially in 2004. According to Sack and Elsasser (2004), there were talks around 2001 that the Treasury might discontinue the TIPS due to the relatively weak demand for TIPS. TIPS outstanding, shown in the bottom panel of Figure 1, began to grow at a faster pace from 2004 onward and now exceeds 400 billion dollars. Figure 2 tells a similar story from the demand side: TIPS transaction volumes grew by sixfold during our sample period and TIPS mutual funds also experienced significant growth.^{6} In view of this institutional history, it is not unreasonable to suppose that TIPS contained a significant liquidity premium at least in its early years and that the liquidity premium have edged lower over time.
A liquidity premium in TIPS can help resolve a seeming inconsistency between survey inflation forecasts (10year SPF survey and Michigan longterm survey) and the 10year TIPS breakeven inflation, all of which are plotted in Figure 3. Recall that the true breakeven inflation, defined as the yield differential between nominal and real bonds of comparable maturities and liquidity features, is the sum of expected inflation and the inflation risk premium, and can be considered as a good measure of the former if the second term is relatively small and does not vary too much over time. However, Figure 3 shows that this is not the case: the TIPS breakeven inflation lied below both measures of survey inflation forecasts almost all the time before 2004.^{7} ^{8} ^{9} Such disparity cannot be attributed solely to the existence of inflation risk premium, as such an explanation would require the inflation risk premium to be mostly negative in the 19992007 period and highly volatile, which stands in contrast with most studies in the literature that find inflation risk premiums to be positive on average and relatively smooth.^{10}
On the other hand, a positive TIPS liquidity premium would push the TIPSbased breakeven inflation below the true breakeven inflation and, if the TIPS liquidity premium exceeds the inflation risk premium in absolute terms, even lower than survey inflation forecasts. Furthermore, part of the volatility of the TIPS breakeven inflation rate may be due to the volatility of the TIPS liquidity premium.
In order to study these issues quantitatively, we need a framework for identifying and measuring the relevant components, including the TIPS liquidity premium, inflation expectations, and inflation risk premium. For this purpose, we use the noarbitrage term structure modeling framework, to which we now turn.
This section details the noarbitrage framework that we use to model nominal and TIPS yields jointly. The noarbitrage approach has the benefit of avoiding the tight assumptions that go into structural, utilitybased models while still allowing term structure variations to be modeled in a dynamically consistent manner by requiring the cross section of yields to satisfy the noarbitrage restrictions.
We assume that real yields, expected inflation and nominal yields are driven by a vector of three latent variables, , which follows a multivariate Gaussian process,
The nominal short rate is specified as
The nominal pricing kernel takes the form
The price level processes takes the form:
A real bond can be thought of as a nominal asset paying realized inflation upon maturity. Therefore, the real and the nominal pricing kernels are linked by the noarbitrage relation
(9)  
(10) 
(11) 
(12) 
By the definition of nominal and real pricing kernels, the time prices of period nominal and real bonds, and , are given by
(17) 
Following the standard literature,^{11} it is straightforward to derive a closedform solution for the bond prices:
(19) 
(20)  
(21) 
Nominal and real yields therefore both take the affine form,
In this model, inflation expectations also take an affine form,
From equations (22)(24), it can be seen that both the breakeven inflation rate, defined as the difference between zero coupon nominal and real yields of identical maturities, and the inflation risk premium, defined as the difference between the breakeven inflation rate and the expected log inflation over the same horizon, are affine in the state variables:
Using Equation (8) we can write the price of a period nominal bond as
(28) 
Given the evidence presented in Section 2 on the existence of a TIPSspecific factor, we allow the TIPS yield to deviate from the true underlying real yield. The spread between the TIPS yields and the true real yield,
We model as containing a stochastic component and a deterministic component. To model the stochastic part, we assume that the investors discount TIPS cash flows by adjusting the true instantaneous real short rate with a positive liquidity spread, resulting in a TIPS yield that exceeds the true real yield by
The instantaneous liquidity spread is given by
(33) 
Appendix B shows that the stochastic part of the TIPS liquidity premium takes the affine form
To accommodate potential nonstationarities associated with the inception of the TIPS market, we also allow for a maturityindependent downwardtrending deterministic component in the liquidity premium, which takes the form
The total liquidity premium on a year TIPS in our "full model" is then the sum of the two components:
(36) 
Besides its tractability, the affineGaussian bond pricing framework used here allows for a flexible correlation structure between the factors. As inflation risk premiums arise from the correlation between the real pricing kernel and inflation, it is important to allow for a general correlation structure. On the other hand, the affineGaussian setup does not capture the timevarying inflation uncertainty and therefore cannot further decompose inflation risk premiums into the part due to timevarying inflation risks and timevarying prices of inflation risk. Nonetheless, the affineGaussian model could still provide a reasonable estimate of inflation risk premium, similar to the way it generates reasonable measures of term premia despite its inability to capture timevarying interest rate volatilities. We view the general affineGaussian model as an important benchmark to be investigated before more sophisticated models can be explored.
Some of the models studied in the earlier literature, such as Pennacchi (1991) and Campbell and Viceira (2001), can be viewed as a special case of this model. For example, Pennacchi (1991)'s model corresponds to a twofactor version of our model with constant market price of risk. Campbell and Viceira (2001) is also a special case of this model, but their real term structure has a lower dimension than the nominal term structure (nominal yields are described by 2 factors and real yields are described by 1 factor). In this paper, we let the real term structure have as many factors as the nominal term structure; if the real term structure is truly lowerdimensional than nominal term structure, we let the data decide on that. A related point is that in a reducedform setup like ours, one cannot make a distinction between real and nominal factors, as correlation effects in the general model make such a distinction meaningless.
To our knowledge, this is the first paper to model liquidity premium in TIPS in a noarbitrage framework. There is a large literature studying the pricing implications of indexed bonds using data from other countries with longer histories of issuing inflation linked securities,^{16} There have also been studies of US real yields and inflation risk premia that use realized inflation^{17} or survey inflation forecasts^{18} without incorporating information from indexed bonds. Due to the short data history, studies using U.S. inflationlinked assets, including TIPS or inflation swaps, are fairly recent and relatively few.^{19} In addition, most of these studies take TIPS at their face value, and typically find a real yield that seems too high as well as inflation risk premium estimates that are insignificant or negative in the early sample when TIPS was first introduced. In contrast, this paper shows that there is a persistent liquidity premium component in TIPS prices, which, if not properly taken account of, will bias the results.
We only impose restrictions that are necessary for achieving identification so as to allow a maximally flexible correlation structure between the factors, which has shown to be critical in fitting the rich behavior in risk premiums that we observe in the data.^{20} In particular, we employ the following normalization:
To summarize, we estimate four models that differ in how TIPS liquidity premium is modeled, including one model that equates TIPS yields with true real yields and assumes zero liquidity premium on TIPS (Model NL), a model with an independent liquidity factor (Model LI), a model allowing the TIPS liquidity premium to be correlated with other state variables (Model LII), and the most general model (Model LIId) in which TIPS liquidity premiums also contain a deterministic trend. Table 2 summarizes the models and the parameter restrictions. Two things are worth noting here: First, as shown in Section 3.5, Models NL, LI and LII can all be considered as special cases of Model LIId. Second, Model NL has a 3factor representation of TIPS yields, while in all other models TIPS yields have a 4factor specification.
We use 3 and 6month, 1, 2, 4, 7, and 10year nominal yields and CPIU data from January 1990 to March 2007. In contrast, our TIPS yields are restricted by data availability and cover a shorter period from January 1999 to March 2007, with the earlier period without TIPS data (19901998) treated as missing observations. Both nominal and TIPS yields, shown in Figure 4, are based on zerocoupon yield curves fitted at the Federal Reserve Board^{22} and are sampled at the weekly frequency, while CPIU inflation is available monthly and assumed to be observed on the last Wednesday of the current month.^{23} As discussed in more details in Appendix A, shortermaturity TIPS yields are affected to a larger degree by the problem of indexation lags. In addition, no more than one TIPS with a maturity of below 5 years existed before 2002, hence nearmaturity zerocoupon TIPS yields cannot be reliably estimated during that period. We therefore only use 5, 7, and 10year TIPS yields in our estimation. Because the models we estimate do not accommodate seasonalities, we use seasonallyadjusted CPI inflation in the estimation. TIPS are indexed to nonseasonallyadjusted CPI; however, our use of seasonallyadjusted CPI is not expected to matter much for the relatively long maturities that we examine.
The sample period 19902007 was chosen as a compromise between having more data in order to improve the efficiency of estimation, and having a more homogeneous sample so as to avoid possible structural breaks^{24} in the relation between term structure variables and inflation. This sample period roughly spans Greenspan's tenure and a little bit of Bernanke's as well.
Results from Kim and Orphanides (2006) suggest that the standard technique of estimating our models using only nominal and TIPS yields and inflation data for a relative short sample period of 19902007 will almost surely run into small sample problems: variables like and may not be reliably estimated, and the estimated model typically generates a path of expected future short rate over the next 5 to 10 years that is too smooth compared to surveybased measures of market expectations.^{25} Therefore, we supplement the aforementioned data with survey forecasts of 3month Tbill yields to help stabilize the estimation and to better pin down some of the model parameters. Note that survey inflation forecast data are not used in the estimation, as we would like to get a measure of inflation expectations from yields data, independently of other sources of information about inflation expectation.
To be specific, we use the 6month and 12monthahead forecasts of the 3month Tbill yield from Blue Chip Financial Forecasts, which are available monthly, and allow the size of the measurement errors to be determined within the estimation. We also use longrange forecast of 3month Tbill yield over the next 5 to 10 years from the same survey, which are available twice a year, with the standard deviation of its measurement error fixed at a fairly large value of 0.75% at an annual rate. This is done to prevent the longhorizon survey forecasts from having unduly strong influence on the estimation, and can be viewed as similar to a quasiinformative prior in a Bayesian estimation.
We denote the shorthorizon survey forecasts by and and the longrange forecast by . The standard deviation of their measurement errors are denoted denoted and , respectively. These surveybased forecasts are taken as noisy measures of corresponding true market expectations. More specifically, we have that for the shortterm survey forecasts,
(40) 
(41) 
(42) 
We rewrite the model in a statespace form and estimate it by the Kalman filter. Denote by the augmented state vector including the log price level, , and the TIPS liquidity factor, . The state equation is derived as Euler discretization of equations (2), (6) and (33) and can be written in a matrix form as
and 
(46) 
We specify the set of nominal yields as , and the set of TIPS yields as , and collect in all data used in the estimation, including the log price level , all nominal yields , all TIPS yields , and 6 monthahead, 12 monthahead, and longhorizon forecasts of future 3month nominal yield:
(47) 
We assume that all nominal and TIPS yields and survey forecasts of nominal short rate are observed with error. The observation equation therefore takes the form
(49) 
Based on the state equation (46) and the observation equation (49), it is straightforward to implement the Kalmanfilter and perform the maximum likelihood estimation. The details are given in Appendix C. Two aspects are worth noting here: first, the log price level is nonstationary, so we use a diffuse prior for when initializing the Kalman filter. Second, inflation, survey forecast and TIPS yields are not available for all dates, which introduces missing data in the observation equation and are handled in the standard way by allowing the dimensions of the matrices and in Equation (49) to be timedependent (see, for example, Harvey (1989, sec. 3.4.7)).
Note that all four versions of our models can be accommodated in the above setup. To implement Model NL without the liquidity premium, one simply set and , and fix , , and at arbitrary values as those variables do not enter the likelihood function of Model NL. To implement Model LI with the independent liquidity factor only, one would fix .
To facilitate the estimation and also to make the results easily replicable, we follow the following steps in estimating all our models:
In this Section, we discuss and compare the empirical performance of the various Models. As we shall see, there are notable differences between the model equating TIPS yields with the true real yields (Model NL) and the models that allow the two sets of yields to differ by a liquidity premium component (Models LI, LII and LIId).
Parameter Estimates
Table 3 reports the parameter estimates for all four models. Four things are worth noting here: First, estimates of parameters governing the nominal term structure are almost identical across the three models; under our setup, these parameters seem to be fairly robustly estimated. In particular, all estimations uncover a very persistent factor with a half life of about 13 years. Note also that all four models exhibit a similar fit to nominal yields and survey forecasts of nominal shortterm interest rates. For example, the nominal yield fitting errors ( ) are fairly small in all four models: except for the 3month yield which has of about 10 basis points, other maturity yields have of 5 basis points or less.
Second, there are notable differences in the estimates of parameters governing the expected inflation process. In particular, the loading of the instantaneous inflation on the second and the most persistent factor is negligible in the model without a TIPS liquidity factor (Model NL) but becomes positive and significant in the three models with a TIPS liquidity factor (Models LI, LII and LIId). As a result, the monthly autocorrelation of oneyearahead inflation expectation is about 0.9 in Model NL but above 0.99 in all other models. As we will see later, the lack of persistence in the inflation expectation process prevents Model NL from generating meaningful variations in longerterm inflation expectations as we observe in the data.
Third, parameter estimates for the TIPS liquidity factor process are significant in both Models LI and LII and assume similar values. The estimated market price of liquidity risk carries the expected negative sign, as one would generally expect any deterioration of liquidity conditions to occur during bad economic times. In Models LII and LIId, the loading of the instantaneous TIPS liquidity premium on each of the three state variables, , is estimated to be indistinguishable from zero; however, a Wald test shows that they are jointly significant.
Finally, the fit to TIPS yields are significantly better in models with a TIPS liquidity factor, as can be seen from the smaller estimates of the standard deviations of TIPS measurement errors. For example, for the 10year TIPS, the fitting errors from the LI and LII models are 6 basis points, while the fitting error from Model NL is 35 basis points. The fitting errors are found to have a substantial serial correlation. For example, in the case of the 5year TIPS, we obtain a weekly AR(1) coefficient of 0.96, 0.91, and 0.91 for Model NL, LI model, and LII model, respectively. The finding of serial correlation in term structure fitting errors are however a fairly common phenomenon, and have been noted by Chen and Scott (1993) and others.
Overall Fit
Panel A of Table 8 reports some test statistics that compare the overall fit of the three models. We first report two information criteria commonly used for model selection, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). Both information criteria are minimized by the most general model, Model LIId.
We also report results from likelihood ratio (LR) tests of the three restricted models, Models NL, LI and LII, against their more general counterparts, Model LI, LII and LIId, respectively. Compared to Model LI, Model NL imposes the restriction . The standard likelihood ratio test does not apply here because the nuisance parameters, , , and , are not identified under the null (Model NL) and appear only under the alternative (Model LI).^{26} Here we follow Garcia and Perron (1996) and calculate a conservative upper bound for the significance of the likelihood ratio test statistic as suggested by Davies (1987). In particular, denote by the vector of nuisance parameters of size , and define the likelihood ratio statistic as a function of :
where is the likelihood value of the alternative model for any admissible values of the nuisance parameters , and is the maximized likelihood value of the null model. For an estimated LR value of , Davies:87:B derives an upper bound for its significance as where represents the Gamma function and V is defined as Garcia and Perron (1996) further assumes that the likelihood ratio statistic has a single peak at , which reduces to . Apply this procedure to testing the null of Model NL against the alternative of Model LI gives an estimate of the maximal p value of essentially zero, hence Model NL is overwhelmingly rejected in favor of Model LI. With the LR statistic estimated as with 1 degree of freedom, we feel confident that using alternative econometric procedures to deal with the nuisance parameter problem is unlikely to overturn the rejection.The LR test of Model LI against Model LII, on the other hand, is fairly standard. Based on the likelihood estimates of the two models, we obtain a LR statistic of
and are able to reject Model LI in favor of Model LII at the level based on a distribution.Finally, Model LII is rejected in favor of the full model, Model LIId, based on the Davies (1987) procedure, with a large LR test value of 433.41.
The estimated standard deviations of TIPS measurement errors reported in the previous section suggest that Model NL has trouble fitting the TIPS yields. This section looks at the fit of TIPS yields and TIPS breakeven inflation across models in more details.
The three left (right) panels of Figures 5 plot the actual and the modelimplied TIPS yields (TIPS breakeven inflation) based on Models NL, LI and LIId, respectively, together with the real yields (the true breakeven inflation) as implied by the three models.^{27} By construction, the modelimplied TIPS yields and the modelimplied real yields coincide under Model NL.
The top left panel of Figures 5 shows that, according to Model NL, the downward path of 10year TIPS yields from 1999 to 2004 is part of a broader decline in real yields since the early 1990's, with real yields estimated to have come down from a level as high as in the early 90's to about around 2003. During the same period, the 10year nominal yield declined from around to a little over . Therefore, Model NL attributes the decline of 10year nominal yield entirely to a lower real yield, leaving little room for lower inflation expectation or lower inflation risk premiums. While there is some empirical evidence suggesting that longterm inflation expectations may have edged down during this period,^{28} it is hard to imagine economic mechanisms that would generate such a large decline in longterm real yields. Furthermore, although Model NL matches the general trend of TIPS yields during this period, it has problem fitting the time variations, frequently resulting in large fitting errors. In contrast, the decline in real yields as implied by Models LI and LIId, plotted in the middle 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 modelimplied 10year breakeven inflation, as shown in the upper right panel of Figures 5. The 10year breakeven rate implied by Model NL, which by construction should equal the 10year TIPS breakeven inflation, appears too smooth compared to its data counterpart and misses most of the shortrun variations in the actual series. The poor fitting of the TIPS breakeven inflation rates highlights the difficulty that the 3factor model has in fitting nominal and TIPS yields simultaneously.^{29} In contrast, the 10year breakeven inflation rates implied by Models LI and LIId, 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 modelimplied and the TIPSbased 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 modelimplied and the TIPSbased breakeven inflation rates is the sum of TIPS liquidity premium and TIPS measurement errors.
To quantify the improvement in terms of the model fit, Panels B and C of Table 8 provide three goodnessoffit statistics for TIPS yields at the 5, 7 and 10year maturities and TIPS breakeven inflation at the 7 and 10year 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 modelimplied 10year TIPS breakeven and the data counterpart rising from to over once we move from Model NL to the other models. The next two statistics are based on onestepahead model prediction errors from the Kalman Filter, , defined in Equation (15) in Appendix C, and are designed to capture how well each model can explain the data without resorting to large exogenous 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 (), defined as the percentage of insample variations of each data series explained by the model:
It is conceivable that a model with more parameters like Model LIId could generate smaller insample fitting errors for variables whose fit 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 modelimplied inflation expectations mimic surveybased 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 realtime, modelfree measure, and hence not subject to model estimation errors or lookahead biases that could affect measures based on insample fitting of realized inflation.^{31}
Panel D of Table 8 reports the three goodnessoffit statistics, CORR, RMSE and , for 1 and 10year 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 statistics at both horizons. This poor fit is especially prominent at the 1year horizon: the RMSE is large, the correlation between the model and survey forecast is essentially 0, and the is highly negative at . In contrast, all other models, which have a liquidity factor, generate a reasonable fit with the survey forecasts at both horizons. The best fit is achieved by Models LII and LIId, both of which generate correlations above and small RMSEs at both horizons and explain a large amount of sample variations in survey inflation forecasts. Models LII and LIId also improve notably upon Model LI, suggesting that some cyclical variations in TIPS yields might not be due to movements in the real yields. Overall, Model LIId does not seem to suffer from an overfitting problem. As we will see from later sections, this model also generate sensible implications for TIPS liquidity premiums and inflation risk premiums, further supporting our conclusion.
A visual comparison of the modelimplied inflation expectations and survey forecasts can help us understand the results in Table 8. The left panels of Figure 6 plot the 1year inflation expectation based on Models NL, LI and LIId, together with the survey forecast. It can be seen that the ModelNLimplied 1year inflation expectation contains a large amount of shortrun 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 comparison, implied 1year inflation expectation based on the other models show a visible downward trend, consistent with the survey evidence. It is interesting that although Models LI and LIId 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 1year inflation expectation implied by Model LIId bears a high resemblance to the 1year survey forecast, the same series implied by Model LI appears to be much more variable than the survey counterpart.
It is also not surprising that Model NL produces a larger RMSEs for 10year inflation expectations than the LI and LII models: the upper middle panel of Figure 6 shows that Model NL completely misses the downward trend in the 10year survey inflation forecast since the early 1990s and implies a 10year 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 ModelNLimplied 10year 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 nearconstancy of the longterm inflation expectation is the most problematic feature of Model NL. Models LI and LIId, on the other hand, produce 10year inflation expectations that are clearly downward trending, though the modelimplied 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 longterm 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 modelimplied inflation risk premiums at the 1 and 10year horizons for the three models under consideration. One immediately notable feature is that ModelNL implies an inflation risk premium, shown in the upper right panel, that is negative and increasing over time in the 19902007 period. In contrast, as mentioned 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/Friedman:84:JF 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 10year inflation risk premium estimates together with the 95% confidence bands for the three models, even after we take into account sampling uncertainties, longterm inflation risk premiums implied by Model NL remain negative over most of the sample period. In comparison, the two models that allow for a liquidity premium, Models LI and LIId, both generate 10year inflation risk premiums that are positive and fluctuate in the 0 to 1% range over the same sample period. The shortterm inflation risk premiums implied by these two models, on the other hand, are fairly small, consistent with our intuition.
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 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 LI and LII, improves upon Model NL in all three aspects and in particular produce longterm inflation expectations that agree quite well with survey forecasts.
Among models allowing a liquidity premium in TIPS yields, Models LII and Model LIId generate shortterm inflation expectations that matches survey counterparts better than Model LI, suggesting it is important to allow for a systematic component in TIPS liquidity premiums.
Finally, a likelihood ratio test rejects Models LII in favor of our preferred model, Model LIId, which features a deterministic trend in TIPS liquidity premium that is designed 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.
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 10year liquidity premiums implied by Models LI and LII, respectively, while the bottom panel shows the the deterministic and the stochastic components of the liquidity premiums based on Model LIId.
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 the fact that the independent liquidity factor is estimated to be very persistent under the riskneutral measure. As can be seen from Table 8, the riskneutral persistence of the liquidity factor, , 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, the persistence parameter under the physical measure, , is not as precisely 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 LI, while under Model LII, the term structure has a mild downwardsloping behavior during the 20012004 period. Technically, a market price of risk on the independent liquidity factor that is on average positive, as is the case in all three models here, would contribute to a downwardsloping term structure of TIPS liquidity premium.^{32} This is in contrast to the standard onefactor interest rate models, where the market price of interest rate risk is typically found to be negative. Although the TIPS liquidity premiums in Models LII and LIId are also driven by the nominal bond factors, , in addition to , a variance decomposition indicates that the TIPSspecific factor drives most of the variations in TIPS liquidity premium. Nonetheless, as we've seen in Section 5, allowing the TIPS liquidity premiums to depend on nominal bond factors seems important in explaining the dynamics of TIPS yields.
Finally, all three models imply that the TIPS liquidity premiums were fairly high (12 %) when TIPS were first introduced, had been on a downward trajectory until around 2004, and have stayed at a relatively low level from 2005 onwards. The deterministic trend implied by Model LIId came down from about 120 basis points in 1999 to below 10 basis points by the end of 2004. After removing the downward trend, the stochastic liquidity premiums appear more stationary and largely vary between 50 and 50 basis points.
The behavior of the liquidity premiums we have seen in Figure 8 seems broadly consistent with the perception that TIPS market liquidity conditions have improved over time. In this section, we examine this issue more closely. One difficulty in this regard is the lack of precise, realtime measures of the TIPS market liquidity. One measure that has been used in the literature^{33} is the 13week moving average of weekly TIPS turnover, defined as the weekly average of daily TIPS transaction volumes divided by the TIPS outstanding at the end of the current month.^{34} As can be seen in the top panel of Figure 3, the turnover in TIPS remained low up to mid 2002 and then rose substantially, suggesting an improvement in the liquidity conditions of the TIPS market in recent years.^{35} The rise coincides roughly with the decline in our TIPS liquidity premium estimates. In particular, all our modelbased TIPS liquidity premiums (Model LI, LII, LIId) have correlations with this measure more negative than 73%.
However, the turnover may be affected by factors other than the liquidity conditions in the TIPS market. For example, there is a large empirical literature documenting a positive contemporaneous correlation between price volatility and trading volumes, independent of current market liquidity conditions.^{36} In particular, as shown in the middle panel of Figure 9, interest rate volatilities declined markedly during the latter half of the sample period, which might have contributed to the drop in TIPS turnover after 2005. Indeed, evidence from TIPS bidask spreads, arguably a better measure of liquidity conditions in the TIPS market, indicates that the early improvement in TIPS liquidity may not have been largely reversed in the most recent sample period, as one might assume based on the rapid decline in TIPS turnover since 2005. For example, two informal surveys of the primary dealers conducted by the Federal Reserve Bank of New York in 2003 and 2007, shown in Table 8, find that the average bidask spreads on TIPS continue to narrow across all maturities during this period.^{37} Unfortunately, a long history of this measure is unavailable.
To quantify the effects of various factors on our estimates of TIPS liquidity premiums, we therefore regress the 5, 7 and 10year TIPS liquidity premiums from Model LIId onto three explanatory variables, the first of which being the TIPS turnover ratio.
Figure 9 plots all three explanatory variables; their correlations with TIPS liquidity premium estimates from Model LIId and with each other are reported in Panel B of Table 8. The three variables are not highly correlated with each other, but all have large correlations with the liquidity premium estimates. The results from this regression are reported in Panel A of Table 8. The coefficients on all three variables are statistically significant and carry the expected sign. In particular, a lower TIPS turnover raises the TIPS liquidity premiums, but the effect will be smaller if the lower turnover is accompanied by a lower volatility.^{39} Together these three variables explain about 80% of the variations in TIPS liquidity premium estimates at all maturities.^{40} It remains an interesting topic for future research to see whether our modelbased measures of TIPS liquidity premiums correlate with other measures of TIPS market liquidity.^{41}
The results in Table 8 suggest one simple way to adjust the TIPS breakeven inflation for the liquidity effects. Consider the difference between the SPF forecast of average inflation over the next ten years, , and the 10year TIPS breakeven inflation, , and regress it on the same three righthandside variables as in Equation (53):^{42}
While the qualitative behavior of TIPS liquidity premium thus seems reasonable, our estimates of does seem large in comparison with that of the corporate bonds. Corporate bonds, including those with the highest credit rating (AAA/AA), tend to trade infrequently at once a day or less; TIPS, in comparison, trade more often than AAA/AA corporate bonds even during the early years when liquidity was the poorest. The bidask spread in TIPS has also been substantially smaller than those of corporate bonds. In this regard, our estimates of TIPS liquidity premium around 1.5% in the early years seems puzzlingly high, in comparison with investmentgrade corporate bonds which typically trade at a liquidity premiums of about 50 basis points.^{43} The high value of the TIPS liquidity premium in the early years may partly reflect a depressed demand for TIPS due to the newness of the security and the relative complexity of TIPS payoff structure. Furthermore, a popular belief that TIPS are taxdisadvantageous for taxable investors^{44} may have further depressed the demand for TIPS.
In this section we examine how much variations in TIPS yields and TIPS breakeven inflation rate can be attributed to variations in TIPS liquidity premiums. Using Equations (2) and (30), we can decompose TIPS yields, , and TIPS breakven inflation, , into different components:
Table 8 reports the variance decomposition results based on Equation (55) and Model LIId estimates, using either the unconditional (Panel A) or the instantaneous (Panel B) variancecovariance matrix of the state variables. A time series plot of the decomposition is shown in Figure 11.
For TIPS yields, real yields dominate TIPS liquidity premiums in accounting for the time variations both unconditionally and instantaneously. In comparison, TIPS liquidity premiums are more important in driving TIPS breakeven inflation, explaining about 23% of its unconditional variations at all three maturities, although expected inflation still accounts for the majority of time variations in TIPS breakeven inflation. The contribution of TIPS liquidity premiums is even larger instantaneously and explains about 43% of the shortrun variations in TIPS breakeven inflation. The last observation suggests that one should be especially cautious in interpreting shortrun variations in TIPS breakeven inflation solely in terms of changes in inflation expectation or inflation risk premiums.
Although it is not the focus of the current paper, our models can also be used to separate nominal yields into real yields, expected inflation and inflation risk premiums:
These results indicate that, at least during our sample period, real yield changes explain more than half of the variations in nominal yields at all maturities. Inflation expectation explains about 40% (20%) of the variations in the 1quarter (10year) nominal yield. Inflation risk premiums account for the remaining 10% of the nominal yield changes. This stands in contrast to previous studies using a longer sample period but not using TIPS yields, which typically find relatively smooth real yields but volatile inflation expectation or inflation risk premiums.^{45} The limited evidence we have so far from TIPS seems to suggest that real yields may also vary considerably over time.
In this paper, we document that there exists a TIPSspecific factor that is important for explaining TIPS yield variations but not as crucial for explaining nominal interest rate movement, and provide evidence that this factor might be reflecting a liquidity premium in TIPS yields.
We then develop a joint noarbitrage term structure model of nominal and TIPS yields incorporating a rich specification of the TIPS liquidity premiums. The main findings can be summarized as the following. First, we find that our estimated liquidity premium was quite large until about 2003 but has come down in recent years, consistent with the common perception that TIPS market liquidity has improved in recent years. Second, our TIPS liquidity premium estimates contain a persistent downward trend in its early years, that is best modeled as a deterministic trend reflecting gradual market acceptance of TIPS as a new debt instrument. Finally, we show that ignoring the liquidity premium components leads to a poor model fit of TIPS yields, TIPS breakeven inflation and survey inflation forecasts.
TIPS breakeven inflation has been increasingly gaining attention as a measure of investors' inflation expectations that is available in realtime and at high frequencies. However, our results raise caution in interpreting movement in TIPS breakeven inflation solely in terms of changing inflation expectations, as substantial liquidity premiums and inflation risk premiums could at times drive a large wedge between between the two. In an encouraging note, the reduced liquidity premium that we find for the most recent period (20052007) raises the possibility that, going forward, the TIPS yields may be a better gauge of the true real yields. However, given that TIPS is less liquid than nominal Treasury securities, we caution that TIPS liquidity premiums might rise again in times of financial market stress. A better understanding of the determinants of TIPS liquidity premiums and the sources of its variation remains an interesting topic for future research.
This appendix is devoted to a more detailed discussion of the TIPS data. Figure A1 shows the smoothed TIPS par yield curves on June 9, 2005 in the top panel and on June 9, 1999 in the bottom panel, together with the actual traded TIPS yields that were used to create the smoothed TIPS paryield and zerocoupon curves. The smoothing is done by assuming that the zerocoupon TIPS yield curve follows the 4parameter NelsonSiegel (1987) functional form up to the end of 2003 and the 6parameter Svensson (1994) functional form thereafter,^{46} and minimizing the fitting error for the actual traded TIPS securities. The substantial increase in the number of points in the top panel reflects the growth of the TIPS market. Note that in 1999 there is essentially one data point on the curve between the 0 and 5 years maturity (corresponding to the 5year TIPS issued in 1997), thus the TIPS term structure in the shortmaturity region of 05 years cannot be determined reliably. With more points across the maturity spectrum in 2005, shorter maturity TIPS yields can be determined more reliably than in 1999.
Still, the analysis of the shortmaturity TIPS are complicated by the indexation lag and seasonality issues. Note that the TIPS payments are indexed to the CPI 2.5 months earlier, thus TIPS yields contain some amount of realized inflation, often referred to as the "carry effect". The yield that is more relevant to policy makers is the one that takes out this realized inflation  the socalled carryadjusted yields, which can be heuristically represented as
As noted in the main text, since data reliability and indexation lags pose larger problems to shortermaturity TIPS, in this paper we focus on longmaturity TIPS yield for which the effects of indexation lag and seasonality are less important. While the analysis of shortermaturity TIPS yields is an important problem in itself, it deserves a fuller treatment elsewhere.^{49} The 5, 7 and 10year carryunadjusted TIPS yields used in this analysis can be viewed as the carrycorrected TIPS yield plus a measurement error, as suggested by Equation (1).
Since is independent of the other state variables in , the first term on the righthand side of Equation (31) can be written as the sum of two components:
(B2) 
The first component can be solved to be
(B3) 
(B4)  
(B5) 
The second component can be shown to take the form
(B7) 
(B10)  
(B11) 
Taken together, we have that
(B12) 
We use the Kalman filter to compute optimal estimates of the unobservable state factors based on all available information. For example, given the initial guess for the state factors , it follows from the state equation (46) that the optimal estimate of the state factor at time is given by
which implies that we carry the error of the initial guess to all subsequent estimations. More generally, we haveThe variancecovariance matrix of the estimation error can be derived as
Given any forecast for , we can compute a forecast for the observable variables based on all time information:
the forecast error of which is given byThe conditional variancecovariance matrix of this estimation error can then be computed as
The next step is to update the equation for the state variables. Before doing this, we need to recover the distribution of The conditional covariance between the forecasting errors for state variables and that for observation variables takes the form of
The conditional joint distribution for and is therefore
We estimate the parameters by maximizing the loglikelihood function
Coefficient: Constant  Coefficient: 3month  Coefficient: 2year  Coefficient: 10year  Adj.  
In Level  3.4637  0.4519  0.6516  0.4384  32.3% 
In Level (standard error)  (0.1138)  (0.0345)  (0.0479)  (0.0346)  
In Weekly Changes  0.0032  0.0110  0.0850  0.5687  59.0% 
In Weekly Changes (standard error)  (0.0025)  (0.0334)  (0.0409)  (0.0366) 
PC  nominal yields only  nominal and TIPS yields 
1st  75.17  71.11 
2nd  93.25  87.26 
3rd  97.44  94.72 
4th  99.36  97.58 
PC1 (nominal and TIPS yield)  PC2 (nominal and TIPS yield)  PC3 (nominal and TIPS yield)  PC4 (nominal and TIPS yield)  
PC1 (nominal yield alone)  0.97  0.21  0.15  0.01 
PC2 (nominal yield alone)  0.12  0.86  0.49  0.02 
PC3 (nominal yield alone)  0.04  0.08  0.20  0.97 
PC4 (nominal yield alone)  0.01  0.03  0.06  0.01 
Model  Restrictions and Idenfications 
Model NL  , , , , and unidentified 
Model LI  , unrestricted, , and unidentified 
Model LII  unrestricted, , and unidentified 
Model LIId  , , unrestricted 
Model NL (parameter estimate)  Model NL (standard error)  Model LI (parameter estimate)  Model LI (standard error)  Model LII (parameter estimate)  Model LII (standard error)  Model LIId: parameter estimate  Model LIId: standard error  
0.8587  (0.5206)  0.8676  (0.5227)  0.8573  (0.4948)  0.8374  (0.4606)  
0.0529  (0.0787)  0.0573  (0.0793)  0.0555  (0.0794)  0.0529  (0.0781)  
1.5219  (0.8048)  1.5321  (0.8040)  1.5394  (0.7729)  1.5672  (0.7574)  
0.3346  (0.3200)  0.3098  (0.3105)  0.3232  (0.3101)  0.3175  (0.3101)  
4.5682  (9.1343)  4.5353  (9.0421)  4.4144  (8.1903)  4.0846  (6.7681)  
0.5524  (0.2581)  0.5449  (0.2522)  0.5494  (0.2538)  0.5362  (0.2469) 
Model NL (parameter estimate)  Model NL (standard error)  Model LI (parameter estimate)  Model LI (standard error)  Model LII (parameter estimate)  Model LII (standard error)  Model LIId: parameter estimate  Model LIId: standard error  
0.0419  (0.0135)  0.0434  (0.0116)  0.0428  (0.0125)  0.0422  (0.0132)  
2.8343  (5.2462)  2.8318  (5.2564)  2.7671  (4.7530)  2.5726  (3.9118)  
0.4825  (0.1249)  0.4797  (0.1239)  0.4809  (0.1248)  0.4675  (0.1207)  
0.6089  (0.0378)  0.6177  (0.0403)  0.6180  (0.0387)  0.6195  (0.0377)  
0.4362  (0.2228)  0.4107  (0.1807)  0.4147  (0.1946)  0.4236  (0.2449)  
0.1818  (0.8732)  0.2933  (0.7752)  0.2447  (0.8142)  0.2049  (0.8544)  
0.0471  (3.3761)  0.4308  (2.8805)  0.2597  (3.1019)  0.1136  (3.3049)  
0.5330  (1.7491)  0.5489  (1.7874)  0.5288  (1.6390)  0.4592  (1.3462)  
1.7508  (4.9179)  1.7894  (4.9932)  1.7104  (4.5060)  1.5341  (3.7315)  
3.7651  (15.9132)  3.8379  (16.1348)  3.6471  (14.3685)  3.1187  (11.0801)  
0.0274  (0.2452)  0.0277  (0.2406)  0.0272  (0.2370)  0.0419  (0.2252)  
0.2865  (0.1339)  0.2854  (0.1318)  0.2849  (0.1330)  0.2752  (0.1326)  
0.6948  (0.8960)  0.6677  (0.8587)  0.6854  (0.8420)  0.6296  (0.6714)  
0.0717  (0.3145)  0.0749  (0.3224)  0.0725  (0.3155)  0.0551  (0.3238)  
0.6171  (0.2355)  0.6329  (0.2429)  0.6303  (0.2397)  0.6314  (0.2404)  
0.6551  (2.1043)  0.6812  (2.1309)  0.6634  (1.9803)  0.5887  (1.7096) 
Model NL (parameter estimate)  Model NL (standard error)  Model LI (parameter estimate)  Model LI (standard error)  Model LII (parameter estimate)  Model LII (standard error)  Model LIId: parameter estimate  Model LIId: standard error  
0.0271  (0.0035)  0.0280  (0.0069)  0.0239  (0.0082)  0.0236  (0.0084)  
0.7843  (2.8227)  0.8836  (1.4332)  0.0746  (0.6436)  0.0729  (0.5890)  
0.0766  (0.1062)  0.2563  (0.0707)  0.2575  (0.1262)  0.2443  (0.1248)  
0.5094  (0.3937)  0.1216  (0.1276)  0.0165  (0.2079)  0.0221  (0.2107)  
0.1724  (0.0651)  0.1176  (0.0720)  0.1050  (0.0763)  0.1076  (0.0757)  
0.0231  (0.0803)  0.0597  (0.0750)  0.0651  (0.0757)  0.0506  (0.0750)  
0.0016  (0.0656)  0.0354  (0.0594)  0.0306  (0.0611)  0.0358  (0.0609)  
0.7795  (0.0288)  0.7279  (0.0241)  0.7144  (0.0245)  0.7144  (0.0245) 
Model NL (parameter estimate)  Model NL (standard error)  Model LI (parameter estimate)  Model LI (standard error)  Model LII (parameter estimate)  Model LII (standard error)  Model LIId: parameter estimate  Model LIId: standard error  
0.6114  (0.0411)  0.6152  (0.0415)  1.2545  (0.0914)  
0.2083  (0.2655)  0.2206  (0.2630)  0.6037  (0.3973)  
0.0218  (0.0113)  0.0157  (0.0115)  0.0003  (0.0105)  
0.3213  (0.6657)  0.2851  (0.5090)  0.0263  (0.3356)  
0.1091  (0.2652)  0.1209  (0.2627)  0.1472  (0.4020)  
0.6765  (1.2459)  2.9521  (5.7532)  
0.0179  (0.1547)  0.2739  (0.1717)  
0.0833  (0.2509)  1.0137  (0.3285)  
1.1871  (0.0310)  
0.0014  (0.0001)  
731467.911  (25.2593) 
Model NL (parameter estimate)  Model NL (standard error)  Model LI (parameter estimate)  Model LI (standard error)  Model LII (parameter estimate)  Model LII (standard error)  Model LIId: parameter estimate  Model LIId: standard error  
0.1005  (0.0026)  0.1012  (0.0027)  0.1012  (0.0027)  0.1013  (0.0027)  
0.0231  (0.0016)  0.0221  (0.0016)  0.0222  (0.0016)  0.0224  (0.0017)  
0.0532  (0.0017)  0.0530  (0.0018)  0.0530  (0.0018)  0.0531  (0.0017)  
0.0000  (140.6008)  0.0000  (59.2489)  0.0000  (50.6589)  0.0000  (27.7162)  
0.0293  (0.0012)  0.0294  (0.0012)  0.0294  (0.0013)  0.0294  (0.0012)  
0.0000  (120.1913)  0.0000  (0.9395)  0.0000  (44.6608)  0.0000  (0.9058)  
0.0489  (0.0018)  0.0490  (0.0019)  0.0490  (0.0019)  0.0489  (0.0018) 
Model NL (parameter estimate)  Model NL (standard error)  Model LI (parameter estimate)  Model LI (standard error)  Model LII (parameter estimate)  Model LII (standard error)  Model LIId: parameter estimate  Model LIId: standard error  
0.4307  (0.0953)  0.0654  (0.0059)  0.0657  (0.0060)  0.0000  (4.8466)  
0.3511  (0.0832)  0.0021  (0.0414)  0.0004  (0.2385)  0.0428  (0.0035)  
0.3578  (0.0802)  0.0647  (0.0060)  0.0643  (0.0059)  0.0520  (0.0038) 
Model NL (parameter estimate)  Model NL (standard error)  Model LI (parameter estimate)  Model LI (standard error)  Model LII (parameter estimate)  Model LII (standard error)  Model LIId: parameter estimate  Model LIId: standard error  
0.1760  (0.0135)  0.1758  (0.0134)  0.1758  (0.0134)  0.1758  (0.0134)  
0.2261  (0.0197)  0.2260  (0.0196)  0.2260  (0.0197)  0.2261  (0.0196) 
Model NL  Model LI  Model LII  Model LIId  
No. of parameters  42  47  50  53 
Log likelihood  53,663.65  55,972.49  55,976.70  56,193.41 
AIC  107,243.30  111,850.97  111,853.40  112,280.81 
BIC  107,041.70  111,625.37  111,613.39  112,026.41 
LR pvalue  0.04 
Model NL  Model LI  Model LII  Model LIId  
5year CORR (in %)  93.14  99.41  99.42  99.53 
5year RMSE  0.43  0.13  0.13  0.11 
5year (in %)  83.93  98.61  98.62  99.06 
7year CORR (in %)  92.99  99.45  99.46  99.50 
7year RMSE  0.36  0.10  0.10  0.10 
7year (in %)  85.96  98.91  98.92  98.93 
10year CORR (in %)  92.52  99.19  99.20  99.42 
10year RMSE  0.37  0.12  0.11  0.10 
10year (in %)  80.90  98.18  98.21  98.76 
Model NL  Model LI  Model LII  Model LIId  
7year CORR (in %)  51.61  97.21  97.24  97.35 
7year RMSE  0.35  0.09  0.09  0.10 
7year (in %)  23.44  94.47  94.54  94.21 
10year CORR (in %)  32.08  94.76  94.80  95.11 
10year RMSE  0.35  0.11  0.11  0.10 
10year (in %)  18.12  88.71  88.85  89.74 
Model NL  Model LI  Model LII  Model LIId  
1year CORR (in %)  2.07  62.94  88.65  88.08 
1year RMSE  0.78  0.58  0.33  0.34 
1year (in %)  52.21  17.45  72.93  70.71 
10year CORR (in %)  73.34  72.72  83.86  83.92 
10year RMSE  0.51  0.40  0.42  0.42 
10year (in %)  17.36  49.46  44.35  42.22 
Less than five years  Five to Ten Years  Above ten years  
2003  1 to 2  2  4 to 16 
Twoyear  Fiveyear  TenYear  TwentyYear  Thirtyyear  
2007  1/2 to 1  1  12  46  610 
Maturity  Coefficients: Constant  Coefficients: Turnover  Coefficients: Imp Vol  Coefficients: ASW Spread  Adjusted 
5year  0.3074  0.4309  0.1196  0.0258  79.52% 
5year (standard error)  (0.0771)  (0.0275)  (0.0095)  (0.0025)  
7year  0.4139  0.3752  0.0824  0.0253  80.95% 
7year (standard error)  (0.0641)  (0.0229)  (0.0079)  (0.0020)  
10year  0.5076  0.3337  0.0465  0.0251  82.30% 
10year (standard error)  (0.0541)  (0.0193)  (0.0067)  (0.0017) 
Liquidity Premiums: 5year  Liquidity Premiums: 7year  Liquidity Premiums: 10year  TIPS Turnover  10Year Imp Vol  
TIPS Turnover  0.7286  0.7547  0.7850  
10year implied volatility  0.5515  0.5098  0.4449  0.1314  
On/off ASW spread  0.7996  0.8189  0.8340  0.6374  0.4742 
Maturity  TIPS yield: real yield  TIPS yield: liq prem  TIPS BEI: inf exp  TIPS BEI: inf risk prem  TIPS BEI: liq prem 
5year  1.1717  0.1717  0.5870  0.1890  0.2240 
5year (standard error)  (0.2836)  (0.2836)  (0.3050)  (0.2386)  (0.3170) 
7year  1.1819  0.1819  0.5659  0.1994  0.2347 
7year (standard error)  (0.2690)  (0.2690)  (0.3065)  (0.2616)  (0.3131) 
10year  1.1910  0.1910  0.5453  0.2090  0.2458 
10year(standard error)  (0.2581)  (0.2581)  (0.3192)  (0.2944)  (0.3095) 
Maturity  TIPS yield: real yield  TIPS yield: liq prem  TIPS BEI: inf exp  TIPS BEI: inf risk prem  TIPS BEI: liq prem 
5year  1.1596  0.1596  0.5447  0.0167  0.4386 
5year (standard error)  (0.2963)  (0.2963)  (0.2473)  (0.2064)  (0.2639) 
7year  1.2285  0.2285  0.5500  0.0239  0.4261 
7year (standard error)  (0.3040)  (0.3040)  (0.2710)  (0.2397)  (0.2597) 
10year  1.3024  0.3024  0.5431  0.0348  0.4221 
10year(standard error)  (0.3073)  (0.3073)  (0.2984)  (0.2740)  (0.2556) 
Note: This table reports the unconditional and the instantaneous variance decompositions of TIPS yields into real yields and TIPS liquidity premiums, and of nominal yields into expected inflation, the inflation risk premiums and the negative of TIPS liquidity premiums, all based on Model LIId estimates. The variance decompositions of TIPS yields are calculated according to
while the variance decompositions of the TIPS breakeven inflation are calculated according toMaturity  real yield  inf exp  inf risk prem 
1quarter  0.5108  0.4156  0.0736 
1quarter (standard error)  (0.2541)  (0.2281)  (0.0927) 
1year  0.5715  0.3497  0.0787 
1year (standard error)  (0.1930)  (0.1843)  (0.0924) 
5year  0.6503  0.2609  0.0888 
5year (standard error)  (0.1486)  (0.1417)  (0.1146) 
10year  0.6715  0.2347  0.0938 
10year (standard error)  (0.1401)  (0.1429)  (0.1362) 
Maturity  real yield  inf exp  inf risk prem 
1quarter  0.7719  0.2252  0.0029 
1quarter (standard error)  (0.1090)  (0.1009)  (0.0312) 
1year  0.7692  0.2172  0.0137 
1year (standard error)  (0.1082)  (0.0915)  (0.0365) 
5year  0.7132  0.2496  0.0372 
5year (standard error)  (0.1231)  (0.1154)  (0.0970) 
10year  0.6892  0.2494  0.0614 
10year (standard error)  (0.1331)  (0.1438)  (0.1345) 
Note: This table reports the unconditional and the instantaneous variance decompositions of nominal yields into real yields, expected inflation, the inflation risk premiums, all based on Model LIId estimates. The variance decomposition is calculated according to
where the results are based on either the unconditional variancecovariance matrix of the state variables (Panel A) or the instantaneous variancecovariance matrix of the state variables, (Panel B). Standard errors calculated using the delta method are reported in parentheses.
The top panel plots gross TIPS issuance broken down by initial maturities of 10, 5, 20 and 30 years. The bottom panel plots TIPS outstanding broken down by remaining maturities, based on data reported in the Treasury's Monthly Statement of the Public Debt (MSPD). 
Top top panel plots the weekly TIPS transaction volumes, defined as 13week moving average of weekly averages of daily TIPS transaction volumes reported by primary dealers in Government Securities Dealers Reports (FR2004). The bottom panels plots number of TIPS mutual funds (left axis) and the total net assets under management (left axis).) 
This chart shows the 10year TIPS breakeven inflation (red line), longhorizon Michigan inflation forecast (blue line), and 10year SPF inflation forecast (black pluses). 
Top top panel plots the 3 and 6month, 1, 2, 4, 7 and 10year nominal yields. The middle panel plots the 5, 7 and 10year TIPS yields. The bottom panels plots the 5 and 7year TIPS breakeven inflation.) 
The panels on the left plot the 10year actual TIPS yields (red), the 10year modelimplied TIPS yields (black) and the 10year modelimplied real yields (blue). The panels on the right plot the 10year actual TIPS breakevens (red), the 10year modelimplied TIPS breakevens (black) and the 10year modelimplied true breakevens (blue). The model estimates are based on Model NL (upper panels), Model LI (middle panels), and Model LIId (lower panels), respectively. 
The left and middle panels plot the 1 and 10year modelimplied inflation expectation, respectively, together with the survey counterparts from Survey of Professional Forecasters (SPF), while the right panels plot the 1 and 10year modelimplied inflation risk premiums. The model estimates are based on Model NL (upper panels), Model LI (middle panels), and Model LIId (lower panels), respectively. 
The three panels plot the modelimplied 10year inflation risk premiums with 2 BHHH standard error bands based on Model NL (top panel), Model LI (middle panel) and Model LIId (bottom panel), respectively. 
The top panel plot the 5, 7 and 10year TIPS liquidity premiums based on Model NL estimates. The bottom two panels plots the same series based on Model LIId estimates, as well as a decomposition of these series into a deterministic component (dashed line in the middle panel) and a stochastic component (bottom panel). 
This chart plots various measures that potentially reflect liquidity conditions in the TIPS market, including the TIPS turnover ratio as defined in Section 6.2 (top panel), implied volatilities from options on the 10year nominal Treasury note futures (middle panel) and the difference between the ontherun and the offtherun 10year Treasuries par asset swap spreads (bottom panel). 
This chart plots the liquidityadjusted 10year TIPS BEI base on Equation (54) (thin blue line) together with the unadjusted series (red line) and the modelimplied true TIPS BEI from Model LIId (thick blue line). 
The top panel decomposes the 10year TIPS yields into the real yield and the TIPS liquidity premiums, while the bottom panel decomposes the 10year TIPS breakeven inflation into the expected inflation, the inflation risk premium and the TIPS liquidity premium, all according to Equation (55). 
This chart decomposes the 1 and 10year nominal yields into real yields, expected inflation and inflation risk premiums according to Equation (56). 

Note: This figure plots 10year carryunadjusted (carryadjusted) TIPS yields in red solid (black dashed) line and 5year carryunadjusted (carryadjusted) TIPS
yields in blue solid (gray dashed) line.
