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Appendix I: Detrending
To make the model stationary, all non-stationary variables in the model should be detrended by by Pt,
First, the amount of nominal bond holdings is detrended as,
where bt and bn,t are the detrended bond holdings for one-period bonds and n-period bonds. Nominal bond holdings are detrended by a price level and long-run income because they are cointegrated with those variables on the balanced growth path as in a standard growth model. In addition, nominal bond holdings should be detrended by the trend inflation,
Note that the n-period bond holdings, Bn,t, and prices, Qπ,t, should be detrended by the trend inflation powered by its maturity,
as in a standard neo-classical growth model. Then, the budget constraint is reformulated by those detrended variables as,
by dividing both sides of the original budget constraint (1) by Pt and
As in the budget constraint, the monetary policy rule is also reformulated by using the de-trended variables as,
by dividing the both sides of the monetary policy rule by trend inflation,
Finally, the Euler equations are detrended. The nominal SDF based on detrended variables,
Finally, the n-period bond prices for risk-neutral agents,
Appendix II: Robustness Check
This appendix conducts the following four robustness checks to verify the validity of the model. First, it examines the case with different values for the CRRA coefficient to see the sensitivity of equilibrium term premiums with respect to the value of risk aversion. Second, it examines the effect of habit formation for household’s consumption. In this case, the household’s value function changes to,
where h is set to 0.25 based on the previous literature. It is an important robustness check because the habit formation is considered one of driving forces for risk premiums in the asset pricing literature. Third, it examines the case with fixed trend inflation. This change may affect the average shape of yield curve because the analysis in the main text shows that the volatility of long term interest rates is quite different between the case of time varying trend inflation and fixed trend inflation. Fourth and finally, it examines what if the monetary policy responds to income gap yt in addition to inflation gap πt. That is, the monetary policy rule in this case changes to,
In all cases, I check the robustness only for the US equilibrium yield curve by changing a part of specifications and examining the marginal change from the baseline result.
Figure 7 shows the result of the robustness checks. All charts in this figure show that the main result is robust to those changes: The model can replicate the positive and sizable term premiums under the estimated inflation and income co-movement. Some comments are in order. First, while the size of term premiums is highly sensitive to the CRRA coefficient as expected, the qualitative result of positive and sizable term premiums is preserved for a relatively wide range of CRRA co-efficients. Second, while the very small effects of habit formation on term premiums are in contrast with the previous finance literature including Wachter (2006), the mechanism behind the small effects of habit formation is similar to Rudebusch and Swanson (2008). As they argue, the effects of habit formation are small if the household can smooth its consumption in response to negative shocks to income. In the face of the negative shock, the household can smooth consumption by adjusting the labor supply in their model and the amount of savings in the present model, both of which leading to very small effects of habit formation. Third, the time varying trend inflation barely affects the size of term premiums because it is very slow-moving with very small volatility, thus generating any risks for consumers over the business cycle. Fourth, when the central bank responds to income gap, the size of term premiums is barely changed but become slightly higher than the baseline. This is because higher interest rates (lower bond prices) suppress consumption growth, thus reinforcing the negative correlation between bond prices and consumption growth.
I would like to thank Francois Gourio, Taisuke Nakata, Hiroatsu Tanaka and staff of the International Monetary Fund for helpful suggestions and comments. I also appreciate the comments of seminar participants at the Federal Reserve Board and 2018 CEF conference. The views expressed here are those of the author and do not necessarily represent the views of the IMF, its Executive Board, or IMF management.
The most actively investigated issue in this literature is the equity premium puzzle. For an extensive survey on this literature, see Cochrane (2017).
Given the negative correlation between consumption growth and inflation observed in most economies, some macro-finance models with exogenous consumption growth can account for positive term premiums (e.g., Piazzesi and Schneider (2007) and Bansal and Shaliastovich (2013)).
As is discussed in IMF (2017), this result implies that the low-for-long economy may be associated with a higher financial stability risk due to the lack of bank profits adequate to build capital buffers, given that the maturity transformation is a main sources of banks profits.
Also, the stylized approach makes it possible to analyze the yield curve in the low-for-long economy. A new Keynesian model faces inflation indeterminacy when interest rates hit the ZLB so often.
This type of “risk premium” for bond holdings is common in a model with exogenous interest rates, typically a small open economy model. See Schmitt-Grohe and Uribe (2003) for more discussion on how to avoid the divergent path in a small open economy model.
The real interest rate and long-term real bond prices are formulated using the real SDF and standard asset pricing formula based on the Euler equations. Those Euler equations, however, can be derived by assuming that real bonds are available for the household and the household optimally chooses the amount of real bond holdings.
See Ngo and Gourio (2016) for more discussion on inflation risk premiums and how they change around the ZLB of nominal interest rates.
Note that the general AR(1) process for gt here includes a random walk process as a special case, and so a random walk process can be chosen as a result of estimation. Furthermore, assuming a random walk process for the non-stationary component of income has a risk to overestimate the role of a non-stationary component. See Quah (1992) for more theoretical discussion on the trend and cycle decomposition assuming that the non-stationary component follows a general ARIMA process.
Here, the correlation between the shocks to a stationary and a non-stationary component of income, the left-bottom and the right-upper component of the covariance matrix, is assumed to be zero. While this assumption is a bit restrictive, it is necessary to identify and decompose the trend and cycle component of income. See Harvey (1985) for more details on this assumption.
The estimation result for ρyπ > 0 seems a bit strange at the first glance because it seems to imply that lagged high inflation causes high income. This interpretation is, however, not appropriate because the VAR in this paper is just a reduced form VAR rather than a structural VAR. Hence, we would need an identification assumption to know the marginal effect of exogenous shift of lagged inflation on income.
I thank the authors for kindly sharing the Matlab code.
An implicit assumption here is that there have been no substantial structural breaks in the bond market to permanently change the level of term premiums, and that all policy changes associated with the bond market, including the quantitative easing after the global financial crisis, are cyclical policy changes.
While the value of CRRA coefficient quantitatively influences the slope of yield curve as is well known, note that the qualitative result of positive and sizable term premiums is preserved for a relatively wide range of CRRA coefficients as shown by a robustness check in Appendix B, and that values of CRRA coefficient enough for replicating the empirically comparable size of term premiums are much more modest and realistic than those in previous literature. Moreover, term premiums in the trend stationary model are almost zero for broad values of CRRA coefficient, suggesting that the performance of the stochastic trend model relative to the trend stationary model does not depend on the value of CRRA coefficient.
See also Bansal and Shaliastovich (2013) for the relationship between long-run risks to exogenous consumption growth and the bond premium puzzle.
This result does not mean that there are little inflation risk premiums. Inflation risk premiums are positive and significant in this model, but there is not large difference in inflation risk premiums for the short- and the long-term interest rates, suggesting that term premiums associated with inflation risk premiums are not large.
The very low inflation responsiveness of UK monetary policy was probably caused by the fact that the monetary policy in the UK was often Treasury-led and subject to political cycles during 1959–1996. Hence, since the Bank of England independence in 1997 and inflation targeting, the parameters of monetary policy response might possibly change and influence the level of term premiums in UK. While the sample since 1997 is too short to discuss the long-term level of term premiums, the effect of such structural breaks in the monetary policy behavior is an interesting topic for future research.
Interestingly, as shown in the robustness check in Appendix B, the level of term premiums is almost the same between the economy with time varying and time invariant trend inflation. See Appendix B for more discussion about the effects of trend inflation on the level of term premiums.