2026-03-17
The previous cluster used survey data to anchor the long-run expectations component of yields. We now turn to the short end of the term structure through Josephine Smith and John Taylor, The Long and the Short End of the Term Structure of Policy Rules. Their abstract states the central empirical claim succinctly: the response of the entire U.S. term structure to inflation and output shifted in the early 1980s, and a change in the monetary-policy rule can explain that shift.
What makes this paper valuable for the series is that it forces us to connect a familiar Taylor-rule intuition to bond yields of many maturities. The short rate reacts directly to inflation and output. Long yields react indirectly through expectations of future short rates and through any change in term premia. The paper derives an equation linking these responses and highlights that the sign is not determined by one force alone.
This post rebuilds that mechanism in the state-space notation of the series. The result is a compact derivation that makes clear why stronger policy reactions can raise the short end and yet flatten or even lower parts of the long end.
Let the short policy rate satisfy a simple rule
i_t = r^\ast + \phi_\pi \pi_t + \phi_y y_t + u_t
where \pi_t is inflation, y_t is the output gap, and u_t is a residual policy disturbance. To keep the notation from colliding with the yield notation, I use i_t for the policy rate and reserve y_t(\tau) for the bond yield of maturity \tau.
The key structural point is that the policy rule determines not just the current short rate but the expected path of future short rates. If inflation and output follow a dynamic system
z_{t+1} = c + F z_t + \varepsilon_{t+1}
with z_t = (\pi_t, y_t, u_t)^\top, then the future policy rate is
i_{t+h} = r^\ast + \ell^\top z_{t+h}
where
\ell = (\phi_\pi, \phi_y, 1)^\top
Therefore
\mathbb{E}_t[i_{t+h}] = r^\ast + \ell^\top \left(F^h z_t + \sum_{j=0}^{h-1}F^j c\right)
The response of the expected future policy path to inflation or output depends on the entire propagation matrix F, not just on the contemporaneous coefficients \phi_\pi and \phi_y.
Ignoring term premia for a moment, the yield on an n-period bond is approximately the average expected future short rate:
y_t^{(n)} \approx \frac{1}{n}\sum_{h=0}^{n-1}\mathbb{E}_t[i_{t+h}]
Substituting the expression above gives
y_t^{(n)} \approx r^\ast + \frac{1}{n}\sum_{h=0}^{n-1}\ell^\top F^h z_t + \frac{1}{n}\sum_{h=0}^{n-1}\ell^\top \sum_{j=0}^{h-1}F^j c
This formula makes the policy-rule transmission channel explicit. The loading of the n-period yield on the macro state is not simply \ell. It is an average of propagated rule coefficients:
b_n^\top = \frac{1}{n}\sum_{h=0}^{n-1}\ell^\top F^h
That one equation explains why the term structure of policy rules is richer than the rule at the short end. A stronger inflation response coefficient has one immediate effect, namely a stronger current short-rate reaction, but it also changes expected future inflation and output through the transition matrix. The long end reacts to both channels.
This is the mathematical origin of the countervailing forces described by Smith and Taylor. A larger \phi_\pi tends to push up the current short rate for a given inflation shock. But if that stronger reaction also dampens future inflation and output, it can lower expected future short rates sufficiently to flatten the long end relative to the short end.
The same logic applies to the output coefficient. A stronger response to the output gap can raise the front end on impact while simultaneously changing the persistence of the macro state in a way that lowers or raises longer yields. The sign depends on the joint dynamics, not on one coefficient in isolation.
This is why policy-rule inference from long yields is delicate. The yield curve does not reveal a Taylor rule in a literal one-step way. It reveals the rule composed with the macro transition system.
Smith and Taylor document a large secular shift in the estimated response of the entire U.S. term structure to inflation and output, with the break occurring in the early 1980s. Their interpretation is that a shift in the monetary-policy rule explains the shift in the term structure. They also investigate the 2002 to 2005 period and discuss its effect on long-term rates.
For this series, the most important takeaway is not a single coefficient estimate. It is the structural reading of the yield curve. The short end reflects the policy rule most directly. The long end reflects the rule only after it has been filtered through the expected macro path. Any empirical implementation that ignores that propagation mechanism is likely to overinterpret the contemporaneous rule coefficients.
A policy-rule term-structure model must choose how much structure to impose on macro dynamics. A fully structural New Keynesian system is one possibility. A reduced linear Gaussian macro state is another. Smith and Taylor are interested in the mapping between rule coefficients and yield responses, so the transition mechanism matters at least as much as the rule itself.
The choice between reduced and structural approaches is not innocuous. A strongly structural model is easier to interpret but harder to estimate and more sensitive to misspecification. A reduced model is more flexible but can blur the meaning of the policy coefficients. For the purpose of this series, a reduced linear state-space model is the right intermediate step because it keeps the algebra visible.
The main econometric difficulty is that changes in long yields can reflect several distinct objects: changes in expected policy rates, changes in inflation expectations, changes in real activity, and changes in term premia. If the model leaves the term premium unconstrained, a policy interpretation can become too loose. If it constrains the term premium too aggressively, the policy rule can be forced to explain movements that it should not explain.
There is also a regime problem. A policy rule estimated over one era may not transport well to another. This is particularly important around the Volcker disinflation and the subsequent Great Moderation. A single stationary law of motion can be a useful approximation, but it should not be mistaken for a literal historical constant.
The term structure of policy rules is the result of averaging propagated monetary-policy responses over the future path of the macro state. That is the derivation worth remembering. It tells us why the long end of the curve is neither a simple mirror of the current short rate nor an object that can be interpreted without a macro transition system.
In the next post, we implement this mechanism in the package. The code will expose a reduced policy-rule term-structure model that separates the macro state, the short-rate rule, and the resulting yield loadings so that we can inspect their interactions directly.