2026-03-17
The final cluster of the planned series turns to Glenn Rudebusch and Tao Wu, A Macro-Finance Model of the Term Structure, Monetary Policy, and the Economy. The abstract of the Federal Reserve Bank of San Francisco version gives the central idea clearly: combine a canonical affine no-arbitrage term-structure specification with standard macroeconomic aggregate relationships for output and inflation.
This is the natural endpoint of the current series arc. The first four posts studied classical short-rate models. Parts 5 and 6 generalized them into an affine state-space framework. Parts 7 to 10 showed how long-run expectations and policy rules can be brought into that framework. Parts 11 and 12 explored richer transition dynamics. A macro-finance model pulls these threads together by letting macro variables and yields live in one joint system.
The price of that unification is complexity. Macro-finance models combine measured macro variables, latent term-structure factors, monetary-policy channels, and no-arbitrage pricing restrictions. The goal of this post is therefore not to reproduce every line of the original derivation. The goal is to expose the minimum state-space structure that makes the economic logic visible.
Let the state vector be
x_t = (\pi_t^\ast, g_t, i_t, z_t)^\top
where \pi_t^\ast is a slow-moving inflation trend or nominal anchor, g_t is an output-gap variable, i_t is the policy-rate component, and z_t is a term-premium or financial factor. This is not the only possible macro-finance state vector, but it is a useful series-level representation because each component has a clear economic role.
The physical transition law is
x_{t+1} = c_P + F_P x_t + \eta_{t+1}
with Gaussian shocks. The important difference from a purely statistical Gaussian term-structure model is that the states now carry macro meaning. The model is no longer content to say that the yield curve is driven by level, slope, and curvature factors with no interpretation. It asks which components correspond to inflation, activity, policy, and risk compensation.
The yields still load affinely on the state:
y_t(\tau) = a_Q(\tau) + b_Q(\tau)^\top x_t + \varepsilon_t(\tau)
But now we also observe macro variables:
m_t = a_M + B_M x_t + \zeta_t
where m_t may contain inflation, the output gap, or the policy rate. This is the defining move of macro-finance term-structure modeling. Yields and macro variables are not estimated in separate silos. They are different observation blocks on the same latent state.
This structure is what allows the model to reinterpret latent finance factors. A factor that looked purely statistical in a yields-only estimation can now be pinned down by its macro loading. Conversely, a macro variable that is noisy or partly unobserved can borrow information from the cross section of yields.
The policy channel is no longer just a short-rate equation sitting outside the term structure. It becomes one part of the transition law and one part of the observation law. In a reduced representation, the policy-rate state evolves with inflation and activity:
i_{t+1} = c_i + \alpha_\pi \pi_t^\ast + \alpha_g g_t + \alpha_i i_t + \varepsilon_{i,t+1}
Then the short end of the curve loads heavily on i_t, while the long end also reflects the persistence of \pi_t^\ast and the financial factor z_t. This is the formal way to express the idea that policy matters for the whole curve, but not exclusively through the current overnight rate.
Rudebusch and Wu emphasize that latent term-structure factors have important macroeconomic and monetary-policy underpinnings. This is one of the most appealing findings in the macro-finance literature because it softens the tension between two styles of modeling. Pure finance models are good at fitting yields but weak on macro interpretation. Pure macro models are interpretable but often too stylized for the cross section of asset prices. A macro-finance state-space model tries to keep the strengths of both.
The same abstract also reports two concrete empirical conclusions: there is no evidence of monetary-policy inertia in the sense of a slow partial adjustment of the policy rate by the Federal Reserve, and both forward-looking and backward-looking elements matter in macro dynamics. These are exactly the kinds of claims that only a joint system can make credibly.
The gains in interpretability come with several difficulties. First, one must decide which macro variables are observed directly and which are latent. Second, the transition law can become crowded very quickly once inflation persistence, output dynamics, policy responses, and financial premia are all included. Third, macro data are revised, and different vintages can materially alter inference. Fourth, no-arbitrage restrictions are helpful but not free; they narrow the model class while potentially amplifying misspecification if the macro block is wrong.
There is also a philosophical difficulty. When a finance factor becomes identified with a macro object, one may gain interpretation but lose flexibility. The model becomes easier to explain and harder to fit. That tradeoff is at the heart of macro-finance term-structure work.
The macro-finance idea can be summarized in one sentence: yields and macro variables are different windows on one joint state. Once that sentence is made mathematical, the rest of the model is a matter of specifying the transition and observation matrices carefully.
In the next post, we implement a reduced macro-finance state-space model in the package. The implementation will expose joint macro and yield observations, a policy-channel decomposition, and a Kalman-filter interface that can handle the combined system while remaining simple enough to inspect.