Letting the Economy Into the Yield Curve: Ang and Piazzesi

For twenty-five years, two groups of clever people studied interest rates and refused to speak to each other.

Finance people built term structure models. Their models were arbitrage-free, mathematically rigorous, and driven by abstract "factors" with no names and no economic meaning. Ask a finance model why the ten-year yield went up and it would say "factor two rose." Ask what factor two is and it would shrug.

Macroeconomists studied interest rates too, as the transmission channel of monetary policy. They ran regressions linking rates to inflation and output. Their variables had names. But their models cheerfully allowed the two-year yield and the three-year yield to imply contradictory things about the future, which is to say they permitted arbitrage, which is to say a trader could have taken them apart for money.

Ang and Piazzesi built the bridge. The paper is technical, but the idea behind it is not.

The problem: the factors have no names

Everyone knows, roughly, why interest rates move. Inflation goes up, the central bank raises rates, and the whole curve responds. The economy weakens, the central bank cuts, the curve responds. This is not controversial. It is on the front page of the newspaper.

Yet the finance models that priced bonds most rigorously had no inflation variable and no growth variable in them at all. They were driven by unobservable latent factors, statistical artefacts extracted from yields themselves. The models were internally consistent and economically mute.

Meanwhile the macro models had the economics but not the discipline. A macroeconomist's vector autoregression, a workhorse tool that regresses a bunch of variables on their own past values, will happily produce forecasts for the two-year and the ten-year yield that are mutually inconsistent, because nothing in the estimation forces them to hang together. And bonds of different maturities are not independent objects. They are claims on the same underlying future, and their prices are locked together by the requirement that no one can construct a free lunch across them.

The key idea via analogy: a family portrait with rules about who can stand where

Ang and Piazzesi take an affine term structure model, the tractable no-arbitrage machinery from Duffie and Kan, and do two things to it.

First, they let real economic variables be the factors. Not just faceless latent factors, but measures of inflation and real economic activity, alongside a couple of unobservable factors to soak up whatever the macro variables miss. The model's drivers now have names you can find in a government statistics release.

Second, and this is the crucial part, they impose no-arbitrage on the whole system. In a standard macro regression, each yield gets its own equation, with its own free coefficients. Ang and Piazzesi refuse. They insist that all the yields be generated by the same underlying factors through the same pricing logic, so that the coefficients in the two-year equation and the coefficients in the ten-year equation are not independent. No-arbitrage ties them together across maturities.

Here is the analogy. Imagine photographing a family and letting each person stand wherever they like. You will get some arrangements that are physically impossible: someone in front of and behind someone else at the same time. Now impose the rule that everyone must be standing on the same floor, in one room, obeying geometry. Suddenly, most arrangements are ruled out. You have far fewer degrees of freedom.

That sounds like a handicap. It is actually the source of the paper's headline result. The restrictions make the forecasts better.

This is counterintuitive and worth dwelling on. Statistically, a restricted model has less freedom to fit the data, so it must fit the historical sample worse. But fitting the sample is not the goal; predicting the future is. An unrestricted macro regression, with dozens of free coefficients and limited data, spends much of its freedom fitting noise. The no-arbitrage restrictions act as a disciplining device: they encode a piece of genuine economic truth (bonds of different maturities are claims on the same future), which stops the model from chasing ghosts. Ang and Piazzesi find that imposing no-arbitrage produces better out-of-sample forecasts than the unrestricted vector autoregressions macroeconomists had been using.

And the macro variables themselves earn their place. Adding inflation and real activity to the model improves forecasts further. Their results indicate that the macro factors account for a substantial share of the movement in yields at the short and middle part of the curve, exactly where monetary policy bites, while the unobservable factors carry most of the weight at the long end, where whatever drives thirty-year yields is evidently not captured by contemporaneous inflation and growth.

Why it mattered

• It founded macro-finance for the yield curve. This paper, more than any other, opened the field of no-arbitrage macro term structure modelling. Rudebusch, Wu, Diebold, Bikbov, Chernov and many others followed, and central banks now run models in this lineage as a matter of routine.
• No-arbitrage as a regularisation device. The finding that theoretical restrictions improve forecasting, rather than merely constraining fit, is a general and important methodological lesson. Economic structure is a form of regularisation, and it is a better-motivated one than an arbitrary penalty term.
• It made yield curve decomposition economically meaningful. Once your factors are inflation and growth, you can ask questions a latent-factor model cannot: how much of the recent rise in ten-year yields is an inflation story, and how much is a term premium story? Central banks care intensely about this. It changes policy.
• It gave the level, slope and curvature factors a possible economic identity. Litterman and Scheinkman had shown three statistical factors explain bond returns; Ang and Piazzesi's work is part of a body of research asking whether those factors are, at bottom, inflation and the business cycle wearing statistical costumes.
• It legitimised the hybrid. After this, saying "my term structure model has no economics in it" or "my macro model permits arbitrage" both became slightly embarrassing.

The honest limitations

• The macro variables are a choice, and other choices exist. Inflation and real activity, measured in a particular way, over a particular sample. Other researchers using other measures, or adding other variables (unemployment, credit spreads, policy expectations), get somewhat different answers about how much the macro factors explain.
• The long end stays mysterious. The most honest reading of the results is that the model explains the part of the curve monetary policy controls and does not explain the part it does not. Whatever drives long-dated yields remains largely captured by unnamed latent factors, which is precisely the problem the paper set out to solve.
• Macro data is slow, revised and stale. Yields update every second. Inflation is published monthly, with a lag, and then revised. A model that conditions on macro data is conditioning on information that arrived late and may later be rewritten, which quietly overstates how usable it would have been in real time.
• It is still an affine model. Everything Dai and Singleton said about the trade-off between stochastic volatility and correlation applies. Everything nonlinear about how rates behave near zero applies. The linear structure is a strong assumption, and the relationship between inflation and rates is unlikely to be linear across a regime shift.
• Structure is only a virtue when it is true. No-arbitrage restrictions improve forecasts because they encode something correct. Impose a restriction that is wrong and you get a confidently mistaken model. The success here is evidence that no-arbitrage is a good restriction, not that restrictions are good in general.
• The sample predates the world we live in. Estimated on decades of conventional monetary policy. The zero lower bound, quantitative easing and forward guidance changed the relationship between the macroeconomy and the curve in ways this vintage of model was never built to represent.

The one-line takeaway

Ang and Piazzesi put inflation and real activity inside a no-arbitrage bond pricing model, and found that forcing all the maturities to hang together, rather than fitting each one freely, made the forecasts better rather than worse, which is how macro-finance term structure modelling began.