A No-Arbitrage Vector Autoregression of Term Structure Dynamics with Macroeconomic and Latent Variables

Summary

Ang and Piazzesi (2003) bridge the macro–vector autoregression (VAR) and affine term structure literatures by building a Gaussian VAR with two observable macro factors (inflation and real activity, constructed as principal components) and three latent yield factors, disciplined throughout by no-arbitrage bond pricing. The model prices the entire yield curve analytically for any maturity, not only the yields included in the VAR. Macro factors explain up to 85% of short- and medium-maturity yield variance; imposing no-arbitrage cross-equation restrictions improves out-of-sample forecasting by ~25% root mean squared error (RMSE) over unrestricted VARs.

Key Claims

• Existing latent-factor term structure models label factors "level/slope/curvature" but never connect them to observable macro variables; existing macro-VARs ignore no-arbitrage and can only price maturities included in estimation.
• Macro factors (inflation, real activity) as first principal components (PCs) of two groups: >70% and >50% of group variance explained by first PC. Inflation correlated 67% with 1-month yield; real activity correlation with yields < 6%.
• Taylor (1993) ordinary least squares (OLS) regression of short rate on contemporaneous macro factors: $R^2$=45%; with 12 lags: $R^2$=53%. Residuals autocorrelation 0.94–0.97, motivating latent factors.
• State vector $X_t = (X_t^{o\prime}, X_t^{u\prime})'$ follows VAR(1) companion form with macro and latent blocks orthogonal (zero off-diagonal blocks in $\Phi$).
• Bond yields are exponential-affine in state: $y_t^n = A_n + B_n' X_t$, where $A_n, B_n$ solve recursive no-arbitrage equations (17) driven by risk-price parameters $\lambda_0, \lambda_1$.
• Two-step maximum likelihood estimation (MLE): (1) OLS for macro dynamics and short-rate macro coefficients; (2) MLE for latent factors and risk prices, treating step-1 parameters as fixed. Single-step MLE produced explosive dynamics.
• Identification: 3 yields "measured without error" (1, 12, 60-month) → solve for 3 latent factors; 2 yields measured with independent and identically distributed (IID) error (3, 36-month).
• Level factor survives: adding macro factors barely changes the level latent factor ($R^2$=99% mapping from Yields-Only to Macro level factor; it proxies the first PC of yields, which macro cannot replicate).
• Slope absorbed by inflation: 49% of Yields-Only slope factor variance explained by macro, mainly inflation (negative loading). High inflation → short rate rises relative to long rate → slope narrows.
• No-arbitrage constraints reduce out-of-sample RMSE ~25% vs. unrestricted VAR; Macro model beats Yields-Only model; both outperform random walk (unlike unconstrained VARs).

Concepts Introduced or Extended

• Affine Term Structure Model — introduced here; pricing kernel, market prices of risk, exponential-affine bond prices, coefficient recursions
• Vector Autoregression — extended: no-arbitrage VAR; cross-equation restrictions from bond pricing improve forecasting
• Impulse Response Function — extended: analytical factor-weight IRFs $B_n(\tau)$ for all maturities; macro shocks have hump-shaped responses
• Variance Decomposition — extended: macro vs. latent factor decomposition; inflation dominates short/medium end; level factor dominates long end

Entities Mentioned

• Andrew Ang
• Monika Piazzesi
• Christopher Sims
• Tao Zha
• Thomas Sargent

Quotes

"Our methodology gives us several advantages over existing empirical VAR approaches. First, it allows us to characterize the behavior of the entire yield curve in response to macro shocks rather than just the yields included in the VAR."

"We find that a significant part of the latent factors implied by traditional models with only latent yield variables can be attributed to macro variables. In particular, 'slope' and 'curvature' factors can be related to macro factors, while the 'level' factor survives largely intact when macro variables are incorporated."

"Imposing the cross-equation restrictions from no-arbitrage helps in forecasting. The improvement in forecasting performance is substantial, generally about 25% of the RMSE and 30% of the MAD for all yields."

My Take

The paper's central contribution is making the affine term structure model estimable alongside a flexible macro VAR, rather than choosing between them. The orthogonality restriction (macro and latent factors independent) is the key simplifying assumption that allows two-step estimation — but it rules out the very feedback from yields to macro variables that makes the yield curve a leading indicator (Estrella-Hardouvelis 1991). The level factor's survival is a clean empirical result: macro variables cannot span the first PC of the yield curve, so an unobserved "level" factor is structurally necessary even in a fully macro-augmented model. The forecasting result (no-arbitrage > unconstrained VAR > random walk) directly motivates imposing cross-equation discipline in any macro-finance VAR.