Campbell-Shiller (1987) Cointegration and Tests of Present Value Models

Summary

Campbell and Shiller apply the then-new econometrics of cointegrated vector autoregressions to test present-value models — models in which a variable equals the expected discounted sum of a future fundamental (stock price and dividends; the long interest rate and future short rates). They show the present-value restriction, combined with a nonstationary $I(1)$ fundamental, implies that the two series are cointegrated with a known cointegrating vector, which turns the theory into testable cross-equation restrictions on a bivariate VAR. On US data they find relatively encouraging results for the rational-expectations theory of the term structure but puzzling (excess-volatility) results for the present-value model of stock prices.

Key Claims

• Present-value model. $Y_t = \theta(1-\delta)\sum_{i=0}^{\infty}\delta^i E_t[y_{t+i}] + c$: a variable is a linear function of the present discounted value of expected future values of a fundamental $y_t$.
• Implies cointegration. If $y_t\sim I(1)$, the present-value model implies $Y_t$ and $y_t$ are cointegrated; the spread $S_t=Y_t-\frac{\theta}{1-\delta}y_t$ is stationary and equals the present value of expected future changes $\Delta y_{t+i}$.
• VAR test. Estimate a bivariate VAR in $(\Delta y_t, S_t)$; the model implies the actual spread should equal the VAR-forecast "theoretical spread" — a set of cross-equation restrictions that can be tested, and the two spreads compared graphically.
• Handles two problems at once. The cointegrated-VAR approach deals with the nonstationarity of the series and with incomplete data on agents' information (the VAR proxies the information set).
• Empirics. Encouraging for the rational-expectations term structure; the present-value model of stock prices is puzzling — actual prices are far more volatile than the dividend-based theoretical value (excess volatility).

Concepts Introduced or Extended

• Cointegration

Entities Mentioned

• John Y. Campbell
• Robert J. Shiller

Quotes

"Application of some advances in econometrics (in the theory of cointegrated vector autoregressive models) enables us to deal effectively with two problems in rational expectations present value models: nonstationarity of time series and incomplete data on information of market participants."

My Take

A model paper for how to take an economic theory to nonstationary data. The key insight is that a present-value relationship is not just suggestive of a long-run link — it is a cointegrating restriction with a coefficient the theory pins down, so the whole apparatus of cointegrated VARs (the Engle-Granger/Johansen machinery) can be brought to bear, and the "spread" becomes a forecast of future fundamental changes you can plot against its VAR-implied value. It rehabilitated Shiller's earlier excess-volatility critique in a form robust to the unit-root objections that had dogged the variance-bounds tests, and the two headline results — term structure roughly OK, stock prices too volatile — set the agenda for a generation of asset-pricing work. The caveats are the usual VAR ones: the test is only as good as the information set the VAR proxies and the assumed constant discount rate, and relaxing the latter is where much later research went.