Dickey-Jansen-Thornton (1991) A Primer on Cointegration with an Application to Money and Income

Summary

A pedagogical survey comparing three methods for testing cointegration — Engle-Granger, Stock-Watson, and Johansen — unified through the vector autoregression (VAR) reparameterization $\psi = (I - A_1 - \cdots - A_p)$ whose rank gives the number of cointegrating vectors. Applied to U.S. quarterly money demand data (1953.2–1988.4), the paper demonstrates that cointegration findings are sensitive to normalization choice (Engle-Granger), aggregation level (Stock-Watson), and interest rate specification (Johansen), and cautions against reading structural meaning into reduced-form cointegrating vectors.

Key Claims

VAR reparameterization. Any VAR( $p$ ) can be rewritten as $\Delta Y_t = \Gamma_1 \Delta Y_{t-1} + \cdots - \psi Y_{t-p} + \varepsilon_t$ where $\psi = I - A_1 - \cdots - A_p$ . The rank of $\psi$ equals the number of cointegrating vectors $k$ ; the system has $n - k$ common stochastic trends.
Engle-Granger test. Estimate the cointegrating regression by ordinary least squares (OLS) for a chosen normalization variable, then apply an augmented Dickey-Fuller (ADF) test to the residuals. Critical values are non-standard. Normalization-sensitive: different choices of the left-hand-side variable can alter test outcomes and cointegrating vector estimates.
Stock-Watson test. Factor-analyze the $\Delta Y_t$ series; directions of largest variance identify the common stochastic trends, while the complementary subspace identifies the cointegrating relations. The number of common trends equals $n - k$ , so testing for trends is equivalent to testing the rank of $\psi$ .
Johansen test. Performs canonical correlation analysis of $\Delta Y_t$ with $Y_{t-p}$ after projecting out lagged differences. The factorization $\psi = \alpha\beta'$ yields cointegrating vectors $\beta$ and loading matrix $\alpha$ ; rank is determined sequentially by trace and max-eigenvalue statistics. Full maximum likelihood (ML) estimation makes this more efficient than Engle-Granger.
Multiple comparison power loss. Sequential rank tests (rank 0, then rank 1, etc.) inflate total type I error and shift critical values far from standard Dickey-Fuller tables. Power decreases as the number of variables grows.
Economic interpretation caveat. Cointegrating vectors estimated from a reduced-form VAR cannot be interpreted as structural equations — they represent linear combinations of the actual structural relations. The Fisher identity $\{1, 1, -1, -1\}$ for $\{\ln M, \ln V^*, \ln P, \ln q\}$ is an economic restriction the reduced form does not automatically recover.
Money demand application. Johansen finds 1 cointegrating vector for {real M1, real income $q$ , interest rate}; income elasticity $\approx 0.68$ – $0.85$ (unity not rejected); interest elasticity significantly negative. M2 and NM1M2 each yield 1 cointegrating (CI) vector. Monetary base with $K$ yields 2 CI vectors with R3M (3-month rate) but only 1 with R10Y (10-year yield), illustrating sensitivity to interest rate choice.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"One must keep in mind that the cointegrating vector estimated from a system of equations is a linear combination of the actual structural equations. In general, it cannot be interpreted as a single structural equation."

My Take

An accessible entry point to cointegration methodology for applied economists. The side-by-side comparison of three competing tests on the same dataset is the paper's main empirical contribution — it exposes real disagreements in practice, not just theoretical asymptotic rankings. The power-loss warning for multiple comparisons is underemphasized in later applied work. The structural interpretation caveat is frequently ignored in empirical money demand studies that cite this very paper.