Common Stochastic Trends

Definition

In a system of $n$ I(1) time series with cointegration rank $r$ , there are exactly $k = n - r$ common stochastic trends — linearly independent I(1) processes that drive the long-run movements of all $n$ variables. The variables are bound together precisely because they share these $k$ trends: any deviation from the long-run equilibria is transitory. The concept provides the permanent-component representation dual to the Vector Error Correction Model (VECM)'s error-correction representation.

Key Ideas

Algebraic fingerprint. In the Wold moving-average representation $\Delta \mathbf{X}_t = \mathbf{C}(L)\boldsymbol{\varepsilon}_t$ , the long-run impact matrix $\mathbf{C}(1) = \sum_{i=0}^\infty \mathbf{C}_i$ has rank $k = n - r$ . Its column span contains the directions of permanent shocks; its left null space is spanned by the $r$ cointegrating vectors $\boldsymbol{\beta}$ .
Beveridge-Nelson generalisation. The single-variable Beveridge-Nelson (1981) decomposition splits $x_t$ into a random-walk trend (the long-run forecast) and a stationary cycle. The multivariate common-trends framework generalises this to $n$ variables sharing $k$ random-walk factors.
Stock-Watson (1988) test. Stock and Watson (1988) proposed testing the number of common trends $k$ (equivalently, the cointegration rank $r = n - k$ ) by examining whether the largest $k$ eigenvalues of a long-run covariance matrix are well separated from zero.
Identification. The $k$ common trends $\boldsymbol{\tau}_t = \mathbf{A}_\perp' \mathbf{X}_t$ are identified up to invertible linear transformations. Economic theory can identify specific trends by restricting the long-run multiplier matrix — as in King-Plosser-Stock-Watson (KPSW, 1991), where balanced-growth theory implies the first column of $\boldsymbol{\Gamma}(1)$ is $(1,1,1)'$ (the balanced-growth shock raises output, consumption, and investment one-for-one in the long run).

How It Works

The Moving-Average Representation

For a cointegrated $n$ -vector $\mathbf{X}_t$ with $r$ cointegrating vectors, the Granger Representation Theorem gives:

$\mathbf{X}_t = \boldsymbol{\mu}t + \mathbf{C}(1)\sum_{s=1}^t \boldsymbol{\varepsilon}_s + \mathbf{C}^*(L)\boldsymbol{\varepsilon}_t + \mathbf{X}_0$

The $k$ -dimensional cumulated innovation process $\boldsymbol{\tau}_t = \mathbf{C}(1)\sum_{s=1}^t \boldsymbol{\varepsilon}_s$ is the common stochastic trend component — a $k$ -dimensional random walk scaled by the columns of $\mathbf{C}(1)$ . The stationary part $\mathbf{C}^*(L)\boldsymbol{\varepsilon}_t$ collects the transitory dynamics.

Since rank $(\mathbf{C}(1)) = k$ , write $\mathbf{C}(1) = \boldsymbol{\beta}_\perp \mathbf{A}_\perp'$ where $\boldsymbol{\beta}_\perp$ ( $n \times k$ ) is the orthogonal complement of the cointegrating matrix $\boldsymbol{\beta}$ and $\mathbf{A}_\perp$ ( $n \times k$ ) is the orthogonal complement of the loading matrix $\boldsymbol{\alpha}$ . Then:

$\mathbf{X}_t^P = \boldsymbol{\mu}t + \boldsymbol{\beta}_\perp \underbrace{\mathbf{A}_\perp' \mathbf{X}_t}_{\boldsymbol{\tau}_t}, \qquad \boldsymbol{\tau}_t = \boldsymbol{\tau}_{t-1} + \boldsymbol{\eta}_t^1$

where $\boldsymbol{\eta}_t^1 = \mathbf{A}_\perp' \boldsymbol{\varepsilon}_t$ are the $k$ permanent innovations. The decomposition $\mathbf{X}_t = \mathbf{X}_t^P + \mathbf{X}_t^S$ into permanent and stationary parts is the multivariate Beveridge-Nelson decomposition.

KPSW (1991) Identification Scheme

King, Plosser, Stock, and Watson (1991) use balanced-growth theory to identify the first permanent shock as a real productivity shock. For the three-variable real system $\mathbf{X}_t = (y_t, c_t, i_t)'$ :

Cointegration rank $r = 2$ (one common trend, $k = 1$ )
Long-run multiplier: $\boldsymbol{\Gamma}(1) = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ (balanced-growth normalisation)
Identification restriction: the permanent shock $\eta_t^1$ is uncorrelated with the two transitory shocks

For the six-variable system ( $k = 3$ trends), the $6 \times 3$ matrix $\mathbf{A}$ is lower triangular with shocks ordered: (i) balanced-growth, (ii) neutral inflation, (iii) real-interest-rate.

Gonzalo-Granger Decomposition

Gonzalo and Granger (1995) provide a general identification of common stochastic trends that does not require economic theory. The $k$ common trends are $W_t = \boldsymbol{\alpha}_\perp' \mathbf{X}_t$ where $\boldsymbol{\alpha}_\perp$ is the orthogonal complement of $\boldsymbol{\alpha}$ . The identifying restriction is that the transitory component $z_t = \boldsymbol{\beta}'\mathbf{X}_t$ does not Granger-cause $W_t$ at very low frequencies. See Cointegration §Gonzalo-Granger for the full decomposition and its relation to the multivariate Beveridge-Nelson decomposition.

Beveridge-Nelson (1981) Univariate Case

For a single I(1) series $x_t$ with AutoRegressive Integrated Moving Average (ARIMA) $(p,1,q)$ representation, the Beveridge-Nelson trend is:

$\tau_t = \lim_{h\to\infty} [E_t x_{t+h} - h \cdot \mu]$

the long-run forecast of $x_{t+h}$ minus its deterministic trend. The Beveridge-Nelson (BN) trend follows a random walk with the same innovation variance as $\Delta x_t$ ; the cycle is $x_t - \tau_t$ , which is I(0). The KPSW multivariate framework generalises this: the $k$ common trends are the multivariate long-run forecasts in the $k$ trend directions.

Why It Matters

The common-stochastic-trends representation provides a structural decomposition of macroeconomic time series into permanent and transitory components — directly answering the Friedman-era question of what fraction of GDP fluctuations is trend versus cycle.
Economic theory (balanced growth, money demand, Fisher parity) generates restrictions on the number and nature of common trends, making the decomposition empirically identifiable without arbitrary normalizations.
KPSW (1991) showed that balanced-growth theory passes its cointegration test, but that the identified balanced-growth trend explains less than half of business-cycle variability in output once nominal variables are included — challenging pure Real Business Cycle (RBC) theories.
The framework underlies trend–cycle decompositions used in potential output estimation, structural fiscal analysis, and medium-term macroeconomic forecasting.

Open Questions

The rotation matrix $\boldsymbol{\Pi}$ in the six-variable KPSW system is not uniquely pinned down by balanced-growth theory alone; the real-interest-rate permanent shock identification requires additional assumptions about money demand.
Structural breaks in the cointegrating relations (e.g., from shifts in the natural rate) can masquerade as permanent components and contaminate the common-trend extraction.
In large systems ( $n > 10$ ), estimating both the number of common trends and their dynamic responses jointly is computationally demanding.