Trend/Cycle Decomposition

Abstract

The decomposition of economic time series is motivated by the idea that distinct forces account for long-term growth, for variation over a time frame associated with the business cycle, and though the seasons. While the latter is typically suppressed by ‘seasonal adjustment’, the issue of how to separate trend from cycle in series such as GDP has been hotly debated since the 1970s and remains unsettled. Surprisingly varied patterns follow from alternative approaches, some placing the bulk of variation into the cycle/trend following a smooth line, others attributing shifts in level to ‘permanent shocks’ to the trend.

Macroeconomists distinguish between the forces that cause long-term growth and those that cause temporary fluctuations such as recessions. The former include population growth, capital accumulation, and productivity change, and their effect on the economy is permanent. The latter are generally monetary shocks such as shifts in central bank policy that affect the real economy through price rigidities that cause output to deviate temporarily from its long-run path. This conceptual dichotomy motivates the decomposition of aggregate output, real GDP, into two components: the trend which accounts for long-term change, and the cycle which is a short-term deviation from trend. While economists no longer believe the ‘business cycle’ to be deterministically periodic, that terminology remains. Seasonal variation could be a third component, though it has been suppressed in ‘seasonally adjusted’ data such as GDP.

This suggests we may express the natural log of GDP (or any other ‘trending’ time series), denoting the observation at time t by ‘y_t’, as follows:

$$ {y}_t={\tau}_t+{c}_t. $$

Here τ_t denotes the value of the trend and c_t the cycle at time t, neither of which is observed directly. Since this single equation cannot be solved directly for the unknown trend and cycle, additional assumptions are required for ‘identification’, a procedure which allows estimates of them to be calculated from the GDP data. The fundamental identifying assumption is that the cycle component is temporary, that it dies out after a sufficiently long time. However, this assumption of ‘stationarity’ or ‘ergodicity’, which distinguishes it from trend, which is permanent, does not suffice by itself to achieve identification. More has to be said about the nature of the trend.

The simplest specification of trend is to make τ_t a linear function of time where the slope is the long-term growth rate. A second identifying assumption is that trend should account for as much of the variation in the data as possible, minimizing the amplitude of the implied cycle. This is achieved by least squares regression of y_t on time and the estimated trend is ${\widehat{\tau}}_t=a+b$time where a and b are estimates of intercept and slope respectively. The implied cycle component is then ${\widehat{c}}_t\equiv {y}_t+{\widehat{\tau}}_t$. Though successful in accounting for a large fraction of the change in GDP over long periods, this approach implies cycles of extraordinary length, well beyond the roughly seven years between recession dates identified by the National Bureau of Economic Research for the United States, and the pattern is contrary to economic intuition (for the United States the 1970s, a decade of poor economic performance, were well above the trend line while the 1990s, a decade of prosperity, were well below trend). A more flexible trend function is clearly called for, but quadratic or higher-order polynomials in time imply unstable paths when extrapolated into the future. Perron (1989) suggested a segmented trend function allowing for an occasional change level or slope to be captured by dummy variables.

A general approach to estimating a flexible and adaptive trend is filtering, where estimated trend is a weighted average of adjacent observations. Here it is the weighting scheme which identifies the components. For example, ${\widehat{\tau}}_t=.25\cdot {y}_{t-1}+.50\cdot {y}_t+.25\cdot {y}_{t+1}$ applies symmetric though unequal weights to the current observations and its immediate neighbours. No filter is perfect in the sense of revealing the actual trend, but a desirable filter is one that extracts as much of the trend as possible from the data. A criterion for choosing a filter would be that it produces cycles having characteristics that match our notions of the business cycle, for example that recessions occur on average about every seven years. A widely used filter that does this is the Hodrick and Prescott (1980), filter which penalizes deviations from trend and changes in trend through a loss function.

The distinction between trend and cycle implies that the forecast of GDP far in the future must be the trend, since the cycle will die away. The approach to trend/cycle decomposition proposed by Beveridge and Nelson (1981) turns this conclusion on its head by proposing that the trend at a date in time be defined as the forecast of the distant future (adjusted for average growth). Specifically, they estimate an autoregressive moving average (ARMA) time series model for the growth rate and compute the forecast of the level into the distant future, adjusting for average growth. The resulting measure of trend shows whether actual GDP is above or below its forecast growth path, the difference being the cycle. Since parameters of the ARMA model are identified, and computation of forecasts is straightforward, the Beveridge–Nelson decomposition is identified. It turns out that the trend component is a random walk with drift regardless of the specific ARMA model, and this accords with the intuition that only unexpected shocks can affect a long horizon forecast.

To obtain the general expressions for the components we rearrange the ARMA model as:

$$ {\displaystyle \begin{array}{l}\varphi (L)\varDelta {y}_t=\theta (L){\upvarepsilon}_t\\ {}\varDelta {y}_t=\psi (L){\upvarepsilon}_t\end{array}} $$

where the average growth rate has been suppressed, the statistical shock ε_t is serially random, Δ denoted first difference, and L is the lag operator, and $\psi (L)=\theta (L)/\varphi (L)$. The growth rate of GDP can be thought of as a weighted history of all past shocks where the coefficient of ε_t−k is ψ_k plus the expected average growth rate μ. It may be shown that an algebraically equivalent expression is

$$ {y}_t=\psi (1)\sum_{k=0}^{\infty }{\varepsilon}_{t-k}-\tilde{\psi}(L){\varepsilon}_t\kern0.5em {\tilde{\psi}}_k=\sum_{j=k+1}^{\infty }{\tilde{\psi}}_j. $$

Note that the first term is the sum of all past shocks each with weight equal to the total effect of all past shocks. The second term may be shown to be a stationary time series with mean zero. Thus the trend is always a random walk regardless of the ARMA model.

For example, growth in US GDP is roughly an AR(1) process with coefficient.25, so the effect of a shock on the trend is ψ(1) = 1/(1 −.25) = 1.33. This illustrates the surprising implication that the trend component may be highly variable; indeed, the results obtained by Beveridge and Nelson imply that variation in observed GDP is largely the result of variation in the trend component and is therefore permanent.

Unobserved components models identify trend and cycle by specifying a separate and specific stochastic process for each. The trend is generally assumed to be a random walk with drift, allowing it to account for long-term growth while permitting it to be shifted by stochastic shocks. The cycle is assumed to be a process that is stationary in the sense of reverting to a mean over time. (The mean of the cycle is zero for symmetric variation around trend, but evidence exits for asymmetric cycles with a negative mean.) This approach was introduced to economics by Harvey (1985) and Clark (1987). An example would be the following:

$$ {\tau}_t={\tau}_{t-1}+\mu +{\eta}_t\kern0.5em {c}_t=\varphi \cdot {c}_{t-1}+{\varepsilon}_t. $$

The parameter μ is the long-term growth rate, the shock η is random and may be positive or negative, the parameter φ measures the persistence of the cycle, and shocks ε drive the cycle. The two shocks are often assumed to be uncorrelated, which reduces the number of parameters to be estimated by one but may also place an unwarranted restriction on the relation between the two components. More generally the cycle process may have a higher-order ARMA specification. Identification of the parameters depends on whether a specific model implies a sufficient number of estimable parameters in the corresponding ARMA reduced form representation of Δy_t (corresponding to identification of simultaneous equation models). Given an identified model and parameter estimates, the estimated trend and cycle may be computed using the Kalman filter.

A useful result is that the random walk trend in the unobserved components model is identified even if its parameters are not identified. Morley et al. (2003) show that the Beveridge–Nelson trend is always the conditional expectation of the trend component given past data. What identifies the trend is the random walk specification for the trend along with the assumption that the cycle process does not persist indefinitely. Thus, the long-horizon forecast reflects only the trend, and such forecasts can always be computed from the reduced form ARMA model.

See Also

• Business Cycle Measurement

• Data Filters

• State Space Models

• Time Series Analysis

• Unit Roots