Trend Analysis

A prominent feature of many time-series variables is that they tend to move upward or downward over time. This tendency may be defined as a trend. Two models frequently are employed to characterize trending data. The first model is a deterministic trend, Yt = λ0 + λ1T + ∊t, where Yt is a time-series, T is time, λ0 and λ1 are PARAMETERS, and ∊t is a STOCHASTIC shock ∼N(0, σ2). If this model generates Yt, the residuals from a REGRESSION of Yt on T constitute a trend-stationary process.

The second model is a stochastic trend, Yt = β0 + β1Yt−1 + ∊t, where Yt−1 is the time-series lagged one period, β0 and β1 are parameters, and ∊t is a stochastic shock. If β0 = 0, and β1 = 1, this model is a RANDOM WALK; otherwise, it is a random walk with drift. Recursive substitution shows that a random walk equals an initial value Y0, plus the sum of all shocks, ∑ ∊i. The random walk with drift also includes β0T, which imparts a deterministic component to the process. First-differencing (i.e., ΔYt = Yt − Yt−1) a random walk or random walk with drift produces a difference-stationary process.

Trending variables are nonstationary because they lack constant means, variances, and covariances. Regressing one nonstationary variable on another poses threats to inference (“spurious regressions”). Thus, it is vital to detect trends (nonstationarity) in time-series data. A trending series has a “characteristic signature” when correlated with lags of itself. Several such correlations constitute an AUTOCORRELATION function (ACF). The ACF for a trending series has large correlations at low lags, and correlations decline slowly at longer lags. Formal statistical tests (UNIT-ROOT tests) also are widely used to detect nonstationarity.

If nonstationarity is suspected, conventional practice is to difference the data (assuming stochastic, not deterministic, trends). To determine if variables move together in the long run, cointegration tests are performed. Cointegrating variables can be modeled in error correction form—for example, ΔYt = β0 + β1ΔXt − α(Yt−1 − λXt−1) + ∊t—where ΔYt is first-differenced Y; ΔXt is first-differenced X; (Yt−1 − λXt−1) is an error correction mechanism for the long-term relationship between Y and X; ∊t is a stochastic shock; and β0,β1,α, and λ are parameters. α measures the speed with which shocks to the system are eroded by the cointegrating relationship between Y and X. α should be negatively signed and less than 1.0 in absolute value.

Differencing to ensure stationarity has been criticized on theoretical and statistical grounds. The claim is that many social science time-series are either “near integrated” (β1 in a random walk model is slightly less 1.0) or “fractionally integrated.” A fractionally integrated series possesses “long memory” and may be nonstationary. Because unit-root tests have low statistical power, they may not detect near or fractionally integrated processes. Estimating fractional differencing parameter (d) in rival ARFIMA (autoregressive, fractionally integrated, moving average) models is a useful way of considering the possibility that near or fractionally integrated processes may be operative. Recent generalizations of the concepts of cointegration and error correction permit one to model fractionally cointegrated processes.

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