1 May 2011 CFA Institute Journal Review

# Consideration of Trends in Time Series (Digest Summary)

1. Johann U. de Villiers

The authors provide working definitions of trends found in economic time series. They show how these definitions can be used to test for trends and to estimate trends. They also discuss how such patterns as breaks, bubbles, or cycles can be analyzed as a sequence of local trends.

It is common to plot an economic data series and spot a trend. It is also common for different analysts looking at the same time series to agree regarding the presence of a trend. Oddly enough, despite this subjective consensus on identifying trends visually, a review of major econometrics texts reveals that there is no generally accepted definition of a trend. The authors aim to provide working definitions of various types of trends to promote clarity and more focused analytical study.

They start by distinguishing between deterministic and stochastic trends. Pure deterministic trends are rare in economic time series, but they could exist in combination with stochastic trends. Strong deterministic trends are monotonically increasing or decreasing, whereas weak deterministic trends could show a zero change between successive data points in some instances. Accelerating and decelerating trends occur when changes between observations are themselves trending. Local trends occur only over a set period of time.

Most economic time series, however, have a random (stochastic) component. In other words, the economic variable is a random variable and has a statistical distribution with a mean, variance, skewness, and so on. These series can also have a trend. A stochastic trend is defined as occurring when one or more aspects of the variable’s distribution (e.g., variance) exhibit a deterministic movement or direction. For example, a random walk has an increasing stochastic trend in its variance but not in its mean because the variance exhibits an increasing deterministic direction. The authors caution analysts that care must be taken to avoid “spurious correlations” when working with two or more data series that contain stochastic trends.

The authors then discuss the algebra of trends to determine what happens when trends are added, subtracted, multiplied, or divided. They also outline various ways to test for trends, starting with fitting a straight line to the data and testing for a statistically significant slope coefficient to detect a trend in mean. They also briefly review tests for identifying trends in variance, including nonparametric and unit root tests.

Some economic analyses of time series focus primarily on cycles (e.g., the business cycle or seasonal cycles). In those instances, it may be appropriate to remove trends from the data to clarify the cyclical components. So, the authors discuss various methods for removing trends from data.

Another area of focus is the difficulty of forecasting in the presence of trends because it is common for trends to break. A break occurs as a consequence of a change in the character of a local trend. Local trends before and after the break can vary, or a trend may occur before but not after the break, or a parameter of the trend can change at the break.

The authors use their definitions of trends to clarify other aspects of time-series data that analysts regularly encounter. For example, they distinguish between breaks and outliers. An outlier is a single observation that does not follow the trend. After the outlier, the trend continues. But after a break, the trend that existed before the break is discontinued and may be replaced by a different trend.

Using their framework, the authors define a “bubble” as consisting of a sequence of two local trends. The first is an initial upward trend. This trend is then followed by a steep downward trend (i.e., crash). The authors focus on technical bubbles in which only the price sequence is considered. A more fundamental analysis of bubbles would include a comparison of asset prices and some fundamental indication of asset value (such as dividends). They conjecture that technical bubbles form when naive investors are enticed to buy assets by a random upward movement in asset prices. By purchasing the assets, the probability increases of further increases in the price. At some point, the bubble breaks, which leads to the steep downward trend (crash).

The authors conclude by discussing methods that can be used to analyze long-term cycles (such as business cycles), which consist of sequences of local trends.

Finally, they caution that trends in economic data should not be considered only in terms of the time series itself. Researchers should always try to relate trends in the data to their underlying driving forces.