Many questions in economics involve long-run or “trend” variation and covariation in time series. Yet, time series of typical lengths contain only limited information about this long-run variation. This paper suggests that long-run sample information can be isolated using a small number of low-frequency trigonometric weighted averages, which in turn can be used to conduct inference about long-run variability and covariability. Because the low-frequency weighted averages have large sample normal distributions, large sample valid inference can often be conducted using familiar small sample normal inference procedures. Moreover, the general approach is applicable for a wide range of persistent stochastic processes that go beyond the familiar I (0) and I (1) models.

INTRODUCTION

This paper discusses inference about trends in economic time series. By “trend” we mean the low-frequency variability evident in a time series after forming moving averages such as low-pass (cf. Baxter and King, 1999) or Hodrick and Prescott (1997) filters. To measure this low-frequency variability we rely on projections of the series onto a small number of trigonometric functions (e.g., discrete Fourier, sine, or cosine transforms). The fact that a small number of projection coefficients capture low-frequency variability reflects the scarcity of low-frequency information in the data, leading to what is effectively a “small-sample” econometric problem. As we show, it is still relatively straightforward to conduct statistical inference using the small sample of low-frequency data summaries.Moreover, these low-frequency methods are appropriate for both weakly and highly persistent processes. Before getting into the details, it is useful to fix ideas by looking at some data.

Figure 1 plots the value of per-capita GDP growth rates (panel A) and price inflation (panel B) for the United States using quarterly data from 1947 through 2014, and where both are expressed in percentage points at an annual rate. The plots show the raw series and two “trends.” The first trend was constructed using a band-pass moving average filter designed to pass cyclical components with periods longer than T/6 ≈ 11 years, and the second is the full-sample projection of the series onto a constant and twelve cosine functions with periods 2T/j for j = 1, …, 12, also designed to capture variability for periods longer than 11 years.