Testing a constant mean (no trend) null hypothesis against an increasing alternative is frequently of interest to the time series analyst. Often a linear function is imposed as the alternative trend, sometimes by default as merely the simplest nonconstant function. This paper studies tests for trends with more general shape-restricted alternatives, which include nondecreasing and convex functions. Shape-restricted alternatives comprise a broad range of trends and may be appropriate when the alternative trend structure is not well understood. As we demonstrate, tests with shape-restricted alternatives can have greater power than traditional tests when the actual trend is misspecified; further, a misspecified trend can lead to incorrect conclusions about the autocovariance memory of a time series. A likelihood ration test statistic is proposed and shown to have a limiting distribution that is a mixture of beta variates. The test is shown to have an asymptotic power of unity. Explicit mathematical examples consider nondecreasing and convex trend structures with autoregressive errors. Application of the methods to several series, including a globally averaged annual temperatures from the Northern Hemisphere, is included.
Inference in Shape-Restricted Regression with Time Series Data