**5. Conclusions**

This paper provides a review of interval-based hypothesis testing methods, which are known under the name of minimum-effect, non-inferiority, and equivalence tests in biostatistics and psychology. Although the first proposal of such a test goes back to Hodges and Lehmann (1954), it has attracted little attention in the business disciplines of science. In the latter, the paradigm of point-null hypothesis has been the major workforce in making statistical decisions and establishing research findings. However, as a number of authors have criticized for many years, the current paradigm has a range of limitations and deficiencies, as discussed in Section 2 of this paper. These problems have become even more apparent in the big data era, where the *p*-value criterion widely and routinely adopted by statistical researchers is no longer usable in making sensible statistical decisions. The consequences are serious, with widespread practice of data-mining (Black 1993), data-snooping (Lo and MacKinlay 1990), *p*-hacking (Harvey 2017), and multiple testing (Harvey et al. 2016), which result in an embarrassing number of false positives as Harvey (2017) puts it. The related empirical evidence is provided by meta-analytic studies conducted by Kim and Ji (2015) and Kim et al. (2018). Even more serious is systematic distortion of published results, such as publication bias (Basu and Park 2014) and replication crisis (Peng 2015). In light of these problems, Rao and Lovric (2016) call for a new paradigm to be in place for statistical testing in the 21st century, with a proposal of interval-based hypothesis testing as a possible solution.

An important point in favor of adopting an interval-based test is the fact that an economic hypothesis cannot be formulated as a point. Rather, it is more sensible when it takes a form of an interval or a neighborhood: see, for example, De Long and Lang (1992), Leamer (1988), and Startz (2014). For example, when a researcher tests for stock market efficiency, she is not testing for a perfect efficiency (as described by a point-null hypothesis), since such a perfect relationship cannot hold economically (Grossman and Stiglitz 1980). More realistically, the researcher is interested in whether the degree of market inefficiency (Campbell et al. 1997) is economically large enough to be concerned (an interval hypothesis). Hence, it makes more sense to consider an interval hypothesis for decision-making in economic or business research.

As we have seen in this paper, an interval-hypothesis can be implemented in a simple and straightforward manner, using the existing instruments of hypothesis testing such as one-tailed test, confidence interval, and non-central distributions. Its main attraction is that the critical values of these tests increase with sample size, overcoming a major deficiency of point-null hypothesis testing. A key requirement of the test is that the researcher should specify an interval of economic significance under the null or alternative hypothesis, preferably before she observes the data. This may require a value judgment depending on contexts, accompanied by a thorough economic analysis on the effect size of the relationship under investigation. This is an integral part of interval-based hypothesis testing, since it has a strong impact on the test outcome and research integrity. It is also highly desirable that the relevant research community establishes a consensus on the range of minimum effect size that matters economically.

We have applied the interval-based tests to economics and finance applications. The first is a test for market efficiency, whether investors' mood has a systematic effect on stock market return. While the effect may appear to show statistical significance under the current point-null paradigm, the minimum-effect tests cannot reject its negligible economic effect. The second is on the empirical validity of asset-pricing models. In contrast to the findings based on point-null hypothesis testing, we find that a class of multi-factor models are empirically valid based on minimum-effect and model equivalence tests. The third is on the degree of persistence of economic time series. A unit root test based on a conventional point-null hypothesis strongly favors the presence of a unit root in many macroeconomic time series such as the real GNP. According to the non-inferiority test, many time series in Nelson–Plosser data set are found to show a degree of persistence of a trend-stationary time series, especially in the real income variables. From these applications, we find that the interval-based tests are applicable to many contentious research problems in the business disciplines of science, shedding new lights on the existing results or stylized facts. We propose that interval-based hypothesis tests be widely adopted in business research, especially in the new era of big data.

**Author Contributions:** J.H.K. conceptualized, developed theory, analyzed data, wrote, and reviewed manuscript. A.P.R. conceptualized, wrote, and reviewed manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
