This article explores a simple property of the Hodrick–Prescott (HP) filter: when the HP filter is applied to a series, the cyclical component is equal to the HP-filtered trend of the fourth difference of the series, except for the first and last two observations, for which different formulas are needed. We use this result to derive small sample results and asymptotic results for a fixed smoothing parameter. We first apply this property to analyze the consequences of a deterministic break. We find that the effect of a deterministic break on the cyclical component is asymptotically negligible for the points that are away from the break point, while for the points in the neighborhood of the break point, the effect is not negligible even asymptotically. Second, we apply this property to show that the cyclical component of the HP filter when applied to series that are integrated up to order 2 is weakly dependent, while the situation for series that are integrated up to order 3 or 4 is more subtle. Third, we characterize the behavior of the HP filter when applied to deterministic polynomial trends and show that in the middle of the sample, the cyclical component reduces the order of the polynomial by 4, while the end point behavior is different. Finally, we give a characterization of the HP filter when applied to an exponential deterministic trend, and this characterization shows that the filter is effectively incapable of dealing with a trend that increases this fast. Our results are compared with those of Phillips and Jin (2015, Business cycles, trend elimination, and the HP filter).

In recent decades, in the research community of macroeconometric time series analysis, we have observed growing interest in the smoothing method known as the Hodrick–Prescott (HP) filter. This article examines the properties of an alternative smoothing method that looks like the HP filter, but is much less well known. We show that this is actually more like the exponential smoothing filter than the HP filter although it is obtainable through a slight modification of the HP filter. In addition, we also show that it is also like the low-frequency projection of Müller and Watson (2018, Econometrica 86, 775–804). We point out that these results derive from the fact that all three similar smoothing methods can be regarded as a type of graph spectral filter whose graph Fourier transform is discrete cosine transform. We then theoretically reveal the relationship between the similar smoothing methods and provide a way of specifying the smoothing parameter that is necessary for its application. An empirical examination illustrates the results.

In this article, we discuss the bootstrap as a tool for statistical inference in econometric time series models. Importantly, in the context of testing, properties of the bootstrap under the null (size) as well as under the alternative (power) are discussed. Although properties under the alternative are crucial to ensure the consistency of bootstrap-based tests, it is often the case in the literature that only validity under the null is discussed. We provide new results on bootstrap inference for the class of double-autoregressive (DAR) models. In addition, we review key examples from the bootstrap time series literature in order to emphasize the importance of properly defining and analyzing the bootstrap generating process and associated bootstrap statistics, while also providing an up-to-date review of existing approaches. DAR models are particularly interesting for bootstrap inference: first, standard asymptotic inference is usually difficult to implement due to the presence of nuisance parameters; second, inference involves testing whether one or more parameters are on the boundary of the parameter space; third, even second-order moments may not exist. In most of these cases, the bootstrap is not considered an appropriate tool for inference. Conversely, and taking testing nonstationarity to illustrate, we show that although a standard bootstrap based on unrestricted parameter estimation is invalid, a correct implementation of the bootstrap based on restricted parameter estimation (restricted bootstrap) is first-order valid. That is, it is able to replicate, under the null hypothesis, the correct limiting distribution. Importantly, we also show that the behavior of this bootstrap under the alternative hypothesis may be more involved because of possible lack of finite second-order moments of the bootstrap innovations. This feature makes for some parameter configurations, the restricted bootstrap unable to replicate the null asymptotic distribution when the null is false. We show that this possible drawback can be fixed by using a novel bootstrap in this framework. For this “hybrid bootstrap,” the parameter estimates used to construct the bootstrap data are obtained with the null imposed, while the bootstrap innovations are sampled with replacement from unrestricted residuals. We show that the hybrid bootstrap mimics the correct asymptotic null distribution, irrespective of the null being true or false. Monte Carlo simulations illustrate the behavior of both the restricted and the hybrid bootstrap, and we find that both perform very well even for small sample sizes.