Autoregressive wild bootstrap inference for nonparametric trends

Abstract

In this paper we propose an autoregressive wild bootstrap method to construct confidence bands around a smooth deterministic trend. The bootstrap method is easy to implement and does not require any adjustments in the presence of missing data, which makes it particularly suitable for climatological applications. We establish the asymptotic validity of the bootstrap method for both pointwise and simultaneous confidence bands under general conditions, allowing for general patterns of missing data, serial dependence and heteroskedasticity. The finite sample properties of the method are studied in a simulation study. We use the method to study the evolution of trends in daily measurements of atmospheric ethane obtained from a weather station in the Swiss Alps, where the method can easily deal with the many missing observations due to adverse weather conditions. JEL classifications: C14, C22.bootstrap samples with a smaller value for . Additionally, it provides a convenient framework for studying the theoretical properties of our method. Specifically, we need → ∞ as n → ∞, such that γ → 1. This is analogous to the block (and dependent wild) bootstrap, where the block size must increase to capture more dependence when the sample size increases. Assumption 7 postulates the formal conditions that needs to satisfy, which imply that γ → 1, but not too fast.Assumption 7. The bootstrap parameter = (n) satisfies → ∞ and / √ nh → 0 as n → ∞.Note that we propose to use a different bandwidthh in Step 1 of the algorithm. This is a common feature in the literature on bootstrap method for nonparametric regression, where by either selecting a larger (oversmoothing) or smaller bandwidth (undersmoothing) than used for the estimator, one can account for the asymptotic bias that is present in the local polynomial estimation, see Hall and Horowitz (2013, Section 1.4) for an extensive literature review. While undersmoothing, such as used in the related paper by Neumann and Polzehl (1998), aims at making the bias asymptotically negligible, oversmoothing aims at producing a consistent estimator of the (non-negligible) bias.Both have advantages and disadvantages, see the extensive discussion in Hall and Horowitz (2013).We follow Bühlmann (1998), also see Härdle and Marron (1991), and consider a solution based on oversmoothing, which we find to work well in practice. After presenting our theoretical results in Section 4, Remark 8 provides an intuition of why oversmoothing allows to consistently estimate the asymptotic bias. 1 We now state the formal conditions thath must satisfy in Assumption 8; one is that h/h → 0 as n → ∞, which ensures the oversmoothing. Assumption 8. The oversmoothing bandwidthh =h(n) satisfies max h , h/h, nh 5h4 → 0 and max h 4 , 1/nh → 0 as n → ∞. Remark 4. There are a number of ways to choose the AWB parameter γ in practice. Using the relation γ = θ 1/ , one can fix θ and choose as a deterministic function of the sample size. Smeekes and Urbain (2014) found in their simulation study that θ = 0.01 paired with = 1.75n ...

A statistical analysis of time trends in atmospheric ethane

“…The time series consists of daily ethane columns recorded under clear-sky conditions between 1986 and 2019 with a total of 2935 data points. Part of the series, from 1994 to 2014, has been analyzed in Franco et al (2015) and Friedrich et al (2020). It is an interesting series to study, since the measurement conditions are very favorable at this location due to high dryness and low local pollution.…”

“…In order to derive theoretical properties of our methods, e.g., (Friedrich et al 2020), one typically assumes that the missing pattern, characterized by {M t }, is independent of the observations. Strictly speaking, in the present case, we cannot exclude a mild dependence between ethane and the missing pattern as ethane's primary sink is oxidation by the hydroxyl radical, which is dependent on solar insolation.…”

“…We allow the probability of observing measurements on a given day to vary over time, which can accommodate for instance seasonal variation and long-term climatic trends. In addition, we assume that the probability of observing a measurement on a given day may be serially dependent, but we need this dependence to decay over time; for the precise meaning we have in mind, please see Friedrich et al (2020), Assumption 4. This ensures that we, over a large enough time span, always have sufficient data available to estimate the trend.…”

“…A second source of missing data-resulting in prolonged periods without observationsmight be instrument failure and/or maintenance, as well as polar nights for stations close to the poles. While instrument failure, if it is not expected to last indefinitely, can be captured by an assumption like Assumption 4 in Friedrich et al (2020) and polar nights can be modeled by varying the probability of missing data, we stress that for prolonged periods without observations, one cannot draw meaningful conclusions. Practically, one needs data around the point of interest to estimate the trend and conduct inference.…”

“…The choice reflects a trade-off: a larger value captures more of the dependence whereas a smaller value allows for more variation in the bootstrap samples. We suggest to follow Friedrich et al (2020) and use γ = θ 1/l with l = 1.75T 1/3 and θ = 0.1. In…”