David A. Dickey

Distribution of the Estimators for Autoregressive Time Series with a Unit Root

Abstract Let n observations Y 1, Y 2, ···, Y n be generated by the model Y t = pY t−1 + e t , where Y 0 is a fixed constant and {e t } t-1 n is a sequence of independent normal random variables with mean 0 and variance σ2. Properties of the regression estimator of p are obtained under the assumption that p = ±1. Representations for the limit distributions of the estimator of p and of the regression t test are derived. The estimator of p and the regression t test furnish methods of testing the hypothesis that p = 1.

Determining the Order of Differencing in Autoregressive Processes

One way of handling nonstationarity in time series is to compute first differences and fit a model to the differenced series unless the differenced series also looks nonstationary. In that case, second- or higher-order differencing is done. To decide if the current degree of differencing is sufficient, one can look at the autocorrelation function for slow decay. A formal statistical test for the need to difference further is available if one is willing to assume that at most one more difference will render the series stationary. In this article, we present a proper sequence of statistical tests that allows the practitioner to handle cases in which a high order of differencing may be needed. The proper sequence is not the traditional sequence, which begins with a test for a single unit root.

Testing for Unit Roots in Seasonal Time Series

Abstract Regression estimators of coefficients in seasonal autoregressive models are described. The percentiles of the distributions for time series that have unit roots at the seasonal lag are computed by Monte Carlo integration for finite samples and by analytic techniques and Monte Carlo integration for the limit case. The tabled distributions may be used to test the hypothesis that a time series has a seasonal unit root.

A Primer on Cointegration with an Application to Money and Income

For some time now, macroeconomists have been aware that many macroeconomic time-series are not stationary in their levels and that many time-series are most adequately represented by first differences.1 In the parlance of time-series analysis, such variables are said to be integrated of order one and are denoted I(1). The level of such variables can become arbitrarily large or small so there is no tendency for them to revert to their mean level. Indeed, neither the mean nor the variance is a meaningful concept for such variables.

Unit Roots in Time Series Models: Tests and Implications

Abstract The decision on whether or not to include a unit root in an autoregressive operator has profound implications. Formal tests for the presence of unit roots give analysts objective guidance in this decision. This article is a practical guide to the use of these tests.

Hypothesis Testing in ARIMA(p, 1, q) Models

Abstract Let the time series {Yt : t ∈ (1, 2, …)} satisfy Yt = ρY t-1 + Z t and Zt + Σ p i=1 a i Zt−1 = et + Σ q j=1 β j et-j, where {e t } is a sequence of normal, independently distributed (NID(0, σ2)) random variables, and y 0 = 0. Associated with the Zt process are the characteristic equations mp + Σ p i=1 aimp-i = 0 and mq + Σ q j=1 βjmq-j = 0, the roots of which are assumed to be less than one in absolute value. Thus, using the notation of Box and Jenkins (1976), we would say Yt is an ARIMA(p, 1, q) process if ρ = 1. Under the assumption that ρ = 1, the limiting distributions of nonlinear least squares regression estimators of the parameters appearing in the preceding model are obtained. Regression t-type statistics for testing the hypothesis that ρ = 1 are discussed. Similar results are obtained for models that allow a nonzero mean. An illustrative example is given.

The Relation of Stroke Admissions to Recent Weather, Airborne Allergens, Air Pollution, Seasons, Upper Respiratory Infections, and Asthma Incidence, September 11, 2001, and Day of the Week

Background and Purpose— Some previous research links stroke incidence to weather, some links strokes to air pollution, and some report seasonal effects. Alveolar inflammation was proposed as the mechanistic link. We present a unified model of time, weather, pollution, and upper respiratory infection (URI) incidence. Methods— We combined existing databases: US Environmental Protection Agency pollution levels, National Weather Service data, counts of airborne allergens, and New York City Health and Hospitals Corporation counts of stroke, asthma, and URI patients. We used autoregressive integrated moving average modeling (a statistical time series modeling technique) with stroke admissions as the response variable and day of week, holidays, September 11th, and other counts and levels as explanatory variables. Results— Using a broad definition of stroke, there were 5.1±2.3 stroke admissions per day: narrowly defined, 4.2±2.1 strokes per day. There are relatively fewer strokes on Sundays (0.50 strokes; P=0.0011), Saturdays (0.62; P<0.0001), Fridays (0.38; P=0.0009) and holidays (0.875; P=0.0016). We found relatively small, independent exacerbating effects of higher air temperature (P=0.0211), dry air (P=0.0187), URIs, (P<0.0001), grass pollen (P=0.0341), sulfur dioxide (SO2; P=0.0471), and suspended particles <10 &mgr;m in size (P=0.0404). These effects are modest: ≤0.6, 0.6, 2.4, 1, 0.9, and 0.7 strokes per day, respectively. We did not find statistically significant exacerbating effects of other variables. Conclusions— We found statistically significant, independent exacerbating effects of warmer, drier air, URIs, grass pollen, SO2, and particulate air pollution. The model supports the theory that links pulmonary inflammation to stroke.

Characterizing ammonia emissions from swine farms in eastern North Carolina: part 1--conventional lagoon and spray technology for waste treatment.

Abstract Ammonia (NH3) fluxes from waste treatment lagoons and barns at two conventional swine farms in eastern North Carolina were measured. The waste treatment lagoon data were analyzed to elucidate the temporal (seasonal and diurnal) variability and to derive regression relationships between NH3 flux and lagoon temperature, pH and ammonium content of the lagoon, and the most relevant meteorological parameters. NH3 fluxes were measured at various sampling locations on the lagoons by a flow-through dynamic chamber system interfaced to an environmentally controlled mobile laboratory. Two sets of open-path Fourier transform infrared (FTIR) spectrometers were also used to measure NH3 concentrations for estimating NH3 emissions from the animal housing units (barns) at the lagoon and spray technology (LST) sites.Two different types of ventilation systems were used at the two farms. Moore farm used fan ventilation, and Stokes farm used natural ventilation. The early fall and winter season intensive measurement campaigns were conducted during September 9 to October 11, 2002 (lagoon temperature ranged from 21.2 to 33.6 °C) and January 6 to February 2, 2003 (lagoon temperature ranged from 1.7 to 12 °C), respectively. Significant differences in seasonal NH3 fluxes from the waste treatment lagoons were found at both farms. Typical diurnal variation of NH3 flux with its maximum value in the afternoon was observed during both experimental periods. Exponentially increasing flux with increasing surface lagoon temperature was observed, and a linear regression relationship between logarithm of NH3 flux and lagoon surface temperature (T l) was obtained. Correlations between lagoon NH3 flux and chemical parameters, such as pH, total Kjeldahl nitrogen (TKN), and total ammoniacal nitrogen (TAN) were found to be statistically insignificant or weak. In addition to lagoon surface temperature, the difference (D) between air temperature and the lagoon surface temperature was also found to influence the NH3 flux, especially when D > 0 (i.e., air hotter than lagoon). This hot-air effect is included in the statistical-observational model obtained in this study, which was used further in the companion study (Part II), to compare the emissions from potential environmental superior technologies to evaluate the effectiveness of each technology.

Symmetric Test for Second Differencing in Univariate Time Series

A test for the null hypothesis that a time series has characteristic equations with two unit roots is presented. The test, based on a standard regression computation, is shown to have good power properties when compared to previously existing tests.

Stress Relaxation of Tufted Carpets and Carpet Components: Analysis of the Tufted Carpet Structure

Dimensional stability of tufted carpets has been a continuing problem in the carpet industry for years. When a tufted carpet is installed by the stretch-in method, it experiences stress relaxation over time which can cause the carpet to buckle, wrinkle and become loose with the only option being a costly re-stretching of the carpet. Analysis of the various components of the tufted carpet composite structure was performed to identify the role each component plays in the phenomenon of stress relaxation. A biaxial loading system was used to test various samples of the primary backing alone, primary backing after tufting (with tufts), secondary backing alone, and the finished carpet after attaching the backings with various binder weights per area. The four variables under consideration included primary and secondary backing constructions, tufting density, and latex weight. A rheological model that includes representations of each component in the carpet structure was developed and will be presented in a following paper.