Forecast Encompassing Tests and Probability Forecasts

We consider tests of forecast encompassing for probability forecasts, for both quadratic and logarithmic scoring rules. We propose test statistics for the null of forecast encompassing, present the limiting distributions of the test statistics, and investigate the impact of estimating the forecasting models’ parameters on these distributions. The small-sample performance of the various statistics is investigated, both in terms of small numbers of forecasts and model estimation sample sizes. Two empirical applications show the usefulness of the tests for the evaluation of recession probability forecasts from logit models with different leading indicators as explanatory variables, and for evaluating survey-based probability forecasts. Probability forecasts ; encompassing tests ; recession probabilities

A laboratory simulation of high altitude photography on three aero emulsions at scales from 1/100,000 to 1/800,000

Author Misnumbered pages, but all are present. Labled 55,57,55,58. Thesis (M.A.)--Boston UniversityThe employment of miniature camera systems in high altitude photography has not been considered practical until very recently. In the past, the quality of lenses and films has been such that long focal length lenses and consequently large formats were necessary te record the desired ground detail. With recent advances in high speed optics and emulsion making, the potentialities of miniature camera systems have greatly increased. It is now possible to construct systems of this type having resolutian capabilities four or five times as great as conventional large format systems. The purpose of this work was to simulate certain aspects of high altitude photography, in order to study: 1) the potential of a 35 mm laboratory camera, 2) the photographic quality and amount of information that could be obtained on three different aero emulsions, 3) the effect of atmospheric haze on 2, and 4) to determine whether more information could be extracted by viewing a projected image of a negative or a print of that negative. An appraisal of the factors influencing the picture quality of aerial photographs was made. Such problems as aircraft motions, camera design and installation, atmospheric effects, and the brightness and contrast of scene detail was investigated. It was concluded that with proper installation and design of the camera system that three main factors influence the quality of high altitude photography of a given target scene. These factors are the quality of the camera lens, the ability of the film to record fine detail, and the amount of atmospheric haze. For this study a high acuity 35 mm laboratory camera was constructed. The lens used with this camera system was a 50 mm f/2.8 Schneider "Xenotar". To provide an extremely fine and wide range of focus this lens was adapted to an interferometer table on which a Leica Focaslide assembly bad been mounted. Leiea If and Leica IIIc camera bodies were used with the system to provide a wide range of shutter speeds. The lens aperture was held constant at f/5.6 throughout the experiment, an aperture at which the lens was essentially diffraction limited. Three Kodak films were used in this study; Micro-File, and two aerial emulsions, SO-1213 and Plus-X Aerecon. MicroFile, a non-aerial film, was included in order to show what could be done with a high resolution film, although in practice its speed is so low as to require extremely fast optical systems. SO-1213, a new experimental film, is one of the finest grained emulsions yet developed for aerial use. Plus-X Aerecon, just recently put into production, has a speed comparable ta Super-XX, but with much better image quality. Atmospheric haze was simulated by placing a diffusing disc inside the lens just behind the iris diaphragm. A circular hole, whose size corresponded with the diaphragm aperture at f/5.6, was bored out of the center of this disc. By opening the diaphragm aperture beyond f/5.6, the diffusing disc added a uniform "layer" of scattered light to the image, thus effectively simulating atmospheric haze. Three aperture settings were used in this work, f/5.6, f/4, and f/2.8. The following haze factors were obtained. at these apertures: 0, 0.14, and 0.36. Haze factor is defined here as the ratio of the haze brightness to the scene high-light brightness. The target scene consisted of four high quality positive transparencies of Boston, Mass. at a scale of 1/20,000. A standard Air Force resolution target was inserted in a corner of one transparency. The target was illuminated by a light panel which had been specially constructed to provide a very even source of diffuse daylight illumination. The brightness range of the illuminated aerial scene was approximately 40:1. Photos were taken at distances such that the resultant scales on the 35 mm negatives were 1/100,000, 1/200,000, 1/400,000, and 1/800,000. The simulated altitudes were 5, 10, 20, and 40 miles, respectively. Thirty-six negatives were obtained using the above system, one for each film, at each scale, and at each haze setting. Care was taken to insure optimum focus. Using a high quality enlarging system, prints of each negative were made. All prints were enlarged to the same scale. The print enlargements for negatives at the fourscales, 1/100,000, 1/200,000, 1/400,000, and 1/800,000, was 3X, 6X, 12X, and 24X, respectively. Three methods were used in evaluating the results; 1) a microscopic examination of the negatives for resolution, 2) a microscopic examination of the prints for recognition and detection of certain scene detail, and 3) a qualitative examination of a greatly enlarged projected image of the negatives. The average resolution in lines per mm for the three films tested using haze factors of 0, 0.14, and 0.36, respectively, was found to be: For Micro-File: 174, 158, and 148; For SO-1213: 107, 93, and 80; and for Plus-X Aerecon: 57, 51, and 45. Microscopic investigation of the prints of the three films at the various scales for detection and recognition of scene detail revealed the following: At a scale of 1/100,000 with Micro-File and SO-1213 objects the size of automobiles could be detected. Aircraft and railroad cars could be easily recognized. The only difference between the two was that detail in the SO-1213 prints was less sharp. At this scale with Plus-X Aerecon automobiles were just barely visible, recognition of aircraft was difficult, and the individual cars of trains could not be seen. At 1/200,000 automobiles were barely detectable on the MicroFile prints. Trains and aircraft were still visible, but classification of aircraft type was questionable. Cars were no longer detectable on the SO-1213 prints and aircraft was barely detectable. With Plus-X Aereeon neither cars nor trains were discernible. Aircraft was just barely detectable. Even small streets could not be seen. At 1/400,000 with Micro-File large aircraft could just be detected and only relatively large buildings could be recognized. Street patterns were still plainly visible. On the SO-1213 prints aircraft and the patterns of small streets were lest. Ships and large buildings could still be seen. At this scale (enlargement of 12 diameters) the prints of Plus-X Aerecon appeared very grainy. Only relatively large structures such as large streets and buildings could be recognized. At 1/800,000 large buildings, bridges, ships, street patterns, wharfs and docks were still distinguishable on the Micro-File prints. With SO-1213 only large buildings and the rough outline of harbor facilities could be recognized. Street patterns were only recognizable in high contrast regions of the scene. The Plus-X Aerecon prints at this scale revealed only very large roadways and buildings. Outlines of harbor facilities were very rough. Under the conditions of this study, relatively small loss of picture quality was caused by introduction of haze. On the Micro-File prints virtually no loss in quality was noticed even with a .36 haze factor. On the SO-1213 prints a slight loss could be detected at .36 haze at the smaller scales in the low contrast regions of the scene. The prints most affected by haze were those of Plus-X Aerecon, the film having the lowest contrast of the three tested, and this was only noticeable at 36% haze. It must be concluded that the majority of detail sizes in the negatives were such that the compression of brightness range by haze did not reduce the detail contrasts below the visual threshold. The high photographic contrasts of these emulsions is therefore valuable. Investigation of projected images of the negatives with a projector of moderate quality revealed that the same amount of information could be obtained as on the prints. However, this method presented the problem of having to view the image from an angle plus the problem of having to interpret a negative image. Results obtained using this method were purely qualitative and its importance is only one of interest. The final results indicate that high acuity miniature camera systems, in conjunction with high resolution emulsiens, offer information gathering capacity comparable to larger, heavier cameras using standard relatively coarse-grained emulsions

A powerful test for linearity when the order of integration is unknown [Revised to become No. 07/06 above]

In this paper we propose a test of the null hypothesis of time series linearity against a nonlinear alternative, when uncertainty exists as to whether or not the series contains a unit root. We provide a test statistic that has the same limiting null critical values regardless of whether the series under consideration is generated from a linear I(0) or linear I(1) process, and is consistent against nonlinearity of either form, being asymptotically equivalent to the efficient test in each case. Finite sample simulations show that the new procedure has good size control and offers substantial power gains over the recently proposed robust linearity test of Harvey and Leybourne (2007).Nonlinearity testing; Wald tests; unit root tests; stationarity tests

Testing for a unit root in the presence of a possible break in trend

In this paper we consider the issue of testing a time series for a unit root in the possible presence of a break in a linear deterministic trend at some unknown point in the series. We propose a break fraction estimator which, in the presence of a break in trend, is consistent for the true break fraction at rate Op(T^-1) when there is either a unit root or near-unit root in the stochastic component of the series. In contrast to other estimators available in the literature, when there is no break in trend, our proposed break fraction estimator converges to zero at rate Op(T^-1/2). Used in conjunction with a quasi difference (QD) detrended unit root test that incorporates a trend break regressor in the deterministic component, we show that these rates of convergence ensure that known break fraction null critical values are applicable asymptotically. Unlike available procedures in the literature this holds even if there is no break in trend (the true break fraction is zero), in which case the trend break regressor is dropped from the deterministic component and standard QD detrended unit root test critical values then apply. We also propose a second testing procedure which makes use of a formal pre-test for a trend break in the series, including a trend break regressor only where the pre-test rejects the null of no break. Both procedures ensure that the correctly sized (near-) efficient unit root test that allows (does not allow) for a break in trend is applied in the limit when a trend break does (does not) occur.Unit root test; quasi difference de-trending; trend break; pre-test; asymptotic power

Simple, Robust and Powerful Tests of the Breaking Trend Hypothesis*

In this paper we develop a simple procedure which delivers tests for the presence of a broken trend in a univariate time series which do not require knowledge of the form of serial correlation in the data and are robust as to whether the shocks are generated by an I(0) or an I(1) process. Two trend break models are considered: the first holds the level fixed while allowing the trend to break, while the latter allows for a simultaneous break in level and trend. For the known break date case our proposed tests are formed as a weighted average of the optimal tests appropriate for I(0) and I(1) shocks. The weighted statistics are shown to have standard normal limiting null distributions and to attain the Gaussian asymptotic local power envelope, in each case regardless of whether the shocks are I(0) or I(1). In the unknown break date case we adopt the method of Andrews (1993) and take a weighted average of the statistics formed as the supremum over all possible break dates, subject to a trimming parameter, in both the I(0) and I(1) environments. Monte Carlo evidence suggests that our tests are in most cases more powerful, often substantially so, than the robust broken trend tests of Sayginsoy and Vogelsang (2004). An empirical application highlights the practical usefulness of our proposed tests.Broken trend, power envelope, unit root, stationarity tests

Robust methods for detecting multiple level breaks in autocorrelated time series

In this paper we propose tests for the null hypothesis that a time series process displays a constant level against the alternative that it displays (possibly) multiple changes in level. Our proposed tests are based on functions of appropriately standardized sequences of the differences between sub-sample mean estimates from the series under investigation. The tests we propose differ notably from extant tests for level breaks in the literature in that they are designed to be robust as to whether the process admits an autoregressive unit root (the data are I(1)) or stable autoregressive roots (the data are I(0)). We derive the asymptotic null distributions of our proposed tests, along with representations for their asymptotic local power functions against Pitman drift alternatives under both I(0) and I(1) environments. Associated estimators of the level break fractions are also discussed. We initially outline our procedure through the case of non-trending series, but our analysis is subsequently extended to allow for series which display an underlying linear trend, in addition to possible level breaks. Monte Carlo simulation results are presented which suggest that the proposed tests perform well in small samples, showing good size control under the null, regardless of the order of integration of the data, and displaying very decent power when level breaks occur.Level breaks; unit root; moving means; long run variance estimation; robust tests; breakpoint estimation

The impact of the initial condition on robust tests for a linear trend

This paper examines the behaviour of some recently proposed robust (to the order of integration of the data) tests for the presence of a deterministic linear trend in a univariate times series in situations where the magnitude of the initial condition of the series is non-negligible. We demonstrate that the asymptotic size and/or local power properties of these tests are extremely sensitive to the initial condition. Straightforward modifications to the trend tests are suggested, based around the use of trimmed data, which are demonstrated to greatly reduce this sensitivity.Trend tests; initial condition; asymptotic local power

Seasonal unit root tests and the role of initial conditions

In the context of regression-based (quarterly) seasonal unit root tests, we examine the impact of initial conditions (one for each quarter) of the process on test power. We investigate the behaviour of the OLS detrended HEGY seasonal unit root tests of Hylleberg et al. (1990) and the corresponding quasi-differenced (QD) detrended tests of Rodrigues and Taylor (2007), when the initial conditions are not asymptotically negligible. We show that the asymptotic local power of a test at a given frequency depends on the value of particular linear (frequency-specific) combinations of the initial conditions. Consistent with previous findings in the non-seasonal case (see, inter alia, Harvey et al., 2008, Elliott and Muller, 2006), the QD detrended test at a given spectral frequency dominates on power for relatively small values of this combination, while the OLS detrended test dominates for larger values. Since, in practice, the seasonal initial conditions are not observed, in order to maintain good power across both small and large initial conditions, we extend the idea of Harvey et al. (2008) to the seasonal case, forming tests based on a union of rejections decision rule; rejecting the unit root null at a given frequency (or group of frequencies) if either of the relevant QD and OLS detrended HEGY tests rejects. This procedure is shown to perform well in practice, simultaneously exploiting the superior power of the QD (OLS) detrended HEGY test for small (large) combinations of the initial conditions. Moreover, our procedure is particularly adept in the seasonal context since, by design, it exploits the power advantage of the QD (OLS) detrended HEGY tests at a particular frequency when the relevant initial condition is small (large) without imposing that same method of detrending on tests at other frequencies.HEGY seasonal unit root tests; initial conditions; asymptotic local power; union of rejections decision rule

Testing for unit roots and the impact of quadratic trends, with an application to relative primary commodity prices

In practice a degree of uncertainty will always exist concerning what specification to adopt for the deterministic trend function when running unit root tests. While most macroeconomic time series appear to display an underlying trend, it is often far from clear whether this component is best modelled as a simple linear trend (so that long-run growth rates are constant) or by a more complicated non-linear trend function which may, for instance, allow the deterministic trend component to evolve gradually over time. In this paper we consider the effects on unit root testing of allowing for a local quadratic trend, a simple yet very flexible example of the latter. Where a local quadratic trend is present but not modelled we show that the quasi-differenced detrended Dickey-Fuller-type test of Elliott et al. (1996) has both size and power which tend to zero asymptotically. An extension of the Elliott et al. (1996) approach to allow for a quadratic trend resolves this problem but is shown to result in large power losses relative to the standard detrended test when no quadratic trend is present. We consequently propose a simple and practical approach to dealing with this form of uncertainty based on a union of rejections-based decision rule whereby the unit root null is rejected whenever either of the detrended or quadratic detrended unit root tests rejects. A modification of this basic strategy is also suggested which further improves on the properties of the procedure. An application to relative primary commodity price data highlights the empirical relevance of the methods outlined in this paper. A by-product of our analysis is the development of a test for the presence of a quadratic trend which is robust to whether or not the data admit a unit root.Unit root test; trend uncertainty; quadratic trends; asymptotic power; union of rejections decision rule

Unit root testing under a local break in trend

It is well known that it is vital to account for trend breaks when testing for a unit root. In practice, uncertainty exists over whether or not a trend break is present and, if it is, where it is located. Harris et al. (2009) and Carrion-i-Silvestre et al. (2009) propose procedures which account for both of these forms of uncertainty. Each uses what amounts to a pre-test for a trend break, accounting for a trend break (the associated break fraction estimated from the data) in the unit root procedure only where the pre-test signals a break. Assuming the break magnitude is fixed (independent of sample size) these authors show that their methods achieve near asymptotically ecient unit root inference in both trend break and no trend break environments. These asymptotic results are, however, somewhat at odds with the finite sample simulations reported in both papers. These show the presence of pronounced "valleys" in the finite sample power functions (when mapped as functions of the break magnitude) of the tests such that power is initially high for very small breaks, then decreases as the break magnitude increases, before increasing again. Here we show that treating the break magnitude as local to zero (in a Pitman drift sense) allows the asymptotic analysis to very closely approximate this finite sample effect, thereby providing useful analytical insights into the observed phenomenon. In response to this problem we propose practical solutions, based either on the use of a with break unit root test but with adaptive critical values, or on a union of rejections principle taken across with break and without break unit root tests. The former is shown to eliminate power valleys but at the expense of power when no break is present, while the latter considerably mitigates the valleys while not losing all the power gains available when no break exists.Unit root test; local trend break; asymptotic local power; union of rejections; adaptive critical values