Business cycle asymmetries: characterisation and testing based on Markov-switching autoregressions

Tests for business cycle asymmetries are developed for Markov-switching autoregressive models. The tests of deepness, steepness, and sharpness are Wald statistics, which have standard asymptotics. For the standard two-regime model of expansions and contractions, deepness is shown to imply sharpness (and vice versa), whereas the process is always nonsteep. Two and three-state models of U.S. GNP growth are used to illustrate the approach, along with models of U.S. investment and consumption growth. The robustness of the tests to model misspecification, and the effects of regime-dependent heteroscedasticity, are investigated.

The heliospheric Hale cycle over the last 300 years and its implications for a “lost” late 18th century solar cycle

A Hale cycle, one complete magnetic cycle of the Sun, spans two complete Schwabe cycles (also referred to as sunspot and, more generally, solar cycles). The approximately 22-year Hale cycle is seen in magnetic polarities of both sunspots and polar fields, as well as in the intensity of galactic cosmic rays reaching Earth, with odd- and even-numbered solar cycles displaying qualitatively different waveforms. Correct numbering of solar cycles also underpins empirical cycle-to-cycle relations which are used as first-order tests of stellar dynamo models. There has been much debate about whether the unusually long solar cycle 4 (SC4), spanning- 1784–1799, was actually two shorter solar cycles combined as a result of poor data coverage in the original Wolf sunspot number record. Indeed, the group sunspot number does show a small increase around 1794–1799 and there is evidence of an increase in the mean latitude of sunspots at this time, suggesting the existence of a cycle ‘‘4b’’. In this study, we use cosmogenic radionuclide data and associated reconstructions of the heliospheric magnetic field (HMF) to show that the Hale cycle has persisted over the last 300 years and that data prior to 1800 are more consistent with cycle 4 being a single long cycle (the ‘‘no SC4b’’ scenario). We also investigate the effect of cycle 4b on the HMF using an open solar flux (OSF) continuity model, in which the OSF source term is related to sunspot number and the OSF loss term is determined by the heliospheric current sheet tilt, assumed to be a simple function of solar cycle phase. The results are surprising; Without SC4b, the HMF shows two distinct peaks in the 1784–1799 interval, while the addition of SC4b removes the secondary peak, as the OSF loss term acts in opposition to the later rise in sunspot number. The timing and magnitude of the main SC4 HMF peak is also significantly changed by the addition of SC4b. These results are compared with the cosmogenic isotope reconstructions of HMF and historical aurora records. These data marginally favour the existence of SC4b (the ‘‘SC4b’’ scenario), though the result is less certain than that based on the persistence of the Hale cycle. Thus while the current uncertainties in the observations preclude any definitive conclusions, the data favour the ‘‘no SC4b’’ scenario. Future improvements to cosmogenic isotope reconstructions of the HMF, through either improved modelling or additional ice cores from well-separated geographic locations, may enable questions of the existence of SC4b and the phase of Hale cycle prior to the Maunder minimum to be settled conclusively.

Tests of sunspot number sequences: 3. Effects of regression procedures on the calibration of historic sunspot data

We use sunspot group observations from the Royal Greenwich Observatory (RGO) to investigate the effects of intercalibrating data from observers with different visual acuities. The tests are made by counting the number of groups RB above a variable cut-off threshold of observed total whole-spot area (uncorrected for foreshortening) to simulate what a lower acuity observer would have seen. The synthesised annual means of RB are then re-scaled to the full observed RGO group number RA using a variety of regression techniques. It is found that a very high correlation between RA and RB (rAB > 0.98) does not prevent large errors in the intercalibration (for example sunspot maximum values can be over 30 % too large even for such levels of rAB). In generating the backbone sunspot number (RBB), Svalgaard and Schatten (2015, this issue) force regression fits to pass through the scatter plot origin which generates unreliable fits (the residuals do not form a normal distribution) and causes sunspot cycle amplitudes to be exaggerated in the intercalibrated data. It is demonstrated that the use of Quantile-Quantile (“Q  Q”) plots to test for a normal distribution is a useful indicator of erroneous and misleading regression fits. Ordinary least squares linear fits, not forced to pass through the origin, are sometimes reliable (although the optimum method used is shown to be different when matching peak and average sunspot group numbers). However, other fits are only reliable if non-linear regression is used. From these results it is entirely possible that the inflation of solar cycle amplitudes in the backbone group sunspot number as one goes back in time, relative to related solar-terrestrial parameters, is entirely caused by the use of inappropriate and non-robust regression techniques to calibrate the sunspot data.

Tests of sunspot number sequences: 1. Using ionosonde data

More than 70 years ago it was recognised that ionospheric F2-layer critical frequencies [foF2] had a strong relationship to sunspot number. Using historic datasets from the Slough and Washington ionosondes, we evaluate the best statistical fits of foF2 to sunspot numbers (at each Universal Time [UT] separately) in order to search for drifts and abrupt changes in the fit residuals over Solar Cycles 17-21. This test is carried out for the original composite of the Wolf/Zürich/International sunspot number [R], the new “backbone” group sunspot number [RBB] and the proposed “corrected sunspot number” [RC]. Polynomial fits are made both with and without allowance for the white-light facular area, which has been reported as being associated with cycle-to-cycle changes in the sunspot number - foF2 relationship. Over the interval studied here, R, RBB, and RC largely differ in their allowance for the “Waldmeier discontinuity” around 1945 (the correction factor for which for R, RBB and RC is, respectively, zero, effectively over 20 %, and explicitly 11.6 %). It is shown that for Solar Cycles 18-21, all three sunspot data sequences perform well, but that the fit residuals are lowest and most uniform for RBB. We here use foF2 for those UTs for which R, RBB, and RC all give correlations exceeding 0.99 for intervals both before and after the Waldmeier discontinuity. The error introduced by the Waldmeier discontinuity causes R to underestimate the fitted values based on the foF2 data for 1932-1945 but RBB overestimates them by almost the same factor, implying that the correction for the Waldmeier discontinuity inherent in RBB is too large by a factor of two. Fit residuals are smallest and most uniform for RC and the ionospheric data support the optimum discontinuity multiplicative correction factor derived from the independent Royal Greenwich Observatory (RGO) sunspot group data for the same interval.