This paper provides a discussion of the developments in econometric modelling that are designed to deal with the problem of spurious Granger causality relationships that can arise from temporal aggregation. We outline the distortional effects of using discrete time models that explicitly depend on the unit of time and outline a remedy of constructing time-invariant discrete time models via a structural continuous time model. In an application to testing for money-income causality, we demonstrate the importance of incorporating exact temporal aggregation restrictions on the discrete time data. We do this by conducting causality tests in discrete time models that: (a) impose the temporal aggregation restrictions exactly, (b) impose the temporal aggregation restrictions approximately, and (c) do not impose these restrictions at all. r 2005 Elsevier B.V. All rights reserved.JEL classification: C32

This paper applies the mildly explosive/multiple bubbles testing methodology developed by Phillips, Shi and Yu (2015a, International Economic Review, forthcoming) to examine the recent time series behaviour of the main six London Metal Exchange (LME) non-ferrous metals prices. We detect periods of mild explosivity in the cash and three-month futures price series in each of copper, nickel, lead, zinc and tin, but not in aluminium. We argue that convenience yield, though the formal counterpart to dividend yield in commodity markets, is not a useful basis on which to assess whether observed explosivity is indicative of bubbles (namely, departures of prices from their fundamental values). We construct other measures that provide evidence that suggests the observed explosivity in the non-ferrous metals market can be associated with tight physical markets.

We analyze the price behaviour of the main precious metals -gold, silver, platinum and palladium -before, during and in the aftermath of the 2007-08 financial crisis. Using the mildly explosive/multiple bubbles technology developed by Phillips, Shi and Yu (2015, International Economic Review 56(4), 1043-1133), we find significant, short periods of mildly explosive behaviour in the spot and futures prices of all four metals. Fewer periods are detected using exchange-rate adjusted prices, and almost none when deflated prices are used.We assess whether these findings are indicative of bubble behaviour. Convenience yield is shown to have little efficacy in this regard, while other fundamentals proxy variables and position data offer only very limited evidence against prices having been anything other than fundamentals-driven. Possible exceptions are in gold in the run-up to the highpoint of the financial crisis, and in silver and palladium around the launch of specific financial products. Some froth, however, is reported and discussed for each metal.JEL classification: C22; D84; G13

This paper provides an analysis of oil prices during and in the aftermath of the Global Financial Crisis, concentrating on the 2007-08 price spike and the 2014-16 price decline. The mildly explosive/multiple bubbles testing strategy by Phillips, Shi and Yu (2015, International Economic Review 56(4), 1043-1133) is used to test for price departures from an underlying stochastic trend and to assess whether any such departures can be explained by fundamentals or other proxy variables. The test dates two significant time periods in both Brent and WTI nominal and real front-month futures prices: a mildly explosive episode during the 2007-08 spike, prior to the peak of the Global Financial Crisis; and a significantly shorter, negative such episode during the 2014-16 price decline, whose commencement is dated around a key OPEC meeting in November 2014. Evidence using other commodity prices points to explanatory factors beyond commodity markets. A global economic activity proxy is found to be decisive in the episode in mid-2008; excess speculation is not. U.S. shale oil production, though contributing to the post-June 2014 price decline, is not seen to have been decisive. Against some recent work tying the CBOE Volatility Index (VIX) to oil futures prices, we find no evidence that the VIX decisively affected oil price levels during the sample period. The results are compared and contrasted with those obtained by Baumeister and Kilian (2016, Journal of the Association of Environmental and Resource Economists 3, 131-158) via a forecasting approach based on a structural vector autoregressive model without financial variables. Taken altogether, the results herein provide new evidence based on formal statistical testing that help resolve a number of recent controversies in the oil price literature.

The exact discrete model satisfied by equispaced data generated by a linear stochastic differential equations system is derived by a method that does not imply restrictions on observed discrete data per se. The method involves integrating the solution of the continuous time model in state space form and a nonstandard change in the order of three types of integration, facilitating the representation of the exact discrete model as an asymptotically time-invariant vector autoregressive moving average model. The method applying to the state space form is general and is illustrated using the prototypical higher order model for mixed stock and flow data discussed by Bergstrom (1986, Econometric Theory 2, 350–373).