Sometimes time-series data derive from an underlying system which can exist in multiple distinct states. Hidden Markov Models (HMMs) are a popular approach in this situation. In a particular state, the data fluctuate with variance characteristic of that state. An N x N transition probability matrix describes the additional dynamics associated with flipping between the N states. The value of N and nature of the unobserved states may or may not be obvious depending on the problem.
Yields on long-term British government debt for the period 1727-2013 are available from the UK debt management office (annual mean yields on perpetual bonds or “consols”). Bond yields change in response to market perceptions of risk. Historically, volatile periods tend to be associated with large attritionalÂ wars, such as Seven Years War (1754-1763),Â Napoleonic wars (1803-1815), World War One (1914-1918) etc.
It is tempting to fit a HMM to the (differenced) gilt yield data. N is found by minimising the Bayesian Information Criterion (BIC).Â It turns out that N=3 is optimal for the differenced time-series (code and graph below).
The modern era of high inflation requires a third underlying state “I” which lasted from 1968-1999.
The R code below returns N=3.
file.in <- “http://joewheatley.net/wp-content/uploads/2014/09/consol.csv”
gilt <- read.csv(file.in)
returns <- diff(gilt$yield)
bics <- sapply(2:10, function(nn) HMMFit(returns, dis=”NORMAL”, nStates=nn)$BIC)
Nearly 40% of the land surface of the earth, orÂ 5 billion hectares, is used for agriculture. Crops are grown on 1.5 billion hectares and there are 3.5 billion hectares ofÂ pasture and meadow.
According to FAO land use statistics, global agricultural land area peaked in 1998. This apparent fall is due to a decrease in area used as permanent pasture and meadow, while the area classed as “arable land and permanent crops” has been relatively static.
FAO also collate annual harvest area data for 178 crop types. Total harvest area of all crops may give a truer picture of the trend in demand for cropland.Â It turns out that global harvest area has continued to increase strongly (8.1 MHa/y i.e. about the area of Ireland per year), even though the nominal cropland area has grown much more slowly (1.4M Ha/y)Â over the past two decades. The increase is due primarilyÂ to oil crops such as soybean (often used as animal feed).
The two measures of cropland area give rise to alternative “global agricultural area” curves shown below. The upper curve suggests that “peak farmland” occurred in 1998, while the lower curve shows no peak. The curves cannot cross becauseÂ harvested area is always lower than the cropland area.
The recent convergence trend between nominal cropland and actual harvest areas indicates increased pressure on croplands. It raises doubts whether peak farmland has really been reached yet.