Applying modern portfolio theory to the analysis of terrorism: computing the set of attack method combinations from which the rational terrorist group will choose in order to maximise injuries and fatalities

In this paper, terrorism is analysed using the tools of modern portfolio theory. This approach permits the analysis of the returns that a terrorist group can expect from their activities as well as the risk they face. The analysis sheds new light on the nature of the terrorist group’s (attack method) choice set and the efficiency properties of that set. If terrorist groups are, on average, more risk averse, the economist can expect the terrorist group to exhibit a bias towards bombing and armed attack. In addition, even the riskiest (from the terrorist group’s point of view) combinations of attack methods have maximum expected returns of less than 70 injuries and fatalities per attack per year

Unit Root Model Selection

Some limit properties for information based model selection criteria are given in the context of unit root evaluation and various assumptions about initial conditions. Allowing for a nonparametric short memory component, standard information criteria are shown to be weakly consistent for a unit root provided the penalty coefficient C_n -> infinity and C_n/n -> 0 as n -> infinity. Strong consistency holds when C_n/(loglog n)^3 -> infinity under conventional assumptions on initial conditions and under a slightly stronger condition when initial conditions are infinitely distant in the unit root model. The limit distribution of the AIC criterion is obtained.AIC, Consistency, Model selection, Nonparametric, Unit root

Challenges of Trending Time Series Econometrics

We discuss some challenges presented by trending data in time series econometrics. To the empirical economist there is little guidance from theory about the source of trend behavior and even less guidance about practical formulations. Moreover, recent proximity theorems reveal that trends are more elusive to model empirically than stationary processes, with the upshot that optimal forecasts are also harder to estimate when the data involve trends. These limitations are implicitly acknowledged in much practical modeling and forecasting work, where adaptive methods are often used to help keep models on track as trends evolve. The paper discusses these broader issues and limitations of econometrics and o.ers some thoughts on new practical possibilities for data analysis in the absence of good theory models for trends. In particular, a new concept of coordinate cointegration is introduced and some new econometric methodology is suggested for analyzing trends and comovement and for producing forecasts in a general way that is agnostic about the specific nature of the trend process. Some simulation exercises are conducted and some long historical series on prices and yields on long securities are used to illustrate the methods.Coordinate instrumental variables, coordinate reduced rank regression, coordinate trend functions, limitations of econometrics, nonstationarity, trend

Bootstrapping Spurious Regression

The bootstrap is shown to be inconsistent in spurious regression. The failure of the bootstrap is spectacular in that the bootstrap effectively turns a spurious regression into a cointegrating regression. In particular, the serial correlation coefficient of the residuals in the bootstrap regression does not converge to unity, so the bootstrap is not even first order consistent. The block bootstrap serial correlation coefficient does converge to unity and is therefore first order consistent, but has a slower rate of convergence and a different limit distribution from that of the sample data serial correlation coefficient. The analysis covers spurious regressions involving both deterministic trends and stochastic trends. The results reinforce earlier warnings about routine use of the bootstrap with dependent data.Asymptotic theory, Bootstrap, Brownian motion, Cointegration, LK representation, Nonstationarity, Residual diagnostics, Unit root

Folklore Theorems, Implicit Maps and New Unit Root Limit Theory

The delta method and continuous mapping theorem are among the most extensively used tools in asymptotic derivations in econometrics. Extensions of these methods are provided for sequences of functions, which are commonly encountered in applications, and where the usual methods sometimes fail. Important examples of failure arise in the use of simulation based estimation methods such as indirect inference. The paper explores the application of these methods to the indirect inference estimator (IIE) in first order autoregressive estimation. The IIE uses a binding function that is sample size dependent. Its limit theory relies on a sequence-based delta method in the stationary case and a sequence-based implicit continuous mapping theorem in unit root and local to unity cases. The new limit theory shows that the IIE achieves much more than bias correction. It changes the limit theory of the maximum likelihood estimator (MLE) when the autoregressive coefficient is in the locality of unity, reducing the bias and the variance of the MLE without affecting the limit theory of the MLE in the stationary case. Thus, in spite of the fact that the IIE is a continuously differentiable function of the MLE, the limit distribution of the IIE is not simply a scale multiple of the MLE but depends implicitly on the full binding function mapping. The unit root case therefore represents an important example of the failure of the delta method and shows the need for an implicit mapping extension of the continuous mapping theorem.Binding function, Delta method, Exact bias, Implicit continuous maps, Indirect inference, Maximum likelihood

Econometric Analysis of Fisher's Equation

Fisher's equation for the determination of the real rate of interest is studied from a fresh econometric perspective. Some new methods of data description for nonstationary time series are introduced. The methods provide a nonparametric mechanism for modelling the spatial densities of a time series that displays random wandering characteristics, like interest rates and inflation. Hazard rate functionals are also constructed, an asymptotic theory is given and the techniques are illustrated in some empirical applications to real interest rates for the US. The paper ends by calculating Gaussian semiparametric estimates of long range dependence in US real interest rates, using a new asymptotic theory that covers the nonstationary case. The empirical results indicate that the real rate of interest in the US is (fractionally) nonstationary over 1934-1997 and over the more recent subperiods 1961-1985 and 1961-1997. Unit root nonstationarity and short memory stationarity are both strongly rejected for all these periods.Fractional integration; hazard rate; long range dependence; real rate of interest; semiparametric estimation; sojourn time; spatial density

Automated Discovery in Econometrics

Our subject is the notion of automated discovery in econometrics. Advances in computer power, electronic communication, and data collection processes have all changed the way econometrics is conducted. These advances have helped to elevate the status of empirical research within the economics profession in recent years and they now open up new possibilities for empirical econometric practice. Of particular significance is the ability to build econometric models in an automated way according to an algorithm of decision rules that allow for (what we call here) heteroskedastic and autocorrelation robust (HAR) inference. Computerized search algorithms may be implemented to seek out suitable models, thousands of regressions and model evaluations may be performed in seconds, statistical inference may be automated according to the properties of the data, and policy decisions can be made and adjusted in real time with the arrival of new data. We discuss some aspects and implications of these exciting, emergent trends in econometrics.Automation, discovery, HAC estimation, HAR inference, model building, online econometrics, policy analysis, prediction, trends