Testing for random walk hypothesis with or without measurement error Tin-Yu Jonathan Hui A thesis submitted in partial fulfilment of the requirement for the degree of Master of Science of Imperial College London and Diploma of Imperial College London September 2012 1 ABSTRACT In biological study it is common to take observations over time and thus there is strong need in developing tools to analyse time series data. Variations of dynamics over time are often modelled for estimation and forecasting. The random walk process in particular is one of the most commonly used models and serves as the null hypothesis in many theories. As a result, this paper adopts the Q-statistic and Variance-Ratio (VR) Test to test the random walk hypothesis and assesses their testing power against AR(1) model. It is shown that both the Q-statistic and VR Test are valid tests in testing against AR(1) and VR has a higher testing power. Testing power decreases when ϕ approaches unity as the AR(1) process becomes less distinguishable from random walks. In population genetics allele frequencies fluctuate along generations, known as genetic drift. If there exist external forces such as selection, the drift will then have a trend and drive the allele to extinction or fixation. This motivates the second test for a trend in random walk. Both Likelihood Ratio Test and modified Variance-Ratio can test for trends within random walk models. It is shown that the power of the two tests increases with the trend-to-standard deviation ratio. In general the Likelihood Ratio Test provides a better power in testing for a trend. In a more realistic scenario, measurement error is assumed in the observations and some newer tests are developed. In this paper the √ ̂( ) statistic and 3-point Variance-Ratio are introduced. The overall testing power of √ ̂( ) statistic is higher in both high and low signal-to-noise ratio. 2 CONTENTS Abstract .............................................................................................................................................................. 2 Contents ............................................................................................................................................................. 3 ZERO: Overview .............................................................................................................................................. 5 ONE: Definitions ............................................................................................................................................. 7 1.1 Introduction .................................................................................................................................... 7 1.2 Random walk 1: Independent and identically distributed increments ................... 7 1.3 Random walk 2: Independent increments .......................................................................... 8 1.4 Random walk 3: Uncorrelated increments ......................................................................... 9 1.5 Section summary ........................................................................................................................... 9 TWO: Testing the random walk hypothesis ...................................................................................... 10 2.1 Introduction .................................................................................................................................. 10 2.2 Q-statistic ....................................................................................................................................... 10 2.3 Variance-Ratio Test .................................................................................................................... 11 2.4 Empirical distributions of test statistics under H0 ......................................................... 12 2.5 Power analysis ............................................................................................................................. 14 2.6 Section summary ......................................................................................................................... 16 THREE: Testing for trend in random walk ......................................................................................... 17 3.1 Introduction .................................................................................................................................. 17 3.2 Likelihood Ratio Test ................................................................................................................. 18 3.3 Modified Variance-Ratio Test ................................................................................................. 19 3.4 Empirical distributions of test statistics under H0 ......................................................... 19 3.5 Power analysis ............................................................................................................................. 20 3.6 Section summary ......................................................................................................................... 22 FOUR: Random walk with measurement error and testing ........................................................ 23 4.1 Introduction .................................................................................................................................. 23 4.2 Behaviour of existing tests under random walk with measurement error .......... 24 4.3 Autocorrelation and variance structure ............................................................................. 27 4.4 Signal to noise ratio .................................................................................................................... 29 3 4.5 √ ̂( ) statistic ......................................................................................................................... 30 4.6: 3-point Variance-Ratio Test ................................................................................................... 34 4.7 Empirical percentiles of test statistics under H0 ............................................................. 36 4.8 Power analysis ............................................................................................................................. 37 4.9 Section summary ......................................................................................................................... 38 FIVE: Trend in random walk with measurement error ................................................................ 39 5.1 Introduction .................................................................................................................................. 39 5.2 Likelihood Ratio test .................................................................................................................. 39 5.3 Power analysis ............................................................................................................................. 40 5.4 Section summary ......................................................................................................................... 41 SIX: Discussions and limitations of this study .................................................................................. 42 SEVEN: Conclusion ...................................................................................................................................... 45 Reference......................................................................................................................................................... 46 Acknowledgements ..................................................................................................................................... 48 4 ZERO: OVERVIEW In many biological experiments data is recorded across a time interval as biologists want to study the dynamics of a situation, in particular, the change in dynamics over time. As a result, the modelling and analysis of time series data becomes important to biological study. The random walk serves the null model in many biological hypothesises. In the Neutral Theory of molecular evolution, the change in allele frequency can be by chance alone without any predetermined way and this process is known as genetic drift (Hartl & Clark, 1997). Wright-Fisher model (Fisher, 1930; Wright, 1931) states that the transition probability follows binomial distribution and the proportions can be approximated by diffusion process, which is a continuous random walk process. In morphological evolution, many variations fall on continuous scale, and random walk models are used to model quantitative traits. Neutral Theory of Ecology (Hubbell, 2001) assumes that biodiversity arises at random and the equivalence between individuals. This result in species within a community follows a random walk in composition. In mathematical ecology, random walks can be used to model population dynamics and animals or cells movement (Codling, Plank & Benhamou, 2008). Furthermore, population viability analysis and extinction risk can be assessed under the assumption of diffusion process (Lande & Orzack, 1988). Apart from biological applications, random walk plays an important role in physical sciences and economics, for instance, in econometrics, geometric Brownian motion is used to model share price movements and lead to the development of derivatives pricing model (Black & Scholes 1973). Many alternative models are raised, such as the niche theory in ecology, density-dependence model (Ricker, 1954)