Lecture 12

# Lecture 12 - Time series processes and realisations With...

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1 Time series - processes and realisations With random sampling the key concepts are population and sample The population can be real (Census) or hypothetical (continuous distribution) Sample statistics are (imprecise) estimates of theunderlying population parameters Random sampling implies: the probability that a particular observation is included does not depend on earlier observations Prob(x t =a) f(x s ), for s t For time series this is not appropriate: the probability of observing a particular value depends on the past history of the series Prob(x t =a) = f(x s ) hence if we treated the single-period observation as the primitive event these events would not be independent Instead we develop the concepts of the ‘stochastic process’ (the population) and ‘realisation’ (the sample) Stochastic processes We can regard the sequence of observations on X {x 1 , x 2 , . ..x T } as a single (T-dimensional) sample point The set of all possible sequences of length T is the (hypothetical) underlying population The population (the mechanism generating all possible sequences) is called a stochastic process A single sample point (an individual sequence) is called a realisation Example : M tosses of a fair coin A sequence (realisation) is {H, T, T, T, H. ...... } or {H, H, H, T, T. ...... } The process (population) generates all such sequences

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2 The aim of time series analysis To make valid inferences about the underlying stochastic process from a single realisation If we have N independent realisations (distinct experiments) we can define ‘ensemble’ estimates of the moments by averaging over realisations μ t = N -1 Σ i x it σ 2 t = (N-1) -1 Σ i (x it - μ t ) 2 for each t Since these averages are based on random sampling (repeated identical experiments) they have desirable properties Suppose instead we have only one realisation, but that the moments do not depend on the observation date E[x 1t ] = μ Var[x 1t ] = σ 2 < Cov[x 1s ,x 1t ] = γ t-s all of these are independent of t ( γ t-s is a function of t-s, not of t or s separately) Split this realisation into N sub-realisations of length T/N:
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Lecture 12 - Time series processes and realisations With...

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