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Lecture 12

# Lecture 12 - Introduction to Econometrics Lecture 12...

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Introduction to Econometrics 12-1 Technical Notes Introduction to Econometrics Lecture 12: Stationary stochastic processes Processes and realisations With random sampling the key concepts are the population and the sample. The population can be real, (as when a Census of Population uses a 10% sample) or hypothetical. A hypothetical population would be generated by a specific probability distribution function: if we say a sample is drawn from a parent population which is N( μ , σ 2 ) then the population could be represented by a normal curve. In this case we cannot measure variables for the whole (infinite) population: but we can still make inferences about the underlying distribution and its parameters from a finite sample. The key feature of random sampling is that each time a particular observation is ‘drawn’ for inclusion in the sample the probability that it is drawn does not depend on earlier drawings Prob(x t =a) f(x s ), for s t This is not true for time series, where the probability of observing a particular value may depend on the past history of the series. For example, if the series depends on its immediate past value Prob(x t =a) = f(x t-1 ) To reintroduce an appropriate concept of ‘randomness’ we need new concepts of population and sample. Suppose we think about a time series of T observations. Then we can write the whole sequence of observations as {x 1 , x 2 , ...x T } Now we can consider this sequence as a single sample point, and the set of all possible sequences of length T which could (hypothetically) have been observed as being the (hypothetical) underlying population. With random sampling the population is described in terms of an underlying probability distribution, and a single sample point is called an observation. With time series sampling the population is described in terms of the underlying stochastic process which generates the sequences, and a single sample point (a sequence) is called a realisation. The general aim of time series analysis is to make inferences about an underlying stochastic process from one (or, rarely, more) realisations of the process Stationarity and ergodicity Usually the data available in economics consists of a single realisation of a time series, so there is only one sample point. If we had N independent realisations we could define what are called ‘ensemble’ estimates of the means and variances for each observation by averaging over realisations using the usual formulae μ t = N -1 Σ i x it σ 2 t = (N-1) -1 Σ i (x it - μ t ) 2 For a given observation t, x it and x jt are independent and so these averages have the conventional desirable properties. Suppose we have only one realisation, but this realisation has the property that its characteristics do not depend on the observation date: specifically that

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