Lecture 13 - Introduction to Econometrics Lecture 13 Linear...

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Introduction to Econometrics 13-1 Technical Notes Introduction to Econometrics Lecture 13: Linear trends and random walks Nonstationary stochastic processes Stationary stochastic processes satisfy the conditions E[x t ] = μ Var[x t ] = σ 2 < Cov[x s ,x t ] = γ t-s where all population parameters are independent of t ( γ t-s is a function of t-s, not of t or s). Hence it is legitimate to treat the (observable) moments calculated over time as estimators of the (unobservable) moments which would be calculated if we could average over realisations. Many economic series obviously do not satisfy these conditions: for example E[GDP 1970 ] > E[GDP 1870 ] where the averaging implied by the expectation E[] is over all (hypothetical) realisations. Such series are nonstationary and cannot be modelled using AR or MA representations. Deterministic nonstationarity The simplest form of nonstationarity arises when the mean of a series is a linear trend (if the series is logarithmic, this corresponds to exponential growth in the level). In this case the series can be written as x t = α + β t + e t where e t is stationary (it could be white noise, or an MA/AR process). In this case x t is nonstationary (as μ t = α + β t is a function of t), but the derived series e t (= x t - α - β t) is stationary. Hence we can analyse the time series properties of x t by detrending it (computing α and β by linear regression). This type of nonstationarity is called deterministic nonstationarity: it can be extended to allow for quadratic or more complex trends. However deterministic nonstationarity is not a satisfactory model for many economic series. Firstly, if any deviations from trend are to persist for more than one period, e t cannot be white noise, and for deviations to match actual experience, the variance of the shock must be quite large. Deterministic trend models also often indicate spurious ‘breaks’ in trend. Deterministic nonstationarity also has unrealistic implications for long-run forecasts of levels or growth rates, which do not accord with our intuitive feelings about the relative uncertainty which should be attached to these. To see why, consider first level forecasts and assume that the parameters α and β are known with certainty. Then at time 0 the uncertainty about any forecast for a future time period s is measured by Var(e s ) = σ 2 , and this is the same however distant the forecast period. If α and β are not known, the uncertainty becomes greater the further the forecast period s is from the mean value observed in the estimation period (this is a
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