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R. Werner Solar Terrestrial Influences Institute - BAS Time Series Analysis by means of inference statistical methods

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Inference statistic analysis of the time series Now: measures about the significance extrapolated trends causal relations between two variables Cross-section analysis: Y is a realization of a stochastic process, for example the errors must have a determined probability distribution Time series analysis: Prognosis for y t+1, the influences of exogenous parameters can be investigated on this basis

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A model that describes probability structures is called stochastic process. The model includes assumptions for the mechanisms generating the observed time series. A general assumption is the stationarity: 4a) Autocovariances with a lag greater than k are assumed zero → moving average models 4b) Autocovariances of a higher order can be calculated by variances of a lower order → autoregressive models weakly stationarity

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Wavelet transformation, MMNR*100 hab data

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Autoregressive (AR) models of order p a t error term: white noise

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AR(1) process

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Theoretical autocorrelation functions (ACF) ACF for AR(1)

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Yule-Walker equations AR(1) AR(2) AR(p)

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2 0 2 φ1φ1 1 0 φ2φ2 AR(2) Conditions of stationarity: In the area under the circle the AR(2) model describes a quasi-cycle process

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z(1)=1 z(2)=2 time

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Step wise calculation of the coefficients from the Yule- Walker equations k=1: k=2: The theoretical PACF of an AR(p) process has values different from zero, only for k=1,2,…,p ! Model identification tool partial autocorrelation function (PACF), as known from the cross-section statistics

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φ 1 =1.7 φ 2 =-0.95 Theoretical autocorrelation functions (ACF) and partial autocorrelation function for an AR(2) process φ 1 =1.7 φ 2 =-0.95

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Yule 1927

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Residues Parameter estimation YuleIn this work φ1φ φ2φ c00

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Distribution of the residues Autocorrelation function of the residues

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Moving-Average (MA) models AR models describe processes as a function of past z values, however as was shown for the AR(2) process z=1.7z t z t-2 the process is forced by the noise a t. (with a theoretical infinite influence of the shocks). Now the idea is: as for the AR-process, to minimize the process parameters of finite series of a t with time lags

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Autocorrelation for a MA(1) process for a MA(2) process

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PACF? θ1θ1 1 0 θ2θ2 МА(2) Invertibility condition: In the area under the circle the MA(2) model describes a quasi-cycle process Empiric ACF is a tool to identification of the MA order

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Invertibility condition: For a MA(1) process we have

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The MA(1) process can be presented by an AR( ) process In general MA(q) process can be presented by an AR( ) process and an AR(p) process can be presented by a MAR( ) process Box-Jenkins Principle: models with minimum number of parameters have to be used

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Other models: ARMA: mixed model of AR and MA ARIMA: autoregressive integrating moving-average model it uses differences of the time series SARIMA: seasonal ARIMA model with constant seasonal figure VARMA : vector ARMA

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Forecast AR(1)

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MA(1) It can be shown that The MA models are not useful for prognosis

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Forecast of the SSN by an AR(9) model

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Dynamical regression Ordinary linear regression: X i may be transformed α and β can be optimally estimated by Ordinary Least Squares (OLS) using the assumptions: Y i is normal distributed, for X i it is not necessary to be stochastically distributed (for ex. can be fixed) 1.E(ε i )=0 2.ε i is not autocorrelated Cov(ε i, ε j )=0 3.ε i is normally distributed 4.Equilibrium conditions

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For time series can be formally written ( i →t ): The assumption of equilibrium is not necessary However: In time series the error term is often autocorrelated -The estimations are not efficient (they have not the minimal variance) - Autocorrelations of X i can be transferred to ε, autocorrelations of ε produce deviations of σ ε from the true value, besides this implicates a not true value of σ β γ autocorr. of the residues λ autocorr. of the predictors

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Simple lag model (models, dynamical by X) Distributed lag model the influence is distributed over k lags for example k=2 The statistical interpretation of β do not make sense

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Therefore more restrictions are needed where the influence decreases exponentially with k. Then the model has only three parameters: α, β 0,δ For the model

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where Using OLS, δ and β 0 can be estimated, and after this β k How do determine the parameters? Koyck transformation

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Similar models Adaptive expectation model Partial adjustment model Models with two or more input variables

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Model with autocorrelative error term We remember, that ε t in lin. regr. has to be N (0,σ). Here ε t is an AR(1) process. Estimation of the regr. coeff. by Cochrane/Orcutt method 1. By OLS estimation of α and β and calculation of the residues e t and estimation of the autocorrelation coeff.

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new regr. equation where Note: to test if ε t is autocorrelated, the Durbin-Watson test can be applied

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e = *t

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Autocorrelation function of the detrended residues

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Partial Autocorrelation function of the detrended residues

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I want to acknowledge to the Ministery of Education and Science to support this work under the contract DVU01/0120 Acknowledgement

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