Time series analysis - lecture 2 A general forecasting principle Set up probability models for which we can derive analytical expressions for and estimate the unknown parameters in that expression Use estimated autocorrelations to select a suitable model class

Time series analysis - lecture 2 The first order autoregressive model: AR(1) The autocorrelation tails off exponentially

Time series analysis - lecture 2 Datasets simulated by using a random number generator and first order autoregressive models  = 0.2  = 0.9

Time series analysis - lecture 2 The first order autoregressive model - prediction

Time series analysis - lecture 2 Weekly SEK/EUR exchange rate Jan Oct 2007

Time series analysis - lecture 2 Daily SEK/EUR exchange rate Jan Oct 2007

Time series analysis - lecture 2 No. registered cars

Time series analysis - lecture 2 Weekly SEK/EUR exchange rate Jan Oct 2007 AR(1) model Final Estimates of Parameters Type Coef SE Coef T P AR Constant Mean

Time series analysis - lecture 2 Daily SEK/EUR exchange rate Jan Oct 2007 AR(1) model Final Estimates of Parameters Type Coef SE Coef T P AR Constant Mean WARNING * Back forecasts not dying out rapidly

Time series analysis - lecture 2 The general auto-regressive model: AR(p) {Y t } is said to form an AR(p) sequence if where the error terms  t are independent and N(0;  )

Time series analysis - lecture 2 The backshift operator B The backshift operator B is defined by the relation It follows that and we can write the AR(p) model:

Time series analysis - lecture 2 The characteristic equation of an AR(p) sequence The equation is called the characteristic equation of an AR(p) sequence If the roots of this equation are all outside the unit circle, then the AR(p) sequence is stationary It returns repeatedly to zero, and the autocorrelations tail off with increasing time lags

Time series analysis - lecture 2 Stationarity of an AR(p) sequence p=1 Characteristic equation: The root is outside the unit circle if p=2 Characteristic equation: The roots of this equation are all outside the unit circle if:

Time series analysis - lecture 2 The moving average model: MA(q) {Y t } is said to form an MA(q) sequence if where the error terms  t are independent and N(0;  )

Time series analysis - lecture 2 The general auto-regressive-moving-average model ARMA(p,q) {Y t } is said to form an ARMA(p,q) sequence if where the error terms  t are independent and N(0;  )

Time series analysis - lecture 2 Typical auto-correlation functions of ARMA(p,q) sequences AR(p): Auto-correlations tail off gradually with increasing time-lags MA(q): Auto-correlations are zero for time lags greater than q ARMA(p,q): Auto-correlations tail off gradually with time-lags greater than q

Time series analysis - lecture 2 Partial auto-correlation The partial auto-correlation between Y t and Y t+m represents the remaining correlation after Y t+1, …, Y t+m-1 have been fixed tt+m

Time series analysis - lecture 2 Typical partial auto-correlation functions of ARMA(p,q) sequences AR(p): Partial auto-correlations are zero for time lags greater than p MA(q): Partial auto-correlations tail off gradually with increasing time-lags ARMA(p,q): Partial auto-correlations tail off gradually with time-lags greater than p

Time series analysis - lecture 2 Consumer price index and its first order differences

Time series analysis - lecture 2 Consumer price index - first order differences

Time series analysis - lecture 2 Consumer price index – predictions using an AR(1) model fitted to first order differences of the original data

Time series analysis - lecture 2 The first order integrated autoregressive model: ARI(1)