Tham khảo tài liệu 'real estate modelling and forecasting by chris brooks and sotiris tsolacos_8', tài chính - ngân hàng, đầu tư bất động sản phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Time series models 235 invertible it can be expressed as an AR tc . a definition of invertibility is therefore now required. The invertibility condition An MA g model is typically required to have roots of the characteristic equation 0 z 0 greater than one in absolute value. The invertibility condition is mathematically the same as the stationarity condition but is different in the sense that the former refers to MA rather than AR processes. This condition prevents the model from exploding under an AR tc representation so that 0 1 L converges to zero. Box shows the invertibility condition for an MA 2 model. Box The invertibility condition for an MA 2 model In order to examine the shape of the pacf for moving average processes consider the following MA 2 process for yt yt Ut 01Ut 1 2ut -2 0 L ut Provided that this process is invertible this MA 2 can be expressed as an AR yt CiLyt i Ut i 1 yt Ciyt-1 c2yt 2 c3yt 3 ut It is now evident when expressed in this way that for a moving average model there are direct connections between the current value of y and all its previous values. Thus the partial autocorrelation function for an MA i model will decline geometrically rather than dropping off to zero after q lags as is the case for its autocorrelation function. It could therefore be stated that the acf for an AR has the same basic shape as the pacf for an MA and the acf for an MA has the same shape as the pacf for an AR. ARMA processes By combining the AR p and MA q models an ARMA p q model is obtained. Such a model states that the current value of some series y depends linearly on its own previous values plus a combination of the current and previous values of a white noise error term. The model can be written ộ L yt ỊM 0 L ut where 0 L 1 01L Ộ2L1 ộpLp and 0 L 1 01L 02L2 0qLq 236 Real Estate Modelling and Forecasting or yt E- 01yt-1 02yt-2 0pyt-p 1ut-1 02Ut-2 Squt-q ut with E ut 0 E u2 Ơ2 E utus 0 t s The characteristics of an
