Linear transformations of vector ARMA processes.

*(English)*Zbl 0555.62072Linear transformations of stochastic processes are used in many ways in economic analyses, for example when linear aggregates or subprocesses are considered. It is demonstrated that a linear transformation of a vector ARMA process is again an ARMA process and conditions for stationarity are given. Three different predictors for a linearly transformed process are compared. Forecasting the original process and transforming the predictions is superior to forecasting the transformed process directly and to transforming univariate predictions of the components of the original process. Conditions for equality of the three different forecasts are provided.

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62M20 | Inference from stochastic processes and prediction |

62P20 | Applications of statistics to economics |

##### Keywords:

linear transformations of vector ARMA procsses; comparison of predictors; conditions for stationarity
Full Text:
DOI

##### References:

[1] | Anderson, O.D., Moving average processes, The Statistician, 24, 283-297, (1975) |

[2] | Anderson, O.D., On a lemma associated with box, Jenkins and Granger, Journal of econometrics, 3, 151-156, (1975) · Zbl 0303.62067 |

[3] | Anderson, O.D., Concerning one of T.W. Anderson’s theorems: A correction to ‘forecasting aggregates of independent ARIMA processes’ by rose, Metron, 36, 99-103, (1978) |

[4] | Anderson, T.W., The statistical analysis of time series, (1971), Wiley New York · Zbl 0225.62108 |

[5] | Ansley, C.F.; Spivey, W.A.; Wrobleski, W.J., On the structure of moving average processes, Journal of econometrics, 6, 121-134, (1977) · Zbl 0357.62071 |

[6] | Box, G.E.P.; Jenkins, G.M., Time series analysis, forecasting and control, (1976), Holden Day San Francisco, CA · Zbl 0109.37303 |

[7] | Chan, W.Y.T.; Wallis, K.F., Multiple time series modelling: another look at the mink-muskrat interaction, Applied statistics, 27, 168-175, (1978) |

[8] | Granger, C.W.J., Investigating causal relations by econometric models and cross spectral methods, Econometrica, 37, 424-438, (1969) · Zbl 1366.91115 |

[9] | Granger, C.W.J.; Morris, M.J., Time series modelling and interpretation, Journal of the royal statistical society, A 139, 246-257, (1976) |

[10] | Hannan, E.J., Multiple time series, (1970), Wiley New York · Zbl 0211.49804 |

[11] | Kohn, R., When is an aggregate of a time series efficiently forecast by its past?, Journal of econometrics, 18, 337-349, (1982) · Zbl 0487.62083 |

[12] | Pierce, D.A.; Haugh, L.D., Causality in temporal systems: characterizations and a survey, Journal of econometrics, 5, 265-293, (1977) · Zbl 0355.62077 |

[13] | Rose, D.E., Forecasting aggregates of independent ARIMA processes, Journal of econometrics, 5, 323-345, (1977) · Zbl 0378.62084 |

[14] | Teräsvirta, T., The invertibility of sums of discrete MA and ARMA processes, Scandinavian journal of statistics, 4, 165-170, (1977) · Zbl 0373.62052 |

[15] | Tiao, G.C.; Guttmann, I., Forecasting contemporal aggregates of multiple time series, Journal of econometrics, 12, 219-230, (1980) · Zbl 0435.62099 |

[16] | Wallis, K.F., Multiple time series analysis and the final form of econometric models, Econometrica, 45, 1481-1497, (1977) · Zbl 0366.62102 |

[17] | Wei, W.W.S.; Abraham, B., Forecasting contemporal time series aggregates, Communications in statistics - theory and methods, A 10, 1335-1344, (1981) · Zbl 0464.62089 |

[18] | Zellner, A.; Palm, F., Time series analysis and simultaneous equation econometric models, Journal of econometrics, 2, 17-54, (1974) · Zbl 0282.90011 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.