Understanding spurious regressions in econometrics.

*(English)*Zbl 0602.62098Regression equations which relate two or more time series represented by integrated random processes of the ARIMA type, frequently have \(R^ 2\) yet display highly autocorrelated residuals indicated by very low Durban- Watson statistics. In such situations the usual signficance tests on the regression coefficients are very misleading.

The present paper develops an asymptotic theory for such regressions, which explains as a special case the spurious regressions where the usual t-ratio significance test is shown to diverge as the sample size gets infinitely large. This asymptotic theory is also extended to multiple regressions where the variables are generated by a general vector integrated process.

The present paper develops an asymptotic theory for such regressions, which explains as a special case the spurious regressions where the usual t-ratio significance test is shown to diverge as the sample size gets infinitely large. This asymptotic theory is also extended to multiple regressions where the variables are generated by a general vector integrated process.

Reviewer: J.K.Sengupta

##### MSC:

62P20 | Applications of statistics to economics |

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

91B84 | Economic time series analysis |

##### Keywords:

cointegrating regressions; F-ratio test; coefficient of determination; Box-Pierce statistic; regression diagnostics; integrated random processes; ARIMA; autocorrelated residuals; asymptotic theory; spurious regressions; t-ratio significance test; multiple regressions; vector integrated process
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##### References:

[1] | Bhargava, A., On the theory of testing for unit roots in observed time series, (1984), London School of Economics London, Mimeo |

[2] | Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201 |

[3] | Box, G.E.P.; Jenkins, G.M., Time series analysis: forecasting and control, (1970), Holden-Day San Francisco, CA · Zbl 0109.37303 |

[4] | Donsker, M.D., An invariance principle for certain probability limit theorems, Memoirs of the American mathematical society, 6, 1-12, (1951) · Zbl 0042.37602 |

[5] | Erdo¨s, P.; Kac, M., On certain limit theorems in the theory of probability, Bulletin of the American mathematical society, 52, 292-302, (1946) · Zbl 0063.01274 |

[6] | Granger, C.W.J.; Newbold, P., Spurious regressions in econometrics, Journal of econometrics, 2, 111-120, (1974) · Zbl 0319.62072 |

[7] | Granger, C.W.J.; Newbold, P., Forecasting economic time series, (1977), Academic Press New York · Zbl 0344.62076 |

[8] | Granger, C.W.J.; Engle, R.F., Dynamic model specification with equilibrium constraints: cointegration and error correction, () · Zbl 0709.62102 |

[9] | Hall, P.; Heyde, C.C., Martingale limit theory and its applications, (1980), Academic Press New York · Zbl 0462.60045 |

[10] | Herrndorf, N., A functional central limit theorem for weakly dependent sequences of random variables, Annals of probability, 12, 141-153, (1984) · Zbl 0536.60030 |

[11] | Ibragimov, I.A.; Linnik, Y.V., Independent and stationary sequences of random variables, (1971), Wolters-Noordhoff Groningen · Zbl 0219.60027 |

[12] | McLeish, D.L., Invariance principles for dependent variables, Zeitschrift fu¨r wahrscheinlichkeitstheorie und verwandte gebiete, 32, 165-178, (1975) · Zbl 0288.60034 |

[13] | McLeish, D.L., A maximal inequality and dependent strong laws, Annals of probability, 3, 829-839, (1975) · Zbl 0353.60035 |

[14] | Plosser, C.I.; Schwert, G.W., Money income and sunspots: measuring economic relationships and the effects of differencing, Journal of monetary economics, 4, 637-660, (1978) |

[15] | Phillips, P.C.B., Time series regression with unit roots, (), forthcoming in Econometrica. · Zbl 0613.62109 |

[16] | Phillips, P.C.B., Understanding spurious regressions in econometrics, () · Zbl 0602.62098 |

[17] | Phillips, P.C.B., Asymptotic expansions in nonstationary vector autoregressions, (), forthcoming in Econometric Theory. · Zbl 0373.62065 |

[18] | Phillips, P.C.B.; Durlauf, S.N., Multiple time series regression with integrated processes, (), forthcoming in Review of Economic Studies. · Zbl 0599.62103 |

[19] | Pollard, D., Convergence of stochastic processes, (1984), Springer-Verlag New York · Zbl 0544.60045 |

[20] | White, H., Asymptotic theory for econometricians, (1984), Academic Press New York |

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