Spurious multivariate regressions under fractionally integrated processes

Abstract

This article studies spurious regression in the multivariate case for any finite number of fractionally integrated variables, stationary or not. We prove that the asymptotic behavior of the estimated coefficients and their t-statistics depend on the degrees of persistence of the regressors and the regressand. Nonsense inference could therefore be drawn when the sum of the degrees of persistence of the regressor and regressand is greater or equal than 1/2. Moreover, the asymptotic behavior from the most persistent regressor spreads to correlated regressors. Thus, the risk of uncovering spurious results increases as more regressors are included. Inference drawn from other test statistics such as the joint $\mathcal{F}$ test, the R-squared, and the Durbin-Watson is also misleading. Finite sample evidence supports our findings.

Keywords:

• Fractional integration
• long memory
• spurious regression
• multivariate regression

Acknowledgments

The authors would like to thank the anonymous referee, as well as Niels Haldrup, Ignacio Lobato, Michael Massmann, and the participants of the Long Memory Conference 2018 for their valuables comments. All remaining errors are ours.

Disclosure statement

The authors state that no financial interest or benefit has arisen from the direct applications of our research.

Notes

1 The sample autocorrelation function for stationary fractionally integrated processes was studied by Hosking (1984); the nonstationary sample autocorrelation function was obtained by Hassler (1997).

2 We write ${\sigma }_i^2$ instead of ${\sigma }_{i,i}$ $\forall i$ to ease notation.

3 Note that both type-I and type-II fractionally integrated processes could be encompassed in Assumption 1 by making ${ϵ}_{z,t}=0,$ for $z=y,x_i,\forall t\le 0,$ for type-II. See Marinucci and Robinson (2000) for further details.

4 Codes for the Monte Carlos analysis are available at: https://github.com/everval/Spurious_Multivariate

5 At each step m, the elements of the m-th column from the $(m+1)$-th row onward become 0.