Bias correction based on AR model in spurious regression

Zhongzhe Ouyang; Ke Liu; Min Lu; Zhongzhe Ouyang; Ke Liu; Min Lu

doi:10.3934/math.2024410

AIMS Mathematics

2024, Volume 9, Issue 4: 8439-8460. doi: 10.3934/math.2024410

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Research article

Bias correction based on AR model in spurious regression

1.
Department of Biostatistics, University of Michigan, MI 48109, USA
2.
School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China
3.
Business School, Hunan Normal University, Changsha 410081, China

Received: 16 January 2024 Revised: 22 February 2024 Accepted: 26 February 2024 Published: 28 February 2024
MSC : 62G05, 62J05, 91B84, 62M10

The regression of mutually independent time series, whether stationary or non-stationary, will result in autocorrelation in the random error term. This leads to the over-rejection of the null hypothesis in the conventional t-test, causing spurious regression. We propose a new method to reduce spurious regression by applying the Cochrane-Orutt feasible generalized least squares method based on a bias-corrected method for a first-order autoregressive model in finite samples. This method eliminates the requirements for a kernel function and bandwidth selection, making it simpler to implement than the traditional heteroskedasticity and autocorrelation consistent method. A series of Monte Carlo simulations indicate that our method can decrease the probability of spurious regression among stationary, non-stationary, or trend-stationary series within a sample size of 10–50. We applied this proposed method to the actual data studied by Yule in 1926, and found that it can significantly minimize spurious regression. Thus, we deduce that there is no significant regressive relationship between the proportion of marriages in the Church of England and the mortality rate in England and Wales.
- spurious regression,
- autoregression,
- bias correction,
- trend
Citation: Zhongzhe Ouyang, Ke Liu, Min Lu. Bias correction based on AR model in spurious regression[J]. AIMS Mathematics, 2024, 9(4): 8439-8460. doi: 10.3934/math.2024410

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Abstract

The regression of mutually independent time series, whether stationary or non-stationary, will result in autocorrelation in the random error term. This leads to the over-rejection of the null hypothesis in the conventional t-test, causing spurious regression. We propose a new method to reduce spurious regression by applying the Cochrane-Orutt feasible generalized least squares method based on a bias-corrected method for a first-order autoregressive model in finite samples. This method eliminates the requirements for a kernel function and bandwidth selection, making it simpler to implement than the traditional heteroskedasticity and autocorrelation consistent method. A series of Monte Carlo simulations indicate that our method can decrease the probability of spurious regression among stationary, non-stationary, or trend-stationary series within a sample size of 10–50. We applied this proposed method to the actual data studied by Yule in 1926, and found that it can significantly minimize spurious regression. Thus, we deduce that there is no significant regressive relationship between the proportion of marriages in the Church of England and the mortality rate in England and Wales.

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