Testing for Common Trends in Nonstationary Large Datasets

Abstract

We propose a testing-based procedure to determine the number of common trends in a large nonstationary dataset. Our procedure is based on a factor representation, where we determine whether there are (and how many) common factors (i) with linear trends, and (ii) with stochastic trends. Cointegration among the factors is also permitted. Our analysis is based on the fact that those largest eigenvalues of a suitably scaled covariance matrix of the data corresponding to the common factor part diverge, as the dimension N of the dataset diverges, whilst the others stay bounded. Therefore, we propose a class of randomized test statistics for the null that the pth largest eigenvalue diverges, based directly on the estimated eigenvalue. The tests only requires minimal assumptions on the data-generating process. Monte Carlo evidence shows that our procedure has very good finite sample properties, clearly dominating competing approaches when no common trends are present. We illustrate our methodology through an application to the U.S. bond yields with different maturities observed over the last 30 years.

Keywords:

• Common trends
• Nonstationary large panels
• Randomized tests

Acknowledgments

We are grateful to the participants to the 1st Italian Workshop on Econometrics and Empirical Economics (Milano, 25–26 January, 2018); the Financial Econometrics Seminar at CREST (Paris, 15 February, 2018); and the Economics Seminar Series at Aarhus University (Aarhus, 1 March, 2018); the Workshop on Big Data in Financial Markets at Cambridge University (Cambridge, 24–25 May, 2018); the Workshop on Macroeconomic and Financial Time Series at Lancaster University (Lancaster, 31 May–1 June, 2018); the EEA-ESEM congress at the University of Manchester (Manchester, 26–30 August, 2019).

Notes

1. https://fred.stlouisfed.org.

2. https://www.treasury.gov/resource-center/economic-policy/corp-bond-yie.

3. We note that, when computing $\widehat{r}$, this is also equivalent to IC3 in Bai and Ng (2002)

4. The chosen value of c is such that it minimizes ${\sum }_{i=1}^N{({\widehat{\lambda }}_{i,2}-{\lambda }_{i,2}(c))}^2$.

5. Data for S_t are available at https://fred.stlouisfed.org. Data for R_t are available at http://www.econ.yale.edu/∼shiller/data.htm.