Matteo Barigozzi, Giuseppe Cavaliere, Lorenzo Trapani
arXiv 29 Jul 2021 · Econometrics
arXiv:2107.13894 · PDF · DOI · OpenAlex · Extracted main text
We study inference on the common stochastic trends in a non-stationary, $N$-variate time series $y_{t}$, in the possible presence of heavy tails. We propose a novel methodology which does not require any knowledge or estimation of the tail index, or even knowledge as to whether certain moments (such as the variance) exist or not, and develop an estimator of the number of stochastic trends $m$ based on the eigenvalues of the sample second moment matrix of $y_{t}$. We study the rates of such eigenvalues, showing that the first $m$ ones diverge, as the sample size $T$ passes to infinity, at a rate faster by $O\left(T \right)$ than the remaining $N-m$ ones, irrespective of the tail index. We thus exploit this eigen-gap by constructing, for each eigenvalue, a test statistic which diverges to positive infinity or drifts to zero according to whether the relevant eigenvalue belongs to the set of the first $m$ eigenvalues or not. We then construct a randomised statistic based on this, using it as part of a sequential testing procedure, ensuring consistency of the resulting estimator of $m$. We also discuss an estimator of the common trends based on principal components and show that, up to a an invertible linear transformation, such estimator is consistent in the sense that the estimation error is of smaller order than the trend itself. Finally, we also consider the case in which we relax the standard assumption of i.i.d. innovations, by allowing for heterogeneity of a very general form in the scale of the innovations. A Monte Carlo study shows that the proposed estimator for $m$ performs particularly well, even in samples of small size. We complete the paper by presenting four illustrative applications covering commodity prices, interest rates data, long run PPP and cryptocurrency markets.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | She, R. and S. Ling (2020) Inference in heavy-tailed vector error correction models | 1.000 | 20 | 6 | 100% |
| 2 | Johansen, S (1991) Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models | 1.000 | 8 | 5 | 100% |
| 3 | Horváth, L. and L. Trapani (2019) Testing for randomness in a random coefficient autoregression model | 1.000 | 7 | 4 | 100% |
| 4 | Caner, M (1998) Tests for cointegration with infinite variance errors | 1.000 | 6 | 3 | 100% |
| 5 | Falk, B. and C.-H. Wang (2003) Testing long-run PPP with infinite-variance returns | 0.874 | 8 | 2 | 100% |
| 6 | Pham, T. D. and L. T. Tran (1985) Some mixing properties of time series models | 0.874 | 5 | 2 | 100% |
| 7 | Aznar, A. and M. Salvador (2002) Selecting the rank of the cointegration space and the form of the intercept using an information criterion | 0.843 | 3 | 3 | 100% |
| 8 | Samorodnitsky, G. and M. S. Taqqu (1994) Stable non-Gaussian random processes | 0.811 | 4 | 2 | 100% |
| 9 | Merikoski, J. K. and R. Kumar (2004) Inequalities for spreads of matrix sums and products | 0.737 | 3 | 2 | 100% |
| 10 | Trapani, L (2016) Testing for (in)finite moments self | 0.737 | 3 | 2 | 100% |
Showing the top 10 of 74 scored citations.