Canonical correlation analysis of stochastic trends via functional approximation

• Department of Statistics, Sapienza University of Rome, Rome, Italy [massimo.franchi@uniroma1.it]

• Department of Economics, University of Bologna, Bologna, Italy [i.georgiev@unibo.it]

• Joint Research Centre, European Commission, Ispra, Italy [paolo.paruolo@ec.europa.eu]

Keywords: Semiparametric inference – cointegration – $I(1)$ – large panels – canonical correlation analysis

1 Paper

The presentation focuses on Franchi et al. [2024]. This abstract lists some related literature, the main contribution and conclusions.

2 Related literature

The analysis of multiple time series with nonstationarity and comovements has been steadily developing since the introduction of the notion of cointegration in Engle and Granger [1987]. Early contributions discuss asymptotic results for diverging sample size $T$, and typical applications considered a cross-sectional dimension $p$ in the low single digits, see e.g. Johansen [1996].

Finite-$T$ properties of estimators and tests were found to worsen (i.e. become increasingly different from the $T$-asymptotic results) as $p$ increases. This motivated the derivation of Barlett-type corrections and bootstrap procedures.

Data with large $p$ and $T$ are nowadays pervasive, and the theory has recently considered extensions for $p$ as large as a few hundreds. Several contributions focus on the vector autoregressive (VAR) model with $p$ and $T$ diverging proportionally.

3 Contribution

The present paper contributes to this literature by providing alternative and novel inferential tools for semiparametric inference on $I(1)/I(0)$ $p$-dimensional vectors $X_t$ that admit the Common Trends (CT) representation

$$X_t=\gamma +\psi {\kappa }^{\prime }\sum _{i=1}^t{\epsilon }_i+C_1(L){\epsilon }_t,t=1,2,\mathrm{\dots },$$

where $L$ is the lag operator, $\gamma$ is a vector of initial values, ${\epsilon }_t$ is a vector white noise, ${\kappa }^{\prime }{\sum }_{i=1}^t{\epsilon }_i$ are $0\le s\le p$ stochastic trends, $\psi$ is a $p\times s$ full column rank loading matrix, and $C_1(L){\epsilon }_t$ is an $I(0)$ linear process.

The proposed approach is based on the empirical canonical correlations between the $p\times 1$ vector of observables $X_t$ and a $K\times 1$ vector of deterministic variables $d_t$, constructed as the first $K$ elements of an orthonormal $L^2[0,1]$ basis discretized over the equispaced grid $1/T,2/T,\mathrm{\dots },1$. In the asymptotic analysis, the cross-sectional dimension $p$ is kept fixed while $T$ and $K$ diverge sequentially, $K$ after $T$.

Monte Carlo simulations provide evidence of reliable performance uniformly in $s$ in systems of dimension $p=10,20,50,100,200,300$ and an empirical analysis of 20 U.S. exchange rates illustrates the methods.

The present approach can be coherently applied to subsets or aggregations of variables. This fact rests on the property, called dimensional coherence, that linear combinations of the observables $X_t$ also admit a common trends representation with a number of stochastic trends that is at most equal to that in the original system.

4 Conclusions

This paper proposes a novel canonical correlation analysis for semiparametric inference in $I(1)/I(0)$ systems via functional approximation. The approach can be applied coherently to panels of $p$ variables with a generic number $s$ of stochastic trends, as well as to subsets or aggregations of variables. This study discusses inferential tools on $s$ and on the loading matrix $\psi$ of the stochastic trends (and on their duals $r$ and $\beta$, the cointegration rank and the cointegrating matrix): asymptotically pivotal test sequences and consistent estimators of $s$ and $r$, $T$-consistent, mixed Gaussian and efficient estimators of $\psi$ and $\beta$, Wald tests thereof, and misspecification tests for checking model assumptions. Monte Carlo simulations show that these tools have reliable performance uniformly in $s$ for small, medium and large-dimensional systems, with $p$ ranging from 10 to 300. An empirical analysis of 20 exchange rates illustrates the methods.