“Wavelet-Based” Early Warning Signals of Financial Stress: An Application to IMF’s AE-FSI

Abstract

In this paper we construct a “wavelet-based” early warning indicator for the IMF financial stress index for advanced economies (FSI-AE) developed in Cardarelli et al. (Financial stress, downturns, and recoveries. IMF Working Papers 09/100, International Monetary Fund, Washington, 2009). Specifically, for each country of the sample we construct a “wavelet-based” early warning indicator of financial stress by selecting, among the individual indicators used in the construction of the IMF financial stress index, those indicators displaying the best leading performance on a “scale-by-scale” basis. The leading properties of each country’s “wavelet-based” early warning indicator for its corresponding financial stress index are evaluated using univariate statistical criteria and a pseudo out-of-sample forecasting exercise. The findings indicate that the “wavelet-based” early warning composite indicator largely outperforms at every horizon any individual financial variable taken in isolation in early detecting financial stress and that the gains tend to increase as the time horizon increases.

Appendix: Basic Notions of Wavelets

Given a stochastic process {X}, if we denote with H = (h ₀, …, h _L−1) and G = (g ₀, …, g _L−1) the impulse response sequence²⁵ of the wavelet and scaling filters h _l , and g _l , respectively, of a Daubechies compactly supported wavelet (with L the width of the filters), when N = L ² we may apply the orthonormal discrete wavelet transform (DWT) and obtain the wavelet and scaling coefficients at the jth level defined as²⁶

$$\displaystyle\begin{array}{rcl} w_{j,t} =\sum \limits _{l=0}\limits ^{L-1}h_{j,l}X_{t-l}& & {}\\ v_{J,t} =\sum \limits _{l=0}\limits ^{L-1}g_{j,l}X_{t-l},& & {}\\\end{array}$$

where h _j, l and g _j, l are the level j wavelet and scaling filters and, due to downsampling by 2^J, we have $\frac{N} {2^{J}}$ scaling and wavelet coefficients.²⁷

The DWT is implemented via a filter cascade where the wavelet filter h _l is used with the associated scaling filter g _l ²⁸ in a pyramid algorithm (Mallat 1989) consisting in an iterative scheme in which, at each iteration, the wavelet and scaling coefficients are computed from the scaling coefficients of the previous iteration.²⁹

However the orthonormal discrete wavelet transform (DWT), even if widely applied to time series analysis in many disciplines, has two main drawbacks: the dyadic length requirement (i.e. a sample size divisible by 2^J),³⁰ and the fact that the wavelet and scaling coefficients are not shift invariant due to their sensitivity to circular shifts because of the decimation operation. An alternative to DWT is represented by a non-orthogonal variant of DWT: the maximal overlap DWT (MODWT).³¹

In the orthonormal Discrete Wavelet Transform (DWT) the wavelet coefficients are related to nonoverlapping differences of weighted averages from the original observations that are concentrated in space. More information on the variability of the signal could be obtained considering all possible differences at each scale, that is considering overlapping differences, and this is precisely what the maximal overlap algorithm does.³² Thus, the maximal overlap DWT coefficients may be considered the result of a simple modification in the pyramid algorithm used in computing DWT coefficients through not downsampling the output at each scale and inserting zeros between coefficients in the wavelet and scaling filters.³³ In particular, the MODWT wavelet and scaling coefficients $\tilde{w}_{j,t}$ and $\tilde{w}_{j,t}$ are given by

$$\displaystyle\begin{array}{rcl} \tilde{w}_{j,t} = \frac{1} {2^{j/2}}\sum \limits _{l=0}\limits ^{L-1}\tilde{h}_{ j,l}X_{t-l}& & {}\\ \tilde{v}_{J,t} = \frac{1} {2^{j/2}}\sum \limits _{l=0}\limits ^{L-1}\tilde{g}_{ J,l}X_{t-l},& & {}\\ \end{array}$$

where the MODWT wavelet and scaling filters $\tilde{h}_{j,l}$ and $\tilde{g}_{j,l}$ are obtained by rescaling the DWT filters as follows³⁴:

$$\displaystyle\begin{array}{rcl} \tilde{h}_{j,l} = \frac{h_{j,l}} {2^{j/2}}& & {}\\ \tilde{g}_{j,l} = \frac{g_{j,l}} {2^{j/2}}.& & {}\\ \end{array}$$

The MODWT wavelet coefficients $\tilde{w}_{j,t}$ are associated with generalized changes of the data on a scale λ _j  = 2^j−1. With regard to the spectral interpretation of MODWT wavelet coefficients, as the MODWT wavelet filter h _j, l at each scale j approximates an ideal high-pass with passband f ∈ [1∕2^j+1, 1∕2^j],³⁵ the λ _j scale wavelet coefficients are associated to periods [2^j, 2^j+1].

MODWT provides the usual functions of the DWT, such as multiresolution decomposition analysis and variance analysis based on wavelet transform coefficients, but unlike the classical DWT it

• can handle any sample size;

• is translation invariant, as a shift in the signal does not change the pattern of wavelet transform coefficients;

• provides increased resolution at coarser scales.³⁶

In addition, MODWT provides a larger sample size in the wavelet variance and correlation analyses and produces a more asymptotically efficient wavelet variance estimator than the DWT.

In addition to the features stated above wavelet transform is able to analyze the variance of a stochastic process and decompose it into components that are associated to different time scales. In particular, given a stationary stochastic process {X} with variance σ _X ² and defined the level j wavelet variance σ _X ²(λ _j ), the following relationship holds

$$\displaystyle\begin{array}{rcl} \sum \limits _{j=1}\limits ^{\infty }\sigma _{ X}^{2}(\lambda _{ j}) =\sigma _{ X}^{2}& & {}\\ \end{array}$$

where σ _X ²(λ _j ) represent the contribution to the total variability of the process due to changes at scale λ _j . This relationship says that wavelet variance decomposes the variance of a series into variances associated to different time scales.³⁷ By definition, the (time independent) wavelet variance for scale λ _j , σ _X ²(λ _j ), is defined to be the variance of the j-level wavelet coefficients

$$\displaystyle\begin{array}{rcl} \sigma _{X}^{2}(\lambda _{ j}) = var\{\tilde{w}_{j,t}^{2}\}.& & {}\\ \end{array}$$

As shown in Percival (1995), provided that N − L _j  ≥ 0, an unbiased estimator of the wavelet variance based on the MODWT may be obtained, after removing all coefficients affected by the periodic boundary conditions,³⁸ using

$$\displaystyle{ \tilde{\sigma }_{X}^{2}(\lambda _{ j}) = \frac{1} {\tilde{N}_{j}}\sum \limits _{t=L_{J}}\limits ^{N}\tilde{w}_{ j,t}^{2} }$$

where $\tilde{N}_{j} = N - L_{j} + 1$ is the number of maximal overlap coefficients at scale j and L _j  = (2 _j − 1)(L − 1) + 1 is the length of the wavelet filter for level j.³⁹ Thus, the jth scale level j wavelet variance is simply the variance of the nonboundary or interior wavelet coefficients at that level (Percival 1995; Serroukh et al. 2000). Both DWT and MODWT can decompose the sample variance of a time series on a scale-by-scale basis via its squared wavelet coefficients, but the MODWT-based estimator has been shown to be superior to the DWT-based estimator (Percival 1995).

Starting from the spectrum Sw _X, j of the scale j wavelet coefficients it is possible to determine the asymptotic variance V _j of the MODWT-based estimator of the wavelet variance (covariance) and construct a random interval which forms a 100(1 − 2p)% confidence interval.⁴⁰

The formulas for an approximate 100(1 − 2p)% confidence intervals MODWT estimator robust to non-Gaussianity for $\tilde{\sigma }_{X,j}^{2}$ are provided in Gençay et al. (2002).⁴¹

Similar to their classical counterparts, we can define the wavelet covariance between two processes X and Y at wavelet scale j as the covariance between scale j wavelet coefficients of X and Y, that is γ _XY (λ _j ) = Cov(w _j, t ^X w _j, t ^Y), and the wavelet correlation between two time series ρ _XY (λ _j ) as the ratio of the wavelet covariance, γ _XY (λ _j ), and the square root of their wavelet variances σ _X (λ _j )and σ _Y (λ _j ) (see Whitcher et al. 1999, 2000). The wavelet correlation coefficient ρ _XY (λ _j ) provides a standardized measure of the relationship between the two processes X and Y on a scale-by-scale basis and, as with the usual correlation coefficient between two random variables, $\left \vert \tilde{\rho }_{XY }(\lambda _{j})\right \vert \leq 1$. Specifically, given the unbiased estimators of the wavelet variances, $\tilde{\sigma }_{X}(\lambda _{j})$ and $\tilde{\sigma }_{Y }(\lambda _{j})$, and covariance, $\tilde{\gamma }_{XY }(\lambda _{j})$, the unbiased estimator of the wavelet correlation for scale j, $\tilde{\rho }_{XY }(\lambda _{j})$, may be obtained by

$$\displaystyle{ \tilde{\rho }_{XY }(\lambda _{j}) = \frac{\tilde{\gamma }_{XY }(\lambda _{j})} {\tilde{\sigma }_{X}(\lambda _{j})\tilde{\sigma }_{Y }(\lambda _{j})}. }$$

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