Theoretical Foundation

The Discrete Wavelet Transform (DWT) decomposes a signal into frequency bands via dilations and translations of mother wavelet \(\psi(t)\) and scaling function \(\phi(t)\):

\[ \psi_{j,k}(t) = 2^{j/2} \psi(2^j t - k) \]

\[ \phi_{j,k}(t) = 2^{j/2} \phi(2^j t - k) \]

For a signal \(x(t)\) of length \(N = 2^J\):

\[ x(t) = \sum_k s_{J,k} \phi_{J,k}(t) + \sum_{j=1}^{J} \sum_k d_{j,k} \psi_{j,k}(t) \]

where \(s_{J,k}\) are smooth coefficients and \(d_{j,k}\) are detail coefficients at scale \(j\).

MODWT Implementation

SignalY uses the Maximal Overlap Discrete Wavelet Transform (MODWT), which:

• Is shift-invariant (unlike DWT)
• Works for arbitrary sample sizes (not just powers of 2)
• Produces aligned multi-resolution analysis (MRA)

Algorithm 1: Wavelet Multi-Resolution Analysis

Require: Signal x, filter type, levels to combine L
1: Apply MODWT: (W₁, ..., W_J, V_J) ← MODWT(x)
2: Compute MRA details: D_j ← MRA(W_j) for j = 1, ..., J
3: Compute smooth: S_J ← MRA(V_J)
4: Combine selected levels: C ← Σ_{j∈L} D_j
5: return (C, S_J, {D_j})

Filter Selection

The package supports Daubechies least asymmetric filters (LA) with \(L\) vanishing moments:

Filter	Length	Recommended Use
la8	8	Economic/financial data
la16	16	Very smooth signals
d4	4	Compact support needed
haar	2	Discontinuous signals

Scale-Period Correspondence

For sampling period \(\Delta t\), level \(j\) captures periods:

\[ T_j \in [2^j \Delta t, 2^{j+1} \Delta t] \]

For annual data (\(\Delta t = 1\) year):

• \(D_3 + D_4\): 8–32 year cycles (business cycle range)
• \(D_5 + D_6\): 32–128 year cycles (long waves)

Sifting Algorithm

EMD decomposes \(x(t)\) into Intrinsic Mode Functions (IMFs) adaptively:

Definition 2.1 (Intrinsic Mode Function). A function \(c(t)\) is an IMF if:

1. The number of extrema and zero-crossings differ by at most one
2. The mean of upper and lower envelopes is zero at all points

Reconstruction Property

\[ x(t) = \sum_{k=1}^{K} c_k(t) + r_K(t) \]

Algorithm 2: EMD Sifting Process

Require: Signal x, max IMFs K
1: r₀ ← x
2: for k = 1, ..., K do
3:     h ← r_{k-1}
4:     repeat
5:         Identify local maxima and minima of h
6:         Interpolate upper envelope e_max, lower envelope e_min
7:         Compute mean envelope: m ← (e_max + e_min)/2
8:         Update: h ← h - m
9:     until h satisfies IMF criteria
10:    c_k ← h
11:    r_k ← r_{k-1} - c_k
12: end for
13: return IMFs {c_k} and residue r_K

State Space Formulation

The Grant-Chan HP filter embeds trend-cycle decomposition in a state space model:

\[ y_t = \tau_t + \psi_t + \varepsilon_t, \quad \varepsilon_t \sim \mathcal{N}(0, \sigma_\varepsilon^2) \]

\[ \Delta^2 \tau_t = \eta_t, \quad \eta_t \sim \mathcal{N}(0, \sigma_\tau^2) \]

\[ \psi_t = \phi_1 \psi_{t-1} + \phi_2 \psi_{t-2} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, \sigma_\psi^2) \]

where \(\tau_t\) is trend, \(\psi_t\) is cycle, and \(\varepsilon_t\) is irregular.

Prior Configurations

The package implements three prior configurations:

1. Weakly Informative:

\[ \sigma_\tau^2, \sigma_\psi^2, \sigma_\varepsilon^2 \sim \text{Inv-Gamma}(0.01, 0.01) \]

\[ \phi_1, \phi_2 \sim \mathcal{N}(0, 10^2) \mathbf{1}_{[\text{stationary}]} \]

2. Informative (calibrated to HP-1600 for annual data):

\[ \sigma_\tau^2 / \sigma_\psi^2 \sim \text{Log-Normal}(\log(1/100), 0.5) \]

3. Empirical Bayes: Hyperparameters estimated from data

Model selection via Deviance Information Criterion:

\[ \text{DIC} = \bar{D}(\theta) + p_D = -2\mathbb{E}[\log p(y|\theta)] + 2p_D \]

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Horseshoe Prior

The standard Horseshoe places:

\[ \beta_j | \lambda_j, \tau \sim \mathcal{N}(0, \lambda_j^2 \tau^2) \]

\[ \lambda_j \sim \mathcal{C}^+(0, 1) \]

\[ \tau \sim \mathcal{C}^+(0, \tau_0) \]

where \(\mathcal{C}^+(0, s)\) denotes half-Cauchy with scale \(s\).

Regularized Horseshoe

Piironen & Vehtari introduce slab regularization via:

\[ \tilde{\lambda}_j^2 = \frac{c^2 \lambda_j^2}{c^2 + \tau^2 \lambda_j^2} \]

with \(c^2 \sim \text{Inv-Gamma}(\nu/2, \nu s^2/2)\) providing a soft ceiling on coefficient magnitude.

Shrinkage Factors

The shrinkage factor \(\kappa_j\) quantifies the degree of shrinkage:

\[ \kappa_j = \frac{1}{1 + \tau^2 \lambda_j^2 / c^2} \]

Proposition 3.1 (Shrinkage Interpretation). As \(\kappa_j \to 0\), \(\beta_j\) is essentially unshrunken (signal). As \(\kappa_j \to 1\), \(\beta_j\) is strongly shrunk toward zero (noise).

Effective Number of Non-Zeros

\[ m_e = \sum_{j=1}^{p} (1 - \kappa_j) \]

Standard PCA

For centered data matrix \(X \in \mathbb{R}^{n \times p}\), PCA solves:

\[ \max_w w^\top X^\top X w \quad \text{s.t.} \quad \|w\| = 1 \]

yielding loadings \(W = [w_1, \ldots, w_k]\) and scores \(Z = XW\).

Bootstrap Significance Testing

Algorithm 3: Block Bootstrap for PCA Loadings

Require: Data X, block length b, replications B
1: Compute original loadings W^(0)
2: for r = 1, ..., B do
3:     Generate bootstrap sample X^(r) via block resampling
4:     Compute loadings W^(r)
5:     Apply Procrustes rotation to align with W^(0)
6: end for
7: For each loading w_jk, compute p-value: (1/B) Σ_r 𝟙[|w^(r)_jk| > |w^(0)_jk|]

Entropy of Loadings

Shannon entropy quantifies loading concentration:

\[ H_k = -\sum_{j=1}^{p} w_{jk}^2 \log(w_{jk}^2) \]

where loadings are normalized so \(\sum_j w_{jk}^2 = 1\).

• Low \(H_k\): Concentrated loadings (few variables dominate)
• High \(H_k\): Diffuse loadings (many variables contribute)

Model

For panel \(X_t \in \mathbb{R}^p\) at time \(t\):

\[ X_t = \Lambda F_t + e_t, \quad e_t \sim \mathcal{N}(0, \Sigma_e) \]

\[ F_t = A_1 F_{t-1} + \cdots + A_q F_{t-q} + \eta_t, \quad \eta_t \sim \mathcal{N}(0, I_r) \]

where \(F_t \in \mathbb{R}^r\) are latent factors, \(\Lambda \in \mathbb{R}^{p \times r}\) are loadings.

Factor Selection via Information Criteria

Bai & Ng propose criteria for selecting \(r\):

\[ IC_1(r) = \log(V(r)) + r \cdot \frac{n + p}{np} \log\left(\frac{np}{n + p}\right) \]

\[ IC_2(r) = \log(V(r)) + r \cdot \frac{n + p}{np} \log(\min(n, p)) \]

\[ IC_3(r) = \log(V(r)) + r \cdot \frac{\log(\min(n, p))}{\min(n, p)} \]

where \(V(r)\) is the sum of squared residuals with \(r\) factors.

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DF-GLS

Apply GLS transformation with \(\bar{\alpha} = 1 - 7/T\) (demeaning) or \(\bar{\alpha} = 1 - 13.5/T\) (detrending), then run ADF on transformed series.

Point Optimal Test

\[ P_T = \frac{S(\bar{\alpha}) - \bar{\alpha} S(1)}{s_{AR}^2} \]