Algorithm 1 Main methodology to identify time-varying lead-lag relationships.

procedureMain step 1: Compute multi-dimensional DTW alignment (Input:  two stationary univariate time series).

Step 1 Z-normalize both time series.

Step 2 Convert univariate time series to multi-dimensional descriptor series

as in shapeDTW [4].

Step 2.1 Pad beginning and ending of time series by repeating the first and last

element $\left(\frac{l-1}{2}\right)$ times, respectively.

Step 2.2 At each original index i, extract the subsequence of length l centred

at i.

Step 2.3 Take each subsequence as vector for describing the respective index.

Step 2.4 Optionally concatenate subsequence with its first difference to

include shape characteristics invariant to the level of the values.

Step 3 Z-normalize each dimension to allow for equal contribution as in

Shokoohi-Yekta et al. [15].

Step 4 Calculate the local cost matrix using dependent multi-dimensional DTW.

Step 5 Perform DTW on the local cost matrix, while optionally relaxing the

boundary condition.

Output DTW distance, optimal warping path, accumulated cost matrix,

local cost matrix, optimal start- and endpoint.

end procedure

procedure Main step 2: Extract lead-lag relationship (Input: optimal warping path, optimal start- and endpoint).

Assumption Relationship of the form $Y\left(j\right)=c+a·X(j-\tau (j\left)\right)+\eta ,$

$with\eta \sim \mathcal{N}(0,{\sigma }_{\eta })$ in similar spirit to Sornette and Zhou [38].

Step 1 If relaxed boundary condition, transfer optimal warping path to

indices of original local cost matrix.

Step 2 Subtract matched indices of the optimal warping path.

Step 3 For each index of Y, take a value that comes last in the optimal warping path.

Output Identified lead-lag relationship.

end procedure