Algorithm 1.

Quantile Regression with Time Series Errors (QUARTS)

Inputs: y, X, q, τ Set ϕ(0) = 0𝖳, ε(0) = 0. Given ϕ(j−1), ε(j−1), let $${\mathrm{y̌}}_{i-q}=y_i-{\varphi }_1^{(j-1)}{\epsilon }_{i-1}^{(j-1)}-\cdots -{\varphi }_q^{(j-1)}{\epsilon }_{i-q}^{(j-1)},\forall i\in \{q+1,\dots ,n\}{\mathit{\beta }}_{\tau }^{(j)}=QR\mathit{\text{fit}}(\mathit{y̌},\mathit{X̃},\tau ){\mathit{\epsilon }}^{(j)}=\mathit{y}-\mathit{X}{\mathit{\beta }}_{\tau }^{(j)}$$ Using QR, fit the regression model ${\epsilon }_i^{(j)}={\varphi }_1{\epsilon }_{i-1}^{(j)}+\cdots +{\varphi }_q{\epsilon }_{i-q}^{(j)}+{\delta }_i$, for i ∈ {q + 1, …, n}. Let ϕ(j) be the estimated AR row vector. Repeat steps 2 and 3 until ϕ(j) and ${\mathit{\beta }}_{\tau }^{(j)}$ converge to steady-state solutions ϕ and βτ, respectively. Return βτ as the final coefficient vector and ϕ as the final AR model coefficient vector.