QCOR Extension

• QCOR is a quantile-based correlation measure that assesses linear associations at a specified quantile, capturing dependence across the distribution.
• The extension includes its partial, autocorrelation, and residual-check counterparts (QPCOR, QPACF, QACF) that aid in model identification and adequacy checks.
• It has practical applications in time-series analysis, finance, and climatology where understanding tail behavior and non-Gaussianity is essential.

Quantile correlation (QCOR) is a statistical measure designed to quantify the strength of linear association between two random variables conditional on a particular quantile of the response. The QCOR, together with the quantile partial correlation (QPCOR), enables the analysis of dependence structure not just in the mean, but throughout the distribution, including in the tails—critical for applications in finance, climatology, and time-series analysis where non-Gaussianity and heterogeneity are prominent. The theoretical framework has led to extensions such as the quantile autocorrelation (QACF) and quantile partial autocorrelation (QPACF) functions, culminating in a systematic approach for model identification, estimation, and adequacy checks in quantile autoregressive (QAR) models (Li et al., 2012).

1. Quantile Correlation: Definition and Sample Estimation

Let $Y$ and $X$ be random variables and fix a quantile level $\tau\in(0,1)$. Define the quantile score function $\psi_\tau(u) = \tau - I(u<0)$. The $\tau$-quantile of $Y$ is $Q_{\tau,Y} = \inf\{y \mid F_Y(y)\geq\tau\}$.

The quantile covariance is

$$qcov_\tau\{Y, X\} = E[\psi_\tau(Y-Q_{\tau,Y})(X - EX)]$$

and the quantile correlation is

$$qcor_\tau\{Y, X\} = \frac{qcov_\tau\{Y, X\}}{\sqrt{\mathrm{var}\{\psi_\tau(Y-Q_{\tau,Y})\} \cdot \mathrm{var}(X)}}.$$

For i.i.d. data $\{(Y_i, X_i)\}_{i=1}^n$, the empirical estimator is

$$\widehat{qcor}_\tau\{Y,X\} = \frac{n^{-1} \sum_{i=1}^n \psi_\tau(Y_i-\widehat Q_{\tau,Y}) (X_i-\bar X)}{\sqrt{(\tau-\tau^2)\, \widehat\sigma_X^2}},$$

with $\bar X = n^{-1}\sum X_i$, $\widehat \sigma_X^2 = n^{-1} \sum (X_i-\bar X)^2$, and $\widehat Q_{\tau,Y}$ the sample $\tau$-quantile [(Li et al., 2012), (1)].

2. Quantile Partial Correlation, QPACF, and QACF

To isolate the linear effect of $X$ on $Y$ at quantile $\tau$ after adjusting for covariates $Z\in\mathbb R^q$, the quantile partial correlation is defined as

$$qpcor_\tau\{Y, X|Z\} = \frac{E[\psi_\tau(Y-\alpha_2-\beta_2'Z)(X-\alpha_1-\beta_1'Z)]}{\sqrt{(\tau-\tau^2)\, \sigma_{X|Z}^2}},$$

where $(\alpha_2,\beta_2')$ solves the $\tau$-quantile regression of $Y$ on $Z$ and $(\alpha_1,\beta_1')$ is the OLS fit of $X$ on $Z$ [(Li et al., 2012), (2)].

For time series $\{y_t\}$ modeled as QAR$(p)$: $$Q_\tau(y_t|\mathcal{F}_{t-1}) = \varphi_0(\tau) + \varphi_1(\tau) y_{t-1} + \ldots + \varphi_p(\tau) y_{t-p}$$ define the quantile partial autocorrelation at lag $k$ as

$$\phi_{kk, \tau} = qpcor_\tau\{y_t, y_{t-k}\mid z_{t,k-1}\},$$

where $z_{t,k-1} = (y_{t-1}, \ldots, y_{t-k+1})'$ [(Li et al., 2012), (7)]. The QPACF provides a cutoff property for true QAR$(p)$ models: $\phi_{pp,\tau}\ne 0$ and $\phi_{kk,\tau}=0$ for $k>p$.

The quantile autocorrelation function of residuals (QACF) is

$$\rho_{k,\tau} = \mathrm{corr}\{\psi_\tau(e_{t,\tau}), e_{t-k,\tau}\}$$

where $$e_{t,\tau} = y_t - \sum_{j=0}^p \varphi_j(\tau) y_{t-j}$$.

3. Asymptotic Theory and Variance Estimation

For both $\widehat{qcor}_\tau$ and $\widehat{qpcor}_\tau$, under suitable regularity (finite moments, smooth densities at the quantile), asymptotic normality holds: $$\sqrt{n}\, \left( \widehat{qcor}_\tau\{Y,X\} - qcor_\tau\{Y,X\} \right)\rightarrow_d N(0, \Omega_1)$$ with $\Omega_1$ expressed explicitly in terms of means, variances, and conditional densities [(Li et al., 2012), (4)]. Variance estimators require plug-in estimates for expectations and conditional densities, which in practice can be obtained by kernel (Nadaraya-Watson) methods.

An analogous result holds for the sample QPACF: $$\sqrt{n} \, \widetilde\phi_{kk,\tau} \rightarrow_d N(0, \Omega_3)$$ where $\Omega_3$ is again computable from second moments and local density estimates [(Li et al., 2012), (10)].

The sample QACF of residuals is jointly asymptotically normal, enabling Ljung–Box-type portmanteau tests for QAR model adequacy.

4. Practical Estimation and Model Selection

Quantile correlation and partial correlation are computed via:

1. Estimating $Q_{\tau,Y}$ by the empirical quantile;
2. Computing $\psi$-scores for each data point;
3. Calculating the empirical covariance with $X$ and normalizing.

For QPCOR, regress $X$ on $Z$ by least squares, $Y$ on $Z$ by quantile regression; form residuals and apply the above recipe.

For QPACF in time series, iterate over lags $k$, fit OLS and quantile regression as above, and compute sample partial correlations. The point at which sample QPACF drops to zero (within confidence bounds) identifies the model order $p$ for QAR(p).

After fitting QAR$(p)$ by quantile regression, residuals are analyzed via QACF; a joint statistic $Q_{BP}(K)=n\sum_{j=1}^K r_{j,\tau}^2$ can be compared against a chi-square with $K-p$ degrees of freedom for model adequacy [(Li et al., 2012), (14)].

Bandwidth selection for local density estimation at the target quantile is a critical aspect: Bofinger (1975) and Hall–Sheather (1988) rules are standard choices.

5. Interpretation, Extensions, and Empirical Application

QCOR$_\tau\{Y, X\}$ measures the linear effect of $X$ on the event $\{Y>Q_{\tau, Y}\}$, with interpretation akin to a Pearson correlation but conditional on the specified quantile. QPCOR$_\tau\{Y, X|Z\}$ further adjusts for $Z$.

In the time-series context, QPACF provides a direct analogue of the PACF for Box–Jenkins model selection but for quantiles. QACF of residuals supports model checking, with a zero pattern suggesting adequacy.

Empirically, on daily Nasdaq composite returns (n=1235, 2002–2007), QAR models fitted via quantile techniques revealed tail-specific serial dependence absent at the median, with distinct autoregressive structures in the lower ($\tau=0.2$) and upper ($\tau=0.8$) quantiles, and model adequacy supported by nonsignificant QACF and portmanteau tests (Li et al., 2012).

6. Summary Table of Core Quantile Dependence Functions

Name	Formula	Role/Interpretation
QCOR	$qcor_\tau\{Y,X\}$	Marginal dependence at quantile
QPCOR	$qpcor_\tau\{Y,X|Z\}$	Conditional dependence at quantile
QPACF	$\phi_{kk,\tau} = qpcor_\tau\{y_t, y_{t-k}|\ldots\}$	Lag-specific AR order selection
QACF	$\rho_{k,\tau}$	Residual dependence model check

7. Significance and Impact

The QCOR and its extensions generalize classical dependence measures to the conditional quantile context, supporting inference and model selection for heterogeneous, non-Gaussian processes. QCOR-based methodology extends the Box–Jenkins paradigm to quantile autoregressive modeling via QPACF and QACF, providing robust tools for the analysis and forecasting of time series with asymmetric or regime-dependent dynamics (Li et al., 2012).