Weighted composite quantile inference for nearly nonstationary autoregressive models

In this paper, we focus on the following nearly nonsationary autoregressive model: \(y_t = q_n y_{t-1}+u_t\) $y_t=q_n{y}_{t-1}+u_t$ , \(t=1,\ldots ,n\) $t=1,\dots ,n$ , where \(q_n=1+c/k_n\) $q_n=1+c/k_n$ with c a non-zero constant and \(\{k_n, n\geqslant 1\}\) $\{k_n,n⩾1\}$ a sequence of positive constants increasing to \(\infty \) $\infty$ such that \(k_n=o(n)\) $k_n=o(n)$ as \(n\rightarrow \infty \) $n\to \infty$ , and \(\{u_t, t\geqslant 1\}\) $\{u_t,t⩾1\}$ is a sequence of independent and identically distributed random variables which are in the domain of attraction of the normal law with zero mean and possibly infinity variance. The weighted composite quantile estimate of \(q_n\) $q_n$ is examined, and the corresponding limiting distributions under the cases of \(c>0\) $c>0$ and \(c<0\) $c<0$ are established. Monte Carlo simulations are conducted to illustrate the theoretical results on finite-sample performance. The simulation results show that the weighted composite quantile estimate method is more robust and efficient than the composite quantile estimate method in terms of bias and accuracy, and we employ this estimator to analyze a real-world data set.