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不稳定自回归过程最小二乘估计的极限分布AI 解读

An autoregressive process $y_n = \beta_1y_{n-1} + \cdots + \beta_py_{n-p} + \varepsilon_n$ is said to be unstable if the characteristic polynomial $\phi(z) = 1 - \beta_1z - \cdots - \beta_pz^p$ has all roots on or outside the unit circle. The limiting distribution of the least squares estimate of $(\beta_1, \cdots, \beta_p)$ is derived and characterized as a functional of stochastic integrals under a $2 + \delta$ moment assumption on $\varepsilon_n$. Up to the present,…

随机回归模型中最小二乘预测量的自适应预测及时间序列应用AI 解读

Herein we consider the asymptotic performance of the least squares predictors $\hat{y}_n$ of the stochastic regression model $y_n = \beta_1 x_{n1} + \cdots + \beta_p x_{np} + \varepsilon_n$. In particular, the accumulated cost function $\sum^n_{k=1} (y_k - \hat{y}_k - \varepsilon_k)^2$ is studied. The results are then applied to nonstationary autoregressive time series. A statistic is also constructed to show how many times one should difference a nonstationary time series…

近非平稳一阶自回归过程的渐近推断AI 解读

A first-order autoregressive process, $Y_t = \beta Y_{t - 1} + \epsilon_t$, is said to be nearly nonstationary when $\beta$ is close to one. The limiting distribution of the least-squares estimate $b_n$ for $\beta$ is studied when $Y_t$ is nearly nonstationary. By reparameterizing $\beta$ to be $1 - \gamma/n, \gamma$ being a fixed constant, it is shown that the limiting distribution of $\tau_n = (\sum^n_{t = 1}Y^2_{t - 1})^{1/2}(b_n - \beta)$ converges to $\mathscr{L}(\gamma)$…