Time series Note
Autogressive model
basic concept
An autoregressive model of order p: xt = ϕ0 +ϕ1xt−1 +ϕ2xt−2 +…+ϕpxt−p +wt
- white noises: all uncorrelated, maen 0, finete varience
- Least squares estimators work if xt is weakly stationary
weakly stationnary
- 均值为常数
- 协方差至与Δt有关
causal stationary
因果平稳过程（Causal stationary）：统计特性不随时间改变，且当前值仅依赖于当前及过去的信息。
非因果平稳过程（Not causal stationary）：统计特性不随时间改变，但当前值可能依赖于未来的信息。
Causal stationary for AR(1)
xt = ϕxt−1 +wt
-
φ <1, weakly stationary, causal -
φ >1, weakly stationary, noncausal -
φ =1, not weakly stationary
Causal stationary for AR(p)
AR(p) 因果平稳的条件： Consider the corresponding polynomial $ϕ(z) = 1−ϕ_1z −…−ϕ_pz^p = Πp(1-r_i^{-1}z)$, where r_i, i = 1,…,p are the roots of ϕ(z). an AR(p) model is causal stationary if and only if |ri| > 1 for all i.
Note:
$ ϕ(z)X_t = w_t $ => $ X_t = ϕ^{-1}(z) w_t $, 因此ϕ(z)必须可逆，也就是ϕ^{-1}(z)展开对应的无穷级数必须收敛
### OLS估计器的渐进性质
- 如果xt是causal stationary，且wt服从正太分布，那么随着样本量n增大，参数的估计变量渐进服从正态分布
ACF function for AR(1)
inverse operator
ϕ^{-1}(z) = φ(z) , φ(z)ϕ(z) = 1 A general technique for finding the coefficients of a linear process is by matching coefficients.
MA
definition: The moving average model of order q, MA(q), is xt = wt +θ1wt−1 +θ2wt−2 +…+θqwt−q, where wt ∼ wn(0,σ2 w) and θ1,…,θq= 0 are parameters.
property: By definition, MA(q) is causal station1ary
- 明显，均值为常数
- ACF：也很好求，且至与Δt有关