### 统计代写|应用时间序列分析代写applied time series analysis代考|Identifying AR Models in Practice

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Partial Autocorrelation Function

The PACF of a time series is a function of its $\mathrm{ACF}$ and is a useful tool for determining the order $p$ of an AR model. A simple, yet effective way to introduce PACF is to consider the following AR models in consecutive orders:
\begin{aligned} r_{t} &=\phi_{0,1}+\phi_{1,1} r_{t-1}+e_{1 t}, \ r_{t} &=\phi_{0,2}+\phi_{1,2} r_{t-1}+\phi_{2,2} r_{t-2}+e_{2 t}, \ r_{t} &=\phi_{0,3}+\phi_{1,3} r_{t-1}+\phi_{2,3} r_{t-2}+\phi_{3,3} r_{t-3}+e_{3 t}, \ r_{t} &=\phi_{0,4}+\phi_{1,4} r_{t-1}+\phi_{2,4} r_{t-2}+\phi_{3,4} r_{t-3}+\phi_{4,4} r_{t-4}+e_{4 t}, \ & \vdots \end{aligned}
where $\phi_{0, j}, \phi_{i, j}$, and $\left{e_{j t}\right}$ are, respectively, the constant term, the coefficient of $r_{t-i}$, and the error term of an $\operatorname{AR}(j)$ model. These models are in the form of a multiple linear regression and can be estimated by the least squares method. As a matter of fact, they are arranged in a sequential order that enables us to apply the idea of partial $F$ test in multiple linear regression analysis. The estimate $\hat{\phi}{1,1}$ of the first equation is called the lag-1 sample PACF of $r{t}$. The estimate $\hat{\phi}{2,2}$ of the second equation is the lag- 2 sample PACF of $r{t}$. The estimate $\hat{\phi}{3,3}$ of the third equation is the lag-3 sample PACF of $r{t}$, and so on.

From the definition, the lag-2 PACF $\hat{\phi}{2,2}$ shows the added contribution of $r{t-2}$ to $r_{t}$ over the $\operatorname{AR}(1)$ model $r_{t}=\phi_{0}+\phi_{1} r_{t-1}+e_{1 t}$. The lag-3 PACF shows the added contribution of $r_{t-3}$ to $r_{t}$ over an $\mathrm{AR}(2)$ model, and so on. Therefore, for an $\operatorname{AR}(p)$ model, the lag- $p$ sample PACF should not be zero, but $\hat{\phi}_{j, j}$ should be close to zero for all $j>p$. We make use of this property to determine the order $p$. Indeed, under some regularity conditions, it can be shown that the sample $\operatorname{PACF}$ of an $\operatorname{AR}(p)$ process has the following properties:

• $\hat{\phi}{p, p}$ converges to $\phi{p}$ as the sample size $T$ goes to infinity.
• $\hat{\phi}_{\ell, \ell}$ converges to zero for all $\ell>p$.
• The asymptotic variance of $\hat{\phi}_{\ell, \ell}$ is $1 / T$ for $\ell>p$.
These results say that, for an $\operatorname{AR}(p)$ series, the sample PACF cuts off at lag $p$.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Information Criteria

There are several information criteria available to determine the order $p$ of an AR process. All of them are likelihood based. For example, the well-known Akaike Infor mation Criterion (Akaike, 1973) is defined as
$$A I C=\frac{-2}{T} \ln (\text { likelihood })+\frac{2}{T} \times(\text { number of parameters }),$$
where the likelihood function is evaluated at the maximum likelihood estimates and $T$ is the sample size. For a Gaussian AR $(\ell)$ model, AIC reduces to
$$\operatorname{AIC}(\ell)=\ln \left(\hat{\sigma}{\ell}^{2}\right)+\frac{2 \ell}{T},$$ where $\hat{\sigma}{\ell}^{2}$ is the maximum likelihood estimate of $\sigma_{a}^{2}$, which is the variance of $a_{t}$, and $T$ is the sample size; see Eq. (1.18). In practice, one computes AIC( $\ell$ ) for $\ell=0, \ldots, P$, where $P$ is a prespecified positive integer and selects the order $k$ that has the minimum AIC value. The second term of the AIC in Eq. (2.13) is called the penalty function of the criterion because it penalizes a candidate model by the number of parameters used. Different penalty functions result in different information criteria.

Table $2.1$ also gives the AIC for $p=1, \ldots, 10$. The AIC values are close to each other with minimum $-5.821$ occurring at $p=6$ and 9 , suggesting that an $A R(6)$ model is preferred by the criterion. This example shows that different approaches for order determination may result in different choices of $p$. There is no evidence to suggest that one approach outperforms the other in a real application. Substantive information of the problem under study and simplicity are two factors that also play an important role in choosing an AR model for a given time series.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Parameter Estimation

For a specified $\operatorname{AR}(p)$ model in Eq. (2.7), the conditional least squares method, which starts with the $(p+1)$ th observation, is often used to estimate the parameters. Specifically, conditioning on the first $p$ observations, we have
$$r_{t}=\phi_{0}+\phi_{1} r_{t-1}+\cdots+\phi_{p} r_{t-p}+a_{t}, \quad t=p+1, \ldots, T,$$
which can be estimated by the least squares method. Denote the estimate of $\phi_{i}$ by $\hat{\phi}{i}$. The fitted model is $$\hat{r}{t}=\hat{\phi}{0}+\hat{\phi}{1} r_{t-1}+\cdots+\hat{\phi}{p} r{t-p}$$
and the associated residual is
$$\hat{a}{t}=r{t}-\hat{r}{I^{*}}$$ The series $\left{\hat{a}{t}\right}$ is called the residual series, from which we obtain
$$\hat{\sigma}{a}^{2}=\frac{\sum{r=p+1}^{T} \hat{a}{t}^{2}}{T-2 p-1}$$ For illustration, consider an AR(3) model for the monthly simple returns of the valueweighted index in Table 2.1. The fitted model is $$r{t}=0.0103+0.104 r_{t-1}-0.010 r_{t-2}-0.120 r_{t-3}+\hat{a}{t}, \quad \hat{\sigma}{a}=0.054$$
The standard errors of the coefficients are $0.002,0.034,0.034$, and $0.034$, respectively. Except for the lag- 2 coefficient, all parameters are statistically significant at the $1 \%$ level.

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## MATLAB代写

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