### 统计代写|应用时间序列分析代写applied time series analysis代考|SIMPLE AUTOREGRESSIVE MODELS

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|SIMPLE AUTOREGRESSIVE MODELS

The fact that the monthly return $r_{I}$ of CRSP value-weighted index has a statistically significant lag-l autocorrelation indicates that the lagged return $r_{t-1}$ might be useful in predicting $r_{t}$. A simple model that makes use of such predictive power is
$$r_{t}=\phi_{0}+\phi_{1} r_{t-1}+a_{l}$$
where $\left{a_{t}\right}$ is assumed to be a white noise series with mean zero and variance $\sigma_{a}^{2}$. This model is in the same form as the well-known simple linear regression model in

which $r_{I}$ is the dependent variable and $r_{t-1}$ is the explanatory variable. In the time series literature, Model (2.6) is referred to as a simple autoregressive (AR) model of order 1 or simply an AR(1) model. This simple model is also widely used in stochastic volatility modeling when $r_{t}$ is replaced by its log volatility; see Chapters 3 and $10 .$

The AR(1) model in Eq. (2.6) has several properties similar to those of the simple linear regression model. However, there are some significant differences between the two models, which we discuss later. Here it suffices to note that an $\mathrm{AR}(1)$ model implies that, conditional on the past return $r_{t-1}$, we have
$$E\left(r_{t} \mid r_{t-1}\right)=\phi_{0}+\phi_{1} r_{t-1}, \quad \operatorname{Var}\left(r_{t} \mid r_{t-1}\right)=\operatorname{Var}\left(a_{t}\right)=\sigma_{a}^{2}$$
That is, given the past return $r_{t-1}$, the current return is centered around $\phi_{0}+\phi_{1} r_{t-1}$ with variability $\sigma_{a}^{2}$. This is a Markov property such that conditional on $r_{t-1}$, the return $r_{t}$ is not correlated with $r_{t-i}$ for $i>1$. Obviously, there are situations in which $r_{t-1}$ alone cannot determine the conditional expectation of $r_{t}$ and a more flexible model must be sought. A straightforward generalization of the AR(1) model is the $\mathrm{AR}(p)$ model
$$r_{t}=\phi_{0}+\phi_{1} r_{t-1}+\cdots+\phi_{p} r_{t-p}+a_{t}$$
where $p$ is a non-negative integer and $\left{a_{t}\right}$ is defined in Eq. (2.6). This model says that the past $p$ values $r_{l-i}(i=1, \ldots, p)$ jointly determine the conditional expectation of $r_{t}$ given the past data. The $\operatorname{AR}(p)$ model is in the same form as a multiple linear regression model with lagged values serving as the explanatory variables.

## 统计代写|应用时间序列分析代写applied time series anakysis代考| AR(1) Model

We begin with the sufficient and necessary condition for weak stationarity of the AR(1) model in Eq. (2.6). Assuming that the series is weakly stationary, we have $E\left(r_{t}\right)=\mu, \operatorname{Var}\left(r_{t}\right)=\gamma_{0}$, and $\operatorname{Cov}\left(r_{t}, r_{t-j}\right)=\gamma_{j}$, where $\mu$ and $\gamma_{0}$ are constant and $\gamma_{j}$ is a function of $j$, not $t$. We can easily obtain the mean, variance, and autocorrelations of the series as follows. Taking the expectation of Eq. (2.6) and because $E\left(a_{t}\right)=0$, we obtain
$$E\left(r_{t}\right)=\phi_{0}+\phi_{1} E\left(r_{I-1}\right)$$
Under the stationarity condition, $E\left(r_{t}\right)=E\left(r_{I-1}\right)=\mu$ and hence
$$\mu=\phi_{0}+\phi_{1} \mu \quad \text { or } \quad E\left(r_{t}\right)=\mu=\frac{\phi_{0}}{1-\phi_{1}}$$

This result has two implications for $r_{t}$. First, the mean of $r_{I}$ exists if $\phi_{1} \neq 1$. Second, the mean of $r_{t}$ is zero if and only if $\phi_{0}=0$. Thus, for a stationary AR(1) process, the constant term $\phi_{0}$ is related to the mean of $r_{t}$ and $\phi_{0}=0$ implies that $E\left(r_{t}\right)=0$.
Next, using $\phi_{0}=\left(1-\phi_{1}\right) \mu$, the AR(1) model can be rewritten as
$$r_{t}-\mu=\phi_{1}\left(r_{t-1}-\mu\right)+a_{t}$$
By repeated substitutions, the prior equation implies that
\begin{aligned} r_{t}-\mu &=a_{t}+\phi_{1} a_{t-1}+\phi_{1}^{2} a_{t-2}+\cdots \ &=\sum_{i=0}^{\infty} \phi_{1}^{i} a_{t-i} \end{aligned}
Thus, $r_{t}-\mu$ is a linear function of $a_{t-i}$ for $i \geq 0$. Using this property and the independence of the series $\left{a_{t}\right}$, we obtain $E\left[\left(r_{t}-\mu\right) a_{t+1}\right]=0$. By the stationarity assumption, we have $\operatorname{Cov}\left(r_{t-1}, a_{t}\right)=E\left[\left(r_{t-1}-\mu\right) a_{t}\right]=0$. This latter result can also be seen from the fact that $r_{t-1}$ occurred before time $t$ and $a_{t}$ does not depend on any past information. Taking the square, then the expectation of Eq. (2.8), we obtain
$$\operatorname{Var}\left(r_{t}\right)=\phi_{1}^{2} \operatorname{Var}\left(r_{t-1}\right)+\sigma_{a}^{2},$$
where $\sigma_{a}^{2}$ is the variance of $a_{l}$ and we make use of the fact that the covariance between $r_{t-1}$ and $a_{t}$ is zero. Under the stationarity assumption, $\operatorname{Var}\left(r_{t}\right)=\operatorname{Var}\left(r_{t-1}\right)$, so that
$$\operatorname{Var}\left(r_{I}\right)=\frac{\sigma_{a}^{2}}{1-\phi_{1}^{2}}$$
provided that $\phi_{1}^{2}<1$. The requirement of $\phi_{1}^{2}<1$ results from the fact that the variance of a random variable is bounded and non-negative. Consequently, the weak stationarity of an AR(1) model implies that $-1<\phi_{1}<1$. Yet if $-1<\phi_{1}<1$, then by Eq. $(2.9)$ and the independence of the $\left{a_{t}\right}$ series, we can show that the mean and variance of $r_{t}$ are finite. In addition, by the Cauchy-Schwartz inequality, all the autocovariances of $r_{t}$ are finite. Therefore, the $\mathrm{AR}(1)$ model is weakly stationary. In summary, the necessary and sufficient condition for the AR(1) model in Eq. (2.6) to be weakly stationary is $\left|\phi_{1}\right|<1$.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Autocorrelation Function of an AR(1) Model

Multiplying Eq. (2.8) by $a_{t}$, using the independence between $a_{t}$ and $r_{t-1}$, and taking expectation, we obtain
$$E\left[a_{t}\left(r_{t}-\mu\right)\right]=E\left[a_{t}\left(r_{t-1}-\mu\right)\right]+E\left(a_{t}^{2}\right)=E\left(a_{t}^{2}\right)=\sigma_{a}^{2}$$

where $\sigma_{a}^{2}$ is the variance of $a_{t}$. Multiplying Eq. $(2.8)$ by $\left(r_{t-\ell}-\mu\right)$, taking expectation, and using the prior result, we have
$$\gamma_{\ell}= \begin{cases}\phi_{1} \gamma_{1}+\sigma_{a}^{2} & \text { if } \ell=0 \ \phi_{1} \gamma_{\ell-1} & \text { if } \ell>0\end{cases}$$
where we use $\gamma_{\ell}=\gamma_{-\ell}$. Consequently, for a weakly stationary AR(1) model in Eq. (2.6), we have
$$\operatorname{Var}\left(r_{t}\right)=\gamma_{0}=\frac{\sigma^{2}}{1-\phi_{1}^{2}}, \quad \text { and } \quad \gamma_{\ell}=\phi_{1} \gamma_{\ell-1}, \quad \text { for } \quad \ell>0$$
From the latter equation, the ACF of $r_{t}$ satisfies
$$\rho_{\ell}=\phi_{1} \rho_{\ell-1}, \quad \text { for } \quad \ell \geq 0$$
Because $\rho_{0}=1$, we have $\rho_{\ell}=\phi_{1}^{\ell}$. This result says that the ACF of a weakly stationary AR(1) series decays exponentially with rate $\phi_{1}$ and starting value $\rho_{0}=1$. For a positive $\phi_{1}$, the plot of ACF of an AR(1) model shows a nice exponential decay.

## 统计代写|应用时间序列分析代写applied time series anakysis代考|SIMPLE AUTOREGRESSIVE MODELS

r吨=φ0+φ1r吨−1+一种l

r吨=φ0+φ1r吨−1+⋯+φpr吨−p+一种吨

## 统计代写|应用时间序列分析代写applied time series anakysis代考| AR(1) Model

μ=φ0+φ1μ 或者 和(r吨)=μ=φ01−φ1

r吨−μ=φ1(r吨−1−μ)+一种吨

r吨−μ=一种吨+φ1一种吨−1+φ12一种吨−2+⋯ =∑一世=0∞φ1一世一种吨−一世

## 统计代写|应用时间序列分析代写applied time series anakysis代考|Autocorrelation Function of an AR(1) Model

Cℓ={φ1C1+σ一种2 如果 ℓ=0 φ1Cℓ−1 如果 ℓ>0

ρℓ=φ1ρℓ−1, 为了 ℓ≥0

## 广义线性模型代考

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## MATLAB代写

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