## 经济代写|计量经济学代写Econometrics代考|EFN508

2023年3月28日
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## 经济代写|计量经济学代写Econometrics代考|Engle’s ARCH test∗

So far we have looked for the presence of autocorrelation in the error terms of a regression model. Engle (1982) introduced a new concept allowing for autocorrelation to occur in the variance of the error terms, rather than in the error terms themselves. To capture this autocorrelation Engle developed the autoregressive conditional heteroskedasticity (ARCH) model, the key idea behind which is that the variance of $u_t$ depends on the size of the squared error term lagged one period (that is $u_{t-1}^2$ ).
More analytically, consider the regression model:
$$Y_t=\beta_1+\beta_2 X_{2 t}+\beta_3 X_{3 t}+\cdots+\beta_k X_{k t}+u_t$$
and assume that the variance of the error term follows an $\mathrm{ARCH}(1)$ process:
$$\operatorname{Var}\left(u_t\right)=\sigma_t^2=\gamma_0+\gamma_1 u_{t-1}^2$$
If there is no autocorrelation in $\operatorname{Var}\left(u_t\right)$, then $\gamma_1$ should be zero and therefore $\sigma_t^2=\gamma_0$. So there is a constant (homoskedastic) variance.
The model can easily be extended for higher-order $\operatorname{ARCH}(p)$ effects:
$$\operatorname{Var}\left(u_t\right)=\sigma_t^2=\gamma_0+\gamma_1 u_{t-1}^2+\gamma_2 u_{t-2}^2+\cdots+\gamma_p u_{t-p}^2$$

Here the null hypothesis is:
$$H_0: \quad \gamma_1=\gamma_2=\cdots=\gamma_p=0$$
that is, no ARCH effects are present. The steps involved in the ARCH test are:
Step 1 Estimate Equation (6.24) by OLS and obtain the residuals, $\hat{u}t$. Step 2 Regress the squared residuals $\left(u_t^2\right)$ against a constant, $u{t-1}^2, u_{t-2}^2, \ldots, u_{t-p}^2$ (the value of $p$ will be determined by the order of $\operatorname{ARCH}(p)$ being tested for).

Step 3 Compute the $L M$-stat $=(n-p) R^2$ from the regression in step 2. If $L M>\chi_p^2$ for a given level of significance, reject the null of no ARCH effects and conclude that ARCH effects are indeed present.

## 经济代写|计量经济学代写Econometrics代考|what is autocorrelation?

We know that the use of OLS to estimate a regression model leads us to BLUE estimates of the parameters only when all the assumptions of the CLRM are satisfied. In the previous chapter we examined the case where assumption 5 does not hold. This chapter examines the effects on the OLS estimators when assumption 6 of the CLRM is violated.

Assumption 6 of the CLRM states that the covariances and correlations between different disturbances are all zero:
$$\operatorname{Cov}\left(u_t, u_s\right)=0 \quad \text { for all } t \neq s$$
This assumption states that the error terms $u_t$ and $u_s$ are independently distributed, termed serial independence. If this assumption is no longer true then the disturbances are not pairwise independent, but are pairwise autocorrelated (or serially correlated). In this situation:
$$\operatorname{Cov}\left(u_t, u_s\right) \neq 0 \text { for some } t \neq s$$
which means that an error occurring at period $t$ may be correlated with one at period $s$. Autocorrelation is most likely to occur in a time series framework. When data are arranged in chronological order, the error in one period may affect the error in the next (or other) time period(s). It is highly likely that there will be intercorrelations among successive observations, especially when the interval is short, such as daily, weekly or monthly frequencies, compared to a cross-sectional data set. For example, an unexpected increase in consumer confidence can cause a consumption function equation to underestimate consumption for two or more periods. In cross-sectional data, the problem of autocorrelation is less likely to exist because we can easily change the arrangement of the data without meaningfully altering the results. (This is not true in the case of spatial autocorrelation, but this is beyond the scope of this text.)

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Engle’s ARCH test∗

$$Y_t=\beta_1+\beta_2 X_{2 t}+\beta_3 X_{3 t}+\cdots+\beta_k X_{k t}+u_t$$

loperatorname ${V a r} \backslash$ left(u_ttright)=Isigma_ $^{\wedge} 2=$ Igamma_0+Igamı u_{t-1 $}^{\wedge} 2$ 如果loperatorname{Var $}$ Neft(u_ttright)
$$\operatorname{Var}\left(u_t\right)=\sigma_t^2=\gamma_0+\gamma_1 u_{t-1}^2$$

loperatorname ${V a r}\left|l e f t\left(u _t t r i g h t\right)=\right| s i g m a _t \wedge 2=$ Igamma_0+Igamı $u_{-}{t-1}^{\wedge} 2$ + Igamma_2 $u_{-}{t-2}^{\wedge} 2+$ lcdots + lgamma_p $\mathrm{u}{-}{\mathrm{tp}}^{\wedge} 2 \operatorname{Var}\left(u_t\right) \gamma_1 \sigma_t^2=\gamma_0$ $\mathrm{ARCH}(p)$ $\operatorname{Var}\left(u_t\right)=\sigma_t^2=\gamma_0+\gamma_1 u{t-1}^2+\gamma_2 u_{t-2}^2+\cdots+\gamma_p u_{t-p}^2$

$$H_0: \quad \gamma_1=\gamma_2=\cdots=\gamma_p=0$$

$$=(n-p) R^2 L M>\chi_p^2$$

## 经济代写|计量经济学代写Econometrics代考|what is autocorrelation?

CLRM 的假设 6 指出不同干扰之间的协方差和相关性都为 零：这个假设指出误差项和独立分布，称为串行独立性。 如果这个假设不再成立，那么干扰就不是成对独立的，而 是成对自相关的（或序列相关的）。在这种情况下: 这意 味着在时间段的错误相关
$\operatorname{Cov}\left(u_t, u_s\right)=0 \quad$ for all $t \neq s$
$u_t u_s$
$\operatorname{Cov}\left(u_t, u_s\right) \neq 0$ for some $t \neq s$
$t s$. 自相关最有可能发生在时间序列框架中。当数据按时 间顺序排列时，一个时期的错误可能会影响下一个 (或其 他) 时期的错误。与横截面数据集相比，连续观察之间很 可能存在相互关联，尤其是当间隔很短时，例如每天、每 周或每月的频率。例如，消费者信心的意外增加可能导致 消费函数方程低估两个或多个时期的消费。在横截面数据 中，自相关问题不太可能存在，因为我们可以很容易地改 变数据的排列，而不会有意义地改变结果。（在空间自相 关的情况下情况并非如此，但这超出了本文的范围。)

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## MATLAB代写

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