# 统计代写|时间序列分析代写Time-Series Analysis代考|STA457H1

## 统计代写|时间序列分析代写Time-Series Analysis代考|Granger causality

Causality is a kind of cause and effect relationship. Granger causality finds the precedence of another variable. If $\mathrm{X}$ and $\mathrm{Y}$ are two economics variables, Granger causality can test if $\mathrm{X}$ variable precedence (occurs before) Y variable or vice versa. Granger causality can measure the causality using probability. Assumption of the Granger causality test is that variables are independent.
Null hypothesis for Granger Causality $\mathrm{H}0: \mathrm{x}(\mathrm{t})$ doesn’t Granger-cause $\mathrm{y}(\mathrm{t})$ $\mathrm{H}_1: \mathrm{x}(\mathrm{t})$ does Granger-cause $y(\mathrm{t})$ Null hypothesis means lagged $\mathrm{x}$-values do not explain the variation in y. ganger causality test can be done for a selected number of lags. Lags are selected according to model order selection method. Before applying this test we should make sure if the time series are stationary and do not have any unit root. Below is the test statistic of Granger causality $$F=\frac{\left(\mathrm{ESS}{\mathrm{R}}-\mathrm{ESS}{\mathrm{UR}}\right) / q}{\mathrm{ESS}{\mathrm{UR}} /(n-k)}$$
If there is a large number of variables and lag orders, then using chitest with likelihood ratio or Wald test is better than using $\mathrm{F}$ test.
The long-run relationship between variables, indicate that there is Granger causality at least for one direction. There are unidirectional and bidirectional Granger Causality.

## 统计代写|时间序列分析代写Time-Series Analysis代考|ARDL bound test

ARDL bound test is based on the assumption that the variables are I(0) or I(1). Variables shouldn’t be I(2). If variables are I(2), we cannot interpret the values of statistics provided by Pesaran et el. Bound test which was developed by Pesaran et al in 2001 can be used to identify long term relationship among the integrated parameters. Long term relationship is identified by comparing Bound $\mathrm{F}$ test against upper and lower critical values provided for different significance levels. (In this book the preferred significance level for all statistical tests is $5 \%$ ). If the $\mathrm{F}$ statistic is higher than critical value I (1) then there is a long run co-integration and the null hypothesis of “no co-integration among variables/no long run relationship among variables” is rejected. If the F statistic is lower than I (0) the model has lack of evidence to identify a long run relationship among the variables.

General ARDL model equation is as below :
\begin{aligned} & y_t=B_0 X_t+B_1 x_{t-1}+B_2 X_{t-2}+\ldots+B_k X_{t-k}+e_t \ & y=\text { Indegenous variable } \ & x=\text { exogenous variable } \ & B=\text { coefficients } \ & \mathrm{t}-\mathrm{k}=\text { lag } \ & \mathrm{c}=\text { crror } \end{aligned}
ARDL (1,1) Model which express both indigenous and exogenous in short run can be represented as below . (In short run y is considered to be stable).
$$y_t=a_0+a_1 y_{t-1}+B_0 x_t+B_2 x_{t-1}+e_t \quad \text { where } t=1,2, \ldots t$$
$\mathrm{y}=$ Indegenous variable
$\mathrm{x}=$ exogenous variable
$a=-1<a<1$ – the speed of adjustment towards the long run model

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|Granger causality

F=(和小号小号R−和小号小号在R)/q和小号小号在R/(n−k)

## 统计代写|时间序列分析代写Time-Series Analysis代考|ARDL bound test

ARDL 绑定测试基于变量为 I(0) 或 I(1) 的假设。变量不应该是 I(2)。如果变量是 I(2)，我们无法解释 Pesaran 等人提供的统计值。Pesaran 等人于 2001 年开发的边界检验可用于识别综合参数之间的长期关系。通过比较 Bound 来确定长期关系F针对不同显着性水平提供的上限和下限临界值进行测试。（在本书中，所有统计检验的首选显着性水平是5%). 如果F统计量高于临界值 I (1) 则存在长期协整，拒绝“变量间无协整/变量间无长期关系”的原假设。如果 F 统计量低于 I (0)，则该模型缺乏证据来确定变量之间的长期关系。

ARDL (1,1) 模型在短期内既表达了本土的又表达了外生的，可以表示如下。（在短期内 y 被认为是稳定的）。

X=外生变量

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