Derive the Autoregressive Polynomial – Engineering Assignment Help

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Assignment Task :
Question 1
(a) A researcher wished to investigate the relationship between two time series, which we denote by x1 and x2.The researcher estimated the following VAR(3),
Dimension
 
(b)
i) Derive the autoregressive polynomial associated with the VAR in (4)
ii) How many roots does the autoregressive polynomial have?
iii) What is the condition for the VAR to be stationary?
(c) The researcher wished to test the null hypothesis that the VAR order is 2 against the alternative that it is 3.
i) What are the restrictions on the coefficients in the equation for dx1 implied by this null hypothesis?
ii) When a sample of 200 observations was used to estimate the restricted and unrestricted models associated with the null hypothesis the following results were obtained:
Hypothesis
 
where LU and LR denote the maximized value of the log-likelihood function for the unrestricted and restricted models respectively. Use this information to test the
null hypothesis that the VAR order is 2 against the alternative that it is 3. In your answer identify the null and alternative hypotheses, the form and sampling distribution of the test statistic, it’s sample and critical values and your test conclusion.
(d) Assume that the VAR order is 2.
i) What are the restrictions on the parameters of the model implied by the null hypothesis that dx2 does not Granger-cause dx1?
ii) When a sample of 200 observations was used to estimate the restricted and unrestricted models associated with the null hypothesis that dx2 does not
 
 
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