The seasonal component of a time series is the same as the cyclical component.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1 (10 marks)

Indicate whether the following statements are TRUE or FALSE and expand on your answer. 

Q1A.

 The seasonal component of a time series is the same as the cyclical component.

 TRUE FALSE

(2 marks)

 Q1B.

 The residual component of a time series represents the part of the data that cannot be explained by the trend or seasonal components.

 TRUE FALSE

(4 marks)

Q1C.

 

The Error Correction Model (ECM) assumes that there is a stable long-run relationship between the variables being modelled.

 

 

 

TRUE FALSE

 

 

(4 marks)

 Question 2 (12 marks)

For each of the questions below, choose the correct answer(s) and provide justification in terms of why the other answer(s) are incorrect.

Q2A. Which of the following are possible consequences of overfitting an ARIMA model?

i. The model may have high bias.

ii. The model may have high out-of-sample variance.

iii. The model may have poor predictive performance.

iv. All of the above.

(4 marks)

Q2B. Which of the following are limitations of using Impulse Response Functions (IRFs) in time series analysis?

i. IRFs are only valid for linear time series models.

ii. IRFs assume that the underlying time series is stationary.

iii. IRFs can provide information about the short-term dynamics of the relationship between variables, but they do not capture longer-term effects or other important aspects of the relationship.

iv. IRFs depend on the specification of the model used to estimate the relationship between variables.

(4 marks)

 Q2C. Considering the following 𝑀𝐴(3) process:

𝑦𝑡 = 𝑢𝑡 − 0.7𝑢𝑡"1 − 0.2𝑢𝑡"$ + 0.4𝑢𝑡"3

Where 𝑢𝑡 is a white noise process with variance equal to 1. What is the value 𝛾𝑘 = 𝑐𝑜𝑣(𝑦𝑡, 𝑦𝑡"𝑘) for 𝑘 = 0?

Provide the correct answer along with the working steps and underlying assumptions used to calculate the value of 𝛾𝑘 = 𝑐𝑜𝑣(𝑦𝑡, 𝑦𝑡"𝑘) for 𝑘 = 0.

i. 𝛾0 = 1.69

ii. 𝛾0 = 0.63

iii. 𝛾0 = 1

iv. 𝛾0 = 0

(4 marks) 

Question 3 (18 marks)

Q3A. Assuming that Assumption 4 (Zero Conditional and Unconditional Mean) holds, what is the value of 𝑐𝑜𝑟𝑟(𝑥, 𝑒)? Provide the working steps and underlying assumptions used to arrive at your answer.

(6 marks)

Q3B. Assess whether the following 𝐴𝑅(1) process satisfies the criteria for covariance stationarity, taking into consideration all three conditions that define a covariance stationary process. Assume that 𝑢𝑡~𝑖. 𝑖. 𝑑(0, 𝜎$), where 𝜎$ = 1. Provide the working

steps and underlying assumptions used to prove if each property holds.

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