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EC3060

Summer Examinations 2021/22

Econometrics 2: Time Series

Section A: Answer ONE question

1. (a) An AR(1) model was estimated on time-series data yt . The residuals from the regression were saved as res 1. Results of a post-estimation specifification test are reported in Table 1. Do the results suggest any specifification problem with the AR(1) model for yt? (10 marks)

(b) Defifine the Partial Autocorrelation Function (PAC). Using the regression results in Table 2 sketch the PAC for yt . (10 marks)

(c) Using the results from parts (a) and (b), what model would you propose for the series yt? What further tests or statistics would you compute to confifirm your intuition? (10 marks)

(i) What assumptions on zt  make this a standard GARCH process? (5 marks)

(ii) What features of conditional variance do GARCH models capture well and what are the main drawbacks of the assumptions made above for modelling high-frequency   financial data? (5 marks)

(iii)  Describe how you would estimate such a model. (15 marks)

(b)  Consider an example of the GARCH(2,1) variant:

gt  = 0y15 + 1y2∈t(2)_1 + 0y2∈t(2)_2 - 0yYgt_1

(i)  Find the ARMA process for the ∈t(2)  associated with this model of conditional variance, taking care to define all terms. (10 marks)

(ii)  Assume that the ∈t(2)  process had been at the unconditional variance for at least two periods, and that the conditional volatility forecast was also equal to its mean prior to the arrival of a shock to conditional volatility in period t + 1 which raised the

volatility 1 unit above the unconditional variance of the process .

Assuming no further shocks, how many periods will it be before the conditional

volatility returns to be less than the unconditional variance + unconditional standard deviation of the process? (15 marks)

Section B: Answer ONE question

3. Consider the process:

yt = c + Φ1yt−1 + Φ2yt−2 + ut   (1)

where yt is an n by 1 vector and E[utu 0 t ] = Σ, a symmetric positive-defifinite matrix.

(a) Find the unconditional mean of the process. Under what conditions does this moment exist? For full marks state the condition in terms of a matrix you can derive from (1). (10 marks)

(b) Show the k period ahead forecast of the process can be written as a weighted average of the unconditional mean and the last observations on the process. (15 marks)

Table 4: Tests of co-integrating rank

(d)  Discuss the identification problem in the VECM, and the nature of the restrictions        needed to achieve just-identification. In an empirical investigation, a researcher imposes a single over-identifying restriction which returns a o2  value of 3.83. Is this additional   restriction a good fit to the data? (10 marks)