ADVANCED TIME SERIES MODELLING
高级时间序列代写 If you attempt more questions than required, only the required number of answers will be marked. Please strike through any answers that you do not
DURATION: 120 MINUTES (2 HOURS)
This paper contains FOUR questions.
Answer TWO questions in total.
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(a) A researcher at the World Bank was investigating if Economic Policy Uncertainty in the UK is non-stationary. He obtained the following results (see Table 1 below):
where Economic Policy Uncertainty is denoted as EPU, C denotes an intercept term, DEPU denotes the first difference of EPU and negative figures in brackets denote a lag of the specified length.
Using the above results, discuss in detail the augmented Dickey-Fuller test, illustrating your answer by testing for the presence of a unit root in EPU at the 1% and 5% level of significance. [25 marks]
(b) Consider the following process for EPU (Economic Policy
EPUt = m + bt + rEPUt–1 + et (Equation 1)
where E[et] = 0 and V[et] = σ2.
(i) Derive E[EPUt] when ρ =1 and m ≠ 0, b ≠ 0.
(ii) Derive V[EPUt] when ρ =1 and b = 0.
(iii) With respect to the results you have derived in (i) and (ii) discuss any implications they have for the properties and the statistical analysis of the variable EPUt. [30 marks]
(c) Provide a detailed account of each of the following tests, clearly defining their underlying motivation and method of application:
(i) The GLS-based Dickey-Fuller unit root test.
(ii) KPSS test for stationarity of a time series. [45 marks]
(a) In what way is Johansen’s cointegration method different from Engle-Granger’s two-step method? Provide a detailed account of the Johansen’s procedure for testing cointegration in economic time series. [50 marks]
(b) An economist at the Bank of England is interested in testing whether Financial Uncertainty (FU) co-evolves with stock return (R) and inflation (P). He applied Johansen procedure to the three variables system (of FU, R, and P), for an effective sample of 300 monthly observations for the USA, which produced the results in Table 2 below:
With reference to the results in Table 2, discuss in detail the Johansen procedure and how it can be employed to examine relationships between economic variables. State the missing terms denoted as ‘X’ and determine how many cointegrating relationships can be detected at the 5% level of significance using the maximal and trace tests. [50 marks]
A financial sector expert at the IMF is interested in investigating if Macroeconomic Uncertainty (MU) affects Bitcoin Price movements (P) in the case of the USA. He ran the following linear regression:
(a) What type of process the expert is looking into by taking Equation 2 and Equation 3 together? What possible reasons the expert might have in mind which give rise to the above processes? [20 marks]
(b) State both econometric and financial economic implications when in Equation 3 above d1 =1. What transformation technique would the expert adopt in this situation to achieve stability in the relationship between current and past variances? [25 marks]
(c) Between ‘conditional variance’ and ‘unconditional variance’, which one is more likely to be useful for producing 1-stepahead and 20-step-ahead volatility forecasts. [10 marks]
(d) Theory suggests that shocks often leave asymmetric impact on financial time series. In relation to this proposition, present the framework and provide an in-depth discussion of the distinguishing properties of Threshold GARCH (TGARCH(1,1)) from a simple GARCH(1,1) process. [30 marks]
(e) The IMF expert wanted to understand if bad news affected variance of Bitcoin returns in the USA more than the good news. He estimated an asymmetric GARCH which produced the following results (in Table 3):
In Table 3, comment on the significance of the each coefficient at 5% levels of significance. Interpret the sign of the asymmetric term. Is it what one would normally expect in a GARCH estimation of stock returns? [15 marks]
(a) A financial analyst is faced with the following two scenarios for stock price increment for a firm:
(i) Yt = Yt-1 + at.
(ii) Yt = 1.097 Yt-1 – 0.97 Yt-2 + at ,.
where Y: stock price increment. a: White noise with zero mean and constant variance σ2=100. t: Time (quarters starting with Q1,1993). Check by an algebraic criterion which one is stationary. [40 marks]
(b) Suppose that an MA(1) model is xt = 10 + wt + 0.7wt-1, where wt∼iidN(0,1). Derive and plot the ACF of this process. [30 marks]
(c) Suppose that an ARMA(1,1) model is given by:
xt = 7 + 0.5 xt-1 + wt + 0.5wt-1,
where wt∼iidN(0,1). Derive and plot the ACF of this process. [30 marks]