The autocorrelation functions (ACFs)

1. (a) Derive the autocorrelation functions (ACFs) for the following two
   variables:
   𝑦𝑡 = 𝜇 + 𝛼1 𝑦𝑡−1 + 𝛼2 𝑦𝑡−2 + 𝜀𝑡
   and:
   𝑥𝑡 = 𝛾 + 𝜀𝑡 + 𝜃1𝜀𝑡−1
   Assume the processes are stationary, and the 𝜀𝑡 are serially
   uncorrelated disturbances.
   What are the key differences between the ACFs?
   (25 marks)
   (b) Explain how the ACF and the partial autocorrelation function
   (PACF) can be used together to identify an appropriate model for a
   time series.
   (25 marks)
   (c) A variable 𝑥𝑡
   is given by:
   𝑥𝑡 = 1.9𝑥𝑡−1 − 0.9𝑥𝑡−2 + 0.5𝑡 + 𝑒𝑡
   ,
   where 𝑒𝑡
   is white noise, and t is a time trend.
   Is the variable I(0), I(1) or I(2)?
   If you took the first difference of the variable, would it be
   stationary? Explain your answer.
   (25 marks)
   (d) Derive the general h-step ahead forecast for a stationary AR(1)
   model with a non-zero mean, and compare this to the h-step
   ahead forecast for an AR(1) with a unit root, and comment on the
   differences between the two. Derive the forecast error variances.
   (25 marks)
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2. Suppose we have the simultaneous equations model considered in
   the lectures,
   𝑄 = 𝛼 + 𝛽𝑃 + 𝛾𝑆 + 𝑢
   𝑄 = 𝜆 + 𝜇𝑃 + 𝜅𝑇 + 𝑣
   which is viewed as simultaneously determining price (P) and output
   (Q), while S and T are exogenous.
   (a) Write down the reduced form equations for P and Q, and hence
   show that the disturbances and explanatory variables in the
   structural form are correlated.
   (25 marks)
   (b) Explain why the parameters of the simultaneous equations
   model cannot be estimated consistently by OLS.
   (25 marks)
   (c) What does it mean for an equation (of a system of
   simultaneous equations) to be over-identified?
   (10 marks)
   (d) Describe in detail a suitable estimation procedure for an overidentified equation, and explain why this estimator is consistent.
   (40 marks)
3. (a) Derive the maximum likelihood (ML) estimators of α, β and σ2
   in the model
   𝑦𝑡 = 𝛼 + 𝛽𝑥𝑡 + 𝜀𝑡
   where εt is a homoscedastic, independently normally distributed
   disturbance term (with variance σ2
   ).
   (25 marks)
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   ICM204 2020/21 A 800
   (b) Derive the OLS estimators of the same three parameters and
   compare the OLS and ML estimators. Are they consistent
   estimators?
   (25 marks)
   (c) Describe how ML can be used to estimate the unknown
   parameters when instead of εt being homoscedastic it is assumed
   to follow an GARCH(1,1) process.
   (25 marks)
   (d) How could we test whether we need to use a GARCH model,
   as opposed to a simpler ARCH model? Describe as fully as you
   can, and mention which testing principle is being employed.
   (25 marks)
4. (a) Explain how a VAR model can be used to investigate
   cointegration. You should show how the VAR model can be rewritten when the variables are cointegrated, and the restrictions
   implied by cointegration on the original VAR.
   (20 marks)
   (b) Outline a test for the number of cointegrating vectors based on
   the VAR model approach.
   (20 marks)

(c) What would we have to assume about the orders of integration
of the individual variables if we found as many cointegrating
vectors as variables in the VAR, and why?
(20 marks)
(d) Do you think (S)VAR models successfully avoid some of the
difficulties that arise in Simultaneous equations models because of
the need to have identifying restrictions? Can we learn about
causal effects without using theory-based restrictions? Discuss as
fully as you can.
(40 marks)
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