The autocorrelation functions
QUESTION
Page 2
ICM204 2020/21 A 800
1. (a) Derive the autocorrelation functions (ACFs) for the following two
variables:
= + 1 −1 + 2 −2 +
and:
= + + 1−1
Assume the processes are stationary, and the 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
is given by:
= 1.9−1 − 0.9−2 + 0.5 +
,
where
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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Page 3
ICM204 2020/21 A800
2. Suppose we have the simultaneous equations model considered in
the lectures,
= + + +
= + + +
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.
Sample Solutions
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Get sample solution
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