TIME SERIES ECONOMETRICS
1. Answer both parts (a) and (b) of this question. Consider the following time series regression model:
(1)
where 
, 
, and are scalar random variables and 
is a constant parameter. We suppose the following conditions:
C1) is an i.i.d. sequence of scalar random variables with zero mean, variance 2
, and finite moments (of any order);
C2) 
and are independent; is an AR(1) process:
¡1 (2)
where is a constant parameter; is an i.i.d. sequence with zero mean, variance
2 , and finite moments (of any order ); and if , the distribution of the initial variable 0 is defined so that becomes strictly stationary, while if ,
0 .
We suppose the availability of a dataset =1 and correct specification of the model (1) and Conditions (C1) (C2).
(a) 
[25 marks] Suppose that
and consider the OLS estimator of 
:
(3)
i. 
Derive the asymptotic distribution of 
. You need to explain each step of your derivation and need to write down the limit (normal) distribution of the normalized estimator, in terms of parameters 
, 2 , and 2 (the same remark applies to part
(b) below).
ii. Given the result in part (a.i), discuss whether you need to implement heteroskedas ticity and autocorrelation consistent (HAC) estimation (i.e., long run variance esti mation) for constructing an asymptotic confidence interval. Provide a specific rea son for your answer.
(b) 
[25 marks] Suppose that and consider estimation of the mean of , , by the following estimator:
i. 
Derive the asymptotic distribution of .
ii. 
Explain how to construct a 95% confidence interval for in practice.
2. Answer both parts (a) and (b) of this question.
We take the same model (1) as in Question 1 and suppose the same conditions (C1) and (C2) with
We also suppose the availability of a dataset 
and the correct specification.
(a) [30 marks]
i. 
Derive the asymptotic distribution of
and 2.
ii. 
Discuss the consistency of .
, in terms of parameters 
, 2 ,
(b) 
[20 marks] Now, an econometrician wants to estimate the regression model. However, she does not have full confidence on the specification (1) of 
and considers the following model:
as well as the following OLS estimators for and :
and 
i. 
Prove the consistency of for 
. You may assume the boundedness of the sequence 
(i.e., for all , ( is some constant independent of ), if it helps your arguments.
ii. Derive/investigate the convergence rate of (for this purpose, it will be helpful to derive the asymptotic distribution).
iii. 
Given the result in part (b.ii), discuss the consistency property of 
for , and compare it with the consistency property of in part (a). Your conclusion here should be different from that in part (a). 
explain why it is different.
3. Answer both parts (a) and (b) of this question.
Let be a time series process that is strong mixing and strictly stationary. We consider the following AR(1) type model:
¡1 ¡1 (4)
where is i.i.d. with being independent of all past variables ¡1 ¡1 ¡2 ¡2 ; and is a continuous function of parametrized by 
(the form of is
known to researchers). We let 
.
(a) [25 marks]
i. 
Given the model (4), write down the conditional probability density function (PDF) of given ¡1 , , and set up the associated (conditional) loglikelihood function (given a set of observations =0), where the PDF of 
is
given by
ii. Let
Suppose that
¡1
0 implies 
0 for some . (5)
In this case, prove that if 
0 
0 0 
, then
is not a constant.
(b) [25 marks]
i. Suppose the correct specification, i.e., (4) is the true data generating mechanism of 
. Then, prove that the condition (5) implies that the true parameter 0 is a unique maximiser of .
ii. 
Suppose that the model (4) is not correctly specified (for example, the AR(1) struc ture is not correct or may not be normally distributed). In this case, 
explain what interpretation we can have for the maximiser of .
4. Answer both parts (a) and (b) of this question.
Let
0 be an vector of variables. We suppose the stationarity of and model it as an autoregressive process of order , VAR 
, given by
1 ¡1 ¡ ,
where is an vector of intercepts, 1 are matrices of unknown parame ters, and is an vector white noise process with mean vector zero and variance matrix
. It is assumed that a sample of observations, =1, is available.
(a) [30 marks] The general form of the Akaike Information Criterion (AIC) is given by
where 
denotes the maximised likelihood function for a VAR 
with estimated parameter vector 
, and 
is the number of estimated parameters in the VAR 
.
i. 
explain how can be used to determine the value of . What issues arise in the choice of (sub) samples when estimating the VAR for different values of ?
ii. If you use the , what property is the selected model expected to have (among candidate models)? Provide explanations.
iii. 
For now, suppose that , (i.e., is the bivariate VAR 
process:
1 ¡1 ), and
1 (6)
Is this a stationary process? 
explain your answer.
iv. 
If your answer is Yes in part (a.iii), write down the moving average representation of in terms of 1, , and , and, given (6), compute the 2step impulse re sponse matrix. If your answer is No, 
explain why such a moving average representation does not exist.
(b) 
[20 marks] The VAR
can be interpreted as the reduced form corresponding to a struc tural VAR (SVAR) of order , given by
0 1 ¡1 ¡ ,
where is an vector of intercepts, 0 
1 are matrices of unknown parameters, and is an vector white noise process with mean vector zero and variance matrix .
i. What condition is required in order to derive the reduced form VAR from the SVAR? Assuming this condition is satisfied, show how the parameters of the structural and reduced form VARs are related.
ii. According to the order condition, how many additional restrictions are required in order to identify the parameters of the SVAR? Provide examples of restrictions that are commonly imposed in practice.
iii. 
Suppose , , and . In which of the following specifications are the
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