1.       Answer both parts (a) and (b) of this question. Consider the following time series regression model:

img1         (1)

img2where img3, img4, and        are scalar random variables and img5 is a constant parameter. We suppose the following conditions:

img6img7C1)        is an i.i.d.  sequence of scalar random variables with zero mean, variance        2

, and finite moments (of any order);

img8C2)  img9 and        are independent;        is an AR(1) process:

img10¡1        (2)

where        is a constant parameter;        is an i.i.d. sequence with zero mean,  variance

img11img12img13img14img152        , and finite moments (of any order ); and if        , the distribution of the initial variable        0  is defined so that        becomes strictly stationary, while if        ,

img16img17img180        .

We suppose the availability of a dataset        =1 and correct specification of the model (1) and Conditions (C1) ­ (C2).

(a)       img19[25 marks] Suppose that

and consider the OLS estimator of img20:



i.       img22Derive the asymptotic distribution of img23. 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 img24, 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)                      img25img26[25 marks] Suppose that        and consider estimation of the mean of  ,        , by the following estimator:

i.       img27Derive the asymptotic distribution of        .

ii.       img28Explain how to construct a 95% confidence interval for        in practice.

2.       Answer both parts (a) and (b) of this question.

img29We 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 img30 and the correct specification.

(a)       [30 marks]

i.       img31Derive the asymptotic distribution of

img32and 2.

ii.       img33Discuss the consistency of        .

, in terms of parameters img34, 2 ,

(b)       img35img36[20 marks] Now, an econometrician wants to estimate the regression model. However, she does not have full confidence on the specification (1) of img37   and considers the following model:

as well as the following OLS estimators for        and :


and img40

img41                img42

i.       img43img44Prove the consistency of for img45. You may assume the boundedness of the sequence img46 (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.       img47img48img49Given the result in part (b.ii), discuss the consistency property of img50 for        , and compare it with the consistency property of in part (a). Your conclusion here should be different from that in part (a). img51 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:

img52img53¡1        ¡1        (4)

img54img55img56img57where        is i.i.d.                with        being independent of all past variables        ¡1        ¡1        ¡2        ¡2        ; and                is a continuous function of        parametrized by img58 (the form of                        is

known to researchers). We let  img59.

(a)       [25 marks]

i.       img60img61img62img63Given 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 img64 is

given by

ii.       Let

Suppose that


img65img66img67    0 implies img680        for some  .        (5)

img69In this case, prove that if        img70   0 img710        0 img72 , then

is not a constant.

(b)       [25 marks]

i.       Suppose the correct specification, i.e., (4) is the true data generating mechanism of img73. Then, prove that the condition (5) implies that the true parameter 0 is a unique maximiser of .

ii.       img74Suppose 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, img75 explain what interpretation we can have for the maximiser of .

4.       Answer both parts (a) and (b) of this question.


0 be an        vector of variables. We suppose the stationarity of and model it as an autoregressive process of order , VAR img77 , given by

img78img79img80img81img82img83img841     ¡1        ¡        ,

img85img86img87img88img89img90where        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

img91img92. 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 img94 denotes the maximised likelihood function for a VAR img95 with estimated parameter vector img96, and img97 is the number of estimated parameters in the VAR img98 .

img99img100i.  img101 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.       img102img103img104img105For now, suppose that        ,        (i.e.,        is the bivariate VAR img106    process:

img107img108img109img1101     ¡1        ), and

img111img1121        (6)

Is this a stationary process? img113 explain your answer.

iv.       img114img115img116If 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, img117 explain why such a moving average representation does not exist.

(b)       img118[20 marks] The VARimg119 can be interpreted as the reduced form corresponding to a struc­ tural VAR (SVAR) of order , given by

img120img121img122img123img1240        1     ¡1        ¡        ,

img125img126img127img128img129img130where is an vector of intercepts, 0 img1311        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.       img132img133img134img135img136img137Suppose        ,  , and        . In which of the following specifications are the



创建时间:2022-05-20 10:44