FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE -MODULE2 Midterm Exam Solutions - March 2015

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1 FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE -MODULE2 Midterm Exam Solutions - March 205 Time Allowed: 60 minutes Family Name (Surname) First Name Student Number (Matr.) Please answer all questions by choosing the most appropriate alternative(s) and/or by writing your answers in the spaces provided. You need to carefully justify and show yourworkinthecaseof open questions. Theremightbemorethanonecorrect answer(s) for each of the multiple choice questions: each selected alternative that is correct will be awarded one point; wrong answers will be penalized with minus 0.5 point. Correct answers not selected and questions that have been left blank will receive zero points. Only answers explicitly reported in the appropriate box will be considered. In the multiple choice case, report your selection by writing one or more of the letters A, B, C, D, E, F in BLOCK CAPITAL LETTERS. No other answers or indications pointing to potential answers will be taken into consideration. In the case of open questions, the maximum number of points is indicated. Section (total weight: 52%) Question. Consider the generic VAR() modelfor variables: X y μ+ A y + u u IID (0 Σ) where the matrices of vector autoregressive coefficients (A A 2..., A ) are full matrices (i.e., with no zero restrictions imposed). Which of the following statements is/are correct: (A) This VAR() modeliswritteninstructuralform. (B) If all the matrices of vector autoregressive coefficients (A A 2..., A ) are diagonal and Σ is diagonal, then the VAR() model consists of a set of independent AR() models. (C) Because its unrestricted nature, this VAR() may be estimated by OLS, equation by equation. (D) If all the matrices of vector autoregressive coefficients (A A 2..., A ) are upper triangular and Σ is upper triangular, then the VAR() model consists of a set of independent AR() models. (E) Its unconditional mean has expression [y ](I A 2 ) μ (F) None of the above. Answer(s) B, C

2 Debriefing: (A) False, this is clearly a reduced form because any contemporaneous effects have been dropped. (B) This is correct because the generic th equation of the VAR() becomes: + X + (C) Correct, as we have seen in the lectures (set 3, slide 20). (D) False (and partially absurd) because the firstequationwillreadasfollows + X X + while being a covariance matrix, Σ cannot be triangular by construction. (E) False on two accounts, because it makes no sense that for a VAR(), the unconditional mean may depend on A only and not A 2,...,A if ; because even though you assume that the appearance of A 2 in the formula is not correct. (F) Because B and C are correct, F cannot be correct. Question.2 Consider the autoregressive process of order (AR()): IID 0 2 Which of the following statements is/are correct: (A) [ ]02+08 (B) [ ]5 2 (C) [ ] 2 (D) [ ] (E) [ 2 ]64 2 (F) None of the above. Debriefing: Answer(s) A, C, D, E (A) Correct, [ ][ ]02+08 as [ ]0 (B) False, [ ] 2 ( 08 2 )(036) 2 (C) Correct, [ ][ ] 2 (D) Correct, [ ]02( 08) (E) Correct, [ 2 ][ ]064[ 2 ] (F) False, because A, B, C, and E are correct. Question.3 2

3 Wold s decomposition states that: (A) Any stochastic process can be expressed as the sum of a deterministic component and a stochastic infinite moving-average component. (B) Any stationary stochastic process can be expressed as the product of a deterministic component and a stochastic moving-average component. (C) Any invertible moving average process can be expressed as the sum of a deterministic component and a stationary stochastic process. (D) Any stationary stochastic process can be expressed as the sum of a mean-reverting process and a stochastic autoregressive component. (E) None of the above. Answer E Debriefing: (A) False, the process must be stationary and the MA does not have to be of infinite order. (B) False, it is no product. (C) False and rather embarrassing. (D) False and rather embarrassing. (E) Correct, as the correct answer was that the theorem claims that any stationary stochastic process can be expressed as the sum of a deterministic component and a stochastic moving-average component, and all additional qualifications in the candidate answers are extraneous and/or incorrect. Question.4 Consider a non-stationary process that contains a deterministic trend: + + IID 0 2 Which of the following statements is/are correct: (A) The process is made stationary by regressing on andthenreplacing with ˆ ˆ ˆ which is white noise under the assumption that the original model was correctly specified. (B) The process is made stationary by taking its first difference. (C) Because it is non-stationary, follows a random walk. (D) If the variable is first-differenced, this originates a non-invertible MA() that therefore cannot be represented as a stationary autoregressive process. (E) None of the above. Answer A, D 3

4 Debriefing: (A) Correct, see slide set 2, slide 24. (B) False and in fact answer D indicates that a few problems may be caused by this practice. (C) False and quite absurd. (D) Correct, because + + ( ) + (E) Because A and D are correct, E cannot be correct. Question.5 Consider the following ARMA(2,) Gaussian process for U.S. stock returns: IID (0 2 ). Indicate which of the following statements is/are correct. (A) The model can be estimated by OLS if and only if it is known ex-ante that 0;inthis case OLS is the same as MLE and therefore the estimates of 0,, 2,and 2 will be the most efficient among all unbiased estimators; when 6 0 the model must be estimated by MLE. (B) The model can be estimated by OLS if and only if it is known ex-ante that 0; in this case OLS is the same as MLE and therefore the estimates of 0,, and 2 will be the most efficient among all unbiased estimators; when and/or 2 60 the model must be estimated by MLE. (C) The model can be estimated by MLE if and only if it is known ex-ante that 0;inthis case MLE is the same as OLS and therefore the estimates of 0,, 2,and 2 will be the most efficient among all unbiased estimators; when 6 0 the model must be estimated by OLS. (D) The model can be estimated by OLS if and only if it is known ex-ante that the MA component is invertible; in this case OLS is the same as GLS and therefore the estimates of 0,, 2,and 2 will satisfy Neyman-Pearson s lemma; when the model must be estimated by GLS. (E) None of the above. Answer(s) A Debriefing: (A) Correct, see slide set 2, slide 7. (B) False, this is the correct answer in A with and OLS flipped around with and 2. (C) False, this is the correct answer in A with MLE and OLS flipped around. (D) False, in fact absurd and completely made up. (E) Because A is correct, E cannot be correct. Question.6 Instrumental variable estimators: (A) They replace endogenous variables on the right-hand side of the structural representation of a system of stochastic equations with fitted OLS values coming from reduced form estimation. 4

5 (B) They replace the endogenous variables on the right-hand side of the structural representation of a system of stochastic equations with other variables that are not correlated with the endogenous variables, but highly correlated with the errors. (C) They replace the endogenous variables on the right-hand side of the structural representation of a system of stochastic equations with other variables that are highly correlated with the endogenous variables, but not correlated with the errors. (D) They solve endogeneity issues in systems of stochastic equations yielding consistent estimators. (E) They solve integrability issues in VAR() yielding mean-square convergent estimators. (F) None of the above. Answer(s) C, D Debriefing: (A) False, this is definition of 2SLS estimators. (B) False, this is C in which the problem is worsened by using IVs, not solved. (C) Correct, see slide set 3, slide 5. (D) Correct, see slide set 3, slide 5. (E) False and completely made up. (F) False, given that C and D above are correct. Question.7 You have performed an Augmented Dickey-Fuller test in find that (ˆ )(ˆ) 2 Then X + (A) Because exceeds the classical, approximate threshold of 2, you will reject the null hypothesis of stationarity in favor of concluding that follows a I() process. (B) Because exceeds the classical, approximate threshold of 2, you will reject the null hypothesis of non-stationarity in favor of concluding that follows a I(0) process. (C) Because exceeds the classical, approximate threshold of 2, you will fail to reject the null hypothesis of stationarity in favor of concluding that follows a I() process. (D) Because exceeds the classical, approximate threshold of 2, you will fail to reject the null hypothesis of non-stationarity in favor of concluding that follows a I(0) process. (E) None of the above. Answer(s) E 0 5

6 Debriefing: (A) False, the null of an ADF test is non-stationarity; the comparison is to be performed with a non-standard critical value that was tabulated by Dickey and Fuller. (B) False, the comparison is to be performed with a non-standard critical value that was tabulated by Dickey and Fuller. (C) False and absurd, as under classical tests, exceeding the critical value implies rejection, not failure to reject. (D) False, because similar to answer B above, apart from the fact under classical tests, exceeding the critical value implies rejection, not failure to reject. (E) Correct. Section 2 (total weight: 26%) Question 2. Consider the following AR() Gaussian process for U.S. stock returns: IID (0 2 ). You know that 0,that[ + ]2 and that [ + ]3 Indicate which of the following is/are correct. (A) 05 and 2 23 (B) 05 and 2 4 (C) 2and 2 4 (D) 05 and 2 52 (E) None of the above. Debriefing: (A) False, see B. (B) Correct because Answer(s) B [ + ] [ + ] (C) False, see B. (D) False, see B. (E) False, given that B above is correct. 6

7 Question 2.2 (6 points) Consider the following VAR() model: Ã! 0 0 IID a (0.5 point) Is this model in structural or reduced form? Carefully justify your answer. Answer. The model is in structural form as can be seen from the presence of the matrix which introduces contemporaneous effects in the system. However, it can be noticed that the matrix is triangular, this meaning that while 2 has a contemporaneous effect on,theopposite does not hold. 2.2b (.5 points) Is the model under-, just-, or over-identified? What type of identification scheme did you apply? What type of ordering between the two endogenous variables does it imply? Carefully justify your answers. Answer. Given that the model has only 2variablesandthe matrix contains one zero restriction we can say that the VAR is just-identified. According to Cholesky s scheme, indeed, in order to have a just-identified scheme is necessary to impose ( 2 )2 With 2,asinthis case, the number of restriction required is (2 2 2)2. Given this identification scheme, we have that while 2 has a contemporaneous effect on, the opposite does not hold, hence 2 is given a higher rank order within the system. 2.2c (2 points) Solve the model and, if possible, write it in reduced-form. Compute means, variances, and the covariance of the reduced-form shocks, to be called [ 2 ] 0. Carefully justify your answer and show your work. Answer. Solving the model, we obtain the following two equations: In order to write in reduced form we have to premultiply both sides by B that is B 2 2 (the determinant is )

8 which gives: ( ) +( ) In order to find means, variances, and the covariance of the reduced-form shocks, to be called [ 2 ] 0, we start by expressing the reduced-form residuals as linear combinations of the structural residuals: [ 2 2 ] 0 as [ ][ 2 ]0 [ 2 ] 0 [ ] [ 2 2 ][ ]+ 2 2[ 2 ] 2 2 [ 2 ] [ 2 ] [ 2 ]4 ( 2 ) [( 2 2 )( 2 )] 2 2 ( 2 ) d (2 points) With reference to this model, explain what an impulse response function will be. How many impulse response functions will you be able to compute in this case and why? What is the total number of impulse response functions that you would be able to compute if you were to also change the identification scheme? To what long-run level will the impulse response functions converge in this specific case and why? Under what conditions on the VAR() will the standard impulse response functions describe the reaction of the endogenous variables to pure shocks affecting the system? Answer. With reference to this model, an impulse response function is the simulation of the effects of one (or more) unit structural shock(s) on the endogenous variables of the VAR, namely and 2 Structural means that we give a unit shock to the structural errors, and 2 In this case, given that the VAR is just-identified we can compute four different impulse response functions (i.e., one per variable and two sets of functions in total): we can give a unit shock to,orto 2. Given that 2 it is not possible to compute more impulse response functions. In this specific case the impulse response functions will converge to zero given that and 2 are zero-mean variables. In order for the standard impulse response function to describe the reaction of the endogenous variables to pure shocks affecting the system, the VAR() must be just-identified. 2.2e ( point) This VAR() model implies two eigenvalues both with module of approximately 0.83; what does this imply for the stationarity of the process? Explain how these eigenvalues are computed. Carefully justify your answer. 8

9 Answer. A linear algebra theorem states that for some matrix A, only if the roots associated to the characteristic equation det(a I )0 lie within the unit circle, i.e., all solutions (eigenvalues) 2 are less than in absolute value, the corresponding VAR() system will be stable. It can be demonstrated that when a VAR() is stable, the first two unconditional moments exist and do not depend on time. Consequently, when a VAR() is stable it is also covariance stationary. In this case, because the two eigenvalues are both less than in absolute value then we can say that the VAR() is stable and stationary. Section 3 (total weight: 22%) Question 3. (5 points) In the following table you have the results of two ARMA(,) model estimation exercises performed in EViews R.Thefirst is obtained by issuing the commands smpl 980: 202:2 equation ar.ls ger stock ret 2ycar() ma() while the second one is obtained by running the commands: smpl 980: 202:2 equation ar.ls ger stock ret 2ycgerstock ret 2y(-) ma() 3.a ( point) Explain the differences in the outputs from the two different regressions and where these derive from despite the model seems to be identical. Carefully justify your answers. Answer. We want to estimate an ARMA(,) process,

10 The two estmation exercises may look identical at firstsightbuttheyaredifferent in fact. The difference lies in the AR component of the process: the first regression is the estimation of ger stock ret 2y on an AR() and a MA() term while the second is the estimation of ger stock ret 2y on a lagged value of itself and an MA() term. While the MA() component is identical in the two estimation exercises, the AR() component is clearly different: when you use the AR() term (as in the first regression) you are not saying that ger stock ret 2y follows an AR() process, but you are saying that the error term follows an AR() process. Starting from the most familiar case, in the second regression you are estimating the following model, + + where + from which, by substituting in,you obtain which is the classical ARMA(,) seen in class. Consequently, the constant can be seen as the constant of an ARMA process, 0. What you estimate in the first equation is + where + + from which, by substituting in,you obtain Clearlythisisdifferent from the regression of ger stock ret 2y on its lagged value and an MA() component. For this reason, cannot be considered as the constant of a standard ARMA(,) model, 0. This explains the difference in the constants of the two regressions. Note: 0.25 bonus points have been given to those able to recognize that after some manipulations, the equation of the first regression can be re-written as: + ( )+ + () ( ) (2) which demonstrates that the constant in the first regression multiplied by ( ) is equal to the intercept of an ARMA (,) process, 0. 3.b (2 points) After setting 2, compute ( ) ( ) ( ) ( ) 2 ( ) ( 2 ) 2 ( ) ( 2 ) ( ) ( ) Carefully justify your answers and show your work. 0

11 Answer. Because we assume a ARMA(,) model for, it follows that: ( ) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( )+ + ( ) ( ) ( ) The correct values to substitute in the equations are: and 2 ( ) bonus points were given to students who have used these formulas and plugged in the correct numbers to calculate results. 3.c (2 points) Derive a test of the hypothesis that the constant in the first regression is equal to ( ) and discuss the statistical evidence that you obtain from this test. Answer. ˆ ( ) (ˆ) where ( ) has been computed from the estimates of the first regression, taking into account that the constant in the table is not equal to the constant of an ARMA (,) process, 0 but to 0 ( ) ( ) 0( ) 0 Given that 0 by construction, we cannot reject the null hypothesis that ˆ ( ). Very generous partial credit has been given also to those who did not take into account this aspect and simply computed in this case ˆ ( ) (ˆ) ( ) Given that is bigger than the classical approximate threshold of 2, we can reject the null hypothesis that ˆ ( ).

12 Question 3.2 With references to the example of a structural VAR() with 2 shown in the lectures, consider the functional relationship between the reduced-form residuals [ ] 0 and the structural residuals [ ] 0. Please indicate which of the following statements is/are correct: (A) While necessary condition for the reduced-form residuals to be uncorrelated is that the contemporaneous matrix coefficients are both zero, 2 2 0,thisisnotsufficient. (B) While sufficient condition for the reduced-form residuals to be the same as the structural residuals is that the contemporaneous matrix coefficients are both zero, 2 2 0,thisisnot necessary. (C) While necessary condition for the reduced-form residuals to be the same as the structural residuals is that the contemporaneous matrix coefficients are both zero, 2 2 0,thisisnot sufficient. (D) While sufficient condition for the reduced-form residuals to be uncorrelated is that the contemporaneous matrix coefficients are both zero, 2 2 0, this is not necessary. (E) None of the above. Answer D Debriefing: (A) False, as this is the correct answer in D with the words necessary and sufficient flipped around. (B) The first part of the statement is correct as implies that when However, in this case the condition is also necessary. (C) False, see B above. (D) Correct because from the slides you can see that

13 so that [ ] [ ] ( 2 2 ) 2 [ {z} ( 2 2 ) 2 [( 2 )( 2 )] 0 2 ( ) 2 2 ( ) ] {z} 0 ( 2 2 ) 2 [ 2( ) 2 2 ( ) 2 ] ( 2 2 ) 2 Therefore while implies that [ ]0 the opposite is not true as may still occur when (say) (E) False, given that D above is correct. 3

14 Financial Econometrics Rules of conduct during exams or other tests. During exams, students must remain quiet and may not use any external support aids, whether paper or digital (e.g. manuals, lecture notes, personal papers, books, publications, cell phones, handheld computers or other electronic devices), if not expressly authorized by the teacher in class. In addition, students may not copy or look at other students exam paper or contact or attempt to contact other people in any way. Students must remain in the classroom for the whole of the time and only for the time needed to finish his or her exam, unless teachers in class give other orders. Students who have questions for the teacher must raise their hand and wait for the examiner to come to them. At the end of the exam, students must return the exam script and the exam paper to the examining faculty member and leave the room. Any breach of these regulations or any other orders given by the faculty member present at the time of the exam will result in the test being cancelled and an official report sent to the Disciplinary Boardinallcases. All disciplinary sanctions will be recorded in the student s academic career. Sanctions greater than a warning will result in forfeiture of benefits for the right to study (scholarships, housing etc.). The Honor Code and detailed regulations for taking exams and other tests are published on the University website Name and Surname (CAPITAL LETTERS) Personal ID Signature: I hereby undertake to respect the regulations described above and undersign my presence at the exam. 4