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Chapter 6

1

In the context of simultaneous equations modelling, which of the following statements is true concerning an endogenous variable?

a)
b)
c)
d)
Correct! In the context of simultaneous equations modelling, the endogenous variables are those whose values are determined within the system. Thus, for the values of the endogenous variables to be determined within the system, there must be an equation for each endogenous variable. That is, the endogenous variables must all be a dependent variable in one equation. By definition, the reduced form equations are those that contain only exogenous variables as independent variables (on the RHS of the equations). Thus c is correct and d is incorrect.Incorrect! In the context of simultaneous equations modelling, the endogenous variables are those whose values are determined within the system. Thus, for the values of the endogenous variables to be determined within the system, there must be an equation for each endogenous variable. That is, the endogenous variables must all be a dependent variable in one equation. By definition, the reduced form equations are those that contain only exogenous variables as independent variables (on the RHS of the equations). Thus c is correct and d is incorrect.Your answer has been saved.
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2

If OLS is applied separately to each equation that is part of a simultaneous system, the resulting estimates will be

a)
b)
c)
d)
Correct! In fact, if OLS is applied separately to each equation that is part of a simultaneous system, the resulting estimates will be both biased and inconsistent. It is fairly easy to show that, since the assumption that E[X'u] = 0 is violated in the context of simultaneous equations, OLS will lead to biased coefficient estimates. However, nothing in this bias will be improved as the sample size increases, and therefore the OLS estimator will be inconsistent as well (although proving its inconsistency is algebraically more complex). Of course, OLS can be used on each equation separately in the sense that the method will lead to some estimates, it is just that OLS cannot be VALIDLY applied.Incorrect! In fact, if OLS is applied separately to each equation that is part of a simultaneous system, the resulting estimates will be both biased and inconsistent. It is fairly easy to show that, since the assumption that E[X'u] = 0 is violated in the context of simultaneous equations, OLS will lead to biased coefficient estimates. However, nothing in this bias will be improved as the sample size increases, and therefore the OLS estimator will be inconsistent as well (although proving its inconsistency is algebraically more complex). Of course, OLS can be used on each equation separately in the sense that the method will lead to some estimates, it is just that OLS cannot be VALIDLY applied.Your answer has been saved.
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3

Which of the following statements are true concerning a triangular or recursive system?

i) The parameters can be validly estimated using separate applications of OLS to

each equation


ii) The independent variables may be correlated with the error terms in other

equations


iii) An application of 2SLS would lead to unbiased but inefficient parameter estimates


iv) The independent variables may be correlated with the error terms in the equations

in which they appear as independent variables

a)
b)
c)
d)

Correct! (i), (ii) and (iii) are all correct. A triangular system is one where the causality between the "endogenous variables" all goes in one direction. Thus, such a system is not a simultaneous system at all, and hence OLS can validly be applied separately to each equation in the system. It thus follows that if a systems technique such as 2SLS is used, this would have been done unnecessarily, and hence an efficiency loss would occur when the reduced-form fitted values are used instead of the actual values. In a triangular system such as that described by equations 21 to 23 in the lecture notes, Y_1 will be uncorrelated with u_1, Y_2 will be uncorrelated with u_2, and Y_3 will be uncorrelated with u_3. This is why OLS can be validly applied. However, in equations 21-23, Y_1, which appears on the RHS of equations 22 and 23, will be correlated with the disturbance term u_1, and Y_2, which appears on the RHS of equation 23, will be correlated with the disturbance term u_2. Thus (ii) is also true, but from the point of view of consistent and unbiased estimation, this will not matter.

Incorrect! (i), (ii) and (iii) are all correct. A triangular system is one where the causality between the "endogenous variables" all goes in one direction. Thus, such a system is not a simultaneous system at all, and hence OLS can validly be applied separately to each equation in the system. It thus follows that if a systems technique such as 2SLS is used, this would have been done unnecessarily, and hence an efficiency loss would occur when the reduced-form fitted values are used instead of the actual values. In a triangular system such as that described by equations 21 to 23 in the lecture notes, Y_1 will be uncorrelated with u_1, Y_2 will be uncorrelated with u_2, and Y_3 will be uncorrelated with u_3. This is why OLS can be validly applied. However, in equations 21-23, Y_1, which appears on the RHS of equations 22 and 23, will be correlated with the disturbance term u_1, and Y_2, which appears on the RHS of equation 23, will be correlated with the disturbance term u_2. Thus (ii) is also true, but from the point of view of consistent and unbiased estimation, this will not matter.

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4

Consider the following system of equations (with time subscripts suppressed and using standard notation)

According to the order condition, the first equation is

a)
b)
c)
d)
Correct! There are 3 equations in this system, so for an equation to be just identified would require there to be 2 variables absent from a given equation. If there are more than 2 variables absent from a given equation, it would be classed as over-identified, while if less than 2 variables are absent, it would be under-identified. The variables in the system in total are Y_1, Y_2, Y_3, X_1, X_2, and X_3. The only variable missing from the first equation is X_3, so this equation is not identified (unidentified).Incorrect! There are 3 equations in this system, so for an equation to be just identified would require there to be 2 variables absent from a given equation. If there are more than 2 variables absent from a given equation, it would be classed as over-identified, while if less than 2 variables are absent, it would be under-identified. The variables in the system in total are Y_1, Y_2, Y_3, X_1, X_2, and X_3. The only variable missing from the first equation is X_3, so this equation is not identified (unidentified).Your answer has been saved.
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5

Consider again the system of equations in question 4. According to the order condition, the second equation is

a)
b)
c)
d)
Correct! There are 3 equations in this system, so for an equation to be just identified would require there to be 2 variables absent from a given equation. If there are more than 2 variables absent from a given equation, it would be classed as over-identified, while if less than 2 variables are absent, it would be under-identified. The variables in the system in total are Y_1, Y_2, Y_3, X_1, X_2, and X_3. Considering the second equation, the variables Y_1 and X_3 are missing, and therefore, the equation is just identified.Incorrect! There are 3 equations in this system, so for an equation to be just identified would require there to be 2 variables absent from a given equation. If there are more than 2 variables absent from a given equation, it would be classed as over-identified, while if less than 2 variables are absent, it would be under-identified. The variables in the system in total are Y_1, Y_2, Y_3, X_1, X_2, and X_3. Considering the second equation, the variables Y_1 and X_3 are missing, and therefore, the equation is just identified.Your answer has been saved.
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6

Consider again the system of equations in question 4. Which estimation method, if any, can be used for the third equation in the system:

i) OLS


ii) 2SLS


iii) ILS

a)
b)
c)
d)
Correct! In fact, all of the variables on the RHS of the third equation are exogenous variables, and thus any of the 3 estimation methods could be used, although both ILS and 2SLS would result in less efficient estimation than a simple application of OLS.

Incorrect! In fact, all of the variables on the RHS of the third equation are exogenous variables, and thus any of the 3 estimation methods could be used, although both ILS and 2SLS would result in less efficient estimation than a simple application of OLS.

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7

The order condition is

a)
b)
c)
d)
Correct! The order condition is a necessary but not sufficient condition for identification. That is, if an equation does not satisfy the order condition, it cannot be identified. On the other hand, if an equation satisfies the order condition, there are some circumstances under which the equation is still not identified. An alternative condition, which is both necessary and sufficient (in other words, it gives you a sure result) is the rank condition. But the rank condition is considerably more complex to evaluate and therefore it was not covered in this course. Fortunately, most systems of equations that we may want to estimate are over-identified so that this is rarely an issue in practice.Incorrect! The order condition is a necessary but not sufficient condition for identification. That is, if an equation does not satisfy the order condition, it cannot be identified. On the other hand, if an equation satisfies the order condition, there are some circumstances under which the equation is still not identified. An alternative condition, which is both necessary and sufficient (in other words, it gives you a sure result) is the rank condition. But the rank condition is considerably more complex to evaluate and therefore it was not covered in this course. Fortunately, most systems of equations that we may want to estimate are over-identified so that this is rarely an issue in practice.Your answer has been saved.
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8

A Hausman test would be used for

a)
b)
c)
d)
Correct! The Hausman test is an exogeneity test - that is, it is used for determining whether a simultaneous framework is needed for a particular variable. It operates by estimating the reduced form equations, obtaining their fitted values then including these fitted values in the structural equations together with the original endogenous variables. If the coefficient estimates on the fitted values are statistically significant, this implies that there is extra information available for that equation from modelling the variables as endogenous than assuming that they are exogenous. Incorrect! The Hausman test is an exogeneity test - that is, it is used for determining whether a simultaneous framework is needed for a particular variable. It operates by estimating the reduced form equations, obtaining their fitted values then including these fitted values in the structural equations together with the original endogenous variables. If the coefficient estimates on the fitted values are statistically significant, this implies that there is extra information available for that equation from modelling the variables as endogenous than assuming that they are exogenous. Your answer has been saved.
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9

Which of the following estimation techniques are available for the estimation of over-identified systems of simultaneous equations?

i) OLS

ii) ILS

iii) 2SLS

iv) IV

a)
b)
c)
d)

Correct! Only 2SLS and IV from this list are validly applicable in the case of over-identified equations. The instruments to be used in an instrumental variables approach could either be some additional exogenous variables thought to be correlated with the included endogenous variables, or they could be the fitted values from estimation of the reduced form equations. In the latter case, 2SLS would be identical to IV. ILS can only be used for calculating the structural form equations from the reduced forms if the equation is just identified, while OLS cannot be used at all for simultaneous systems.

Incorrect! Only 2SLS and IV from this list are validly applicable in the case of over-identified equations. The instruments to be used in an instrumental variables approach could either be some additional exogenous variables thought to be correlated with the included endogenous variables, or they could be the fitted values from estimation of the reduced form equations. In the latter case, 2SLS would be identical to IV. ILS can only be used for calculating the structural form equations from the reduced forms if the equation is just identified, while OLS cannot be used at all for simultaneous systems.

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10

Which of the following are advantages of the VAR approach to modelling the relationship between variables relative to the estimation of full structural models?

i) VARs receive strong motivation from financial and economic theory


ii) VARs in their reduced forms can be used easily to produce time-series forecasts


iii) VAR models are typically highly parsimonious


iv) OLS can be applied separately to each equation in a reduced form VAR

a)
b)
c)
d)

Correct! First, VARs, like ARMA models, are "theory-poor" - that is, they are usually based only very loosely on economic or financial theory. The approach of VAR modelling is usually rather to throw all of the variables in and see which have significant impacts. This contrasts with structural models, where theory will play a much stronger part in determining which variables should appear in which equations. It is true that VARs can easily be used to produce time-series forecasts. This arises since, in their reduced or "standard" forms, VARs only have lagged values of variables on the RHS. This removes the problem with standard structural models that you have to forecast the future values of the explanatory variables in order to be able to produce forecasts of the explained variable(s). Producing one-step or multi-step ahead forecasts with VARs is simply an exercise in iterating with the conditional expectations operator (as we did with ARMA models). An important disadvantage of VARs is that they are usually NOT parsimonious, i.e. they usually contain many variables. This can use up degrees of freedom, reduce out-of-sample forecasting ability, and make interpretation of the VAR even more difficult than it otherwise would have been. OLS can indeed be applied separately to each equation in a reduced-form VAR since the equations contain only pre-determined (lagged) variables on the RHS and thus they are not really simultaneous equations.

Incorrect! First, VARs, like ARMA models, are "theory-poor" - that is, they are usually based only very loosely on economic or financial theory. The approach of VAR modelling is usually rather to throw all of the variables in and see which have significant impacts. This contrasts with structural models, where theory will play a much stronger part in determining which variables should appear in which equations. It is true that VARs can easily be used to produce time-series forecasts. This arises since, in their reduced or "standard" forms, VARs only have lagged values of variables on the RHS. This removes the problem with standard structural models that you have to forecast the future values of the explanatory variables in order to be able to produce forecasts of the explained variable(s). Producing one-step or multi-step ahead forecasts with VARs is simply an exercise in iterating with the conditional expectations operator (as we did with ARMA models). An important disadvantage of VARs is that they are usually NOT parsimonious, i.e. they usually contain many variables. This can use up degrees of freedom, reduce out-of-sample forecasting ability, and make interpretation of the VAR even more difficult than it otherwise would have been. OLS can indeed be applied separately to each equation in a reduced-form VAR since the equations contain only pre-determined (lagged) variables on the RHS and thus they are not really simultaneous equations.

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11

How many parameters will be required to be estimated in total for all equations of a standard form, unrestricted, tri-variate VAR(4), ignoring the intercepts?

a)
b)
c)
d)
Correct! A tri-variate VAR is one where there will be three equations, each for a variable, and an unrestricted VAR(4) is one containing 4 lags of each variable in each equation. Thus, each equation will have 4 lags of 3 variables (i.e. 12 parameters to estimate per equation), for each of the 3 equations = 12 ( 3 = 36 parameters in total across all 3 equations.Incorrect! A tri-variate VAR is one where there will be three equations, each for a variable, and an unrestricted VAR(4) is one containing 4 lags of each variable in each equation. Thus, each equation will have 4 lags of 3 variables (i.e. 12 parameters to estimate per equation), for each of the 3 equations = 12 ( 3 = 36 parameters in total across all 3 equations.Your answer has been saved.
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12

Which one of the following statements is true concerning VARs?

a)
b)
c)
d)
Correct! As with ARMA models, one of the limitations of VAR modelling is that it is hard to interpret the coefficient estimates, since there will typically be a number of lags of each variable in each equation. This problem is confounded by the fact that the coefficient estimates will typically change sign from one lag to another, so that it is not clear whether one variable has a positive or negative impact on another overall. It is also likely to be the case that only some of the lags are statistically significantly different from zero, making it rather hard to see what actually is the relationship between the variables implied by the VAR. It is true that VARs often produce good forecasts in practice, and better than simultaneous structural models that could be used. The reason for this forecasting accuracy of VARs is not entirely clear, but could result in the lack of restrictions forced on the model to make the equations identified, as would have to be done for simultaneous structural models. Finally, it is not necessarily true that all components of a VAR have to be stationary if the only use required of the estimated model is for forecasting. Since non-stationarity of any of the components of a VAR will not affect the consistency or unbiasedness of coefficient estimation, we can still obtain good forecasts from a VAR with no-stationary components. Incorrect! As with ARMA models, one of the limitations of VAR modelling is that it is hard to interpret the coefficient estimates, since there will typically be a number of lags of each variable in each equation. This problem is confounded by the fact that the coefficient estimates will typically change sign from one lag to another, so that it is not clear whether one variable has a positive or negative impact on another overall. It is also likely to be the case that only some of the lags are statistically significantly different from zero, making it rather hard to see what actually is the relationship between the variables implied by the VAR. It is true that VARs often produce good forecasts in practice, and better than simultaneous structural models that could be used. The reason for this forecasting accuracy of VARs is not entirely clear, but could result in the lack of restrictions forced on the model to make the equations identified, as would have to be done for simultaneous structural models. Finally, it is not necessarily true that all components of a VAR have to be stationary if the only use required of the estimated model is for forecasting. Since non-stationarity of any of the components of a VAR will not affect the consistency or unbiasedness of coefficient estimation, we can still obtain good forecasts from a VAR with no-stationary components. Your answer has been saved.
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13

Suppose that two researchers, using the same 3 variables and the same 250 observations on each variable, estimate a VAR. One estimates a VAR(6), while the other estimates a VAR(4). The determinants of the variance-covariance matrices of the residuals for each VAR are 0.0036 and 0.0049 respectively. What is the values of the test statistic for performing a test of whether the VAR(6) can be restricted to a VAR(4)?

a)
b)
c)
d)

Correct! The likelihood ratio statistic will be given by the number of observations multiplied by the difference between the logs of the determinants of the variance-covariance matrices for the restricted and unrestricted models. The determinant of the variance-covariance matrix for the restricted model (the VAR(4)) will be the bigger of the two. The calculation is thus LR = 250 ( [log(0.0049) - log(0.0036)] = 77.07.

Incorrect! The likelihood ratio statistic will be given by the number of observations multiplied by the difference between the logs of the determinants of the variance-covariance matrices for the restricted and unrestricted models. The determinant of the variance-covariance matrix for the restricted model (the VAR(4)) will be the bigger of the two. The calculation is thus LR = 250 ( [log(0.0049) - log(0.0036)] = 77.07.

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14

Consider again the VARs that were discussed in question 13. What is the number of degrees of freedom for the critical value for testing the restriction?

a)
b)
c)
d)
Correct! The critical value will be a chi-squared, with degrees of freedom equal to the total number of restrictions placed on all equations. In the notation used in the lectures, the df will be given by the square of the number of variables in the system (3 squared), multiplied by the number of lags restricted (2). In other words, in total there will be 2 lags of each of the 3 variables restricted in each of the 3 equations = 18 restrictions. The critical value from the chi-squared table is 31.5 at the 1% level so that this restriction is clearly not supported by the data. We would conclude that it is preferable to estimate a VAR(6). Incorrect! The critical value will be a chi-squared, with degrees of freedom equal to the total number of restrictions placed on all equations. In the notation used in the lectures, the df will be given by the square of the number of variables in the system (3 squared), multiplied by the number of lags restricted (2). In other words, in total there will be 2 lags of each of the 3 variables restricted in each of the 3 equations = 18 restrictions. The critical value from the chi-squared table is 31.5 at the 1% level so that this restriction is clearly not supported by the data. We would conclude that it is preferable to estimate a VAR(6). Your answer has been saved.
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15

Suppose now that a researcher wishes to use information criteria to determine the optimal lag length for a VAR. 500 observations are available for the bi-variate VAR, and the values of the determinant of the variance-covariance matrix of residuals are 0.0336, 0.0169, 0.0084, and 0.0062 for 1, 2, 3, and 4 lags respectively. What is the optimal model order according to Akaike's information criterion?

a)
b)
c)
d)

Correct! The formula to compute the values of Akaike's criterion is log(det((sigma-hat)) + 2k/T, where log(det((sigma-hat)) is the natural log of the determinant of the variance-covariance matrix of residuals, k is the total number of parameters in total across both equations, and T is the number of observations (500). If the number of lags is 1, there will be one lag of each of the variables plus an intercept in each equation, i.e. a total of 6 parameters to estimate. For 2 lags, there will be 2 lags of 2 variables plus an intercept in each equation, i.e. a total of 10 parameters. For 3 and 4 lags, there will be 14 and 18 parameters respectively. The values of the criterion are as follows:

1 lag: log(0.0336) + (2 ( 6/ 500) = -3.369

2 lags: log(0.0169) + (2 ( 10 / 500) = -4.040

3 lags: log(0.0084) + (2 ( 14 / 500) = -4.724

4 lags: log(0.0083) + (2 ( 18 / 500) = -4.719

Since the information criterion is minimised (i.e. in this case is most negative) at 3 lags, this is the optimal VAR size. Notice, though, that there is little to choose between the VAR(3) and the VAR(4), which have very similar logs of the determinants of the variance-covariance matrices of residuals and also very similar values of the information criteria.

Incorrect! The formula to compute the values of Akaike's criterion is log(det((sigma-hat)) + 2k/T, where log(det((sigma-hat)) is the natural log of the determinant of the variance-covariance matrix of residuals, k is the total number of parameters in total across both equations, and T is the number of observations (500). If the number of lags is 1, there will be one lag of each of the variables plus an intercept in each equation, i.e. a total of 6 parameters to estimate. For 2 lags, there will be 2 lags of 2 variables plus an intercept in each equation, i.e. a total of 10 parameters. For 3 and 4 lags, there will be 14 and 18 parameters respectively. The values of the criterion are as follows:

1 lag: log(0.0336) + (2 ( 6/ 500) = -3.369

2 lags: log(0.0169) + (2 ( 10 / 500) = -4.040

3 lags: log(0.0084) + (2 ( 14 / 500) = -4.724

4 lags: log(0.0083) + (2 ( 18 / 500) = -4.719

Since the information criterion is minimised (i.e. in this case is most negative) at 3 lags, this is the optimal VAR size. Notice, though, that there is little to choose between the VAR(3) and the VAR(4), which have very similar logs of the determinants of the variance-covariance matrices of residuals and also very similar values of the information criteria.

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16

Consider the following bivariate VAR(2) model:

Which one of the following conditions must hold for it to be said that Granger causality runs from y1 to y2 only?

a)
b)
c)
d)
Correct! b is correct. For Granger causality to run from y1 to y2 but not the other way around implies that coefficients on lags of y1 should be significant in the equation for y2, but that the coefficients on lags of y2 should not be significant in the equation for y1. Therefore the d coefficients should be statistically significant and the b coefficients should be statistically insignificant. Notice that it doesn't affect the presence or otherwise of Granger causality if the b or c coefficients are either significant or insignificant.Incorrect! b is correct. For Granger causality to run from y1 to y2 but not the other way around implies that coefficients on lags of y1 should be significant in the equation for y2, but that the coefficients on lags of y2 should not be significant in the equation for y1. Therefore the d coefficients should be statistically significant and the b coefficients should be statistically insignificant. Notice that it doesn't affect the presence or otherwise of Granger causality if the b or c coefficients are either significant or insignificant.Your answer has been saved.
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17

Consider again the VAR model of equation 16. Which of the following conditions must hold for it to be said that there is bi-directional feedback?

a)
b)
c)
d)
Correct! d is correct. Bi-directional causality implies that Granger-causality flows in both directions - i.e. that y1 causes y2 and that y2 causes y1. This implies that both the b and the d coefficients must be statistically significant. Again, it does not affect the presence or otherwise of Granger causality if the a and c coefficients are significant or not - this is irrelevant.Incorrect! d is correct. Bi-directional causality implies that Granger-causality flows in both directions - i.e. that y1 causes y2 and that y2 causes y1. This implies that both the b and the d coefficients must be statistically significant. Again, it does not affect the presence or otherwise of Granger causality if the a and c coefficients are significant or not - this is irrelevant.Your answer has been saved.
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18

Which of the following statements is true concerning variance decomposition analysis of VARs?

i) Variance decompositions measure the impact of a unit shock to each of the variables on the VAR

ii) Variance decompositions can be thought of as measuring the proportion of the forecast error variance that is attributable to each variable

iii) The ordering of the variables is important for calculating impulse responses but not variance decompositions

iv) It is usual that most of the forecast error variance for a given variable is attributable to shocks to that variable

a)
b)
c)
d)

Correct! (ii) and (iv) only are correct. It is the impulse response functions that measure the impact of a unit shock to each of the variables on the VAR, not variance decompositions, whilst the answer (ii) is the definition of a variance decomposition. Impulse responses and variance decompositions offer alternative ways of looking at essentially the same thing - which variables are the most important driving factors in each equation, given their interconnectedness across the equations. The ordering of the variables is important in both impulse responses and variance decompositions. This arises in practice, since the VAR residual terms are likely to be correlated with one another, and therefore it is necessary to compute "orthogonalised" impulse responses and variance decompositions. In the bivariate case, these will attribute the entire common component in the residuals to the first of the variables. Finally, it is true that most of the forecast error variance for a given variable will typically attributable to shocks to that variable. Another way of saying this is that in VAR models, the dependent variable is usually more strongly affected by its own lagged values than by the lagged values of the other variables in the system.

Incorrect! (ii) and (iv) only are correct. It is the impulse response functions that measure the impact of a unit shock to each of the variables on the VAR, not variance decompositions, whilst the answer (ii) is the definition of a variance decomposition. Impulse responses and variance decompositions offer alternative ways of looking at essentially the same thing - which variables are the most important driving factors in each equation, given their interconnectedness across the equations. The ordering of the variables is important in both impulse responses and variance decompositions. This arises in practice, since the VAR residual terms are likely to be correlated with one another, and therefore it is necessary to compute "orthogonalised" impulse responses and variance decompositions. In the bivariate case, these will attribute the entire common component in the residuals to the first of the variables. Finally, it is true that most of the forecast error variance for a given variable will typically attributable to shocks to that variable. Another way of saying this is that in VAR models, the dependent variable is usually more strongly affected by its own lagged values than by the lagged values of the other variables in the system.

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