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HOMEWORK ECONOMETRICS CHAPTER 6 Cao Thanh Hằng BAFNIU13025 Trương Nguyễn Thùy Oanh BAFNIU13071 Question 1: Consider the
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HOMEWORK ECONOMETRICS CHAPTER 6 Cao Thanh Hằng BAFNIU13025 Trương Nguyễn Thùy Oanh BAFNIU13071 Question 1: Consider the following simultaneous equations system y1t =α0+α1y2t +α2y3t +α3X1t +α4X2t +u1t (6.94) y2t =β0+β1y3t +β2X1t +β3X3t +u2t (6.95) y3t =γ0+γ1y1t +γ2X2t +γ3X3t +u3t (6.96) a. Derive the reduced form equations corresponding to (6.94)–(6.96). Renumbering:
y1t 0 1 y 2 t 2 y 3t 3 X 1t 4 X 2 t u1t y 2 t 0 1 y 3t 2 X 1t 3 X 3t u2 t y 3t 0 1 y1t 2 X 2 t 3 X 3t u3t
(1) (2) (3)
Take equation (1), and substitute in for y3t, to get
y1t 0 1 y 2 t 2 ( 0 1 y1t 2 X 2 t 3 X 3t u3t ) 3 X 1t 4 X 2 t u1t
y1t 0 1 y 2 t 2 0 2 1 y1t 2 2 X 2 t 2 3 X 3t 2 u3t 3 X 1t 4 X 2 t u1t Gathering terms in y1t on the LHS:
y1t 2 1 y1t 0 1 y 2 t 2 0 2 2 X 2 t 2 3 X 3t 2 u3t 3 X 1t 4 X 2 t u1t
y1t (1 2 1 ) 0 1 y 2 t 2 0 2 2 X 2 t 2 3 X 3t 2 u3t 3 X 1t 4 X 2 t u1t (4) Substituting into (2) for y3t from (3).
y 2 t 0 1 ( 0 1 y1t 2 X 2 t 3 X 3t u3t ) 2 X 1t 3 X 3t u2 t Removing the brackets
y 2 t 0 1 0 1 1 y1t 1 2 X 2 t 1 3 X 3t 1u3t 2 X 1t 3 X 3t u2 t (5)
Substituting into (4) for y2t from (5),
y1t (1 2 1 ) 0 1 ( 0 1 0 1 1 y1t 2 X 2 t 1 3 X 3t 1u3t 2 X 1t
3 X 3t u2 t ) 2 0 2 2 X 2 t 2 3 X 3t 2 u3t 3 X 1t 4 X 2 t u1t Taking the y1t terms to the LHS:
y1t (1 2 1 1 1 1 ) 0 1 0 1 1 0 1 2 X 2 t 1 1 3 X 3t 1 1u3t 1 2 X 1t
1 3 X 3t 1u2 t 2 0 2 2 X 2 t 2 3 X 3t 2 u3t 3 X 1t 4 X 2 t u1t Gathering like-terms in the other variables together:
y1t (1 2 1 1 1 1 ) 0 1 0 1 1 0 2 0 X 1t (1 2 3 ) X 2 t (1 1 2 2 2 4 ) X 3t (1 1 3 1 3 2 3 ) u 3t (1 1 2 ) 1 u2 t u1t (6 )
(1 2 1 1 1 1 ) Multiplying all through equation (3) by
:
y 3t (1 2 1 11 1 ) 0 (1 2 1 11 1 ) 1 y1t (1 2 1 11 1 )
2 X 2 t (1 2 1 11 1 ) 3 X 3t (1 2 1 11 1 ) u3t (1 2 1 11 1 ) (7)
y1t (1 2 1 11 1 ) Replacing
in (7) with the RHS of (6),
0 1 0 1 1 0 2 0 X 1t (1 2 3 ) y 3t (1 2 1 11 1 ) 0 (1 2 1 11 1 ) 1 X 2 t (1 1 2 2 2 4 ) X 3t (1 1 3 1 3 2 3 ) u3t (1 1 2 ) 1u2 t u1t
2 X 2 t (1 2 1 11 1 ) 3 X 3t (1 2 1 11 1 ) u3t (1 2 1 11 1 )
(8) Expanding the brackets in equation (8) and cancelling the relevant terms
y 3t (1 2 1 11 1 ) 0 1 0 11 0 X 1t (1 2 1 1 3 ) X 2 t ( 2 1 4 ) X 3t ( 11 3 3 ) u3t 11u2 t 1u1t (9)
(1 2 1 1 1 1 ) Multiplying all through equation (2) by
:
y 2 t (1 1 1 1 1 2 ) 0 (1 1 1 1 1 2 ) 1 y 3t (1 1 1 1 1 2 )
2 X 1t (1 1 1 1 1 2 ) 3 X 3t (1 1 1 1 1 2 ) u2 t (1 1 1 1 1 2 ) (10)
y 3t (1 2 1 11 1 ) Replacing
in (10) with the RHS of (9),
0 1 0 11 0 X 1t (1 2 1 1 3 ) y 2 t (1 1 1 1 1 2 ) 0 (1 1 1 1 1 2 ) 1 X 2 t ( 2 1 4 ) X 3t ( 3 11 3 ) u3t 11u2 t 1u1t
2 X 1t (1 1 1 1 1 2 ) 3 X 3t (1 1 1 1 1 2 ) u2 t (1 1 1 1 1 2 )
(11) Expanding the brackets in (11) and cancelling the relevant terms
y2t (1 1 1 ( 1 12 ) 0 02 1 1 0 1 10 X 1t 1 1 3 2 22 1 ) X 2 t ( 1 2 1 14 ) X 3t ( 1 3 3 32 1 ) 1u3t u2 t (1 2 1 ) 1 1u1t (12) From (6),
0 1 0 1 1 0 2 0 (1 2 3 ) ( 2 2 4 ) X 1t 1 1 2 X 2t (1 2 1 1 1 1 ) (1 2 1 1 1 1 ) (1 2 1 1 1 1 ) (1 1 3 1 3 2 3 ) u ( 2 ) 1 u2 t u1t X 3t 3t 1 1 (1 2 1 1 1 1 ) (1 2 1 1 1 1 ) y1t
(13) From (12),
y2 t
0 02 1 1 01 10 ( 1 1 3 2 22 1 ) X 1t (1 1 11 12 ) (1 1 11 12 )
( 1 2 1 14 ) X (1 1 11 12 ) 2 t
( 1 3 3 32 1 ) u u (1 2 1 ) 1 1u1t X 3t 1 3t 2 t (1 1 11 12 ) (1 1 11 12 ) (14) From (9),
y 3t
0 1 0 11 0 (1 2 1 1 3 ) ( 2 1 4 ) X 1t X (1 2 1 11 1 ) (1 2 1 11 1 ) (1 2 1 11 1 ) 2 t
( 11 3 3 ) u 11u2 t 1u1t X 3t 3t (1 2 1 11 1 ) (1 2 1 11 1 ) (15) All of the reduced form equations (13)-(15) in this case depend on all of the exogenous variables, which is not always the case, and that the equations contain only exogenous variables on the RHS, which must be the case for these to be reduced forms. b. What do you understand by the term ‘identification’? Describe a rule for determining whether a system of equations is identified. Apply this rule to (6.94–6.96). Does this rule guarantee that estimates of the structural parameters can be obtained? The term “identification” refers to whether or not it is in fact possible to obtain the structural form coefficients (the, , and ’s in equations (1)-(3)) from the reduced form coefficients (the ’s) by substitution. An equation can be over-identified, justidentified, or under-identified, and the equations in a system can have differing orders of identification. If an equation is under-identified (or not identified), then we cannot obtain the structural form coefficients from the reduced forms using any technique. If it is just identified, we can obtain unique structural form estimates by back-substitution, while if it is over-identified, we cannot obtain unique structural form estimates by substituting from the reduced forms. There are two rules for determining the degree of identification of an equation: the rank condition, and the order condition. The rank condition is a necessary and sufficient condition for identification, so if the rule is satisfied, it guarantees that the equation is indeed identified. The rule centers on a restriction on the rank of a submatrix containing the reduced form coefficients, and is rather complex and not particularly illuminating, and was therefore not covered in this course.
The order condition can be expressed in a number of ways, one of which is the following. Let G denote the number of structural equations (equal to the number of endogenous variables). An equation is just identified if G-1 variables are absent. If more than G-1 are absent, then the equation is over-identified, while if fewer are absent, then it is not identified. Equation (1): X3t only is missing, so the equation is not identified. Equation (2): y1t and X2t are missing, so the equation is just identified.
Equation (3): y2t and X1t are missing, so the equation is just identified. However, the order condition is only a necessary (and not a sufficient) condition for identification, so there will exist cases where a given equation satisfies the order condition, but we still cannot obtain the structural form coefficients. Fortunately, for small systems this is rarely the case. c. Describe a method of obtaining the structural form coefficients corresponding to an overidentified system A tempting response to the question might be to describe indirect least squares (ILS), that is estimating the reduced form equations by OLS and then substituting back to get the structural forms; however, this response would be WRONG, since the question tells us that the system is over-identified. A correct answer would be to describe either two stage least squares (2SLS) or instrumental variables (IV). Either would be acceptable, although IV requires the user to determine an appropriate set of instruments and hence 2SLS is simpler in practice. 2SLS involves estimating the reduced form equations, and obtaining the fitted values in the first stage. In the second stage, the structural form equations are estimated, but replacing the endogenous variables on the RHS with their stage one fitted values. Application of this technique will yield unique and unbiased structural form coefficients. Question 2: Consider the following system of two equations y1t =α0+α1y2t +α2X1t +α3X2t +u1t (6.97) y2t =β0+β1y1t +β2X1t +u2t (6.98) a. Explain, with reference to these equations, the undesirable consequences that would arise if (6.97) and (6.98) were estimated separately using OLS. A glance at equations (6.97) and (6.98) reveals that the dependent variable in (6.97) appears as an explanatory variable in (6.98) and that the dependent variable in (6.98) appears as an explanatory variable in (6.97). The result is that it would be possible to show that the explanatory variable y2t in (6.97) will be correlated with the error term in that equation, u1t, and that the explanatory variable y1t in (6.98) will be correlated with the error term in that equation, u2t. Thus, there is causality from y1t to y2t and from y2t to y1t, so that this is a simultaneous equations system. If OLS were applied separately to each of equations (6.97) and (6.98), the result would be biased and inconsistent parameter estimates. That is, even with an infinitely large number of observations, OLS could not be relied upon to deliver the appropriate parameter estimates.
b. What would be the effect upon your answer to (a) if the variable y 1t had not appeared in (6.98)? If the variable y1t had not appeared on the RHS of equation (6.98), this would no longer be a simultaneous system, but would instead be an example of a triangular system (see question 3). Thus it would be valid to apply OLS separately to each of the equations (6.97) and (6.98). c. State the order condition for determining whether an equation which is part of a system is identified. Use this condition to determine whether (6.97) or (6.98) or both or neither are identified. The order condition for determining whether an equation from a simultaneous system is identified was described in question 1, part (b). There are 2 equations in the system of (6.97) and (6.98), so that only 1 variable would have to be missing from an equation to make it just identified. If no variables are absent, the equation would not be identified, while if more than one were missing, the equation would be over-identified. Considering equation (6.97), no variables are missing so that this equation is not identified, while equation (6.98) excludes only variable X2t, so that it is just identified. Question 3: Consider the following vector autoregressive model
Where yt is a p × 1vector of variables determined by k lags of all p variables in the system, ut is a p × 1 vector of error terms, β0 is a p × 1 vector of constant term coefficients and βi are p × p matrices of coefficients on the ith lag of y. (a)If p=2, and k=3, write out all the equations of the VAR in full, carefully defining any new notation you use that is not given in the question. p=2 and k=3 implies that there are two variables in the system, and that both equations have three lags of the two variables. The VAR can be written in long-hand form as:
y1t 10 111 y1t 1 211 y 2t 1 112 y1t 2 212 y 2t 2 113 y1t 3 213 y 2t 3 u1t y 2t 20 121 y1t 1 221 y 2t 1 122 y1t 2 222 y 2t 2 123 y1t 3 223 y 2t 3 u 2t 10 y1t u1t , yt , ut 20 y2t u2 t
0 where
, and the coefficients on the lags of yt
are defined as follows: ijk refers to the kth lag of the ith variable in the jth equation. This seems like a natural notation to use, although of course any sensible alternative would also be correct. (b)Why have VARs become popular for application in economics and finance, relative to structural models derived from some underlying theory? The most important point is that structural models require the researcher to specify some variables as being exogenous (if all variables were endogenous, then none of the equations would be identified, and therefore estimation of the structural equations would be impossible). This can be viewed as a restriction (a restriction that the exogenous variables do not have any simultaneous equations feedback), often called an “identifying restriction”.. Under a VAR, all the variables have equations, and so in a sense, every variable is endogenous, which takes the ability to cheat (either deliberately or inadvertently) or to mis-specify the model in this way, out of the hands of the researcher. Another possible reason why VARs are popular in the academic literature is that standard form VARs can be estimated using OLS since all of the lags on the RHS are counted as pre-determined variables. Further, a glance at the academic literature which has sought to compare the forecasting accuracies of structural models with VARs, reveals that VARs seem to be rather better at forecasting (perhaps because the identifying restrictions are not valid). Thus, from a purely pragmatic point of view, researchers may prefer VARs if the purpose of the modelling exercise is to produce precise point forecasts. (c) Discuss any weaknesses you perceive in the VAR approach to econometric modelling. The most important of these criticisms is that VARs are atheoretical which means they use very little information form economic or financial theory to guide the model specification process. The result is that the models often have little or no theoretical interpretation, so that they are of limited use for testing and evaluating theories. Second, VARs can often contain a lot of parameters. The resulting loss in degrees of freedom if the VAR is unrestricted and contains a lot of lags, could lead to a loss of efficiency and the inclusion of lots of irrelevant or marginally relevant terms. Third, it is not clear how the VAR lag lengths should be chosen. Finally, the very tools that have been proposed to help to obtain useful information from VARs, i.e. impulse responses and variance decompositions, are themselves difficult to interpret