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Estimation of a system of linear dynamic asset demand equations
Economics Letters 8 (I 98 I) 355-360 North-Holland Publishing Company
355
ESTIMATION OF A SYSTEM OF LINEAR DYNAMIC ASSET DEMAND EQUATIONS Computational Considerations B.F. HUNT Unrorrsrty
Received
*
of New South
12 November
Wales, Kensrngton,
N.S. W., Austruha
1981
The error covariance matrix of a system of linear dynamic asset demand equations is of less than full rank. This property requires that system be modified before estimation. We show that a knowledge of the structure of the asset equations can be the basis for a favourable reduction in the size of the problems of estimation and inference. It is possible to obtain computational savings while retaining all the properties of the original system.
The singularity of the error covariance matrix of a system of asset demand equations requires that some modification is necessary before the equations are suitable for estimation by a systems estimator. This paper presents a form of modification which has the attractive feature of being computationally convenient while preserving the important properties of the original system. Static linear asset demand equations can be derived from a process of maximization of negative exponential utility function under a balance sheet constraint. ’ The portfolio of asset demand equations, at time t can be written as
xi = (z;, W,), y, is a (k X 1) vector of assets demanded, B is a matrix of coefficients; W, is scalar portfolio size variable, z, is a (( p - 1) X 1) vector of other explanatory variables. In order to preserve
where
(k Xp)
* I would like to acknowledge that the comments and suggestions of Ron Bewley and Mark Upcher were particularly helpful to the completion of this note. ’ See for example Parkin (I 970).
0165-1765/82/0000-0000/$02.75
0 1982 North-Holland
the balance sheet constraint it is necessary that I’B = (0.. 0, l), where I is a vector of units. It has been shown, Christofides (1976), Hunt and Upcher (1979) that if there exists quadratic disequilibrium costs and adjustment costs, the static system, (1) is transformed to a first-order dynamic system yl”
(2)
=Bxt,
AY, =D(Y;
-Y,-,),
(3)
where yt* is a (k X 1) vector of desired asset levels, B is a (k X p) matrix of long-run coefficients and D is a full (k X k) matrix of adjustment coefficients. D is consistent with the balance sheet constraint in that I’D = I’. Combining (2) and (3) and dropping the assumption that the equations hold exactly gives y,=DBx,+(Z-D)y,_,+u,, = cx, fAy,_,
+ u,,
(4) (5)
where C = DB, A = Z - D and u, is a (k X 1) vector of disturbances. It is assumed that u, - N(0, a,). It should be emphasized that economic theory does not directly provide assertions about the properties of C or A but rather provides assertions as to the nature of the coefficient matrices D and B. In other words we are not interested in testing direct inferences on the nature of C or A, but rather in using C and A to provide inference on B and D. The fact that B and D are not directly obtained is not a great drawback as provided we are able to obtain consistent estimates of ,C, (6) and A, (a), then we can deduce consistent estimates of D, (D = Z-A) and B, (g = (I --a)-‘c). Moreover we are able to test theory pertaining to B and D as the standard errors of the element of B and D are available from the covariance matrix of the parameters of (5) 9,:, [Valentine (1975) and Hunt and Upcher (1979)]. Q2, = Q2, and 9, =j’QA : J, where J’=(C’(Z-A’)-‘@(Z-A)-‘:Z@(Z-A)-‘). A very important property of the system of dynamic portfolio equations is its stability or lack of stability. Intuitive economic sense supported by economic theory [Hunt and Upcher (1979)] suggest that the characteristic roots of D should lie between zero and unity. Again it is a
comparitively straightforward procedure to obtain the standard errors of the characteristic roots of D from 9,. It follows from the above discussion that point estimates and variances of the elements of B and D and the characteristic roots of D can be obtained from a d and 8, : c. Direct estimates of r? and A^ can be obtained by directly applying a single equation estimator to (5), however this method has considerable drawbacks. Direct estimation is unsatisfactory as it does not produce an estimate of the full covariance matrix fiA : c necessary to calculate standard errors of the long-run coefficients. To achieve a satisfactory estimate fiA : c a systems estimator must be used. Unfortunately a systems estimator cannot be applied to eq. (5) as these equations have a covariance matrix of residuals, L?,,which is singular. The singularity devices from the balance sheet constraint implicit in (4) which dictates that Z’u, = 0 = f’Oll = 0. The singularity of 3, requires that the estimates of e and 2 be obtained indirectly if a systems estimator is to be employed. This can be achieved by dropping one of the equations, say the last, of (4). Consider a partitioning of (4) into the first (k - 1) equations and the last equation,
(6) The dimensions of the various components of (4) are as follows: Y”, D,,, B;, and u” are all ((k 1) X 1; Y’, D,,, B,,, w and u’ are all DA> Bm scalars; D,, and I0 are ((k - 1) X (k - 1)); B,, is ((k - 1) X (p - 1)) z is ((P - 1) X 1). We assume that the covariance matrix of u, has only one zero root, originating from the consistency restrictions. Thus it is possible to apply a systems estimation to the first (k - 1) equations of (4),
Y0= m,B,, f4,B,,D,,B,* +(Z’-D,,-D,,)
+wM[Z,],
1 +u”.
I’0 [ Y’ 1-I
(7)
Systems estimation of (7) is feasible as the assumption of a single zero root in 3, is equivalent to assuming that variance of u’, E(uOuo’) = L?:, is of full rank. Estimation of eq. (7) will yield consistent estimates for the coefficients on zl, wr, ylo and yr2. However, because the coefficient matrix on the lagged dependent variables (y’v’),_, is not longer square it is convenient to further modify (6) by substituting for y,‘, in (7). The adding up restrictions imply that
(8) where I is k - 1 row vector. Substituting
+(zO
-D,,
+D,,I’)yp_,
(8) in (7)
+up.
Comparing the sets of equations modified set of equations (9) has
(9)
(6) with (9) it is obvious
that the
(9
one less equation, a square adjustment coefficient matrix on a reduced endogenous variables, y,%, , (iii) an additional explanatory variable w,_ , . (ii)
set of lagged
The important question is, does the modified eq. (9), have the same properties as the original eq. (6) in respect of the eigenvalues of the adjustment matrix, and of the long-run coefficients? Concentrating on the roots of the adjustment matrix of (6). By performing elementary operations on partitioned I - D we can transform it, firstly by substracting each of the first k - 1 rows from the final row and then secondly, by subtracting the resultant final column from each of the first k - 1 rows, IO-D,,
-42 These
-D,,
’
II"-D,,
I 1- Da, I>I0 elementary
operations
-D,,I 0
ho-D,,
I-0 I
I
fD,,I'
I
do not alter the eigenvalues
-D,,'
O
I.
of the Z - D
matrix.
Examining
the characteristic (Z” -D,,
I(Z-D)-AZl=
equation
+ D,,I’)
-AZ0
0
=I-Xll(Z”
of Z - D -D,, -A
-D,,
+D,,I’)-xZ”j.
(10)
It can be seen from (10) that the values of h which satisfy /(ID) -XZI=O also satisfy I(Z o - D,, + D,,I’) - AZ0 I except for h = 0. Thus the adjustment matrix of (9), ( I0 - D,, + D,,I’) has the same eigenvalue as the adjustment matrix of (6) Z - D except for the zero root. Examining (6) it can be seen that the long-run equation is
(11) It can be demonstrated that long-run solution for the attenuated set of equations (9) is identical to the first k - 1 equation of (11). Long-run equilibrium requires that yt = y,_ , and wc = wtP,. Substituting these longrun conditions into (9) gives
ltD1241DI ,4, + D,,B,, -D,*) Now adding up restrictions imply Substituting these restrictions in r:
=(D,,
-D,,Z’)P(P,,
YP=(Bdd
[;][+“P
-Z&Z’)
[:lt+uo.
that, B,, = -I’B,,
B,,(D,,
-Z&Z’)
(12) + B,, = 1 = I’B,,.
4,)
[;lt+“Pq (13)
Eq. (12) demonstrates that the long-run coefficients of the attenuated equations (9) are identical to the long-run coefficients of the first (k - 1) equations of the entire portfolio. We can express the reduced system (9) as yF=D”BoX,+(lO-D’)yL,+u;,
(14)
where Do = D - D,,I’ and xi = (z;, w,, wt_ ,). 2 The link beltkeen the full and the modified set of asset equations can be further explored by a partitioned inverse of the full adjustment matrix. It can be shown that (D” - D121’) = (DO)-‘, where D” and D12 are elements of D - ‘. Many of the burdensome computations associated with inference on the elements of B have been alleviated with introduction by Bewley (1979) of a new formulation of (4). His formulation involves a transformation of (4) to provide the pseudo-structural form, y,=PAy,+Bx,+v, where P=I-A-’ and v=A-‘u. Estimation of (15) by a suitable simultaneous equation estimator has the advantage that the estimates of B are directly observable. Bewley’s full system (15) can be reduced in a similar manner to that described above by dropping an equation and substituting for the change in the last asset. The resulting attenuated system again provides computational economies over the full formulation with elements (and functions of) A. This note has shown that a knowledge of the structure of dynamic portfolio equations can be the basis for a favourable reduction in the size of problem of estimation and inference. By utilizing restrictions within the assets equations it has been possible to provide computational savings while retaining all the properties of the original system.
References Bewley. R.A., 1979, The direct estimation of the equilibrium model, Economics Letters 3. Christofides, L.N., 1976, Quadratic costs and multi-asset Applied Economics. Hunt, B.F. and M.R., Upcher, 1979, Generalized adJustment Economic Papers, Dec. Parkin, M., 1970, Discount house portfolio and debt selection, Valentine, T.J., 1975, Adjustments in employment, overtime the Australian economy, Economic Record 5 I.
’ The long-run meaning.
coefficients
on w, and
w_
,
must be added
response partial
in a linear dynamic
adjustment
equations,
of asset equations,
Australian
Review of Economic Studies. and inventory investiment in
together
before
they have any
355
ESTIMATION OF A SYSTEM OF LINEAR DYNAMIC ASSET DEMAND EQUATIONS Computational Considerations B.F. HUNT Unrorrsrty
Received
*
of New South
12 November
Wales, Kensrngton,
N.S. W., Austruha
1981
The error covariance matrix of a system of linear dynamic asset demand equations is of less than full rank. This property requires that system be modified before estimation. We show that a knowledge of the structure of the asset equations can be the basis for a favourable reduction in the size of the problems of estimation and inference. It is possible to obtain computational savings while retaining all the properties of the original system.
The singularity of the error covariance matrix of a system of asset demand equations requires that some modification is necessary before the equations are suitable for estimation by a systems estimator. This paper presents a form of modification which has the attractive feature of being computationally convenient while preserving the important properties of the original system. Static linear asset demand equations can be derived from a process of maximization of negative exponential utility function under a balance sheet constraint. ’ The portfolio of asset demand equations, at time t can be written as
xi = (z;, W,), y, is a (k X 1) vector of assets demanded, B is a matrix of coefficients; W, is scalar portfolio size variable, z, is a (( p - 1) X 1) vector of other explanatory variables. In order to preserve
where
(k Xp)
* I would like to acknowledge that the comments and suggestions of Ron Bewley and Mark Upcher were particularly helpful to the completion of this note. ’ See for example Parkin (I 970).
0165-1765/82/0000-0000/$02.75
0 1982 North-Holland
the balance sheet constraint it is necessary that I’B = (0.. 0, l), where I is a vector of units. It has been shown, Christofides (1976), Hunt and Upcher (1979) that if there exists quadratic disequilibrium costs and adjustment costs, the static system, (1) is transformed to a first-order dynamic system yl”
(2)
=Bxt,
AY, =D(Y;
-Y,-,),
(3)
where yt* is a (k X 1) vector of desired asset levels, B is a (k X p) matrix of long-run coefficients and D is a full (k X k) matrix of adjustment coefficients. D is consistent with the balance sheet constraint in that I’D = I’. Combining (2) and (3) and dropping the assumption that the equations hold exactly gives y,=DBx,+(Z-D)y,_,+u,, = cx, fAy,_,
+ u,,
(4) (5)
where C = DB, A = Z - D and u, is a (k X 1) vector of disturbances. It is assumed that u, - N(0, a,). It should be emphasized that economic theory does not directly provide assertions about the properties of C or A but rather provides assertions as to the nature of the coefficient matrices D and B. In other words we are not interested in testing direct inferences on the nature of C or A, but rather in using C and A to provide inference on B and D. The fact that B and D are not directly obtained is not a great drawback as provided we are able to obtain consistent estimates of ,C, (6) and A, (a), then we can deduce consistent estimates of D, (D = Z-A) and B, (g = (I --a)-‘c). Moreover we are able to test theory pertaining to B and D as the standard errors of the element of B and D are available from the covariance matrix of the parameters of (5) 9,:, [Valentine (1975) and Hunt and Upcher (1979)]. Q2, = Q2, and 9, =j’QA : J, where J’=(C’(Z-A’)-‘@(Z-A)-‘:Z@(Z-A)-‘). A very important property of the system of dynamic portfolio equations is its stability or lack of stability. Intuitive economic sense supported by economic theory [Hunt and Upcher (1979)] suggest that the characteristic roots of D should lie between zero and unity. Again it is a
comparitively straightforward procedure to obtain the standard errors of the characteristic roots of D from 9,. It follows from the above discussion that point estimates and variances of the elements of B and D and the characteristic roots of D can be obtained from a d and 8, : c. Direct estimates of r? and A^ can be obtained by directly applying a single equation estimator to (5), however this method has considerable drawbacks. Direct estimation is unsatisfactory as it does not produce an estimate of the full covariance matrix fiA : c necessary to calculate standard errors of the long-run coefficients. To achieve a satisfactory estimate fiA : c a systems estimator must be used. Unfortunately a systems estimator cannot be applied to eq. (5) as these equations have a covariance matrix of residuals, L?,,which is singular. The singularity devices from the balance sheet constraint implicit in (4) which dictates that Z’u, = 0 = f’Oll = 0. The singularity of 3, requires that the estimates of e and 2 be obtained indirectly if a systems estimator is to be employed. This can be achieved by dropping one of the equations, say the last, of (4). Consider a partitioning of (4) into the first (k - 1) equations and the last equation,
(6) The dimensions of the various components of (4) are as follows: Y”, D,,, B;, and u” are all ((k 1) X 1; Y’, D,,, B,,, w and u’ are all DA> Bm scalars; D,, and I0 are ((k - 1) X (k - 1)); B,, is ((k - 1) X (p - 1)) z is ((P - 1) X 1). We assume that the covariance matrix of u, has only one zero root, originating from the consistency restrictions. Thus it is possible to apply a systems estimation to the first (k - 1) equations of (4),
Y0= m,B,, f4,B,,D,,B,* +(Z’-D,,-D,,)
+wM[Z,],
1 +u”.
I’0 [ Y’ 1-I
(7)
Systems estimation of (7) is feasible as the assumption of a single zero root in 3, is equivalent to assuming that variance of u’, E(uOuo’) = L?:, is of full rank. Estimation of eq. (7) will yield consistent estimates for the coefficients on zl, wr, ylo and yr2. However, because the coefficient matrix on the lagged dependent variables (y’v’),_, is not longer square it is convenient to further modify (6) by substituting for y,‘, in (7). The adding up restrictions imply that
(8) where I is k - 1 row vector. Substituting
+(zO
-D,,
+D,,I’)yp_,
(8) in (7)
+up.
Comparing the sets of equations modified set of equations (9) has
(9)
(6) with (9) it is obvious
that the
(9
one less equation, a square adjustment coefficient matrix on a reduced endogenous variables, y,%, , (iii) an additional explanatory variable w,_ , . (ii)
set of lagged
The important question is, does the modified eq. (9), have the same properties as the original eq. (6) in respect of the eigenvalues of the adjustment matrix, and of the long-run coefficients? Concentrating on the roots of the adjustment matrix of (6). By performing elementary operations on partitioned I - D we can transform it, firstly by substracting each of the first k - 1 rows from the final row and then secondly, by subtracting the resultant final column from each of the first k - 1 rows, IO-D,,
-42 These
-D,,
’
II"-D,,
I 1- Da, I>I0 elementary
operations
-D,,I 0
ho-D,,
I-0 I
I
fD,,I'
I
do not alter the eigenvalues
-D,,'
O
I.
of the Z - D
matrix.
Examining
the characteristic (Z” -D,,
I(Z-D)-AZl=
equation
+ D,,I’)
-AZ0
0
=I-Xll(Z”
of Z - D -D,, -A
-D,,
+D,,I’)-xZ”j.
(10)
It can be seen from (10) that the values of h which satisfy /(ID) -XZI=O also satisfy I(Z o - D,, + D,,I’) - AZ0 I except for h = 0. Thus the adjustment matrix of (9), ( I0 - D,, + D,,I’) has the same eigenvalue as the adjustment matrix of (6) Z - D except for the zero root. Examining (6) it can be seen that the long-run equation is
(11) It can be demonstrated that long-run solution for the attenuated set of equations (9) is identical to the first k - 1 equation of (11). Long-run equilibrium requires that yt = y,_ , and wc = wtP,. Substituting these longrun conditions into (9) gives
ltD1241DI ,4, + D,,B,, -D,*) Now adding up restrictions imply Substituting these restrictions in r:
=(D,,
-D,,Z’)P(P,,
YP=(Bdd
[;][+“P
-Z&Z’)
[:lt+uo.
that, B,, = -I’B,,
B,,(D,,
-Z&Z’)
(12) + B,, = 1 = I’B,,.
4,)
[;lt+“Pq (13)
Eq. (12) demonstrates that the long-run coefficients of the attenuated equations (9) are identical to the long-run coefficients of the first (k - 1) equations of the entire portfolio. We can express the reduced system (9) as yF=D”BoX,+(lO-D’)yL,+u;,
(14)
where Do = D - D,,I’ and xi = (z;, w,, wt_ ,). 2 The link beltkeen the full and the modified set of asset equations can be further explored by a partitioned inverse of the full adjustment matrix. It can be shown that (D” - D121’) = (DO)-‘, where D” and D12 are elements of D - ‘. Many of the burdensome computations associated with inference on the elements of B have been alleviated with introduction by Bewley (1979) of a new formulation of (4). His formulation involves a transformation of (4) to provide the pseudo-structural form, y,=PAy,+Bx,+v, where P=I-A-’ and v=A-‘u. Estimation of (15) by a suitable simultaneous equation estimator has the advantage that the estimates of B are directly observable. Bewley’s full system (15) can be reduced in a similar manner to that described above by dropping an equation and substituting for the change in the last asset. The resulting attenuated system again provides computational economies over the full formulation with elements (and functions of) A. This note has shown that a knowledge of the structure of dynamic portfolio equations can be the basis for a favourable reduction in the size of problem of estimation and inference. By utilizing restrictions within the assets equations it has been possible to provide computational savings while retaining all the properties of the original system.
References Bewley. R.A., 1979, The direct estimation of the equilibrium model, Economics Letters 3. Christofides, L.N., 1976, Quadratic costs and multi-asset Applied Economics. Hunt, B.F. and M.R., Upcher, 1979, Generalized adJustment Economic Papers, Dec. Parkin, M., 1970, Discount house portfolio and debt selection, Valentine, T.J., 1975, Adjustments in employment, overtime the Australian economy, Economic Record 5 I.
’ The long-run meaning.
coefficients
on w, and
w_
,
must be added
response partial
in a linear dynamic
adjustment
equations,
of asset equations,
Australian
Review of Economic Studies. and inventory investiment in
together
before
they have any