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Efficiency of least-squares-estimation of polynomial trend when residuals are autocorrelated
Economics Letters 016%1765/94/$07.00
45 (1994) 267-271 0 1994 Elsevier
267 Science
B.V. All rights
reserved
Efficiency of least-squares-estimation of polynomial trend when residuals are autocorrelated Ralf Busse, Roland Jeske, Walter Kriimer* Department of Statistics, University of Dortmund, Received Accepted
Dortmund
50, Germany
17 February 1993 18 September 1993
We derive are stationary JEL
P.O. Box 50 05 00, D-46@
lower bounds AR( 1).
classification
for the efficiency
of OLS relative
to GLS in the polynomial
trend
model
where
disturbances
: C22
1. Introduction
Regressing data on polynomial trends is common practice in various branches of empirical economics [see, for example, Shiller (1981) and the subsequent variance bounds literature, where logs of share prices are often regressed on time, and many other applications in empirical finance and economics]. More often than not, the disturbances in such regressions are acknowledged to be serially dependent, but estimation is still done by OLS. Therefore there appears to be interest in the relative efficiency of OLS in polynomial regressions, yr=Po+P*t+...+PKfK+U,
(t=l,.
..,T),
(I)
where the disturbances U, are stationary but not necessarily serially independent. It has long been known [see, for example, Grenander (1954)] that in this model OLS is in general asymptotically as efficient as GLS when sample size T tends to infinity, but lower bounds for the relative efficiency of OLS in finite samples are only known for very special cases. Below we assume stationary AR(l) disturbances u,, where U,=P%l+eI>
lPl
(2)
and where the disturbance covariance matrix V is given by ui, = IJ:~‘~-“. The covariance matrix of the OLS coefficient estimator b = (X’X))‘Xy (where x,; = t’, t = 1,. . . , T; i = 0, . . , K) is then cov(& = (X’X)_‘x’vx(x’x)-‘, and the covariance- matrix of the GLS estimator j!?= (x’v-‘x)-‘x’v-’ y is cov(p”) = (X’V-‘X)-l. The relative efficiency of OLS, defined as * Corresponding SSDI
author.
0165-1765(94)00427-4
268
R. Busse et al. I Economics
e(p)
Letters
45 (1994) 267-271
det(cov(P”))
=
(3)
’
det(cov( p))
is then a function of the disturbance autocorrelation p Grenander (1954) tends to 1 for any given p in (-1, l), but bound for finite T. Extending Chipman et al. (1968), Chipman (1979), and lower bounds by first showing that the efficiency function components, and by applying this factorization to determine quadratic trends.
2. Factorization
of the efficiency
In view of the expressed as
multiplicative
(given T and K), which in view of which has no obvious non-trivial lower Kramer (1980, 1982), we derive such e(p) factors in two somewhat simpler lower bounds for e(p) for linear and
function property
of determinants,
the efficiency
function
e(p)
can be
det(X’X)* c(p) =
det(X’VX)
(4)
’
det(X’V’X)
It is not affected by linear transformations of the regressors, so we can without use the procedure from Draper and Smith (1981, p. 275) to make the columns This leaves the first column L = (1, . . . , 1) as it is, gives a second column . ,T-3,T-1)
F+(-(T-l),-(T-3),.. (a centered
linear
trend),
72 = i2 _
(T
+
loss of generality of X orthogonal.
a third l)i
column
72 with ith position
CT+ ltT + 2, ,
+
and so on, with the columns of the transformed regressor matrix being in turn symmetric (columns 1,3, . . .) or skew symmetric (columns 2,4, . . .). This, however, implies that after the transformation both X’VX and X’V -‘X have zeros in all positions (i, j) where i - j is odd, so det(X’VX)
= det(X(“‘VX”‘)
det(X’z”VX’2’)
(5)
and det(X’V’X)
= det(X(‘)V-‘Xc”)
is the relative
=
,
all odd and Xc2) comprises all even columns e(p) factors into e(p) = e(‘)( p)e”‘( p), where
where X”’ comprises matrix X. Therefore, e”‘(p)
det(X(‘)‘V’X(‘))
(6) of the transformed
det(X”“X”‘)2 det(X”“VX”’
efficiency
in a model
(7)
det(X”)‘V-‘X(l)) comprising
regressor
only the odd columns
and where
269
R. Busse et al. I Economics Letters 45 (1994) 267-271
eC2)( p) =
det(X(2)‘X(2))2
(8)
det(X(*)‘VX(*)) det(X(*)‘VIX(*))
is the relative efficiency of OLS in a model comprising only the even columns of X (with all the columns made orthogonal). This factorization immediately produces a non-trivial lower bound for the relative efficiency of OLS in the case of a linear trend: from Chipman et al. (1968), we have e”‘(p) 2 0.877, and from Chipman (1979), we have e’*‘(p) 2 0.753, so e(p) 2 0.877 X 0.753 = 0.660.
(9)
This lower bound is not sharp, however, as we demonstrate below. of e(p) does not For quadratic and more general polynomial trends, the factorization immediately produce non-trivial lower bounds from known results, but is still a useful stepping stone, as again we demonstrate below.
3. Sharp efficiency
bounds for linear and quadratic
trends
Below we index efficiency functions by sample size and degree of polynomial. We first consider a linear trend y, = PO + P,t + u,, where the efficiency function factors into e,,(p)
= e,,,(p).
e”,(d
(10)
y
with
(11) Using the expressions in Chipman et al. (1968, p. 1240) and Chipman individual factors, we therefore have e,,,(p)
=
(1979, p. 120) for the
CT- W4(T+ U2(1+P)*(l-d’ ’ STbk,(PP,(PY*(P)
(12)
where fr(p) = -6pT+‘[(T-
1)~ - (T + l)]’ - (T3 - T)p4 + 2(T* - l)(T-
+ 12(T2 + l)p2 - 2(T2 - l)(T + 3)p + (T3 - T) ) gr(p) = (Tk,(P) = T(L -
3)(T-
P’) -
2)p2 - 2(7-- 3)(T + l)p + T(T + 1) ) 2p(l-
P’) >
3)p3 (13) (14) (15)
and I,(p)=T-(T-2)p.
(16)
Along the lines of Chipman (1979), one can demonstrate by straightforward but tedious calculations that e,,,(p) has a unique minimum at some p T E (0,l) which decreases with T. This minimum can be expressed as
270
R. Busse et al. I Economics
eT,l 1 i where
-$ 1 ,
z: = T( 1 - p:),
It is easily
Letters 45 (1994) 267-271
checked
(17)
and where
we have from the monotonicity
that
that
2,(2):=IillxeT~,
1 -f C
1 7
1
= (z + 2)(2 - 1 + fez)(z2 has a unique
of the minimum
minimum
+ 62 + 12) (z” - 3.z2 + 12 - 3e-‘(2
in (0, m) at z = 2.893873,
t?,(2.893873)
so the minima
+ 2)2)
of the efficiency
= 0.661850,
functions
(19) tend to (20)
which is at the same time the desired lower bound. Similarly, we derive a sharp lower bound for the relative quadratic trend. Here, the efficiency function factors into
efficiency
of OLS in the case of a
where det[(i:r2)‘(i:r2)] e’(p)
= det[(i:72)‘V(i:72)] =
det[(i:r’)‘V’(i:r’)] + 1)2(T + 2)2(1 + p)(l - p)’
(T - 2)(T - 1)T2(T
7
&(P)g”,(P)
(22)
with g”=(p) = -12(T-
2)(T-
1)(T2 - 2T + 2)pT+4
+ 12(T2 - 4)[(T - 1)(3T - 2)~ - (T + 1)(3T + 12(T + l)(T + T(T2 - l)(T’
+ 2)]~~-~
+ 2)(T2 + 2T + 2)pT+’ - 4)~’ - 3(T - 4)(T2 - 1)(T2 - 4)p4
+ 2(T + 2)(T4 - 20T3 + 35T2 - 1OOT + 24)~~ + 2(T - 2)(T4 + 20T3 + 35T2 + 1OOT + 24)~~ - 3(T + 4)(T2 - 1)(T2 - 4)~ + T(T2 - 1)(T2 - 4) , and h,(p)
= - (T - 4)(T - 3)(T - 2)p3 + 3(T - 4)(T - 3)(T + 2)p2 - 3(T - 4)(T + l)(T
+ 2)~ + T(T + l)(T
+ 2) .
(23)
R. Busse et al. I Economics
Letters 45 (1994) 267-271
271
These expressions might look complicated, but are straightforwardly derived from computing the various determinants in Eq. (22). In the analogy to the linear case, it can be shown that e,,,(p) has a unique minimum p; in (0, l), which is likewise decreasing as T increases. Again defining 2; = T( 1 - p r ), we therefore have (24) with gym
e T.2 12 Z =
(z’
+
62 + 12)(z3 - 32’ + 12 - 3e-‘(2 + 2)2) 1
’ (z’ + 122’ + 6Oz + 120) 1 ’ (z4 - 6z3 + 1202 - 360 + 6e-‘(z3 + 10~’ + 402 + 60)) ’
(25)
This latter function has a unique minimum in (0, co) at z * = 3.194907, so the minima of the efficiency functions tend to e;(3.194907) = 0.490671 , which is the desired lower bound.
References Chipman, J.S. 1979, Efficiency of least squares estimation of linear trend when residuals are autocorrelated, Econometrica 47, 115-128. Chipman, J.S., K.R. Kadiyala, A. Madansky and J.W. Pratt, 1968, Efficiency of the sample mean when residuals follow a first-order stationary Markoff process, Journal of the American Statistical Association 63, 1237-1246. Draper, N.R. and H. Smith, 1981, Applied regression analysis, 2nd. edn. (Wiley, New York). Grenander, U., 1954, On the estimation of regression coefficients in the, case of autocorrelated disturbances, Annals of Mathematical Statistics 25, 252-272. Kramer, W., 1980, Finite sample efficiency of Ordinary Least Squares in the Linear Regression Model with autocorrelated errors, Journal of the American Statistical Association 75, 1005-1009. Kramer, W., 1982, Note on estimating linear trend when residuals are autocorrelated, Econometrica 50, 1065-1067. Shiller. R.J., 1981, Do stock prices move too much to be justified by subsequent changes in dividends?, American Economic Review 71, 421-436.
45 (1994) 267-271 0 1994 Elsevier
267 Science
B.V. All rights
reserved
Efficiency of least-squares-estimation of polynomial trend when residuals are autocorrelated Ralf Busse, Roland Jeske, Walter Kriimer* Department of Statistics, University of Dortmund, Received Accepted
Dortmund
50, Germany
17 February 1993 18 September 1993
We derive are stationary JEL
P.O. Box 50 05 00, D-46@
lower bounds AR( 1).
classification
for the efficiency
of OLS relative
to GLS in the polynomial
trend
model
where
disturbances
: C22
1. Introduction
Regressing data on polynomial trends is common practice in various branches of empirical economics [see, for example, Shiller (1981) and the subsequent variance bounds literature, where logs of share prices are often regressed on time, and many other applications in empirical finance and economics]. More often than not, the disturbances in such regressions are acknowledged to be serially dependent, but estimation is still done by OLS. Therefore there appears to be interest in the relative efficiency of OLS in polynomial regressions, yr=Po+P*t+...+PKfK+U,
(t=l,.
..,T),
(I)
where the disturbances U, are stationary but not necessarily serially independent. It has long been known [see, for example, Grenander (1954)] that in this model OLS is in general asymptotically as efficient as GLS when sample size T tends to infinity, but lower bounds for the relative efficiency of OLS in finite samples are only known for very special cases. Below we assume stationary AR(l) disturbances u,, where U,=P%l+eI>
lPl
(2)
and where the disturbance covariance matrix V is given by ui, = IJ:~‘~-“. The covariance matrix of the OLS coefficient estimator b = (X’X))‘Xy (where x,; = t’, t = 1,. . . , T; i = 0, . . , K) is then cov(& = (X’X)_‘x’vx(x’x)-‘, and the covariance- matrix of the GLS estimator j!?= (x’v-‘x)-‘x’v-’ y is cov(p”) = (X’V-‘X)-l. The relative efficiency of OLS, defined as * Corresponding SSDI
author.
0165-1765(94)00427-4
268
R. Busse et al. I Economics
e(p)
Letters
45 (1994) 267-271
det(cov(P”))
=
(3)
’
det(cov( p))
is then a function of the disturbance autocorrelation p Grenander (1954) tends to 1 for any given p in (-1, l), but bound for finite T. Extending Chipman et al. (1968), Chipman (1979), and lower bounds by first showing that the efficiency function components, and by applying this factorization to determine quadratic trends.
2. Factorization
of the efficiency
In view of the expressed as
multiplicative
(given T and K), which in view of which has no obvious non-trivial lower Kramer (1980, 1982), we derive such e(p) factors in two somewhat simpler lower bounds for e(p) for linear and
function property
of determinants,
the efficiency
function
e(p)
can be
det(X’X)* c(p) =
det(X’VX)
(4)
’
det(X’V’X)
It is not affected by linear transformations of the regressors, so we can without use the procedure from Draper and Smith (1981, p. 275) to make the columns This leaves the first column L = (1, . . . , 1) as it is, gives a second column . ,T-3,T-1)
F+(-(T-l),-(T-3),.. (a centered
linear
trend),
72 = i2 _
(T
+
loss of generality of X orthogonal.
a third l)i
column
72 with ith position
CT+ ltT + 2, ,
+
and so on, with the columns of the transformed regressor matrix being in turn symmetric (columns 1,3, . . .) or skew symmetric (columns 2,4, . . .). This, however, implies that after the transformation both X’VX and X’V -‘X have zeros in all positions (i, j) where i - j is odd, so det(X’VX)
= det(X(“‘VX”‘)
det(X’z”VX’2’)
(5)
and det(X’V’X)
= det(X(‘)V-‘Xc”)
is the relative
=
,
all odd and Xc2) comprises all even columns e(p) factors into e(p) = e(‘)( p)e”‘( p), where
where X”’ comprises matrix X. Therefore, e”‘(p)
det(X(‘)‘V’X(‘))
(6) of the transformed
det(X”“X”‘)2 det(X”“VX”’
efficiency
in a model
(7)
det(X”)‘V-‘X(l)) comprising
regressor
only the odd columns
and where
269
R. Busse et al. I Economics Letters 45 (1994) 267-271
eC2)( p) =
det(X(2)‘X(2))2
(8)
det(X(*)‘VX(*)) det(X(*)‘VIX(*))
is the relative efficiency of OLS in a model comprising only the even columns of X (with all the columns made orthogonal). This factorization immediately produces a non-trivial lower bound for the relative efficiency of OLS in the case of a linear trend: from Chipman et al. (1968), we have e”‘(p) 2 0.877, and from Chipman (1979), we have e’*‘(p) 2 0.753, so e(p) 2 0.877 X 0.753 = 0.660.
(9)
This lower bound is not sharp, however, as we demonstrate below. of e(p) does not For quadratic and more general polynomial trends, the factorization immediately produce non-trivial lower bounds from known results, but is still a useful stepping stone, as again we demonstrate below.
3. Sharp efficiency
bounds for linear and quadratic
trends
Below we index efficiency functions by sample size and degree of polynomial. We first consider a linear trend y, = PO + P,t + u,, where the efficiency function factors into e,,(p)
= e,,,(p).
e”,(d
(10)
y
with
(11) Using the expressions in Chipman et al. (1968, p. 1240) and Chipman individual factors, we therefore have e,,,(p)
=
(1979, p. 120) for the
CT- W4(T+ U2(1+P)*(l-d’ ’ STbk,(PP,(PY*(P)
(12)
where fr(p) = -6pT+‘[(T-
1)~ - (T + l)]’ - (T3 - T)p4 + 2(T* - l)(T-
+ 12(T2 + l)p2 - 2(T2 - l)(T + 3)p + (T3 - T) ) gr(p) = (Tk,(P) = T(L -
3)(T-
P’) -
2)p2 - 2(7-- 3)(T + l)p + T(T + 1) ) 2p(l-
P’) >
3)p3 (13) (14) (15)
and I,(p)=T-(T-2)p.
(16)
Along the lines of Chipman (1979), one can demonstrate by straightforward but tedious calculations that e,,,(p) has a unique minimum at some p T E (0,l) which decreases with T. This minimum can be expressed as
270
R. Busse et al. I Economics
eT,l 1 i where
-$ 1 ,
z: = T( 1 - p:),
It is easily
Letters 45 (1994) 267-271
checked
(17)
and where
we have from the monotonicity
that
that
2,(2):=IillxeT~,
1 -f C
1 7
1
= (z + 2)(2 - 1 + fez)(z2 has a unique
of the minimum
minimum
+ 62 + 12) (z” - 3.z2 + 12 - 3e-‘(2
in (0, m) at z = 2.893873,
t?,(2.893873)
so the minima
+ 2)2)
of the efficiency
= 0.661850,
functions
(19) tend to (20)
which is at the same time the desired lower bound. Similarly, we derive a sharp lower bound for the relative quadratic trend. Here, the efficiency function factors into
efficiency
of OLS in the case of a
where det[(i:r2)‘(i:r2)] e’(p)
= det[(i:72)‘V(i:72)] =
det[(i:r’)‘V’(i:r’)] + 1)2(T + 2)2(1 + p)(l - p)’
(T - 2)(T - 1)T2(T
7
&(P)g”,(P)
(22)
with g”=(p) = -12(T-
2)(T-
1)(T2 - 2T + 2)pT+4
+ 12(T2 - 4)[(T - 1)(3T - 2)~ - (T + 1)(3T + 12(T + l)(T + T(T2 - l)(T’
+ 2)]~~-~
+ 2)(T2 + 2T + 2)pT+’ - 4)~’ - 3(T - 4)(T2 - 1)(T2 - 4)p4
+ 2(T + 2)(T4 - 20T3 + 35T2 - 1OOT + 24)~~ + 2(T - 2)(T4 + 20T3 + 35T2 + 1OOT + 24)~~ - 3(T + 4)(T2 - 1)(T2 - 4)~ + T(T2 - 1)(T2 - 4) , and h,(p)
= - (T - 4)(T - 3)(T - 2)p3 + 3(T - 4)(T - 3)(T + 2)p2 - 3(T - 4)(T + l)(T
+ 2)~ + T(T + l)(T
+ 2) .
(23)
R. Busse et al. I Economics
Letters 45 (1994) 267-271
271
These expressions might look complicated, but are straightforwardly derived from computing the various determinants in Eq. (22). In the analogy to the linear case, it can be shown that e,,,(p) has a unique minimum p; in (0, l), which is likewise decreasing as T increases. Again defining 2; = T( 1 - p r ), we therefore have (24) with gym
e T.2 12 Z =
(z’
+
62 + 12)(z3 - 32’ + 12 - 3e-‘(2 + 2)2) 1
’ (z’ + 122’ + 6Oz + 120) 1 ’ (z4 - 6z3 + 1202 - 360 + 6e-‘(z3 + 10~’ + 402 + 60)) ’
(25)
This latter function has a unique minimum in (0, co) at z * = 3.194907, so the minima of the efficiency functions tend to e;(3.194907) = 0.490671 , which is the desired lower bound.
References Chipman, J.S. 1979, Efficiency of least squares estimation of linear trend when residuals are autocorrelated, Econometrica 47, 115-128. Chipman, J.S., K.R. Kadiyala, A. Madansky and J.W. Pratt, 1968, Efficiency of the sample mean when residuals follow a first-order stationary Markoff process, Journal of the American Statistical Association 63, 1237-1246. Draper, N.R. and H. Smith, 1981, Applied regression analysis, 2nd. edn. (Wiley, New York). Grenander, U., 1954, On the estimation of regression coefficients in the, case of autocorrelated disturbances, Annals of Mathematical Statistics 25, 252-272. Kramer, W., 1980, Finite sample efficiency of Ordinary Least Squares in the Linear Regression Model with autocorrelated errors, Journal of the American Statistical Association 75, 1005-1009. Kramer, W., 1982, Note on estimating linear trend when residuals are autocorrelated, Econometrica 50, 1065-1067. Shiller. R.J., 1981, Do stock prices move too much to be justified by subsequent changes in dividends?, American Economic Review 71, 421-436.