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A joint test for serial correlation and random individual effects
Statistics & Probability North-Holland
Letters
11 (1991) 277-280
March
A joint test for serial correlation individual effects
1991
and random
Badi H. Baltagi and Qi Li Texas A &M University, Department
of Economrcs,
College Station,
TX 77843-4228,
USA
Received March 1990 Revised May 1990
Abstract: This paper derives a simple lagrange multiplier (LM) test which jointly tests the presence of random individual effects and serial correlation. This test is an extension of the Breusch and Pagan (1980) LM test. It is computationally simple and requires only the OLS residuals. It should prove useful for panel data applications where both serial correlation and random individual effects are suspect.
Keywords:
Lagrange
multiplier
test, error components,
Breusch-Pagan
1. Introduction In panel data applications, the error component model, popularized by Balestra and Nerlove (1966) is by far the most widely used specification in economics. This model has been recently extended to take into account serial correlation in the remainder disturbance term by Lillard and Willis (1978). One can test for the existence of the random effects, assuming there is no serial correlation, using the lagrange multiplier (LM) test developed by Breusch and Pagan (1979, 1980). Also, one can test for serial correlation, assuming there are no random effects, using the LM test derived in Godfrey (1978), Breusch (1978) and Breusch and Pagan (1980). This paper considers the question of jointly testing for serial correlation and the random individual effects. In this context, it is important to note that Bhargava, Franzini and Narendranathan (1982) modified the Durbin-Watson statistic to test for serial correlation when the individual effects are assumed fixed. When these individual effects are assumed random, this paper shows that a simple lagrange 0167-7152/91/$03.50
0 1991 - Elsevier Science Publishers
test, serial correlation.
multiplier test can be derived which jointly tests for serial correlation and the random effects. As expected, this LM test is much simpler to compute than the likelihood ratio test which requires the estimation of the unrestricted model, see Breusch and Pagan (1980) Engle (1984) and the recent monograph by Godfrey (1989) for other econometric examples that show the easy applicability of the LM test. In this case, the LM computation requires OLS residuals only, and the resulting test statistic should prove useful for panel data applications.
2. The model and the LM test Consider the following panel data regression: Y,, = x:,/I = 24l, )
i=l,2
,...,
N, t=1,2
,...,
T, (1)
where p is a k x 1 vector of regression coefficients including the intercept, i denoting individuals and t denoting time-periods. Following Lillard and
B.V. (North-Holland)
277
Volume11. Number3
STATISTICS & PROBABILITY LETTERS
Willis (1978) follows:
we model the disturbance terms as
u,, = lJ, + V,I
(2)
with ~1,- IIN(0, 0,‘) and v,~= ~y,,,-r + E,.~, I P I < 1 and E,, - IIN(0, u,‘). The pi’s are independent of the Y,,‘s, and vlo - N(0, uE2/(1- p2)). This is a one-way error component model with individual effects and a serially correlated remainder term. It is well established, see for e.g. Kadiyala (1968), that (l-pz)1’2
0
10
-P :
c=
0
‘.’
0
0
0
'.'
0
0
0
::
.
.
0
0
0
.. .
ip
;
(j
0
0
0
'.'
0
-p
1
u*rl = /Yqu,r
= 41
+ Q>
- P)Z% + G&l
(3)
for i = 1,. . _, N, where (Y= {(l + p)/(l - p) . The remaining observations are transformed, as expected, u,T = (% - P%,,-1) =p,(l--p)+~,~,
t=2 ,...,
T, i=l,...,
where eT is a vector of ones of dimension T, IN is an identity matrix of dimension N, Z-L’= (pr,. .., Pi) and v’= (v,,,. . ., vlT, . . ..vN1... ., >.
(IN@
C)u=
(I,@
Ce,)p+
=(I_p)(ZN8e~)~+(ZN~C)Y 278
Ce, = (1 - p)eF.
u,‘(l - p)“[ I,-$8
e;eF’]
+
u:( IN
8
IT)
since (IN @ C)E(vv’)( Z, @ C’) = e,‘(Z, This can be re-written as ~*=d%,2(1-p)2[z,@
eFeF’/d’]
(6) @ Z,).
+ cf2( Z, @ IT) (7)
=o~(ZN@a~;)+o,‘(ZN@E;)
(8)
where u,’ = d2q2(1 - p)2 + oF2 is the first characteristic root of sZ* of multiplicity N and u,” is the second characteristic root of Q* of multiplicity N(T - l), see Baltagi and Li (1990). Therefore, f2*P=(~~)P(ZN@~;)+(u~)P(ZN@E;)
(9)
where p is any arbitrary scalar. p = - 1 obtains the inverse, while p = - i obtains Q*-“2. D= E(uu’) is related to tin* by sZ* = (IN @ C)a(ZN @ C’) and (Cl =/g, lZN@Cl = lCIN and I a* I = (#(T-l) (CT,‘)“. Therefore, the log likelihood function can be written as: L(P,
e;> p, eE2) = const.-
:N(T-
1) log uE2
;N log(1 - p2)
- ;N log[ u,’ + u;d*(l
u=(Z,@e,)p++
=
=
+
In vector form:
u*
and
3* = E(u*u*‘)
N. (4)
‘NT
= ((Y, ek_,)
where d2 = e;‘eF = a* + (T - 1). This replaces eFe$ by its idempotent counterpart JF = eFeF’/d2. Replacing I, by EF +.?f, where Ea = I, - J,” and collecting terms with the scme matrices, see Wansbeek and Kapteyn (1982,1983), one gets the spectral decomposition of G*:
will transform the usual AR(l) model into a serially uncorrelated regression with independent observations. The initial observation for each individual i, will be transformed as follows:
= \il(p’,
where e; Therefore,
March 1991
(IN@ C)y (5)
-&*‘a
*-I u*
- p)‘] (IO)
where a*-’ is given by (9). Following Breusch and Pagan (1980) we let e = cu,2, p, uE2)‘. The information matrix will be block diagonal between the e and p parameters, and since the hypothesis tested H,; p = 0 and U* = 0 involves only the B parameters, the part of tie iniormation matrix corresponding to j3 will be
Volume
11, Number
STATISTICS
3
& PROBABILITY
ignored in computing the LM statistic, see equation (7) of Breusch and Pagan (1980, p. 241). LM = fi,;&; ‘fii
(II)
where & = (aL/a0)(6) is a 3 x 1 vector of partial derivatives of the likelihood function with respect to each element of 8, evaluated at the restricted mle 6. Jl, = E[ --a2L/ae ae’] is the part of the information matrix corresponding to 8, and J;, is J,, when the null hypothesis is true, evaluated at the restricted mle 6. Under the null hypothesis, the variance-covariance matrix reduces to Q* = 52 = a,‘&. and the restricted mle of p is poLs, so are the OLS residuals and that ii = y - x’Bo,, 6’F = ii'ii/NT. In fact,
I
$(l-
fi,=
F
1991
statistic reverts back to (NT)A2/2(T - 1) reported in Breusch and Pagan (1980). Similarly, if uF2= 0 and one is testing H,: p = 0, then 0 = (p, u,‘)’ and we ignore the first element of fii and the first row and column of J;,. In this case, the LM statistic becomes [NT2/(T - 1)] ~(A’fi_,,‘C’fi)~ = NT2fi2/(T-1). In conclusion, the LM statistic for the joint test H,: p = 0 and up2= 0, given in (12), involves an interaction term (4AB) in addition to the familiar A2 and B2 terms. This clearly demonstrates the importance of carrying out this joint test whenever both serial correlation and random individual effects are suspect.
E
NT(fi'K,/ii'E) 0
The authors would like to thank referee for his helpful comments.
2(T-1)$
T 2&4
T 0
and the LM statistic
NT2
LM = 2(T _ l)(T
0
References
’
1I
is given by
_ 2) [ A2 + 4AB + 2TB2]
(12) and is asymptotically null hypothesis.
distributed
[c’(I,,,czJJ~)c/c’G]
as xi
under
the
and
B=ii'L,,'ii'ii. A is the familiar
an anonymous
I
2(T-l)$ 1
A = l-
March
Acknowledgement
ii’(r,C%&-)ii/ii’ii)
T &=f$
LETTERS
term used in testing ep2= 0, see Breusch and Pagan (1980), while B = p is the familiar estimate of p used in testing p = 0. These terms can be easily computed from the OLS residuals. If p = 0, and one is testing H,: u,‘= 0, the same derivation applies except 0 = (a:, a,‘)’ and we ignore the second element of d, and the second row and column of Jy,. In this case, the LM
Balestra, P. and M. Nerlove (1966), Pooling cross-section and time-series data in the estimation of a dynamic model: the demand for natural gas, Econometrica 34, 585-612. Baltagi, B. and Q, Li (1990), A transformation that will circumvent the problem of autocorrelation in an error component model, J. of Economerr~s, forthcoming. Bhargava, A., L. Franzini and W. Narendranathan (1982), Serial correlation and the fixed effects model, Reu. Econom. Stud. 49, 533-549. Breusch, T.S. (1978), Testing for autocorrelation in dynamic linear models, Aurtral. Econom. Papers 17, 334-355. Breusch, T.S. and A.R. Pagan (1979), A simple test for heteroscedasticity and random coefficient variation, Econometrica 47, 1287-1294. Breusch, T.S. and A.R. Pagan (1980), The Lagrange multiplier test and its applications to model specification in econometrics, Rev. Econom. Stud. 47, 239-253. Engle, R.F. (1984). Wald, likelihood ratio, and Lagrange multiplier tests in econometrics, in: Z. Griliches and M.D. Intrilligator, eds., Handbook of Econometrics (North-Holland, Amsterdam). Godfrey, L.C. (1978), Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables, Econometrica 46, 12931302. Godfrey, L.C. (1989). Muspecification Tests in Econometrics (Cambridge Univ. Press, Cambridge). 279
Volume
11, Number
3
STATISTICS
& PROBABILITY
Lillard, L.A. and R.J. Willis (1978), Dynamic aspects of eaming mobility, Econometrica, 46, 985-1012. Wansbeek, T. and A. Kapteyn (1982), A simple way to obtain the spectral decomposition of variance components models for balanced data, Comm. Statist. All, 2105-2112.
280
LETI-ERS
March
1991
Wansbeek, T. and A. Kapteyn (1983), A note on spectral decomposition and maximum likelihood estimation of ANOVA models with balanced data, Statisf. Probab. Letf. 1, 213-215.
Letters
11 (1991) 277-280
March
A joint test for serial correlation individual effects
1991
and random
Badi H. Baltagi and Qi Li Texas A &M University, Department
of Economrcs,
College Station,
TX 77843-4228,
USA
Received March 1990 Revised May 1990
Abstract: This paper derives a simple lagrange multiplier (LM) test which jointly tests the presence of random individual effects and serial correlation. This test is an extension of the Breusch and Pagan (1980) LM test. It is computationally simple and requires only the OLS residuals. It should prove useful for panel data applications where both serial correlation and random individual effects are suspect.
Keywords:
Lagrange
multiplier
test, error components,
Breusch-Pagan
1. Introduction In panel data applications, the error component model, popularized by Balestra and Nerlove (1966) is by far the most widely used specification in economics. This model has been recently extended to take into account serial correlation in the remainder disturbance term by Lillard and Willis (1978). One can test for the existence of the random effects, assuming there is no serial correlation, using the lagrange multiplier (LM) test developed by Breusch and Pagan (1979, 1980). Also, one can test for serial correlation, assuming there are no random effects, using the LM test derived in Godfrey (1978), Breusch (1978) and Breusch and Pagan (1980). This paper considers the question of jointly testing for serial correlation and the random individual effects. In this context, it is important to note that Bhargava, Franzini and Narendranathan (1982) modified the Durbin-Watson statistic to test for serial correlation when the individual effects are assumed fixed. When these individual effects are assumed random, this paper shows that a simple lagrange 0167-7152/91/$03.50
0 1991 - Elsevier Science Publishers
test, serial correlation.
multiplier test can be derived which jointly tests for serial correlation and the random effects. As expected, this LM test is much simpler to compute than the likelihood ratio test which requires the estimation of the unrestricted model, see Breusch and Pagan (1980) Engle (1984) and the recent monograph by Godfrey (1989) for other econometric examples that show the easy applicability of the LM test. In this case, the LM computation requires OLS residuals only, and the resulting test statistic should prove useful for panel data applications.
2. The model and the LM test Consider the following panel data regression: Y,, = x:,/I = 24l, )
i=l,2
,...,
N, t=1,2
,...,
T, (1)
where p is a k x 1 vector of regression coefficients including the intercept, i denoting individuals and t denoting time-periods. Following Lillard and
B.V. (North-Holland)
277
Volume11. Number3
STATISTICS & PROBABILITY LETTERS
Willis (1978) follows:
we model the disturbance terms as
u,, = lJ, + V,I
(2)
with ~1,- IIN(0, 0,‘) and v,~= ~y,,,-r + E,.~, I P I < 1 and E,, - IIN(0, u,‘). The pi’s are independent of the Y,,‘s, and vlo - N(0, uE2/(1- p2)). This is a one-way error component model with individual effects and a serially correlated remainder term. It is well established, see for e.g. Kadiyala (1968), that (l-pz)1’2
0
10
-P :
c=
0
‘.’
0
0
0
'.'
0
0
0
::
.
.
0
0
0
.. .
ip
;
(j
0
0
0
'.'
0
-p
1
u*rl = /Yqu,r
= 41
+ Q>
- P)Z% + G&l
(3)
for i = 1,. . _, N, where (Y= {(l + p)/(l - p) . The remaining observations are transformed, as expected, u,T = (% - P%,,-1) =p,(l--p)+~,~,
t=2 ,...,
T, i=l,...,
where eT is a vector of ones of dimension T, IN is an identity matrix of dimension N, Z-L’= (pr,. .., Pi) and v’= (v,,,. . ., vlT, . . ..vN1... ., >.
(IN@
C)u=
(I,@
Ce,)p+
=(I_p)(ZN8e~)~+(ZN~C)Y 278
Ce, = (1 - p)eF.
u,‘(l - p)“[ I,-$8
e;eF’]
+
u:( IN
8
IT)
since (IN @ C)E(vv’)( Z, @ C’) = e,‘(Z, This can be re-written as ~*=d%,2(1-p)2[z,@
eFeF’/d’]
(6) @ Z,).
+ cf2( Z, @ IT) (7)
=o~(ZN@a~;)+o,‘(ZN@E;)
(8)
where u,’ = d2q2(1 - p)2 + oF2 is the first characteristic root of sZ* of multiplicity N and u,” is the second characteristic root of Q* of multiplicity N(T - l), see Baltagi and Li (1990). Therefore, f2*P=(~~)P(ZN@~;)+(u~)P(ZN@E;)
(9)
where p is any arbitrary scalar. p = - 1 obtains the inverse, while p = - i obtains Q*-“2. D= E(uu’) is related to tin* by sZ* = (IN @ C)a(ZN @ C’) and (Cl =/g, lZN@Cl = lCIN and I a* I = (#(T-l) (CT,‘)“. Therefore, the log likelihood function can be written as: L(P,
e;> p, eE2) = const.-
:N(T-
1) log uE2
;N log(1 - p2)
- ;N log[ u,’ + u;d*(l
u=(Z,@e,)p++
=
=
+
In vector form:
u*
and
3* = E(u*u*‘)
N. (4)
‘NT
= ((Y, ek_,)
where d2 = e;‘eF = a* + (T - 1). This replaces eFe$ by its idempotent counterpart JF = eFeF’/d2. Replacing I, by EF +.?f, where Ea = I, - J,” and collecting terms with the scme matrices, see Wansbeek and Kapteyn (1982,1983), one gets the spectral decomposition of G*:
will transform the usual AR(l) model into a serially uncorrelated regression with independent observations. The initial observation for each individual i, will be transformed as follows:
= \il(p’,
where e; Therefore,
March 1991
(IN@ C)y (5)
-&*‘a
*-I u*
- p)‘] (IO)
where a*-’ is given by (9). Following Breusch and Pagan (1980) we let e = cu,2, p, uE2)‘. The information matrix will be block diagonal between the e and p parameters, and since the hypothesis tested H,; p = 0 and U* = 0 involves only the B parameters, the part of tie iniormation matrix corresponding to j3 will be
Volume
11, Number
STATISTICS
3
& PROBABILITY
ignored in computing the LM statistic, see equation (7) of Breusch and Pagan (1980, p. 241). LM = fi,;&; ‘fii
(II)
where & = (aL/a0)(6) is a 3 x 1 vector of partial derivatives of the likelihood function with respect to each element of 8, evaluated at the restricted mle 6. Jl, = E[ --a2L/ae ae’] is the part of the information matrix corresponding to 8, and J;, is J,, when the null hypothesis is true, evaluated at the restricted mle 6. Under the null hypothesis, the variance-covariance matrix reduces to Q* = 52 = a,‘&. and the restricted mle of p is poLs, so are the OLS residuals and that ii = y - x’Bo,, 6’F = ii'ii/NT. In fact,
I
$(l-
fi,=
F
1991
statistic reverts back to (NT)A2/2(T - 1) reported in Breusch and Pagan (1980). Similarly, if uF2= 0 and one is testing H,: p = 0, then 0 = (p, u,‘)’ and we ignore the first element of fii and the first row and column of J;,. In this case, the LM statistic becomes [NT2/(T - 1)] ~(A’fi_,,‘C’fi)~ = NT2fi2/(T-1). In conclusion, the LM statistic for the joint test H,: p = 0 and up2= 0, given in (12), involves an interaction term (4AB) in addition to the familiar A2 and B2 terms. This clearly demonstrates the importance of carrying out this joint test whenever both serial correlation and random individual effects are suspect.
E
NT(fi'K,/ii'E) 0
The authors would like to thank referee for his helpful comments.
2(T-1)$
T 2&4
T 0
and the LM statistic
NT2
LM = 2(T _ l)(T
0
References
’
1I
is given by
_ 2) [ A2 + 4AB + 2TB2]
(12) and is asymptotically null hypothesis.
distributed
[c’(I,,,czJJ~)c/c’G]
as xi
under
the
and
B=ii'L,,'ii'ii. A is the familiar
an anonymous
I
2(T-l)$ 1
A = l-
March
Acknowledgement
ii’(r,C%&-)ii/ii’ii)
T &=f$
LETTERS
term used in testing ep2= 0, see Breusch and Pagan (1980), while B = p is the familiar estimate of p used in testing p = 0. These terms can be easily computed from the OLS residuals. If p = 0, and one is testing H,: u,‘= 0, the same derivation applies except 0 = (a:, a,‘)’ and we ignore the second element of d, and the second row and column of Jy,. In this case, the LM
Balestra, P. and M. Nerlove (1966), Pooling cross-section and time-series data in the estimation of a dynamic model: the demand for natural gas, Econometrica 34, 585-612. Baltagi, B. and Q, Li (1990), A transformation that will circumvent the problem of autocorrelation in an error component model, J. of Economerr~s, forthcoming. Bhargava, A., L. Franzini and W. Narendranathan (1982), Serial correlation and the fixed effects model, Reu. Econom. Stud. 49, 533-549. Breusch, T.S. (1978), Testing for autocorrelation in dynamic linear models, Aurtral. Econom. Papers 17, 334-355. Breusch, T.S. and A.R. Pagan (1979), A simple test for heteroscedasticity and random coefficient variation, Econometrica 47, 1287-1294. Breusch, T.S. and A.R. Pagan (1980), The Lagrange multiplier test and its applications to model specification in econometrics, Rev. Econom. Stud. 47, 239-253. Engle, R.F. (1984). Wald, likelihood ratio, and Lagrange multiplier tests in econometrics, in: Z. Griliches and M.D. Intrilligator, eds., Handbook of Econometrics (North-Holland, Amsterdam). Godfrey, L.C. (1978), Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables, Econometrica 46, 12931302. Godfrey, L.C. (1989). Muspecification Tests in Econometrics (Cambridge Univ. Press, Cambridge). 279
Volume
11, Number
3
STATISTICS
& PROBABILITY
Lillard, L.A. and R.J. Willis (1978), Dynamic aspects of eaming mobility, Econometrica, 46, 985-1012. Wansbeek, T. and A. Kapteyn (1982), A simple way to obtain the spectral decomposition of variance components models for balanced data, Comm. Statist. All, 2105-2112.
280
LETI-ERS
March
1991
Wansbeek, T. and A. Kapteyn (1983), A note on spectral decomposition and maximum likelihood estimation of ANOVA models with balanced data, Statisf. Probab. Letf. 1, 213-215.