- Email: [email protected]
Mean lag in general error correction models
Economics Letters 143 (2016) 107–110
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Mean lag in general error correction models Peter Fuleky a,b,∗ , Luigi Ventura c a
UHERO, University of Hawaii, United States
b
Department of Economics, University of Hawaii, United States
c
Department of Economics and Law, Sapienza, University of Rome, Italy
article
info
Article history: Received 25 January 2016 Accepted 30 March 2016 Available online 12 April 2016 JEL classification: B41 C18 C22 C32 C50
abstract Most of the empirical literature inappropriately applies Hendry’s (1995) mean lag formula – which he derived for first order autoregressive distributed lag models under the assumption of a homogeneous long-run equilibrium – to error correction models that have complex lag structures and lack long-run homogeneity. We derive an expression for the mean lag in general error correction models without imposing the assumption of a homogeneous equilibrium. In addition, we quantify the bias due to the incorrect use of Hendry’s (1995) formula. © 2016 Elsevier B.V. All rights reserved.
Keywords: Mean lag Autoregressive distributed lag model Error correction model
1. Introduction The mean lag is a summary measure of the lag structure of dynamic models. It can be used to estimate the average delay in the transmission of shocks, such as the passthrough of income shocks to consumption, oil price shocks to gas prices, or market interest rates to retail rates, among others. A large number of empirical studies have resorted to an explicit mean lag formula published by Hendry (1995, p. 215, Eq. (6.53)). He derived it for the first order autoregressive distributed lag (ADL(1, 1)) model and the associated error correction (EC ) model under the assumption of a homogeneous equilibrium relationship. However, the formula is invalid in cases when the lag structure is more complex or the long-run homogeneity assumption does not hold. Nonetheless, we found a number of studies that use the formula inappropriately under these more general conditions, including Chong and Liu (2009); De Bondt (2005); Charoenseang and Manakit (2007); De Graeve et al. (2007); Leibrecht and Scharler (2008, 2011); Leszkiewicz-Kedzior and Welfe (2014); Scholnick (1996), among others. We fill a gap in the literature by deriving an expression for the mean lag in general EC models without imposing the assumption
of a homogeneous equilibrium. In addition, we evaluate the bias of the mean lag estimate arising from inappropriately imposing longrun homogeneity. 2. General form of the mean lag In this section we derive the mean lag in a general, nonhomogeneous, relationship. A general autoregressive distributed lag, or ADL(p, q; n), model can be written as yt = c +
p i=1
α(L)yt = c +
βk,j xk,t −j + ϵt or
k=1 j=0 n
(1)
βk (L)xk,t + ϵt , p
q
j where ϵt ∼ IID, α(L) = 1 − i=1 αi Li and βk (L) = j=0 βk,j L are lag polynomials, and n is the number of exogenous variables in the model. Regressors with varying lag lengths can be readily accommodated at the cost of further notational complexity. Rearrange Eq. (1) to obtain the reduced form equation
c
α(L)
∗
http://dx.doi.org/10.1016/j.econlet.2016.03.028 0165-1765/© 2016 Elsevier B.V. All rights reserved.
q n
k=1
yt = Correspondence to: 2424 Maile Way, Saunders Hall 508, Honolulu, HI 96822, USA. Tel.: +1 808 956 7840. E-mail address: [email protected] (P. Fuleky).
αi yt −i +
= c∗ +
+
1
n
α(L)
k=1
n k=1
βk (L)xk,t +
wk (L)xk,t + ut ,
ϵt α(L) (2)
108
P. Fuleky, L. Ventura / Economics Letters 143 (2016) 107–110
β (L)
k where wk (L) = α( = j=1 wk,j Lj . The ‘‘weight’’ associated with L) ∂y lag j of variable xk , wk,j = ∂ x t , captures the effect of xk,t −j on yt .
∞
k,t −j
Hendry (1995, p. 215) defined the mean lag as ∞
µk =
jwk,j
j =0
∞
= wk,j
∂wk (L) wk (1) ∂L
1
L=1
j =0
βk′ (L) βk (L)α ′ (L) = − wk (1) α(L) α(L)2 L =1 ′ 1 βk (L) α ′ (L) β ′ (1) α ′ (1) = wk (L) − = k − , wk (1) βk (L) α(L) L=1 βk (1) α(1) 1
model, the mean lag defined in Eq. (3) takes the following form
β1 + 2β2 + 3β3 α1 + 2α2 + 3α3 + β0 + β1 + β2 + β3 1 − α1 − α2 − α3 κω + b0 + b1 + b2 1 + κ − a1 − a2 = − κω κ ω(a1 + a2 − 1) + b0 + b1 + b2 = . (8) κω If κ ̸= 0, ωk ̸= 0, then consistent estimation of the parameters θ = (κ, ω, a1 , a2 , b0 , b1 , b2 )′ in Eq. (7) allows us to obtain a consistent estimate of the mean lag, µ ˆ = µ(θˆ ), and its variance ∂µ(θˆ ) ∂µ(θˆ ) ˆ Var(µ) ˆ = ∂θ ′ Var(θ ) ∂θ , where Var(θˆ ) is the covariance matrix µ=
of coefficients estimated in Eq. (7).
Before generalizing this result to an EC (p − 1, q − 1; n) model
(3) where z ′ = ∂∂ zL . Note, the mean lag associated with variable xk does not depend on the coefficients of the other variables xl , l ̸= k. In Eq. (3), wk (1) represents the long run impact of xk on y. n Consequently, yt − c ∗ − k=1 wk (1)xk,t captures a deviation from the long-run equilibrium between the dependent variable y and regressors x1 . . . xn . Following the steps outlined in Section 2.1 of Banerjee et al. (1993), the ADL(p, q, n) model in Eq. (1) can be transformed into an EC (p − 1, q − 1; n) model
1yt = κ yt −1 − c ∗ −
n
ωk xk,t −1 +
k=1
n
bk,0 1xk,t
k =1
1yt = κ yt −1 − c − ∗
n
ωk xk,t −1 +
k=1
+
q −1 n
p−1
ai 1yt −i
i =1
bk,j 1xk,t −j + ϵt ,
(9)
k=1 j=0
we make the following set of assumptions: Assumption 1. The variables y, x1 . . . xn entering models (1) and (9) are either jointly stationary, n or cointegrated with a stationary equilibrium error yt − c ∗ − k=1 ωk xk,t .
(4)
Assumption 2. The error in Eqs. (1) and (9), ϵt , is independently and identically distributed and is independent of the variables x1 . . . xn .
p−1 j 1 aj L with j= q j aj = − i=j+1 αi , bk (L) = j=1 bk,j L with bk,j = − i=j+1 βi , and p − 1 and q − 1 stand for the maximum lag lengths of 1y and 1x,
Assumption 3. The parameters in Eq. (9), θ = (κ, ω1 . . . ωn , a1 . . . ap−1 , b1,0 . . . b1,q−1 . . . bn,0 . . . bn,q−1 )′ , are estimated consistently with an estimator that has an asymptotically normal distri-
+ a(L)1yt +
n
bk (L)1xk,t + ϵt ,
k=1
where κ = −α(1), ωk = wk (1), bk,0 = βk,0 , a(L) =
p
q −1
respectively. By convention, a term does not enter the summation if the lower limit exceeds the upper limit. The models described by Eqs. (1) and (4) are isomorphic. Example 1. Transformation of the ADL(3, 3; 1) model
√
bution
Assumption 4. κ ̸= 0, ωk ̸= 0 for k ∈ {1 . . . n}. The mean lag, µk (θ ), is a continuous function of θ and is continuously differentiable with respect to θ . Proposition 1. Under Assumptions 1–4 the mean lag estimator
yt = c + α1 yt −1 + α2 yt −2 + α3 yt −3 + β0 xt + β1 xt −1
+ β2 xt −2 + β3 xt −3 + ϵt ,
ωˆ k
(5)
µ ˆ k = µk (θˆ ) =
yields the following EC (2, 2; 1) model
1yt = −(1 − α1 − α2 − α3 ) yt −1 − 1 − α1 − α2 − α3 β0 + β1 + β2 + β3 − xt −1 + β0 1xt 1 − α1 − α2 − α3 − (α2 + α3 )1yt −1 − α3 1yt −2 − (β2 + β3 )1xt −1
√
(6)
which can be estimated in a simplified form
1yt = κ yt −1 − c ∗ − ωxt −1 + b0 1xt + a1 1yt −1 + a2 1yt −2 + b1 1xt −1 + b2 1xt −2 + ϵt .
p−1
q −1
aˆ i − 1
i=1
+
j =0
κˆ ωˆ k
bˆ k,j
,
(10)
is consistent and asymptotically normally distributed
c
− β3 1xt −2 + ϵt ,
d
T (θˆ − θ ) −→ N (0, Var(θ )).
(7)
The expression in brackets represents the equilibrium error. The coefficients estimated in (7) can be mapped back to the ones in (5) and (6) with β0 = b0 , β1 = b1 − b0 − κω, β2 = b2 − b1 , β3 = −b2 , α1 = 1 +κ + a1 , α2 = a2 − a1 , α3 = −a2 . Hence, for the EC (2, 2; 1)
d
T (µk (θˆ ) − µk (θ )) −→ N
0,
∂µk (θ ) ∂µk (θ ) Var (θ ) . ∂θ ′ ∂θ
(11)
Proposition 1 extends the results obtained in Example 1 to a general EC (p − 1, q − 1; n) model. The details of the proof are provided in an Online Supplement on the first author’s homepage. Remark 1.1. If the variables y, x1 . . . xn are cointegrated, then the ′ elements of the cointegrating √ vector, ω = (ω1 . . . ωn ) , are ˆ − ω) = op (1). As a result, the estimated super-consistently: T (ω Var(θ ) components associated with the cointegrating vector, ω, converge to zero and do not contribute to the asymptotic variance of the mean lag estimator in (11). The Supplement contains a more detailed exposition of this issue (see Appendix A). Perhaps due to previous unavailability of the formula presented in Eq. (10), some researchers have ignored the lag structure of
P. Fuleky, L. Ventura / Economics Letters 143 (2016) 107–110
109
Fig. 1. Bias and relative bias of the mean lag estimate arising from an inappropriate assumption of homogeneity in an EC (p − 1, q − 1; n) model with
q−1 j=0
p−1 i=1
ai = 0.6,
bk,j = 0.2 and κ = −0.4.
Table 1 Impact of imposing long-run homogeneity in the literature. Study A B C
Item
ˆ µ( ¯ θ)
ˆ µ(θ)
Relative bias
Overnight Maturity over 2 years IB TD6-12 Footnote 8
1.58 0.97 0.26 7.09 2.90
1.24 0.50 0.14 2.63 2.20
27% 94% 78% 169% 39%
Note: Illustration of bias in the mean lag estimate due to an inappropriate assumption of long-run homogeneity. Panels (A)–(C) refer to the following published results: (A) Table 8 in De Bondt (2005), (B) Table 6 in Charoenseang and Manakit (2007), (C) page 501 in Leibrecht and Scharler (2008). Each of these studies ˆ defined in Eq. (15) in place of µ(θ) ˆ defined in Eq. (12) to incorrectly used µ( ¯ θ) estimate the mean lag.
their EC models when estimating the mean lag (see for example De Graeve et al., 2007; Leibrecht and Scharler, 2011). Corollary 1.1. For the frequently used ADL(1, 1; n) and the associated EC (0, 0; n) model, expression (10) simplifies to
ˆ = µk (θ)
bˆ k,0 − ω ˆk
κˆ ωˆ k
.
(12)
An error correction model is considered to be homogeneous if, for each k, the xk and y variables move one-for-one in equilibrium. Homogeneity implies ωk = wk (1) = 1 or α(1) = βk (1), leading to a special form of the mean lag
β ′ (1) − α ′ (1) β ′ (1) − α ′ (1) µ ¯k = k = k . α(1) βk (1)
(13)
Corollary 1.2. The mean lag estimator associated with variable xk in a homogeneous EC (p − 1, q − 1; n) model takes the form p−1
ˆ = µ ¯ k (θ)
q−1
aˆ i − 1 +
i =1
j =0
κˆ
bˆ k,j
,
(14)
which simplifies to
ˆ = µ ¯ k (θ)
bˆ k,0 − 1
κˆ
(ωk − 1)
q −1
bk,j
j=0
µ ¯ k (θ ) − µk (θ ) =
(16)
κωk
and the relative bias
µ ¯ k (θ ) − µk (θ ) = µk (θ )
(ωk − 1)
q −1
.
(15)
for a homogeneous EC (0, 0; n) model. Expression (15) is equivalent to the formula derived by Hendry (1995, p. 215, Eq. (6.53)). Although this formula does not hold for general, non-homogeneous, relationships between xk and y, it has been inappropriately used in place of Eq. (12) by several
bk,j
j =0
ωk
p−1
(17)
q −1
ai − 1
+
i=1
3. Mean lag under long-run homogeneity
researchers. The list of studies that relied on Eq. (15) despite nonunity ωk coefficients includes Chong and Liu (2009); De Bondt (2005); Charoenseang and Manakit (2007); De Graeve et al. (2007); Leszkiewicz-Kedzior and Welfe (2014); Leibrecht and Scharler (2008, 2011); Scholnick (1996), among others. In Table 1, we illustrate the impact of imposing long-run homogeneity in some of these studies. Expressions (10) and (14) allow us to quantify the bias in the mean lag estimate arising from an inappropriate assumption of homogeneity. The bias
bk,j
j =0
vanish as ωk → 1, but diverge otherwise. Fig. 1 illustrates the p−1 magnitude of the bias and relative bias for ωk ∈ (−2, 2), i=1 ai = 0.6,
q −1 j =0
bk,j = 0.2, and κ = −0.4.
4. Concluding remarks Researchers have been inappropriately using Hendry’s (1995) mean lag formula – which he derived for homogeneous EC (0, 0, 1) models – in more complex settings. We fill a gap in the literature by deriving an expression for the mean lag in general EC models, and show that using the incorrect formula can have a sizable impact on the estimated delay in the transmission of shocks. The ADL and EC models discussed above can be viewed as components of vector autoregressive (VAR) and vector error correction (VEC ) models, respectively. Consequently, the presented results are also valid for the mean lags of variables in individual equations of VAR and VEC models. Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2016.03.028.
110
P. Fuleky, L. Ventura / Economics Letters 143 (2016) 107–110
References Banerjee, A., Dolado, J., Galbraith, J., Hendry, D., 1993. Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. Oxford University Press. Charoenseang, J., Manakit, P., 2007. Thai monetary policy transmission in an inflation targeting era. J. Asian Econ. 18 (1), 144–157. Chong, B.S., Liu, M.-H., 2009. Islamic banking: interest-free or interest-based? Pac.Basin Finance J. 17 (1), 125–144. De Bondt, G.J., 2005. Interest rate pass-through: Empirical results for the euro area. Ger. Econ. Rev. 6 (1), 37–78.
De Graeve, F., De Jonghe, O., Vander Vennet, R., 2007. Competition, transmission and bank pricing policies: Evidence from belgian loan and deposit markets. J. Banking Finance 31 (1), 259–278. Hendry, D., 1995. Dynamic Econometrics. Oxford University Press. Leibrecht, M., Scharler, J., 2008. Reconsidering consumption risk sharing among oecd countries: some evidence based on panel cointegration. Open Econ. Rev. 19 (4), 493–505. Leibrecht, M., Scharler, J., 2011. Borrowing constraints and international risk sharing: evidence from asymmetric error correction. Appl. Econ. 43, 2177–2184. Leszkiewicz-Kedzior, K., Welfe, A., 2014. Asymmetric price adjustments in the fuel market. Cent. Eur. J. Econ. Model. Econom. 2, 105–127. Scholnick, B., 1996. Asymmetric adjustment of commercial bank interest rates: evidence from malaysia and singapore. J. Int. Money Finance 15 (3), 485–496.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Mean lag in general error correction models Peter Fuleky a,b,∗ , Luigi Ventura c a
UHERO, University of Hawaii, United States
b
Department of Economics, University of Hawaii, United States
c
Department of Economics and Law, Sapienza, University of Rome, Italy
article
info
Article history: Received 25 January 2016 Accepted 30 March 2016 Available online 12 April 2016 JEL classification: B41 C18 C22 C32 C50
abstract Most of the empirical literature inappropriately applies Hendry’s (1995) mean lag formula – which he derived for first order autoregressive distributed lag models under the assumption of a homogeneous long-run equilibrium – to error correction models that have complex lag structures and lack long-run homogeneity. We derive an expression for the mean lag in general error correction models without imposing the assumption of a homogeneous equilibrium. In addition, we quantify the bias due to the incorrect use of Hendry’s (1995) formula. © 2016 Elsevier B.V. All rights reserved.
Keywords: Mean lag Autoregressive distributed lag model Error correction model
1. Introduction The mean lag is a summary measure of the lag structure of dynamic models. It can be used to estimate the average delay in the transmission of shocks, such as the passthrough of income shocks to consumption, oil price shocks to gas prices, or market interest rates to retail rates, among others. A large number of empirical studies have resorted to an explicit mean lag formula published by Hendry (1995, p. 215, Eq. (6.53)). He derived it for the first order autoregressive distributed lag (ADL(1, 1)) model and the associated error correction (EC ) model under the assumption of a homogeneous equilibrium relationship. However, the formula is invalid in cases when the lag structure is more complex or the long-run homogeneity assumption does not hold. Nonetheless, we found a number of studies that use the formula inappropriately under these more general conditions, including Chong and Liu (2009); De Bondt (2005); Charoenseang and Manakit (2007); De Graeve et al. (2007); Leibrecht and Scharler (2008, 2011); Leszkiewicz-Kedzior and Welfe (2014); Scholnick (1996), among others. We fill a gap in the literature by deriving an expression for the mean lag in general EC models without imposing the assumption
of a homogeneous equilibrium. In addition, we evaluate the bias of the mean lag estimate arising from inappropriately imposing longrun homogeneity. 2. General form of the mean lag In this section we derive the mean lag in a general, nonhomogeneous, relationship. A general autoregressive distributed lag, or ADL(p, q; n), model can be written as yt = c +
p i=1
α(L)yt = c +
βk,j xk,t −j + ϵt or
k=1 j=0 n
(1)
βk (L)xk,t + ϵt , p
q
j where ϵt ∼ IID, α(L) = 1 − i=1 αi Li and βk (L) = j=0 βk,j L are lag polynomials, and n is the number of exogenous variables in the model. Regressors with varying lag lengths can be readily accommodated at the cost of further notational complexity. Rearrange Eq. (1) to obtain the reduced form equation
c
α(L)
∗
http://dx.doi.org/10.1016/j.econlet.2016.03.028 0165-1765/© 2016 Elsevier B.V. All rights reserved.
q n
k=1
yt = Correspondence to: 2424 Maile Way, Saunders Hall 508, Honolulu, HI 96822, USA. Tel.: +1 808 956 7840. E-mail address: [email protected] (P. Fuleky).
αi yt −i +
= c∗ +
+
1
n
α(L)
k=1
n k=1
βk (L)xk,t +
wk (L)xk,t + ut ,
ϵt α(L) (2)
108
P. Fuleky, L. Ventura / Economics Letters 143 (2016) 107–110
β (L)
k where wk (L) = α( = j=1 wk,j Lj . The ‘‘weight’’ associated with L) ∂y lag j of variable xk , wk,j = ∂ x t , captures the effect of xk,t −j on yt .
∞
k,t −j
Hendry (1995, p. 215) defined the mean lag as ∞
µk =
jwk,j
j =0
∞
= wk,j
∂wk (L) wk (1) ∂L
1
L=1
j =0
βk′ (L) βk (L)α ′ (L) = − wk (1) α(L) α(L)2 L =1 ′ 1 βk (L) α ′ (L) β ′ (1) α ′ (1) = wk (L) − = k − , wk (1) βk (L) α(L) L=1 βk (1) α(1) 1
model, the mean lag defined in Eq. (3) takes the following form
β1 + 2β2 + 3β3 α1 + 2α2 + 3α3 + β0 + β1 + β2 + β3 1 − α1 − α2 − α3 κω + b0 + b1 + b2 1 + κ − a1 − a2 = − κω κ ω(a1 + a2 − 1) + b0 + b1 + b2 = . (8) κω If κ ̸= 0, ωk ̸= 0, then consistent estimation of the parameters θ = (κ, ω, a1 , a2 , b0 , b1 , b2 )′ in Eq. (7) allows us to obtain a consistent estimate of the mean lag, µ ˆ = µ(θˆ ), and its variance ∂µ(θˆ ) ∂µ(θˆ ) ˆ Var(µ) ˆ = ∂θ ′ Var(θ ) ∂θ , where Var(θˆ ) is the covariance matrix µ=
of coefficients estimated in Eq. (7).
Before generalizing this result to an EC (p − 1, q − 1; n) model
(3) where z ′ = ∂∂ zL . Note, the mean lag associated with variable xk does not depend on the coefficients of the other variables xl , l ̸= k. In Eq. (3), wk (1) represents the long run impact of xk on y. n Consequently, yt − c ∗ − k=1 wk (1)xk,t captures a deviation from the long-run equilibrium between the dependent variable y and regressors x1 . . . xn . Following the steps outlined in Section 2.1 of Banerjee et al. (1993), the ADL(p, q, n) model in Eq. (1) can be transformed into an EC (p − 1, q − 1; n) model
1yt = κ yt −1 − c ∗ −
n
ωk xk,t −1 +
k=1
n
bk,0 1xk,t
k =1
1yt = κ yt −1 − c − ∗
n
ωk xk,t −1 +
k=1
+
q −1 n
p−1
ai 1yt −i
i =1
bk,j 1xk,t −j + ϵt ,
(9)
k=1 j=0
we make the following set of assumptions: Assumption 1. The variables y, x1 . . . xn entering models (1) and (9) are either jointly stationary, n or cointegrated with a stationary equilibrium error yt − c ∗ − k=1 ωk xk,t .
(4)
Assumption 2. The error in Eqs. (1) and (9), ϵt , is independently and identically distributed and is independent of the variables x1 . . . xn .
p−1 j 1 aj L with j= q j aj = − i=j+1 αi , bk (L) = j=1 bk,j L with bk,j = − i=j+1 βi , and p − 1 and q − 1 stand for the maximum lag lengths of 1y and 1x,
Assumption 3. The parameters in Eq. (9), θ = (κ, ω1 . . . ωn , a1 . . . ap−1 , b1,0 . . . b1,q−1 . . . bn,0 . . . bn,q−1 )′ , are estimated consistently with an estimator that has an asymptotically normal distri-
+ a(L)1yt +
n
bk (L)1xk,t + ϵt ,
k=1
where κ = −α(1), ωk = wk (1), bk,0 = βk,0 , a(L) =
p
q −1
respectively. By convention, a term does not enter the summation if the lower limit exceeds the upper limit. The models described by Eqs. (1) and (4) are isomorphic. Example 1. Transformation of the ADL(3, 3; 1) model
√
bution
Assumption 4. κ ̸= 0, ωk ̸= 0 for k ∈ {1 . . . n}. The mean lag, µk (θ ), is a continuous function of θ and is continuously differentiable with respect to θ . Proposition 1. Under Assumptions 1–4 the mean lag estimator
yt = c + α1 yt −1 + α2 yt −2 + α3 yt −3 + β0 xt + β1 xt −1
+ β2 xt −2 + β3 xt −3 + ϵt ,
ωˆ k
(5)
µ ˆ k = µk (θˆ ) =
yields the following EC (2, 2; 1) model
1yt = −(1 − α1 − α2 − α3 ) yt −1 − 1 − α1 − α2 − α3 β0 + β1 + β2 + β3 − xt −1 + β0 1xt 1 − α1 − α2 − α3 − (α2 + α3 )1yt −1 − α3 1yt −2 − (β2 + β3 )1xt −1
√
(6)
which can be estimated in a simplified form
1yt = κ yt −1 − c ∗ − ωxt −1 + b0 1xt + a1 1yt −1 + a2 1yt −2 + b1 1xt −1 + b2 1xt −2 + ϵt .
p−1
q −1
aˆ i − 1
i=1
+
j =0
κˆ ωˆ k
bˆ k,j
,
(10)
is consistent and asymptotically normally distributed
c
− β3 1xt −2 + ϵt ,
d
T (θˆ − θ ) −→ N (0, Var(θ )).
(7)
The expression in brackets represents the equilibrium error. The coefficients estimated in (7) can be mapped back to the ones in (5) and (6) with β0 = b0 , β1 = b1 − b0 − κω, β2 = b2 − b1 , β3 = −b2 , α1 = 1 +κ + a1 , α2 = a2 − a1 , α3 = −a2 . Hence, for the EC (2, 2; 1)
d
T (µk (θˆ ) − µk (θ )) −→ N
0,
∂µk (θ ) ∂µk (θ ) Var (θ ) . ∂θ ′ ∂θ
(11)
Proposition 1 extends the results obtained in Example 1 to a general EC (p − 1, q − 1; n) model. The details of the proof are provided in an Online Supplement on the first author’s homepage. Remark 1.1. If the variables y, x1 . . . xn are cointegrated, then the ′ elements of the cointegrating √ vector, ω = (ω1 . . . ωn ) , are ˆ − ω) = op (1). As a result, the estimated super-consistently: T (ω Var(θ ) components associated with the cointegrating vector, ω, converge to zero and do not contribute to the asymptotic variance of the mean lag estimator in (11). The Supplement contains a more detailed exposition of this issue (see Appendix A). Perhaps due to previous unavailability of the formula presented in Eq. (10), some researchers have ignored the lag structure of
P. Fuleky, L. Ventura / Economics Letters 143 (2016) 107–110
109
Fig. 1. Bias and relative bias of the mean lag estimate arising from an inappropriate assumption of homogeneity in an EC (p − 1, q − 1; n) model with
q−1 j=0
p−1 i=1
ai = 0.6,
bk,j = 0.2 and κ = −0.4.
Table 1 Impact of imposing long-run homogeneity in the literature. Study A B C
Item
ˆ µ( ¯ θ)
ˆ µ(θ)
Relative bias
Overnight Maturity over 2 years IB TD6-12 Footnote 8
1.58 0.97 0.26 7.09 2.90
1.24 0.50 0.14 2.63 2.20
27% 94% 78% 169% 39%
Note: Illustration of bias in the mean lag estimate due to an inappropriate assumption of long-run homogeneity. Panels (A)–(C) refer to the following published results: (A) Table 8 in De Bondt (2005), (B) Table 6 in Charoenseang and Manakit (2007), (C) page 501 in Leibrecht and Scharler (2008). Each of these studies ˆ defined in Eq. (15) in place of µ(θ) ˆ defined in Eq. (12) to incorrectly used µ( ¯ θ) estimate the mean lag.
their EC models when estimating the mean lag (see for example De Graeve et al., 2007; Leibrecht and Scharler, 2011). Corollary 1.1. For the frequently used ADL(1, 1; n) and the associated EC (0, 0; n) model, expression (10) simplifies to
ˆ = µk (θ)
bˆ k,0 − ω ˆk
κˆ ωˆ k
.
(12)
An error correction model is considered to be homogeneous if, for each k, the xk and y variables move one-for-one in equilibrium. Homogeneity implies ωk = wk (1) = 1 or α(1) = βk (1), leading to a special form of the mean lag
β ′ (1) − α ′ (1) β ′ (1) − α ′ (1) µ ¯k = k = k . α(1) βk (1)
(13)
Corollary 1.2. The mean lag estimator associated with variable xk in a homogeneous EC (p − 1, q − 1; n) model takes the form p−1
ˆ = µ ¯ k (θ)
q−1
aˆ i − 1 +
i =1
j =0
κˆ
bˆ k,j
,
(14)
which simplifies to
ˆ = µ ¯ k (θ)
bˆ k,0 − 1
κˆ
(ωk − 1)
q −1
bk,j
j=0
µ ¯ k (θ ) − µk (θ ) =
(16)
κωk
and the relative bias
µ ¯ k (θ ) − µk (θ ) = µk (θ )
(ωk − 1)
q −1
.
(15)
for a homogeneous EC (0, 0; n) model. Expression (15) is equivalent to the formula derived by Hendry (1995, p. 215, Eq. (6.53)). Although this formula does not hold for general, non-homogeneous, relationships between xk and y, it has been inappropriately used in place of Eq. (12) by several
bk,j
j =0
ωk
p−1
(17)
q −1
ai − 1
+
i=1
3. Mean lag under long-run homogeneity
researchers. The list of studies that relied on Eq. (15) despite nonunity ωk coefficients includes Chong and Liu (2009); De Bondt (2005); Charoenseang and Manakit (2007); De Graeve et al. (2007); Leszkiewicz-Kedzior and Welfe (2014); Leibrecht and Scharler (2008, 2011); Scholnick (1996), among others. In Table 1, we illustrate the impact of imposing long-run homogeneity in some of these studies. Expressions (10) and (14) allow us to quantify the bias in the mean lag estimate arising from an inappropriate assumption of homogeneity. The bias
bk,j
j =0
vanish as ωk → 1, but diverge otherwise. Fig. 1 illustrates the p−1 magnitude of the bias and relative bias for ωk ∈ (−2, 2), i=1 ai = 0.6,
q −1 j =0
bk,j = 0.2, and κ = −0.4.
4. Concluding remarks Researchers have been inappropriately using Hendry’s (1995) mean lag formula – which he derived for homogeneous EC (0, 0, 1) models – in more complex settings. We fill a gap in the literature by deriving an expression for the mean lag in general EC models, and show that using the incorrect formula can have a sizable impact on the estimated delay in the transmission of shocks. The ADL and EC models discussed above can be viewed as components of vector autoregressive (VAR) and vector error correction (VEC ) models, respectively. Consequently, the presented results are also valid for the mean lags of variables in individual equations of VAR and VEC models. Appendix A. Supplementary data Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2016.03.028.
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P. Fuleky, L. Ventura / Economics Letters 143 (2016) 107–110
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