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Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d

Global stability of dynamic models Wojciech Grabowski, Aleksander Welfe ⁎ University of Lodz, 41 Rewolucji 1905r., 90-214 Lodz, Poland

a r t i c l e

i n f o

a b s t r a c t

Article history: Accepted 25 October 2010

A necessary and sufﬁcient condition for global stability of dynamic models is summability to one of the longrun elasticities and cointegration. The short-run coefﬁcients automatically satisfy the homogeneity condition. A relevant restriction has to be imposed in the parameter estimation process; otherwise, the ratios of variables appearing in the model will tend to either inﬁnity or zero, which is economic nonsense in most cases. This conclusion is particularly important for the ECM or VEC (SVEC) models that decompose behavior into long and short-run parts. © 2010 Elsevier B.V. All rights reserved.

JEL classiﬁcation: C32 C51 Keywords: Dynamic models Global stability Equilibrium

1. Introduction Let us assume that the long-run relation between the variables Yt = [Y1t... YMt] is given by the following log-linear function: M

y1t = β0 + ∑ βm ymt + dt θ + υt ; m=2

t = 1; 2; …

ð1Þ

where Y1t, …, YMt are integrated variables, ymt = ln Ymt, βm N 0 stand for the structural parameters, dt = [d1t... dVt] is the vector V of deterministic variables, θ = [θ1... θV]T – the vector of parameters, υt – the whitenoise error term, m = 1,..., M. Variables are usually expected to increase in such a way that the Y1t relations for m ∈ {2, …, M} will be ﬁnite and positive for all Ymt observations. Since Y1t, …, YMt are stochastic variables, global stability Y1t ratio in probability: is expected to occur for each Ymt 1 Y1t b b a ≥1−ε: ð2Þ ∀ ∃ inf P a Ymt ε N 0a N 0 t The above keeps the system described by the dynamic econometric model in balance. This can be illustrated using the import function of ﬁnal goods MC, where the explanatory variables are personal consumption, C, and government consumption, G. All MC/C, MC/G and C/G relations must stabilize, because it would be economic nonsense for them to take values close to 0 or inﬁnitely large values. The same reasoning applies to the shares of personal consumption or investment in national income that Klein and Kosobud (1961) called the great ratios. It should be added that the relations between models' variables do not ⁎ Corresponding author. E-mail address: [email protected] (A. Welfe). 0264-9993/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2010.10.011

have to meet the stability condition in every instance. For example, in the money demand function mt = pt + β1yt + β2it + ξt the ratios have no economic interpretation, so they are free to take any value. Therefore, in order to ensure global stability, the conditions to be met by the model parameters have to be deﬁned. 2. Conditions of global stability in probability When model (1) has two variables (M = 2), Eq. (1) can be written as follows: Y ln 1t = β0 + ðβ2 −1Þ lnY2t + dt θ + υt ; Y2t

t = 1; 2; …

ð3Þ

If β2 − 1 ≠ 0, then, given the integration of the variable Y2t, a unit Y1t to Y2t show global stability in probability. To demonstrate that the condition is also sufﬁcient, let us write the Eq. (3) for β2 = 1 in the following way: value of the parameter β2 is a necessary condition for the ratio

Y ln 1t = β0 + dt θ + υt ; t = 1; 2; … Y2t

ð4Þ

Let us note that the deterministic variables meet the condition: ∃ ∀ jβ0 + dt θjb S:

ð5Þ

S t

Using the Chebyshev's inequality: ∀ ∀ P ðjυt jbRÞ≥1− t RN0

κ R2

ð6Þ

W. Grabowski, A. Welfe / Economic Modelling 28 (2010) 782–784

where ∀ Varðυt Þ = κ, and the formula (5), we obtain:

where

t

∀ ∀ P ð−L b β0 + dt θ + υt bLÞ≥1− t RN0

κ ; R2

ð7Þ

M

Y κ ∀ ∀ P −Lb ln 1t bL ≥1− 2 ; t RN0 Y2t R

ð8Þ

or: expð−LÞb

t RN0

Y1t κ b expðLÞ ≥1− 2 : Y2t R

ð9Þ

Y The above formula means that for β2 = 1 the 1t ratio meets condition Y2t of the global stability in probability. This probability will tend towards unity following growing R, i.e. the expansion of the interval. For a model containing more than two variables (M N 2), the Eq. (1) can be written as: M

y1t −ymt = β0 + ðβm −1Þymt + ∑ βj yjt + dt θ + υt : j= 2 j≠m

ð10Þ

= β0 + ∑ βj yjt −ymt + M

j= 2 j≠m

j =2 j≠m

∑ βm −1 ymt + dt θ + υt

∀

i =2 j =2 i≠j j≠m i≠m

Therefore, homogeneity and cointegration of the variables are Y1t ratios globally stable. If either Ymt of the two conditions is not met, then the ratios between the model's variables can grow inﬁnitely, exceeding in the long run an arbitrarily large value with probability 1, which leads to economic nonsense. necessary and sufﬁcient to make the

3. Global stability of a dynamic model As far as the univariate analysis is concerned, the Eq. (1) can be taken as the ﬁrst step of the Engle and Granger procedure (1987). Let us consider the simplest model with S lagged variables and one independent variable (see Sargan (1964), Banerjee et al. (1993) and Juselius (2006)): S

S−1

s=2

s=0

Δyt = α0 + ðα1 −1ÞECTt−1 + ∑ αs ECTt−s + ∑ ϑs Δxt−s + ξt ð15Þ

ð12Þ

n≠m

m;n≥2

κ ˜ mn;pq , respectively, and amn is a constant.

m;n;p;q≥2

Y1t is not globally stable in probability, Ymt M is integrated. However, if ∑ βm = 1, then

If ∑ βm ≠1, then the ratio because the variable Ymt

s=2

s=0

where ϕi = ϑi − ϑi − 1 (i = 0,..., S) and ϑ− 1 = ϑS = 0. Recursive substitution for N periods (N being a large natural number) produces:

m=2

the Eq. (10) can be transformed to read: M y1t −ymt = β0 + ∑ βj yjt −ymt + dt θ + υt : j =2 j≠m

ð13Þ

N+1

S

N+s

s=1

s=2

h=s

yt = yt−N−1 + ðα1 −1Þ⋅ ∑ ECTt−s + ∑ αs ∑ ECTt−h + N

minðs;SÞ

s=0

p=0

+ ∑ xt−s ⋅ where ζmnt and υt are uncorrelated for each m ∈ {2, …, M} and for each n ∈ {2, …, M}, and ζmnt is a white-noise error term whose variances and covariances are ∀ varðζmnt Þ = κ ˜ 2mn and ∀ cov ζmnt ; ζpqt = M

S

ð16Þ

ðymt −ynt Þ = amn + ζmnt ;

m=2

S

yt = yt−1 + α0 + ðα1 −1ÞECTt−1 + ∑ αs ECTt−s + ∑ ϕs xt−s + ξt

m=2

Let us assume that the variables on the right-hand side of the Eq. (1) are cointegrated: t m;n≥2

M

∑ βi βj κ ˜ im;jm

!

M

ð11Þ

∀

M

where ECT stands for the error correction term. The above equation can be equivalently rewritten as:

or, alternatively: y1t −ymt

2 2

τm = κ + ∑ βj κ ˜ jm + ∑

where L = R + S, which substituted into the Eq. (4) ultimately produces:

∀ ∀ P

783

∑ ϕp

!

N+S

+

∑

s=N+1

xt−s ⋅

S

!

∑ ϕp :

p = s−N

ð17Þ

The condition for exogenous shocks to disappear in the long run is ∂y lim t + N = 0 or lim ðϕ0 + ϕ1 + … + ϕS Þ = 0. Because the exN→∞ N→∞ ∂xt pression is not determined by N, the condition ϕ0 + ϕ1 + … + ϕS = 0 is met by deﬁnition. The conclusion holds true also for a model with multiple explanatory variables. In other words, summability to one of the long-run elasticities contained in the error correction term and the Y cointegration of the variables are sufﬁcient for the 1t ratios to be Ymt globally stable. The short-run elasticities do not have to meet any additional conditions. 4. Monte Carlo investigation of global stability

Y1t is not globally From the above formula it follows that the ratio Ymt stable in probability if the condition (12) is not met. M

Function homogeneity ( ∑ βm = 1) and cointegration (see m=2

formula (12)) are both necessary and sufﬁcient conditions for global stability to occur, because applying the Chebyshev's inequality and the Eqs. (7) and (13) we obtain: 0

1 M M B C Y τm ∀ PB −S−R + ∑ βj ajm b ln 1t b S + R + ∑ βj ajm C A≥1− R2 t @ Y mt j=2 j=2 j≠m

j≠m

ð14Þ

The Monte Carlo investigations were aimed to demonstrate computationally that homogeneity and cointegration are both Y1t necessary and sufﬁcient conditions for the ratios' stability. Ymt In the ﬁrst experiment, two integrated variables {y1t} and {y2t} were generated using the following schemes: y1t = y1t − 1 + ut, ut ~ N(0, 1) and y2t = α + βy1t + εt, εt ~ N(0, 1). The values taken by the parameter β were 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0, while α was set to unity. The sample ranged from 50 to 500 observations. The function T Y1t Y ∑ 1 N 1000∪ 1t b0:001 Y2t Y2t DðT; βÞ = t = 1 T

ð18Þ

784

W. Grabowski, A. Welfe / Economic Modelling 28 (2010) 782–784

Table 1 Percentages of the extreme ratios in a model with two variables. β T

0.5

0.6

0.7

0.8

0.9

1

50 100 200 300 500

0.5717 0.7858 0.8931 0.9289 0.9572

0.4603 0.7304 0.8653 0.9105 0.9461

0.2572 0.6163 0.8080 0.8723 0.9233

0.0413 0.3353 0.6665 0.7780 0.8666

0.0011 0.0040 0.1530 0.4188 0.6485

0 0 0 0 0

Y1t ratios in a T − element Y2t sample for the given value of the parameter β, where Yjt = exp(yjt), j = 1, 2, t = 1, …, T. The series were generated 100 000 times and D (T, β) was computed, giving Di(T, β), i = 1, …, 100 000 for each replication. The average share of the extreme values corresponding to a speciﬁc number of observations and different β values is given by deﬁnes the percentage of the extreme

100000

∑ Di ðT; βÞ

DðT; βÞ =

i=1

ð19Þ

100 000

Y1t stabilizes only for β = 1 (see Table 1). If Y2t β ≠ 1, then the percentage of the extreme ratios expands following the increasing number of observations. This result conﬁrms the analytical derivations. In the second experiment, M = 3 variables were considered. The {y1t} process was generated as y1t = 1 + β2 y2t + β3y3t + ξt, ξt ~ N(0, 1), while {y2t} and {y3t} that are I(1) were generated as y3t = α + y2t + εt, εt ~ N(0, 1) and y2t = y2t − 1 + ut, ut ~ N(0, 1). The parameter α was set to unity. Three combinations of the β parameters satisfying the homogeneity condition were considered: It is clear that the ratio

– β2 = 0.5, β3 = 0.5 – β2 = 0.4, β3 = 0.6 – β2 = 0.2, β3 = 0.8,

– β2 = 0.4, β3 = 0.4, – β2 = 0.4, β3 = 0.2, – β2 = 0.25, β3 = 0.25.

DðT; β2 ; β3 Þ =

t =1

Y1t Y N 1000∪ 1t b0:001 Y2t Y2t T

ð20Þ

Table 2 Percentages of the extreme ratios in a model with three variables satisfying the cointegration condition (12).

50 100 200 300 500

β2 = 0.4

β2 = 0.2

β3 = 0.6

β3 = 0.8

0.7434 0.8730 0.9361 0.9573 0.9741

0.7810 0.8908 0.9456 0.9631 0.9785

0.8315 0.9154 0.9578 0.9695 0.9830

and 100000

∑ Di ðT; β2 ; β3 Þ

DðT; β2 ; β3 Þ =

i=1

100000

ð21Þ

Y Even though the variables cointegrate, the 1t ratio stabilizes only if Y2t β2 + β3 = 1 (see Table 2). In the third case, three integrated variables were considered, but {y2t} and {y3t} did not cointegrate, because y2t = y2t − 1 + ut, ut ~ N(0, 1), and y3t = 1 + y3t − 1 + Δy2t + εt, εt ~ N(0, 1). The ﬁrst variable was generated as follows: y1t = 1 + β2y2t + β3y3t + ξt, ξt ~ N(0, 1). The parameters β were set to be: – β2 = 0.5, β3 = 0.5, – β2 = 0.4, β3 = 0.6, – β2 = 0.2, β3 = 0.8. Y1t The empirical results provided in Table 3 show that the ratio Y does not stabilize, even if the homogeneity condition is satisﬁed. 2t The results of the Monte Carlo experiments fully corroborate the analytical derivations – homogeneity and cointegration are necessary and sufﬁcient conditions for global stability in probability. Acknowledgements

References

Analogously, the percentages of the extreme ratios for the speciﬁc numbers of observations and different variants were found as follows

T

50 100 200 300 500

β2 = 0.5 β3 = 0.5

We are deeply grateful to Marek Kałuszka for the very stimulating discussions we held together and his valuable suggestions. We would also like to thank Stephen Hall, Aleksander Weron, Agnieszka Wyłomańska, Joanna Janczura and two anonymous referees for their constructive criticism on the earlier version of this paper. Any errors are the sole responsibility of the authors. Financial support from the KBN Grant N111 032 32/3723 is gratefully acknowledged.

and three others, which do not add up to unity:

∑1

Table 3 Percentages of the extreme ratios in a model with three variables not satisfying the cointegration condition (12).

β2 = 0.5

β2 = 0.4

β2 = 0.2

β2 = 0.4

β2 = 0.4

β2 = 0.2

β3 = 0.5

β3 = 0.6

β3 = 0.8

β3 = 0.4

β3 = 0.2

β3 = 0.2

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0.0001 0.0005 0.0070 0.0187 0.0563

0.0069 0.0331 0.1014 0.1303 0.2591

0.0414 0.1212 0.2417 0.3108 0.4115

Banerjee, A., Dolado, J.J., Galbraith, J.W., Hendry, D.F., 1993. Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press, Oxford. Engle, R.F., Granger, C.W.J., 1987. Co-integration, and error correction: representation, estimation and testing. Econometrica 55, 251–276. Juselius, K., 2006. The cointegrated VAR model. University Press, Oxford. Klein, L.R., Kosobud, R.F., 1961. Some econometrics of growth: great ratios of economics. Quarterly Journal of Economics 173–198. Sargan, J.D., 1964. Wages and prices in the United Kingdom: a study in econometric methodology. In: Hart, P.E., Mills, G., Whitaker, J.K. (Eds.), Econometric Analysis for National Economic Planning. Butterworths Scientiﬁc Publications, London.