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On the identifiability of Euler equation estimates under
q
saddlepath stability
*
Bernd Lucke , Christian Gaggermeier
¨ ¨
Universitat Hamburg, Fachbereich Wirtschaftswissenschaften, Institut f ur Wachstum und Konjunktur, Von-Melle-Park 5,
D-20146 Hamburg, Germany
Received 27 June 2000; accepted 12 October 2000
Abstract
In all but the most trivial settings Euler equation estimation in saddlepath stable systems is faced with a fundamental identification problem: the Euler equation allows for an unstable root (of its characteristic equation), while the data used to estimate the Euler equation also obey the transversality condition, which rules out unstable roots. Thus, even if the model is true, then the data are completely uninformative with respect to the unstable root of the Euler equation. But ignorance of the unstable root implies that the parameters are not identified if the relationship between parameters and roots is one to one. We illustrate the issue using a linearized Euler equation and present an application with OECD consumption data. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Euler equation estimates; Transversality condition; Real business cycle models
JEL classification: C22; E21
1. Introduction
Today’s dynamic macroeconomic theory relies heavily on models of intertemporally optimizing agents, whose behavior is characterized by the necessary conditions of suitably formulated maxi-mization problems. Often, estimating the structural parameters is feasible only for estimating the first order conditions, cf. Wickens (1995). Among these, Euler equation estimation is widely used to retrieve key parameter values.
q
Former versions of this paper were distributed under the more exciting but less precise title ‘Don’t estimate Euler equations!’.
*Corresponding author. Tel.: 149-40-2838-2080; fax: 149-40-2838-6314.
E-mail address: [email protected] (B. Lucke).
In this paper, we argue that a fundamental problem of Euler equation estimation lies in the fact that if the model is true, then the data used to estimate the Euler equation are the solution to the complete set of first order conditions including, in particular, the transversality condition. The Euler equation to be estimated represents only a subset of the restrictions the optimal trajectory is required to meet. Specifically, Euler equations in saddlepath stable systems contain an explosive root, while the observed data (which fulfil the transversality condition) do not. Hence the explosive root cannot be estimated from the data and the identifiability of parameters related to this root is not ensured.
We also show that the problem is absent in the textbook time separable case, where the change in consumption is unrelated to lagged consumption. This is Hall’s (1978) random walk scenario. While the unit root is unstable, it is not explosive and does hence not violate the transversality condition. Empirically, however, the change in consumption is often correlated to lagged consumption, so that momentary utility cannot be viewed as time separable, possibly due to habit persistence, durable goods, or liquidity constraints. In this case, Euler equation estimates of some of the deep parameters will be nonsensical. As an illustration we show that estimates of the representative agents’s discount rate (which should be between 0 and 1) will tend to infinity due to this problem.
The body of this paper is organized as follows: in Section 2 we propose a very general representative consumer model and derive log linear approximations to selected first order conditions. (We work with log linear approximations here since the solutions to linear systems of difference equations are well known. For nonlinear Euler equations the problem is probably at least as severe, and certainly less well analytically tractable.) Further, we derive simple properties of solutions to the Euler equation and compare these to the analogous properties of solutions to the complete set of first order conditions. We argue that saddlepath stability induces an identifiability problem that renders Euler equation estimates grossly biased or causes estimation algorithms to break down. We illustrate our point in an empirical application with consumption data of selected OECD countries in Section 3. Section 4 concludes.
2. The model
Consider, for simplicity, a non-growing economy, in which a representative household maximizes expected utility,
`
t
E0
F
O
b u(?)G
(1)t50
subject to the budget constraint
Ct1At115w Lt t1(11r )A .t t (2)
Here, C is consumption, L is the household’s labor supply, A is real wealth, and r and w are thet t t t t interest rate and the real wage in period t.b[(0, 1) is a discount factor. We assume that factor prices are stochastic, for instance due to productivity shocks as the source of uncertainty.
We assume standard properties for momentary utility u(?), time invariance in particular. Typically,
In the following, we will study the time non-separable case u5u(C , Ct t21) to allow for habit
1
persistence or durable consumption goods.
Differentiating the Lagrangean with respect to C and At t11 yields the first order conditions
t t11
lt5b u (C , C1 t t21)1b E [u (Ct 2 t11, C )]t (3)
and
lt5E [(1t 1rt11)lt11] (4)
where partial derivatives are indicated by indices. The transversality condition is of the usual form
limltAt1150. (TVC)
t→`
We combine (3) and (4) to get the Euler equation
u (C , C1 t t21)1bE u (Ct
f
2 t11, C )tg
5bE (1tf
1rt11) u (Cs
1 t11, C )t 1bu (C2 t12, Ct11)dg
(5)2
which is a nonlinear stochastic difference equation of odd order. If the model is true, then the observed consumption time series will obey Eq. (5) — and this fact is precisely the idea of Euler equation estimation. Unfortunately, as we will show, the parameters are not identified unless momentary utility is time separable, i.e. u2;0.
To illustrate this claim, we would need the solution of (5). Since this is not known, we will follow
3
the standard approach in the literature and study a linearized version of (5). We will argue that the transversality condition eliminates the unstable root of the difference equation, so that the observed data, which also obey the transversality condition, are uninformative with respect to the unstable root. Not knowing the unstable root, however, it is impossible to retrieve all structural parameters. We further argue that nothing of this line of reasoning is special to linear models, except for the fact that the general solution of the linear equation is known. The key point in the argument is not the linearity
4
but the saddle path property — which should be identical for nonlinear Euler equations.
We linearize the Euler equation about the steady state values C and 11r. As the procedure is
standard (cf. King et al. (1988)), we do not give the details here. After some calculus, we obtain
u11 u22 u11 u22 1 1
It is not difficult to show that all of our arguments remain valid if higher consumption lags (up to Ct2k, say) or other contemporaneous variables affect momentary utility. The textbook time separable case is, of course, contained as a special case in the non-separable set-up we study.
2
This odd order property holds for arbitrary k in the general utility function u(C , . . . , Ct t2k). 3
Of course, if momentary utility is quadratic then the linearized equation and the true Euler equation coincide. Thus in this case our argument is immune against criticisms of the quality of log-linear approximation as advanced e.g. by Carroll (1997).
4
5
where ´t12 collects the expectational errors.
The following Euler equation properties (EEP) of the (linearized) Eq. (6) are noteworthy:
EEP 1: the lag polynomial in log consumption has a unit root, oai51. EEP 2: the order of the lag polynomial in log consumption is odd.
EEP 3: the coefficient of the highest log consumption lag, a3, is larger than one.
While the first property dates back to Hall’s (1978) seminal paper and comes as no surprise, the
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very robust properties EEP 2 and EEP 3 have not received similar attention.
Let us study the solution of the homogenous part of (6). The lag polynomial is of degree three, i.e. there are three rootsm1, m2, andm3 which govern the dynamic properties of the general solution. One root, by EEP 1, is equal to one, and at least one more root is explosive (i.e. smaller than one in absolute value) by EEP 3 and by virtue of the following.
k j
Lemma. Let p (z)k 512oj51a z be a polynomial defined on the complex numbers withj uaku.1. Then
there is at least one root of the polynomial with absolute value smaller than one.
k k
The general solution to the homogenous part of (6) is given by
t t
1 1
] ]
ln Ct5cnst11cnst2
S D
m 1cnst3S D
m (7)2 3
where cnst , cnst , cnst are arbitrary constants and1 2 3 m1 is assumed to be the unit root. Without loss of generality, let us further assume that m3 be inside the unit circle. Thus (7) is generally explosive, unless cnst equals zero.3
It is well known that every explosive trajectory given by (7) violates the transversality condition
7
(TVC). Hence it is necessary for utility maximization to set cnst350, which immediately determines the initial value of the optimal consumption trajectory. Thus the optimal (and observed) consumption stream satisfies
5
Note that I have used ln (11rt11)¯rt11, (11r)b 51, and u215u12 in deriving (6). 6
These properties refer to the homogenous part of (6) only. This is why we could allow momentary utility to depend on other contemporaneous variables without affecting EEP 1–EEP 3. Note that the properties continue to hold if momentary utility depends on further lagged consumption terms: u5u(C , . . . , Ct t2k). The coefficient of the highest consumption lag is
2k
then given by a112k5b , which is larger than one for k.0. Only in the time separable case of k50 do we get
a 5 a1 112k51; i.e. consumption is a random walk. 7
Note that the unit root is not explosive and does hence not violate the transversality condition. This is easy to see: (TVC) states that the product of wealth and the present value shadow price converge to zero. Standard analysis of such models reveals that the product of wealth and the current value shadow price converges to a constant if the homogenous part of the central difference equation features a unit root, but no explosive roots. Since the present value multiplier equals the
t
t
1
]
ln Ct5cnst11cnst2
S D
(8)m2
which happens to be the general solution of a second-order homogenous difference equation with a unit root, i.e. an equation of the form
ln Ct5aln Ct211(12a)ln Ct22. (9)
Ifm2 is stable (the usual case), then 12a is smaller than one in absolute value, i.e. (9) is stable. Note that any time series generated from (9) also obeys (the homogenous part of) (6), but asymptotically the estimate of a3 from this time series would be zero and hence inconsistent.
Thus, the dynamical properties of trajectories which solve the full system of first order conditions (the Euler equation and the transversality condition) are very different from the dynamical properties of general solution trajectories of the Euler equation alone. In the (saddlepath) case in which just one root of (6) is explosive, we can sum up the full system properties (FSP) as:
FSP 1: the lag polynomial in log consumption has a unit root. FSP 2: the order of the lag polynomial in log consumption is even.
FSP 3: the coefficient of the highest log consumption lag is smaller than one.
If, in rare circumstances, more than one root of the lag polynomial in the linearized Euler equation lies inside the unit circle, then properties FSP 1 and FSP 3 are unchanged. Property FSP 2 does not necessarily hold, however: all that can be said instead is that the order of the lag polynomial in log consumption is definitely smaller than the order suggested by the Euler equation.
The jeopardies of Euler equation estimation should now be clear: so long as the transversality condition and its implications for initial conditions are neglected, efforts to estimate (6) will fall prey to an identifiability problem: by writing the homogenous part of (6) in growth rates, ct[ln Ct2ln
Ct21, we have
ct5sa121 cd t211sa11a221 cd t22. (10)
After some algebra, we can write (10) in terms of its roots m1 and m2 as
m21m3 1
]]] ]]
ct5 m m ct212m m ct22. (11)
2 3 2 3
Apparently, estimating a1 and a2 in (10) from data generated by (8) is tantamount to estimating m2
andm3 in (11) from data generated by (8). But the latter task is obviously impossible, since (8) does not contain any information about the instable rootm3. Hencem3 is not identified and any attempt to estimate it is doomed to failure. Not knowing m3, however, it is impossible to retrieve a1 and a2.
crash when the Hessian becomes singular, or it will stop with a biased estimate ofb when the cost in terms of the implied coefficients a1 and a2 (which also depend on b) becomes too large.
3. Empirical findings
We illustrate the point made in the previous section in an analysis of consumption data of 26 OECD countries. The data are yearly and range from 1960 to 1996, they are taken from the OECD System of National Accounts. We confine the analysis here to univariate methods, since the inclusion of further conditioning variables hardly changes the qualitative properties of the lag polynomial in consumption. For extended results see Gaggermeier and Lucke (1999). We add a constant for regressions in first differences to account for the fact that real world data are generated by growing economies.
We initially test the unit root property of log consumption which is a common feature of both the Euler equation and the solution to the full system of first order conditions. We apply standard augmented Dickey–Fuller tests (ADF-tests), where the regression equation contains a linear trend in order to account for the upward trend in consumption under the alternative. The results are suppressed here, but available on request. They indicate that the unit root hypothesis cannot be rejected at the 5% level of significance for any of the countries.
Next we test whether EEP 2 or FSP 2 are supported by the data. We impose the unit root property to exploit the fact that any integrated AR(p) process
p
ln Ct5a01
O
ailn Ct2i1´t i51can be rewritten as an AR(p21) process in first differences:
p21
Dln Ct5a01
O
fiDln Ct2i1´t i51p
where fi5 2oj5i11aj. This transformation yields an equation in stationary variables so that inference in OLS regressions is standard.
Note that in first differences EEP 2 implies a lag polynomial of even order, while FSP 2 implies
odd order, unless the unit root is the only unstable root, i.e. the lag order in first differences is zero.
We begin by evaluating the Schwarz (1978) information criterion (SC) for the first differences of log consumption to get an impression of the appropriate lag orders, cf. Table 1.
With the notable exception of Germany, all suggested lag lengths are either zero or odd. While we find evidence for only one country in which the appropriate lag length may be even, precisely half of the countries show evidence for odd and nonzero lag lengths, which are incompatible with EEP 2.
To complement this rather mechanical investigation, we estimate simple autoregressions by OLS. To focus on the main issue, we again neglect conditioning variables like interest rates and labor and estimate an autoregression in consumption only. (For extended results, the interested reader is referred to our discussion paper (Gaggermeier and Lucke, 1999).) Starting from a total of five lags and successively deleting insignificant regressors we aim at finding the most parsimonious representation with white noise residuals. The estimation output of this exercise is presented in Table 2.
Table 1
Schwarz information criterion for consumption growth rates
Lag length 0 1 2 3 4 5 Preferred
lag order
Australia 28.62 28.55 28.46 28.38 28.30 28.33 0
Austria 27.91 27.81 27.71 27.64 27.55 27.46 0
Belgium 27.76 27.89 27.77 27.78 27.77 27.66 1
Canada 27.54 27.75 27.63 27.52 27.41 27.32 1
Denmark 27.11 27.03 26.97 26.89 27.01 27.04 0
Finland 26.70 26.85 26.77 26.68 26.70 26.59 1
France 28.48 28.73 28.64 28.64 28.53 28.42 1
Germany 27.79 28.19 28.32 28.29 28.20 28.08 2
Greece 27.38 27.58 27.53 27.59 27.48 27.39 3
Iceland 25.31 25.23 25.25 25.14 25.04 24.95 0
Ireland 26.78 26.76 26.68 26.62 26.51 26.40 0
Italy 27.45 27.64 27.53 27.55 27.53 27.42 1
Japan 27.11 27.18 27.08 27.07 26.96 27.85 1
Luxembourg 27.82 27.98 27.87 27.79 27.70 27.59 1
Mexico 26.37 26.32 26.21 26.10 26.99 26.87 0
Netherlands 27.64 28.20 28.11 28.00 27.92 27.89 1 New Zealand 26.99 26.97 26.95 26.84 26.77 26.67 0
Norway 27.30 27.24 27.13 27.07 27.00 26.88 0
Portugal 25.73 25.69 25.58 25.55 25.56 25.58 0
South Korea 27.00 27.27 27.22 27.12 27.01 27.05 1
Spain 27.28 27.77 27.67 27.56 27.47 27.43 1
Sweden 27.65 27.72 27.61 27.50 27.43 27.32 1
Switzerland 28.16 28.55 28.52 28.45 28.33 28.22 1
Turkey 25.70 25.60 25.51 25.47 25.49 25.43 0
United Kingdom 27.37 27.43 27.39 27.30 27.25 27.14 1
USA 28.17 28.14 28.10 27.99 27.87 27.76 0
which is again Germany. In eight cases the appropriate lag order is zero, which indicates the absence of explosive roots in the Euler equation, i.e. the general solution of the Euler equation does not violate the transversality condition.
Since we imposed the unit root (and accordingly deleted the linear trend from the regressions), it is not too surprising to find some cases in which the lag structure of the growth rate autoregressions is in disaccord with the lag structure of the ADF-regressions. There are discrepancies in eight (out of 26) cases: five of these concern decisions of whether lag order zero or lag order one is appropriate. Hence, either way the specifications are compatible with the maintained hypothesis that observed data reflect the full set of first order conditions. In the other three cases (Belgium, Greece, and Italy), both the ADF test and the growth rate autoregression suggests nonzero and odd lag orders, all of which are compatible with FSP 2, but incompatible with EEP 2.
Table 2
Greece 0.005 0.395 – 0.405 0.871 Odd (0.006) (0.151) (0.153) 0.697
Iceland 0.030 0.273 Zero
(0.011) 0.254
Ireland 0.018 0.306 0.457 Odd
(0.007) (0.168) 0.738
Italy 0.009 0.371 – 0.247 0.445 Odd (0.008) (0.151) (0.150) 0.267
smaller than one and significantly so. Thus EEP 3 is not at all recognizable in the data, and any attempt to take advantage of such a Euler equation property in estimation is sure to fail. The evidence thus suggests that neglect of the transversality condition (as it is inherently done in many attempts to estimate Euler-equations) fundamentally misspecifies the dynamics of observed real world time series
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even if the underlying model is true. .
4. Conclusions
Inference based on Euler equation estimates may be grossly misguided, since the implications of the transversality condition are not properly taken into account. We demonstrated this fact for the case of a linearized Euler equation, since the solution theory for difference equations linear in variables is well developed and well known. Nothing suggests that the problem pointed out in a linear setting will be less severe in non-linear set-ups. Moreover, the transversality condition is just one necessary condition which supplements the Euler equation. Other restrictions on the general solution of a Euler equation may be expressed in other necessary conditions, and these, too, would be ignored in a single equation set-up. So as long as the implications of the full set of first order conditions (including, e.g., Legendre conditions) are not well understood, limited information methods of estimation are not advisable.
References
Carroll, C.D., 1997. Death to the log-linearized consumption Euler equation! (And very poor health to the second-order approximation). NBER, Cambridge, MA, Working paper no. 6298.
Gaggermeier, C., Lucke, B., 1999. Does the representative consumer fulfil Euler equation restrictions?. University of Hamburg, Discussion paper.
Hall, R.E., 1978. Stochastic implications of the life cycle-permanent income hypothesis: theory and evidence. Journal of Political Economy 89, 974–1009.
King, R.G., Plosser, C.I., Rebelo, S.T., 1988. Production, growth, and business cycles. I. The basic neoclassical model. Journal of Monetary Economics 21, 195–232.
Schwarz, G., 1978. Estimating the dimension of a model. Annals of Statistics 6, 461–464.
Wickens, M.R., 1995. Real business cycle analysis: a needed revolution in macroeconometrics. The Economic Journal 105, S1637–1648.
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