Can we Identify the Fed s Preferences?

Transcription

1 an we Identify the Fed s references? Jean-Bernard hatelain y Kirsten Ralf z November 8, 6 Abstract The number of reduced form parameters is larger with Ramsey optimal policy under commitment than without commitment, although the number of structural parameters, including central bank preferences, is the same. Hence, at least one of the two equilibria is over-identi ed or under-identi ed. The reference new- Keynesian model is under-identi ed with Ramsey policy (one identifying equation missing) and hence under-identi ed for optimal policy without commitment (three identifying equations missing). A pre-test for the reference new-keynesian model lead to reject optimal policy without commitment and (optimal) simple rule for the period 96 to 6 for the U.S. Fed. Estimating a structural VAR for Ramsey policy during Volcker-Greenspan period, the Fed s preferences (weight of the volatility of the output gap) and the new-keynesian hillips curve output gap structural parameter are not statistically di erent from zero at the 5% level. JEL classi cation numbers: 6, 6, E43, E44, E47, E5, E58. Keywords: Identi cation, Optimal policy, ommitment, Discretion, Simple rule, entral bank preferences, Monetary olicy Transmission hannel, New-Keynesian model. Introduction an we identify central bank preferences in the case of Ramsey optimal policy under commitment or of optimal policy without commitment? Do optimal policies eliminate the identi cation issues of simple Taylor rules raised by ochrane ()? In this paper, Ramsey optimal policy under commitment faces identi cation issues similar to vector auto-regressive (VAR) models (Sims (98) and ochrane ()). Optimal policy without commitment faces identi cation issues similar to reduced form simple rule models (ochrane ()). This paper proposes a pre-test of the di erent spanning conditions of Ramsey optimal policy under commitment versus optimal policy without commitment and (optimal) simple rules. The principle is to test the number of linearly independent observed variables predicted by each of the equilibria for any DSGE model. In a second step, entral Bank preferences and monetary policy transmission mechanism are tested for the model which We thank Stéphane Gauthier and Georges Overton for very helpful insights. y aris School of Economics, Université aris I antheon Sorbonne, ES, entre d Economie de la Sorbonne, 6- Boulevard de l Hôpital aris ede 3. jean-bernard.chatelain@univparis.fr z ESE International Business School, rue Setius Michel, 755 aris, Kirsten.Ralf@esce.fr.

2 passed the pre-test, using the full-information structural VAR related to maimal number of non-collinear observable variables predicted by the model. Distinct identi cation restrictions are requested for each model of optimal policy. The largest number of noncollinear variables in a stationary VAR estimation of commitment (i.e. the dimension of its stable subspace) is larger than the one of optimal policy without commitment. Hence, the number of reduced form parameters is larger in optimal policy with commitment than without commitment. But the number of structural parameters, including central bank preferences, is the same in optimal policy with commitment or without commitment. At least one of the two equilibria is necessarily over- or under-identi ed. This pre-test and test, taking into account identi cation restrictions, can be performed on any DSGE model. We pre-test and test Ramsey optimal policy under commitment versus optimal policy without commitment with the reference new-keynesian model (Gali (5, chapter 5) and larida, Gali and Gertler (999)) with eplicit closed form solutions. Optimal policy without commitment and the related (optimal) simple rule is rejected by pre-test with respect to Ramsey optimal policy for pre-volcker Fed and for Volcker-Greenspan Fed. What is more, because of the lower number of non-collinear variables in optimal policy without commitment than in commitment, three identifying equations are missing in optimal policy without commitment. In the case of Ramsey optimal policy, Fed s preferences and monetary policy transmission channel structural parameters are identi ed when the Fed s discount factor is eogenously given, because one identifying equation is missing. The nal step is to test the statistical signi cance of the Fed s preferences and monetary policy transmission mechanism in the case of Ramsey optimal policy. There are many estimations of central banks preferences when in ation is assumed to be a predetermined variable or a forward-looking variable, with or without commitment, with hillips curve or new-keynesian hillips curve e.g. echetti and Ehrmann (), Ozlale (3), astelnuovo and Surico (4), astelnuovo (6), Juillard et al. (6), Adjemian and Devulder (), Givens (), Levieuge and Lucotte (4), Matthes (5), aez-farrell (5). Söderlind s (999) seminal paper proposed a maimum likelihood estimation using the Kalman lter of the Hamiltonian system of Ramsey optimal policy and optimal policy without commitment. But this estimation method does not take into account the di erent stable subspace restrictions of each optimal policy. These estimates t out-of-equilibrium non-stationary paths violating Blanchard and Kahn s (98) determinacy condition. Assuming the new-keynesian hillips curve monetary policy transmission mechanism, the estimation of Gali (5, chapter 5) and larida, Gali and Gertler (999) Ramsey optimal policy rule responding to the lagged value of the policy instrument and to a non-observable auto-regressive cost-push shock faces a well-known identi cation problem (Griliches (967), Feve, Matheron and oilly (7)). In order to circumvent this identi cation problem, we substitute the non-observable auto-regressive cost-push shock by the observable policy instrument. We estimate a full-information structural VAR of the policy rule and of the policy transmission mechanism on in ation, with white-noise perturbations without auto-regressive components. The estimate of the relative weight of output in the Volcker-Greenspan Fed s loss function is not statistically di erent from zero. For the transmission mechanism, the estimate of the e ect of the output gap on in ation in the new-keynesian hillips curve is not statistically di erent from zero in commitment. Although this last result has been found

3 with limited-information single-equation estimates of the new-keynesian hillips curve in Mavroeidis, lagbord-möller and Stock (4), the full-information two-equations structural VAR of Ramsey optimal policy imply more constraints on parameters than a singleequation new-keynesian hillips curve. Section presents Ramsey optimal policy estimation method. Section 3 presents the optimal policy without commitment. Section 4 pre-tests and tests Ramsey optimal policy versus optimal policy without commitment for pre-volcker Fed and for Volcker- Greenspan Fed. Section 5 evaluates the robustness to misspeci cation of Ramsey optimal policy versus optimal policy without commitment. The last section concludes. The Monetary olicy roblem: The ase of an E cient Steady State Gali s (5, chapter 5) reference model for optimal policy with or without commitment considers the case of an e cient steady state. The welfare losses eperienced by the representative household are, up to a second-order approimation, proportional to: ( v( ; u ) ma f t; tg E X + ) t t + t () where t represents the welfare-relevant output gap, i.e. the deviation between (log) output and its e cient level. t denotes the rate of in ation between periods t and t. u t denotes a cost-push shock. denotes the discount factor. E t denotes the epectation operator. v( ; u ) denotes the optimal value function. oe cient > represents the weight of the uctuations of the marginal cost of the rm (measured by the output gap) relative to in ation in the loss function. oe cient > is the relative cost of the changing the policy instrument with respect to the costs of uctuations of the policy target, which is in ation. It is given by where is the coe cient on the marginal cost of the rm t in the New Keynesian hillips curve, and is the elasticity of substitution between goods. More generally, and stepping beyond the welfare-theoretic justi cation for (), one can interpret as the weight attached by the central bank to deviations of output from its e cient level (relative to price stability) in its own loss function, which does not necessarily have to coincide with the household s. A structural equation relating in ation and the welfare-relevant output gap can be derived leading to the new-keynesian hillips curve: t t E t [ t+ ] + t + u t where >, < < () The central bank minimizes () subject to the sequence of constraints given by (). The cost push shock u t includes an eogenous auto-regressive component: u t u t + " u;t where < < and " u;t i.i.d. normal N ; u where denotes the auto-correlation parameter and " t is identically and independently distributed (i.i.d.) according to a normal distribution with constant variance u. (3) 3

4 . Ramsey Optimal olicy under ommitment A policymaker with a mandate for a new policy regime revised on date t (corresponding to a structural break in econometrics) is bound to commit with commitment to an optimal policy plan from the current date until a given known date T where the mandate of this policy regime is revised again. Her Lagrangian includes a sequence of Lagrange multipliers t+. L X t t + t + t+ ( t t t+ ) E tt t The law of iterated epectations has been used to eliminate the condition epectations that appeared in each constraint. Because of the certainty equivalence principle for determining optimal policy in the linear quadratic regulator including additive normal random shocks (Simon (956)), the epectations of random variables u t are set to zero and do not appear in the Lagrangian. The program includes given initial u and nal boundary conditions for the predetermined forcing variable variable lim t!+ t u t. It also includes optimal initial and nal boundary values of the forward-looking variable in ation. These transversality conditions minimize the optimal value of the central bank s loss function at the initial and the nal date: (4) lim t ; u t ) t t predetermined for t f; T g, t t for t f; T g T ; u T ) T T!+ T T, lim T lim t!+ t!+ T if T! + (6) We follow Gali (5) and we consider the limit case where the revision for a new policy regime happens in the in nite horizon. Di erentiating the Lagrangian with respect to the policy instrument (output gap t ) and to the policy target (in ation t ) yields the rst order t ) t t ) t + t+ t (8) ) and (9) that must hold for t ; ; ::: where, because the in ation Euler equation corresponding to period is not an e ective constraint for the central bank choosing its optimal plan in period. The former commitment to the value of the policy instrument of the previous period is not an e ective constraint. The policy instrument is predetermined at the value zero at the period preceding the commitment. ombining the two optimality conditions to eliminate the Lagrange multipliers yields the optimal initial anchor of forward in ation on the predetermined policy instrument : 4

5 () and the central bank s Euler equation for the periods following period, for t ; ; 3::: t t t : () The central bank s Euler equation links recursively the future or current value of central bank s policy instrument t to its current or past value t, because of the central bank s relative cost of changing her policy instrument is strictly positive >. This non-stationary Euler equation adds an unstable eigenvalue in the central bank s Hamiltonian system including three laws of motion of one forward variable (in ation t ) and of two predetermined variables (u t ; t ) or (u t ; t ). Ljungqvist and Sargent (, chapter 9) seek the stationary equilibrium process using the augmented discounted linear quadratic regulator (ADLQR) solution of the Hamiltonian system (Anderson, Hansen, McGrattan and Sargent (996)) as an intermediate step. Using the method of undetermined coe cients, this solution seeks optimal negative-feedback rule parameters F (F ; ; F u; ) satisfying the in nite horizon transversality conditions which restricts the policy instrument to be eactly correlated with private sectors variables: t F ; t + F u; u t : () Ljungqvist and Sargent (, chapter 9) ADLQR intermediate step basis vectors ( t ; u t ) of the stable subspace or Ljungqvist and Sargent (, chapter 9), nal step basis vectors ( t ; u t ) or Gali s (5, chapter 5) basis vectors ( t ; u t ) include the nonobservable predetermined cost-push shock u t in their VAR() within the Hamiltonian system (H). We suggest using the basis vectors ( t ; t ) of the stable subspace for the VAR() representation within the Hamiltonian system, using the mathematical equivalence of systems of equations for t ; ; 3::: : with: 8 >< (H) >: 8 ><, >: t+ u t+ t+ t+ A + BF t t t (A + BF ) u t + t F ; t + F u; u t and u given M (A + BF) M t t F ; F u; t u t F u; t and u given F ; M t u t F u; " t + M with M F u; F ; " t (3) 5

6 (see appen- F u; is eliminated using di ): F u; ( ) F u; F ; and F ; M (A + BF) M with: ( ) F ; ( ) F ; + ( ) ( ) ( ) + ( ) ; ; F + + s where the two invariant stable eigenvalues of the stable subspace are denoted by Gali (5) and (appendi ). The other representations of the VAR() including the non-observable cost-push shocks u t amounts to estimate the VAR() as a partial adjustment in ation or output gap equation with serially correlated cost-push shocks u t. These equations face a classic problem of identi cation and multiple equilibria, because the auto-correlation of the dependent variable and of the disturbances are competing to model persistence (Griliches (967), Blinder (986), McManus et al. (994), Fève, Matheron oilly (7), appendi ). We eliminate the non-observable serially correlated cost-push shock u t with a change of basis vectors ( t ; t ) including observable variables. Hence, we are able to t a structural VAR() with the assumption of white noise shocks instead of serially correlated shocks (Sims (98)). Structural parameters are estimated with feasible generalized non-linear least squares for a system of equations. Theory-based constraints on the four reduced form parameters of the matri M (A + BF) M imply that only three structural parameters can be identi ed:,, F ; or,, or, () ; () for a given value of the discount factor : () ) () F ; F ; () : (4) If initial values of in ation and of the policy instrument (in deviation from their equilibrium values) were perfectly measured at the date of commitment, the ratio would be over-identi ed by the optimal initial anchor of forward in ation on the predetermined policy instrument equation: : (5) The semi-reduced form cost-push shock rule parameter F u; requires an identi cation restriction, for eample, setting a value for (see appendi ): F u; () F ; < : (6) The standard error u of cost-push shock is computed using the standard error of residuals "; of the output gap rule equation in the VAR(). It requires an identi cation restriction, because it depends on F u; : 6

7 u () "; F u; () : (7) The standard error of the measurement of the in ation equation (which is theoretically predicted to be zero) and its covariance with the cost push shock F u; u are also available. One identifying equation is missing in order to identify the remaining four structural parameters ( ; ; ; u ) and the negative feedback rule parameter F u;. We set an identi cation restriction on the discount factor to a given value: :99 or in the estimations. One of the reason why an identi cation restriction is required for commitment is that the AR() cost-push shock u t is not observable. The usual practice of DSGE modelers is to include a number of AR() processes equal to the number forward variables, i.e. all prices and ows of quantities variables in their model. The larger the number of non-observable AR() processes, the more likely identifying structural parameters of Ramsey optimal policy under commitment (including central bank preferences) would require additional identi cation restrictions.. Optimal olicy without ommitment Gali (5, chapter 5) considers the case of optimal policy without commitment. It is a time-consistent solution which is the same if the policymaker s optimizes only at the rst period or if he optimizes at each period (Oudiz and Sachs (985)). "That case is often refered in the literature as optimal policy under discretion" (Gali (5)). A big issue with such a wording is that "discretion" refers to opposite predictions than the one of this particular optimal policy model. The prediction of the model is that the policy-maker sticks to a time-consistent e-ante and e-post restricted rule where the policy instrument responds only to current in ation or only to the current non-observable cost-push shock with a perfect correlation. Such a strict rule is the opposite of policymaker s discourse on discretion, being the ability to be eible because new problems arise that could not be anticipated. The Ramsey policy allows a less-restricted, more- eible rule, where the policy instruments responds to its lagged value in addition to current in ation or current cost-push shock. In practice, this rule has more room for eibility and adjustment due to misspeci cation. For this reason, we refer to "optimal policy without commitment" in order to avoid an opposite interpretation of "discretionary policy". The central bank minimizes its loss function subject to the new-keynesian hillips curve and subject to two additional constraints. Both the private sector and the central bank know that their best response rule reacts only to the contemporary predetermined state variable u t at all periods t, with time-invariant rule parameters N D and F u;d to be optimally chosen: t N D u t t F u;d u t F ;D t with F ;D F u;d N D The entral follows a more restrictive policy rule than in commitment. To avoid a confusion between rules and discretion, we use the label of "optimal policy without commitment" instead of "discretion" for this equilibrium. With commitment, the policy rules depends also on lagged values of the policy instrument, and not only on predetermined 7

8 non-observable shock or on in ation. Substituting the private sector s in ation rule and the policy rule in the loss function: ma f; tg +X E t t + t ma t ff u;d ;N Dg The central bank rst order condition is: N D + Fu;D u N u;d + F u;d F ;D F u;d N u;d Substituting the private sector s in ation rule and the policy rule in the in ation law of motion leads to the following relation between N D on date t, N D;t+ and F u;d : t E t [ t+ ] + t + u t ) N D u t N D;t+ u t + F u;d u t + u t N D N D;t+ + F u;d + In the reference Oudiz and Sachs (985) dynamic Nash equilibrium, the central bank foresees that N D;t+ N D in its optimization (see appendi): N D F u;d + F ;DN D + u;d The endogenous rule parameters are increasing function of the central bank cost of changing the policy instrument. They are bounded by limit values of ]; +[: < t;d ( ) N D ( ) u t + ( ) + < N < t;d F u;d ( ) u t + ( ) + < < t;d F ;D ( ) F u;d t;d + N D ( ) < < For an in nite cost of changing the policy instrument! +, we label this equilibrium as "laissez-faire" because two policy rule parameters are both equal to zero F ;D F u;d. The policy instrument t is set to zero at all dates: it is eliminated in the model. It corresponds to the maimal initial response of in ation (in absolute values) to cost-push shock Nu t u t for optimal policy without commitment. For the limit case of a zero cost of changing the policy instrument (! ), the policy instrument (output gap) has its largest response to cost-push shock u so that the policy target (in ation) does not respond to the cost-push shock (N D is zero). The policy instrument (the output gap) t is eactly negatively correlated (F ;D < ) with the policy target (in ation) t. When increasing the central bank s preferences ( ) for the relative cost of changing the output gap from zero to in nity, the strictly negative 8

9 rule parameter F ;D increases from minus in nity to zero. There is one stable eigenvalue and one unstable eigenvalue: < < < D F ;D < +: (8) The welfare loss of optimal policy without commitment v D as a proportion of the limit maimal value of the welfare loss with the largest volatility of in ation (laissez-faire) v LF turns to be equal to the ratio of in ation under optimal policy without commitment to in ation under laissez-faire. It increases from zero to one when the cost of changing the policy instrument increases from zero to in nity: < v D v LF N D + Fu;D ( ) N ( ) + N D N t;d t;lf <.3 Optimal Simple Rule corresponds to Optimal olicy without ommitment Simple rule assume that there is no policy-maker loss function nor welfare function and that the policy instruments are forward variables. For simple rule, when in ation and the policy instrument are forward variables, only the cost-push auto-regressive shock is a predetermined variable with an eogenous stable eigenvalue. Blanchard and Kahn s (98) determinacy condition imply that the controllable eigenvalue, indeed by S for "simple rule": S F ;S should be unstable (j S j > ). This implies constraints on the values of the in ation rule parameter F ;S. Because of the simple rule stable subspace is of dimension one, omitting the identifying restriction F u;s would imply that both rule parameters F ;S and F u;s are not identi ed. roposition : In Gali s (5) model, an optimal simple rule minimizing the central bank loss function is a reduced form of optimal policy without commitment. roof: For a given monetary policy transmission mechanism (; ; ; u ), a simple rule with a strictly negative in ation parameter F ;S, forcing an unstable eigenvalue S i ; + h by positive feedback, is the reduced form of optimal policy without commitment with a unique central bank preference parameter given by: F ;S F ;D < ) F ;S The remaining cases of simple rules with positive rule parameter F ;S ; [ h h + ; + forcing an unstable eigenvalue S ; [ ] ; [ by positive feedback do not minimize a central bank loss function in optimal policy without commitment. For F ;S ;, these simple rules imply a jump of in ation larger than in laissez-faire (N S > N). For F ;S + ; +, these simple rules imply a jump of in ation with an opposite sign with respect to laissez-faire (N S < < N). Those simple rule solution are never optimal simple rule nor reduced form of optimal policy without commitment. When the policy instrument is a forward variable, simple rule parameter F ;S is never the reduced form rule parameter of the commitment in ation rule parameter F ; forcing a stable eigenvalue S ]; [ by negative feedback, where the policy instrument is predetermined. Q.E.D. (9) 9

10 Sargent and Ljunqgvist s () LQR intermediate step allows a direct comparison between the reduced form in ation rule parameters F ;D of optimal policy without commitment (respectively F ; of commitment), which is a ne negative function of the eigenvalue D (respectively ): F ;D and D F ;D + () F ; and s () Figure plots the eigenvalue D of optimal policy without commitment (and respectively the eigenvalue of commitment) as non-linear decreasing (respectively increasing) function of the relative cost of changing the policy instrument for the estimated parameters :995, :34 for a given :99 of the commitment model during Volcker-Greenspan s Fed starting 979q3-6q (see estimation section, with estimated :856 and 4:55). For a minimal cost of changing the policy instrument, the eigenvalue tends to zero for commitment and D tends to in nity for optimal policy without commitment. For an in nite cost of changing the policy instrument: the eigenvalue tends to one for commitment and D tends to > for optimal policy without commitment. Figure plots in ation rule parameter F ;D of optimal policy without commitment (and respectively F ; of Ramsey optimal policy) as non-linear decreasing (respectively increasing) function of the relative cost of changing the policy instrument for the same estimated parameters than for gure (with estimated F ; :447 and 4:55). For a minimal cost of changing the policy instrument, the in ation rule parameter F ; tends to and F ;D tends to minus in nity for optimal policy without commitment. For an in nite cost of changing the policy instrument, the in ation rule parameter F ; tends to ( ) for commitment and F ;D tends to zero for optimal policy without commitment. Figures and : Eigenvalues and in ation rule parameter F function of for commitment (solid line) and optimal policy without commitment (dash line) Eigenvalue 3 Rule parameter ost of changing policy instrument ost of changing policy instrument Table summarizes the opposite properties of commitment with respect to optimal policy without commitment and simple rule when in ation is a forward variable. Table : Ramsey optimal policy, optimal policy without commitment and Simple Rule Stable Subspace 4 6

11 olicy redetermined in ation t Forward in ation t predetermined: u t ; t stable subspace dim forward: t predetermined: u t ; t or t stable subspace dim forward: t ommitment ]; +[ F ; ]; +[ F ; Optimal olicy without commitment Simple rule ; ]; [ negative feedback predetermined: u t ; t stable subspace dim forward: t ]; +[ F ; ; ]; [ negative feedback predetermined: u t ; t stable subspace dim forward: t ] ; [ ; + F negative feedback. ; ]; [ negative feedback predetermined: u t stable subspace dim forward: t ; t ]; +[ F ;D i ] ; [ h ; F u;d D ; + positive feedback predetermined: u t stable subspace dim forward: i t h ; t h h S ; + [ ; [ ] ; [ F ;S ] ; [ [ ; positive feedback F u;s [ + ; + Time-consistent Ramsey optimal policy when zero gains from commitment. For very small or very large values of, optimal policy without commitment and commitment paths and hence central bank losses are identical. There are no gains from commitment because Ramsey policy is already time-consistent without commitment in these limit cases. Optimizing at later periods leads to a negligible deviation from the optimal path chosen at the initial period. When!, in ation t tends to zero at all dates for any initial value of the cost-push shock u for both commitment and optimal policy without commitment. For commitment, the policy rules are F ; F ; and F u;. For optimal policy without commitment, the policy rule parameter F ;D! and F u;d. When! +, t tends to laissez-faire in ation u t at all dates for any initial value of the cost-push shock u for both commitment and optimal policy without commitment. For commitment, the policy rules are F ; F u;. For optimal policy without commitment, the policy rule parameters are both equal to zero (laissez-faire equilibrium) F ;D F u;d. The distinct policy rule parameters for optimal policy without commitment versus commitment re ects that although the equilibrium paths of optimal policy without commitment and commitment are the same and evolve in a subspace of dimension one, there eists a two-dimension stable subspace where the commitment path is surrounded by stable out-of-equilibrium paths which does not eist for optimal policy without commitment and for laissez-faire. Timeless-perspective commitment assumes that the in ation jump occurred for eample periods before, even if the cost-push shock u occurs now. This is equivalent to consider as the current in ation jump is, the value of in ation periods ahead. Hence, in ation jump is very small and very close to the long run value of in ation. Timeless-perspective commitment is equivalent to a time-consistent optimal policy with a very low cost of changing the policy rate and with a maimal volatility

12 of the policy instrument. Timeless- perspective simulation raise two issues. First, it is a theoretically inconsistent optimization. The ad hoc near-zero jump of in ation, which would be consistent with a large volatility of the policy instrument related to nearzero cost of changing the policy instrument (! ) is usually assumed without a large volatility of the policy instrument, corresponding to a non-negligible cost of changing the policy instrument. Second, it assumes away tests of real world structural breaks of new commitment. Timeless-perspective commitment simulations eplains Volcker s structural break in by Martin chairman of the Fed during A re-test of Fed s Optimal olicy without ommitment Optimal policy without commitment is described by a permanent anchor of in ation on the output gap and by two AR() processes of in ation and of the output gap. Variables, such as in ation ( t + ) are not computed as deviations of equilibrium (already denoted t ). Estimates of equilibrium values ( ; ) are then sample mean values found in the estimates of intercepts. The reduced form optimal policy without commitment policy rule to be tested (which corresponds to a permanent anchor of in ation on the output gap) allows to estimate the reduced form optimal policy without commitment rule parameter F ;D : t + F ;D ( t + ) + ( F ;D ) + " ;t with () " ;t for all dates, R and F ;D < The simple correlation between the output gap and in ation provides another estimate of the optimal policy without commitment rule parameter F ;D r "; "; <. It is equal to the one found using the ratio of standard errors of residuals of the AR() estimations for in ation and for the output gap : F ;D "; "; only if the following condition is satis ed: r. Testing optimal policy without commitment against commitment amounts to test the perfect negative correlation between the output gap and in ation: r. Because of test of a simple correlation eactly equal to cannot be performed, we can perform a one-sided test of a composite null hypothesis of a simple correlation very close to minus one: H ;No commitment : r < :99 (3) However, this non-perfect correlation may be due to measurement errors. Hence, the key test of optimal policy without commitment versus commitment is a test of the autocorrelation of residuals of the output gap policy rule function of in ation. In optimal policy without commitment, errors should not be auto-correlated. If they are autocorrelated, this alternative hypothesis suggests that at least the lagged policy instrument is missing in the regression of the policy rule, which is eactly a reduced form of the Ramsey optimal policy rule (which also depends on in ation): H ;No commitment : "; for " ;t "; " ;t + t with t i.i.d. (4) Additionally, the two AR() process for in ation and the output gap are:

13 t + ( t + ) + ( ) + N D " u;t with " u;t i.i.d. (5) t + ( t + ) + ( ) + F ;D N D " u;t with " u;t i.i.d. (6) If the two previous tests did not reject the null hypothesis, we can perform a third test that the auto-correlation coe cients are identical for in ation and for the output gap: H ;No commitment : (7) If this hypothesis is not rejected, the AR() estimates identify the auto-correlation parameter of the non-observable cost-push shock:. The ratio of the standard errors of residuals of each AR() estimations of in ation and output gap provides another estimate of F ;D, (if r ) with a negative sign restriction predicted by theory, and grounded by positive feedback: F ;D "; "; (8) The variance "; of perturbations of the in ation AR() process is: "; N D ";u ) N D "; ";u Unfortunately, the cross equations covariance "; between the residuals of both AR() process of in ation and of the output gap does not allow to identify either the private sector parameter N D anchoring in ation on the cost-push shock or the variance of the cost-push shock ";u. The simple correlation between the two residuals is predicted to be eactly negatively correlated (r "; ): "; "; "; "; ";u (9) ";u "; "; < : (3) Finally, only one structural parameter related to the cost-push shock and one reduced form parameter F ;D are identi ed. However, there remain four structural parameters parameters. For eogenous central bank preferences, these parameters are ; ; ; ";u. For a welfare loss function, as, these parameters are ; ; ; ";u where is the elasticity of substitution between goods. It is not possible to identify at least one of these four parameters separately, because the identi ed parameter F ;D does not depend only on one of these four structural parameters: F ;D : (3) Three identifying equations are missing in the case of optimal policy without commitment instead of one identifying equation in the case of commitment. In the case of endogenous central bank preferences (welfare loss function case,) it is not possible to identify the elasticity of substitution between goods. In the case of eogenous central bank preferences, it is not possible to disentangle eogenous central bank preferences from the monetary transmission mechanism parameter (the identical discount rate is identical to central banks and to the private sector). We cannot disentangle whether an estimated impulse response function is obtained by a large cost of changing the policy instrument and a large marginal e ect of the policy instrument on the policy target 3

14 or by a low cost with a large response of the instrument in the policy rule and a low marginal e ect of the policy instrument on the policy target. This is unfortunate because the main value added of the estimation of optimal optimal policy without commitmentary policy with respect to the estimation of a reduced form positive feedback simple-rule parameter F ;D is to estimate central bank preferences. Finally, the tests of reduced form parameters of bivariate VAR() of optimal policy without commitment versus commitment are not feasible. The eact multicollinearity (eact correlation) between regressors (current output gap and current in ation) imply a bivariate VAR() with in nite coe cients with denominator including the term r equal to zero: t+ t+ + + t ND + t F ;D N D " t (3) The optimal policy without commitment equilibrium predicts that out-of-equilibrium behavior corresponds to a non-stationary bivariate VAR including one unstable eigenvalue D and one stable eigenvalue, which cannot be estimated. By contrast, the stationary structural VAR() of output gap and in ation with optimal policy under commitment allows to identify a larger number of structural parameters. t+ t+ a c b t + d t Fu; Two additional reduced form parameters (b; c) are available, because the stable subspace of the VAR process is of dimension with optimal policy under commitment instead of dimension with time-consistent optimal policy without commitmentary policy. " t 3 re-tests and tests of the Fed s preferences The annualized quarter-on-quarter rate of in ation and the congressional budget o ce (BO) measure of the output gap are taken from Mavroeidis () online appendi (details in appendi 5). The pre-volcker sample covers the period 96q to 979q and the post-volcker sample runs until 6q. The period of aul Volcker s tenure is 979q3 to 987q. The period of Alan Greenspan s tenure is 987q3 to 6q. A rst issue is to de ne the quarter corresponding to the beginning of aul Volcker s commitment. According to Gali (5, chapter 5), this structural break of commitment is related either to the beginning of a permanent anchor of forward in ation on output gap in the case of optimal policy without commitment or to an initial anchor (jump) of forward in ation on output gap in the case of commitment. larida, Gali and Gertler () and Mavroeidis () consider the beginning of aul Volcker s mandate 979q3 as a structural break. Givens () considers 98q as a structural break, after 98 fall of in ation and before the 98 recession. Matthes (5) estimation of the private sectors beliefs regarding central bank regimes also points to 98q as a structural break. Table presents summary statistics before and after the 979q3 and 98q structural breaks. Table : Summary statistics of in ation and output gap for 979q3 and 98q breaks. 4

15 Break obs. var. mean min ma before79 78 t 4:39 :59 :79 (:7) before79 78 t :47 (:59) before8 88 t 4:86 (:9) before8 88 t : (:59) after79 8 t 3:8 (:3) after79 8 t : (:7) after8 98 t :64 (:8) after8 98 t :3 (:) 4:97 6: :59 :79 4:97 6: :64 :93 7:95 3: :64 5:6 7:95 3: Standard deviations in parentheses. The mean of in ation and of output gap are lower after Volcker than before Volcker. Ecluding the period 979q3 to 98q4, in particular the sharp disin ation which occurred during 98 ( gure 3), the standard error of in ation decreases by half from.3 to.8. Table 3: re-tests of in ation and output gap permanent anchor correlation for 979q3 and 98q breaks. Break obs r Low 95% r p R F ;D c "; before79 78 :3 : < : : :3 (:) after79 88 :4 :3 < : :6 : (:9) before8 8 :3 :4 < : :9 :3 (:9) after8 98 :4 :53 < : :6 :78 (:8) :3 (:56) :5 (:53) :4 (:35) :3 (:5) :9 (:4) :9 (:5) :9 (:4) :89 (:5) The pre-tests of the null hypothesis of a quasi perfect negative correlation H : r < :99 between observed in ation and observed output gap are rejected. The test uses Fisher s Z transformation using the procedure corr with the software SAS. The threshold of the composite null hypothesis :99 is far away from the 95% single tail con dence interval, where the lowest 95% con dence limit reported in table 3 is at most equal to :53 for the period after 98q4 ( gure 4). The opposite null hypothesis H : r ( t ; t ) is not rejected before 979q3. optimal policy without commitment predicts a perfect correlation for the anchor of in ation epectations with the output gap. If the optimal policy without commitment equilibrium occurred before 979q3, we do not reject the null hypothesis H : r (E t ( t ) ; t ) that the rational epectations of in ation are orthogonal to observed in ation. The pre-tests of the null hypothesis of the auto-correlation of residuals H : "; are strongly rejected, with a point estimate at least equal to :89 ( gure 5). These tests gives a hint of model misspeci cation. They suggest an omitted lagged policy instrument in the policy rule. When it is included in commitment policy rule, the R increases from 6% (table 3, last line) to 93% (table 5, last line) beginning in 98q. Figure 3. In ation and output gap anchor correlation. Table 4 investigates the auto-correlation and unit roots of in ation and output gap. Table 4: breaks. Auto-correlation and AR() statistics for 979q3 and 98q 5

16 Break obs. var. r R c " " DF before79 78 t :86 :74 :88 :6 :38 : :55 :4 (:6) (:3) (:) before79 78 t :93 :86 :93 (:4) before8 88 t :88 :78 :88 (:5) before8 88 t :9 :85 :93 (:4) after79 8 t :89 :79 :85 (:4) after79 8 t :94 :88 :94 (:3) after8 98 t :64 :4 :59 (:7) after8 98 t :96 :9 :95 (:3) :3 (:) :63 (:8) :3 (:) :4 (:6) :7 (:8) :6 (:) : (:7) :99 :6 (:) :35 :9 (:) :3 :3 (:) :93 :7 (:9) :7 :34 (:9) :83 : (:) :6 :35 (:9) : :33 :34 :3 :7 :37 :4 : : :5 : : :7 :35 The output gap and in ation are highly auto-correlated (respectively.93 and.86), ecept when in ation ecludes the 98 disin ation for the period after 98q4. For the period 98q to 6q, the in ation auto-correlation coe cient falls in the 95% con dence interval :6 :4 and it is statistically di erent from the output gap autocorrelation coe cient in the 95% con dence interval :95 :6 ( gures 4 and 5). As the optimal policy without commitment equilibrium predicts that the auto-correlation of the output gap and of in ation should be the same, this is an additional test against optimal policy without commitment, which holds for the period 98q to 6q. There is a negative auto-correlation of residuals " for in ation and a (statistically signi cant at the 5% level) positive auto-correlation of residuals for the output gap. The column DF reports the p-value of the Dickey-Fuller test of unit root with one lag without trend. The column reports the p-value of the hillips-erron test of unit root, which takes into account auto-correlation, with one lag without trend. The null hypothesis of a unit root is rejected for in ation after 979q and after 98q4. The commitment bivariate VAR takes into account the lagged cross-correlation between in ation and the output gap (table 5). Table 5: In ation and output gap cross-correlogram for 979q3 and 98q breaks and Granger ausality. Break obs. GW r ( t ; t ) r tt r ( t+ ; t ) GW before79 78 :3 : :3 :3 :5 before8 88 : :3 :4 :5 :9 after79 8 : :33 :3 :4 :99 after8 98 : :47 :4 :33 :34 There are larger correlation from lagged in ation to output gap than from lagged output gap to in ation, although those correlation coe cient are far beyond auto-correlation coe cients ecept after 98q for the auto-correlation of in ation. Although they are related to the partial correlations of the VAR() of table 6, we report the p-values of Granger causality Wald test that the cross-variables coe cient is zero in the two columns GW in table 5, for comparison with simple cross-correlations. Even though the simple correlation of in ation with lagged output gap increased for periods more and more recent in table 5, their partial e ect in the VAR including lagged in ation is small enough to reject Granger causality of the output gap on in ation, in particular for the Volcker-Greenspan period. By contrast, Granger causality from in ation to the output gap is not rejected at the 5% level. Table 6: In ation and output gap unconstrained VAR() for 979q3 and 6

17 98q breaks. Break obs. var. t t c " R dr ; jj before79 78 t :9 :89 :9 :54 :6 :75 : (:6) (:6) (:3) (:) :8i before79 78 t :9 (:4) :9 (:4) :4 (:) : (:) :87 : :9 before8 88 t :89 (:5) before8 88 t :8 (:4) after79 8 t :85 (:4) after79 8 t :8 (:3) after8 98 t :56 (:7) after8 98 t :7 (:5) :6 (:6) :9 (:4) : (:4) :9 (:3) :4 (:4) :9 (:3) :55 (:9) :39 (:) :43 (:6) :7 (:) :9 (:) :38 (:4) : (:) :7 (:) :7 (:9) :9 (:9) :7 (:) :3 (:9) :79 : :9 :7i :85 : :9 :79 : :85 :89 : :9 :4 : :54 :9 : :93 Table 6 presents the unconstrained VAR() results in order to compare them with commitment structural VAR() in table 7 and table 8. The second equation of the VAR() is a representation of the optimal policy rule. The increase of R of the VAR() including cross-correlations with respect to the R obtained with the AR() process is at most of % for each equations. This is re ected in the impulse response functions which are not statistically signi cant for cross-correlation even in the most favorable case of after 98q4 with a parameter :7 for the output gap policy rule. The auto-correlation coe cients for each equations are nearly identical to the ones of AR() equations. They are not negligible: the p-value of the Lagrange multiplier test of auto-correlation of order of residuals of both equations are for each period in chronological order: :, :4, : and :. One does not reject the auto-correlation of residuals at the 5% level, with very low p-value for the Volcker-Greenspan period. This auto-correlation of residuals test gives a rst hint of model misspeci cation of the commitment model during the Volcker- Greenspan period. The policy target, in ation, does not depend on the policy instrument of Gali s (5) reference model (the output gap) according to the t-test after 979q and after 98q4. There is no Granger causality of output gap on in ation in the reduced form VAR(). The imaginary components of the eigenvalues (which are small) contradicts the assumption of an eogenous real auto-correlation coe cient for the cost-push shock before 979q and before 98q4. Before Volcker, one cannot identify structural parameters of commitment even for a given discount factor. The estimations does not converge for the constrained VAR() using non-linear estimation for before Volcker s periods. Table 7: In ation and output gap structural VAR() reduced form (ecluding periods with conjugate comple roots). Break obs. var. t t c c c u R RU U after79 8 t :85 :85 after79 8 t :64 :84 after8 98 t :56 :56 after8 98 t :84 :7 :9 : :999 :97 :3 :3 :48 (:6) :98 (:) :86 (:) :43 (:6) :7 (:) :9 (:) :793 :79 :857 (:54) :888 :89 :995 (:4) :46 :4 :56 (:79) :5 :59 :38 :99 :9 : :933 :9 (:4) (:4) (:5) Table 7 compares the reduced form parameters of the structural VAR() with the :85 :9 :54 7

18 unconstrained VAR() estimates (estimates without brackets below the reduced form parameters of in ation and output gap). The in ation equation does not change. The constraints modify the VAR() parameter estimate towards a unit root for the output gap (the parameter shifts from :93 to ) along with a small decline of the e ect of lagged in ation on the output gap and with a fall of the constant. R are very close for the structural and the unconstrained output gap equation. Table 7: Tests of Ramsey optimal policy structural parameters for given discount factor (ecluding periods with conjugate comple roots). Break F ; () () F u; () u () after79 :995 (:4) after79 after8 after8 :995 (:4) : (:5) : (:5) :857 (:54) :857 (:54) :56 (:79) :56 (:79) :447 (:9) :447 (:9) :89 (:) :89 (:) 3:375 (6:67) 3:375 (6:67) 6:79 (3:93) 6:79 (3:93) 5 :3 (:33) 5 :99 :34 (:34) :55 :35 (:7) :55 :99 :354 (:76) (opposite sign?). 4:96 (5:447) 4:55 (5:73) 5:64 (9:) 5:84 (9:44) 3:7 :9 :86 :4 :436 :56 :43 :583 5 is not in the con dence interval of Table 7 presents estimates of structural parameters, with small changes for estimates with two given values for the discount factor or :99. Estimates are found using three structural VAR estimations for (; ; F ; ), ; ; and (; () ; ()). We check that point estimates satisfy the theoretical constraints of appendi. The cost-push shock faces a unit root. Because of the fall of the auto-correlation of in ation from :857 starting 979q3 to :56 starting 98q, estimates of other parameters have very di erent magnitude for both periods. The new-keynesian hillips curve parameter is not signi cantly di erent from zero at the 5% level, as in many of Mavroeidis, lagbord-möller, Stock (4) limited-information single-equation estimations. The Fed s preference parameter is not signi cantly di erent from zero at the 5% level, as well as the reduced form rule parameter F ;. However, the ratio is statistically signi cant when starting the estimation in 979q3. Although the goodness of t indicators of the commitment structural VAR() is far better than optimal policy without commitment, the monetary policy transmission mechanism (the new-keynesian hillips curve parameter) and the policy-maker preferences are not statistically di erent from zero at the 5% level. One can compare two observationally equivalent representations of the reduced form of the policy rule of optimal policy under commitment for the period after 979q3, with the rule parameter F u; () identi ed with the identi cation restriction :99. The rst one includes the reduced form parameters of the structural VAR() of observable variables. The second one is the ADLQR intermediate representation including the nonobservable cost-push shock: t :995 t :64 t :86" u;t and t :447 t :86u t for :99 (33) Although both representations look di erent, in particular with the opposite sign of the policy rule parameter related to current or lagged in ation, they are observationally equivalent within the Hamiltonian system of equations taking into account the stable subspace constraint. By contrast, the reduced form policy rule for optimal policy without commitment would be for the period after 979q3: t : t (34) 8

19 The key di erence between reduced form rules of commitment versus optimal policy without commitment is that the policy instrument responds to two epected variables in the commitment stable subspace of dimension two and to one epected variable in the optimal policy without commitment stable subspace of dimension one. 4 Robustness to Misspeci cation The initial anchor (commitment) or the permanent anchor (optimal policy without commitment) of in ation on the output gap raises the issue of robustness of this anchor to misspeci cation related to the various measurement errors of the output gap and of in ation. First, gross domestic product (GD) implicit price de ator may di er from various measures of core in ation which is the target of central banks. Substracting the changes in the quality of goods in the increase of prices is not done with a perfect accuracy. National accountants may change the scope of GD over time, which also changes the price de ator. For eample, they included investment in intangible assets after 6. The current measure of output does not correspond to the nal measure after revisions by national accountants up to three years after. Output gap can be computed using di erent methods leading to distinct values, in particular when there is uncertainty on structural breaks on the trend of productivity growth, as in 973 and in 7. Figure 7 (respectively gure 8) represents impulse response functions of output gap and in ation after a small cost-push shock u % for optimal policy without commitment (resp. commitment) using commitment point estimates of Volcker-Greenspan 979q3-6q period ( 4:55, :99, :34 and :995). Figures 7 and 8 plots two out-of-equilibrium impulse responses for an evil-agent :% taking into account a small measurement error of the initial date output gap. In the case of optimal policy without commitment for a measurement error of the initial output gap :%, the evil-agent out-of-equilibrium optimal policy without commitment path leads to depression in a year coupled to hyperin ation in two years ( gure 7). For a measurement error of the initial output gap + :%, the evil-agent out-of-equilibrium optimal policy without commitment path leads to boom in four quarters coupled with de ation in ve quarters. On gure 8, evil-agent out-of-equilibrium commitment in ation path are converging with the optimal path in three years, with near-zero in ation in ve years. Because the policy instrument linearly responds to the near-unit-root cost-push shock, the convergence towards equilibrium of out-of-equilibrium and equilibrium paths is very long. These gures shows that for plausible evil-agent measurement errors of the in ation anchor, commitment is a pre-condition for robust-control optimal policy (Hansen and Sargent (8)), unless the behavior of a good-little-evil-agent is strictly limited in order to only operate in the stable subspace of optimal policy without commitment (Giordani and Söderlind (4)). 5 onclusion This paper proposes pre-tests of Ramsey optimal policy versus optimal policy without commitment and (optimal) simple rules. The principle is to test the number of linearly independent observed variables predicted by each equilibria for any DSGE model. Then, as a second step, entral Bank preferences and monetary policy transmission mechanism are tested for the model which passed the pre-test. 9