3.4 Estimation of Policymakers' Preferences using Dynamic Programming
3.4.3 Optimal Control and GMM versus Dynamic Programming
policymaker problem, and the GMM estimation with an optimal control solution, yeld the same answers to the questions motivating this research.
Both set-ups suggest that the sample period should be partitioned, and react well to the partition at 1985:IV. Both suggest that it is only possible to identify an equilibrium real interest rate in the second sub-sample, and broadly agree in their estimate (4.5 - 4.6). Both frameworks precisely estimate very similar inflation targets for the 1986:I-2001:II period (2.7 - 2.6), and fail in doing so for 1972:I-1985:IV. Both clearly indicate that the Euro Area policy regime of 1986:I-2001:II, has been one of strict inflation targeting with interest rate smoothing. Both calculate a decrease in the standard error of the residuals of their optimizing equations slightly above 45 percent, after 1986 - which is remarkable, given the differences between the Euler equation and the optimal linear policy rule. And both agree in their estimation of the fall in the standard error of the Phillips curve residuals, after 1986 - precisely 41 percent for both.
As reviewed above, there is less agreement between the estimates obtained from these two approaches for the US case. First, the inflation targets estimated for the most recent regime are 2.63 percent in Favero and Rovelli (2001) and 1.38 percent in Dennis (2001). Second, and most particularly, while Favero and Rovelli find a statistically significant weight of the output gap in the Fed's loss function, Dennis finds a point estimate of that weight that is much larger but he can not rule out the hypothesis that the true coefficient is zero. In view of the similarity of our broad results across both methods - except for the divergence discussed below - we conclude that the differences
77 We have also checked the sensitivity of our results to the timing assumption in the model. Specifically,
we have estimated the model assuming that the central bank watches inflation and the gap up until period (t-1), when deciding his policy for period (t). This gives an optimal state contingent linear policy rule with inflation and the gap lagged one period in comparison to the specification in the text - resembling the Taylor rule used in Cogley and Sargent (2001). Our estimation results show an increase in the lag of optimal interest rates to actual rates, but the essence of the econometric results is unchanged.
between Favero and Rovelli results and Dennis' can originate from the disparities in their empirical procedures discussed in section 3.1 above.
Coming back to the Euro Area case, the Euler equation estimated by GMM fits the first and second moments of interest rate somewhat worse than the optimal state- contingent linear policy rule estimated with FIML. The root of the mean square error (RMSE) of fitted rates is 0.62 for the Euler equation, and is 0.46 in the optimal-policy- rule estimated by FIML. The standard deviation of the series of fitted interest rates is 3.11 in GMM and 2.63 in FIML, against a sample standard deviation of 2.65.
Figure 3.7 – Actual versus Fitted Interest Rate, 1986:I-2001:II, Loss Function: SITIRS, Dynamic Programming and FIML, and Optimal Control and GMM
FIML GMM ACTUAL 0 2 4 6 8 10 12 14 1986 Q1 1986 Q3 1987 Q1 1987 Q3 1988 Q1 1988 Q3 1989 Q1 1989 Q3 1990 Q1 1990 Q3 1991 Q1 1991 Q3 1992 Q1 1992 Q3 1993 Q1 1993 Q3 1994 Q1 1994 Q3 1995 Q1 1995 Q3 1996 Q1 1996 Q3 1997 Q1 1997 Q3 1998 Q1 1998 Q3 1999 Q1 1999 Q3 2000 Q1 2000 Q3 2001 Q1
Notes: GMM fitted interest rates obtained by dynamically solving the IS-Phillips-Euler system, using the coefficient estimates obtained for the sample period 1986:I-2001:II;
FIML fitted interest rates computed as observed interest rates minus residuals of estimated optimal policy rule for the sample period 1986:I-2001:II.
Figure 3.7 shows that the interest rates fitted by the Euler equation (GMM) and by the optimal policy reaction rule (FIML) are not dramatically different. By construction, they are not, however, strictly comparable: the series of GMM fitted
interest rates is obtained by dynamically solving the estimated IS-Phillips-Euler system, assuming perfect knowledge of the inflation rate at 3 and 4 quarters ahead.78
One important difference that is apparent in the figure is that the rates fitted with the FIML and dynamic programming approach tend to lag actual and GMM fitted rates. The contemporaneous correlation between the interest rates fitted by the two methods is 0.966, whereas it increases to 0.978 when the GMM interest rates are lagged once. This reflects the dominance of the auto-regression element in the optimal policy rule, which is associated to the high estimate of the interest rate smoothing weight obtained with the dynamic programming approach: 1.999, versus 0.014 in Euler-GMM.
We come now to the heart of the difference between the results in sections 3.3 and 3.4: the estimate of µ. The difference already exists - and is even quantitatively more important - between results for the US in Favero and Rovelli (2001) and Dennis (2001). When comparing his results to those of Favero and Rovelli, Dennis suggested that the difference appears to stem from two facts. First, GMM models the interest rate changes - in the Euler equation - while the interest rate equation in FIML is estimated in levels. Second, GMM implies a truncation of the policy horizon - in practice, it assumes that δ =0, for all quarters - while FIML considers the infinite horizon optimization problem when estimating the policymakers' optimal reaction function. This could be important, for Dennis, as it takes long and variable lags for monetary policy to impact on the real economy. Soderlind et al. (2002) also points out this second reason as the main explanation for the divergences in results.
5 ≥
i
There are also econometric differences possibly affecting the results. From certain points of view, the FIML approach is more restrictive and sensitive than GMM, as it requires the assumption of normality of the residuals of the structural system, while GMM depends only on a set of orthogonality conditions and not on probabilistic assumptions - see Wooldridge (2001). Also, FIML may be more sensitive in that it adjusts the coefficient estimates to improve the fit of the equation with worse mean square errors of the system. On the other hand, GMM is more sensitive to non-
78 If the fitted interest rate from the GMM approach was computed as that of the FIML framework -
observed series minus interest rate equation estimation residuals - we would obtain a fitted series observationally equivalent to the actual series. This is caused by the small magnitude of the Euler equation residuals, when estimated - as in this research - using actual future values of inflation in place of its expected values.
stationarity of the moment conditions - Hamilton (1994, page 424) -, than FIML is to non-stationary time-series in the system. Also, GMM could suffer more from small- sample problems, which especially difficult the estimation of the variance-covariance matrix when the moments are serially correlated as in our case.79 It is not clear, at this stage, the net effect of all these econometric particularities.
Favero and Milani (2001), and Castelnuovo and Surico (2001), have offered evidence somewhat suggestive that the interest-rate-smoothing puzzle could be solved by some consideration of the model uncertainty faced by policymakers, using Granger's (2000) thick modeling approach.80
Another hypothesis is that the puzzle could be caused by the fact that the Euler/GMM framework uses actual future values of inflation, in place of expectations, while the dynamic-programming/FIML approach uses only actual current and lagged state variable values. If policy inertia is caused by expectations errors - the excessive smoothness of inflation forecasts being passed on to the policy rates path - it could happen that the Euler/GMM framework generates lower interest rates smoothing weights estimates. If this hypothesis is true, then the estimates of the degree of optimal policy inertia from both methods would only converge if inflation expectations were replaced, in the Euler equation, by expectations available to policymakers in real-time. We have no chance, however, of testing such an hypothesis, at least for the time being.
Finally, we explore another possible explanation. The dynamic programming/FIML approach is based on the lagrangean method of solution to the optimisation problem, which, as Chow (1997, page 25) notes, always finds an optimal control function, even when the system does not reach a steady state. Now, the state vector converges to an equilibrium if and only if the matrix governing the dynamics of Xt under optimal control has all its characteristic roots smaller than unity in absolute
value - see Ljungqvist and Sargent (2000, chapter 4). From equations (3.22) and (3.16) above, we have
79 See Hansen et al (1996), Anderson and Sorensen (1996), Burnside and Eichenbaum (1996), Christiano
and Den Haan (1996), Canova (1999b), Florens et al. (2001), and Wooldridge (2001) on the finite-sample problems of GMM.
80 In subsequent research, Castelnuovo and Surico (2002) fix µ = 0.2 - the standard value assumed by
Rudebusch and Svensson (1999, 2002) - and estimate the inflation and gap variability weights minimising the distance of the interest rate fitted by the optimal state-contingent linear rule to that fitted by the unconstrained estimate of the rule.
C A Bi A AX A Xt+1= 0−1 t + 0−1 t + 0−1 ⇔Xt+1 =A~Xt +B~it +C~ C~ ) g GX ( B~ X A~ Xt 1= t + t + + ⇔ + ⇔ Xt+1 =(A~+B~G)Xt +(B~g +C~)
Hence, the optimal dynamic system for Xt will only be stable - Xt will converge
to a unique stationary distribution - if the maximum absolute value of the eigenvalues of matrix )(A~+B~G is strictly smaller than unity. Some of the studies elsewhere in the literature check for this condition. For instance, studying data of the 11 EMU member- states within a dynamic programming framework close to ours, Aksoy et al. (2002) report maximum eigenvalues of 0.99 for all countries.
In studies that use dynamic programming to compute the optimal linear policy rule, given estimates of the structural parameters of the model, roots so close to unity do not create problems. However, when dynamic programming is used together with non- linear estimation, such proximity to non-stationarity may create numerical problems.
Table 3.10 shows that the maximum value in modulus of the characteristic roots of the optimal dynamic matrix for the state vector is almost always numerically undistinguished from one, in this study. The eigenvalue that is further away from unity is the one of the strict inflation targeting regime estimated for 1986:I-2001:II, in which case the maximum root of the system is 0.983. Interestingly, this is, among our estimates, the case where the loss weights are most precisely estimated and, indeed, have more reasonable point estimates.
Table 3.10 – Maximum Absolute Value Of Eigenvalues Of Matrix (A+BG)
1973:I - 2001:II 1973:I - 1985:IV 1986:I - 2001:II
Loss: ∆ t t t x i L π , , 0.996 0.992 1.007 ∆ t t i L π , 0.995 0.991 0.983
Values reported are the maximum of the characteristic roots, in modulus, of matrix (A~+B~G), which is the optimal dynamics of the state vector given by the solution to the dynamic optimization problem
As before, we finalize this section by checking whether the results would be different if the estimation period is restricted to 1986:I-1995:II, to assess if the well
identified monetary regime beginning in 1986 has significantly changed in 1995. We find that the greater absolute value of the eigenvalues of the optimal state-vector dynamics is 0.985, for that period, which is very much close to the maximum absolute value of the characteristic roots for 1986:I-2001:II.81
In the end, we consider the interest rate smoothing question an unsolved puzzle. We have offered arguments that seem to suggest that neither GMM results with optimal control, nor FIML results with dynamic programming, should be considered superior, with our present knowledge of the problem. Yet, we have shown that there are reasons to cast some doubts on the numerical results of FIML estimation based on the lagrangean method of solution to the dynamic programming problem. Fortunately, the results from both methods are qualitatively identical, so our conclusions in this research seem to be reasonably robust.