According to Taylor's rule, monetary policy can be characterized in terms of the response of interest rates to a weighted sum of inflation and output variation. The reason for this is that the weights in efficient Taylor rules are functions of the policy authority's behavioral parameters, which are generally neither observable nor amenable to estimation. Estimation of the representative comparisons for the six countries included in this study is undertaken in Section 4.
Second, the empirical evidence on the importance of expected future inflation as a determinant of the current inflation rate is mixed, and the results appear to be quite sensitive to the estimation method used.
Derivation of the Efficiency Conditions
The effective region for γ consists of all values of γ that satisfy (18) with respect to α1, α2, and β1. The effective region for γ can be obtained from (18) by allowing λ to vary from zero to infinity for all possible values of δ (i.e. 0< δ <1).13 The limiting value of the effective region associated with λ = 0 is easy to determine. It will be found below that γ =α2(α1+β1)/α21 is the lower bound of the efficiency region for γ when β1 is positive.14 The boundary value of the efficiency region as λ approaches infinity is somewhat more difficult to determine.
Due to the complexity of the expression for ¯K given by (13), a general characterization of the efficiency range of this expression is not clear.
Estimated Efficient Ranges
In the context of the model employed here, efficiency also requires that all the generalized Taylor rule coefficients fall within their individual efficient ranges. For Italy, the quality of the estimation results steadily deteriorated as the sample was extended after 1992. Perron's (1989) Model A captures the impact of the oil price shocks in the form of a shift in the average of the inflation and/or interest rate processes in France, Italy, the United Kingdom and the United States.
Details of the variable definitions, unit root tests, and the estimation results can be found in Appendix 3. The lower bounds of the effective areas shown in the second column of this table were calculated as ˆ. The third column of Table 4 gives the 95% confidence interval for the estimated lower bound of the effective range for each country.20.
20The 95% confidence interval for ˆγ was obtained by using the asymptotic standard error of the estimator ˆγ= ˆα2(ˆα1+ ˆβ1)/αˆ21 and the critical value tc = 1.96. The construction of these efficient series is therefore based on the assumption that changes in the value of λ have no impact on the estimated values of the parameters needed to calculate them. The validity of the calculated efficient ranges clearly depends on the extent to which the Lucas critique represents a significant empirical problem in this study.
The first step involves careful modeling of the individual processes that generate the variables included in the estimating equation. Details of the estimated marginal processes and the invariance test results are provided in Appendix 3.
Efficient Classes of Rules
The results reported in Table 4 indicate that the pure price rule (γ = 0) is included in the 95% confidence interval for Canada and Germany. According to (23), efficient weights for pure price rules are independent of the values of λ and δ. Effective pure price weights, along with their 95% confidence intervals, are reported in Table 5 for Canada and Germany.
Pure price rules are excluded from the feasible set of benchmark rules for all other countries in the sample.). Because the coefficients α1, α2, β1 and β2 are generally positive, (23) suggests that the effective weight on inflation variation in a pure price rule must be negative. It is also clear that an efficient pure price rule ensures that the real interest rate is negatively related to the output gap.
Moreover, (25) can be used to describe the conditions under which pure price rules will be efficient. Substituting the estimated values of α1, α2, and β1 into (25) and letting δ vary from 0 to 1 identifies the range of λ values for which efficient pure pricing rules are feasible. The values of λ for which pure price rules are efficient are shown in the last two columns of Table 5.
As is the case with pure price rules, nominal income rules are efficient only for special combinations of the policy authority's preference parameter and the discount rate. 21The fact that the 95% confidence intervals for the values of λ associated with efficient pure price rules and nominal income rules overlap for Canada and Germany indicates that there is a range of λ values for which these countries can choose to use benchmark rules for pure prices or use nominal incomes. .
Estimated Interest-Rate Rules
A striking feature of the results reported in Table 7 is that the point estimate of g1 associated with the best interest rate rule is negative for Canada, France, Italy, and the United Kingdom; however, positive values for ˆg1 are excluded from the 95% confidence interval for Italy only. In Table 8, positive values for ˆg1 are excluded from the 95% confidence intervals for Canada, France, and the United Kingdom.22 In the literature, a negative ˆg1 is generally seen as evidence that the monetary authority has allowed the real interest rate to fall during inflationary periods and . 22 There is some evidence that the sign of the inflation response coefficient ˆg1 is quite sensitive to changes in the specification of the interest rate rule used to signal monetary policy.
For example, Clarida, Gal'ı, and Gertler (1998) use a policy rule where the interest rate responds to expected future inflation and obtain positive point estimates of g1 for the countries in their sample. The real interest rate is usually defined as the difference between the nominal interest rate in a given period and expected, rather than concurrent, inflation. To obtain estimates of the relative weight γ, the country-specific estimates of g1 and g2 can be used.
Comparing the interval estimates for ˆγ with the efficient ranges given in Table 4 shows that France, Germany, Italy and the United States satisfy the necessary condition for efficiency given by (18) in that some or all of the estimated 95 % interval for ˆγ is contained in the 95% confidence interval for the efficient range. Point estimates of the effective values of g1, g2, g31 and g32, together with their 95% confidence intervals are reported in Table 10. A summary of the results obtained by comparing the characteristics of each country's estimated real and effective interest rate rule is presented in Table 11.
For the purposes of this analysis, a parameter is considered efficient as the intersection between the 95% confidence interval for the efficient parameter range (for ˆg1, ˆg2 and ˆγ) or the efficient parameter value (for ˆg31 and ˆg32) and the 95% confidence interval for the parameter value associated with the implemented interest rule is not empty. The second possibility is that the introduced interest rate rule has the correct form, but conflicts with one or more of the efficiency criteria derived in paragraph 3. The source of the inefficiency before that period appears to have been a tendency on the part of the Federal Reserve to overreact to German inflation.
In the model used in this article, a policy authority's choice of interest rate policy depends on the behavioral parameters δ and λ.
Conclusion
First, the fact that several coefficient estimates for Germany and Italy fail the invariance test indicates that the comparative ranges for the Taylor rule coefficients for these two countries, as well as the conclusions drawn from these estimates, are unreliable. Second, although there is evidence that expectations of future inflation may not be important in determining the inflation rate in the United States, it is not clear whether this is the case for the other countries in this study. Because the comparative Taylor rule is sensitive to the underlying structure of the model, an efficient rule based on the traditional Phillips curve may not be an appropriate measure for some countries in the sample.
Finally, during the sample period, European monetary authorities were in the process of transitioning to a single currency. European monetary policy was limited by repeated speculative attacks on individual currencies and also by some provisions of the Maastricht Treaty. Although quadratic loss functions are the most common way to represent the objectives of policy authorities, the results of this study suggest that loss functions of this form may not describe the objectives of European monetary authorities very well.
On the other hand, the results obtained here provide some indirect support for using a quadratic loss function to describe the Federal Reserve's policy objectives.
Determination of k 1 and k 2
Using (6) to substitute V(πt+2|t+1) into (5) and taking the derivative of the expression in parentheses with respect to πt+1|t gives. A.3) Differentiation of the assumed solution for V(πt+1|t), given by (6), with respect to πt+1|t yields.
Limiting values of k 1 and k 2
Only the positive root of (A.7) is a solution fork2 because, from (A.6), k2 must equal 1 for all non-zero values of δ and α2 when λ= 0; this condition is not met by the negative root. Now C and D are constants, so the denominator on the right-hand side of (A.14) is given by.
Details of Empirical Procedures
The invariance tests performed using the procedure described in Section 4.2 of the main text are summarized in Tables A3.1 and A3.2. The calculated values of the test statistic and the distribution of the statistic under the null hypothesis are reported. An asterisk added to the test statistic in Table A3.2 indicates rejection of the null hypothesis that the estimated parameter value is invariant at the 5% significance level.
In those cases where visual inspection of the data suggested the presence of structural changes, the procedure of Perron (1989) was used. Perron has shown that the critical values of the test statistic depend on the time period in which the structural break occurs. The year of the break and the proportion λ of the total observations occurring before the break are given in the third column of Table A3.3.
The absence of an entry in this column indicates that there was no apparent break in the data over the sample period. The test statistics obtained on the basis of Augmented Dickey-Fuller tests and Perron's test procedure are given in the last column of the table under the heading ADF/ADFP. It is clear from the reported results that all the variables used in the estimation of (3) and (4) are I(0) at a level of significance of at least 10%.
In Table A3.3, the significance of the test statistic at the 1% and 5% level is indicated by ** and *, respectively. The presence of a deterministic time trend was rejected at the 5% significance level for all variables.
