APPENDIX III The output gap estimates of the US and Germany 131
3.1 Estimation results
All models were estimated using maximum likelihood method ultimately based on Hamilton (1994). The estimations were carried out using MSVAR package for Ox, which applies the EM algorithm. If the economic relationships to be modelled could be adequately characterized by a linear model, there is no sense in estimating a non-linear model, like the Markov switching model. Thus, the estimations were started by comparing the adequacy of the linear model against the Markov switching model using both the conventional AIC and SBC information criteria as well as the LR tests. Using the LR tests is actually problematic, since the parameters p and 11 p appear only in the MS model but not in the linear model. The 22 information matrix of the model with two states would then be singular if the null hypothesis of only one state holds. Accordingly, the asymptotic distribution of the LR test is not known11. The problem was handled by using the conservative χ2(1) distribution. The LR tests still always suggested that the hypothesis of a linear model should be rejected at 99 % confidence level. The information criteria also suggested estimating a non-linear model.
The number of states in the Markov switching model was chosen to be two for the models for both the Fed and the Bundesbank. Actually, the information criteria suggested models with three states, but these models turned out to be too difficult to interpret, mainly because of the very ragged shapes of the line graphs of the smoothed probabilities of the regimes. The Markov switching modelling also allows for estimation of the error term variance separately for each of the states, which is clearly an advantage in our estimation. The assumption of a constant variance is not likely to hold for short-term interest rates as it is the case for also many other financial time series. The estimated parameter values for the policy rules of the Fed and the Bundesbank are presented in Tables 3.1 and 3.2 below. The timing of the regimes is represented by the time paths of the filtered, smoothed and predicted probabilities of the two regimes that are shown in Appendix II. The filtered probability is the optimal inference on the state variable at time t using only the information up to time t, i.e. Pr(st =mYt)). The smoothed probability refers to the optimal inference on the state variable at time t using the
optimal inference on the state variable at time t using only the information up to time t-1, that is, Pr(st =mYt−1).
Table 3.1 Coefficient estimates for the policy rules of the Fed.
Model 1 Model 2 Model 3 Model 4
State 1 State 2 State 1 State 2 State 1 State 2 State 1 State 2
Const. 1,99 0,77 6,16 10,55 1,94 0,89 5,98 10,79
(3,87) (2,89) (41,36) (22,84) (3,92) (3,8) (43,77) (22,08)
Infgap 0,88 1,79 0,86 2,55 0,92 1,75 0,21 1,94
(9,39) (31,82) (5,88) (7,06) (9,29) (36) (1,66) (4,88)
Ygap 0,61 0,15 0,98 0,37 0,82 0,25 1,22 0,67
(7,63) (3,39) (10,26) (2,64) (8,79) (4,37) (10,66) (3,92)
Model 5 Model 6 Model 7 Model 8
State 1 State 2 State 1 State 2 State 1 State 2 State 1 State 2
Const. 1,29 0,04 1,61 2,73 0,97 1,75 1 3,13
(5,14) (0,05) (5,97) (2,1) (4,57) (1,23) (4,53) (2,37)
Interest 0,75 0,73 0,77 0,76 0,72 0,6 0,84 0,7
(20,36) (6,55) (18,94) (6,52) (19,88) (4,19) (23,64) (5,67)
Infgap 0,11 0,45 -0,08 0,65 0,17 0,42 -0,15 0,57
(3,37) (2,07) (1,48) (1,72) (4,51) (1,64) (2,42) (1,72)
Ygap 0,34 0,13 0,35 0,22 0,42 0,31 0,35 0,36
(8,14) (1,64) (7,71) (2,14) (9,22) (2,28) (6,54) (2,71)
Model 9 Model 10 Model 11 Model 12
State 1 State 2 State 1 State 2 State 1 State 2 State 1 State 2
Const. 4,05 4,52 6,18 9,85 0,98 0,48 1,22 0,68
(12,89) (8,45) (38,83) (39,98) (4,85) (0,56) (5,32) (0,82)
Interest 0,82 0,94 0,83 0,95
(27,28) (10,64) (24,29) (10,93)
Infgap 0,31 1,03 1,88 3,16 0,08 0,05 0,04 -0,01
(4,69) (10,82) (8,91) (14,97) (2,93) (0,37) (0,55) (0,05)
Ygap 0,23 -0,21 0,9 -0,1 0,28 0,19 0,3 0,19
(2,76) (2,67) (10,2) (1,27) (7,85) (2,1) (7,4) (2,16)
The different specifications are numbered according to Table 2.1. The figures in the parenthesis show the absolute values of the t-values of the coefficient estimates.
11 See e.g. Kajanoja (2001), p. 117.
Model 1 Model 2 Model 3 Model 4
The different specifications are numbered according to Table 2.2. The figures in the parenthesis show the absolute values of the t-values of the coefficient estimates.
It can be seen in Table 3.1 that for the Fed, the statistically significant coefficients during both regimes were of the “right” sign, that is, positive. It is more difficult to give the results for the Bundesbank a sensible interpretation, because the parameter estimates are insignificant in many of the models and sometimes they even have wrong (negative) signs. This is the case especially for the coefficient on the inflation gap in the models with the interest rate smoothing term. The transition probabilities between the two estimated regimes show that both the Fed’s and the Bundesbank’s regimes are fairly persistent. When the economy has reached either of the two states, the probability for staying at that state is more than 90 %.
the two central banks seems to have been driven by a pure “inflation nutter ” regime during the sample period, since this kind of regime should be fully absorbing with the transition probabilities very close to unity. This follows from the obvious fact that once a central bank has committed to very strict policy against inflation, abandoning that regime would be very costly in terms of the lost credibility.
Considering then the robustness of the estimation results on our alternative sets of explanatory variables, both the timing and the parameter values of the Fed’s regimes seem to be somewhat sensitive on the exact specification of the model, particularly on whether the specification contains lagged interest rate term or not. The finding is in line Kozicki (1999) and Cerra et al.
(2000) who found the estimation results to be remarkably sensitive on the way the inflation and output gaps are measured.
The estimation results for the Bundesbank seem to be even more sensitive on the specification.
As in the case of the Fed, the parameter estimates are again particularly sensitive on the inclusion of the lagged interest rate term in the model. In the specifications with the interest rate smoothing term, the estimates of both the constant term and the inflation gap coefficient get significantly lower values than in the models without the interest rate smoothing. In addition, if the models are divided into models with and without the lagged interest rate term, the variation in the estimation results within the two sort of specifications is greater in the case of the Bundesbank than in the case of the Fed.
The coefficient estimates of the inflation and output gaps reveal that the monetary policy of the Fed clearly seems to have been switching between two regimes during our sample period.
The first of the regimes is characterized by a stronger response to the inflation gap, while a stronger response to the output gap is associated with the second of the regimes. This finding also seems to be robust to the model specification, although the difference between the two regimes seems to be less clear in the specifications with the interest rate smoothing term. It is possible to also find some differences between the two regimes of the Bundesbank, although the differences between the regimes are not very clear. The coefficients for both the inflation
parameters to the model slightly more often during the regime two.