3. Parametric and Nonparametric VAR Models
3.3. Testing for the Fisher Effect
The time-detrended Fisher Effect holds statistically if the cumulative orthogonalized dynamic multiplier ratios converge to one as g – the lag length of the impulse response function – goes to infinity. The orthogonalized MA(∞) coefficients are used to form the nonparametric conditional orthogonalized impulse response functions. For instance, the form of the orthogonalized dynamic multiplier of the (t s+ ) response of the (r')th regressand caused by a shock to the rth regressand at time t is:
( )
( )r t st r r r
d y M
dw x
′ +
= ′ . (7)
4 The VAR has also been estimated using different window widths for each equation of the VAR, but the results did not statistically differ.
The Nonparametric Time-Detrended Fisher Effect 73 The gth cumulative dynamic multiplier ratio is referred to as Γg, where Γg is the sum of the responses of detrended inflation caused by a shock to detrended nominal interest rates
(Mπi)to the sum of the responses of detrended nominal interest rates caused by a shock to detrended nominal interest rates ( )Mii . Specifically for this paper, for each sample period, the gth ratio of nonparametric orthogonalized cumulative dynamic multipliers, which is denoted as Γg, is of the general form of:
, 0
, 0 g
i s s
g g
ii s s
M M Γ = π
=
= ∑
∑ , (8)
where g = 1, 2, . . . ,∞.
For both the parametric and nonparametric models, up to one hundred lag lengths of the impulse response functions are calculated in order to determine whether the dynamic multiplier ratios converge to unity. This is needed since potential non-stationarity can mistakenly indicate an early appearance of the Fisher Effect in the medium run, which may break down in the very long run. The convergence to unity signifies that the changes in detrended nominal interest rate are being matched by the changes in detrended inflation, which indicates that the Fisher Effect holds.
Three different methodologies to compute the orthogonal dynamic multipliers are presented in order to test for the Fisher Effect. One is based on the parametric VAR and the orthogonal dynamic multiplier ratios described in Equation (8). The other two methods are based on the nonparametric VAR. Methods 2 and 3 can be computed using the median nonparametric and the average nonparametric coefficients. The three techniques are as follows:
METHOD 1: The parametric VAR of detrended nominal interest rates and inflation is estimated equation-by-equation, in order to obtain the sum of the orthogonalized dynamic multipliers of the responses of inflation divided by the responses of nominal interest rates to a shock in nominal interest rates.
METHOD 2: The orthogonal dynamic multiplier ratios of the Fisher Effect are obtained from the local nonparametric estimation of the VAR using the median of the T estimated coefficients for each regressor as a measure of central tendency. The median coefficients are used to form the MA(∞) coefficients, which are then used to form the dynamic multipliers.
The error terms from the T local nonparametric equations of the VAR are used to obtain the
(2 2× ) unconditional variance-covariance matrix of the error terms. The Choleski decomposition can then be calculated in order to form the orthogonalized impulse responses – the orthogonal dynamic multipliers.
METHOD 3: The orthogonal dynamic multiplier ratios of the Fisher Effect are obtained from the nonparametric estimation of the VAR. The median of T estimated coefficients for each regressor is obtained. The median nonparametric coefficients are used to form the
Heather L. R. Tierney 74
MA(∞) coefficients that comprises the dynamic multiplier ratios of the Fisher Effect. As indicated by Equation (9), the error terms of each equation in the VAR are then obtained from the regression:
_
np Y X np med
ε = − β , (9)
which is then used to obtain the (2 2× ) variance-covariance matrix of the error terms. Once the variance-covariance matrix of the error terms is obtained, the Choleski decomposition can then be calculated to form orthogonal dynamic multipliers.5
The average, orthogonal, nonparametric dynamic multiplier ratios of the time-detrended Fisher Effect are obtained by replacing the median nonparametric coefficients in the formation of the MA(∞) version of the VAR(p) with the average nonparametric coefficients.
If the orthogonal dynamic multiplier ratios of the Fisher Effect converge to unity and are within the 95% bootstrapped confidence band, the Fisher Effect statistically holds. For each test of the Fisher Effect, the bootstrapped confidence band is constructed from the empirical density based on five thousand iterations of re-sampling with replacement. For each run, the VAR is estimated, and the test for the Fisher Effect is formed using the bootstrapped data with the average used to construct the confidence band.
By definition, the Fisher Effect means that the real interest rate is constant in the long run and is not impacted by either the short-term movements of nominal interest rates or inflation.
Hence, if the Fisher Effect holds, the convergent movement of nominal interest rates should match the convergent movement of inflation. Intuitively, where δ is some constant, that is:
(10)
(11)
(12) For monetary policy purposes, examining Equation (12) can be an informative tool for
inflation-targeting regimes, since an inequality indicates that inflation and nominal interest rates are in disequilibria. If the orthogonal dynamic multiplier ratios are greater than unity, the
5 In most instances, the cumulative multiplier ratio of Equation (8) converges to approximately the same level regardless of whether Method 2 or Method 3 as is shown in Tables 6A and 6B.
The Nonparametric Time-Detrended Fisher Effect 75 cumulative responses of inflation are greater than the cumulative responses of nominal interest rates, which would act as a signal to the monetary authorities that anti-inflationary measures in the form of tight monetary policies might be needed. This is particularly the case if the ratio exceeds unity by a preset amount. Alternatively, if the orthogonal dynamic multiplier ratios are less than unity, then the monetary authorities might consider implementing loose monetary policies that could bring inflation and nominal interest rates back into synchronization, given that the Fisher Effect holds in the long run.