184 L. Coppock, M. Poitras International Review of Economics and Finance 9 2000 181–192
Table 1 Estimates of Eq. 1
1 II
III IV
Variable OLS
OLS, omit Brazil Two-step BI
Krasker-Welsch BI Inflation
1.01 0.667
0.784 0.633
0.09 0.059
0.089 0.053
Inflation variance 20.328
20.136 20.294
20.105 0.101
0.056 0.174
0.050 Intercept
2.65 5.36
4.88 5.54
1.21 0.69
0.81 0.59
R
2
0.851 0.880
0.713 0.898
ow
i
40 39
36.72 36.18
Parentheses contain standard errors. The reported R
2
applies to the weighted dependent variable. Indicates significant difference from zero at the 0.01 level.
Indicates significant difference from zero at the 0.05 level.
such as Brazil, Bolivia, and Peru; these extreme observations are likely candidates for outliers since empirical regularities that exist under conditions of moderate inflation
may break down under conditions of hyperinflation. The magnitude of the hyperinfla- tion observations could permit them in a small sample to exert considerable leverage
on the parameter estimates. The results in Table 1 show that individual observations can in fact have a pivotal effect on the estimates.
The first two columns of Table 1 present Ordinary Least Squares OLS estimates of Eq. 1. The estimates in column I use the full data set, while those in column II
omit Brazil, the highest inflation country. The results using the full data set support those of Duck 1993. The point estimate of b
, the marginal effect of inflation, lies quite close to one, and we cannot reject the hypothesis of a point-for-point Fisher
effect. In contrast, column II shows that omitting Brazil reduces the estimate of b significantly below one at the .01 level. Hence, the statistical inference on the Fisher
hypothesis hinges on whether the sample includes Brazil. Other high inflation countries also exert substantial influence on the estimates.
12
To limit the effects of influential outliers, we want to apply a formal technique rather than to arbitrarily discard observations. The bounded-influence BI techniques
presented by Welsch 1980, and Krasker and Welsch 1982, implement a weighting scheme to yield estimates that do not pivot on a small subset of the data. Welsch
1980 demonstrates a two-step BI technique, while Krasker and Welsch 1982 present an iterative procedure. To explore the robustness of results across techniques, the
next section presents both two-step and iterated BI estimates.
3. Bounded-influence methods and results
The BI technique amounts to a kind of weighted least squares with the weights w
i
satisfying [Eq. 2]. S
i
w
i
y
i
2 x
i
bˆx9
i
5 0. 2
L. Coppock, M. Poitras International Review of Economics and Finance 9 2000 181–192 185
The BI technique defines the weights according to [Eq. 3] w
i
5 min {1, at
i
}, 3
where a is a constant chosen to set the bound on the influence, and t
i
is a measure of the influence of observation i.
13
The two-step Welsch and iterative Krasker-Welsch techniques use somewhat different methods to measure the influence, t
i
. In either case, however, the definition of t
i
consists of a quadratic function dx
i
of the explana- tory variables that measures the “leverage” of observation i, multiplied by a standard-
ized measure of the estimated residual for observation i. Thus, we have [Eq. 4]
14
t
i
5 dx
i
|y
i
2 x
i
bˆs|. 4
The two-step procedure performs OLS in the first stage and weighted least squares in the second stage. The Krasker-Welsch iterative procedure updates the estimates
of b, w
i
, and s at each step. The procedure converges to a unique solution, and the resulting estimator is consistent and asymptotically normal. The Krasker-Welsch
estimator is also asymptotically efficient among BI estimators, given that an efficient BI estimator exists.
15
The BI technique does not provide a panacea for all possible econometric pathologies, but the method does yield inferences robust to small sample
changes and likely mitigates the effects of omitted variables and data reporting errors. Given the size of our sample, the BI technique insures that any subsample of meaning-
ful size should yield essentially the same inferences.
Columns III and IV of Table 1 display BI estimates of Eq. 1. In contrast to the OLS results, the BI estimates of b
lie significantly below one at the .01 level, indicating rejection of the Fisher hypothesis. The rejection is especially strong in light of the
fact that full adjustment with interest taxation implies b greater than one. The results
contradict those of Duck 1993, and demonstrate the importance of bounding the influence of the outliers in these data. While the results do not support a point-for-
point Fisher effect, the estimates do indicate a significant partial effect.
16
Table 2 lists the downweighted observations and the values of the weights, w
i
, for the squared residuals. The two techniques can generate different weights because
they define the influence differently. Note that the set of downweighted observations almost exclusively includes high-inflation countries; the two-step procedure down-
weights five of the six highest-inflation observations, and the iterative procedure downweights six of the nine highest. In particular, both methods strongly downweight
Brazil, suggesting that this observation does not conform to the same empirical model as do the bulk of the data.
In the following section we present a brief discussion of some leading theoretical explanations for the failure to observe a full Fisher effect. We then extend the empirical
model to examine a theory that attributes the source of partial adjustment to the liquidity properties of financial assets.
4. Sources of partial adjustment
