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Misspecification bias in models of the effect of inflation uncertainty
Economics Letters North-Holland
325
38 (1992) 325-329
Misspecification bias in models of the effect of inflation uncertainty George
Davis
Miami Unil~ersity, Oxford OH, USA
Bryce Kanago Northern Illinois University, DeKalb IL, USA Received Accepted
15 October 1991 20 December 1991
In this paper we show that the appropriate measure of inflation uncertainty is relative measure. Specifically, is the standard deviation of inflation relative to the gross expected rate of inflation. Empirical studies of the effects of inflation uncertainty have misspecified their models by not using the relative measure. The bias introduced is equivalent to omitting a relevant variable. We derive the asymptotic bias introduced by this error, and estimate its magnitude for the United States and eight high-inflation countries. Estimates of the effect of inflation uncertainty from high-inflation countries, and cross country studies that include them are likely to suffer from a considerable negative bias. We conclude that future studies should employ this relative measure.
1. Introduction There is a large empirical literature on the effects of inflation uncertainty on variables of economic interest. Among these, Mullineaux (1980) found that increases in inflation uncertainty reduce various measures of real activity, Levi and Makin (1979) concluded that greater inflation uncertainty lowers real interest rates, and Holland (1986) showed that increased inflation uncertainty leads to a greater degree of wage indexation. These results are important since they bear directly on the issues of the costs of inflation and the proper conduct of monetary policy. All of these studies included a measure of the variance, standard deviation, or the root mean squared error of inflation as an explanatory variable in their regression analysis. In this paper we use a two-period model with stochastic inflation to show that this standard specification is incorrect. The correct variable is not the standard deviation of inflation or some other proxy alone, but is instead a relative measure of inflation uncertainty similar to a coefficient of variation. ’ The error caused by misspecifying the measure of inflation uncertainty is equivalent to omitting a relevant variable from the analysis. We derive the asymptotic bias introduced by this error, and estimate the bias for the United States and eight high-inflation countries. We find that the size of the bias in the high-inflation countries is likely to be large. The last result will be particularly relevant for cross country studies of the effect of inflation uncertainty. Correspondence to: George Davis, Department of Economics, School of Business Administration, Miami University, 208 Laws Hall, Oxford, OH 45056, USA. ’ Below we show that the relative measure is given by a/1(1 + X), where c is the standard deviation of inflation and ?i is the mean of inflation. 0165-1765/92/$05.00
0 1992 - Elsevier
Science
Publishers
B.V. All rights reserved
326
G. Daub, B. Kanago / The effect of inflation uncertainty
The paper is organized as follows. In the next section we argue that the appropriate measure of inflation uncertainty is a relative one. In Section 3 we present the misspecification error as an omitted variables problem, and calculate the asymptotic bias in the estimate of the effect of inflation uncertainty. Section 4 contains the empirical results. We end with some concluding remarks.
2. Relative inflation
uncertainty
We use the model presented by Driffill, Mizon and Ulph (1990) to formally show the appropriateness of the relative measure of inflation uncertainty. ’ Our strategy is to use the result in Davis (1989) to show that the expected utility of an agent depends on the relative measure of inflation uncertainty. It then follows directly that the decisions of the agent also depend on the relative measure of inflation uncertainty. We assume that the future price level, and hence inflation, is stochastic. The Driffill-Mizon-Ulph model considers a single household who chooses C,, C,, M, and B to maximize
subject to P,C, +M+B
= W-P,T(C,,
P,C,=M+B(l
M/P,),
+R),
where C, is consumption in period j, A4 is nominal money holdings, B nominal bond holdings, R is the nominal rate of interest, and T gives the transactions costs associated with real purchases. The transaction costs are used to motivate a demand for money. We have modified their specification by excluding real bond holdings. This is inessential to our argument, but does simplify notation and the presentation. We assume that P, is stochastic, and that the household maximizes expected utility. The two constraints above may be used to substitute for C, in the utility function.The result given by
y=(w-T(C,,m)-m-C,)(l+R)+m, where r is the rate of inflation (rr = (P, - P,)/P,), m = M/P, b = B/P, and w = W/P. From the result in Davis (1989) it follows that the expected utility of second-period consumption may be written as E[prL(~/l(I+r))]
=.n(y/(I
or
2 See Davis (1989) for an intuitive
illustration.
+*)I
a/(1+*)>
a(3)/(I
+F),...)
327
G. Dacis, B. Kanago / The effect of inflation uncertain@
where CTis the standard deviation of inflation and g(j) is the by [E(r - +)j]‘/j. The standard deviation of inflation, and expected utility only relative to the gross rate of inflation, 1 ’ For choices of m, C,, and b depend on relative uncertainty. balances may be written as
3. Specification The typical
jth root moment of inflation defined other root moments as well, enter + +. It now follows that the optimal example, the demand for real money
error regression
equation
yt=a+put+yz,+E
used to study the effect of inflation
uncertainty
is given by
(1)
f’
where y, is a measure of real economic activity, the interest rate, or some other variable of interest, z, is a vector of other explanatory variables, and E, is a random error term. For a recent sample, see Holland (1988) where y, is real GNP growth, and z, includes measures of wage indexation, and inflation surprise. We have argued that the appropriate specification is not (l), but instead Y, = Q! +
P(q/(l
+ r,>> +YZ,
+
l:.
(2)
If eq. (2) is the correct specification, then estimates of p from eq. (1) are biased. nature of the bias rewrite eq. (2) in the equivalent form:
To see the
(3) Estimating eq. (1) instead of (3) amounts to omitting the variable -a,~,/(1 + rTTt)which has the coefficient p, For simplicity ignore the terms in the vector zI, and focus on the relationship between the omitted variable and a,. In the present context the standard formula [see, e.g., Maddala (1988, p. 122)] yields an estimate p given by
(4) where X, = a,rr,/(l + .sr,), and the superscript d indicates the variable is in deviations from mean. Taking probability limits in (4) yields the following result for the asymptotic bias estimate plim p^=p(l
-cov[a,
Recalling that x = a~/(1 bias of fi can be written as
3 The rest of the specification
p^.
X]/var[a]).
+ r> and the formula
is standard.
its
E[xyl = E[xl . E[yl + cov[x, yl, the asymptotic
328 Table 1 The determinants
G. Davis, B. Kanago / The effect of inflation uncertainty
of the bias. ‘zh
Source
Measures
Mean Covar Total
of inflation
uncertainty
Livingston
Michigan-SRC
ARCH
0.026 - 0.57 - 0.031
0.055 0.079 0.134
0.042 0.053 0.095
’ Mean is the sample mean of r/(1 + a), ~-/cl + x), and dividing the difference by b The Livingston data was provided by the ARCH measure is based on an inflation measure itself is based on 4 lagged values (1982) for details. The Michigan-SRC data
and Covar is calculated by taking the sample covariance of (a - @)a and the sample variance of (r. Federal Reserve Bank of Philadelphia for the period 1946.4 to 1989.4, the forecasting equation that contains 4 lagged values of inflation. The ARCH of squared residuals and is estimated for the period 1960.1 1989.3. See Engle is an updated version of the Juster and Comment (1980) series.
The sign and magnitude of the bias of p^ is determined covariation between inflation and the standard deviation standard deviation of inflation.
4. Empirical
by the average rate of inflation, and the of inflation relative to the variance of the
results
In this section we estimate the asymptotic bias caused by misspecifying the regression equation. We first report these estimates for three commonly used measures of inflation uncertainty for the United States. We then turn our attention to the magnitude of the bias in the high-inflation countries. For the United States the most often used measures of inflation uncertainty are taken from the Livingston survey data, the Michigan Survey Research Center data, and Engle’s ARCH measure. The asymptotic bias for each of these measures is calculated in table 1. The bias is about 10% for both the Michigan data and the ARCH measure. The total bias for the Livingston measure is negative and smaller in absolute value since the covariation term for the Livingston data is, somewhat surprisingly, negative and relatively large. Since the estimates in table 1 are not large, the effect of this bias on earlier work for the U.S. is also probably not large.
Table 2 The bias for eight high-inflation
countries.
a
Country
Bias
Bolivia Brazil Chile Columbia Israel Peru Turkey Uruguay
0.29 0.33 0.29 0.17 0.35 0.29 0.22 0.30
a Bias is the sample mean of rr/(l+ P). Data is for 1972 through Statistics, Supplement Series no. 12, 1986.
1985, and is taken from the IMF’s International Financial
G. Davis, B. Kanago / The effect of inflation uncertainty
329
For other countries measures of (T are not readily available. To examine the nature of the bias we calculated the mean of rr,/(l + rr,) for eight high-inflation countries. The sample period runs from 1972 through 1985, and was taken from the IMF’s International Financial Statistics. The results are given in table 2. The biases here are considerable. For Israel it is nearly 35%, and it exceeds or is very near 30% for five other countries. Therefore, the bias for studies of these countries, or for cross country studies that include them, will be quite large. 4
5. Conclusion In this paper we have argued that the appropriate measure of inflation uncertainty is a relative measure. Earlier studies typically used only the level of the standard deviation of inflation (or some proxy for it> in their empirical specifications. This amounts to omitting a relevant variable from the regression equation and results in biased estimates of the effect of inflation uncertainty. We computed the bias for nine countries to show that it is significant in some countries. This strongly suggests that in future work, especially cross country comparisons, a relative measure of inflation uncertainty should be used.
References Davis, G., 1989, Relative uncertainty, Economics Letters 29, 307-310. Davis, G. and B. Kanago, 1991, Contract duration, inflation uncertainty, and the welfare effects of inflation, Working paper (Miami University, Oxford, OH). Driffill, J., G. Mizon and A. Ulph, 1990, Costs of inflation, in: B. Friedman and F. Hahn, eds., Handbook of monetary economics Vol. II (North-Holland, New York) 1013-1066. Engle, R., 1983, Estimates of variance of U.S. inflation based upon ARCH model, Journal of Money, Credit and Banking 15, Aug., 286-301. Holland, A.S., 1986, Wage indexation and the effect of inflation uncertainty on employment: An empirical analysis, American Economic Review 76, March, 235-244. Holland, A.S., 1988, Indexation and the effect of inflation uncertainty on real GNP, Journal of Business 61, 213-228. Juster, T. and R. Comment, 1980, A note on the measurement of price expectations, Mimeo. (University of Michigan Research Center). Levi, M. and J. Makin, 1979, Phillips, Friedman, and the measured impact of inflation on interest, Journal of Finance 34, 35-52. Maddala, G., 1988, Introduction to econometrics (MacMillan, New York). Mullineaux, D., 1980, Unemployment, industrial production, and inflation uncertainty in the United States, Review of Economics and Statistics 62, 163-169. 4 This will be true so long as it is not offset
by a strong
negative
correlation
between
inflation
and its standard
deviation.
325
38 (1992) 325-329
Misspecification bias in models of the effect of inflation uncertainty George
Davis
Miami Unil~ersity, Oxford OH, USA
Bryce Kanago Northern Illinois University, DeKalb IL, USA Received Accepted
15 October 1991 20 December 1991
In this paper we show that the appropriate measure of inflation uncertainty is relative measure. Specifically, is the standard deviation of inflation relative to the gross expected rate of inflation. Empirical studies of the effects of inflation uncertainty have misspecified their models by not using the relative measure. The bias introduced is equivalent to omitting a relevant variable. We derive the asymptotic bias introduced by this error, and estimate its magnitude for the United States and eight high-inflation countries. Estimates of the effect of inflation uncertainty from high-inflation countries, and cross country studies that include them are likely to suffer from a considerable negative bias. We conclude that future studies should employ this relative measure.
1. Introduction There is a large empirical literature on the effects of inflation uncertainty on variables of economic interest. Among these, Mullineaux (1980) found that increases in inflation uncertainty reduce various measures of real activity, Levi and Makin (1979) concluded that greater inflation uncertainty lowers real interest rates, and Holland (1986) showed that increased inflation uncertainty leads to a greater degree of wage indexation. These results are important since they bear directly on the issues of the costs of inflation and the proper conduct of monetary policy. All of these studies included a measure of the variance, standard deviation, or the root mean squared error of inflation as an explanatory variable in their regression analysis. In this paper we use a two-period model with stochastic inflation to show that this standard specification is incorrect. The correct variable is not the standard deviation of inflation or some other proxy alone, but is instead a relative measure of inflation uncertainty similar to a coefficient of variation. ’ The error caused by misspecifying the measure of inflation uncertainty is equivalent to omitting a relevant variable from the analysis. We derive the asymptotic bias introduced by this error, and estimate the bias for the United States and eight high-inflation countries. We find that the size of the bias in the high-inflation countries is likely to be large. The last result will be particularly relevant for cross country studies of the effect of inflation uncertainty. Correspondence to: George Davis, Department of Economics, School of Business Administration, Miami University, 208 Laws Hall, Oxford, OH 45056, USA. ’ Below we show that the relative measure is given by a/1(1 + X), where c is the standard deviation of inflation and ?i is the mean of inflation. 0165-1765/92/$05.00
0 1992 - Elsevier
Science
Publishers
B.V. All rights reserved
326
G. Daub, B. Kanago / The effect of inflation uncertainty
The paper is organized as follows. In the next section we argue that the appropriate measure of inflation uncertainty is a relative one. In Section 3 we present the misspecification error as an omitted variables problem, and calculate the asymptotic bias in the estimate of the effect of inflation uncertainty. Section 4 contains the empirical results. We end with some concluding remarks.
2. Relative inflation
uncertainty
We use the model presented by Driffill, Mizon and Ulph (1990) to formally show the appropriateness of the relative measure of inflation uncertainty. ’ Our strategy is to use the result in Davis (1989) to show that the expected utility of an agent depends on the relative measure of inflation uncertainty. It then follows directly that the decisions of the agent also depend on the relative measure of inflation uncertainty. We assume that the future price level, and hence inflation, is stochastic. The Driffill-Mizon-Ulph model considers a single household who chooses C,, C,, M, and B to maximize
subject to P,C, +M+B
= W-P,T(C,,
P,C,=M+B(l
M/P,),
+R),
where C, is consumption in period j, A4 is nominal money holdings, B nominal bond holdings, R is the nominal rate of interest, and T gives the transactions costs associated with real purchases. The transaction costs are used to motivate a demand for money. We have modified their specification by excluding real bond holdings. This is inessential to our argument, but does simplify notation and the presentation. We assume that P, is stochastic, and that the household maximizes expected utility. The two constraints above may be used to substitute for C, in the utility function.The result given by
y=(w-T(C,,m)-m-C,)(l+R)+m, where r is the rate of inflation (rr = (P, - P,)/P,), m = M/P, b = B/P, and w = W/P. From the result in Davis (1989) it follows that the expected utility of second-period consumption may be written as E[prL(~/l(I+r))]
=.n(y/(I
or
2 See Davis (1989) for an intuitive
illustration.
+*)I
a/(1+*)>
a(3)/(I
+F),...)
327
G. Dacis, B. Kanago / The effect of inflation uncertain@
where CTis the standard deviation of inflation and g(j) is the by [E(r - +)j]‘/j. The standard deviation of inflation, and expected utility only relative to the gross rate of inflation, 1 ’ For choices of m, C,, and b depend on relative uncertainty. balances may be written as
3. Specification The typical
jth root moment of inflation defined other root moments as well, enter + +. It now follows that the optimal example, the demand for real money
error regression
equation
yt=a+put+yz,+E
used to study the effect of inflation
uncertainty
is given by
(1)
f’
where y, is a measure of real economic activity, the interest rate, or some other variable of interest, z, is a vector of other explanatory variables, and E, is a random error term. For a recent sample, see Holland (1988) where y, is real GNP growth, and z, includes measures of wage indexation, and inflation surprise. We have argued that the appropriate specification is not (l), but instead Y, = Q! +
P(q/(l
+ r,>> +YZ,
+
l:.
(2)
If eq. (2) is the correct specification, then estimates of p from eq. (1) are biased. nature of the bias rewrite eq. (2) in the equivalent form:
To see the
(3) Estimating eq. (1) instead of (3) amounts to omitting the variable -a,~,/(1 + rTTt)which has the coefficient p, For simplicity ignore the terms in the vector zI, and focus on the relationship between the omitted variable and a,. In the present context the standard formula [see, e.g., Maddala (1988, p. 122)] yields an estimate p given by
(4) where X, = a,rr,/(l + .sr,), and the superscript d indicates the variable is in deviations from mean. Taking probability limits in (4) yields the following result for the asymptotic bias estimate plim p^=p(l
-cov[a,
Recalling that x = a~/(1 bias of fi can be written as
3 The rest of the specification
p^.
X]/var[a]).
+ r> and the formula
is standard.
its
E[xyl = E[xl . E[yl + cov[x, yl, the asymptotic
328 Table 1 The determinants
G. Davis, B. Kanago / The effect of inflation uncertainty
of the bias. ‘zh
Source
Measures
Mean Covar Total
of inflation
uncertainty
Livingston
Michigan-SRC
ARCH
0.026 - 0.57 - 0.031
0.055 0.079 0.134
0.042 0.053 0.095
’ Mean is the sample mean of r/(1 + a), ~-/cl + x), and dividing the difference by b The Livingston data was provided by the ARCH measure is based on an inflation measure itself is based on 4 lagged values (1982) for details. The Michigan-SRC data
and Covar is calculated by taking the sample covariance of (a - @)a and the sample variance of (r. Federal Reserve Bank of Philadelphia for the period 1946.4 to 1989.4, the forecasting equation that contains 4 lagged values of inflation. The ARCH of squared residuals and is estimated for the period 1960.1 1989.3. See Engle is an updated version of the Juster and Comment (1980) series.
The sign and magnitude of the bias of p^ is determined covariation between inflation and the standard deviation standard deviation of inflation.
4. Empirical
by the average rate of inflation, and the of inflation relative to the variance of the
results
In this section we estimate the asymptotic bias caused by misspecifying the regression equation. We first report these estimates for three commonly used measures of inflation uncertainty for the United States. We then turn our attention to the magnitude of the bias in the high-inflation countries. For the United States the most often used measures of inflation uncertainty are taken from the Livingston survey data, the Michigan Survey Research Center data, and Engle’s ARCH measure. The asymptotic bias for each of these measures is calculated in table 1. The bias is about 10% for both the Michigan data and the ARCH measure. The total bias for the Livingston measure is negative and smaller in absolute value since the covariation term for the Livingston data is, somewhat surprisingly, negative and relatively large. Since the estimates in table 1 are not large, the effect of this bias on earlier work for the U.S. is also probably not large.
Table 2 The bias for eight high-inflation
countries.
a
Country
Bias
Bolivia Brazil Chile Columbia Israel Peru Turkey Uruguay
0.29 0.33 0.29 0.17 0.35 0.29 0.22 0.30
a Bias is the sample mean of rr/(l+ P). Data is for 1972 through Statistics, Supplement Series no. 12, 1986.
1985, and is taken from the IMF’s International Financial
G. Davis, B. Kanago / The effect of inflation uncertainty
329
For other countries measures of (T are not readily available. To examine the nature of the bias we calculated the mean of rr,/(l + rr,) for eight high-inflation countries. The sample period runs from 1972 through 1985, and was taken from the IMF’s International Financial Statistics. The results are given in table 2. The biases here are considerable. For Israel it is nearly 35%, and it exceeds or is very near 30% for five other countries. Therefore, the bias for studies of these countries, or for cross country studies that include them, will be quite large. 4
5. Conclusion In this paper we have argued that the appropriate measure of inflation uncertainty is a relative measure. Earlier studies typically used only the level of the standard deviation of inflation (or some proxy for it> in their empirical specifications. This amounts to omitting a relevant variable from the regression equation and results in biased estimates of the effect of inflation uncertainty. We computed the bias for nine countries to show that it is significant in some countries. This strongly suggests that in future work, especially cross country comparisons, a relative measure of inflation uncertainty should be used.
References Davis, G., 1989, Relative uncertainty, Economics Letters 29, 307-310. Davis, G. and B. Kanago, 1991, Contract duration, inflation uncertainty, and the welfare effects of inflation, Working paper (Miami University, Oxford, OH). Driffill, J., G. Mizon and A. Ulph, 1990, Costs of inflation, in: B. Friedman and F. Hahn, eds., Handbook of monetary economics Vol. II (North-Holland, New York) 1013-1066. Engle, R., 1983, Estimates of variance of U.S. inflation based upon ARCH model, Journal of Money, Credit and Banking 15, Aug., 286-301. Holland, A.S., 1986, Wage indexation and the effect of inflation uncertainty on employment: An empirical analysis, American Economic Review 76, March, 235-244. Holland, A.S., 1988, Indexation and the effect of inflation uncertainty on real GNP, Journal of Business 61, 213-228. Juster, T. and R. Comment, 1980, A note on the measurement of price expectations, Mimeo. (University of Michigan Research Center). Levi, M. and J. Makin, 1979, Phillips, Friedman, and the measured impact of inflation on interest, Journal of Finance 34, 35-52. Maddala, G., 1988, Introduction to econometrics (MacMillan, New York). Mullineaux, D., 1980, Unemployment, industrial production, and inflation uncertainty in the United States, Review of Economics and Statistics 62, 163-169. 4 This will be true so long as it is not offset
by a strong
negative
correlation
between
inflation
and its standard
deviation.