ARE EXPECTED INFLATION RATES AND EXPECTED REAL RATES NEGATIVELY CORRELATED? A LONG-RUN TEST
OF THE MUNDELL-TOBIN HYPOTHESIS
Keshab Shrestha University of Regina Sheng-Syan Chen Yuan Ze University Cheng-few Lee Rutgers University Abstract
Some empirical evidence suggests that the expected real interest and expected inflation rates are negatively correlated. This hypothesis of negative correlation is sometimes known as the Mundell-Tobin hypothesis. In this arti-cle we reinvestigate this negative relation from a long-term point of view using cointegration analysis. The data on the historical interest rate on T-bills and the inflation rate indicate that the Mundell-Tobin hypothesis does not hold in the long run for the United States, the United Kingdom, and Canada. We also obtain similar results using the real interest rate on index-linked gilt traded in the United Kingdom.
JEL Classifications: C32, G12, G15
I. Introduction
As Mishkin (1992) points out, the relation between the level of interest rates and inflation is one of the most studied topics in financial economics. Most of the studies concentrate on the test of the Fisher effect (Fisher 1930), which states that the nominal interest rate fully reflects the expected inflation rate. Several studies demonstrate a negative correlation between the expected inflation rate and the expected real rate (Fama and Gibbons 1982; Mishkin and Simon 1995). Such a
The authors wish to thank Jim Barth, Drew Dahl, an anonymous referee, and seminar participants at the 1997 FMA meetings for helpful comments and suggestions. Any remaining errors are the authors.
negative relation between the expected inflation rate and the expected real interest rate is sometimes referred to as the Mundell-Tobin hypothesis. Some researchers (Carmichael and Stebbing 1983) even argue that the nominal interest rate is constant, and thus, the real interest rate adjusts to the changes in the expected inflation rate. This phenomenon (of perfect negative correlation) is sometimes known as the inverted Fisher hypothesis.
Several theoretical explanations are offered for this negative correlation. Mundell (1963) and Tobin (1965) argue that higher inflation leads to a portfolio shift out of nominal assets and into real assets, thus pushing down the return on real assets. Fama and Gibbons (1982) relate the expected real rate to the productivity in the economy and use capital expenditure process and the money market equilibrium to explain the negative correlation. Others, such as Stulz (1986), use the liquidity premium concept.
This negative correlation has important investment implications. Because the real interest rates are usually represented by real returns on financial assets such as T-bills, the negative correlation implies that such financial assets cannot be considered as a good inflation hedge.
The empirical studies that find the negative correlation do not use cointe-gration analysis to distinguish between the long-run (stationary) relation and the spurious relation among nonstationary time series. Many studies perform cointe-gration analysis in testing the Fisher effect (Moazzami 1991; Mishkin 1992). How-ever, to our knowledge, none performs cointegration analysis to investigate the negative relation between the expected inflation series and the expected real rate series.
In this article we perform cointegration tests to analyze the negative relation for the United States, the United Kingdom, and Canada. Cointegration analysis re-quires the estimation of the expected inflation and expected real interest rate series. We use three approaches in estimating these two series. The first two approaches, which are model based, use the Fisher equation and returns on T-bills to estimate the two series. The third approach uses the observed real yield on the index-linked gilt and is not based on any specific model such as the Fisher equation. Cointegration analysis is appropriate because both the expected inflation and expected real rate series are found to be nonstationary. When using the returns on T-bills, we find that the sample correlation between the two series is negative and significant for all three countries. However, the two series are not cointegrated; that is, a long-run negative relation does not exist between these two series. This indicates that T-bills can be regarded as a long-term inflation hedge. When using the index-linked gilt, we find that the sample correlation is positive but not statistically signifi-cant. Furthermore, the real interest rate is not cointegrated with the inflation rate. This also supports the long-run independence of the real interest rate and inflation rate.
II. The Models
We consider three approaches in testing the long-run relation between the expected inflation rate and the expected real interest rate. Because the expected inflation rate and the expected real interest rates are not directly observed, we need to use some specific models to estimate these two series. The three approaches differ from one another in terms of how these two series are estimated. Two of these approaches use the Fisher equation and the last approach uses the observed real yield to maturity on index-linked gilt traded in the United Kingdom to estimate the expected real interest rate.
Fisher Equation with Mean-Reverting Expected Real Interest Rate The generalized version of the Fisher equation can be written as:1
Rt = rt+ πte+ 0.5vart(πt)− γ covt(ct,πt) , (1a)
where
Rt = nominal interest rate for period t;
rt = expected real interest rate for period t;
πe
t = expected inflation rate for period t;
πt = actual inflation rate for period t;
ct = rate of change of consumption for period t; and
γ = coefficient of relative risk aversion.
To simplify the Fisher equation, the last two terms on the right-hand side of equation (1a)—that is, 0.5 vart(πt)− γ covt(ct,πt)—are replaced by a constant
that leads to the following version of the Fisher equation:2
Rt = rt+ πte+ φ, (1b)
whereφ = 0.5 vart(πt)− γ covt(ct,πt) represents the risk premium.
1See Shome, Smith, and Pinkerton (1988) and Evans and Wachtel (1990, 1992) for a more complete
discussion of the equation.
2This is consistent with the fact that Shome, Smith, and Pinkerton (1988) find neither the conditional
variance (vart(πt)) nor the covariance (covt(ct,πt)) to be significantly different from zero. Crowder and
Hoffman (1996) use similar argument in replacing the last two terms by a constant. Fama and Gibbons (1982), Mishkin (1995), and Mishkin and Simon (1995) use the simplified version of the Fisher equation withφ equal to zero.
The nominal interest rates on financial assets are known at the beginning of a period. Therefore, this variable can be regarded as an observable variable. How-ever, the other two variables are not directly observable. We assume the following relation between the observed inflation rateπt and the expected inflation rateπte:
πt = πte+ ut, (2)
where ut(the unexpected component of the observed inflation) is assumed to be an
independently and normally distributed process with mean zero and varianceσ2; that is, ut ∼ i.i.d.N(0,σ2). For equation (2) to hold, it is sufficient that the investors’
expectations of inflation are rational. As to the expected real rate, we assume that it follows a mean-reverting process as follows:
rt − rt−1= α (ρ − rt−1)+ vt, (3)
where the process vtis also assumed to be a zero mean independent normal process
with variance equal toλσ2, that is, v
t ∼ i.i.d.N(0,λσ2). Furthermore, vtis assumed
to be independent of ut. In this case,ρ is the long-run expected real interest rate and
α measures the speed of adjustment. This model of the expected real interest rate is a general model that can characterize wide varieties of interest rate processes. If α = 0, the process characterized by equation (3) is a random-walk process. With α = 1, the model allows the real rate to be a constant mean process, and with α = 1
andλ = 0 the process represented by equation (3) becomes a constant real rate
process. The mean-reversion model of interest rate is popular among researchers (Cox, Ingersoll, and Ross 1985; Hull and White 1990).
Equations (1b)–(3) can be rewritten as follows:
πt = −rt+ Rt − φ + ut, (4)
rt = αρ + (1 − α) rt−1+ vt. (5)
Because both the parameterφ and the expected real interest rate series are unob-servable, they cannot be separately identified. Therefore, we consider the following model that will replace the set of equations (4) and (5):
πt = −rt∗+ Rt + ut, (6)
rt∗= αρ∗+ (1 − α) rt∗−1+ vt, (7)
where rt∗ = rt + φ and ρ∗= (ρ + φ). We resolve the problem of identification by
estimating (ρ + φ) rather than estimating ρ and φ separately. Because our main objective is to analyze the cointegrating relation between the expected inflation and expected real rate series, this does not affect our result.
The parameters of the model (θ = (α, ρ∗,λ, σ2)) can be estimated using the Kalman filtering technique that involves the use of the following recursive equations: ˆrt∗+1|t = αρ + (1 − α) rt∗|t−1+ Kt|t−1ηt, (8) ηt = πt + ˆrt∗|t−1− Rt, (9) Kt|t−1 = (1 − α) Pt|t−1(−1) Pt|t−1+ σ2 −1 , (10) Pt|t−1= [(1 − α) + Kt|t−1]Pt|t−1(1− α) + λσ2, (11) ht|t−1= Pt|t−1+ σ2, (12)
where ˆrt∗+1|tdenotes the prediction of rt∗+1given the information available up to and including period t,ηtrepresents the innovation process, and Kt|t−1is known as the
Kalman gain matrix. Similarly, Pt|t−1and ht|t−1are known as the estimation error
covariance matrix and innovation covariance matrix, respectively. The innovation processηtis a serially independent normally distributed process with variance given
by ht|t−1. Thus, the log likelihood function (apart from the irrelevant constant) is
given by: L (θ) = T t=1 L(t,θ) = −1 2 ln|ht|t−1| + η2 t ht|t−1 . (13)
The maximum likelihood estimates can be obtained using the iterative technique suggested by Berndt et al. (1974). This involves the following iterative procedure:
θn = θn−1+ [Q(θn−1)]−1∂ L(θ n−1) ∂θ , (14) where Q (θ) = T t=1 ∂ L(t, θ) ∂θ ∂ L(t, θ) ∂θ ,
andθnis the estimated vector at the nthiteration. Various hypotheses can be tested
using the consistent estimator of the covariance matrix given by [Q(θ)]−1. The Kalman filter technique also provides the estimate of the real rate and inflation series. Once these series are estimated, we use the unit root and cointegration tests on these series to find the long-run relation between these series.
If the real interest rate is a nonstationary process, the Kalman filter estima-tors do not have the standard properties (e.g., consistency, asymptotic normality,
etc.).3 We address this issue in two ways. First, empirical distributions of the Kalman filter estimators are obtained using simulation with 5,000 replications. Alterna-tively, we assume that the real interest rate follows a random-walk process rather than a mean-reverting process. Once we assume a random-walk process, we do not need to use Kalman filter. This leads us to the second approach, which is discussed next.
Fisher Equation with Random Walk Expected Real Interest Rate
In this model, the relation between the observed inflation rateπt and the
expected inflation rateπte is still given by equation (2); however, we assume that the real interest rate follows a random-walk process given by:
rt = rt−1+ υt, (15)
where the processυt is assumed to be a zero mean normal process with variance
equal toλ21σ2; that is,υt ∼ i.i.d.N(0,λ21σ
2). Furthermore, we use a different version
of the Fisher equation given by:
πt = −rt + β Rt − φ + ut. (16)
Note that equation (16) becomes equation (4) if we impose a restriction that β is equal to one. Several researchers use this version of the Fisher equation when testing the Fisher hypothesis (Fama and Gibbons 1982; Mishkin and Simon 1995). As before, we can simplify the model as follows:
πt = −rt∗+ β Rt + ut, (17)
rt∗ = rt∗−1+ υt, (18)
where rt∗ = rt + φ. The parameters of this model as well as the expected inflation
and expected real rate series can be estimated using the exact maximum likelihood estimation technique proposed by Shrestha (1988).4
Observed Real Interest Rate on Index-Linked Gilt
The two previous approaches use the Fisher equation and some assumptions regarding the process the real interest rate follows to estimate the expected real
3We thank an anonymous referee for pointing this out.
4Note that when we replace a with 1 in Shrestha’s (1988) model, we end up with the model represented
interest rate process. However, such complicated models are not required if we use the real yields to maturity on the index-linked gilts traded in the United Kingdom to estimate the real interest rate process. In this article we assume that the relation between the expected real interest rate rt and the real yield to maturity on the
index-linked gilt rgt is given by:
rgt = rt+ ζt, (19)
whereζt is a white noise process. In other words, the expected real interest rate and
real yield to maturity differ by a stationary white noise process. Again, we assume that the relation between the observed inflation rateπt and the expected inflation
rateπteis given by equation (2).
It is clear from equation (19) that if the expected real interest rate is a random-walk process, the yield to maturity on the index-linked gilt will also be a random-walk process. Similarly, it is clear from equation (2) that if the expected inflation rate follows a random-walk process, the observed inflation rate will follow a random-walk process. Furthermore, a stationary relation between the expected series (e.g., expected inflation rate and expected real rate) implies a stationary rela-tion between their observed counterparts (e.g., observed inflarela-tion rate and observed yield to maturity on the index-linked gilt). Therefore, if a long-run (cointegrating) relation does not exist between the observed inflation rate and the yield to matu-rity on index-linked gilt, we can conclude that the long-run relation between the expected real rate and expected inflation rate does not exist.
III. Long-Run Test Procedures
Once the expected inflation and expected real rate series are estimated, we need to perform the long-run (cointegrating) analysis. This involves the unit root and cointegration tests. To test for the presence of unit root in the expected real rate rt, we estimate the following two regression equations:
rt = α0+ α1rt−1+ p j=1 γjrt− j+ εt, (20) rt = δ0+ δ1rt−1+ δ2t+ p j=1 γjrt− j+ εt. (21)
Null hypothesis Test statistic (a) α1= 0 N ˆα1 (b) α1= 0 t-ratio (c) α0= α1= 0 F -test1 (d) δ1= 0 N ˆα1 (e) δ1= 0 t-ratio (f) δ0= δ1 = δ2= 0 F -test2 (g) δ1= δ2 = 0 F -test3
Critical values are obtained from Dickey and Fuller (1981). In this article, the Phillips and Perron (1988) test, which allows a more general structure of the error term, is used to test for the unit root. The unit root for the expected inflation series is similarly carried out.
The cointegration test can be performed using either the two-step proce-dure suggested by Engle and Granger (1987) or the maximum likelihood estimator suggested by Johansen and Juselius (1990). In this article we use both types of cointegration tests.
IV. Empirical Results
Fisher Equation with Mean-Reverting Expected Real Interest Rate
The model given by equations (6) and (7) is estimated for the United States, the United Kingdom, and Canada using interest rate on three-month T-bills and the Consumer Price Index (CPI) inflation rate. The data are obtained from the International Monetary Fund’s International Financial Statistics. The rates used are quarterly rates and cover from the first quarter of 1964 to the fourth quarter of 1994. All the rates are continuously compounded quarterly rates. The summary statistics for the nominal interest rates and observed inflation rates are given in Table 1.
TABLE 1. Summary Statistics on Nominal Interest Rate, Observed Inflation Rate and Observed Real Interest Rate (Percent per Quarter).
Variable Mean Std. Dev. Minimum Maximum U.S. interest rate 1.6799 0.6938 0.7205 3.7919 U.K. interest rate 2.3419 0.8099 0.9448 4.1733 Canadian interest rate 2.0777 0.9116 0.7612 5.2651 U.S. inflation rate 1.2752 0.8433 −0.9485 3.9105 U.K. inflation rate 1.8995 1.6046 −1.4399 7.2932 Canadian inflation rate 1.3479 0.9130 −0.8413 3.7264 U.S. real rate 0.4048 0.7871 −1.3441 2.7440 U.K. real rate 0.4424 1.4888 −4.7140 3.5148 Canadian real rate 0.7298 0.9752 −1.8954 2.9480
TABLE 2. Coefficient Estimates.
United States United Kingdom Canada Parameter Estimate Std. Error Estimate Std. Error Estimate Std. Error Panel A. Fisher Equation with Mean-Reverting Expected Real Interest Rate
α 0.0739 0.0331 0.0429 0.0357 0.0557 0.0279
ρ∗ 0.4071 0.4339 0.4985 0.6363 0.7535 0.5091
σ2 0.2318 0.1414 1.1866 0.1732 0.3631 0.1230
λ 0.2143 0.0312 0.0626 0.0425 0.1547 0.0549 Panel B. Fisher Equation with Random Walk Expected Real Interest Rate
β 0.5081 0.1566 0.9364 0.2124 0.3115 0.1229
λ1 0.4663 0.0957 1.0967 0.0316 0.5626 0.0480
σ 0.4354 0.0361 0.2242 0.1739 0.2903 0.0477 Note: Panel A contains the results of the estimation of the parameters of the model given by equations (6) and (7) using the Kalman filtering technique. Panel B contains the results of the estimation of the parameters of the model given by equations (17) and (18).
The parameter estimates of the mean-reverting model are reported in Table 2, Panel A. All the estimates are as expected. As mentioned before, we cannot use the standard t-test to test for the random walk for the expected real inter-est rate series. To perform the tinter-est, empirical distributions based on the hypothesis that the expected real rate follows a random-walk process are generated. A brief discussion on the empirical distribution is provided in the Appendix. The empirical distributions are summarized in Table 3, Panel A. The 95% confidence interval for α is equal to (–0.0.13, 0.0996). The estimates of α for all three countries lie inside the confidence interval, indicating that the null hypothesis of random walk for the expected real interest rate cannot be rejected. This is also consistent with the unit root test performed on the expected real interest rate series. The result of the unit root test is discussed later.
Note that the estimate of λ is low for the United Kingdom. This low estimate, together with the insignificant α, may indicate that the expected real interest rate for the United Kingdom is constant. Because the conventional standard error (reported in Table 2) cannot be used here, empirical distributions based on the hypothesis that the expected real rate is constant (i.e., α = 0 and λ = 0) are generated. The empirical distributions are summarized in Table 3, Panel B. The 95% confidence interval forλ is equal to (0.0, 0.0013). Because the estimate of 0.0626 lies outside this interval, the expected real interest rate is not constant for the United Kingdom. Therefore, we conclude that the expected real interest rate follows a random-walk process for all three countries.
When the parameters of the model are estimated using the Kalman filter, the real interest rate and the expected inflation rate series are also simultaneously
TABLE 3. Empirical Distribution Based on 5,000 Replications.
Percentile α ρ∗ λ σ2
Panel A. Distribution Under the Null Hypothesis of Random Walk Real Interest Rate
0.5% −0.0180 −2.1872 0.2652 0.0029 1.0% −0.0159 −1.9297 0.3036 0.0034 2.5% −0.0130 −1.5790 0.4013 0.0042 5.0% −0.0108 −1.2455 0.4715 0.0050 95.0% 0.0718 2.3890 3.0232 0.0141 97.5% 0.0996 2.7112 3.8496 0.0152 99.0% 0.1435 3.1333 5.2989 0.0162 99.5% 0.1821 3.3541 6.2556 0.0170 Theoretical value 0.0000 0.5000 1.0000 0.0100 Mean 0.0170 0.5132 1.3305 0.0095 Std. dev. 0.0517 1.1055 0.9311 0.0043 Skewness 15.9772 0.0960 2.9742 19.7944 Standardized kurtosis 362.7502 −0.1533 14.1616 735.8063 Panel B. Distribution Under the Null Hypothesis of Constant Real Interest Rate
0.5% −0.0365 0.4661 0.0000 0.0068 1.0% −0.0305 0.4742 0.0000 0.0070 2.5% −0.0178 0.4791 0.0000 0.0074 5.0% 0.0006 0.4828 0.0000 0.0078 95.0% 0.0012 0.5163 0.0001 0.0124 97.5% 0.0020 0.5197 0.0013 0.0129 99.0% 0.0704 0.5242 0.0119 0.0134 99.5% 0.1311 0.5273 0.0386 0.0138 Theoretical value 0.0000 0.5000 0.0000 0.0100 Mean 0.0028 0.4999 0.0008 0.0099 Std. dev. 0.0296 0.0105 0.0126 0.0014 Skewness 17.2301 −0.2288 30.4560 0.1764 Standardized kurtosis 374.5887 0.8951 1181.7734 0.5948
estimated.5The estimates of the correlation coefficients between the real rate and inflation rate are –0.5088, –0.7170, and –0.3365 for the United States, the United Kingdom, and Canada, respectively. These negative and significant correlations are consistent with the correlation of –0.9 reported by Mishkin and Simon (1995) for Australia. The evidence at this point seems to support the Mundell-Tobin hypothesis of negative relations between the expected real rate and expected inflation rate for the three economies considered. However, as mentioned earlier, such negative
5Actually, the Kalman filtering provides the estimate of the real interest rate plus the constant risk
premium series (i.e., rt∗= rt+ φ). However, the unit root test and cointegration test performed on rt∗is
valid for rtbecause a constant term added to a series does not alter the stochastic nature of the series. This is
TABLE 4. Unit Root Test for the Expected Real Rate and the Expected Inflation Rate from the Mean-Reverting Model.
Test
Country (c)1 (d) N ˆα1 (e) t-ratio (f)2 (g)3
Expected real rate
Unites States 1.974 −8.322 −2.033 1.380 2.068 United Kingdom 1.188 −8.174 −2.113 1.523 2.278 Canada 1.513 −12.230 −2.620 2.350 3.506 Expected inflation rate
Unites States 4.165 −15.206 −2.859 2.796 4.188 United Kingdom 2.527 −9.648 −2.283 2.079 3.119 Canada 3.069 −11.748 −2.472 2.338 3.508 Critical value N= 100, 5% 4.71 −20.7 −3.45 4.88 6.49 Note: The test statistics are calculated based on the estimation of equations (20) and (21) for the expected real interest rate and the expected inflation rate series.
relations between two series may not imply the negative long-run relations if the two series are nonstationary. The main objective of this article is to test whether the Mundell-Tobin hypothesis holds in the long run. This involves the unit root test and the cointegration analysis.
To perform the unit root test, equations (20) and (21) are estimated using the estimated real rate and expected inflation rate series. The test statistics are calculated following Phillips and Perron (1988).6 The results are summarized in
Table 4.
First, consider the F-test statistics 3. For all three countries, the null
hypothesis of a unit root and no trend cannot be rejected for either the real rate or the inflation rate. The tests (d) and (e) also support the null hypothesis of a unit root for all three countries and both series. To test for nonzero drift, we perform the 2test. None of the expected real rate and expected real rate series seems to have
nonzero drift. These results of random walk without drift are confirmed by the1
test based on equation (20). These results of random walk for the expected real rate are consistent with the random-walk test (α = 0) performed earlier.
To make sure the expected real rate and expected inflation rate series do not have more than one unit root, the same tests are performed on the first difference of the series. Although not reported, the results indicate that all of the first differenced series are stationary. This confirms that each of the six series has a single unit root. Because each of the series is a unit-root series, it is clear that the negative correlation may represent a spurious negative relation or a long-run equilibrium
TABLE 5. Engle-Granger Cointegration Test with the Expected Real Rate as the Dependent Variable.
Test United States United Kingdom Canada Critical Value at 5% Panel A. Fisher Equation with Mean-Reverting Expected Real Interest Rate
(a) N ˆα −6.7185 −7.3053 −5.0221 −20.50 (b) t-ratio −1.8473 −2.0137 −1.5983 −3.37 (c) N ˆα −6.1924 −11.760 −9.3748 −21.50 (d) t-ratio −1.6920 −2.2134 −2.1773 −3.42 Panel B. Fisher Equation with Random Walk Expected Real Interest Rate
(a) N ˆα −7.2595 −7.3047 −7.5979 −20.50 (b) t-ratio −1.9333 −2.0545 −2.0057 −3.37 (c) N ˆα −7.5915 −7.1932 −12.1340 −21.50 (d) t-ratio −1.9517 −1.6631 −2.4472 −3.42 Note: The tests (a) and (b) are the tests with no trend in cointegrating regression and tests (c) and (d) are those with trend in cointegrating regression. The critical values are obtained from Phillips and Ouliaris (1990).
relation. Thus, to determine whether the negative relation is a long-run stationary relation (among nonstationary variables), we use the same estimated series and test for the existence of the cointegrating relation between the real rate and expected inflation.
First, the cointegration test as suggested by Engle and Granger (1987) with the expected real rate as the dependent (left-hand side) variable is performed and the results are summarized in Table 5, Panel A. The results show that the real rate and the expected inflation rate are not cointegrated. The same conclusion of no cointegration is obtained using the expected inflation rate as the dependent variable in the cointegrating regression. Therefore, we can conclude that the negative correlation between the real rate and the expected inflation rate is not a stationary long-run relation.
To verify the result, we perform the Johansen and Juselius (1990) tests. The results of the trace test and lambda max test are summarized in Table 6, Panel A, where the critical values are obtained from Osterwald-Lenum (1992). Both the trace test and the lambda max test indicate there is no cointegration between the expected real interest rate and the expected inflation rate. This is the same result obtained using the Engle-Granger test.
Fisher Equation with Random Walk Expected Real Interest Rate
The same data (on T-Bills and CPI inflation) are used to estimate the random-walk model given by equations (17) and (18) for the United States, the United Kingdom, and Canada. The parameter estimates are summarized in Table 2, Panel B. The sample correlations between the real rate and the inflation rate are
TABLE 6. Johansen-Juselius Cointegration Tests of the Expected Real Rate and the Expected Inflation Rate.
No. of Cointegrating Relation United States United Kingdom Canada Critical Value at 5% Panel A. Fisher Equation with Mean-Reverting Expected Real Interest Rate
Trace test
0 18.475 12.631 15.196 19.96
1 5.017 4.282 2.635 9.24
Lambda max test
0 13.458 8.349 12.561 15.67
1 5.017 4.282 2.635 9.24
Panel B. Fisher Equation with Random Walk Expected Real Interest Rate Trace test
0 10.822 15.469 14.792 19.96
1 4.698 3.262 3.118 9.24
Lambda max test
0 6.124 12.207 11.674 15.67
1 4.698 3.262 3.118 9.24
Panel C. Observed Real Interest Rate on Index-Linked Gilt Trace test
0 17.009 19.96
1 7.099 9.24
Lambda max test
0 9.910 15.67
1 7.099 9.24
–0.8522, –0.7390, and –0.9140 for the United States, the United Kingdom, and Canada, respectively. Note that these correlations between the real rate and the inflation rate are considerably more negative than the correlations from the mean-reverting model.
The unit root tests, reported in Table 7, indicate that the expected inflation rate and the expected real rate consist of a single unit root. The Engle-Granger cointegration tests reported in Table 5, Panel B, indicate that the negative relation is
TABLE 7. Unit Root Test for the Expected Inflation Rate from the Random Walk Model.
U.S. Expected U.K. Expected Candian Expected Critical Value Test Inflation Rate Inflation Rate Inflation Rate N= 100, 5%
(c)1 2.3664 1.2550 0.9209 4.71
(d) N ˆα1 −8.0512 −6.3197 −4.1360 −20.70
(e) t-ratio −2.1758 −1.8582 −1.4834 −3.45
(f)2 1.9809 1.6187 1.6040 4.88
not a long-run cointegrating relation. These results are supported by the trace test and lambda max test reported in Table 6, Panel B.
Observed Real Interest Rate on Index-Linked Gilt
We use the real yield to maturity on 2,011 index-linked gilt and inflation in CPI from October 1986 to October 1996 to perform the analysis discussed in section II.7 The sample correlation between the real rate (on index-linked gilt) and the inflation rate is 6.185%, which is not significantly different from zero. The unit root tests (not reported) indicate that both series consist of a single unit root. However, both the trace test and the lambda max test (see Table 6, Panel C) indicate there is no cointegration between the real rate and the inflation rate. Note that the expected real rate is assumed to differ from the observed real rate series by a stationary error (see equation (19)). Similarly, the expected inflation rate is assumed to differ from the observed inflation rate by a stationary error as described by equation (2). Therefore, the lack of cointegration between the observed inflation rate and real rate implies there is no long-run relation between the expected inflation rate and the expected real rate. This is consistent with our earlier results.
The results from all three approaches indicate there is no cointegrating relation between the expected real interest rate and the expected inflation rate. Therefore, the Mundell-Tobin hypothesis does not seem to hold in the long run.
V. Summary and Conclusions
In this article, we use three approaches to analyze the cointegrating relation between the expected real interest rate and the expected inflation rate. The first approach is based on the Fisher equation, in which the expected real interest rate is assumed to follow a mean-reverting process. The Kalman filter technique is used to estimate the parameters of the model. The second approach is also based on the Fisher equation, in which the expected real interest rate is assumed to follow a random-walk process. The parameters are estimated using the exact maximum likelihood method proposed by Shrestha (1988). The third approach is based on the observed real yield on index-linked gilt traded in the United Kingdom. The first two approaches use the quarterly data on T-bills and CPI inflation for the United States, the United Kingdom, and Canada. The last approach uses monthly data on the real yield to maturity on 2,011 index-linked gilt and the CPI inflation rate.
7The real yield to maturity used here is based on 2,011 index-linked gilt traded in the United Kingdom.
We find that the expected real rate of interest is not constant over time. Furthermore, the empirical results suggest that the expected real rate consists of a single unit root. The same is true for the expected inflation rate; that is, each of the expected inflation series for all three economies follows a unit-root process.
Based on the first two approaches, we find negative and significant sample correlations between the real rate and the expected inflation rate series for all three countries. However, these two series are not cointegrated for all three countries. This implies that the negative relation is not a long-run stationary relation; hence, the Mundell-Tobin hypothesis does not hold in the long run. Therefore, T-bills can be considered as a long-run inflation hedge.
When the real interest rate on index-linked gilt is used, the sample corre-lation between the expected real rate and the expected infcorre-lation rate is positive but insignificant. Again, the cointegration analysis indicates no cointegrating relation between these two series, indicating that the Mundell-Tobin hypothesis does not hold true in the long run.
Appendix
Here we briefly describe the simulation procedure used to generate the em-pirical distribution. In the simulation, we need to generate three unit root processes representing nominal interest rate Rt, expected real interest rate rt, and expected
inflation rateπe
t that satisfy the Fisher equation. The three unit root series are
gen-erated using the following three-variable dynamic stochastic system:
x1t = u1t
x1t − x2t = u2t
x1t+ x2t − x3t = u3t, (A.1)
where the series u1t, u2t, and u3tare generated by:
(1− L)u1t = ε1t
(1− L)u2t = ε2t
u3t = ε3t, (A.2)
Finally, the innovation seriesε1t,ε2t, andε3tare generated as independent and
nor-mal processes,εit∼ N(0, 0.01), i = 1, 2, 3. To make the series have a positive mean,
3.5 is added to the nominal interest rate x2tand 2.5 is added to the inflation rate.
For each replication and given sample size T, T + 100 observations are generated with the initial values of u1t, u2t, and u3t being set equal to zero. Then, the first
100 observations are discarded to reduce the effect of the initial conditions. The remaining T observations are used to estimate the parameters of the model using the Kalman filtering technique as discussed in the text. The entire process is repeated 5,000 times.
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