Cointegration tests of the Fisher Hypothesis with variable trends in the world real interest rate.

Terrell, Dek

I. Introduction

After seventy years, the Fisher hypothesis has proven to be one of the most durable and influential theories in economics. Yet after years of debate and testing, the empirical accuracy of the proposition remains in question. The Fisher hypothesis simply states that a one point increase in inflation leads to a one point increase in the nominal interest rate leaving real interest rates unchanged. This relationship does not rule out the possibility of other variables influencing the real interest rate or variable trends in the real interest rate arising from sources other than inflation. This paper merges the literature on real interest rate parity with tests of the Fisher hypothesis to control for variable trends in the real interest rate.

From the beginning, tests of the Fisher hypothesis yielded mixed results. Studies such as Fama [15], Carr, Pesando and Smith [7], Cargill [6], Levi and Makin [28], Peek and Wilcox [37], Hoover [24], and Mishkin [33] have supported inflation neutrality and the Fisher hypothesis. However, other studies present evidence against the hypothesis [43; 31; 32; 3; 20; 42; 8]. Beginning with Nelson and Plosser [36], evidence of variable trends in both inflation and nominal interest rates began to build. Several recent papers address this issue and use cointegration tests to explore the long-run relationship between inflation and interest rates.

Unfortunately, the cointegration tests also yield mixed results. For example, Rose [3<] rejects cointegration between inflation and interest rates, while Arkins [1] tests support cointegration. Not surprisingly, MacDonald and Murphy [29] and Wallace and Warner [45] find that the results of cointegration tests are sensitive to time period and country. The failure of cointegration tests to find a stationary combination of nominal interest rates and inflation does not imply the absence of a long-run equilibrium relationship between the variables or necessarily reject the Fisher hypothesis. Instead these results may indicate the need for a richer model specification.

The presence of a variable trend in real interest rates offers one plausible explanation for the ambiguity of these cointegration tests. The real interest rate, which is equal in the long-run to the return on capital, can be influenced by both temporary factors (such as fiscal or monetary policy) or permanent factors (such as technological shift parameters and permanent tax rate changes). King, Plosser, Stock, and Watson [27] supply evidence from the U.S. economy that the real interest rate is related to business cycle phenomena and is nonstationary. In a multicountry study, Bosner-Neal [5] reports that monetary regime shifts influence the real interest rate and Rose [38] shows that the real interest rate is unstable in the U.S. and other OECD economies.

In addition to evidence on the univariate properties of real interest rates, theory and empirical evidence indicate a link between real interest rates across countries. Studies such as Mishkin [31; 36], Cumby and Obstfeld [11], Cumby and Mishkin [12], Merrick and Saunders [30], Gaab, Granziol and Horner [21], Modjtahedi [34], and Dutton [13] find correlations of real rates across economies. Cumby and Mishkin for instance find that "there is strong evidence that there is a positive relationship between movements in the U.S. real interest rate and those in Europe." Using cointegration tests, Modjtahedi [34] investigates the long-run relationship between real interest rates and finds cointegration between interest rates across countries. The existence of a common variable trend in real interest rates across countries suggests a specification of the Fisher equation that includes this variable trend.

This paper tests the Fisher hypothesis using quarterly data from Canada, France, Germany, U.K., Japan, and Italy over the period 1973-1989. Our analysis includes the U.S. real interest rate to account for variable trends in the world real interest rate. We analyze long-run relationships between inflation, nominal interest rates, and the U.S. real interest rate in each of the six countries. This paper explores these relationships using the multivariate cointegration methodology proposed by Johansen [25] and Johansen and Juselius [26]. The Johansen approach allows a test of the Fisher hypothesis in a trivariate framework and avoids drawbacks of the Engle-Granger regression methodology. Unlike the static Engle-Granger approach, the Johansen approach allows for dynamic interrelationships among variables, simple tests of restrictions, and tests for the number of cointegrating vectors.

II. A Basic Model of Interest Rates

The Fisher hypothesis asserts that a one point rise in inflation leads to a one point rise in nominal interest rates. Tests of this assertion are complicated by the fact that innovations unrelated to inflation affect both real and nominal interest rates and that all three series appear non-stationary. We derive an empirical model based on real interest parity, which allows for tests of the Fisher hypothesis. Consider the Fisher equation for country j:

[Mathematical Expression Omitted]

where

[i.sup.j] = the nominal interest rate in country j,

[[Pi].sup.ej] = the expected inflation rate in country j, and

[r.sup.ej] = the expected real interest rate in country j.

Equation (1) describes a long-run relationship between domestic nominal interest rates, expected inflation and ex ante domestic real interest rates in country j. Tests of the Fisher hypothesis often proceed based on the assumption that the real interest rate is constant over time. Recently, King, Plosser, Stock, and Watson [27], Bosner-Neal [5], and Modjtahedi [34] provided evidence that the real interest rate varies over time and is non-stationary. We account for non-stationary real interest rates based on the equilibrium relationship between real interest rates across countries:

[Mathematical Expression Omitted]

where [r.sup.ew] denotes the expected world real interest rate.

Equation (2) combines real interest parity with the possibility that inflation effects real interest rates. If real interest rates are equalized across countries, then [[Beta].sub.0] = 0 and [[Beta].sub.1] = 1. The parameter [[Beta].sub.0] may deviate from zero as a result of a risk premium. Likewise transaction costs or differences in tax rates across countries may cause [[Beta].sub.1] to differ from one, and in the extreme case of a closed economy [[Beta].sub.1] may equal zero. Empirical evidence supports a relationship between real interest rates across countries, but suggests that [[Beta].sub.1] lies between zero and one [10; 21; 30]. Modjtahedi [34] corroborates the existence of such a relationship, by finding evidence of cointegration among real interest rates across countries. This result of cointegrated real interest rates suggests that all permanent innovations to the countries real interest rate stem from innovations in the world real interest rate, and thus u(t) denotes a stationary moving average process.

Equation (2) also allows for the possibility that the expected domestic inflation rate influences the domestic real interest rate. If [Gamma] = 0, as the Fisher hypothesis asserts, equation (2) reduces to the standard real interest parity equation. However, several authors point to reasons for non-neutrality of inflation. Mundell [35] and Tobin [44] present a model in which higher expected inflation reduces the demand for real balances, encouraging agents to shift into short-term assets and reducing their return. The lower returns then stimulate economic activity.

Fama and Gibbons [16] propose an alternative model that generates a similar violation of inflation neutrality. Contrary to the Mundell-Tobin approach, Fama and Gibbons assert that the inverse relationship between expected inflation and lower real rates of return is a consequence, not a cause of higher real activity. In the Fama and Gibbons model, money supply and expected inflation are endogenous variables that depend on consumption-investment allocation decisions. Both the Mundell-Tobin and Fama-Gibbons models predict [Gamma] [less than] 0.

On the other hand, Darby [12] and Feldstein [17] note that the failure of tax codes to index interest income implies that an increase in the before-tax real interest rate is required to offset additional tax liabilities from inflation, and thus [Gamma] [greater than] 0. Two problems make tests of the Feldstein-Darby hypothesis difficult. First, the Mundell-Tobin effect may offset any tax effects and influence [Gamma]. Second, the appropriate tax rate is difficult to obtain. Although MacDonald and Murphy [29] use an average tax rate from Tanzi [43], Gandolfi [28] and Rose [39] argue that the choice of a relevant tax rate is difficult and, due to capital gains and depreciation allowances, the tax effect is likely to be small. For these reasons, we concentrate on tests of the Fisher hypothesis excluding tax rates rather than the Feldstein-Darby proposition.

Combining equations (1) and (2) yields the long-run relationship:

[Mathematical Expression Omitted]

where [[Beta].sub.2] = 1 - [Gamma].

The Fisher hypothesis asserts [[Beta].sub.2] = 1 and the assumptions in (1) and (2) imply a stationary error in (3) and cointegration. Non-cointegration indicates the presence of permanent innovations in the real interest rate not common to the rest of the world. Using the U.S. real interest rate to proxy for world real interest rates, non-cointegration may indicate imperfect capital markets or high costs of transactions between the domestic economy and the U.S. Note that if the model excludes the world real interest rate, which contains a variable trend, nominal interest rates and inflation are not cointegrated.

If [[Beta].sub.1] = 1 and [[Beta].sub.2] = 1, real interest rates move one-for-one and, given the assumption that u(t) is a stationary moving average, the real interest rate differential is stationary. Modjtahedi's [34] tests for stationarity of real interest differentials corresponds to hypothesis tests of this proposition in our model.

All equations above contain unobserved expectations of inflation and real interest rates. Assuming no systematic errors in expectations, we use a simple model relating expected values of interest rates and inflation to observed inflation and interest rates:

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [[Pi].sup.j] and [r.sup.j] denote ex post inflation and real interest rates in country j.

Substituting equation (4) into the other equations normally leads to the classic errors in variables problem and inconsistent parameter estimates. However permanent shocks dominate any temporary stochastic errors in the case of cointegrated variables, which allows for consistent estimation even with the presence of the errors in variables problem [40; 41]. Therefore, we proceed to cointegration tests using observed values of inflation and real interest rates based on the substitution for expected values of the variables as given in equation (4).

We test the importance of the world real interest rate using Johansen's multivariate co-integration methodology. Define a vector Z[prime] = [[i.sup.j][[Pi].sup.j][r.sup.w]]. Using this notation equation (3) may be written as the vector autoregression:

[Z.sub.t] = [summation of] [[Pi].sub.[Tau]][Z.sub.t - [Tau]] where [Tau] = 1 to k + [[Epsilon].sub.t] (5)

where [[Epsilon].sub.t] is distributed N(0, [Sigma]). Johansen's methodology requires the reparameterization:

[Delta][Z.sub.t] = [summation of] [[Gamma].sub.[Tau]][Delta][Z.sub.t - [Tau]] where [Tau] = 1 to k + [[Gamma].sub.k][[Z.sub.t - k], 1] + [[Epsilon].sub.t] (6)

where [[Gamma].sub.k] = -I + [[Pi].sub.1] + ... + [[Pi].sub.k].

The matrix [[Gamma].sub.k] represents the long-run effects of the innovation vector [[Epsilon].sub.t] on Z and has rank equal to m, the number of distinct cointegrating vectors. The number of parameters minus the rank of [[Gamma].sub.k] supplies the number of unit roots in the vector representation of the series and the number of stochastic trends needed to characterize the series. Thus if [[Gamma].sub.k] is of full rank, all variables in the system are stationary and the system contains no unit roots. If the rank of [[Gamma].sub.k] is zero then no stationary linear combination of domestic inflation, domestic nominal interest rates, and world real interest rates exists, and the series are characterized by three stochastic trends. Theory and the empirical model discussed above suggest that two permanent stochastic trends may completely characterize the three series; thus one cointegrating vector exists.

In addition to estimation of the rank of [[Gamma].sub.k], Johansen's methodology permits estimation of the cointegrating vector(s) using the decomposition [Alpha][[Beta][prime], [[Beta].sub.0]][prime] = [[Gamma].sub.k], where [Beta] denotes a 3 x m matrix of cointegrating vectors and [Alpha] is a 3 x m matrix of error correction parameters. Assuming [TABULAR DATA FOR TABLE I OMITTED] a single cointegrating vector, the normalized cointegrating vector supplies estimates of the parameters from equation (3).

III. Data/Preliminary Tests

The tests described above require nominal interest rates and inflation data for each country, and a measure of the world real interest rate. The nominal interest rate data (i) consist of quarterly call money rates for Canada, France, Italy, Germany, Japan, U.K., and the U.S. We compute quarterly inflation data ([Pi]) as the annualized percentage change of the end of quarter consumer price index for each country, and use the ex-post U.S. real interest rate, calculated as the quarterly call money rate minus inflation, as the real interest rate. The use of the U.S. real rate for the world real rate is based on the large country assumption and appears commonly in the literature [34].(1) All data series are obtained from the Citibase Macroeconomic database for the period 1973:1 through 1989:4.

Table I presents results of the augmented Dickey-Fuller test for unit roots for all series. In all cases, the tests fail to reject the null hypothesis of a unit root in nominal interest rates, inflation, and the real interest rate at the 5% level for all countries. Additional tests reject unit roots in differences at the 5% level for both inflation and interest rates. Based on these results, we proceed with an empirical model expressing interest rates and inflation as I(1).

IV. Johansen Tests

The empirical model described in section II predicts cointegration between nominal interest rate inflation, and the U.S. real interest rate. Combined with the failure to reject a unit root in tt real interest rate for each country, the equation also predicts that no stationary combination of [TABULAR DATA FOR TABLE II OMITTED] inflation and nominal interest rates exists when the U.S. real interest rate is excluded from the model. We explore these predictions using the Johansen rank test for cointegration in two sets of models, one set including all three variables and a second excluding the U.S. real interest rate.

Table II presents Johansen rank tests for cointegration. Column 4 presents Johansen statistics for the full model given in equation (3). Column 3 contains rank tests of the traditional Fisher equation and excludes the U.S. real interest rate in the Johansen tests. A comparison of these results reveals striking differences. Excluding the U.S. real interest rate, the Johansen results fail to reject non-cointegration in five of the six countries at the 5% significance level. Across the entire sample, we conclude that the results provide little evidence in favor of cointegration between nominal interest rates and inflation.

The rank tests in column 4, which include the U.S. real interest rate, stand in stark contrast to those in column 3. We reject non-cointegration in favor of a single cointegrating vector at the 5% level in four of six countries.(2) In Japan non-cointegration is rejected at the 10% significance level and the critical value of 19.76 indicates that we would reject non-cointegration at near the 10% level of significance for Canada. When the U.S. real interest rate is included, the Johansen tests provide strong evidence in favor of cointegration between the nominal interest rate, inflation, [TABULAR DATA FOR TABLE III OMITTED] and the U.S. real interest rate. Thus inclusion of the U.S. real rate removes the variable trend and restores the long term relationship between inflation and nominal interest rates. Further, the rank tests fail to supply any evidence of more than one cointegrating vector in any country.

The results of this exercise are consistent with Blanchard and Summers [4], Cumby and Mishkin [10], Merrick and Saunders [30], and Barro and Sala-i-Martin's [2] conjectures of shifts in the world interest rate over this time period. Burro and Sala-i-Martin [2] report several shifts in the world real interest rate since 1959, and that the U.S. real interest rate moved similarly to the average for eight other major countries.(3) Both Summers [42] and Barro and Sala-i-Martin [2] argue that high expected stock market profitability during the period 1981-86 served as the fundamental cause of high real interest rates over that time period. Although our methodology cannot identify sources of interest rate shifts, our results confirm the presence of world-wide permanent shifts during the period 1973-1989.

Table III reports the parameter estimates from the Johansen procedure and hypothesis tests. Columns 2 and 3 find [[Beta].sub.0] near zero and [[Beta].sub.1] near one for most countries included in the sample. The general pattern of estimates across countries lends itself to interpretation. These results suggest that the U.S. real interest rate exerts a strong influence on nominal interest rates in Canada, U.K., and Italy with weaker effects on those in Germany, Japan, and France. This may reflect the close U.S. ties to Canada and U.K., or differences in restrictions regarding movements of capital and lower transactions costs in some countries included in the sample.

The point estimates of [[Beta].sub.2] in column 3 range from a prediction of a .6 point rise in nominal interest rates for every one point increase in inflation in Japan to a 1.48 point rise in Germany. Comparing results across countries reveals that [[Beta].sub.2] lies below one for five of six countries, a pattern consistent with the Mundell-Tobin and Fama-Gibbons models. Column 4 presents a formal test of the Fisher hypothesis, which is imposed through the restriction [[Beta].sub.2] = 1 (with the coefficient of nominal interest rates normalized to -1). This chi-square test statistic rejects the restriction at the five percent level in Italy, France, and Germany, and at the ten percent level in Japan.

Columns 6-8 of Table III present tests of restrictions involving [[Beta].sub.0] and [[Beta].sub.1]. Consistent with the rank tests, the results in column 6 suppl} a strong rejection of the exclusion of real world interest rates from the model. As in previous studies, evidence on risk premiums and real interest equalization vary across country. Column 7 presents tests of the hypothesis [[Beta].sub.0] = 0. These tests find no evidence of a risk premium in Canada, Germany, or U.K., but strongly reject the null in favor of a risk premium for France. Column 8 contains tests of real interest equalization. The results vary across the countries, rejecting equalization for France, Germany, and Japan. Overall our results are consistent with most previous studies of real interest rates, supplying strong evidence of a relationship between real interest rates across countries, although weaker evidence in favor of equalization of rates.

Column 9 contains tests for one-for-one movements between domestic interest rates and interest rates in the U.S., the restrictions [[Beta].sub.1] = 1 and [[Beta].sub.2] = 1. This test is identical to Modjtahedi's [34] test for stationarity of real interest differentials. The test statistics reject this hypothesis at the 10% level in all countries and at the 5% level in four of six countries. These results support the findings of Mishkin [32] and Cumby and Mishkin [10], but vary from Dutton [13] and Modjtahedi [34].(4)

V. Conclusion

This paper conducts tests for cointegration between real word interest rates, inflation, and nominal interest rates in Canada, France, Italy, Germany, U.K., and Japan and supplies tests of the Fisher hypothesis and real interest parity allowing for the presence of a variable trend in world real interest rates. Results indicate that the sensitivity of past cointegration tests to the sample country and time period are explained to a large extent by the presence of a variable trend in world real interest rates. Johansen trace tests indicate that cointegration exists between nominal interest rates, inflation and the U.S. real interest rate in five of six countries, but fail to reject non-cointegration in five of six countries (at the 5% level) considered if the traditional Fisher equation is tested with the U.S. real interest rate excluded from the model. These test statistics provide robust evidence of an equilibrium relationship between word real interest rates, domestic inflation and nominal interest rates that cannot be detected in models excluding world real interest rates.

1. We tested the use of the U.S. real rate versus the German rate in our model using the non-nested specification tests of Cox [9], Fisher and McAleer [18], and Godfrey and Pesaran [23]. The tests reject the use of the German rate in favor of the specification using the U.S. real rate in three of five countries and are inconclusive in the remaining two countries.

2. This finding indicates that a long-run equilibrium relationship exists between domestic nominal interest rates, inflation, and the U.S. real interest rate in at least four of six countries.

3. Barro and Sala-i-Martin [2] find a correlation coefficient of .73 between the U.S. real interest rate and the average real rate for those countries for the period 1959-1989.

4. Differences in price indices and interest rates may explain the deviations. Dutton uses a price index composed only of traded goods, relevant for trade decisions, while our work uses the interest rate relevant for the agent's borrowing and lending decisions (see Mishkin [32] for related discussion). Both Dutton [13] and Modjtahedi [34] use monthly offshore Eurocurrency rates rather than the call money rates used in this paper. Eurocurrency rates are more likely to represent the international return on savings; whereas call money rates represent the domestic return on savings and investment.

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