4 Level and Volatility of U.S. Interest Rates: Roles of Expected Inflation, Real Rates, and Taxes

Vito Tanzi
Published Date:
June 1984
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This paper analyzes the major forces affecting the behavior of interest rates, placing particular emphasis on the unusually high levels and variability of rates in recent years. The approach is eclectic in the sense that strong prior views are not allowed to rule out consideration of any possible avenue of investigation that might help to explain unusual interest rate movements.


As Irving Fisher theorized a long time ago, the market interest rate (hereinafter referred to as the interest rate) on a security maturing in t periods of time is approximately the sum of an expected real rate (hereinafter referred to as the real rate) and the level of inflation expected over t periods. A change in the level or the volatility of interest rates should thus arise from (a) a change in the level or the volatility of the real rate, (b) a change in the level or the volatility of expected inflation, or (c) a change in the impact that unit changes in either or both of these variables have on interest rates.

In fact, it has been difficult to explain fully the behavior of interest rates for two reasons. First, econometric models estimated for a particular sample period have tended to perform poorly outside of the sample period. Second, since both expected inflation and the real rate are unobservable variables, attribution of movements in interest rates to either of these components has depended crucially on the use of proxies to represent their behavior. The latter problem has been highlighted recently, when efforts to slow money growth while actual and expected fiscal deficits are growing have resulted in higher real rules, slowdowns in economic activity, and currency appreciation. Furthermore, contrary to a considerable body of economic theory, it appears that the real rate may be quite responsive over more than the very short run (say, one quarter) to monetary and fiscal policy actions and/or to fluctuations in economic activity.

The possible relationship between the configuration of monetary and fiscal policy and the real rate has had a significant impact on analysis of costs attributed to measures aimed at controlling inflation (see Makin (1982 b)), Based on conventional macroeconomic theory, it would be considered unusual to hear calls for tax increases or cuts in government spending in the midst of a serious recession. Yet such calls have been heard with increasing frequency since the onset of the U.S. recession in the summer of 1981; and, indeed, in the summer of 1982, there was a major tax increase in the United States. The usual arguments for pump-priming measures aimed at ending recessions have become less convincing amid widespread concern that fiscal deficits, both actual and projected, have been responsible for the high real rates that have depressed U.S. expenditure on new plant and equipment, housing, and durables. The possible “crowding-out” of private investment by large government borrowing requirements to finance large fiscal deficits is at the core of the ongoing debate over the effects of a tight monetary and loose fiscal configuration of macroeconomic policy.

Another element to be considered in the investigation of interest rate movements is the role played by taxes. Changes in actual or perceived tax rates on interest income from financial instruments, relative to tax rates on incomes from alternative assets, should affect the relationship between nominal interest rates and real interest rates, given the level of expected inflation. This means that when actual or perceived marginal tax rates change, empirical models estimated over a given sample period may break down outside of that period even when the other variables needed to explain interest rates are identified and accurately predicted.

Empirical investigations of interest rate behavior through the mid-1970s by Tanzi (1980 a) and Levi and Makin (1978) suggest that investors tended to adjust interest rates to insulate, to a large extent, real rates from the effects of expected inflation but not of income taxes. This “fiscal illusion” (Tanzi (1980 a)) may be expected to have disappeared for several reasons. First, the high rates of inflation in recent years would inevitably make the effect of taxes on real rates obvious to most investors. These effects are far less obvious when inflation is low. Second, these tax effects were discussed in several well-known articles, such as those by Darby (1975), Feldstein (1976), and Tanzi (1976), Third, as inflation rates climbed, a combination of “bracket-creep” and higher interest rates tended to enlarge the absolute gap between before-tax and after-tax real interest incomes while simultaneously enhancing the attractiveness of returns on real assets that were subject only to low capital gains tax rates (and only upon realization) or to no tax at all, as is true for many collectibles and antiques (Tanzi (1982 a and 1982 b)), As expected inflation rose, expected after-tax real interest rates quickly became negative when allowance was made for taxation of nominal interest earnings.

Section II of this paper briefly describes a framework for proximate analysis of interest rate behavior. Section III considers in more detail the role of expected inflation. A theoretical framework for analysis of real rates is developed in Section IV. Section V presents results of some empirical tests. Section VI discusses remaining puzzles concerning behavior of interest rates. Section VII presents some concluding remarks and summarizes suggestions for future investigation.


The Fisher equation describing an interest rate, i, in terms of a real rate, r. and expected inflation π is written as1

where the subscript t denotes time. When taxes are considered, the aim is to define as the investor’s objective an after-tax real rate r* written as

where τ denotes the perceived marginal tax rate on nominal interest income. Transposing equation (2) to place it, On the left-hand side, the result is

Equation (3) summarizes the proximate determinants of the interest rate discussed earlier. Movements in i can be decomposed into movements in after-tax real returns, changes in expected inflation, and/or changes in tax rates that alter the impact on i of given changes in r* or π. These three sources of movements in i, together with some modifications that arise from more detailed consideration of determinants of the after-tax real rate, will be explored in turn. It is, however, immediately evident from equation (3) that for a given level of expected inflation, the provisions of the U.S. Economic Recovery Tax Act of 1981 will elevate observed interest rates, since accelerated depreciation allowances should raise the expected after-tax real returns on investment projects. However, the impact of such a rise in r* would be somewhat diminished by reduced marginal individual income tax rates, particularly in higher tax brackets, which would lower τ.

As for interest rate volatility, an expression for the variance of interest rates based on equation (3) is given by

where σπ2 denotes the variance of anticipated inflation, σr*2 the variance of the after-tax real rate, and ρrπ* the coefficient of correlation between r* and π. Notice that when taxes are ignored, as in equation (1), the variance of i is written as

The effects of considering tax rates are evident when equations (4) and (5) are compared. Ignoring the effects of possible correlation between r* and π (ρrπ*=0 for now), the volatility of after-tax real rates unambiguously produces more volatility in i, since for any τ > 0, [1/(1 – τ) ]2 > 1. (Given τ = 0.35, [1/(1 – τ) ] 2 = 2.37.) Given an average tax rate of 35 percent on interest income, a rise of 1 percent in the variance of the after-tax real rate raises the variance of i by 2.37 times the effect of a rise in the variance of the real rate when taxes are ignored, as in equation (5). Furthermore, the higher is τ, the greater will be the volatility of i, ceteris paribus. Of course, this argument assumes that there is no fiscal illusion, so that tax effects are fully recognized by investors.2 Uncertainty over future tax policy can have a powerful impact on the observed volatility of i. equation (4) suggests also that the effects of a rise in the variance of expected inflation on the observed variance of i will be magnified.

It will be seen in the detailed discussion of real rates that theoretical considerations (the Mundell-Tobin effect) suggest a negative correlation between expected inflation and the real rate (ρrπ*<0). Strong evidence for this effect is found in Fama and Gibbons (1982) and Makin (1983). Therefore, the variance of i, with or without taxes, is reduced (though the variance is greater with taxes than without).3

Consideration of the proximate sources of interest rate movements suggests a number of avenues for an explanation of high and volatile interest rates. These include determinants of inflationary expectations, after-tax real rates, and possible effects of changes in actual or perceived marginal tax rates on financial assets and alternative assets.


Survey data on inflationary expectations provide a rich source of information on the outlook regarding the level and stability of the value of nominal contracts whose prices determine interest rates. As already discussed, expected inflation is a major determinant of interest rates. Since expectations about inflation are necessarily predictions, it is also relevant to consider the implications for interest rate behavior of the dispersion of such views about their mean value as well as of the symmetry of such dispersion above and below the mean. It is also useful to consider the variance and skewness of expected inflation.

The variance across expectations of inflation held by survey respondents may either be taken at face value as an index of the dispersion of views on the outlook for inflation or as a measure of uncertainty about inflation. The latter concept relates to the uncertainty attached to the single-valued forecast given by a respondent who is asked simply to describe his expectation regarding some future price level relative to today’s. An individual may give the same response, say πx shown in Figure 1, at two points in time. However, distribution A in Figure 1 represents a forecast given with more certainty, and hence with a higher probability (PA) attached to πx than the forecast represented by distribution B (with probability PB). It should be kept in mind that πx is the most likely outcome for both distributions.

Figure 1.Normal Probability Distributions of Expected Future Prices

Uncertainty about inflation both across survey respondents and by individual investors may be linked as follows. If investors form their own expectations regarding inflation by sampling the outlook of forecasters, as does the Livingston survey of inflationary expectations,4 they will be more uncertain as to the outlook for inflation when they discover an increase in the dispersion of outlooks across forecasters. Based on this reasoning, the variance of the Livingston survey of inflationary expectations is used as a measure of the uncertainty about inflation.

The sign of the impact of changes in inflation uncertainty on observable interest rates is not clear. It operates through an impact on the equilibrium after-tax real rate, which adjusts to equilibrate real savings and investment.5 A rise in uncertainty about inflation will cause risk-averse investors who are contracting to pay money to finance projects to reduce their investments because of the increased uncertainty regarding the real value of contractual payments. Cukierman and Wachtel (1982) have shown that relative price uncertainty tends to increase with inflation uncertainty; such increased uncertainty will, in turn, reduce real investment, which, after all, represents commitments to a given expected set of relative prices. Downward pressure on the real investment schedule will, ceteris paribus, cause the expected after-tax real rate to fall. At the same time, however, on the other side of the market, risk-averse savers who are contracting to receive dollars will, because of increased uncertainty regarding the real value of contractual receipts, reduce the supply of funds at given levels of income and of real balances. This will, ceteris paribus, cause the expected after-tax real rate to rise. The net impact on the expected after-tax real rate of the negative shift in real investment and real savings schedules is uncertain. If the negative impact on investment dominates, greater uncertainty about inflation depresses the equilibrium after-tax real rate and thereby the nominal rate. The reverse holds true if the negative impact of uncertainty about inflation on saving dominates. Results reported below, although statistically weak, suggest that greater uncertainty about inflation tends to depress short-term nominal interest rates, suggesting that the negative impact on investment dominates the negative impact on savings.6

It is also possible that the asymmetry (skewness) of views about the outlook for inflation may affect interest rates. Suppose that the probability distribution for a typical respondent X is as described by A in Figure 2; in other words, the respondent considers πx the most likely outcome, but he also considers that outcomes that imply expectations lower than πx are more likely than outcomes implying expectations higher than πx. Assume that, while the most likely outcome remains πx, the shape of the probability distribution changes from A to B—that is, from negatively skewed to positively skewed, so that outcomes implying inflation higher than πx become more likely than outcomes implying inflation lower than πx. Although the reply that the respondent would give to the pollster would remain unchanged (and equal to πX), his attitude as a lender or borrower would certainly change. As a lender, he would now expect to receive a higher rate of interest to compensate him for the higher risk. As a borrower, he would be willing to pay a higher rate. The net effect would be an increase in the market rate of interest. Therefore, given the average level and variance of expected inflation, the more positively skewed are such expectations, the higher is the rate of interest likely to be.7

Figure 2.Asymmetrical Probability Distributions of Expected Future Prices

Since 1981, two important considerations have been paramount in the minds of sophisticated investors. First, as the money supply was being expanded at a slower pace, the expectation was that the rate of inflation would fall. Second, as a growing fiscal deficit loomed large on the horizon, one significant probability was that the Federal Reserve Board would, at some point, reverse its policy in order to accommodate the fiscal deficit with monetary expansion. Under these circumstances, it seems safe to assume that, for many observers, while the point estimate of inflation (πX in Figure 2) was falling, that fall was accompanied by a positive change in the skewness of the probability distribution describing expected inflation (such as the one represented by the change from A to B in Figure 2). As a consequence, a lower observed π might not be accompanied by as great a fall in the nominal rate of interest as might have been expected. Therefore, this effect would be reflected in a higher real rate. Unfortunately, we do not have data that would permit an empirical verification of this effect, so it will have to be left in the realm of theoretical speculation.8

While there is no information about the skewness of the probability distribution for each of the respondents in the Livingston survey sample, there is information about the variance across survey respondents on expected inflation. Table 1 reports the mean level and standard deviation of inflation forecasts from 1973 through the end of 1981 for the Livingston survey. The December 1981 forecast, of course, covers the first half of 1982. The steady rise of the mean until mid- 1980 is consistent with steady upward pressure on the level of interest rates until that time. After that, and particularly during 1981, the 3.5 percent drop in the mean level of expected inflation clearly suggests that if real (after-tax) rates had remained constant, by December 1981 short-term interest rates should have been between 3.5 percent and 5.5 percent below what they were in December 1980. In fact, the rate on six-month treasury bills fell by about 2.5 percent over that period, implying that after-tax real rates rose considerably during 1981. Overall, the behavior of anticipated inflation suggests that, ceteris paribus, interest rates should have come down during 1981 by more than they actually did. Further, the rise in rates early in 1982 seems to be somewhat at variance with the slowing of actual inflation rates and the appearance of further survey data suggesting a drop in longer-term expected inflation.9

Table 1.Means and Standard Deviation of Six-Month U.S. Consumer Price Index Forecasts
Standard Deviation
Month and YearMeanCross-sectionOver time
June 19734.001.311.34 (June 1973 to December 1975)
December 19735.172.10
June 19747.122.38
December 19747.702.32
June 19755.642.12
December 19755.841.38
June 19765.301.300.66 (June 1976 to December 1978)
December 19765.231.81
June 19775.921.36
December 19775.991.23
June 19786.401.57
December 19786.971.75
June 19798.312.351.46 (June 1979 to December 1981)
December 197910.142.37
June 198010.672.57
December 198010.512.58
June 19818.862.83
December 19816.962.21
Source: Livingston survey data at annual rates.
Source: Livingston survey data at annual rates.

Table 1 also suggests that the volatility of expected inflation over time has been considerably higher since mid-1979 than during the period of comparable length from mid-1976 through the end of 1978. The standard deviation of expected inflation rates during the later period was 1.46 percent, or more than twice the 0.66 percent during the earlier period. This rise may well have contributed to the rise in the volatility of interest rates since 1979. The rise in the volatility of expected inflation likely reflects the rise in the volatility of U.S. money growth rates since October 1979, when new operating procedures were adopted by the Federal Reserve Board.

It is also clear from Table 1 that cross-sectional uncertainty about expected inflation has risen steadily since 1977. This phenomenon, as discussed above, may be linked to a rise or fall in after-tax real rates.


A strict version of the Fisherian relationship between interest rates and inflation assumes that the rate of interest rises pari passu with the rate of inflation. In other words, it assumes that the real rate of interest is constant. This version received a strong boost when a particularly influential study by Fama (1975) failed to reject the joint hypothesis of constancy of the real rate and rationality of inflation forecasts. A later study by Nelson and Schwert (1977) argued that Fama’s test of that joint hypothesis was not sufficiently powerful. After applying more powerful tests, these authors concluded that the data permitted rejection of the constant real rate hypothesis. Mishkin (1981) argued that Fama’s failure to reject constancy of the real rate might alternatively be viewed as an artifact of the sample period he employed (first quarter 1953 through second quarter 1973).

The recent behavior of interest rates is difficult to explain without recourse to the hypothesis that the real rate has fluctuated considerably. A relevant question then becomes how to explain movements in the real rate. Those movements have been substantial, particularly since 1979 (see Chart 1).

Chart 1.Expected Real Return on Three-Month U.S. Treasury Bills, 1960–82

(Quarterly averages less Livingston Survey estimates of expected inflation)

Studies by Levi and Makin (1978), Makin (1982 b), Mishkin (1981), Peek (1982), Tanzi (1980 a), and others have singled out many factors that may cause, at least in the short run, changes in the after-tax real rate of interest. Among these, the following deserve specific mention: (a) expected inflation itself; (b) the stage of the business cycle; (c) unanticipated changes in the fiscal deficit; (d) taxes; (e) unanticipated changes in the money supply; and (f) uncertainty about the level of inflation.

It has been speculated above on ways in which uncertainty about inflation might affect the after-tax real rate. And the role of taxes has already been discussed and the hypothesis has simultaneously been advanced that up to the mid-1970s there was too little tax effect because of a “fiscal illusion,” which has progressively disappeared since then. This disappearance would, of course, be translated into an increase in the impact of changes in expected inflation on nominal interest rates.

Also operating through the measured impact of expected inflation on the nominal interest rate is the well-known Mundell-Tobin effect. Under the Mundell-Tobin effect, the real rate can be affected by changes in expected inflation. A rise in expected inflation causes a shift out of money balances and into real capital, thereby depressing the marginal product of capital and the equilibrium real rate. This is the Tobin effect. Mundell (1963) describes a similar phenomenon whereby a rise in anticipated inflation depresses equilibrium real cash balances, in turn increasing the steady-state level of flow savings owing to the real balance effect. Equilibrium is restored by means of a lower real interest rate, which increases the level of investment until it equals the higher level of savings. This effect, operating as it does on the steady-state rate of saving, is not expected to be subsequently reversed in the absence of a further change in the rate of expected inflation.

The impact of the Mundell-Tobin effect on the relationship between expected inflation and nominal rates of taxes can best be understood with the aid of a structural model that determines the equilibrium value of the after-tax real rate. This approach also helps to clarify the role of uncertainty about inflation, money surprises, and surprise fiscal deficits in determining observable nominal interest rates. The structural model presented here, which extends the model developed in Makin (1983), yields a reduced-form equation for the after-tax real rate. The resulting expression for the after-tax real rate can then be substituted into a Fisher equation describing the observable nominal rate in terms of the after-tax real rate and expected inflation.

The structural equations are expressed in the familiar IS-LM format with some modifications, along with an expression for real income (output) in terms of a distributed lag on money surprises as derived by Blinder and Fischer (1981). The log of non-income-induced expenditure is written as


Equation (6) describes non-income-induced expenditure. As such, it is negatively tied to the after-tax real rate and to uncertainty about inflation and is positively affected by any real exogenous shock to expenditure that is unrelated to other right-hand variables in equation (6). Expected inflation produces a negative impact on investment owing to the depressing impact on corporate profits arising from historical cost depreciation rules noted by Feldstein (1976) and Feldstein and Summers (1978).10GAP is a measure of capacity utilization, or the stage of the business cycle, first employed by Tanzi (1980 a). As GAP rises, so does pressure on existing capacity, signaling a need for more investment. In effect, GAP captures an accelerator effect on investment.

The log of the sum of real saving, taxes, and imports is written as


Equilibrium in the money sector is written as

Equation (8) takes the after-tax nominal interest rate as the opportunity cost of holding money.

The supply side of the model represents real income (output) as


Finally, the Fisher equation is written as equation (3)


Equations (6) through (9) and equation (3) can now be used to solve for rt*. Setting equation (6) equal to equation (7), substituting from equation (8) for real balances and from equation (9) for real output, and substituting the resulting expression for rt* into equation (3) yield a reduced-form equation for the nominal interest rate in terms of a constant term, expected inflation, money surprises, uncertainty about inflation, exogenous demand disturbances, GAP, time, and an error term.11

The coefficients on the interest rate equation (10) are functions of the underlying structural parameters defined in equations (3) and (6) through (9). It is useful to note that the measured impact upon nominal interest of each of the explanatory variables in equation (10) also depends upon the effective marginal tax rate, τ, on interest income. While the underlying structural parameters are unidentified (they cannot be measured based on empirical estimation of equation (10)), this framework is still useful for three reasons. First, it shows clearly why it may be that even with nonzero tax rates applied to interest income, the measured coefficient on expected inflation will be less than [1/(1 - τ)] and may even be less than unity. Second, it clearly shows that the measured impact on interest rates of all explanatory variables is conditional on the tax rate, τ. Since that rate may vary over time, it suggests a reason for changes over time in the fit of many interest rate equations. Finally, the derivation of equation (10) makes clear the theoretical basis for a negative relationship between expected inflation and the expected after-tax real rate.13

The impact of expected inflation on the nominal interest rate reflects a combination of four factors: (1) the Fisher effect, whereby the nominal interest rate rises by the full amount of a rise in expected inflation; (2) the tax effect, whereby the nominal interest rate must rise by more than the rise in expected inflation to maintain a constant expected after-tax real return; (3) the Mundell-Tobin effect, captured in equations (7) and (8), whereby a rise in expected inflation decreases equilibrium real cash balances, in turn increasing the steady-state level of flow savings owing to the real balance effect, with equilibrium being restored by means of a lower after-tax real interest rate, which raises investment to the higher level of savings; and (4) the Feldstein-Summers effect, whereby a rise in anticipated inflation depresses expected after-tax profits and causes investment to fall. In sum, tax effects move the coefficient above unity, while the Mundell effect and the Feldstein-Summers effect both push it below unity. Typical parameter values for τ and λ1 indicate an expected value of 0.75 for the coefficient describing the impact of expected inflation on the nominal interest rate.14 Even though tax effects by themselves tend to push the coefficient above unity, the combined depressing impact of the Mundell effect and the Feldstein-Summers effect may result in a net impact below unity. The general equilibrium approach employed here resolves the apparent “mystery” regarding a less than unitary impact of expected inflation on interest rates when taxes are considered.

The hypothesized negative impact of money surprises on the real rate arises from their positive impact on real income, which, in turn, increases real saving and requires a decrease in the real rate to produce an equilibrating increase in real investment. This effect outweighs the simultaneous upward pressure on the real rate that results from excess demand for real balances associated with increased real income. (See footnote 14.) The effect of money surprises may be contemporaneous, or it may persist over a number of periods owing either to stickiness or, more rigorously, to attempts to restore desired inventory stocks. (See Blinder and Fischer (1981).)

It is important to distinguish between the real income impact of a money surprise described here and an expectations effect like that reported by Mishkin (1981). Mishkin reports a positive relationship between quarterly money surprises and end-of-period short-term interest rates. The result arises, in Mishkin’s view, from a positive impact of a money surprise on expected inflation. In contrast, this study employs period-average short-term rates as a dependent variable in order to capture the real income impact under way during the quarter, before comparison of an actual with an anticipated money supply gives rise to an expectations effect. A fuller discussion of Mishkin’s results and their relationship with the results obtained here is contained in Makin (1982 c). An alternative liquidity rationale for a negative relationship between money surprises and short-term rates is discussed in Makin (1982 b) and Khan (1980).

The impact of uncertainty about inflation on the equilibrium aftertax real rate is ambiguous, as was discussed earlier. The negative impact of uncertainty about inflation on real investment is measured by α2 in equation (6). The negative impact on real saving of uncertainty about inflation is measured by γ3 in equation (7). The ambiguous impact on the interest rate is given by λ3 in equation (10).

The impact of exogenous shifts in aggregate demand on the aftertax real rate is unambiguously positive. If there is an exogenous upward shift in aggregate demand, the after-tax real rate must rise to “crowd out” private investment in order to restore commodity market equilibrium. The model represented by equations (6) through (10) makes it clear that tests of the possible impact of fiscal deficits on interest rates cannot be conducted by inserting a measure of the actual fiscal deficit directly into an interest rate equation. Since tax proceeds rise with income, the built-in portion of deficits is endogenous and typically countercyclical. Interest rates are typically procyclical; therefore, the coefficient on the actual deficit (measured as a positive number) term in the interest rate equation will be downwardly biased and possibly negative.15

Expected future deficits have been identified by some as a source of higher interest rates. However, such effects ought to be confined to long-term rates. Measurement of the impact of expected future deficits on long-term rates is confounded by the cyclical biases just discussed, together with the fact that few actual time series on anything like a comprehensive measure of expected future deficits exist for years before 1980. Some analysts contend that the impact on interest rates of fiscal deficits arises only from their impact on longer-run inflationary expectations. Others suggest that the expected “crowding out” that large fiscal deficits imply for credit markets will raise real rates and thereby raise nominal rates. But for all of these longer-run concepts, measurement presents a serious difficulty.

One way to avoid these difficulties is to test the impact on interest rates of unanticipated movements in the fiscal deficit.16 This approach purges the deficit of its systematic component, which—as noted above—tends to bias downward the deficit’s measured impact on interest rates. Further, given period-average short-term rates as the dependent variable, it is possible, as with money surprises, to capture the impact on interest rates of higher-than-expected sales of government securities during the quarter. This impact should occur before the end of the quarter, when comparison of an actual with an anticipated fiscal deficit may give rise to an expectations effect. More specifically, a surprise increase in the deficit may cause market participants to expect higher money growth and, therefore, higher inflation. But if this expectations effect is already captured in the expected inflation term, the surprise deficit will appear to have no additional explanatory power. The use of a period-average interest rate as a dependent variable, as noted, avoids this problem of apparent redundancy of fiscal deficits in an interest rate equation. It is expected that a surprise deficit will raise the period-average interest rate.

Once a measure is obtained of the impact on interest rates of a surprise intraquarter rise in the fiscal deficit, expressed in terms of basis points per billion dollars per quarter, some idea of the cumulative effect over a year of a rise in predicted deficits can be obtained. The presumption is that if forecast fiscal deficits over a year rise by, say, $100 billion, the instant impact on interest rates is equivalent to the present discounted compound impact of $25 billion per quarter in fiscal deficit surprises over the coming year. Based on estimates to be reported below, a $100 billion rise in the estimated annual deficit distributed as a surprise of $25 billion per quarter over four quarters would raise three-month treasury bill rates by 40 basis points.

The GAP variable, as defined above, is positively related to the interest rate. A rise in GAP or pressure on capacity produces an accelerated effect on investment, which, ceteris paribus, requires a higher expected after-tax real rate to maintain equilibrium (Tanzi (1980 a)).

After consideration of all these factors, it is clear from equation (10) that regression of nominal interest on a constant, a surprise deficit, a money surprise, GAP, a measure of uncertainty about inflation, and expected inflation ought to (a) test the hypothesized positive impact on the after-tax real interest rate of an exogenous shock to aggregate demand (measured by an unanticipated deficit); (b) test the hypothesized negative impact of a money surprise on the after-tax real rate by checking to see if the coefficient on the surprise is significantly less than zero;17 (c) test the hypothesized negative impact of expected inflation on after-tax real interest by checking to see if the coefficient on expected inflation is significantly below [1/(1 – τ)]; (d) measure the net impact of uncertainty about inflation on the after-tax real rate; and (e) test the impact of GAP on the after-tax real rate. Contemporaneous and lagged money surprises may depress the real interest rate insofar as they increase real output above its natural level.


Some empirical tests that attempt to incorporate several of the above hypotheses are reported below. These tests, however, will not be able to capture some of those hypotheses, such as the disappearance of fiscal illusion. As a prelude to those tests, it is, perhaps, useful to report some results of the simplest possible test for the Fisherian relationship, such as that reflected in an equation of the type it = rt + βπt, where all the symbols have the same meaning as above. The results obtained are of some interest.

(1) The worst period for a test of this simple Fisherian relationship insofar as proximity to one of the coefficients on expected inflation is concerned is between the late 1950s and the mid-1970s. For this period, the coefficient of expected inflation β is below 0.70.18

(2) If one keeps, say, 1958 as the initial period and extends the period beyond 1975 to 1981, β rises to well over 0.80.

(3) If one keeps the terminal year at 1981 but moves the initial year beyond 1958, the coefficient of π changes little up to the mid-1960s, but then it starts rising. For the period after 1970, the coefficient of π is significant and substantially exceeds unity, which is consistent with what one would expect from the partial equilibrium framework with taxes.

(4) The results are about the same whether one uses 3-month, 6- month, or 12-month treasury bills.

These findings obtained by estimating the basic Fisher equation suggest that either security markets do not fully reflect changes in anticipated inflation or else significant movements in the after-tax real rate have, in varying degrees over separate subperiods, distorted inferences drawn from estimating the basic Fisher equation.19 In the light of the theory of interest rate behavior developed in Section IV, the latter possibility seems most likely.20

Results of estimating interest rate equations suggested by equation (10) in Section IV are presented in Table 2. Particular attention is given to implications of proper modeling of residuals made possible by the use of transfer function procedures discussed in Box and Jenkins (1970). An attempt is made to check for the possible atrophy since 1979 of fiscal illusion.

Table 2.Interest Rate Equations: Various Time Periods1





Full Period

The dependent variable in all equations is the three-month treasury bill rate. All equations are estimated as transfer functions. Residuals are modeled by an ARMA(1,1) model. (The t-statistics, which are shown in parentheses, are all above 5.0.) White noise test significance levels for the first 24 residuals are, respectively, 0.49, 0.21, 0.11, and 0.26 for equations (2.1) through (2.4).

Anticipated inflation is based on Livingston survey data on six-month expected inflation. Interpolation is employed to obtain a quarterly series.

Money surprises are measured as residuals from an ARMA (0,8) model of money (M1) growth. For a full discussion of this procedure and of alternatives, see Makin (1983).

Surprise deficits are measured as residuals from a univariate time-series model of the government finance deficit measured in billions of U.S. dollars at an annual rate (line 80 of the U.S. country pages in the Fund’s monthly publication, International Financial Statistics).

GAP is calculated from quarterly data on U.S. real gross national product capacity estimated by the U.S. Council of Economic Advisers and actual U.S. real gross national product.

The time variable in equation (10) is captured by the noise model of residuals estimated simultaneously with the coefficients on explanatory variables in the interest rate equation employing the transfer function estimation procedure.

The dependent variable in all equations is the three-month treasury bill rate. All equations are estimated as transfer functions. Residuals are modeled by an ARMA(1,1) model. (The t-statistics, which are shown in parentheses, are all above 5.0.) White noise test significance levels for the first 24 residuals are, respectively, 0.49, 0.21, 0.11, and 0.26 for equations (2.1) through (2.4).

Anticipated inflation is based on Livingston survey data on six-month expected inflation. Interpolation is employed to obtain a quarterly series.

Money surprises are measured as residuals from an ARMA (0,8) model of money (M1) growth. For a full discussion of this procedure and of alternatives, see Makin (1983).

Surprise deficits are measured as residuals from a univariate time-series model of the government finance deficit measured in billions of U.S. dollars at an annual rate (line 80 of the U.S. country pages in the Fund’s monthly publication, International Financial Statistics).

GAP is calculated from quarterly data on U.S. real gross national product capacity estimated by the U.S. Council of Economic Advisers and actual U.S. real gross national product.

The time variable in equation (10) is captured by the noise model of residuals estimated simultaneously with the coefficients on explanatory variables in the interest rate equation employing the transfer function estimation procedure.

The fit of the equation for the full period (equation (2.1) in Table 2) is displayed in Chart 2. Actual and predicted values listed in Table 3 indicate that the model tracks interest rates well within the sample period. Use of the transfer function procedure implies that the goodness of fit reflects explanatory power both of the independent variables employed and of the past history of interest rates.21

Chart 2.Three-Month U.S. Treasury Bill Rate, 1960–82

Table 3.Actual Versus Predicted Values of the Three-Month U.S. Treasury Bill Rate1

Predicted values based on equation (2.1) in Table 2.

Predicted values based on equation (2.1) in Table 2.

Initial estimation of the equations just described resulted in heteroscedastic error terms. A Park-Glejser test strongly supported the hypothesis that error variances grew over time. Therefore, in all results reported here, variables have been divided by the positively trended series on expected inflation to adjust for heteroscedasticity.

A number of conclusions emerge from Table 2. First, the coefficient on expected inflation is in all cases below unity. All of the subsample estimates of that coefficient lie within one standard error of the fullperiod result. Second, the money surprise variable produces the hypothesized negative impact on interest rates.22 The GAP variable produces the hypothesized positive impact on interest rates in a manner consistent with results reported by Tanzi (1980 a).23

The impact on interest rates of an exogenous shock to aggregate demand, as measured by an unanticipated increase in the fiscal deficit, is of particular interest. The model presented in Section IV suggests a positive relationship that would represent the potential “crowding out” of private investment that has been widely discussed in connection with large projected U.S. deficits after passage of the Economic Recovery Tax Act of 1981. Of course, here it is suggested that only deficits differing from those projected subsequently to passage of that legislation will produce an impact on interest rates beyond that embodied in projections made at the time of its passage.

Results reported in Table 2 for all sample periods suggest that a surprise increase in the deficit at an annual rate of $10 billion during a quarter (an actual $2.5 billion surprise during the quarter) would raise the interest rate on three-month treasury bills by four basis points.24 This effect may seem small, but it should be remembered that this is the impact on the short-term bill rate only. Deficit “surprises” of $20 billion to $40 billion a quarter have not been uncommon since 1981, and surprises of this magnitude imply annual rates of $80 billion to $160 billion, which, in turn, would raise the short-term rate by 32 to 64 basis points. The impact on longer-term rates could be larger.

Recall that discussion of the expected coefficient on expected inflation suggested that tax effects tend to push it above unity and that the Mundell-Tobin and Feldstein-Summers effects tend to push it below unity. The reported results below unity are consistent with our hypothetical predicted value of 0.75 based on “reasonable” parameter values.

The significant negative impact on interest rates of the money surprise term is consistent with reported findings in a number of related studies. Levi and Makin (1978) report that output growth decreases interest rates, and Makin (1982 a) reports that surprise money growth increases output growth. Therefore, it is to be expected that both variables would be negatively correlated with interest rates. The findings are also consistent with the hypothesis that surprise money growth produces the liquidity effects discussed in Makin (1982 b) and Khan (1980).

Uncertainty about inflation was not significant in the presence of all of the other variables. This is, of course, theoretically possible, since uncertainty about inflation depresses both saving and investment schedules. Indeed, results reported here suggest that it is not possible to reject the hypothesis that the net impact on the equilibrium aftertax real rate of these shifts is zero.

This result differs from findings reported in Levi and Makin (1978) and Makin (1983) of a significant negative coefficient on uncertainty about inflation. The reason for the different finding reported here may be linked to the presence of the GAP variable, which was not included in the other studies. A rise in GAP produces a significant positive impact on the interest rate. At the same time, it is negatively correlated with the measure of uncertainty about inflation σ2 (ρ = −0.32). The GAP variable is likely proxying for σ2, with a rise in GAP associated with lower inflation uncertainty. No simple economic explanation for this association comes readily to mind, but it does reconcile results reported here with the finding reported elsewhere of a significant negative relationship between interest rates and inflation uncertainty. A rise in GAP may, in addition to its shift impact on investment, proxy for a drop in σ2, which, in turn, is associated with higher interest rates.

Compared with the simple Fisher equation, results discussed at the beginning of this section and reported in Table 2 suggest that the impact of expected inflation on interest has been remarkably stable over a number of subperiods. It appears that the variability of that impact discovered in a number of investigations of the simple Fisher equation is due to time-varying bias on estimates drawn from equations that have omitted significant explanatory variables.


Many aspects of interest rate behavior are still not well understood, particularly the behavior in recent years. Although the inflation and inflationary expectations fell sharply during 1982, both short-term and long-term rates remained high, particularly during the first half of that year. The inconsistency of this experience with what the present model would have predicted is clear from Table 4. Interest rates for 1982 were badly unpredicted by the model when the model parameters (estimated using data for 1960–81) were used jointly with actual values of exogenous variables for 1982.

Table 4.Actual Versus Predicted U.S. Treasury Bill Rate and Velocity Growth Surprises, 1982





One possible explanation for this result may be the very unusual behavior of monetary velocity during 1982. M1 velocity grew at an average annual rate of 3.2 percent from 1950 to 1982, although growth rates usually fell somewhat during contractions. But the drop in actual velocity during 1982 was a remarkable 4.8 percent. Based on a quarterly time series of M1 velocity behavior for 1960–81, growth of velocity during 1982 should have been about 2.95 percent. The unanticipated drop in annual velocity growth over the year was 7.75 percent, distributed over the quarters as shown in Table 3.

If, as some have suggested, including the Federal Reserve Board (see the testimony of Chairman Paul A. Volcker before the U.S. Senate Committee on Banking, Housing, and Urban Affairs on February 16, 198325), the collapse in velocity growth during 1982 represented a large and unpredictable increase in money demand, then excess money demand may have been responsible for the very high interest rates during much of 1982. The drop in interest rates during the last half of the year may have reflected some relief of that excess demand condition resulting from the rapid acceleration of money growth during that period. The annual growth rate of money (M1) was 1.5 percent during the first half of 1982 and 15.1 percent during the second half.

It is tempting to attribute the persistence of high short-term interest rates during much of 1982, even in the face of lower expected inflation, to the unexpectedly sharp drop in velocity that occurred at the same time. However, coincidence does not necessarily imply causality, and it is reasonable to ask if a stable relationship has existed between unanticipated movements in velocity and interest rates over a longer period and in the presence of the other explanatory variables in equation (10). As a matter of fact, unexpected velocity growth has, contrary to expectations, a weak positive impact on interest rates when included as an explanatory variable in equations like those reported in Table 2, either for the 1960–81 or the 1960–82 sample period.26 When interest rates are regressed on anticipated inflation together with velocity shocks, the velocity shocks again have a significant positive impact on nominal interest rates and the explanatory power is comparable to that of equations reported in Table 2.

The inconsistency between the negative association between velocity shocks and interest rates during 1982 and the positive association typical of the 1960–81 sample period can be explained as a manifestation of the identification problem first encountered by Henry Schultz in 1938. As Schultz discovered, where demand is highly volatile relative to supply, price and quantitative measures map out a positive relationship. During 1982, a positive shock to money demand dominated money supply shifts, and the sharp drop in velocity reflected an excess money demand shift which caused interest rates, controlled for the level of expected inflation, to rise. Alternatively, during most of the postwar period, including 1960–81, shocks to money supply have dominated shocks to money demand. As a result, a sharp drop in velocity usually reflects a positive shock to money supply, which, in turn, produces a simultaneous negative impact on interest rates while liquidity effects dominate expectation effects. The result, consistent with findings reported in Table 2 on the effects of unexpected money growth, is a dominant positive association between velocity shocks and interest rates during much of the postwar period.

In short, most of the shocks to monetary equilibrium which have caused short-term rates to move above or below levels implied by changes in expected inflation, at least during the 1960–81 sample period, have been supply shocks. In contrast, 1982 was characterized by dominance of a (positive) demand shock. Such a demand shock prevented short-term interest rates from falling by as much as expected inflation. Once the situation was alleviated, short-term rates fell roughly by as much as expected inflation. The expected real rate on three-month treasury bills fell from 7.09 percent during the second quarter of 1982 to 3.25 percent during the fourth quarter of 1982.


This paper has attempted to demonstrate a need to expand the simple Fisherian view whereby changes in interest rates are explained largely by changes in expected inflation. The need for this expansion became particularly evident during the early 1980s. Our measure of expected inflation dropped from 10.5 percent per annum during the fourth quarter of 1980 to 7.6 percent per annum during the third quarter of 1981. Over the same period, average yields on three-month treasury bills rose from 13.6 percent to 15.1 percent. Some explanation for this apparent discrepancy in terms of the results reported here may be useful.

The failure of interest rates to display a sustained drop during 1981, as the expected rate of inflation fell steadily, resulted from a combination of forces. During the first quarter of 1981, some downward pressure was exerted on interest rates, but a sustained fall was prevented by a rise in economic activity. Our measure of excess capacity (minus GAP) fell from 5.5 percent during the fourth quarter of 1980 to 4.3 percent during the first quarter of 1981. Our estimates suggest that this change alone would add about 25 basis points to short-term rates.

Rates remained high during the second quarter of 1981 owing, among other things, to a shift to unexpectedly tight money. (See Makin (1982 b) for a fuller discussion.) This shift, by itself, raised short-term rates by about 24 basis points, according to our estimates. Unexpectedly tight money persisted into the third quarter of 1981, during which passage of the Economic Recovery Tax Act of 1981 also added to the upward pressure on rates. Short-term rates were 2.61 percentage points above the level predicted by our interest rate equation for the third quarter of 1981, suggesting that some exogenous shock pushed up rates. This was the largest positive residual during the 20 years covered by our sample, and it seems to be attributable to fundamental changes in the outlook for the cyclical pattern of deficits attributable to passage of the Economic Recovery Tax Act of 1981. (This factor is discussed further below.)

In the fourth quarter of 1981, there was a sharp fall in short-term rates. This fall was attributable to, among other things, (a) a large increase in excess capacity (29 basis points); (b) a large positive surprise in money growth (36 basis points); (c) a surprise surplus during that quarter (16 basis points); and (d) a drop in inflationary expectations (48 basis points). The actual fall of 330 basis points was greater than the 129 basis points indicated here, but the discrepancy is considerably reduced by accounting for the effects of variables other than expected inflation. In practice, our noise model, or the past history of the three-month treasury bill rate itself, accounts for all but 69 basis points of the remaining actual fall in rates.

During December 1981 and January 1982, there was a sharp acceleration of money growth accompanied by a rise in interest rates. This would be contrary to our prediction if part of the sharp increase had come as a surprise. It must be remembered, however, that the sharp increase coincided with the first appearance of reports of sharply higher projections of U.S. fiscal deficits totaling $338.7 billion for fiscal years 1982 through 1984. These projections, which were revised upward even further during 1982, suggest another reason for persistently high interest rates during the first half of 1982. If these figures materialize, they may well break the traditional countercyclical pattern historically followed by U.S. fiscal deficits. If the U.S. economy expands during 1982–84, a rise in fiscal deficits will coincide with attempts by the private sector to increase borrowing.

Traditionally, in the expansionary phase of the cycle, there has been a drop in fiscal deficits. The perception of a change in the cyclical pattern of deficits has kept interest rates high since the end of 1981, whereas normally the dramatic drop in inflation and expectations about future inflation would have lowered rates. Viewed in this way, it may be that the surprise increase in money growth during December 1981 and January 1982, in fact, reduced the upward pressure on rates caused by the likelihood of procyclical U.S. fiscal deficits coupled with the then prevailing expectation that the U.S. economy would recover during the second quarter of 1982.27

Finally, it is possible that an atrophy of fiscal illusion resulting in a rise in perceived tax rates employed to calculate real after-tax returns has tended to increase observed pretax market rates. Such a rise in perceived tax rates would magnify the impact on observed pretax interest rates of rises in expected after-tax real rates.

The high volatility of interest rates since 1979 is explained fairly straightforwardly by the sharp rise in the volatility of expected inflation (the variance of π rose from 0.44 during June 1976–December 1978 to 2.13 during June 1979–December 1981) and by the likely increase in volatility of expected after-tax real rates resulting from increased uncertainty about the cyclical pattern of U.S. fiscal deficits and from pressures created by adherence to monetary targets. The increased variance of expected inflation likely reflects, in part, the increased volatility of money growth since 1979. This is disquieting in view of the Federal Reserve System’s stated goal of reducing volatility of money growth under its new operating procedures, but its impact on uncertainty about the outlook for inflation was quite predictable. Persistent success in targeting aggregates ought to reduce sharply the volatility of expected inflation. The net result will be more stable nominal rates, even given some higher level of real rate volatility.

Overall interest rates have remained high since 1980, although progress in lowering expected inflation would normally have brought reductions, because a number of forces have acted to raise after-tax real rates since that time. Interest rates have been highly volatile in response to the increased uncertainty about the outlook for inflation and for expected real rates.

These findings suggest a need for further investigation in a number of areas. Can effects of tax policy on interest rates be considered separately from the effects of other variables? Analysis by Peek (1981) suggests that this may be possible. How significant a factor is the perceived structural change in the cyclical behavior of U.S. fiscal deficits in pushing up real rates? Is there evidence that fiscal illusion has moderated and, if so, by how much? Might it be expected to reappear given a sharp reduction in expected inflation? If higher real rates persist owing to “crowding out” associated with persistently enlarged fiscal deficits, what will be the effects on private savings, on international capital flows, and on private investment? What are the implications of increased uncertainty about expected inflation? Finally, and perhaps most important, what are the implications for the world economy of the unusual behavior exhibited by U.S. interest rates since 1979? More specific questions include possible effects on observed and real exchange rates, effects on worldwide economic growth and capital formation, and implications for the form of the international monetary system.


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The interaction term r, πt is ignored Here, since it is relatively small for the United States.

Thus if it is true that in the past two to three years fiscal illusion has disappeared, equation (4) would go a long way toward explaining the greater volatility of interest rates in this period than in earlier periods.

The effect of a rise in σr*2 on σr2 (where, for convenience σr*2σπ2) is given by


when tax rates are considered and, ignoring taxes, by


Forτ=0.35andρrπ*=0.25, the first of these equations equals 3.55 while the second equals 2.6.

This survey is conducted by the Philadelphia Inquirer under the guidance of Joseph Livingston and compiled semiannually by the Federal Reserve Bank of Philadelphia.

This discussion is drawn from Makin (1983).

Hartman and Makin (1982) employ a utility-maximizing framework in a two-period model to develop an alternative rationale for the proposition that uncertainty about inflation has a negative impact on the nominal rate. The approach yields a definition of the after-tax real rate


which implies


In equation (10′), the coefficient on σt2 is negative even if λ3 is negative and less than unity in absolute value, with the impact of τ2 on savings dominating its impact on investment. In short, though the expectation of a negative impact of σ2 on i is enhanced, it may not be due to the impact on savings versus investment but rather to the alternative definition of the after-tax real rate given by equation (2′).

See also Fama (1976).

Furthermore, the argument is likely to be more important for financial assets of longer maturity than for those of short maturity.

During February 1982, Manufacturer’s Hanover Trust reported a drop since late 1981 of about 2 percent in expected inflation over the coming five years, from the 9 percent range to the 7 percent range.

In the second half of 1982, rates on short-term treasury bills fell to levels broadly consistent with a tax-adjusted Fisher hypothesis.

The depressing impact on corporate profits of actual inflation may be offset by a reduction in the real value of corporate debt only to the extent that actual inflation is unanticipated. But expected inflation will not result in a lower value of corporate debt, since it will be reflected in a higher nominal interest rate demanded by lenders and paid by borrowers in the face of an anticipated depreciation of money against commodities. Expected inflation will, however, depress expected after-tax profits under existing depreciation rules.

It is assumed that money surprises are independent of the error term in equation (10).


λ2 is positive, since γl, the elasticity of real saving plus imports with respect to real income, is unity, given a constant ratio of savings plus taxes and imports to income, while βl, the elasticity of demand for real balances with respect to real income, and λ2 the (elasticity) real balance effect on saving plus imports, are both fractions.

This phenomenon may also arise in open economies where inventory behavior results in inertia of commodity prices. See Criswell (1983).

Parameter values which make i/π=0.75 areα1=0.5;α4=0.2;γ2=0.2;β2=0.5;τ=0.33.

This is confirmed by results reported in Section V. For a thorough discussion of government deficits and aggregate demand, see Feldstein (1982).

Another way could be to use the full-employment budget surplus. This was tried in place of the unanticipated deficit, and the results are reported below. The full-employment budget surplus did not enter significantly into the estimated interest rate equation.

Besides Mishkin’s (1982) expectations effect, some investigators—including Grossman (1981), Engel and Frankel (1982), and Roley (1982)—have found a short-run “policy expectations effect,” whereby a positive difference between a consensus forecast and the announced weekly increase in the money supply causes short-run interest rates to rise. The result is seen to follow from an anticipated tightening of Federal Reserve policy in response to excessive money growth.

This finding, like Mishkin’s, is not inconsistent with a finding that prior to operation of an expectations effect, when an actual money number materializes that can be compared with a forecast, an income or liquidity effect occasioned by money growth above its anticipated path will depress interest rates. The dependent variable in the “policy expectations effect” studies is the change in three-month treasury bill yields from 3:30 p.m. to 5:00 p.m. on Friday afternoon; this is done in order to isolate a pure expectations effect. Detection of an income or liquidity effect in these studies would require regressing the average interest rate from 5:00 p.m on Friday of the previous week to 3:30 p.m. on the current Friday on the current Friday’s money surprise. A negative interest impact via an income or liquidity effect ought to lead the money surprise. These issues are discussed further in Makin (1982 c).

This was a period of accelerating inflation.

They may also indicate that it is not just the rate of inflation but also the change in that rate that may be significant. For example, the Mundell-Tobin effect may be tied to accelerating inflation rather than to the rate of inflation itself.

Summers (1982) has argued, however, that nominal interest rates do not adjust by the full amount implied by the Fisher hypothesis as modified to allow for marginal tax rates on interest earnings. His results, based on both pre- and post-World War II data, arise from equations that employ actual inflation rates in place of expected inflation and that generally do not include variables to control for movements in the expected real rate.

Estimation employing ordinary least squares with adjustment for serial correlation of residuals yields nearly identical results. This is not surprising, since this procedure is nearly equivalent to a transfer function with an (autoregressive) AR (1) noise model and equations in Table 2 are estimated employing an (autoregressive moving average) ARM A (1,1) model.

Money surprises are measured as residuals from an ARMA (0,8) model of money (M1 growth. For a full discussion of this procedure and of alternatives, see Makin (1983).

The time variable in equation (10) is captured by the noise model of residuals estimated simultaneously with the coefficients on explanatory variables in the interest rate equation using the transfer function methodology.

Estimation employing the actual deficit in place of the surprise deficit gives


As noted earlier, the countercyclical deficit combined with the procyclical interest rates tends to bias downward the estimated coefficient on the deficit. The result also suggests an enhanced negative impact of inflation uncertainty σt2 on the interest rate when GAP is excluded from the estimated equation (see below).

Federal Reserve Bulletin (Washington), Vol. 69 (March 1983), pp. 167–74.

Unexpected velocity growth was employed in place of surprise growth of the money supply, since velocity growth is defined as GNP growth less money supply growth and therefore is likely to be highly correlated with money growth.

A sharp drop in interest rates did materialize by August 1982, when it became clear that recovery of the U.S. economy was to be delayed and when monetary policy began to become more accommodative.

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