Pattern-Based Inflation Expectations and the U.S. Real Rate of Interest

Transcription

1 Pattern-Based Inflation Expectations and the U.S. Real Rate of Interest Tobias F. Rötheli* Department of Economics University of Erfurt Nordhäuser Strasse 63 PF D Erfurt Germany Tel: Fax: January 2014 JEL-Classification: E43, D84, E31, C83, C91 Key Words: Real interest rate, behavioral model of inflation expectations, inflation risk premium Abstract: This study applies the concept of pattern-based inflation expectations to the measurement of U.S. real rates of interest. The measure of expected inflation builds on a laboratory-based survey of expectations. We show how our measures of real interest rates differ from measures based on the Michigan survey of inflation expectations. In econometric estimates analyzing the determinants of real interest rates we find no evidence of an effect of the heterogeneity of inflation expectations. However, higher uncertainty regarding expected inflation tends to increase the real rate of interest. We show that this risk premium may account for findings suggesting the non-stationarity of the real rate of interest. * Comments on pattern-based inflation expectations from participants of seminars at the Federal Reserve Bank of Atlanta and the Swiss National Bank and financial support by the German Research Foundation (DFG) are gratefully acknowledged.

2 2 1. Introduction This article applies a new method of modeling inflation expectations to the measurement of the real rate of interest. Following Rötheli (2011) we experimentally elicit inflation expectations in the laboratory by showing subjects an encompassing set of cases of the course of the price level. These cases (patterns) are stylized short time sequences of how the price level evolves in a country like the U.S. The task of subjects is to assess the price level oneyear and five-year ahead given that they only see a visual display of the values of the last four periods (years) of the consumer price index. This approach builds on experimental findings that subjects base their projections mostly on the history of the time series to be forecast (Bloomfield and Hales, 2002, Roos and Schmidt, 2011) and thereby strongly rely on simple patterns in the series (Feldman, 1963, Eggleton, 1982, Rötheli, 1998). In the laboratory-based survey subjects are asked about their point prediction as well as the interquartile range of their predictions. The answers to the former question allow the time-series modeling of expected inflation whereas the latter responses allow a quantification of the uncertainty of inflation expectations. Furthermore, computing expected inflation for each subject leads to a measure of the heterogeneity of inflation expectations. The detailed steps of the lab-based method and the quantification of historical inflation expectations are described in a companion paper (Rötheli, 2013) and will be summarized in the next section. The econometric analysis reported contributes to the literature on the determinants of U.S. real interest rates. Our approach stands in the tradition of studies that use survey measures of inflation expectations (e.g., De Bondt and Bange, 1992, Kaliva, 2008) and complements studies that rely on the returns on nominal and inflation indexed bonds (Söderlind, 2011, Haubrich et al., 2012). The estimates reported here shed light on the question regarding the time-series properties of the real rate of interest. Starting with Rose (1988) empirical

3 3 researchers have debated whether the real interest rate is a stationary variable or not. In a recent contribution using a variety of measures for real rates and a number of statistical tests Norrbin and Smallwood (2011) give a detailed account of the difficulties of deciding whether real rates are mean-reverting or whether they exhibit the unit root property. This study suggests that a risk premium compensating investors for the uncertainty of future inflation may help to resolve the issue of stationarity. 2. The data We begin with the experimentally elicited data. 1 The series for expected inflation used in this study is based on the average of the answers of 45 subjects who are shown 22 different cases of the CPI. The concrete cases shown are the 22 stylized patterns that are sufficient to comprehensively trace CPI data for the U.S. after Subjects answers for each of these patterns are applied to historical data according to the following procedure: In a first step the stylized patterns and the related responses of subjects are scaled to the size of the historical changes. In a second step we compute a similarity measure for each (overlapping) historical 4-year window of CPI data with each of the experimentally shown patterns. In the third step the properly scaled lab-based expectations are weighted according to their similarity with the given historical sequence to generate the expected inflation rate. Finally, this procedure is applied sequentially for each month of the sample period. Note that in these computations of expected inflation we only use data available up to the month just before the given month. Hence, by way of an example, the computed expected inflation as of January 1951 is based on the December values of the CPI from 1950, 1949, 1948, 1947 as well as on the experimentally elicited answers. The described routine is used both for subjects projections 1 All details of the procedure presented are described in Rötheli (2013).

4 4 one year and five years into the future. The resulting one-year and the five-year ahead expected inflation are displayed in figure 1. Figure 1 about here Turning to interest rate measures we rely on the 12-month Treasury constant maturity rate and, for the longer perspective, on the five-year Treasury constant maturity rate. Figure 2 shows these two nominal interest rates for the U.S. The data for these bond yields is available starting in 1962M1. We use end-of-month observations in order to avoid spurious autoregression in the econometric estimates that follow. Subtracting the level of pattern-based inflation expectations (i.e., the inflation premium) from the nominal interest rate gives the exante real rate of interest. The resulting one-year and five-year real rates are shown in figure 3. We find that the long-term real rate is less volatile than the short-term rate. The standard deviation of monthly changes in real rates is for the one-year rate and for the five year rate. For comparison with the new measures of real rates figure 4 shows real interest rates calculated using the University of Michigan survey of inflation expectations for the 12 month and the 5 year horizon. 2 One distinction between the two approaches of computing real rates is evident when comparing the series in figures 3 and 4. Pattern-based real rates can be computed for a much longer period. For the one-year horizon the Michigan data on expected inflation start only in 1978M1 and for the five-year horizon in 1990M4. Furthermore, the Michigan survey-data based rates are more often and more strongly negative than are the pattern-based real rates. This finding holds not only when considering ex-ante rates based on the Michigan survey but also many other measures of ex-ante rates. The series constructed by Dotsey et al. (2003), for example, suggest that real rates were negative over a substantial 2 As is common in the literature we use the series of the median expectations of the Michigan survey data. This is justified with the empirical magnitude of outliers in the polled data.

5 5 portion of the 1970s. Although negative real rates are not a logical impossibility longer spans of negative rate appear implausible. 3 Figures 2 to 4 about here In figure 5 we show the pattern-based measures of uncertainty concerning future inflation for the two horizons considered. These measures are computed based on subjects projections of the upper and the lower bound of the interquartile range. Our measure of the uncertainty of expected inflation for any given period is the average of the individual differences between the upper and the lower bound of expected inflation. The apparent fact that the uncertainty concerning short-term expectations is higher than that regarding the longer-term inflation expectations has to be seen against the background of the definition of this measure: we document the uncertainty of annual inflation over the next five years. If we were to measure uncertainty of inflation over five years we would have to multiply our measure by a factor of five. In figure 6 we show the heterogeneity measures of pattern-based inflation expectations. Our measures of heterogeneity of inflation expectations build on the data of individual subjects and individually estimated series of short- and long-run expected inflation. Specifically, for both horizons we compute 45 series of expected inflation and use the interquartile range as our measure of heterogeneity of inflation expectations. Here, it is noteworthy that the interquartile range of annualized five-year ahead expectations tends to be higher than for the one-year ahead inflation expectations. Figures 5 and 6 about here 3 A scenario where market equilibrium determines a negative ex-ante real rate of interest is the situation with a high risk of collapse in the stock market and in consumption (see DeLong and Magin, 2009).

6 6 3. Estimation of interest rate equations We start our estimation with an encompassing specification taking into account the possible e autocorrelation of real rates as well as effects stemming from the uncertainty ( π, uncer t ) and e, heterr heterogeneity ( π t ) of inflation expectations. A cyclical effect is captured by y t 1 which stands for the lagged growth rate of real GDP. Equation (1) shows the specification with which we begin our estimation. r t = α + e, uncer e, heter 0 + α1 rt 1 + α 2 yt 1 + α 3 π t + α 4 π t ε t (1) For the estimates we have to take into consideration that the measure of uncertainty tends to be correlated with the computed series of expected inflation, that is, the measure of the inflation premium. As a result, using ordinary least squares would lead to a biased estimate of the effect of uncertainty on the real rate of interest. Hence, we have to use an instrumented series for the uncertainty of expectations. Appendix 1 explains the procedure of instrumenting the uncertainty of expected inflation. Table 1 shows the estimates for the one-year real interest rate. The top line gives the results for the encompassing specification of equation (1). The second line shows the specification which results after deleting explanatory variables whose inclusion is not justified according to Wald-tests. For the one-year rate this means deleting the heterogeneity measure. Lines 3 and 4 give results of the parsimonious specification that cover roughly equal portions of the entire sample period from 1962 to 2012 (that is from 1962 to 1987 and from 1988 to 2012). Table 2 offers corresponding findings for the five-year real interest rate. Tables 1 and 2 about here The results for the one-year rate indicate that the uncertainty of expected inflation has a significantly positive effect on the real rate of interest. This indicates that for the case of

7 7 short-run interest rate the presence of a positive inflation risk premium can be documented. The two estimates for the subsamples indicate that the role of this risk premium might have diminished over time. However, a Chow test does not reject the hypothesis (p-value of 0.49) that the estimated coefficients are stable over the two sub-periods. The same holds for the estimates concerning the five-year real rate (p-value of 0.89). The estimates for short- and long-run real rates replicate the procyclical effect of real output documented by Dotsey et al. (2003). The heterogeneity of expectations appears to have no significant effect on real interest rates. The documented result concerning the risk premia in real interest rates is in line with results reported by Söderlind (2011). However, I find a significant effect of the uncertainty of future inflation only for the one-year interest rate and not for five-year interest rate. For the shortterm rate the reported effect of inflation risk holds promise to explain the findings of nonstationarity of the real rate documented in the literature. We assess whether a risk premium for uncertainty of future inflation can explain the time-series property of the real rate by testing for stationarity the real one-year rate ( r ) as well as the one-year rate net of the effect of the uncertainty of inflation expectations ( r netofrisk ). For the calculation of the adjusted rate we use a version of (1) without the lagged term r t 1. This is justified on the conjecture that a change in inflation risk should have an immediate effect on the interest rate. The estimated coefficient for the uncertainty term is instead of the of table 1. 4 Hence, the adjusted real rate is netofrisk r t rt = π. (2) e, uncer t 4 The average of our measure of risk compensation for the uncertainty of future inflation for the period 1962 to 2012 is (standard deviation 0.005). This suggests a sizable effect of inflation uncertainty on the interest rate.

8 8 Table 3 shows the results of various tests applied to the measure of the short-run real rate (in column 2) and to the real rate net of the inflation risk premium (in column 3). Going through the results we note that in 4 out of 6 cases the test statistic indicating stationarity is higher for netofrisk r than it is for r. In 5 out of 6 cases the test statistic of the adjusted rate signals stationarity at least at the 5 percent level of significance whereas for the unadjusted rate only 3 out of 6 statistics indicate significance at this level. Hence, the risk premium compensating holders of bonds for the uncertainty of future inflation helps to explain the findings of nonstationarity of the real rate of interest. 5 Table 3 about here 4. Conclusions This article applies the concept of pattern-based inflation expectations to the measurement of ex-ante U.S. real interest rates. We show that our measures of real rates of interest differ from measures based on the Michigan survey of expected inflation. Notably, our measures are less often negative and can be calculated farther back in history. Concerning the determinants of the real rate and its time-series properties we find the following: First, the procyclical effect of real output is verified for both the one-year and the five-year ahead real rate of interest. Second, we identify a significantly positive effect of the uncertainty of inflation expectations for the one-year rate. Analyzing the real interest rate net of this risk premium suggests that the market adjustment for inflation uncertainty could be the reason for results indicating a nonstationarity of the real rate. Finally, we find no sign of an effect of the heterogeneity of inflation expectations on real interest rates. 5 Obviously, given that the uncertainty measure does not help explain the course of the longer-term interest rate, here we cannot provide an explanation for the downward trend in the real five-year interest rate apparent in recent years.

9 9 References Bloomfield, Robert and Jeffrey Hales (2002), Predicting the Next Step of a Random Walk: Experimental Evidence of Regime-Shifting Beliefs, Journal of Financial Economics, 65, De Bondt Werner F. M and Mary M. Bange (1992), Inflation Forecast Errors and Time Variation in Term Premia, Journal of Financial and Quantitative Analysis, 27 (4), DeLong, J. Bradford and Konstantin Magin (2009), The U.S. Equity Return Premium: Past, Present, and Future, Journal of Economic Perspectives, 23, (1), Dotsey, Michael, Carl Lantz, and Brian Scholl (2003), The Behavior of the Real Rate of Interest, Journal of Money, Credit, and Banking, 35 (1), Eggleton, Ian R. C. (1982), Intuitive Time-Series Extrapolation, Journal of Accounting Research, 20, Feldman, Julian, (1963), Simulation of Behavior in the Binary Choice Experiment. In E. A. Feigenbaum, J. Feldman (Ed.), Computers and Thought. (McGraw-Hill, New York) pp Haubrich, Joseph, Pennacchi, George and Peter Ritchken (2012) Inflation Expectations, Real Rates, and Risk Premia: Evidence from Inflation Swaps, Review of Financial Studies, 25 (5), Kaliva, Kasimir (2008), The Fisher Effect, Survey Data and Time-Varying Volatility, Empirical Economics, 35 (1), Norrbin, Onsurang and Aaron D. Smallwood (2011), Mean Reversion in the Real Interest Rate and the Effects of Calculating Expected Inflation, Southern Economic Journal, 78 (1), Roos, Michael W. and Ulrich Schmidt (2011), The Importance of Time-Series Extrapolation for Macroeconomic Expectation, German Economic Review, 13 (2), Rose, Andrew K. (1988), Is the Real Interest Rate Stable? Journal of Finance, 43 (5), Rötheli Tobias F., 2011, Pattern-Based Expectations: International Experimental Evidence and Applications in Financial Economics, Review of Economics and Statistics, 93 (4), Rötheli Tobias F., 2013, Pattern-Based Inflation Expectations, SSRN Working Paper: Söderlind, Paul, 2011, Inflation Risk Premia and Survey Evidence on Macroeconomic Uncertainty, International Journal of Central Banking, 7 (2),

10 10 Appendix 1: Instrumenting the measure of the uncertainty of expected inflation The instrumentation of the uncertainty measure is necessary because the uncertainty measure and expected inflation (i.e., the inflation premium which is part of the left-hand variable in our real interest rate estimates) are contemporaneously correlated. The negative correlation between these variables comes from the following effect: a decline in the inflation rate (often concurrent with a break in a trending pattern of the CPI) leads to a decline in the expected inflation rate but also leads to an increase in inflation uncertainty. The experimental data clearly indicate a negative correlation between the expected change in the series and the uncertainty measure (i.e., the interquartile range). As a result, a regression relating patternbased inflation expectations to a constant and the uncertainty measure yields a negative and significant coefficient (standard error ). In an OLS estimate for the real interest rate this negative effect would bias the result in the direction of a positive effect of uncertainty on the real rate. In fact, even in the absence of any factual effect of uncertainty on the nominal rate this correlation would lead to an OLS-estimate of a positive effect of on the real rate. Clearly, this would just be a statistical artifact. We can avoid this estimation bias by noting that the observed correlation between the value of inflation expectations and uncertainty is due to fact that the computation of both the expected inflation and its uncertainty for any given period (month) is based on the observed value of the CPI in the previous period. We circumvent this problem by instrumenting the CPI value for the calculation of the latter variable with the fitted CPI-values from the following NKPCtype estimated inflation equation where P and Ydev stand for the CPI and for the deviation of t t-1 ln t-2 log real GDP from a linear trend ln( P ) = ln( P ) ( P ) e ( P ) ln( P ) ln( P ) ( Ydev ) ln t-3 t-12 t-1 ln t-1 with an 2 R of 0.46 and a D.W. of When we base the computations of the uncertainty of expected

11 11 inflation on this instrumented CPI-series we find that the simultaneity issue is resolved. In a regression relating pattern-based inflation expectations to a constant and this instrumented measure of uncertainty of expectations we find a positive and non-significant coefficient of (standard error ).

12 12 Figure 1: Annualized pattern-based expected 1-year and 5-year ahead inflation expectations 0.10 PB-Expected inflation over the next year PB-Expected anual inflation over the next 5 years Figure 2: Nominal interest rates 0.18 One-year tbill rate Five-year tbill rate

13 13 Figure 3: Real interest rates based on pattern-based inflation expectations 0.14 One-year real tbill rate Five-year real tbill rate Figure 4: Real interest rates based on the Michigan survey of inflation expectations Michigan survey one-year real tbill rate Michigan survey five-year real tbill rate

14 14 Figure 5: PB-uncertainty measures for 1-year and 5-year ahead inflation expectations 0.12 Uncertainty of one-year ahead inflation expectations Uncertainty of five-year ahead inflation expectations Figure 6: PB measures of heterogeneity of 1-year and 5-year ahead inflation expectations 0.08 Heterogeneity of one-year inflation expectations Heterogeneity of annualized five-year inflation expectations

15 15 Table 1: Regression results for the one-year real interest rate Sample period α 0 α 1 α 2 α 3 α 2 4 R D.W. Std. Err * * ** ** (0.0003) (0.0690) (0.0825) (0.1125) (0.1475) * * ** ** (0.0003) (0.0694) (0.0835) (0.1040) ** ** (0.0006) (0.0827) (0.1171) (0.1624) ** * (0.0003) (0.0591) (0.0865) (0.1251) **, * indicate significance at the 1% and 5% level of significance, respectively. Numbers in parentheses are standard errors of estimated coefficients. Table 2: Regression results for the five-year real interest rate Sample period α 0 α 1 α 2 α 3 α 2 4 R D.W. Std. Err * ** (0.0002) (0.0547) (0.0563) (0.1777) (0.1794) * ** (0.0002) (0.0550) (0.0559) * (0.0004) (0.0743) (0.0756) * * * (0.0002) (0.0576) (0.0783) **, * indicate significance at the 1% and 5% level of significance, respectively. Numbers in parentheses are standard errors of estimated coefficients.

16 16 Table 3: Test for stationarity for the one-year real interest rate and the inflation-risk compensated one-year real interest rate Test Variable 1-year real rate 1-year real rate adjusted for inflation risk Augmented Dickey-Fuller * Dickey-Fuller GLS ** Phillips-Perron Kwiatkowski-Phillips- Schmidt-Shin ** ** Elliott-Rothenberg- Stock ** * Ng Perron * * **, * indicate significance at the 1% and 5% level of significance, respectively.