Forecasting the real interest rate

Forecasting the Real Interest Rate

DONNA J. FLETCHER and 0. DAVID GULLEY

ABSTRACT A method first employed by Mishkin to forecast the ex ante real interest rate involves regressing the ex post real rate on various explanatory variables. The fitted values from this regression are taken to be the ex ante real interest rate series. We argue that this method does not reproduce how forecasters actually generate ex ante real rates because the method generates in-sample predictions and uses information unavailable to forecasters at the time of the forecast. Our paper develops several regression-based forecasting models to generate alternative ex ante real rates using only information available at the time of the forecast. Unlike Mishkin, we update our model with new information as time passes and generate out-of-sample forecasts. We show that the parameters of the model change over time, yielding results different from Mishkin. While Mishkin’s technique is found to yield better forecast diagnostic statistics, we contend that our models more closely approximate how forecasters operate. If so, then our ex ante interest rates will be closer to the true, unobservable ex ante rates. Finally, as an application of our technique, we consider the debate between the real business cycle theorists and the monetarist school on the effects of money on output. Recent studies of this question have argued that money growth may be a poor proxy of monetary policy. Accordingly, we use two other measures of monetary policy more consistent with the actual operating procedures of the Federal Reserve. Our results lend some support to the monetary models.

INTRODUCTION

Given its importance in allocating resources over time and across sectors of the economy, the ex ante real rate has been studied extensively by economists. In particular, much work has been done to generate measures of the ex ante real rate. There are three methods to obtain a measure of this rate. The first method is to survey forecasters and obtain their expectations of the price level/inflation rate over the relevant time horizon. The ex ante real rate is then calculated using the Fisher equation. Mishkin (198 1) and others have pointed out that there are doubts about the quality/rationality of these forecasts because forecast errors are often serially correlated, correlated across forecasters, or correlated with available information, such as money supply growth. Keane and Runkle (1990) re-examine this issue and find that once various factors have been accounted for, price forecasts of the ASA-NBER forecasters are indeed rational. Darin and Hetzel (1995) examine four different survey forecasts of inflation and conclude that because the forecasts “display the same broad patterns,” they are useful in calculating ex ante real interest rates. In any event, we choose to focus on the third method described below because it is commonly used in the literature.

Donna J. Fletcher l Department of Finance, Bentley College, Waltham, MA 02154. e-mail: [email protected]. 0. David Gulley l Department of Economics, Bentley College, Waltham, MA 02154. e-mail: [email protected].

North American Journal of Economics & Finance 7(l): 55-76 Copyright 0 1996 by JAI Press Inc. ISSN 1062-9408 All rights of reproduction in any form reserved

56 FLETCHER and GULLEY

Th& second method uses changes in securities prices to calculate the expected inflation

rate. The ex ante real interest rate is then recovered from these predictions, again using the

Fisher equation. Since this method requires that changes in term premia be properly accounted for, recovering measures of expected inflation has been problematic. Using a

novel methodology, Kandel, Ofer, and Sarig (1996) employ Israeli nominal and index

bonds to construct ex ante real interest rates. Svensson (1994) discusses how forward rates

can be used to calculate expected inflation rates, which in turn can be used to construct ex

ante real interest rates. Finally, Mishkin (198 1, 1984) and Huizinga and Mishkin (1986) argue that the ex ante

real interest rate can be generated by regressing the ex post real rate on various explanatory

variables. The series of fitted values from this regression is taken to be the ex ante real rate

series. We argue that such a method does not reproduce how forecasters use regression

models to generate an ex ante real rate because the technique uses information unavailable

to forecasters at the time the prediction is made. In other words, Mishkin’s method yields

in-sample predictions of the real interest rate. A true ex ante real interest rate is generated

with an out-of-sample prediction. Our paper develops three regression-based forecasting models to generate ex ante real

rates using only information available at the time of the forecast. Like Mishkin, we

regress the ex post real rate on various explanatory variables. Unlike Mishkin, we update

our model with new information as time passes using a least-squares learning framework.

We believe that updating more accurately reflects how forecasts are constructed. We

show that the parameters of the model change over time, yielding results different from

Mishkin. While Mishkin’s technique is found to yield better forecast diagnostic statistics, we con-

tend that our models more closely approximate how forecasters operate. If so, then our ex

ante interest rates will be closer to the true, unobservable ex ante rates. As an application of our forecasting model, we examine the explanatory power of

monetary policy on the expected real interest rate. Recent studies in this area have

revealed that money growth may be a poor indicator of monetary policy. Thus, follow-

ing suggestions made by Diba and Oh (1991) and Eichenbaum and Evans (1995), we

use the spread between the federal funds rate and a long term interest rate and the ratio

of nonborrowed to total reserves, respectively, as alternative measures of monetary pol-

icy. Our results lend some support to the notion that money does affect the real interest

rate, and therefore, output.

MISHKIN’S METHODOLOGY

The basis for Mishkin’s methodology is straight-forward and follows Mishkin (1984). For a one-period asset, the Fisher equation requires that in equilibrium:

eprr, = i, -pr = rrt - e, (1)

where eprrt is the ex post real rate of interest from t-l to t, i, is the continuously compounded nominal interest rate, J+ is the actual inflation rate, rr, is the ex ante real interest rate, and e, is the inflation forecast error. The assumption of rational expectations (i.e., that

the difference between the actual and predicted value of eprr, is a serially uncorrelated random error term) implies that

Forecasting the Real Interest Rate 57

rrI = E(eprr,IZ_,) (2)

where Zr_l represents all information available to forecasters at the time of the forecast.

Equation (2) indicates that the ex ante real rate is the expected value of the ex post real rate

conditional on available information. Let Xr_l represent the observable elements of It_]. The linear projection of rrt on X,-l is:

P(rr,lX,_l) = X,_,B

Hence:

rrI = X,_,B + ut (4)

While Equation (4) cannot be estimated, it can be substituted into Equation (1) yielding:

eprr, = X,_,B + ut - e, (5)

Equation (5) can be estimated. The estimate of B, Bols, can then be used to calculate the

ex ante real interest rate:

rrl = X B I-, ols (6)

Mishkin notes that although B,l, is a consistent estimator of B, Bols will have higher stan-

dard errors than B (the variance is increased because the ex post real rate is composed of the

ex ante rate plus the negative of the inflation forecast error). He also points out that the

variables in Xr_l need not cause rrt and that rational expectations imply that the regression

residuals should be white noise. These and other econometric concerns are taken into con-

sideration in our model discussed below.

LIMITATIONS OF MISHKIN’S METHODOLOGY

Mishkin’s technique has become widely used to generate ex ante real interest rates. Mish-

kin (1981, 1984, 1988), Huizinga and Mishkin (1986), Walsh (1988) and Bonser-Neal

(1990) are but some of the papers that employ this methodology. The goal of our research

is to extend this technique to allow for updating of the information set and learning on the

part of forecasters. We believe this extension is important for several reasons. First, we argue that the aforementioned implementations of Mishkin’s estimation tech-

nique are not in the true spirit of forecasting, since forecasters can only use information

that is available to them at the time of the forecast. For example, suppose a researcher is

using monthly data and is interested in the sample period 1959: 1 to 1992: 12. Equation (5)

is estimated over the full sample period and rr, for, say, 1990: 1 is generated using Equa- tion (6). Note that rrt is calculated with information unavailable to forecasters at 1989: 12,

the time of a one period ahead forecast. Specifically, information from 199O:l to 1992:12

is employed. This gives the researcher (who is looking backward) an advantage over the forecaster (who is looking forward).’ True forecasting requires out-of-sample prediction.

As previously implemented, Mishkin’s technique yields in-sample predictions.*


Second, Mishkin’s technique does not allow learning to occur over time. For example, a researcher looking backward would certainly attempt to model the impact of the OPEC

shock onreal interest rates. This is because the researcher knows when the shock began, its magnitude, and (thanks to other researchers) how the shock may have affected various

macroeconomic variables. However, the forecaster, who possibly has never dealt with such a situation, has none of this information. The forecaster must use whatever information is

available to determine if and how such a shock will affect the real interest rate. Only then

may some energy-related variable be included in the forecasting model. Thus, the compo- sition of the forecaster’s Xr_r matrix can change over time as the forecaster receives new information. Hence, the forecaster’s predictions may be significantly different from those

of the researcher.3

EXTENSION OF MISHKIN’S METHODOLOGY

For the reasons explained above, we contend that Mishkin’s technique should be modified to yield an ex ante, out-of-sample measure of the real interest rate. The modification

involves the use of several least-squares learning frameworks, which update the information set on a period-by-period basis. New coefficients are estimated each period, and these are used to generate a one-period ahead forecast of the real rate.

While this type of updating technique has not been implemented with real interest

rates,” other researchers have applied updating techniques to problems involving

expected inflation. Caskey (1985) examines the Livingston inflation forecasts using a

Bayesian-updating technique implemented with a Kalman filter. He finds that the weight placed on past inflation and money supply growth by the Livingston panel increases over time. For example, the weight placed on lagged money growth increased dramatically

following the October 1979 change in monetary policy. The changes in the weights are

reflected in the Bols vector. He argues that the reweighting occurred because forecasters

were updating their models to account for the new information concerning expected inflation and that this learning process (and not the exclusion of relevant information or the

misuse of information) accounts for the observed serially correlated forecast errors of the

Livingston panel. Holland (1993) also uses a Bayesian updating process to generate a measure of expected

CPI inflation. Using semi-annual data (to coincide with the Livingston forecasts), he

includes a constant, two lagged values of money growth, and six lagged values of inflation. Like Caskey (1985), Holland finds that the coefficients on lagged money growth and inflation increase over time.5 Moreover, examination of his results indicates that the regression- based measure of expected inflation tends to underpredict inflation to a lesser extent than

does the Livingston forecast during the mid-1970s. Therefore, the regression-based measure of the ex ante real rate will not be as persistently negative as the real rate generated by the Livingston forecasts.

Barro and Sala-i-Martin (1990) calculate expected inflation for various OECD countries (including the United States) with an ARMA model that is updated each period. The ex ante real interest rate for each country is computed from the inflation forecast and then combined into a weighted world real interest rate. This rate is found to be related to its own lagged value, and lagged values of the relative price of oil, stock returns, inflation, money growth, the ratio of investment to GDP, and fiscal policy indicators. The authors reach a similar conclusion with their estimation of the expected inflation rate for the United States.


They find that a continuously updated ARMA model yields inflation forecasts that are less

likely to generate negative real interest rates6 We argue, then, that the nonupdating technique of Mishkin departs from the spirit of

forecasting and does not allow for learning on the part of forecasters, either in the compo- sition of the Xr_l matrix or in the values of B,lS.7

Caskey (1985), Barro and Sala-i-Martin (1990), and Holland (1993) help provide the basis for the use of a least-squares learning framework to generate a direct measure of the

ex ante real interest rate. To use this framework, we must decide on the updating technique(s) to employ and the variables to include in the regressions. We employ three differ-

ent updating methods. First, following Holland (1993), we use a regression model where observations are added each period, the regression re-estimated, and the prediction made.

Second, we use a rolling regression model that deletes one observation from the beginning of the information set as one new observation is added.8 Compared to the first updating

method, this second one places more weight on more recent observations. Finally, in the

spirit of Barro and Sala-i-Martin, we use an ARMA model of the ex post real rate that, like the first model above, is updated each period.’

It is important to emphasize that if the coefficients of Equation (5) are stable over time,

the real rates generated by Mishkin’s technique and ours will be the same. However, there is considerable evidence to suggest that the coefficients are not stable. Huizinga and Mish-

kin (1986) find that the Fed’s changes in operating policy in 1979 and 1982 had a significant effect on the real rate and the coefficients in the estimating equation.” Thus, they

estimate separate regressions across these breakpoints. Of course, from a researcher’s point of view, this is the correct procedure. But such a technique may not capture how the fore-

caster is adjusting to these (and other) shifts. Our updating methods attempt to capture this

learning process. Further, as reported above, Mullineaux (1980), Caskey (1985), and Hol- land (1993) all find (using different estimation techniques and time periods) that the coef-

ficients used to generate an ex ante inflation rate change over time. It is reasonable to hypothesize that this finding can be applied to the coefficients used to generate ex ante real interest rates.

DATA AND SAMPLE PERIODS

Before proceeding with the empirical analysis, a description of the sample periods and data

is in order. Following Mishkin and others, we use monthly data. Our sample period is from 1959: 1 to 1992: 12. We assume that the forecaster begins making out-of-sample forecasts for 1970:1, so that Xr_l initially contains information up through 1969:12. We begin with

1970 because the 1970s and 1980s saw the largest fluctuations in inflation, nominal interest

rates, and ex post real interest rates. As time passes, Xt_, is updated with additional obser-

vations and a new forecast is made. The dependent variable is the ex post real interest rate, eprrt, which is calculated by com-

puting eprr, = i, - p,, where i, is the continuously compounded nominal interest rate, and p,

is the continuously compounded inflation rate. The end-of-month one- and three-month Treasury bill rates are used for the nominal interest rate. The inflation rate is calculated using the price indexes described below. For example, to calculate eprr, for March, the annualized percentage change in the price index from February to March is subtracted from the annualized one-month T-bill rate as observed on February 28. To calculate eprr, for March with the three-month Treasury Bill, the annualized percentage change in the price index from

FLETCHER and GULLEY

February to May is subtracted from the annualized three-month T-bill rate as observed on February 28. The timing of the variables follows that of Mishkin (1990, 1992). l1

We use four price indexes to calculate the inflation rate, which of course yields four

alternative real rates. The appropriate real rate is a function of the question at hand. For example, if one wishes to find the relevant real interest rate for the household sector, the

CPI is a reasonable choice to use for measuring the inflation rate. If one wishes to find the real rate for the agricultural sector, then the relevant price index may be the farm products

component of the PPI, as in Mishkin (1988). We employ these two price indexes, along with the PPI for the manufacturing sector (to measure the real rate for manufacturing production) and the Commodity Research Bureau (CRB) index for commodity prices.

As previously mentioned, variables in Xr_l need not cause rrt, yet the choice of the variables included in Xr_l by researchers is always specified with the idea of causation in mind.

For example, theory predicts that the supply and demand for loanable funds, monetary and

fiscal policy, supply shocks, and other factors will affect the real interest rate. Yet, Mishkin (1981, 1984) and Huizinga and Mishkin (1986) employ variables such as real GNP growth, industrial production, the unemployment rate, investment, and the federal budget deficit

with little success. Moreover, Mishkin (198 1) finds that lagged inflation is statistically sig-

nificant, but lagged money growth is not. In Mishkin (1984), the Xr_l matrix includes the nominal interest rate, lagged inflation and money growth, as well as a polynomial in time.

Lagged inflation and the polynomial (used to proxy for missing variables) are found to be

significantly related to the real rate. Lagged money growth yields mixed results. Huizinga and Mishkin (1986) use the nominal interest rate, lagged inflation, and lagged values of the

relative price of energy and find that these are statistically significant. Using three-month time periods, Bonser-Neal (1990) also employs the current nominal interest rate, lagged

inflation, and lagged relative price of energy. Diba and Oh (1991) specify the real rate as an

AR process and find mixed results concerning the impact of money growth on the real rate. Based on these findings, our general forecasting model in 1970 includes the nominal

interest rate, lagged inflation, lagged money supply growth, and a supply shock variable in Xt_]. Since the first oil shock did not occur until 1973, forecasters in 1970 would likely not

include a supply shock variable, such as the relative price of energy. However, we include such a variable so as to pinpoint the months in which it has an influence on the real interest

rate.‘* We use the relative price of energy as our supply shock variable (following Huiz- inga and Mishkin 1986) calculated as the log of the ratio of the PPI fuel component to the

overall PPI. ’ 3 As mentioned above, it is important to include only information known to forecasters at

the time of the forecast in Xr_]. For example, using monthly data, it might seem reasonable to include, say, February’s inflation rate to forecast March’s ex ante real interest rate. How-

ever, since February’s price indexes are not reported until mid-March, February’s inflation rate is not known to the forecaster. So, either a forecast of February’s inflation rate or Jan- uary’s actual inflation rate should be included in X*-l. For simplicity, we include the inflation rate lagged two periods. We follow a similar procedure with all relevant variables, which insures that only information known at the time of the forecast is exploited.14

Table 1 provides summary statistics. As can be seen, the mean values of the ex post rates are very similar. The most striking feature of the table is the difference in behavior between CPI inflation rates and the other inflation rates: CPI inflation varies little compared to inflation rates in the farm, manufacturing, and commodity sectors of the economy. Items that comprise the CPI (household goods) often tend to change prices less frequently than do

Forecasting the Real interest Rate 61

TABLE 1. Summary Statistics

Variable

TBILLl

TBILL3

CPIINFl

FRMINFl

MFGINFl

CRBINFl

CPIINF3

FRMINF3

MFGINF3

CRBINF3

CPIEPRR 1

FRMEPRRl

MFGEPRR 1

CRBEPRR 1

CPIEPRR3

FRMEPRR3

MFGEPRR3

CRBEPRR3

SHOCK

Ml

SPREAD

NBRRATIO

Mean

5.82953759

SE Min. Max.

2.74936556 1.430388 15.79057

6.29225671 2.86621697

4.39180429 3.5022403

2.73922326 29.91738358

4.52186174 21.6914229

2.49729995 25.68452109

4.40375579 2.9925693 1

2.04965896 13.65049995

3.36020502 12.39047509

1.90538425 14.07489046

1.4377333 3.06266987

3.09031433 30.2030856

1.30767585 21.43634473

3.33223764 26.02462245

1.88850093 2.60180544

4.24259775 14.22541673

2.93205 169 12.14404549

4.38687246 14.72981891

-0.54135648 0.30448464

5.88380263 18.79926352

0.65669118 1.77214638

0.95736052

2.134843

-5.508809

-83.25112785

-113.11119661

-80.04781425

-3.793825

-46.61443472

-65.15879413

-36.32466259

-11.012298

-240.87532573

-91.32092364

-151.45087961

-4.983296

-60.34241978

-55.78352125

-69.33815155

-0.90946477

-5 1.14575569

-6.51

16.0278

19.30421

249.16723773

97.80486664

159.74279161

14.57849

67.36950478

63.44556825

76.36523655

12.202436

90.77746385

119.80482361

87.54503425

10.913501

55.16572372

72.27847013

46.69720789

0.04000533

53.48123745

3.85

0.69237558 0.99856855

Notes: All variables (except NBRRATIO) are seasonally unadjusted and (with the exception of SHOCK and NBRRATIO)

expressed in continuously compunded annual percentages. TBILL is the Treasury Bill rate. “1” and “3” indicate one- and

three-month annualized rates. CPI is the CPI-U, MFG is the PPI for manufacturers. FRM is the farm component of the

PPI, and CRB spot index of commodities. INF is the rate of inflation. EPRR is the ex post real interest rate, which is cal-

culated using the Fisher equation. SHOCK is the logged ratio of the fuel component of the PPI to the overall PPI. Ml is

an annualized growth rate. SPREAD is the difference between the IO-year T-note and the federal funds rate. NBRRATIO

is the ratio of nonborrowed to total reserves. The sample period is 1959: l-1992: 12. Frederic Mishkin provided the data

for TBILLl, TBILL3, CPIINFl, and CPIINF3 through 1990. TBILL data for 1991-1992 are from the CRSP tapes. All

other variables arc from the Citibank Data Tapes:

Variable Citibank Label

PPI PW

Fuel component of the PPI PWFUEL

PPI for manufacturing PW2410

CRB Index PSCCOM

Farm products in PPI PWFA

CPI-u PZUNEW

Ml Money Supply FZMl

IO-Year T-Note FYGTlO

Federal Funds Rate FYFF

Nonborrowed reserves FMRNBA

Total reserves FMRRA


items that comprise the other indexes, which are made up mainly of goods whose prices are determined on a daily basis in markets. These differing adjustment processes may have implications for how real interest rates calculated with these indexes behave.

There is some evidence for this supposition. As will be discussed further below, Huiz- inga and Mishkin (1986) specify a very different model of the ex post real rate using CPI data from that of Mishkin (1988), who uses the farm component of the PPI to measure inflation in the farm sector. For illustrative purposes, the three-month ex post real interest

rates using CPI and farm PPI data are presented in Figures 1 and 2. Note the dramatic fluctuations of the farm PPI ex post rate relative to the real rate calculated with the CPI-U. The behavior of the ex post real rate is thus dependent upon the behavior of the price index used to measure inflation.

Finally, as a first step in the process of generating ex ante real interest rates, we examine the time series properties of the variables employed in our regressions. The nominal interest rates all exhibit unit roots, as does the energy shock variable. The inflation rates are stationary, as are the ex-post real rates and the money supply growth rate. Thus, cointegration/ error correction methodology is not appropriately employed here. l5

Here we are estimating:

Simple Updating Model

rrt = Xt_‘B + e, (7)

I5

10

5

0

ii

8 -5

%

-10

-15

-20

E :pI

Figure 1. Month Ex Post Real Rate: CPI


100

50

0

-50

2 -100

Y !

-150

-200

-250 1 I I, I I I I I I I I I I I I I I I, I I I I I I I, I I I I I I, I I I I I , I I,

1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 I988 ! 992

Figure 2. Month Post Real Rate: Farm PPI

with OLS. Following Holland (1993) we do not place restrictions on the coefficients

because these “ . ..are combinations of parameters, including those of the monetary policy reaction function, which can take on any value” (p. 42). While he estimates equations for

ex ante inflation, we argue that his reasoning is also appropriate for our model since by

forecasting the real rate, we are indirectly generating a forecast of inflation. Thus, we do

not impose Koyck or other restrictions on the lag structure. In order to evaluate the accuracy of our forecasting model, we estimate in-sample fore-

casting equations based on the independent variables employed by several prior studies. Using CPI data, we estimate the equations specified in Huizinga and Mishkin (1986). The

independent variables are a constant, the nominal interest rate (measured using the one- or three-month T-Bill rate), two lagged values of inflation, and the supply shock variable

described above. The authors split their sample into three subsamples: 1953:1-1979:10, 1979: 11-1982: 10, and 1982:11-1984: 12 to account for the two shifts in monetary policy.

We simply extend the last subsample period to 1992:12.16‘

To evaluate the accuracy of both the in-sample and out-of-sample forecasts of the real interest rate, we employ four diagnostic statistics: The mean absolute deviation (MAD), the root mean square error (RMSE), the mean absolute percentage error (MAPE), and Theil’s U. The comparisons are shown in Table 2. The ex ante forecasts we construct with the simple updating model are generally not as accurate as the forecasts generated by Mishkin’s in-sample technique. This is not surprising for several reasons. First, in-sample forecasting

should be more accurate than out-of-sample forecasting. Second, the forecasts of a

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Forecasting the Real /r&vest Rate 65

researcher should be better than (or at least as good as) those of a forecaster. Again, this is because of the previously discussed advantages of a researcher. Yet, we believe our methods more closely approximate those of forecasters. Thus, our ex ante real rates will be closer to the true, unobserved ex ante rates.

It is also interesting to note that there are large differences in the forecast diagnostics between the four real rates. Clearly, forecasting the CPI real rate is less fraught with diffi- culty than is forecasting the farm PPI real rate. This finding is likely due to the fact that the farm PPI has a much higher variance than the CPI.

Table 2 also presents the results using a three-month horizon. As can be seen, the forecast statistics for the three-month horizon are dramatically improved relative to those for the one-month horizon. Recall from Table 1 that the three-month data display far less variation than the one-month data. Thus, forecasting accuracy should be (and is) improved with three-month data. l7

Rolling-Regression Model By simply updating the regression estimates in the “Simple Updating Model” section of

this paper, we are implicitly giving relatively old data (the 1960s) equal weight with new data (the 1990s) when making forecasts during, say, the 1990s. A simple way to give additional weight to more recent data is to estimate forecasting equations with a fixed number of observations so that when one observation is added to the end of the sample period, an observation is dropped off the beginning of the sample period.

IO.0

2.5

0.0

5

8 -2.5

%

-5.0 80

u -7.5 *

1 I

-10.0 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992

Figure 3. In Sample vs. Rolling Forecasts: CPI


We choose to use a lo-year time span, so that the sample period always contains 120

observations. Thus, the first regression uses the sample period 1960:1-1969:12 to make a

forecast for 1970: 1. As above, we include a constant, the T-Bill rate, the second and third

lagged values of the inflation rate, and the second lagged values of the energy shock vari-

able and the growth rate of Ml. The results using one-month rates are also presented in Table 2. As can be seen, the

rolling regression model provides the best forecasts for CPI data and manufacturing PPI

data. However, the simple updating model does better with farm PPI data and CRB data. Figure 3 compares the forecasts of the rolling regression model using CPI data to those

made by Mishkin and Huizinga’s (1986) model. While the correlation between the two

forecasts is high, they do diverge on several occasions. This may be because we use vari-

ables lagged two or more periods, while Huizinga and Mishkin (1986) use variables lagged

one or more periods. Thus, our models will take longer to “catch up” to changes in the ex

post rate. Finally, Table 2 presents the nearly identical results using three-month rates. Similar to

the results using one-month data, the rolling regression model provides the best forecasts

for CPI and manufacturing PPI data.

ARMA Updating Model A very common forecasting technique is the use of an ARIMA model. The first step in

ARIMA model building is to examine the stationarity of the variables at hand. As dis-

cussed in the “Data and Sample Periods” section of this paper, tests indicate that all four ex

post real rates are stationary. Since differencing is not required, an ARMA model is esti-

mated. Further, examination of the autocorrelation and partial autocorrelation functions

indicates that an AR model using the second and third lags should be adequate for all four

ex post rates. As before, only information available at the time of the forecast is eligible for

inclusion in the model. The forecast statistics are presented in Table 2 for the one- and three-month rates. Of

the three models, the ARMA model produces the best forecast statistics for the farm PPI

data and the CRB data. But again, the in-sample forecasts are superior in diagnostic

terms. To evaluate the ex ante rates generated with the ARMA models, we regressed these on

past inflation, monetary policy variables (discussed further below), and the oil shock vari-

able. We find that these variables generally are statistically significant, which implies that

the ARMA forecasts can be improved by incorporating information on these variables.

Specifically, we find that increases in the reserves ratio and the growth rate of Ml are asso-

ciated with lower ex ante real interest rates. Changes in the interest rate spread do not seem

to impact the ex ante rate.‘”

EVALUATION AND APPLICATION OF THE MODELS

Table 3 presents summary statistics for the one- and three-month ex ante rates estimated

with CPI data. The ex post rates are presented for comparison. For the one-month rates, the

sample means of the updating and rolling regression models are not statistically different

than the ex post rate. Notice also that the distributions of the three ex ante rates appear to be

different based on the tests for skewness and kurtosis.


TABLE 3. Forecast Evaluation Statistics: Alternative Money Measures

MAD RMSE MAPE u Ml

Updating 2.404 3.115 8.622 0.960

Rolling* 2.328 3.075 8.004 0.948

SPREAD

Updating 2.378 3.068 8.006 0.825

Rolling* 2.274 2.996 6.832 0.806

NBRRATIO

Updating 2.394 3.107 7.853 0.835

Rolling* 2.279 3.003 6.698 0.807

In-sample 1.841 2.417 7.336 0.745

Notes: *Indicates the out-of-sample model with the best forecast error statisitics. All of the models use a one-month horizon.

The monetary policy indicators are the growth rate of Ml (Ml), the interest rate spread (SPREAD), and the ratio of non-

‘borrowed to total reserves (NBRRATIO). In-sample is Mishkin’s in-sample technique for generating ex ante real interest

rates, again used here for a benchmark. MAD is the mean absolute deviation, MAPE is the mean absolute percentage

error. RSME is the root mean square error. and U is Theil’s U. All forecasts are computed over the period 1970: I-

1!+92:12 using the methods described in the text.

0.75

0.50

0.25

0.00

-0.25

-0.50

Figure 4. T-Bill: CPI

FLETCHER and GULLEY

-0.06

-0.24

-0.30

-0.36

-0.42

12

6

0

-6

-12

-18

-24

-36

I I I I I I I I I I I I I1 I I I I I II I I

1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992

Figure 5. Second Lag of Inflation: CPI

I I I I I, I I I I I I I1 I1 I I Ia II I

1970 1972 1974 1976 1978 1980 1982 I984 1986 1988 1990 1992

Figure 6. Energy Shock: CPI


The sample means for the three-month rates are all statistically different from the ex post rate. Further investigation indicates that the severe underprediction of real rates in the early

to mid-1980s is responsible for the difference. Here, though, the distributions of the three ex ante rates are generally skewed and (with the exception of the updating regression

model) display kurtosis. Thus, while each model generates different ex ante real interest rates, each technique

allows for learning on the part of forecasters. Such learning will be reflected by changes in

the coefficients on the forecasting regressions. For example, Figures 4-7 present graphs of some of the coefficients of the rolling

regression model estimated with the CPI using one-month data. As can be seen, many of the coefficients undergo dramatic shifts in 1973-1974. In addition, Figure 4 shows the increasing economic significance given to the current T-Bill rate in the early 1980s when such rates were reaching their modern-day peaks. The statistical significance of the T-Bill

rate also increased significantly during this time period. Figure 5 indicates the second lag of inflation appears to provide less information as the 1980s and 1990s pass. Figure 6 indicates that the supply shock coefficient is highly variable in the early 1970s. This variable is statistically significant during the mid-1970s, the mid-1980s and the early 1990s.

Thus, during these times, DRI’s “on/off” supply shock variable would be “on.” Finally, Figure 7 shows that the second lag of the money supply provides little informa-

tion with which to forecast the real rate since the coefficient is statistically insignificant. It is on the significance of this last variable that we focus in applying our forecasting model.

0.012

0.006

-0.000

-0.006

-0.024

! I I I I I1 T I I I I I I I I1 I I I I I I

1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992

Figure 7. Ml: CPI


Although we do not formally test for Granger causality, the insignificance of the second lag of the money supply as a determinant of the expected real interest rate is unfavorable to

conventional monetary models of the business cycle. These models hold that monetary pol- icyinfluences output in the short run via (among other channels) the real interest rate. But as various authors have pointed out, money growth is only one of many ways to measure monetary policy. Thus, we will focus on an alternative monetary policy variable as an

application of our model. Diba and Oh (199 1) suggest that the spread between the Federal funds rate and other

interests rates is a good indicator of monetary policy and helps predict output growth. We re-estimate our model using the yield on the lo-year Treasury bond less the federal funds

rate (SPREAD) as a proxy for monetary policy, replacing money growth. Forecast statistics appear in Table 4 for both the simple updating and rolling regression models, using the CPI as the price index. Interestingly, the forecast statistics improved under both modeling

techniques compared to when the growth rate of Ml was used, with the best results again occurring under the rolling regression methodology.

In Table 4 we also show the forecast statistics for our model when the ratio of nonborrowed to total reserves (NBRRATIO) is used as the monetary policy indicator, as employed by Eichenbaum and Evans (1995). Again, the forecasting diagnostics improve

for our model under both updating techniques when this proxy is used. Figures 8 and 9 show the coefficients of the SPREAD and NBRRATIO variables,

respectively, for the rolling regression model, using one-month data. In general, using the SPREAD variable as a proxy for monetary policy lends support to the monetarists’ view that money affects output. The coefficient is statistically significant until 1974, is not significant from 1974 to 1979, is again significant from 1980 to 1989, and then provides little information from 1990 forward. It is also interesting to note the sign change: the coefficient is positive between 1975 and 1991 and negative before and after this period.”

The ratio of nonborrowed reserves to total reserves, on the other hand, provides little

TABLE 4. Summary Statistics for Ex Ante Real Interest Rates

Method Sample Mean SE Skewness Kurtosis

One Month

Updating 1.596 2.511 0.694 0.175

Rolling 1.458 2.508 0.623 0.124

ARMA 1.816* 1.922 0.001 0.172

Ex-post 1.424 3.467 0.744 0.039

Three Month

Updating o.f330* 2.296 0.011 0.16

Rolling 0.702* 2.134 0 0

ARMA 1.258* 1.271 0 0

Ex-post 1.941 3.059 0.049 0.858

Nutes: *Statistically different from ex post sample mean at the 5% level. The sample means and standard errors are in percent. The reported values for skewness and kurtosis are the p-values

from hypothesis tests conducted on the ex ante real rates. The null hypotheses are no skewness or no kuttosis. The ex

ante rates are generated using the methods (updating, rolling, and ARMA) dscribed in the text. The ex post rates are included for comparison puropses. The forecasts are computed over the period 1970: l-1992:12.

Forecasting the Real interest Rate

1.0

I I I I I I I I , , , , , , , , , , ,

1970 1972 1974 1976 1978 1980 1982 1964 1986 1988 1990 1992

0.5

0.0

-0.5

-1.0

I5

IO

5

0 I

-5

-10

-15

-20 \

-25

Figure 8. Spread: CPI

I I I I I / , I , , , , , , , , , , , ,

1970 1972 1974 1976 1978 1980 1982 1984 1986 I9E8 1990 1332

Figure 9. NBRRATIO: CPI

FLETCHER and GULLEY

information to our forecasting model in that the coefficient is never statistically signifi-

cant. Even the shift of the Fed to targeting nonborrowed reserves during the period 1979 to 1982 does not appear to affect the coefficient on this variable. The results of this model

thus appear to agree with those of Mishkin (1981) who finds that money growth is statistically insignificant, and are contrary to the those of Eichenbaum and Evans (1995), who

find that the ratio of nonborrowed to total reserves has a statistically significant impact on the exchange rate.

Based on the above results, it does appear to matter what proxy is used for monetary policy in evaluating the effects of money on output. Clearly, use of the spread between the federal funds rate and long term interest rates lends support to the monetarist view,

while using the ratio of nonborrowed to total reserves as a proxy for monetary policy does not.

SUMMARY AND CONCLUSIONS

The purpose of this paper was to extend Mishkin’s in-sample forecasting technique to an out-of-sample technique. We employed four different measures of inflation to generate

four different real interest rates. We then used three different forecasting methods (updated regression, rolling regression, and ARMA) to calculate one-period ahead fore-

casts. These forecasts were then compared to Mishkin’s and to each other with four forecast evaluation statistics. Not surprisingly, we found that Mishkin’s method yielded better forecasts in terms of accuracy when compared to all three of our out-of-sample tech-

niques. We also found that the rolling regression and ARMA models generally outper-

formed the simple updating regression technique. However, while Mishkin’s method yields more accurate forecasts, we believe our methods yield forecasts that are closer to the true ex ante interest rates because we are actually forecasting, not simply calculating residuals. As such, we argue our method represents a step forward in measuring ex ante

real interest rates. Application of our model to studying the effects of monetary policy on output indicates

that the choice of the proxy for monetary policy is very important to the results of the empirical testing. More work in this area needs to be done before the debate between the

monetarist school and the real business cycle theorists can be resolved.

Acknowledgments: The authors would like to thank Frederic Mishkin for kindly supplying us with data, Scott Sumner for insightful discussions, and an anonymous referee for very helpful suggestions. Funding for this project was provided by a Bentley College Summer Research Grant. All errors remain our own.

APPENDIX

This Appendix presents the regression results for the in-sample forecasts of the real interest rate. NOMINAL INTEREST is the one-month T-Bill rate, INFLATION{ } is the lagged value of the inflation rate, and SHOCK is the log of the ratio of the fuel component of the PPI to the overall PPI. Note that in several cases, the Q statistics are statisti-

cally different from zero. This finding arises due to 12th order serial correlation. Mishkin (1981) indicates that such autocorrelation is not of much concern.

6 3 B

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FLETCHER and GULLEY

NOTES

1. Of course, when B,[, is constant over time, rrr generated by the researcher and by the forecaster will be identical. We return to this issue below.

2. The reasoning above is similar to that found in the literature on the construction of hedges for financial instruments. For example, Myers and Thompson (1989) calculate allegedly ex ante hedge ratios by using information that is only available ex post. This type of static hedge has increasingly been supplanted by dynamic hedge ratios that are calculated with information updated period-by- period. See Kroner and Sultan (1993).

3. Christ (1993) notes that prior to the OPEC shock, most macroeconometric models did not include variables/equations to measure energy shocks.

4. Several authors have employed time-varying parameter techniques to examine the real rate. See, for example, Garbade and Wachtel(l978) and Antoncic (1986). However, these researchers do not use out-of-sample forecasting to generate the ex ante real rate. In addition, the real rate is restricted to follow a random walk. Litterman and Weiss (1985) find that the real rate is a more general AR process.

5. Mullineaux (1980) also finds the coefficients on lagged inflation and money growth generally increase during the 1966 to 1977 period.

6. The idea of negative ex ante interest rates is somewhat controversial. Darin and Hetzel(l995) argue that the final move to fiat money, combined with the oil shocks and a monetary policy targeting economic growth and the unemployment rate, changed the process that generated inflation in the 197Os, resulting in persistent forecast errors. If this argument is valid, ex ante real rates during the mid-1970s may have been positive. Barro and Sala-i-Martin (1990) also discuss negative ex ante real interest rates.

7. Cecchetti (1992) employs Mishkin’s technique using out-of-sample forecasts as a robustness check of his ARMA models of inflation. He finds that his conclusion that deflation was anticipated during the Great Depression is robust to changes in the estimation procedure. Thus, our methodology parallels that of a time-varying parameter estimation of expected inflation. The difference is that we are estimating the ex ante real rate.

8. Figlewski (1994) employs the rolling regression model in his attempts to forecast the volatility of various financial assets.

9. Other techniques that allow parameters to vary over time include the Kalman filter and variable parameter estimation.

10. Bonser-Neal(l990) finds similar results. Schwert (1986) finds less evidence for such a shift

by using slightly different methodology. 11. Frederic Mishkin kindly provided interest rate and CPI data through 1990. 12. In its forecasting model, DRI uses an on/off supply shock variable to measure the extent to

which an energy shock affects economic variables (telephone conversation with David Wyss, Chief Financial Economist at DRI). Thus, our method allows us to discover the months when the supply shock variable would be “on.”

13. All data are seasonally unadjusted in keeping with the true spirit of forecasting. See Table 1 for a more complete description of the data.

14. In future work, we will take advantage of the fact that the CRB index is computed on a daily basis, which will allow us to exactly match inflation rates with nominal interest rates. Such matching will allow better computation of ex post real interest rates.

15. These results are available on request. It would seem inconsistent that the nominal interest rate is nonstationary, while both the inflation and real interest rates are stationary. There may be several reasons for this result. First, the Fisher equation may not perfectly describe the relationship between the three series. Second, the Fisher equation should be thought of as a long run relationship that need not hold at every moment in time. Finally, it should be noted that unit-root test results for inflation and ex post real interest rates can be sensitive to the choice of the sample period. See Gul- ley (1995).

Forecasting the Real Merest Rate 75

16. The Appendix describes the regression specifications and results for each of the four infla-

tion indices. We also include the second lag of the growth rate of Ml for reasons we discuss below. 17. Figlewski (1994) reports similar results in his attempts to forecast the volatility of various

financial assets. He finds that “...forecast errors are substantially lower for long term than for short term volatility forecasts” (p. 29).

18. These results are available on request. 19. The reason(s) for the changes in significance and sign are unknown. We speculate that the

dramatic shifts in the slope of the yield curve from the 1970s to the 1980s higher and more variable inflation in the 1970s the final move toward flexible exchange rates, the oil shocks, and the changes

in monetary policy operating procedures all play a roll in the coefficient changes.

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Barro, Robert J., and Xavier Sala-i-Martin. 1990. “World Real Interest Rates.” Pp. 15-61 in NBER Macroeconomics Annual, edited by Olivier Jean Blanchard and Stanley Fischer. Cambridge, MA:

The MIT Press. Bonser-Neal, Catherine. 1990. “Monetary Regime Changes and the Behavior of Ex Ante Real Inter-

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Caskey, John. 1985. “Modeling the Formation of Price Expectations: A Bayesian Approach.” Amer-

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Christ, Carl F. 1993. “Assessing Applied Econometric Results.” [Federal Reserve Bank of St. Louis]

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to Monetary Policy on Exchange Rates.” Quarterly Journal of Economics 110: 975-1009. Figlewski, Stephen. 1994. “Forecasting Volatility Using Historical Data.” Stem School of Business,

New York University, Working Paper S-94-13. Garbade, Kenneth, and Paul Wachtel. 1978. “Time Variation in the Relationship Between Inflation

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Mishkin, Frederic S. 1984. “The Real Interest Rate: A Multi-Country Empirical Study.” The Cana-

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Myers, Robert J., and Stanley R. Thompson. 1989. “Generalized Optimal Hedge Ratio Estimation.” American Journal of Agricultural Economics 71: 858-868.

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Documents

Forecasting the real interest rate