There has been considerable interest in modeling the evolution and interactions of nominal interest rates. Models of the term structure of interestrates fall into two categories: those based upon arbitrage argu- ments and those based on a general equilibrium formulation. In the former category, single-factor models of the term structure of interest rates have been proposed by Merton (1973), Vasicek (1977), Dothan

(1978), Schaefer and Schwartz (1978), and many others, Multifactor term structure models have been proposed by Richard (1987), Brennan and Schwartz (1979), Langetieg (1980), Schaefer and Schwartz (1984), and Heath et al. (1992). Models representing a complete general equi- librium specification of the term structure have been put forth by Cox, Ingersoll and Ross (1985a,b), Longstaffand Schwartz (1992) and many others. Chan, Karolyi, Longstaffand Sanders (1992) and Broze, Scaillet, and Zakoian (1995) provided empirical comparisons of the adequacy of the models' explanation of the data. Both theoretical and empirical results from these two branches of research suggest that term struc- ture models that allow yield nonlinearity can provide additional insight and explanatory power for the modelling of equilibrium interest rates.

This paper models nonlinearities in the evolution of the condi- tional means of U.S. Treasury securities’ yield changes for various maturities and attempts to exploit those nonlinearities to improve fore- casting performance. Our approach is nonstructural and univariate as it relies on the historical behavior of individual securities’ yield series to model nonlinearities. It is also nonparametric as we do not impose maintained hypotheses of smoothness on the regression function; in- stead, we let the data determine the regression function. To detect whether nonlinearity characterizes the securities' yields, we apply the BDS test to the series. We also investigate the low-frequency proper- ties of securities yields by subjecting the series to unit root tests that consider both integer and fractional orders of integration.

We are primarily interested in the out-of-sample forecasting accu- racy of our nonparametric model as measured by root mean square error (RMSE). We employ the locally weighted regression (LWR) estimation method, a local-fitting methodology, to provide a regression surface and carry out prediction, We compare the forecasting performance of the nonparametric fit to the performance of two benchmark linear models: an autoregressive (AR) model and a random-walk-with-drift (RW) model. The overall evidence shows improved forecastibility with nonparametric predictors out of sample over best linear predictors for most U.S. interest rate series.

The plan of the paper is as follows. In section 2 we present the nonparametric method. Data and diagnostic tests are presented in section 3, while the empirical estimates are reported in section 4. Finally, we con- clude in section 5 with a summary of our results and suggestions for fu- ture research.

We attempt to uncover nonlinear relationships in U.S, T-bill and bond yields using the nonparametric locally weighted regression (LWR) method. LWR is a nearest-neighbor estimation technique, first intro- duced by Cleveland (1979) and further developed by Cleveland and Devlin (1988), and Cleveland, Devlin, and Grosse (1988). Itis a way of estimating a regression surface through a multivariate smoothing pro- cedure, fitting a function of independent variables locally and in a mov- ing-average manner. Suppose that the regression function is given by y= a(x)+er, Pela (1)

Where x,- Gu... Xpe)is a 1x p vector of (weakly) exogenous ex- planatory variables, g(-) isa smooth function mappingR? — R, and

& sis an independent and identically distributed disturbance with mean zero and variance go 2.

LWR is a numerical algorithm that describes howé(x" }the esti- mate of g at the specific value of x*, is estimated, Let be a smoothing constant such that0 < / <1, and let g,= int (f-7), where int(-) extracts the integer part ofits argument. The LWR uses the “window” of q, obser- vations nearest x*, where proximity is defined using the Euclidean dis- tance. In the LWR algorithm, the conditional mean is estimated from a weighted least squares regression of y on x for the relevant g ,observa- tions. More specifically, given a point x* called the current state, rank the x;'s by Euclidean distance from x*. Let |}||measure Euclidean dis- tance; then the Euclidean distance from x* to its%g / nearest neighbors is we al, x4,) = if (ie, = | (2)

Each of the q, nearest neighbors are inversely weighted by their Euclidean distance from the current state. Let w, = -u, where

[ie x4 ga

Our data set consists of a variety of short- and long-term U.S. Treas- ury interest rates: monthly observations on the Federal Funds rate, 3- month, 6-month, and 12-month U.S. Treasury bill yields, and yields on 5- year and 10-year Treasury bonds. The sample period is 1957:01 to 1993:12 for all series except for the 6-month Treasury-bill yield, for which it is 1959:01 to 1993:12, The 3- and 6-month yields are percentage annual rates obtained from the secondary market. The 1-, 5-, and 10-year yields are constant maturity percentage annual rates, All data series are obtained from the Citibank data base. As we are primarily interested in the out-of- sample forecasting performance of the nonparametric method, we reserve the last 48 observations (1989:01 to 1993:12) from each yield-change series for forecasting purposes. Diagnostic tests presented below are ap- plied to the estimation sample series, excluding the post-sample observa- tions.

Table 1 presents selected summary statistics for first-differenced yield series. The means for all series are not statistically different from zero and all series exhibit dependence in the third and fourth cumulants. All yield-change series are negatively skewed with the exception of the 5- year yield which is positively skewed. They are all characterized by fat tails, i. e., leptokurtosis. However, the presence of skeweness and kurtosis is much stronger for the short term as opposed to longer term interest rates. As expected, yield changes for short maturities exhibit greater vari- ability than those for longer maturities.

We first investigate the low-frequency properties of the yield se- ries. To do so, we apply the Phillips-Perron tests (PP) (Phillips (1987), Phillips and Perron (1988)) to both levels and first differences of our sample series, with results presented in Table 2. Inference is robust to the order of serial correlation allowed in the data. All PP tests fail to reject the unitroot null hypothesis in the yield series but strongly reject the unit root null in yield changes. The PP test results therefore strongly support the hypothesis ofa single unit root in the yield series, Given the low power of standard unit root tests against fractional alternatives (Diebold and Rudebusch (1991)) we apply the semi-nonparametric procedure suggested by Geweke and Porter-Hudak (GPH, 1983) to the yield series, The GPH test avoids the knife-edged /(1) and /(0) distinction in the PP test by al- lowing the integration order to take on any real value (fractional integra- tion). Table 3 reports the empirical estimates for the fractional differencing parameter. We find no evidence in support of the fractional alternative for any of our sample series. We therefore conclude that all yield series are integrated processes of order one and subsequently apply our analy- sis to yield changes.'

To obtain some preliminary evidence regarding the presence of nonlinearities, we perform the test suggested by Brock, Dechert, and Scheinckman (BDS, 1987) to yield changes and filtered yield changes. The BDS test tests the null hypothesis of independent and identical dis- tribution (i.1.d.) in the data against an unspecified departure from iid. A rejection of the i.i.d. null hypothesis in the BDS test is consistent with some type of dependence in the data, which could result from a linear stochastic system, a nonlinear stochastic system, or anonlinear determin- istic system. Under the null hypothesis, the BDS test statistic asymptoti- cally converges to a standard normal variate. However, Monte Carlo simulations by Brock, Hsieh, and LeBaron (1991) and Hsieh and LeBaron (1988) suggest that the asymptotic distribution is a poor approximation to the finite sample distribution when there are fewer than 500 observa- tions, and is not appropriate when applied to the standardized residuals of ARCH models.*

Table 4 reports the BDS test statistics for three sets of data: the yield-change series and two prewhitened versions created with autoregressive and autoregressive conditionally heteroscedastic (ARCH) model filters, Going from the shorter to longer maturity yield series, the AR orders chosen on the basis of AIC are 13, 20, 19, 19, 6, and 22, respectively (the maximum order allowed is 24), and the ARCH orders are 3, 4, 2, 2, 2, and 4, respectively.’ We applied the BDS test to these three sets of series for embedding dimensions of m=2,3,4 and 5. For each m,€ is set to 0.5 and 1.0 standard deviations (c) of the data. We use the quantiles from the small sample simulations reported by Brock et al, (1991) as approximations to the finite-sample critical values of our BDS statistics. The i.i.d. null hypothesis is overwhelmingly rejected in all cases for yield changes, When the BDS test is applied to the AR- filtered series we still obtain strong rejections of the i.i.d. null hypoth- esis suggesting that linear dependence in the first moments does not fully account for rejection of i.i.d. in yield changes. When applied to the ARCH model, the obtained residuals are standardized by their esti- mated conditional standard deviations and the BDS test is applied to these standardized residual series. The standardized residuals appear to be 1.1.d. suggesting no neglected nonlinearity in the series after prop- erly accounting for time variation in their second moments. However. the estimated ARCH effects could be proxying nonlinearity in the conditional mean of our series.‘ As we are interested in uncovering nonlinear structure in the conditional mean of yield-change series, we now turn to modeling nonlinearities in their first moments by means of local- fitting methodology.

In this section we estimate a nonlinear model for the yield-change se- ries and compare its ex ante forecasting performance to that of bench- mark linear models. In modeling nonlinearities in the conditional mean of our sample yield-change series we allow for a nonparametric functional form in the relationship. Employing nonparametric regression in the esti- mation process has the following advantages: (i) it reduces the possibility of model misspecification, (ii) it provides a versatile method of exploring a general relationship, and (iii) it can serve as a diagnostic tool in sug- gesting simple parametric formulations of the regression relationship. The nonparametric method employed is locally weighted regression (LWR), as specified in section 2 above.

We compare the LWR forecasts to those obtained by estimating two standard linear models: an autoregressive model (AR) anda random- walk-with-drift model (RW). The last 48 observations (1989:01 to 1993:12) from each yield-change series are reserved for forecasting pur- poses. The first twenty-four yield changes (1957:02 to 1959:02; 1959:02 to 1961:02 for the 6-month yield changes) are used for model initializa- tion. Therefore, for forecasting purposes the sample period 1959:03 (1961:03 for the 6-month rate) to 1988:12 is the training set and the sample period 1989:01 to 1993:12 is the test set. The out-of-sample fore- casting horizon is one-step ahead, with the competing forecasting models estimated over the training set and applied over the test set to generate genuine one-step-ahead out-of-sample forecasts. The criterion for fore- casting performance is root mean square error (RMSE).‘ The AR or- der for the linear model is chosen on the basis of AIC, as described in section 3.

Tables 5 through 10 report the out-of-sample forecasting perform- ance of the LWR model and its linear counterparts for our six sample series.° To ensure robustness of our evidence, we report the forecast- ing performance of nonlinear autoregressions of order one through six and for varying window sizes.’ Comparing the forecasting perform- ance of the linear models, we observe that the AR model outperforms the RW model for all yield series except for 3-month and 12-month maturities, We therefore restrict ourselves to comparisons of the fore- casting performance between the nonparametric model and the AR model.

On the basis of the RMSE forecasting criterion, the LWR model outperforms the AR model across the different lag structures and win- dow sizes considered. Only ina very limited number of cases is the AR fit better than the LWR fit and these primarily concentrate on the 5-year yield changes. The percentage reductions in RMSE of the LWR fit over the AR fit range from 1.08% to 15.54% for the Federal Funds rate, 6.79% to 20.34% for the 3-month maturity, 2.75% to 13.40% for the 6-month maturity, 7.71% to 21.14% for the 12-month maturity, 0.002% to 5.54% for the 5-year maturity, and 1.16% to 7.20% for the 10-year maturity. The improvements in forecasting performance appear to be greater for the short-term as opposed to longer-term yield changes. The consistency in the forecasting improvements of the nonparametric specification with respect to autoregression order and window size enhances the view that the estimated nonlinearities are not a statistical artifact, but rather cap- ture essential aspects of the data generating process.

In order to formally evaluate model performance, we apply the forecast comparison test of Granger and Newbold (1986, pp. 278-280) to test the hypothesis that there is no difference in the forecasting accu- racy between the linear and nonlinear models. Given that the nonparametric fit generally achieves a lower RMSE, we test the null hypothesis of no difference in the forecasting performance between the AR and LWR mod- els against the one-sided alternative that the LWR model has superior forecasting performance. Table 11 reports the Granger-Newbold test re- sults. For the 3-month, 6-month, 12-month and 10-year yield-change se- ries, the forecasting improvements obtained by the nonparametric model are statistically significant. For the 5-year yield series, we cannot reject the null hypothesis of no difference in the forecasting accuracy of the two competing models. This is not surprising given that the RMSE values attained by the LWR method were not uniformly lower than those at- tained by the AR method, and in cases when they were lower, they were only marginally so. Finally, for the Federal Funds rate, the LWR method attains a lower RMSE in 51 out of 54 cases considered, but fails to dem- onstrate superiority in a statistically significant sense. Overall, the test results show that the LWR model's forecast performance is statistically superior to that of the AR model for most of our sample series. This enhances the view that the LWR method isa superior modeling strategy for these benchmark interest rate series.

We provide evidence that the nonparametric fit generated by the LWR method results in significant improvements, compared to benchmark (linear) models, in out-of-sample U.S. short- and long-term interest rate forecasts, The superior performance of the LWR methodology is ro- bust to autoregression order and window size. The evidence provided es- tablishes the presence of significantnonlinear mean predictability in U.S. interest rates, as suggested by theoretical findings. Since it appears that the dynamical behavior of US interest rates is very local in nature, letting the data determine the regression function pays dividends in terms of improvements in the forecasting accuracy of interest rate models, Although the LWR estimation method has failed to successfully predict stock re- turns (Hsieh (1991), LeBaron (1988)) and exchange rates (Diebold and Nason (1990), Meese and Rose (1990, 1991), Mizrach (1992)) it ap- pears to be very useful in modeling conditional mean changes in interest rate series." In addition to nonlinearities in variances and possibly higher moments, significant nonlinearities in the mean clearly exist for U.S. in- terest rates, and LWR appears to successfully capture those nonlinearities.

Our results can be extended in several ways. First, the usefulness of the LWR estimation method as a forecast generating mechanism for tmultiple-step-ahead forecasting horizons should be investigated. Second, our positive results invite the use of alternative nonparametric methods to be employed as forecasting tools for U.S. interest rates. Finally, an obvious avenue of future research is to apply the LWR methodology to interest rate series from other industrial countries.