Out-of-sample forecasts and nonlinear model selection with an example of the term structure of interest rates.

1. Introduction

Much of the current interest in nonlinear time-series models stems from the large literature documenting the asymmetric behavior of many macroeconomic variables. Neftci (1984), Ashley and Patterson (1989), and Scheinkman and LeBaron (1989) find asymmetric adjustment in U.S. unemployment, production, and employment over the course of the business cycle. Similarly, Tinsley and Krieger (1997) find that negative deviations from trend production are larger than the positive ones and that price levels more readily increase than decrease. Beaudry and Koop (1993) find that negative innovations to gross domestic product (GDP) are much less persistent than positive ones, and Rhee and Rich (1995) find asymmetric effects of monetary shocks on output and expected inflation.

A number of papers have tried to explicitly model the type of nonlinearity present in the data. For example, Terasvirta and Anderson (1992) use a smooth transition autoregressive model to show that industrial production in 13 OECD countries responds more sharply to negative shocks than to positive shocks. Sichel (1993) and Ramsey and Rothman (1996) examine whether business cycle troughs are more pronounced than peaks and whether contractions are steeper than expansions. Potter (1995) modeled changes in real U.S. gross national product (GNP) as a threshold adjustment process and found that the post-1945 U.S. economy is significantly more stable than the pre-1945 U.S. economy. Shen and Hakes (1995) applied a threshold autoregressive model to the reaction function of the central bank of Taiwan and found that central bank responses depend on the severity of inflation. Ball and Mankiw (1994) presented a menu-cost model with positive trend inflation to show that prices respond more strongly to positive shocks than to negative shocks, and Maravall (1983) used a bilinear model to estimate and forecast the Spanish exchange rate.

A potential problem with such nonlinear estimates involves the possibility of overfitting. Although the Akaike information criterion (AIC) and Schwartz Bayesian criterion (SBC) were designed to combat the problem of overfitting by adding a penalty term for each estimated parameter, nonlinearity adds another dimension to the problem. (1) A search across functional forms and the various parameterizations of each is likely to yield some nonlinear model that "fits" a particular data set especially well. However, Rothman (1998) shows that the "best" nonlinear specification of the U.S. unemployment rate does not provide the best out-of-sample forecasts.

The problem is complicated by the fact that tests for nonlinearity are not particularly good at determining the precise form of the nonlinearity. For example, the Lagrange multiplier (LM) tests for nonlinearity reviewed in Granger and Terasvirta (1993) have the null of linearity against an alternative hypothesis specifying a particular type of nonlinear adjustment. It is quite possible that an LM test for threshold adjustment and an LM test for bilinear adjustment are both supported by the same data set. As a result, papers such as McCracken (2000) argue that inference using the out-of-sample characteristics of a model can be superior to in-sample inference.

The aim of this article is to examine the extent to which out-of-sample forecasts can help select the appropriate nonlinear specification. We focus on a small sample in order to consider a circumstance in which asymptotic methods may not provide useful guidelines concerning model selection. For similar reasons, we emphasize threshold autoregressive (TAR) specifications because they contain an unidentified nuisance parameter under the null hypothesis of linearity. It is shown that standard in-sample estimation procedures typically lead to overfitting in that they select a nonlinear model when the true data-generating process is linear. In contrast, out-of-sample forecasts tend to select the linear specification. When the data are generated from a threshold autoregressive model, the results are somewhat mixed. Some form of threshold process usually has the lowest AIC, SBC, and/or mean square prediction error (MSPE). However, for some parameterizations, a linear model will tend to produce the lowest MSPE.

2. The Time-Series Models

Let {[y.sub.t]} be a time-series of interest and suppose that the objective is to forecast the subsequent realization of the series conditional on the current and past observations. In particular, let

[y.sub.1] = f([y.sub.t]i, [[epsilon].sub.t-j]; i = 1,...p, j = 1,...,q) + [[epsilon].sub.1], (1)

where [[epsilon].sub.t] is a zero-mean white-noise disturbance.

The conditional mean of [y.sub.t+1] is given by

E([y.sub.t+1]\[y.sub.t+1-i], [[epsilon].sub.t+1-j]; i = 1,...,p, j = 1,...,q)

= f([y.sub.t+1-i], [[epsilon].sub.i+1-j]; i = 1,...,p, j = 1,...,q). (2)

The econometric problem is that the functional form of f(.) along with the magnitudes of the various parameters need to be estimated from the data. Autoregressive moving-average (ARMA) models, threshold autoregressive (TAR) models, exponential autoregressive (EAR) models, bilinear autoregressive (BL) models, and generalized autoregressive (GAR) models are popular functional forms for modeling economic data. In order to set the stage for our Monte Carlo experiment, we briefly review each of these functional forms.

Autoregressive Moving Average (ARMA) Models

The standard ARMA(p, q) model has the form (2)

[y.sub.t] = [[alpha].sub.0] + [summation over (p/i=1)] [[alpha].sub.i][y.sub.t-i] + [[epsilon].sub.t] + [summation over (q/i=1)] [[beta].sub.i][[epsilon].sub.t-i]. (3)

ARMA(p, q) models have been extensively analyzed and popularized by Box and Jenkins (1976), where model specification, estimation, and diagnostic checking were analyzed. The main econometric problem is to determine the lag lengths p and q and then estimate the parameters [[alpha].sub.i] and [[beta].sub.i]. If all [[beta].sub.i] = 0, the ARMA model is a pure autoregressive (AR) model of order p. The key point to note is that the ARMA model is linear; all values of [y.sub.t-i] and [[epsilon].sub.t-i] are raised to the power 1 and there are no cross-products of the form of [y.sub.t-i][[epsilon].sub.t-j] or [y.sub.t-i][y.sub.t-j]. (3)

Threshold Autoregressive (TAR) Models

The threshold autoregressive models developed by Tong (1983, 1990) allow for a number of different regimes with a separate autoregressive model in each regime. In our Monte Carlo experiments, we focus on the simple two-regime TAR model, (4)

[y.sub.t] = [I.sub.t] [[alpha].sub.10] + [summation over (p/i=1)] [[alpha].sub.1i][y.sub.t-i]] + (1 - [I.sub.t]) [[[alpha].sub.20] + [summation over (p/i=1)] [[alpha].sub.2i][y.sub.t-i]] + [[epsilon].sub.t], (4)

where [I.sub.t] is the Heaviside indicator function such that

[I.sub.t] = {1 if [y.sub.t-1] [greater than or equal to] [tau]

{0 if [y.sub.t-1] < [tau]. (5)

In regime 1, [y.sub.t-1] [greater than or equal to] so that [I.sub.t] = 1, (1 - [I.sub.t]) = 0, and [y.sub.t] = [[alpha].sub.10] + [[alpha].sub.11][y.sub.t-1] + ... + [[alpha].sub.1p][y.sub.t-p] + [[epsilon].sub.t]. In regime 2, [y.sub.t-1] < [tau] so that [y.sub.t] = [[alpha].sub.20] + [[alpha].sub.21][y.sub.t-1] + ... + [[alpha].sub.2p][y.sub.t-p] + [[epsilon].sub.t]. Although {[y.sub.t]} is linear in each regime, the possibility of regime switching means that the entire {[y.sub.t]} sequence is nonlinear.

The momentum threshold autoregressive (M-TAR) model used by Enders and Granger (1998) allows the regime to change according to the first-difference of {[Y.sub.t-1]}. Hence, Equation (5) is replaced with

[I.sub.t] = {1 if [DELTA][y.sub.t-1] [greater than or equal to] [tau]

{0 if [DELTA][y.sub.t-1] < [tau]. (6)

It is argued that the M-TAR model is useful for capturing situations in which the degree of autoregressive decay depends on the direction of change in {[y.sub.t]}. Enders and Granger (1998) and Enders and Siklos (2001) show that interest rate adjustments to the term-structure relationship display M-TAR behavior. It is important to note that, for the TAR and M-TAR models, if all [[alpha].sub.1i] = [[alpha].sub.2i], the TAR and M-TAR models are equivalent to an AR(p) model.

If [tau] is known, the estimation of the TAR and M-TAR models is straightforward. Simply form the variables [y.sup.+.sub.t-i] = [I.sub.t][y.sub.t-i] and [y.sup.-.sub.t-i] = (1 - [I.sub.t])[y.sub.t-i] and estimate Equation (4) using ordinary least squares (OLS). (5) The lag length p can be determined as in an AR model. When [tau] is unknown, Chan (1993) shows how to obtain a superconsistent estimate of the threshold parameter. For a TAR model, the procedure is to order the observations from smallest to largest such that

[y.sup.0.sub.1] < [y.sup.0.sub.2] < [y.sup.0.sub.3] ... < [y.sup.0.sub.T]. (7)

For each value of [y.sup.0.sub.j], let [tau] = [y.sup.0.sub.j], set the Heaviside indicator according to Equation (5), and estimate an equation in the form of (4). The regression equation with the smallest residual sum of squares contains the consistent estimate of the threshold. In practice, the highest and lowest 10% of the {[y.sup.0.sub.j]} values are excluded from the grid search to ensure an adequate number of observations on each side of the threshold. For the M-TAR model, (7) is replaced by the ordered first differences of the observations.

Exponential Autoregressive (EAR) Models

EAR models were examined extensively by Ozaki and Oda (1978), Haggan and Ozaki (1981), and Lawrance and Lewis (1980). The form of the EAR model that we use in our Monte Carlo study is

[y.subl.t] = [[alpha].sub.0] + [summation over (p/i=1)] [[theta].sub.i][y.sub.t-i] + [[epsilon].sub.i], (8)

where [[theta].sub.i] = [[alpha].sub.i] + [[beta].sub.i] exp(-[gamma][y.sup.2.sub.t-1]) and [gamma] > 0 is the smoothness parameter.

In the limit as [gamma] [right arrow] 0 or [infinity], the EAR model becomes an AR(p) model because each [[theta].sub.i] is constant. Otherwise, the EAR model displays nonlinear behavior. For example, Equation (8) can capture a situation in which the periods surrounding the turning points of a series (i.e., periods in which [y.sup.2.sub.t-1] will be extreme) have different degrees of autoregressive decay than other periods.

Bilinear Autoregressive (BL) Models

The general form of the bilinear model BL(p, q, r, s) is

[y.sub.t] = [[alpha].sub.0] + [summation over (p/i=1)] [[alpha].sub.i][y.sub.t-i] + [[epsilon].sub.t] + [summation over (q/i=1)] [[beta].sub.i][[epsilon].sub.t-i] + [summation over (r/i=1)] [summation over (s/j=1)] [c.sub.ij][y.sub.t-i][[epsilon].sub.t-j]. (9)

Bilinear models are a natural extension of ARMA models in that they add the crossproducts of [y.sub.t-i] and [[epsilon].sub.t-j] to account for nonlinearity. If all values of [c.sub.ij] equal zero, the bilinear model reduces to the linear ARMA model. Priestley (1980) argues that bilinear models can approximate any reasonable nonlinear relationship. In our Monte Carlo work, we restrict 0< p [less than or equal to] 2 and r [less than or equal to] 2 and q and s to be no greater than 1.

Generalized Autoregressive (GAR) Models

The general form of a GAR model is

[y.sub.t] = [[alpha].sub.0] + [summation over (p/i=1)] [[alpha].sub.i][y.sub.t-i] + [summation over (q/i=1)] [summation over (r/j=1)] [summation over (s/k=1)] [summation over (u/l=1)] [[beta].sub.ijkl][y.sup.k.sub.t-i][y.sup.l.sub.t-j] + [[epsilon].sub.t], (10)

where p, q, r, s, and u are integers that are greater or equal to 1.

GAR models extend AR models by adding various powers of lagged values and cross-products of [y.sub.t-i]. Mittnik (1991) introduced the GAR model as an autoregressive analogue of the discrete Volterra series. Since GAR models are linear in their parameters, they can be estimated using OLS.

3. A Monte Carlo Study of Overfitting

In order to determine the extent of overfitting using standard in-sample and out-of-sample goodness-of-fit measures, we performed two different Monte Carlo experiments. The first experiment consisted of the following steps.

Step 1. We selected values for [[alpha].sub.0], [[alpha].sub.1], and [[alpha].sub.2] and generated an AR(2) series with 100 simulated observations of the form [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.t][y.sub.t-1] + [[alpha].sub.2][y.sub.t-2] + [[epsilon].sub.t]. The {[[epsilon].sub.t]} were pseudorandom numbers drawn from a normal (0, 1) distribution and the series was initialized by drawing [y.sub.1] and [y.sub.2] from a normal distribution with mean [[alpha].sub.0]/(l - [[alpha].sub.1] - [[alpha].sub.2]. Very little substance is lost by reporting results using only the five parameter sets: [[alpha].sub.0] = 1 and values of the autoregressive parameters such that ([[alpha].sub.1], [[alpha].sub.2]) = (0.5, 0.4), (1.2, -0.5), (0.0, 0.9), (0.9, 0.0), and (0.5, 0.0). For some parameterizations, [[alpha].sub.2] = 0 so that the AR(2) model reduces to an AR(l).

Step 2. We estimated the simulated {[y.sub.t]} series as an autoregressive process using the true value of p (i.e., p = 1 or 2). We recorded the AIC and SBC as measures of in-sample goodness of fit. Because the estimated model is consistent with the actual data-generating process, these values serve as a benchmark for the other estimates.

Step 3. We pretended we did not know the data-generating process and estimated the simulated {[y.sub.t]} series using each of the various linear and nonlinear models discussed above. Our specification search allowed for the following maximum orders: AR(3), ARMA(2, 1), TAR(3), M-TAR(3), EAR(3), and BL(2,1,2,1). For the OAR model, we used orders such that p, q, and r are all less than or equal to 3 and s + u [less than or equal to] 3.

To avoid being arbitrary in selecting the order of any model, we used a mechanized search procedure. For the AR, TAR, M-TAR, EAR, and GAR models, we created all possible variables (e.g., [y.sub.t-1][y.sub.t-2], [y.sup.2.sub.t-1], [I.sub.t][y.sub.t-1], ...) and used the standard stepwise regression procedure to find the order providing the "best fit" within that class of model. (6) Candidate variables were entered into the regression equation from the most to the least significant. A variable was allowed to remain in the regression if it were significant at the 5% level and any variable was dropped if it were not significant at the 5% level. (7) We needed to use a slightly different procedure for the ARMA and BL models because both have unobserved components. For these two classes of models, we estimated the full model and eliminated all coefficients that were not significant at the 5% significance level.

We recorded the value of the AIC and the value of the SBC for the model chosen from each category. The logic of the procedure is that, in extremely large samples, the AIC and the SBC should be smallest for an autoregressive model with p = 2 as compared with p = 1 or 3 and/or q = 1. Similarly, the EAR model entails the unnecessary estimation of the [[beta].sub.i], the bilinear model entails the unnecessary estimation of the [c.sub.ij], and the GAR model entails the unnecessary estimation of the [[beta].sub.pquv].

Because the true model is nested within these three specifications, the AIC and SBC values found in this step should equal those found in step 2. However, for any value of j, the TAR and M-TAR models entail the estimation of [[alpha].sub.1j] and [[alpha].sub.2j] instead of the one coefficient [[alpha].sub.j]. Given that it is also necessary to estimate [tau], we should expect the AIC and the SBC using the overparameterized TAR and M-TAR specifications to be greater than those found in step 2. (8)

Step 4. We use rolling regressions to obtain one-step-ahead forecasts from each of the estimated models and obtained the mean square prediction error (MSPE) from each. (9) Specifically, for each type of model, we first set T = 50, estimated an equation as in step 3 (so that the order may change), and obtained the one-step-ahead forecast for time period T + 1. Next, we updated T by one period, re-estimated the equation as in step 3, and obtained an additional one-step-ahead forecast. Repeating this process through the end of the sample period gave us 50 out-of-sample forecasts. The MSPE was then calculated for each model.

Step 5. We repeated steps 1-4 a total of 1000 times. Thus, we obtained the AIC, SBC, and MSPE of 1000 estimates of an AR(2) process estimated as a true AR(2) and as the best fitting AR, ARMA, TAR, M-TAR, EAR, GAR, and BL models.

Results from the First Experiment

The means of the AIC, SBC, and MSPE for each of the models are reported in Table 1 for various values of [[alpha].sub.0], [[alpha].sub.1], and [[alpha].sub.2]. Figures 1-3 show the cumulative distributions of the AIC, SBC, and MSPE for the specific parameter set [[alpha].sub.0] = 1.0, [[alpha].sub.1] = 0, and [[alpha].sub.2] = 0.90. Because the qualitative results are not sensitive to the parameter values, we can save a considerable amount of space by reporting only this set. For comparison purposes, the standard errors of the means are also reported in the table.

Properties of the AIC

The striking result for the AIC values in Table 1 is that the average value of the AIC is smaller for the TAR and the M-TAR models than for any of the other models [including the true AR(2) model]. For example, for [[alpha].sub.0] = 1.0, [[alpha].sub.1] = 0.5, and [[alpha].sub.2] = 0.4, the average value of the AIC over the 1000 Monte Carlo replications was 442.6899 using the M-TAR model and 445.5328 using the true AR(2) model. This is in spite of the fact that, for any given value of p, the M-TAR model entails twice the number of estimated coefficients (plus the value of [tau]). In fact, compared with most other models, the estimated AR(2) model has a relatively large average value of the AIC. Figure 1 shows that, for any given percentile, the AIC is always lowest for one of the two threshold models. Also note that the AIC values for the other models tend to cluster together.

Properties of the SBC

For the five sets of parameters reported in the values for the SBC of Table 1, the mean of the SBC is always smallest for the GAR or for the EAR model. Because the SBC penalizes additional parameters more than the AIC, it is not too surprising the SBC tends to be large for the TAR and MTAR models. This pattern is reinforced by the distributions shown in Figure 2. For any given percentile, the SBC values for the TAR and M-TAR models are noticeably greater than those for the true AR(2), ARMA, AR(p), BL, EAR, and GAR models. Nevertheless, the linear models still do not perform as well as the more general BL, EAR, and GAR models. The point is that the in-sample measures of goodness of fit tend to select a nonlinear model over the true linear model.

Properties of the MSPE

The situation is quite different when we consider the MSPE. As shown in the values for the MSPE of Table 1, the average value of the MSPE is always lowest for the true AR(2) model. The AR(p), ARMA, and BL specifications averaged smaller MSPEs than the other models. Thus, these three models tend to forecast better than the models that are not linear in {[y.sub.t]}. As shown in Figure 3, for any given percentile, the MSPE for the AR(2) model is always the smallest and those for the TAR and M-TAR models are far above the others. The linear AR(p) and ARMA models have a lower MSPE than any nonlinear models except, at some percentiles, the BL model. Thus, the out-of-sample forecasts tend to select the correct form of the model.

Results from the Second Experiment

In contrast with experiment 1, it is possible that the researcher is actually working with a nonlinear series. As such, we modified step 1 to allow for a nonlinear data-generating process. We report results using a TAR data-generating process because this model fit particularly well in experiment 1. The TAR model is also of interest because each segment of the model is actually a linear AR model. For various values of [[alpha].sub.10], [[alpha].sub.11], [[alpha].sub.20], [[alpha].sub.21], and [tau], we generated 1000 series of the form

[[gamma].sub.t] = {[[alpha].sub.10] + [[alpha].sub.11][[gamma].sub.t-1] + [[epsilon].sub.t] if [[gamma].sub.t-1] [greater than or equal to] [tau] [[alpha].sub.20] + [[alpha].sub.21][[gamma].sub.t-1] + [[epsilon].sub.t] if [[gamma].sub.t-] < [tau] (11)

We report results for the following five parameterizations:

   [[alpha].sub.10]  [[alpha].sub.11]  [[alpha].sub.20]

1        0.5               0.95              0.9
2        0.3               0.95              3.2
3        0.4               0.80              1.2
4        0.4               0.95              4.2
5        1.2               0.80              1.4

   [[alpha].sub.21]  [tau]  [[micro].sup.-]  [[micro].sup.+]

1        0.9          9.5         9.0             10.0
2        0.6          7.0         6.0              8.0
3        0.6          2.5         2.0              3.0
4        0.3          7.0         6.0              8.0
5        0.3          4.0         2.0              6.0

Figure 4 illustrates the type of threshold processes under consideration. Note that each parameterization is such that there are two long-run equilibrium values; [[mu].sub.+] is the long-run equilibrium when {[[gamma].sub.t-1]} is above the threshold and [[mu].sup.-] is the long-run equilibrium when {[[gamma].sub.t-1]} is below the threshold. Note that, for these parameterizations, the threshold is chosen to be between the long-run equilibrium values of the two regimes.

For parameterization 1, the difference between [[alpha].sub.11] and [[alpha].sub.21] is small and the two equilibria are relatively close together (9.0 and 10.0, respectively). Parameterizations 2 and 3 slightly modify the difference between [[alpha].sub.11] and [[alpha].sub.21] and the difference between the equilibrium values. Parameter set 5 has a large discrepancy between the two long-run equilibrium values, and both parameterizations 4 and 5 have large discrepancies between [[alpha].sub.11] and [[alpha].sub.21]. Hence, parameter set 1 yields a model that is closer to being linear than do parameter sets 4 and 5.

Each of the five processes was estimated by the TAR model in Equation (11) using only one lag of {[Y.sub.t-1]}. We also estimated each as a general AR(p) model, an ARMA(p, q) model, a TAR model with p lags, an M-TAR with p lags, a GAR model, a bilinear model, and an EAR model. As in the first experiment, the true form of the TAR model is estimated in order to serve as a benchmark. All other specifications are estimated as described in steps 2-5 above. The means of the AIC, SBC, and MSPE for each type of model are reported in Table 2 and the distributions of the AIC, SBC, and MSPE for parameter set 4 are shown in Figures 5-7.

The results are similar to that of experiment 1. As shown in Table 2, the TAR(p) models have the smallest average AIC among all the models for all parameterizations. The general TAR(p) model fits the data better than estimates using the true lag length. Thus, the AIC is smallest for one of the threshold models [i.e., the TAR(p) or M-TAR(p)] regardless of whether the true data-generating process is an AR(2) or a TAR(1). Regarding the other models, the average AIC is smallest for the true TAR functional form for parameter sets 4 and 5 but not for the other parameter sets. The SBC does not fare better than the AIC. For parameter set 5, the average SBC is smallest for the TAR(l) functional form. Otherwise, the average SBC is smallest for the GAR or the TAR(p) functional forms. The average MSPE is lowest for the true TAR(l) model using parameter sets 2, 4, and 5. For parameter sets 1 and 3, the average MSPE is smallest for the linear AR(p) model. However, when there is a substantial degree of nonlinearity as in pa rameter sets 4 and 5, the average MSPE is smallest for the true TAR(1) model. For parameter set 4, the EAR model has a smaller average MSPE than all models but the true TAR(l). For parameter set 5, the TAR(p) model has a smaller MSPE than all but the true TAR(1). Hence, the out-of-sample forecasts suggest the presence of a linear model when the data-generating process is linear and the presence of a nonlinear model when the true data-generating process exhibits a substantial amount of nonlinear behavior.

The distributions of the AIC, SBC, and MSPE for parameter set 4 shown in Figures 5-7 reveal the same pattern. For this parameter set, both the AIC and SBC are smallest for the TAR(p) model over the correct specification at every percentile. However, for all percentiles, the MSPE of the TAR(1) model is lower than that for any of the other specifications (except the true specification).

4. An Example Using the Term Structure

There is substantial support for the claim that interest rate adjustments toward the long-run term-structure relationship are asymmetric. An argument based on Guirauis (1994) is that the Federal Reserve is more concerned about a positive discrepancy between the actual and target inflation rate than a negative discrepancy. Hence, an increase in the long-term interest rate (representing increases in inflationary expectations) leads the Fed to quickly increase the Funds rate. If inflationary expectations (and long-term rates) decline, the Federal Reserve has little inclination to raise inflation by following a policy of monetary ease. As such, positive discrepancies between the long-term and the Fed Funds rate are less persistent than negative ones. Alternatively, Clark, Laxton, and Rose (2001) argue that optimal policy responses are nonlinear when positive aggregate demand shocks are more inflationary than negative shocks are more inflationary than negative shocks are disinflationary. A positive demand shock th at raises long-term interest rates will be followed by a sharp increase in the Fed Funds rate. Negative demand shocks will not elicit an equivalent policy response by the Federal Reserve.

A number of papers provide evidence that long-term and short-term interest rates are cointegrated such that the error-correction model displays some form of TAR adjustment. For example, Anderson (1997) estimates a smooth transition error-correction model of the U.S. treasury bill market. Alternatively, Balke and Fomby (1997) estimate the adjustment using the TAR specification, while Enders and Granger (1998), Enders and Siklos (2001), and Shin and Lee (2001) use the M-TAR specification. Because these papers do not perform any out-of-sample forecasting, it seems reasonable to apply the methodology used in the previous section to the interest rate differential. We obtained the natural logarithms of the monthly yields of the Fed funds rate ([r.sub.S]) and 10-year interest rate on U.S. government securities ([r.sub.L]) from the CD-ROM version of the International Financial Statistics over the 1979:10-1997:04. The Fed Funds rate (i.e., the short rate) is our instrument of monetary policy and the 10-year yield (i.e ., the long rate) is our indicator of inflationary expectations. As in Enders and Siklos (2001), our sample begins in 1979:10 due to changes in Federal Reserve operating procedures.

Define [Y.sub.t] as the difference between [r.sub.L] and [r.sub.S] in time period t. Although visual inspection of Figure 8 indicates that the spread may be I(1), Enders and Siklos reject the null hypothesis of a unit root in {[y.sub.t]} over this period. Hence, we proceed by estimating the spread in levels (and not first differences). (10) Moreover, we mimic the type of TAR and M-TAR models of the spread used in Enders and Granger (1998) and Enders and Siklos (2001). For the TAR model, they set the indicator function according to whether [Y.sub.t-1] is greater or less than zero. Similarly, for the M-TAR model, they set the indicator function according to whether [DELTA][y.sub.t-1] is greater or less than zero. Hence, positive discrepancies from the mean value of the spread exhibit a different degree of persistence than do negative discrepancies. For notational purposes, we denote threshold models estimated in this fashion with a 0 appearing in parentheses; hence, TAR(0) and M-TAR(0) refer to threshold models where the threshold is constrained to be zero.

In -Sample Performance

Table 3 reports the estimates of the various models of the interest rate spread. The ARMA(1, 1) fits better, both in-sample and out-of-sample, than other ARMA(p, q) or AR(p) models. In order to conserve space, we report the estimates of the ARMA(1, 1) model but not the other linear models. Respectively, the AIC and SBC of the ARMA(1, 1) model are -44.14 and -34.14. The estimated residual autocorrelation function and the Ljung--Box Q statistic suggest that the model is adequate in the sense that there is little linear structure in the residuals. However, the in-sample measures of goodness of fit do not select the ARMA(1, 1). The last column of the table (labeled Ratio) reports the standard error of the ARMA(1, 1) divided by the standard error of the associated nonlinear model. Moreover, the AIC for the ARMA( 1, l)is larger than that of any other estimated model and the SBC selects the ARMA(1, 1) only over the M-TAR model.

As measured by the AIC and the SBC, the TAR model with a consistent estimate of the threshold fits better than the model using a zero value of the threshold. Both of these models fit better than the M-TAR of M-TAR(0) models. In fact, the AIC selects the TAR model over all other models and the SBC selects the TAR model over all models but the GAR model.

Out-of-Sample Performance

Next, we obtained a total of 111 one-step-ahead out-of-sample forecasts beginning with 1988:2. For each class of model, we used the order that was found over the full sample period but allowed the specific parameter values to change from one period to the next. For each set of out-of-sample forecasts, we measure the bias as

bias = [summation over (n/t=1)] ([y.sub.t] - [y.sub.t])/n,

where [y.sub.t] is the forecasted value of [y.sub.t] and n is the number of forecasts.

For each model, the bias and the ratio of the MSPE to that of the ARMA(1, 1) are shown in Table 4. The bias for the ARMA(l, 1) model is in the midrange of that for all the other models. However, there is no evidence of significant forecast bias for any class of model. As shown in Table 4, the absolute value of the t-statistic for the null hypothesis bias = 0 is always less than 1.96.

The MSPE of the ARMA(l, 1) model also ranks in the middle range. Note that the GAR model forecasts better than any of the other models. The TAR(0), M-TAR, and bilinear models also generate lower MSPEs than the ARMA(1, 1) model. Although the TAR model had superior in-sample goodness-of-fit measures, it fits very poorly out-of-sample. In contrast, the GAR model has the lowest MSPE and the lowest value of the SBC. Moreover, the AIC selected only one other model over the GAR model. We consider this to be strong evidence in favor of the GAR model over the TAR and M-TAR specifications previously used for the spread. Nevertheless, when we applied Mizrach's (1995) test, we were not able to reject the null hypothesis that the MSPE from the GAR model is statistically different from that of any of the other models. Hence, it is an open question as to which model is actually the most appropriate.

The TAR and GAR models make for an interesting comparison because the "skeleton" of each tells a very different story regarding the behavior of spread. In brief, the skeleton refers to the model without the error term and captures the properties of the underlying nonlinear difference equation. For example, the skeleton of the TAR model with the consistent estimate of the threshold is

[y.sub.t] = 1.189[y.sub.t-1] - 0.202[y.sub.t-2] if [y.sub.t-1] [greater than or equal to] - 0.083 (12)

[y.sub.t] = -0.134 + 1.100[y.sub.t-1] - 1.294[y.sub.t-2] + 0.840[y.sub.t-3] - 0.390[y.sub.t-4] if [y.sub.t-1] < 0.083. (13)

Equations (12) and (13) indicate that there are two equilibrium values for the spread. When [y.sub.t-1] [greater than or equal to] -0.083, the spread will slowly gravitate toward the attractor of zero and exhibit near random-walk behavior. (11) Instead, when [y.sub.t-1] < -0.083, the attractor is -0.18 [-0.134 / (1 - 1.1 + 1.294 - 0.84 + 0.39) [approximately equal to] -0.18]. Because the sum of the autoregressive coefficients is negative, the approach toward the attractor tends to be oscillatory. The skeleton of the estimated GAR model is

[y.sub.t] = 1.242[y.sub.t-1] - 0.338[y.sub.t-2] - 0.611[y.sup.3.sub.t-1] + 0.540[y.sub.t-1] [y.sub.t-2]. (14)

Equation (14) is a cubic equation containing the three equilibrium values: 0.0, 0.2467, and 0.6372. However, it can be shown that the equilibrium surrounding 0.2467 is unstable; as such, there are only two attractors. For example, if both [y.sub.t] and [y.sub.t-1] are below 0.2647, the subsequent values of the spread will gravitate toward zero, and if both are above 0.2647, the spread will gravitate toward 0.6372. Hence, the TAR and GAR models find a zero value for an attractor. However, the alternative attractor of 0.6372 seems more plausible than the value of -0.18 found by the TAR model. If you look at Figure 8, it should be clear that the GAR model also captures the high-spread period that prevailed in the early 1990s.

5. Conclusions

One difficulty in fitting a nonlinear model to an economic timeseries is to ascertain the precise nature of the nonlinearity. In-sample measures tend to overfit the data and Lagrange multiplier tests may support various forms of nonlinearity. It was shown that out-of-sample forecasting can aid in selecting the appropriate specification of the nonlinearity. To examine the nature of various nonlinear estimates, we generated a large number of series from second-order autoregressive processes. The best nonlinear model was compared with the best linear model using in-sample and out-of-sample criteria. In small samples, if the data-generating process is linear, we verified the well-known result that the standard goodness-of-fit measures typically lead to a specification that overfits the data. Instead, the mean square prediction error tends to select the correct linear specification. The experiment was repeated using a nonlinear data-generating process. The in-sample and out-of-sample properties of various linear a nd nonlinear models were again compared. The AIC tends to select a linear model and the SBC fares only slightly better. Instead, the mean square prediction error tends to select a nonlinear model when there is a substantial amount of nonlinearity in the data-generating process. An example was provided using the term structure of interest rates. Both the in-sample and out-of-sample measures of fit tended to select the nonlinear GAR functional form.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Table 1

Means and Standard Errors of the AIC, SBC, and MSPE for Experiment 1

[[alpha].sub.1]      [[alpha].sub.2]    True        ARMA         BL

Values for the AIC
 0.5                        0.4       445.5328    445.4386    445.6017
                                       (0.466)     (0.471)     (0.480)
 1.2                       -0.5       446.2637    446.0124    446.0132
                                       (0.461)     (0.461)     (0.472)
 0.0                        0.9       444.8585    444.5904    444.5625
                                       (0.466)     (0.467)     (0.466)
 0.9                        0.0       444.2697    445.1879    445.7982
                                       (0.466)     (0.542)     (0.575)
 0.5                        0.0       444.4369    444.0585    444.3319
                                       (0.466)     (0.466)     (0.475)
Values for the SBC
 0.5                        0.4       453.2569    452.2976    453.0503
                                       (0.466)     (0.470)     (0.483)
 1.2                       -0.5       453.9878    453.9271    454.4274
                                       (0.461)     (0.461)     (0.475)
 0.0                        0.9       450.0080    449.5004    449.9231
                                       (0.466)     (0.467)     (0.470)
 0.9                        0.0       449.4191    450.5768    451.7123
                                       (0.466)     (0.543)     (0.582)
 0.5                        0.0       449.5863    449.7615    450.7172
                                       (0.466)     (0.466)     (0.477)
Values for the MSPE
 0.5                        0.4         1.0664      1.1125      1.1086
                                       (0.00708)   (0.00772)   (0.00798)
 1.2                       -0.5         1.0572      1.0719      1.0726
                                       (0.00689)   (0.00708)   (0.00745)
 0.0                        0.9         1.0440      1.0769      1.0737
                                       (0.00683)   (0.00710)   (0.00711)
 0.9                        0.0         1.0497      1.0985      1.1077
                                       (0.00674)   (0.00878)   (0.00963)
00.5                        0.0         1.0371      1.0533      1.0524
                                       (0.00666)   (0.00690)   (0.00695)

[[alpha].sub.1]         AR          EAR         GAR         TAR

Values for the AIC
 0.5                 445.6361    445.6199    444.4124    442.9117
                      (0.470)     (0.469)     (0.472)     (0.480)
 1.2                 446.1329    445.7208    447.3273    445.2952
                      (0.461)     (0.462)     (0.470)     (0.473)
 0.0                 444.6688    443.8898    443.8698    442.0580
                      (0.467)     (0.470)     (0.468)     (0.477)
 0.9                 444.0751    443.4854    443.0844    441.8941
                      (0.468)     (0.468)     (0.466)     (0.478)
 0.5                 444.0712    443.7558    443.2822    442.1763
                      (0.466)     (0.468)     (0.474)     (0.466)
Values for the SBC
 0.5                 452.4848    452.4763    451.3873    455.8985
                      (0.469)     (0.468)     (0.471)     (0.488)
 1.2                 453.9136    453.6509    455.0282    462.0489
                      (0.461)     (0.463)     (0.469)     (0.482)
 0.0                 449.4681    448.7998    448.8493    454.3626
                      (0.467)     (0.470)     (0.468)     (0.479)
 0.9                 448.998     448.5344    448.0923    453.0477
                      (0.467)     (0.468)     (0.466)     (0.483)
 0.5                 449.4498    449.2683    448.9517    453.8475
                      (0.466)     (0.467)     (0.466)     (0.481)
Values for the MSPE
 0.5                   1.1208      1.1220      1.1222      1.2804
                      (0.00775)   (0.00774)   (0.00780)   (0.01070)
 1.2                   1.0749      1.0978      1.2152      1.2824
                      (0.00712)   (0.00750)   (0.00963)   (0.01044)
 0.0                   1.0796      1.0841      1.0969      1.2874
                      (0.00712)   (0.00725)   (0.00737)   (0.01135)
 0.9                   1.0841      1.0885      1.1006      1.2454
                      (0.00712)   (0.00720)   (0.00735)   (0.00988)
00.5                   1.0558      1.0793      1.0904      1.2169
                      (0.00694)   (0.00735)   (0.00757)   (0.00922)

[[alpha].sub.1]       M/TAR

Values for the AIC
 0.5                 442.6889
                      (0.471)
 1.2                 444.8783
                      (0.475)
 0.0                 441.9651
                      (0.478)
 0.9                 441.9085
                      (0.477)
 0.5                 442.2259
                      (0.476)
Values for the SBC
 0.5                 455.3307
                      (0.486)
 1.2                 460.2725
                      (0.479)
 0.0                 454.1924
                      (0.481)
 0.9                 453.7624
                      (0.476)
 0.5                 454.6335
                      (0.483)
Values for the MSPE
 0.5                   1.2695
                      (0.00982)
 1.2                   1.2389
                      (0.00954)
 0.0                   1.2772
                      (0.01054)
 0.9                   1.2334
                      (0.00965)
00.5                   1.2044
                      (0.00891)

Each cell reports the mean value of the AIC, SBC, or MSPE along with the
standard error of the mean. The standard error of the mean is in
parentheses.

The underlined numbers are the smallest among all the models for the
AIC, SBC, and MSPE for each parameter vector.

The bold numbers are the smallest MSPEs except the true AR(2) models.

Table 2

Means and Standard Errors of the AIC, SBC, and MSPE for Experiment 2

Parameter Set          True       ARMA        BL         AR

Values for the AIC
 1                   446.0258   445.1133   445.5999   444.0961
                      (0.463)    (0.527)    (0.573)    (0.461)
 2                   446.1787   447.8633   448.3674   447.0731
                      (0.462)    (0.494)    (0.535)    (0.463)
 3                   445.6452   444.7521   445.0513   444.5751
                      (0.463)    (0.458)    (0.476)    (0.457)
 4                   446.0276   452.5516   452.5603   452.2506
                      (0.464)    (0.482)    (0.531)    (0.474)
 5                   449.3015   464.6882   464.8412   464.3222
                      (0.444)    (0.515)    (0.610)    (0.483)

Values for the SBC
 1                   456.3246   449.9461   450.8858   448.4500
                      (0.463)    (0.526)    (0.578)    (0.459)
 2                   456.4775   453.5251   454.9998   452.3152
                      (0.461)    (0.490)    (0.537)    (0.462)
 3                   455.9440   450.4577   451.5653   449.9408
                      (0.463)    (0.456)    (0.478)    (0.456)
 4                   456.3265   457.8941   459.1310   457.1658
                      (0.464)    (0.483)    (0.537)    (0.473)
 5                   459.6004   469.7784   471.2084   469.0236
                      (0.444)    (0.515)    (0.612)    (0.483)

Values for the MSPE
 1                    1.0932     1.0867     1.0980     1.0687
                     (0.00758)  (0.00959)  (0.01140)  (0.00706)
 2                    1.0750     1.1122     1.1172     1.1014
                     (0.00807)  (0.00813)  (0.00874)  (0.00757)
 3                    1.0656     1.0573     1.0577     1.0565
                     (0.00697)  (0.00679)  (0.00697)  (0.00678)
 4                    1.0822     1.1727     1.1710     1.1697
                     (0.00745)  (0.00862)  (0.01019)  (0.00832)
 5                    1.0983     1.2937     1.2910     1.2929
                     (0.00748)  (0.01057)  (0.01262)  (0.00979)

Parameter Set           EAR        GAR        TAR       M-TAR

Values for the AIC
 1                   443.5589   443.1783   441.4928   441.8919
                      (0.462)    (0.462)    (0.474)    (0.470)
 2                   445.9150   445.9794   441.7442   444.8020
                      (0.466)    (0.465)    (0.468)    (0.470)
 3                   443.9089   443.5974   441.4967   442.7876
                      (0.459)    (0.460)    (0.473)    (0.461)
 4                   450.5393   450.4730   442.6870   448.6649
                      (0.473)    (0.476)    (0.475)    (0.476)
 5                   462.2699   462.0072   449.1756   460.6419
                      (0.485)    (0.494)    (0.449)    (0.484)

Values for the SBC
 1                   448.0852   447.7690   452.2628   453.2310
                      (0.460)    (0.460)    (0.479)    (0.471)
 2                   451.5382   451.2653   452.7459   456.8362
                      (0.466)    (0.463)    (0.474)    (0.470)
 3                   449.5604   449.1356   452.8589   455.2080
                      (0.457)    (0.457)    (0.458)    (0.476)
 4                   455.9462   455.6018   453.1687    46.0426
                      (0.473)    (0.473)    (0.482)    (0.475)
 5                   467.3730   467.2828   460.4863   471.9192
                      (0.484)    (0.489)    (0.455)    (0.484)

Values for the MSPE
 1                    1.0707     1.0845     1.2366     1.2272
                     (0.00702)  (0.00734)  (0.01035)  (0.01010)
 2                    1.1108     1.1212     1.2117     1.2638
                     (0.00764)  (0.00789)  (0.00978)  (0.01022)
 3                    1.0800     1.0868     1.2042     1.2163
                     (0.00724)  (0.00748)  (0.00954)  (0.00886)
 4                    1.1640     1.1771     1.2156     1.3385
                     (0.00819)  (0.00863)  (0.00979)  (0.01165)
 5                    1.2807     1.3060     1.2558     1.4461
                     (0.00976)  (0.01055)  (0.01052)  (0.01502)

Each cell reports the mean value of the AIC, SBC, or MSPE along with the
standard error of the mean. The standard error of the mean is in
parentheses.

The underlined numbers as the smallest among all the models for the AIC,
SBC, and MSPE for each parameter vector.

The bold numbers are the smallest MSPEs except the true AR(2) models.

Table 3

Estimated Models of the Spread

Model Type          Estimated Model (a)             AIC     SBC    Ratio

ARMA(1, 1)    [y.sub.t] = 0.246 + 0.940            -44.14  -34.14
              [y.sub.t - 1] + 0.288 [epsilon.sub.
              t - 1] (2.64) (41.63) (4.16)
TAR           [y.sub.t] = 1.189[y.sub.t - 1] -     -80.13  -53.47  0.798
(consistent)  0.202[y.sub.t - 2] if [y.sub.t - 1]
              [greater than or equal to] -0.083
              (16.73) (-2.86)
              [y.sub.t] = -0.134 + 1.100[y.sub.t
              - 1] - 1.294[y.sub.t - 2] + 0.840
              [y.sub.t - 3] - (-4.46) (6.22)
              (-5.05) (3.23)
TAR(0)        [y.sub.t] = 0.990[y.sub.t - 1] if    -62.68  -42.68  0.887
              [y.sub.t - 1] [greater than or
              equal to] 0 (75.84)
              [y.sub.t] = -0.049 + 1.165
              [y.sub.t - 1] - 0.988[y.sub.t - 2]
              + 0.718[y.sub.t - 3] - 0.289
              [y.sub.t - 4] if [y.sub.t - 1] < 0
              (-2.90) (9.23) (-4.88) (3.03)
              (-1.98)
M-TAR         [y.sub.t] = 0.038 + 0.918            -48.42  -28.42  0.950
(consistent)  [y.sub.t - 1] if [DELTA][y.sub. t -
              1] [greater than or equal to]
              0.0309 (3.49) (32.43)
              [y.sub.t] = 1.267[y.sub.t - 1] -
              0.580[y.sub.t - 2] + 0.306[y.sub.t
              - 3] if [DELTA][y.sub.t - 1] <
              0.0309 (11.99) (-3.88) (3.38)
M-TAR(0)      [y.sub.t] = 0.032 + 0.918[y.sub. t   -53.05  -36.39  0.939
              - 1] if [DELTA][y.sub.t - 1]
              [greater than or equal to] 0 (3.92)
              (37.54)
              [y.sub.t] = 1.279[y.sub.t - 1] -
              0.635[y.sub.t - 2] + 0.356[y.sub.t
              - 3] if [DELTA][y.sub.t - 1] < 0
              (11.10) (-3.81) (3.47)
GAR           [y.sub.t] = 1.242[y.sub.t - 1] -     -69.53  -56.20  0.874
              0.338[y.sub.t - 2] - 0.611 -
              [y.sup.3.sub.t - 1] + 0.540[y.sub.t
              - 1][y.sub.t - 2] (18.58) (-5.09) 9
              (-5.04) (6.03)
EAR           [y.sub.t] = 1.336 exp(-[y.sup.2.     -51.08  -37.75  0.949
              sub.t - 1] + [0.850 - 1.378 exp
              (-[y.sup.2.sub.t - 1])][y.sup.t - 2
              ] + 0.165[y.sub.t - 3] (17.73)
              (8.57) (-13.28) (2.44)
BILINEAR      [y.sub.t] = 1.461[y.sub.t - 1] -     -65.14  -48.44  0.88
              (0.465[y.sub.t - 2] - 1.864[y.sub.t
              - 1][[epsilon].sub.t - 1] + 0.959
              [y.sub.t - 1][[epsilon].sub.t - 2]
              (23.75) (-7.65) (-12.38) (7.99)

(a)t-Statistics are in parentheses.

Table 4

Bias and MSPE Ratios of Different Models for the Spread

Model        Bias (a)  MSPE Ratio (b)

ARMA(1, 1)   -0.005
            (-0.99)
TAR          -0.008        1.216
            (-1.54)       (0.92)
TAR(0)       -0.005        0.965
            (-1.04)       (0.97)
M-TAR         0.001        0.916
             (0.21)       (0.85)
M-TAR(0)     -0.001        1.005
            (-0.23)       (0.99)
GAR          -0.001        0.763
            (-0.26)       (0.81)
EAR          -0.007        1.102
            (-1.41)       (0.91)
BILINEAR     -0.004        0.856
            (-0.96)       (0.93)

(a)The t-statistic is in parentheses.

(b)MSPE ratios are the ratios of MSPEs the nonlinear models to that
of the ARMA model; the numbers in parentheses are the p-values for
Mizrach's (1995) robust forecast comparison statistic, testing that
the MSPE ratio equals 1.

Received April 2001; accepted April 2002.

(1.) The forms of the AIC and SBC we use are AIC = T In(RSS) + 2 k and SBC = T In(RSS) + k In(T), where RSS is the residual sum of squares, T is the number of usable observations, and k is the number of estimated parameters. For threshold models with a consistent estimate of the threshold. k incorporates the estimated threshold parameter.

(2.) We assume that the degree of differencing is appropriate to render {[y.sub.t]} stationary.

(3.) The ARMA model has unobserved components and the likelihood function is nonlinear in the values of [[alpha].sub.1] and [[beta].sub.i]. Because there is no closed-form solution for the maximum likelihood estimates, nonlinear search methods, such as the Simplex or Bemdt-Hall-Halt-Hausman (BHHH) algorithms, are needed to find the parameter estimates that maximize the likelihood function. Nevertheless, we refer to the ARMA model as being linear because the estimated model is linear in the various [y.sub.t-i] and [[epsilon].sub.i].

(4.) The lag lengths of the two regions of the TAR model may differ. For simplicity, p denotes the longest lag length. Moreover, in the general TAR model, the indicator can be set according to the value of [y.sub.t-d]. We consider only the case of d = 1.

(5.) To allow for the different intercepts in the TAR models, we included [l.sub.t] and (1 - [I.sub.t]) as regressors.

(6.) An alternative procedure would be to estimate every possible model and to use the one with the lowest AIC, SBC, and MSPE. Because we needed 50 out-of-sample forecasts for each class of model, this procedure was excessively time intensive for the TAR and M-TAR models. For each T, we also had to search over alt possible values of [tau] for the TAR and M-TAR models.

(7.) For the TAR and M-TAR models, we performed the stepwise estimation across all allowable values of [tau]. Hence, for each [tau], we created the variables [I.sub.t.], (I - [I.sub.t]), [I.sub.t][y.sub.t-1], [I - [I.sub.t][y.sub.t-1]], ... and performed the stepwise estimations. This procedure gave a regression for each value of [tau] containing only coefficients that were significant at the 5% level. From this set of regressions, we selected the "best" regression (and the associated value of [tau]) using the AIC and the "best" using the SBC.

(8.) Clearly, setting [tau] equal to zero does not yield a linear specification. Hence, our specification search beginning with the TAR or M-TAR models cannot he pared down to the linear AR(2) model.

(9.) Most nonlinear models are not especially useful for multistep-ahead forecasts. Consider the simple GAR model [y.sub.t] = [[beta].sub.1211[y.sub.t-1][y.sub.t-2] + [[epsilon].sub.t]. Given that [y.sub.t], [y.sub.t-1], and [y.sub.t-2] are observed, it is straightforward to forecast [y.sub.t] as [[beta].sub.1211[y.sub.t][y.sub.t-1]. However, recursive multistep-ahead forecasts are not possible because [E.sub.t][y.sub.t+3] = [[beta].sub.1211[E.sub.t][y.sub.t+2][y.sub.t+1] [not equal to] [[beta].sub.1211[E.sub.t][y.sub.t+2][E.sub.t][y.sub.t+1]. As in Rothman (1998), it is common to linearize a nonlinear model in order to obtain multistep-ahead forecasts. As such, we confine ourselves to one-step-ahead forecasts.

(10.) The results in this section are qualitatively similar to using first differences of the spread. To conserve space, we do not report the results here. Details are available from the authors.

(11.) Chen and Tsay (1991) discuss conditions such that one regime of a TAR process can have a unit root. Also, as discussed in Koop, Pesaran, and Potter (1996), the skeleton alone is not sufficient to characterize the impulse responses of a nonlinear time-series equation.

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Yamei Liu *

Walter Enders +

* Decision Science Group, 3CCOC, 6400 Las Colinas Building, Irving, TX 75039, USA; E-mail [e-mail].

+ Department of Economics, Finance and Legal Studies, Culverhouse College of Business, University of Alabama, Tuscaloosa, AL 35487, USA; E-mail [e-mail]; corresponding author.

We thank two anonymous referees for their helpful suggestions.