A Threshold Model of Real U.S. GDP and the Problem of Constructing Confidence. Intervals in TAR Models

Transcription

1 A Threshold Model of Real U.S. GDP and the Problem of Constructing Confidence Intervals in TAR Models Walter Enders Department of Economics, Finance and Legal Studies University of Alabama Tuscalooosa, AL Barry Falk Department of Economics Iowa State University Ames, IA and Pierre L. Siklos Department of Economics Wilfrid Laurier University Waterloo, ON N2L 3C5 CANADA Abstract We estimate real U.S. GDP growth as a threshold autoregressive process, and construct confidence intervals for the parameter estimates. However, there are various approaches that can be used in constructing the confidence intervals. Specifically, standard-t, bootstrap-t, and bootstrap-percentile confidence intervals are simulated for the slope coefficients and the estimated threshold. However, the results for the different methods have very different economic implications. We perform a Monte Carlo experiment to evaluate the various methods. JEL Classifications: C12, C22, E17 KeyWords: Bootstrap, Threshold Autoregressive, Confidence Intervals

2 1 I. Introduction Threshold autoregressive (TAR) models are popular, providing a straightforward, economically appealing, and econometrically tractable nonlinear extension of the linear autoregressive (AR) model. TAR models are particularly suited for time-series processes that are subject to periodic shifts due to regime changes. Examples of their application include Burgess (1992), Cao and Tsay (1992), Enders and Granger (1998), Galbraith (1996), Hansen (1997), Krager and Kugler (1993), Potter (1995), and Rothman (1991). The consensus opinion seems to be that the growth rate of real US GDP is a nonlinear process, perhaps of the threshold autoregressive variety (e.g., Potter 1995). Several papers, such as Potter (1995), presume that regimes with negative GDP growth behave very differently from positive regimes. However, there are a number of reasonable alternatives for conducting inference when working with a threshold autoregressive (TAR) process. We one aim of this paper is to show that the different methods of constructing confidence intervals in a TAR model have very different implications for the behavior real GDP. Although there have been a number of important developments in the asymptotic theory for estimation and inference in TAR models [e.g., Hansen (1997) and Chan and Tsay (1998), Gonzalo and Wolf (2001)], there has been relatively less research about the finite sample properties of these procedures. There are a number of ways to perform inference in TAR models and the second of our aims is to explore the small-sample properties of the alternative methods. One complicating factor is the need to know if the process is continuous at the threshold. The issue is important as a comparison of Hansen (1997) and Chan and Tsay (1998) indicates that the distributions relevant for testing a continuous (C-TAR) model are different from those for a discontinuous (D-TAR) model. Whereas economic analysis may predict the existence of

3 2 different regimes, it may not be clear whether a C-TAR or a D-TAR model is most appropriate. Enders and Siklos (2004) show that it is the combined values of the intercepts, threshold and autoregressive coefficients that determine whether the model is continuous at the threshold. Although a C-TAR model can be viewed as a restricted version of a D-TAR model, their work shows that conventional testing does not permit the investigator to test the restriction since it is not pivotal. To address some of these these issues, we apply Monte Carlo methods to study the small sample coverage properties of confidence intervals for the slope coefficients and the threshold parameter in the class of first-order, stationary, two regime self-exciting threshold autoregressive models (i.e., stationary TAR(1) models). The procedures we consider include approaches suggested by asymptotic theory and bootstrap procedures. Since a general-to-specific modeling strategy would lead one to opt for a discontinuous TAR model we show that confidence intervals are poorly estimated under a variety of conditions. Indeed, we show that methods used to estimate confidence intervals of the threshold perform poorly in both the continuous and discontinuous cases. As a result, the appropriate way to conduct inference in TAR models is unclear and our results cast doubt on some standard methods. These results are applicable to the study of real US GDP growth. Although this series has been subjected to more careful scrutiny than any other in the macroeconometric literature, there is no a priori way of knowing whether it behaves as a C-TAR or a D-TAR process. We show that some methods of constructing confidence intervals indicate that regimes with negative GDP growth behave very differently from positive regimes. However, depending on the method used to construct the confidence intervals, it may not be possible to rule out a positive value for the threshold. As a result, concluding that there are different degrees of persistence in positive

4 3 versus negative regimes may be problematic throwing into doubt some widely held beliefs concerning the properties of US real GDP growth. II. The TAR(1) Model The TAR(1) model can be formulated as follows: (1) y t = (α 0 + α 1 y t-1 )I t + (β 0 + β 1 y t-1 ) (1 I t ) + ε t t = 1,, T where: I t is the indicator function defined in terms of the threshold parameter τ: I t 1 = 0 if if y y t 1 t 1 > τ τ and: the ε t s are i.i.d. (0, σ 2 ) random variables. We assume that σ 2 < and that the autoregressive parameters in (1) satisfy stationarity conditions. Sufficient conditions for stationarity are: 0 < α 1 < 1 and 0 < β 1 < 1. Petrucelli and Woolford (1984) provide further discussion of conditions for stationarity and ergodicity. This model has sometimes been studied and applied under the assumption that the threshold parameter is known (and equal to 0). We will also consider the inference problem for the following constrained version of (1): (2) y t = τ + α 1 (y t-1 - τ)i t + β 1 (y t-1 τ)(1 I t ) + ε t t = 1,,T where: τ denotes the thresholds parameter, α 1 and β 1 are slope coefficients that satisfy stationarity conditions, and I t and ε t are stochastic processes defined as in (1). Model (2) implies that y t has a unique long-run equilibrium, which is equal to the threshold parameter. Short-run dynamics depend on whether the system is above or below the long-run equilibrium. In some applications, this version of the model may be a more natural representation of the TAR model than version (1). Note that, in contrast to version (1), version (2) implies that y t is continuous in the neighborhood of the threshold. In the remainder of the

5 4 paper we will refer to the discontinuous TAR model given by (1) as the D-TAR(1) model and we will refer to the continuous TAR model given by (2) as the C-TAR(1) model. Extensions of these models allowing for unit roots and higher order autoregressive terms have been considered in the theoretical and applied literature. Note that if α 1 = β 1 (and in (1), α 0 = β 0 ) the TAR(1) model collapses to an AR(1) model. However, testing this restriction complicated by the failure of τ to be identified under the null hypothesis. 1 The ordinary least squares estimator of the α s and β s in (1) [and (2)] is asymptotically efficient when the threshold parameter is known or when it is unknown but is replaced by a consistent estimator. Thus, the nonlinear least squares estimator produces asymptotically efficient estimates of the intercepts and slope coefficients. In particular, conditioning on τ or a consistent estimator of τ, the least squares t-statistics associated with α o, α 1, β 0, and β 1 converge in distribution to standard normal random variables. Consequently, the standard textbook approach to confidence interval construction provides intervals with asymptotically correct coverage. However, even in the special case of the linear AR(1) model, the OLS t-statistics for the slope coefficient will not be approximately normal or even approximately pivotal in small samples, particularly for values close to the unit root boundary. Monte Carlo simulations by Hansen (1999) illustrates the poor finite sample performance of normal confidence intervals, bootstrap-t, and bootstrap-percentile confidence intervals for the AR(1) model. We will apply Monte Carlo simulations to evaluate the finite sample performance of these intervals for the slope coefficients in (1) and (2) when the threshold is known and when the threshold is estimated by nonlinear least squares.

6 5 Confidence interval construction for the threshold parameter τ has been considered by Hansen (1997) for model (1) and by Chan and Tsay (1998) for model (2). Both procedures involve inverting a likelihood ratio statistic constructed from the nonlinear least squares estimator. Hansen s procedure is based on the limiting distribution of the likelihood ratio statistic for model (1), a nonstandard distribution derived by Hansen (1996). Chan and Tsay (1998) show that the limiting distribution of the likelihood ratio statistic is chi-square for model (2). The difference in the asymptotic behavior of the likelihood ratio statistic according to whether model (1) or model (2) is assumed is what motivates our consideration of both models. We apply Monte Carlo simulations to evaluate the finite sample performance of these procedures as well as intervals constructed from bootstrapped distributions of these (asymptotically pivotal) statistics. For those cases where the threshold parameter is known or is replaced with a consistent estimator (i.e., the nonlinear least squares estimator), we will examine the finite sample properties of the following types of confidence intervals for the slope parameters: intervals constructed from the t-statistics for α 1 and β 1 using the student-t distribution; bootstrap-t confidence intervals (which assume that the t-statistics are approximately pivotal though not necessarily student-t); bootstrap-percentile intervals. 2 Confidence intervals for the threshold parameter itself will be constructed from inversion of the likelihood ratio statistic using the asymptotically correct critical values and using its bootstrapped distribution.

7 6 III. Confidence Intervals a. Confidence Intervals for the Slope Coefficients: Known Threshold For each parameterization of the TAR model, 1000 realizations of y 1,, y T were generated for T = 50, 100 and The threshold parameter τ was always set to 0, the initial value y 0 was set to the unconditional mean of the process (= 0) and the ε t s were drawn from the standard normal distribution. For each realization, the model was fitted using ordinary least squares. The OLS t-statistics were used to construct confidence intervals with nominal coverage equal to 0.9, 0.95, and 0.99 for each of the two slope coefficients using student-t distributions for the t-statistics. The OLS estimates were used to construct 1000 bootstrap samples, which were used to construct the bootstrap-percentile and bootstrap-t intervals with nominal coverage equal to.9,.95, and.99. Actual coverage percentages were computed as the proportion of times over 1000 realizations that the true slope coefficients fell into each type of constructed interval. The following parameterizations of the D-TAR(1) model were used: α 1 = 0, β 0 = 0, α 0 = {0.3, 0.6, 0.9}, and β 1 = {0.3,0.6,0.9,0.95}. These include but expand the parameterizations considered by Hansen (1996). The parameterizations used for the C-TAR(1) model were: (α 1,β 1 ) = {(0.3,0.3), (0.3,0.6), (0.3,0.9), (0.3,0.95), (0.6,0.6), (0.6,0.9), (0.6,0.95), (0.9,0.9), (0.9,0.95), (0.95,0.95)}. The results for the D-TAR model and the C-TAR model are summarized in Tables 1 and 2, respectively, for T equal to 100 and nominal coverage equal to Results for other values of T (50,250) and nominal coverage (.95,.99) were not qualitatively different from those for T = 100 and 90-percent nominal coverage and so are not reported here. Tables 1 and 2 report two characteristics of the Monte-Carlo confidence intervals: the actual coverage attained and a measure of the symmetry of the intervals. The symmetry measure

8 7 was constructed as follows. An ideal 90-percent confidence interval should have coverage of 90- percent with the true parameter falling below the lower bound of the interval five percent of the time and falling above the upper bound of the interval five percent of the time. Let B denote the actual percentage of times that the parameter falls below the lower bound of the interval and let A denote the actual percentage of times it falls above the upper bound. Then the symmetry measure we construct is S = [(A 5) 2 + (B 5) 2 ] 1/2. Tables 1 and 2 tell essentially the same story. Standard confidence intervals constructed using the student-t distribution display coverage very close to the target coverage. Moreover, our symmetry measure indicates no undue bias towards asymmetry. For the D-TAR model the coverage ranges from 87.4 percent to 91.5 percent and in most cases is between 89.0 and 91.0 percent. For the C-TAR model, the coverage ranges from 86.7 percent to 90.6 percent (and from 88.6 percent to 90.6 percent when the (.95,.95) parameterization is excluded). Bootstrap-t intervals also work reasonably well. Though they tend to be too small, the bootstrap-t intervals are generally more symmetric that the normal intervals. Based upon our symmetry measure an argument can be made that the bootstrap-t intervals are better than the normal intervals, particularly for the C-TAR case. The bootstrap-percentile intervals are not very well behaved for either model and with respect to either measure. In summary, our Monte Carlo analysis suggests that the normal approximation and the bootstrap-t approaches to confidence interval construction work well in the stationary C-TAR and D-TAR models with known threshold. The overall coverage of the normal confidence intervals is very close to the nominal coverage in most cases, although these intervals tend to be asymmetric. The overall coverage of the bootstrap-t intervals is slightly downward biased but the errors are more symmetric than are the errors for normal intervals.

9 8 b. Confidence Intervals for the Slope Coefficient: Unknown Threshold Next we constructed confidence intervals for the slope coefficients in the D-TAR and C- TAR models under the assumption that the threshold parameter is unknown. The Monte-Carlo design was the same as that used in the preceding section with the following two modifications. First, for each realization of the y t process the threshold parameter was estimated along with the intercept and slope coefficients, using the nonlinear least squares estimator. 4 Second, for each realization of the y t process, the bootstrap samples used to construct the bootstrap-percentile and bootstrap-t intervals were generated using the estimated threshold rather than the true threshold and the threshold parameter was re-estimated (along with the intercept and slope coefficients) for each bootstrap sample. These simulations were very time-consuming because of the need to apply the nonlinear least squares estimator to each of the bootstrap samples. Therefore, we restricted our attention to a single sample size (100), a single nominal coverage value (90-percent), and a smaller set of parameter combinations than we considered in the known-threshold case. In the data-generating process, the threshold parameter τ was set to zero, the slope coefficient α 1 was set to 0.3, and the slope coefficient β 1 was sequentially selected from {0.6, 0.9, 0.95}. For the D- TAR model (1) we set the intercepts α 0 and β 0 equal to 0 and 0.9, respectively. Intervals for the Slope Coefficients in the D-TAR Model The simulated coverage properties of the 90-percent nominal confidence intervals for the slope coefficients are presented in Table 3 (D-TAR) and Table 4 (C-TAR). We also report the measure of interval symmetry that we introduced earlier. These results can be compared to those presented in Table 2 for the slope coefficients in the D-TAR model with known threshold. Rows

10 9 2-4 of Table 1 use the same combination of slope parameters to generate the simulated data as those used to construct Table 3. In 13 of the 18 cells in Table 3, the coverage probabilities are farther from the nominal probability than in the corresponding cells of Table 1. This holds for all six of the intervals constructed using the normal approximation. In two of the six cells for intervals constructed using the bootstrap-percentile approach, the coverage probability actually improved when the threshold was estimated. These improvements came in the intervals for α 1 when β 1 was set to 0.90 (93.7 percent to 92.8 percent) and when β 1 = 0.95 (94.3 percent to 90.9 percent). Coverage probabilities of all three intervals for β 1 constructed from the bootstrap-t improved in moving from the known threshold to the unknown threshold: β 1 = 0.6 (from 88.1 to 91.3), β 1 = 0.9 (from 86.6 to 90.5), and β 1 = 0.95 (from 83.5 to 88.8). However, the symmetry measure improved in only two of the 18 cells: the bootstrap-percentile interval for α 1 when β 1 = 0.9 (from 2.55 to 2.52) and the bootstrap-t interval for β 1 when β 1 = 0.95 (from 1.77 to 5.22). Looking more closely at the intervals constructed using the normal approximation, the coverage probability ranged from 90.0 to 91.4 in the known threshold case (rows 2 4 in Table 1). For the unknown threshold case, the coverage probability ranged from 71.2 to The normal approximation, which provided the best coverage among the three procedures for the known threshold, suffered the greatest reduction in performance in moving from the known threshold to the unknown threshold. It is clearly dominated by the bootstrap-t procedure in the unknown threshold case. The bootstrap-t provided coverage probabilities for β 1 that are very close to 90 percent. These intervals are actually closer to 90-percent than the corresponding intervals constructed with the known threshold, though they tend to be less symmetric. When β 1 is close to one (i.e.,

11 or 0.95) the intervals for α 1 have poorer coverage (tending to undercover) and are less symmetric when the threshold is unknown. However, in the case of the unknown threshold, the bootstrap-t intervals for α 1 and β 1 provide better coverage than the intervals constructed from the normal approximation. The bootstrap-percentile intervals for α 1 have very good coverage (90.6, 92.8, and 90.9), outperforming the other procedures in this regard and improving from the known threshold case. However, as in the known threshold case, the bootstrap-percentile intervals for β 1 perform poorly (absolutely and relatively) when β 1 is large. We conclude that for the D-TAR model with unknown threshold, none of the three procedures performs well for both slope coefficients across the range of parameters we considered. However, the normal approximation performed most poorly and, in terms of average performance, the bootstrap-t intervals performed best. Intervals for the Slope Coefficients in the C-TAR Model The simulated coverage properties of the 90-percent nominal confidence intervals for the slope coefficients in the C-TAR model with unknown threshold are presented in Table 4. These results can be compared to those presented in Table 2 for the slope coefficients in the C-TAR model with known thresholds. Rows 2 4 of Table 2 use the same combination of slope parameters to generate the simulated data as those used to construct Table 4. For each parameter combination, the coverage probabilities for the normal intervals and the bootstrap-t intervals worsened in moving from the known to the unknown threshold case. The coverage of the normal intervals ranged from 88.0 to 90.1 in the known threshold case (rows 2-4 of Table 2) and ranged from 62.5 to 80.9 in the unknown threshold case. The coverage of the bootstrap-t intervals ranged from 85.3 to 87.9 in the known threshold case and from 77.4 to 84.8

12 11 in the unknown threshold case. Thus the coverage of the bootstrap-t intervals, which was not quite as good as the normal intervals in the known threshold case, did not deteriorate as much as the normal intervals in moving to the unknown threshold. The bootstrap-percentile interval coverage for β 1 improved considerably in the unknown threshold case [from 85.4 to 92.8 (β 1 = 0.6), from 80.6 to 88.3 (β 1 = 0.9), and from 74.5 to 90.8 (β 1 = 0.95)], providing good coverage and better coverage than the coverage provided by the other two procedures. The bootstrap-percentile intervals for α 1, which under-covered in the known threshold case, over-covered in the unknown threshold case. The magnitudes of the under-coverage of the intervals with known threshold (87.9, 84.7, and 82.1 for β 1 = 0.6, 0.9, 0.95, respectively) are comparable to, but slightly larger than, the magnitudes of the corresponding intervals with unknown threshold (95.9, 97.9, 98.4). However, the bootstrappercentile intervals for α 1 are more symmetric in the unknown threshold case. We conclude that for the C-TAR model with unknown threshold, none of the three procedures performs well for both slope coefficients across the range of parameters we considered. However, among these procedures the intervals constructed from the normal approximation performed most poorly. The two bootstrap procedures provided comparable coverage errors and symmetry measures for α 1 ; the bootstrap-t intervals under-covering by roughly the same amount as the over-coverage of the bootstrap-percentile intervals. The bootstrap-percentile intervals for β 1 performed well and better than the corresponding bootstrap-t intervals. c. Confidence Intervals for the Threshold Parameter Hansen (1997) derived the (non-standard) asymptotic distribution of the least-squares estimator of the threshold parameter and the likelihood ratio statistic for inference concerning the

13 12 threshold parameter in the D-TAR model. Critical values for these distributions are tabulated in Hansen (1997). Chan and Tsay (1998) studied the asymptotic distribution of the least squares estimator of the threshold parameter and the likelihood ratio statistic in the C-TAR model, showing that the asymptotic distribution of the likelihood ratio statistic is a chi-square with one degree of freedom. In this section of the paper we evaluate the finite sample performance of three procedures to construct confidence intervals for the threshold parameter τ in the D-TAR and C-TAR models. The first procedure is to invert the likelihood ratio statistic using the asymptotic critical values. That is, the likelihood ratio statistic to test the null hypothesis that τ = τ 0 is LR(τ 0 ) = T[σ 2 (τ 0 ) σ 2 (τ)]/σ 2 (τ) where: T is sample size and Tσ 2 (τ) is the sum of squared residuals from the regression model (1) or (2) fit by OLS conditional on τ. The δ-percent confidence interval for τ found by inversion of the likelihood ratio statistic is Γ(δ) = {τ: LR(τ) < C(δ)} where C(δ) is the δ-level critical value from the asymptotic distribution of LR(τ). Following Hansen (1997) we use the convexified region Γ * (δ) = [τ 1 τ 2 ], where τ 1 and τ 2 are the minimum and maximum elements of Γ(δ), respectively. The second procedure is identical to the first except that it uses the bootstrapped distribution of LR(τ) to determine the critical value C(δ). The third procedure is the bootstrap percentile distribution constructed as the values of τ falling within the (1 δ)/2 and 1 δ+(1 δ)/2 percentiles of the bootstrap distribution of τ. The results are presented in Tables 5 and 6 for the D-TAR and C-TAR models, respectively. They were obtained as part of the Monte Carlo experiments used to construct the confidence intervals for the slope coefficients in the TAR models with unknown threshold. For

14 13 the bootstrap procedures, the estimated threshold from each simulated data sample was used to generate the bootstrap samples and re-estimated for each bootstrap sample to simulate the bootstrap distributions of the least-squares estimator of τ and the likelihood ratio statistic LR(τ). 5 According to Table 5, none of the three procedures performs satisfactorily. Hansen s (1997) asymptotic approximation under-covers, providing reasonably good coverage when β 1 = 0.6 but not for β 1 = 0.9 or β 1 = The bootstrap-percentile procedure provides good coverage when β 1 = 0.9, but not for β 1 = 0.6 (with actual coverage equal to 100-percent) or β 1 = 0.95 (with actual coverage equal to 68.6 percent). Inverting the bootstrapped distribution of the likelihood ratio statistic provided confidence intervals that never excluded the true threshold parameter for any of the three cases considered. The poor performance of the bootstrap-lr procedure is somewhat surprising since the likelihood ratio statistic is asymptotically pivotal. The coverage probabilities of nominal 90-percent confidence intervals for the threshold parameter τ in the C-TAR model are supplied in Table 6. Inversion of the bootstrapped distribution of the likelihood ratio statistic provides reasonably good coverage in all three cases: 88-percent, 93-percent, and 91.9-percent for β 1 equal to 0.6, 0.9, and 0.95, respectively. The actual coverage of the bootstrap-percentile interval for τ when β 1 is equal to 0.6 (89.7) is virtually the same as the nominal coverage, though the intervals tend to be asymmetric. For values of β 1 close to one the bootstrap-percentile intervals for τ over-cover, with coverage probabilities approximately equal to Coverage probabilities of the intervals constructed from the asymptotic distribution of the likelihood ratio statistic are 78.4 percent (β 1 = 0.6), 84.3 percent (β 1 = 0.9), and 83.3 (β 1 = 0.95).

15 14 IV. Confidence Intervals for TAR Estimates of U.S. GDP The aim of this section is to compare the various methods for constructing confidence intervals for the threshold and slope parameters for real U.S. GDP. The time path of the logarithmic change in real U.S. GDP (y t ) over the 1947:Q1 to 2003:Q4 sample period is shown as the solid line in Figure 1. It is possible that there are more nonlinear estimates of this series than any other macroeconomic time series. For example, Potter (1995) modeled and fit the logarithmic difference of real U.S. GNP (not GDP) through 1990:Q4 to a threshold autoregressive model under the assumption that the threshold is known and equal to zero. 6 a. Linear and TAR Models of GDP We began by estimating the GDP growth rate as an AR(p) process beginning with p = 8. If the t-statistic for lag p was insignificant at the five-percent significance level, we reduced p by one and re-estimated the model. The resulting AR(1) model is: (3) y t = y t-1 + ε t aic = rss = (6.85) (5.31) When we used the aic to select the lag length, we found: 7 (4) y t = y t y t y t-3 + ε t (5.97) (4.75) (1.70) ( 1.80) aic = , rss = ρ 1 = 0.02 ρ 2 = 0.00 ρ 3 = 0.05 ρ 4 = 0.07 ρ 5 = 0.10 ρ 6 = 0.01 ρ 7 = 0.05 ρ 8 = 0.05 Q(4) = 0.75 Q(8) = 0.68 Q(12) = 0.35 Q(16) = 0.27 Q(20) = 0.39 An F-test for the restriction that the coefficients on y t-2 and y t-3 are jointly equal to zero has a prob value of Since the two models are quite similar but the aic selects the AR(3)

16 15 specification, we report results using (4). Details using the shorter lag length are virtually identical to those reported here and are available from the authors. We fixed the lag length at 3 and estimated TAR models with delay parameters of 1, 2, and 3. Since we are interested in forming confidence intervals on the threshold parameter, we used the least-squares estimate of the threshold instead of fixing the threshold at zero. Given that we have a relatively large sample size of 224 usable observations, we searched for the threshold within the middle 80% of the observations. We found that the least squares principle selected a delay parameter of 2--the aic values with delay parameters of 1, 2, and 3 are , and , respectively. Our estimated TAR model in the form of (4) is: (5) y t = I t [ y t y t y t-3 ] (4.47) (4.56) (1.36) ( 1.51) + (1 I t )[ y t y t y t-3 ] + ε t ( 1.14) (1.31) ( 2.75) ( 1.31) I t 1 if yt = 0 if yt 1 < aic = , rss = ρ 1 = 0.02 ρ 2 = 0.01 ρ 3 = 0.03 ρ 4 = 0.06 ρ 5 = 0.10 ρ 6 = 0.04 ρ 7 = 0.04 ρ 8 = 0.03 Q(4) = 0.89 Q(8) = 0.84 Q(12) = 0.30 Q(16) = 0.12 Q(20) = 0.12 Notice that TAR model (5) has a smaller value for the aic than the linear model and that the diagnostics for the residuals are about the same as for the linear model. The estimated threshold of is shown as the dashed line in Figure 1. It is interesting that the value of the threshold is near zero and that the high-growth regime (i.e., τ > ) is more persistent than the low-growth regime. These results are roughly in line with those reported by

17 16 Potter (1995, Table II). The issue is to determine the confidence intervals surrounding these parameter estimates. b. Confidence Intervals for the Threshold Figure 2 shows the value of the LR-statistic as a function of the potential threshold values where: 2 2 σ ( τ 0 ) σ ( τ ) (6) LR ( τ 0 ) = T 2 σ ( τ ) and: T = number of observations, σ 2 ( τ ) = variance estimate at the potential threshold τ 0 and σ 2 ( τ ) = variance estimate using the consistent estimate of the threshold. It should be clear from the figure that the change in LR is quite pronounced around the estimated threshold τ = Hansen (1997) shows that under the null hypothesis τ 0 = τ, the 90%, 95% and 99% critical values for LR(τ 0 ) are 5.94, 7.35 and 10.59, respectively. These critical values are drawn as the horizontal dashed lines drawn in the figure. The 90% critical value contains the continguous interval through Thus, using this confidence interval, it is possible to reject the null hypothesis that the threshold is zero. The issue is a bit more complicated using a 95% or 99% confidence interval since the threshold range is disjoint. Whenever the confidence region not continuous, Hansen (1997) suggests the conservative procedure of using the entire interval (i.e, the convexifed confidence interval). 8 As such, using these more conventional confidence levels, it is not possible to rule out the possibility that the threshold is positive. Specifically, if we use the point estimate of τ = , the confidence intervals implied from asymptotic the approximations are:

18 17 Asymptotic Confidence Intervals for τ Low High 90% % % We bootstrapped equation (5) 2500 times and retained the 2500 bootstrap estimates of the threshold. Retaining only the middle 90%, 95% and 99% of the ordered threshold estimates yielded the bootstrap percentile confidence intervals shown below. Note that these intervals are far larger than the confidence intervals reported above and always span τ = 0. In fact, the 99% confidence interval constructed using the percentile method spans nearly all of the data set. Percentile Confidence Intervals for τ Low High 90% % % Finally, since LR(τ 0 ) is an asymptotically pivotal statistic LR(τ 0 ), we also bootstrapped the 90%, 95% and 99% critical values of its distribution and used these to construct the confidence intervals for τ. The 90% (and, therefore, 95% and 99%) critical values were so large that all of the data points that were candidate threshold values fell into each of these intervals. Consequently, the threshold confidence intervals implied by this procedure went from the 10-th percentile of the data ( ) to the 90-th percentile of the data ( ). This is consistent with our simulation indicating that the actual coverage of the 90% bootstrap-lr intervals for the threshold in the D-TAR model were 100% for each parameter configuration we considered. Bootstrap-LR Confidence Intervals for τ

19 18 Low High 90% % % The point is that it is not possible to claim that the threshold for real US GDP growth is negative. Only a 90% confidence interval using an asymptotic approximation allows us to rule out the possibility of a positive threshold. Hence, the data does not support the assertion that the time-series properties of negative growth rates behave differently from positive growth. c. Confidence Intervals for the Slope Parameters The two first-order slope coefficients are and with t-statistics of 4.56 and 1.31, respectively. Given that we have 224 usable observations, the standard errors are and Since we use a consistent estimate of the threshold, it is possible to conduct inference on the slope parameters using the normal distribution. The 90%, 95% and 99% confidence intervals for the two slope coefficients (called α 1 and β 1 ) using the normal approximation are reported in the left portion of Table 7. We bootstrapped equation (5) 2500 times and retained the 2500 bootstrap estimates of the two first-order slope coefficients. Retaining only the middle 90%, 95% and 99% of the ordered slope coefficients yielded the percentile confidence intervals reported in the middle portion of Table 7. For each bootstrapped series, we also constructed the bootstrap t- statistic for the null hypothesis α 1 = and β 1 = This bootstrap t-statistic allows us to back-out the confidence intervals reported in the right portion of Table 7. Unlike the confidence intervals for τ, the normal approximation and the bootstrap-t methods yield similar confidence intervals. Nevertheless, there are some interesting differences. Note that the percentile method yielded 90%, 95% and 99% confidence intervals for β 1, the firstorder slope coefficient in the low growth regime, that fully contain the confidence intervals

20 19 constructed from the two other methods. Also, the confidence intervals for α 1 and β 1 from the normal approximation are almost always contained within those constructed from the bootstrap-t, the exception involving the third digit the confidence interval of α 1. Thus, the percentile method appears to be the most conservative method and the normal approximation appears to be the least conservative method to obtain the confidence intervals. The percentile method seems to exacerbate the effect of poorly estimated coefficients. d. The C-TAR Model Since there is no a priori way of knowing whether real GDP growth is a C-TAR or a D- TAR process, we estimated y t as a continuous threshold process in the form of (2). Nevertheless, the C-TAR model does not clarify the asymmetric nature of real GDP growth. Consider: y t = τ + I t [ 0.310(y t-1 τ) y t-2 τ) 0.062(y t-3 τ) ] (3.01) (2.70) ( 0.55) + (1 I t )[ 0.341(y t-1 τ) y t-2 τ) 0.133(y t-3 τ) ] + ε t (3.91) (0.255) ( 1.62) τ = I t 1 = 0 if if y t 1 y t 1 τ < τ aic = ssr = Notice that the C-TAR model does not fit the data as well as the D-TAR model. Moreover, the two first-order autoregressive coefficients are quite similar and the threshold is large. At an annual rate, the threshold of 3.9% exceeds that average growth rate of real GDP (i.e., 4* = ). Figure 3 shows the value of the LR-statistic constricted as in (6) as a function of the potential threshold values. Notice that the 90%, 95% and 99% confidence intervals are relatively tight. In particular:

21 20 Asymptotic Confidence Intervals for τ in the C-TAR Model Low High 90% % % Using a 90% or a 95% confidence interval, the confidence intervals from the asymptotic and percentile methods do not even overlap. Consider: Percentile Confidence Intervals for in the C-TAR Model τ Low High 90% % % Finally, as in the D-TAR model, the confidence intervals for the bootstrap-t span the entire range of potential thresholds. 9 V. Summary and Conclusions Monte Carlo methods were applied to study the finite-sample performance of standard regression approaches to confidence interval construction in threshold autoregressive models. More specifically, intervals based upon asymptotic approximations and bootstrap (t and percentile) intervals were generated for the coefficients in the stationary, first-order, selfexciting, threshold autoregressive model. Interval coverage probabilities and interval symmetry were used to measure the quality of the various procedures. When the true threshold parameter is known, bootstrap-t intervals and classical t-intervals for the slope coefficients work well enough to be recommended in practice. The classical intervals tend to provide slightly better coverage but the bootstrap-t intervals tend to be more symmetric. Bootstrap-percentile intervals tend to be very erratic, performing very well for some parameters and very poorly for others.

22 21 When the true threshold is unknown and is estimated along with the slope parameters, none of the procedures provide intervals for the slope coefficients with good coverage properties over the full range of parameters considered. Standard-t intervals performed especially poorly in this case. Confidence intervals for the threshold parameter itself were constructed by inversion of the asymptotic distribution of the likelihood ratio statistic, by inversion of the bootstrap distribution of the likelihood ratio statistic, and by the bootstrap-percentile method. None of the procedures perform satisfactorily across the full range of parameter values for discontinuous TAR(1) models. Interestingly, the bootstrap-lr procedure generated overly large confidence intervals with 100% actual coverage for each parameterization of the D-TAR(1). This suggests that the bootstrap-lr procedure may not be very useful in D-TAR models. However, the bootstrap-lr procedure worked reasonably well and better than the other procedures for constructing confidence intervals for the threshold parameter in the C-TAR(1) models we considered. The bottom line, as Hansen (1997), Gonzalo and Wolf (2001), have shown for the asymptotic case, is that inference is difficult in TAR models. We applied these procedures to obtain confidence intervals for the slope and threshold coefficients in a discontinuous TAR model of real GDP growth rates, a model similar to the one estimated by Potter (1995) in which he assumed that the threshold growth rate is zero. When the threshold is estimated by nonlinear least squares and the asymptotic distribution of Hansen s (1997) LR statistic is applied, the confidence interval for the estimated threshold excludes zero at the 90% level, but not the 95% or 99% levels. Moreover, the bootstrap-lr procedure produces intervals so large that all candidate thresholds are included in these intervals. This is in line with

23 22 our simulation results. Intervals for the slope coefficients appear to be more stable across the three procedures considered. The main message of the paper is that when the threshold parameter is unknown, asymptotic and bootstrap approximations of finite sample distributions do not lead to satisfactory confidence intervals for slope or threshold parameters in stationary TAR models. Since inference in a TAR model is problematic, caution must be exercised in applications attempting to conduce inference in threshold models.

24 23 NOTES 1. For further discussion of this issue, see Andrews (1994), Andrews and Ploberger (1994), and Hansen (1996). 2. Bootstrap confidence intervals are discussed in detail in Efron and Tibshirani (1993). The grid-bootstrap approach described in Hansen (1999) and the bias-corrected bootstrap approach described in Efron and Tibshirani (1993) were designed to improve the performance of the bootstrap-t and bootstrap-percentile methods, respectively. We did not consider these approaches for practical reasons. Our Monte Carlo experiments were very time-consuming when the threshold parameter was treated as unknown because of the nonlinear estimation problem that had to be solved for each bootstrap sample. Evaluating the grid-bootstrap and bias-corrected approaches would have added enough complexity to have made this study impractical. 3. In simulating TAR processes, it is possible that the constructed series never crosses the true threshold. This turned out to be especially true for values of β 1 equal to 0.9 and Unless there are two observations on each side of the threshold it is impossible to fit a TAR model to the data. In practice, researchers searching for an unknown threshold typically discard the largest and smallest 10 or 15 percent of the ordered data from their search. If one of our simulated series did not contain at least three points on each side of the threshold, it was discarded and replaced with another simulated series. We applied this rule throughout this study, including the bootstrap simulations. 4. This is the estimation procedure used in Chan (1993) and Hansen (1997). The meaningful candidates for the threshold are the observed values of the data series. 5. Note that the bootstrapped values of the likelihood-ratio statistic can be negative since the estimated threshold for any given bootstrap sample can generate a smaller sum of squared residuals than the sum of squared residuals obtained from the estimated threshold fit to the original sample. 6. In addition, Potter (1995) estimates the model for each regime separately to allow for heteroskedasticity across regimes, which is straightforward when the threshold is assumed to be known. 7. We report the aic as: T*log(rss) + 2*k where: T = number of usable observations, rss = residual sum of squares, and k = number of parameters estimated. When we estimate a TAR model, k reflects the fact that the threshold parameter is estimated. We let ρ i denote the residual autocorrelation for lag i and Q(n) denote the prob-value for the Ljung-Box statistic for the null hypothesis that the autocorrelations through lag n are statistically different from zero. 8. It may be that there is a second threshold. However, our exploratory checks for a second threshold did not reveal any compelling evidence for a three-regime model. 9. We do not confidence intervals for the autoregressive coefficients since α 1 is so similar to β 1.

25 24 REFERENCES Andrews, D.W.K. and W. Ploberger (1994). Optimal tests when a nuisance parameter is present only under the alternative, Econometrica, 64: Burgess, S.M. (1992). Nonlinear dynamics in a structural model of employment, Journal of Applied Econometrics, 7: S101-S118. Chan, K.S. (1993). Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model, The Annals of Statistics, 21: Chan, K.S. and R.S.Tsay (1998). Limiting properties of the least squares estimator of a continuous threshold autoregressive model, Biometrika, 45: Cao, C. Q. and Tsay, R.S. (1992). Nonlinear time-series analysis of stock volatilities, Journal of Applied Econometrics, 7: S165-S185. Efron, B. and R.J. Tibshirani (1993). An Introduction to the Bootstrap, New York: Chapman and Hall. Enders, W. and P.L. Siklos (2004), D-TAR Versus C-TAR Models? Modeling the Dynamics of Inflation, in R. Becker and S. Hurn (Eds.), Advances in Economics and Econometrics: Theory and Applications (Edward Elgar, forthcoming). Enders, W. and C.W.J. Granger (1998). Unit-root tests and asymmetric adjustment with an example using the term structure of interest rates, Journal of Business and Economic Statistics, 16: Galbraith, J.W. (1996). Credit rationing and threshold effects in the relation between money and output, Journal of Applied Econometrics, 11: Gonzalo, J. and M. Wolf (2001), Subsampling Inference in Threshold Autoregressive Models, working paper, Universidad Carlos III, Madrid, September. Hansen, B.E. (1996). Inference when a nuisance parameter is not identified under the null hypothesis, Econometrica, 64: Hansen, B.E. (1997). Inference in TAR models, Studies in Nonlinear Dynamics and Econometrics, 1: Hansen, B.E. (1999). The grid bootstrap and the autoregressive model, The Review of Economics and Statistics, 81: Krager, H. and P. Kugler (1993). Non-linearities in foreign exchange markets: a different perspective, Journal of International Money and Finance, 12: Petrucelli, J. and S. Woolford (1984). A threshold AR(1) model, Journal of Applied Probability, 21: Potter, S.M. (1995). A nonlinear approach to U.S. GNP, Journal of Applied Econometrics, 2:

26 Rothman, P. (1991). "Further Evidence on the Asymmetric Behavior of Unemployment Rates Over the Business Cycle," Journal of Macroeconomics, 13:

27 26 TABLE 1 Confidence Intervals for the Slope Coefficients in the D-TAR(1) Model with Known Threshold Nominal Coverage = 90-percent T= 100 α 1 = 0, β 0 = 0, τ = 0 Normal Approx BS Percentile BS - t α 0 = β 1 = α 1 β 1 α 1 β 1 α 1 β (1.35) 90.0 (1.41) 90.8 (2.61) 92.1 (3.38) 89.3 (0.61) 88.8 (1.02) (1.56) (1.56) (1.14) (1.39) (2.50) (3.11) (2.86) (1.66) (2.42) (1.56) (2.15) 90.6 (2.02) 90.0 (1.70) 91.4 (2.34) 89.3 (1.43) 87.5 (2.31) 89.0 (3.19) 90.2 (3.26) 90.0 (1.27) 89.3 (2.52) 90.6 (2.02) 91.5 (2.56) 90.2 (2.55) 93.7 (4.16) 94.3 (4.20) 90.6 (2.72) 88.0 (4.20) 90.9 (5.06) 92.5 (4.66) 89.5 (3.06) 89.6 (3.69) 92.2 (4.12) 92.9 (3.91) 90.3 (6.12) 81.6 (14.3) 72.8 (22.8) 92.8 (3.81) 91.8 (5.94) 77.5 (18.2) 71.7 (23.8) 92.5 (4.66) 90.1 (7.00) 77.2 (18.4) 71.7 (23.8) 89.7 (0.30) 87.7 (2.02) 87.2 (3.01) 88.5 (1.17) 86.9 (2.28) 87.9 (1.25) 87.7 (1.64) 89.7 (0.22) 88.7 (1.04) 89.6 (0.76) 89.5 (0.64) 88.1 (1.63) 86.6 (3.93) 83.5 (5.22) 88.7 (1.20) 89.2 (2.33) 85.7 (3.59) 84.4 (4.08) 89.0 (0.82) 88.6 (1.61) 83.8 (4.75) 86.9 (2.19) Notes: The top line in each cell is the actual coverage from the 1000 Monte Carlo simulations. The number in parentheses is a symmetry measure calculated as: [(x-5)**2 + (y-5)**2]**.5, where x is the proportion of times that the true α 1 (β 1 ) lies below the lower bound of the nominal 90-percent confidence interval and y is the proportion of times that the true α 1 (β 1 ) lies above the upper bound of the nominal 90-percent confidence interval.

28 27 TABLE 2 Confidence Intervals for the Slope Coefficients in the C-TAR(1) Model with Known Threshold Nominal Coverage = 90-percent T= 100 τ = 0 Normal Approx BS Percentile BS - t α 1 = β 1 = α 1 β 1 α 1 β 1 α 1 β (1.21) (1.12) (4.08) (6.27) (1.64) (1.22) (1.63) 89.9 (1.63) 87.9 (6.05) 85.4 (9.09) 87.4 (1.97) 87.0 (1.97) (1.77) 90.1 (1.49) 84.7 (10.8) 80.6 (14.9) 87.9 (1.63) 85.9 (3.20) (3.42) 88.6 (3.67) 82.1 (13.8) 74.5 (21.0) 86.6 (2.40) 85.3 (4.90) (2.69) 90.2 (2.55) 84.7 (9.81) 85.8 (10.1) 88.7 (1.12) 88.4 (1.21) (3.33) 88.8 (3.36) 79.7 (15.8) 77.9 (17.7) 87.6 (1.79) 87.5 (1.88) (3.62) 89.6 (3.41) 79.3 (16.2) 75.5 (20.1) 87.5 (1.77) 88.8 (1.90) (5.58) 88.8 (5.58) 71.2 (24.3) 68.7 (26.3) 87.7 (2.40) 88.9 (0.92) (4.60) 90.1 (4.60) 69.6 (25.8) 64.3 (31.1) 89.3 (0.50) 86.2 (3.60) (6.98) 88.8 (6.71) 60.3 (35.1) 67.0 (28.4) 87.8 (2.41) 89.5 (1.21) See Note to Table 1.

29 28 TABLE 3: Confidence Intervals for the Slope Coefficient in the D-TAR(1) Model with Unknown Threshold Nominal Coverage = 90-percent T= 100 α 0 = 0, β 0 = 0.9, α 1 = 0.3, τ = 0 Normal Approx BS Percentile BS - t β 1 = α 1 β 1 α 1 β 1 α 1 β (8.37) (10.63) (2.86) (6.27) (3.02) (2.80) (8.41) (8.10) 76.6 (18.82) 71.2 (23.34) 92.8 (2.52) 90.9 (4.50) 76.5 (19.16) 60.6 (34.76) 83.2 (8.09) 83.1 (11.54) 90.5 (2.92) 88.8 (1.77) See Note to Table 1. TABLE 4: Confidence Intervals for the Slope Coefficient in the C-TAR(1) Model with Unknown Threshold Nominal Coverage = 90-percent T= 100 α 1 =0.3, τ = 0 Normal Approx BS Percentile BS - t β 1 = α 1 β 1 α 1 β 1 α 1 β (13.40) (10.79) (5.08) (3.57) (4.40) (9.07) (19.88) (23.84) See Note to Table (12.47) 79.8 (15.25) 97.9 (5.78) 98.4 (5.98) 88.3 (8.36) 90.8 (6.53) 84.4 (4.16) 82.1 (5.59) 84.8 (9.25) 84.1 (10.55)

30 29 TABLE 5: Confidence Intervals for the Threshold Parameter in the D-TAR(1) Model Nominal Coverage = 90-percent T= 100 α 0 = 0, β 0 = 0.9, α 1 = 0.3, τ = 0 Asymptotic BS Percentile BS LR Approximation β 1 = (3.62) (18.40) (29.30) (8.68) 68.6 (26.87) See Note to Table 1. TABLE 6: Confidence Intervals for the Threshold Parameter in the C-TAR(1) Model Nominal Coverage = 90-percent T=100 α 1 = 0.3, τ = 0 Asymptotic BS Percentile BS LR Approximation β 1 = (14.09) (8.08) (5.38) 89.7 (7.29) 96.2 (4.87) 94.6 (3.28) 88.0 (8.46) 93.0 (4.87) 91.9 (4.65)

31 TABLE 7: Confidence Intervals for the Slope Coefficients of the TAR Model of GDP Growth Rates Normal Approximation Percentile Bootstrap t low high low high low high α 1 90% % % β 1 90% % %

32 Figure 1: Quarterly Growth Rate of Real U.S. GDP percent per quarter GDP Growth Threshold

33 14 Figure 2: Confidence Intervals for the Threshold Likelihood Ratio Threshold LR Statistic 90% 95% 99%