Extended Tests for Threshold Unit Roots and Asymmetries in Lending and Deposit Rates

Transcription

1 Extended Tests for Threshold Unit Roots and Asymmetries in Lending and Deposit Rates Walter Enders Junsoo Lee Mark C. Strazicich Byung Chul Yu* February 5, 2009 Abstract Enders and Granger (1998) develop threshold unit root tests to test for a unit root in the presence of possible asymmetric adjustment paths observed in many economic time series. We build on this pioneering work by extending their tests in several important directions. In particular, we examine a more flexible model specification where the deterministic terms and short-run dynamics, as well as the persistent parameters, can differ in each regime. Using this more flexible model specification, we consider several other extensions. First, we adopt a percentile threshold parameter that can take on a range of values between 0 and 1. Second, we consider a supreme type test to estimate the threshold parameter from the data. Third, we describe conditions where a threshold parameter can be consistently identified under the null and alternative hypotheses. Asymptotic properties are provided and finite sample properties are examined in simulations. An empirical example is provided to test for asymmetries in the spread of the U.S. prime lending rate minus the 3-month certificate of deposit rate. Overall, we find significant asymmetric effects in two regimes. Keywords: Asymmetric Effects, Threshold Unit Root Tests, Interest Rates JEL Classifications: C220, E400 Walter Enders, Professor, Department of Economics, Finance and Legal Studies, University of Alabama, Tuscaloosa, AL , Phone (205) , wenders@cba.ua.edu; Junsoo Lee, Professor, Department of Economics, Finance and Legal Studies, University of Alabama, Tuscaloosa, AL , Phone (205) , jlee@cba.ua.edu; Mark C. Strazicich, Associate Professor, Department of Economics, Appalachian State University, Boone, NC , Phone (828) , strazicichmc@appstate.edu; Byung Chul Yu (corresponding author), Professor, Department of International Trade, Dong-A University, Busan, Korea, Phone (82-51) , bcu@dau.ac.kr.

2 1. Introduction Many important economic variables have been observed to display asymmetric adjustment paths. To address this, Enders and Granger (1998, EG hereafter) develop new threshold unit root tests in two model specifications. Their first specification is based on the threshold autoregressive (TAR) model developed by Tong (1983), while the second is a momentum threshold autoregressive (MTAR) model where the change rather than the level of the threshold variable is adopted. Their tests have been widely employed in empirical work and provide perhaps the first formal tests for threshold unit roots with relevant critical values. In this paper, we build on this pioneering work of EG by extending their tests in several important directions. In particular, we examine a more flexible model specification where the deterministic terms and short-run dynamics, as well as the persistent parameters, can differ in each regime. Using this more flexible model specification, we consider several other extensions. First, we allow the threshold parameter to take on a range of values while noting that the distribution of the threshold unit root test statistic will depend on the particular threshold parameter. In EG, the model specifications assume a zero value of the threshold variable in the most basic model and a zero mean value of the detrended threshold variable in the models with deterministic terms. While a zero value of the threshold variable can fit well in many empirical works, there may be cases where the threshold value can take on other values. To accommodate this, we explicitly treat the threshold parameter as a value to be estimated from the data. Since it will be tedious to construct critical values with different threshold values that can in theory vary between - and +, we transform the threshold value into a percentile 1

3 parameter that can take on a standard range between 0 and 1 in each case. Since the threshold test depends on the threshold parameter, this outcome is convenient and allows us to adopt a wide range set of threshold values that might otherwise not be feasible. The asymptotic distribution confirms this outcome and relevant critical values are provided at different percentile values. In general, the value of the threshold parameter is unknown a priori and must be estimated from the data. This situation leads to the so-called Davies (1987) problem when the threshold parameter is identified only under the stationary alternative. One popular approach in such cases is to consider a supreme type F-test to jointly estimate the threshold parameter and test the null hypothesis of a unit root. Such treatment has been popular in the literature, since it allows us to overcome the so-called Davies (1987) problem. However, what if a threshold parameter can be clearly identified a priori or can be consistently confirmed using common search procedures? Since the critical values of the supreme F-test are in general larger than the corresponding F-test with a known threshold value, confusion can arise about what critical value to employ. In such cases, if a threshold value is already confirmed to exist, it does not seem desirable to use critical values from the supreme F-test just. Instead, if the threshold parameter is known to exist and/or can be consistently estimated prior to testing, it would be more desirable to treat the threshold value as given and use critical values that utilize this information. Otherwise, lower power is expected. Given this situation, we examine conditions where a threshold parameter can be consistently identified. One such condition is apparent if the deterministic terms and/or short-run dynamics differ in each regime. This condition does not involve the persistent 2

4 parameter which we test on for the unit root hypothesis, but it provides a case where the threshold parameter can be identified and estimated from the differences in the deterministic terms and/or short-run dynamics over different regimes. If so, a threshold parameter is known to exist and the Davies (1987) problem will not occur in testing for the threshold unit root hypothesis. Moreover, under these conditions, the threshold parameter can be identified even if the persistent parameters are symmetric and there is a unit root is each regime. As a result, a threshold unit root test with a known threshold parameter will be valid under the null and alternative hypotheses. We therefore conclude that the supreme type test be adopted only when one can first demonstrate that the deterministic terms and/or short-run dynamics do not differ in two regimes. This outcome provides further reasons to adopt a more general model specification when performing threshold unit roots tests. We further believe that similar advantages can be extended to other nonlinear time series models where a Davies problem might occur. Our paper proceeds as follows. In Section 2, we define and discuss our extended threshold models and testing procedures for different threshold unit root tests. Asymptotic properties are provided. In Section 3, we report Monte Carlo simulations to examine finite sample properties of our tests. In Section 4, an empirical example is provided to test for asymmetric behavior in the U.S. prime lending rate minus the 3- month certificate of deposit rate. Concluding remarks are provided in Section Threshold Unit Root Tests and Extensions We begin by describing the threshold models considered by EG. To specify the asymmetric adjustment process of a time series, Enders and Granger (1998) initially consider the following model: 3

5 Δy t = I t ρ 1 y% t 1 + (1-I t )ρ 2 y% t 1 + p δ j Δy t-j + e t, (1) j=1 where y% t 1 = y t-1 % is the demeaned or detrended series, and % is the estimated dt 1 dt 1 deterministic term. That is, d t-1 = a 0 for a model with drift and d t-1 = a 0 + a 1 (t-1) for a model with drift and trend. In the basic model with no deterministic term, d t-1 is empty so that % = y t-1, where y% t 1 can be described as a shock or deviation. I t is defined as a yt 1 Heaviside indicator function that depends on whether the threshold variable is the level variable y t-1, or its change Δy t-1. Specifically, I t is defined for the TAR model as I t = 1 when y% t 1 0, and I t = 0 otherwise, (2a) and for the MTAR model as I t = 1 when Δ% yt 1 0, and I t = 0 otherwise. (2b) In the above, the threshold parameter is assumed to be zero in the basic model with no deterministic terms. In more general models with a drift and trend, the threshold parameter is the mean zero value of the detrended threshold variable; obviously, the mean of the detrended residual series of y% t 1 or Δ% yt 1 is zero when the residual comes from the regression with a constant term. To test the null of a unit root, EG consider an F-statistic to test the hypothesis that ρ 1 = ρ 2 = 0 in (1) and provide relevant critical values for the models in (2a) and (2b). In the present paper, we modify and extend the models of EG as follows. For the TAR model, we consider the indicator function: I t = 1 when y t-1 c, and I t = 0 otherwise, (3a) where c is the threshold variable. For the MTAR model, we consider the indicator function: 4

6 I t = 1 when Δy t-1 c, and I t = 0 otherwise. (3b) Note that the level, or change in the level, describing the threshold variable in (3a) and (3b) is not a detrended series as in (2a) and (2b). While one can still utilize a detrended series y% t 1 or Δ% yt 1 as in (2a) and (2b), this is a matter of preference and does not affect the asymptotic distribution of our test statistics. If desired, we can allow for a delay parameter d by using y t-d or Δy t-d in (3a) and (3b). While we let d = 1 in our simulation experiments, the value of d can be selected to maximize the fit of the underlying model. In this paper, we consider a fully unconstrained threshold model by extending equation (1) as follows: p p Δy t = I t [a 0 + a 1 t +ρ 1 y t-1 + δ 1j Δy t-j ] + (1-I t )[b 0 + b 1 t +ρ 2 y t-1 + δ 2j Δy t-j ] + e t. (4) j=1 j=1 We want to test the following hypotheses: H 0 : ρ 1 = ρ 2 = 0 against H a : ρ 1 < 0 or ρ 2 < 0, (5) where ρ 1 = ρ 2 = 0 implies that y t is a non-stationary unit root and there is no asymmetry in the long-run persistent effects. In the special case where a 0 = b 0, a 1 = b 1, and δ 1j = δ 2j, j=1,..,p, the basic model of EG emerges as in equation (1), implying that the deterministic terms and short-run dynamics are the same in each regime. These restrictions can be formally tested from the data. In what follows, we will denote the existence of asymmetric deterministic terms and/or short-run dynamics as Condition 1, implying that the general model in (4) should be adopted. On the other hand, if Condition 1 does not hold then the original EG specification is preferred. We next allow for a threshold parameter that separates the sample data into two regimes. Using the threshold variable y t-1, consider the following normalization 5

7 1 1/2 I t = I(y t-1 > c) = I( σ T y > 1 1/2 ) = I( t 1 σ T 1 1/2 c σ T y > t c* ), (6) where c * 1 1/2 = σ T c is the normalized threshold parameter and σ 2 = T -1 E( y t-1 ) 2. Since c * can take on any value of the threshold variable and differ in different data samples, it is not feasible to provide relevant critical values at all possible values of c *. We instead 1 1/2 * consider the sorted threshold variable, σ T y t 1. Obviously, the sorted threshold parameter cannot be greater or less than the maximum or minimum value of the 1 normalized threshold variable y t-1 *. Let * y ( ) t 1 τ = c * be the τ-th percentile of the empirical distribution of y t-1 *, such that P[ 1 1/2 σ T y t 1 * c * 1 1/2 * ] = P[ σ T y t 1 yt * ( ) 1 τ ] = τ. Then, we can show that 1 1/2 I t = I( σ T y t 1 > yt * ( ) 1 τ ) I(W(r) > c * ) = I(W * (r) > τ), (7) where W * (r) is the Brownian motion using the sorted data corresponding to the usual Brownian motion W(r), defined on r [0,1]. The threshold parameter is thus transformed into a percentile parameter τ, defined over the interval 0 and 1, and the asymptotic distribution of the threshold unit root test depends only on the percentile parameter. Alternatively, in the MTAR model where the threshold variable is Δy t-1, we define the sorted threshold variable Δy t-1 *, where Δy t-1 * (τ) = c is the τ-th percentile of the empirical distribution of Δy t-1 *. Then, we can show that I t = I(Δy t-1 > c) = I(σ 1 1 Δy t-1 > σ 1 1 Δy t-1 * (τ)) I(U(r) > τ), (8) where U(r) is a process having a uniform distribution on [0,1] and σ 1 2 = T -1 E( Δy t-1 ) 2. As a result, the threshold parameter is transformed into a more manageable percentile parameter with one set of critical values. 6

8 To test the unit root null hypothesis, we consider an F type test for the joint hypothesis ρ 1 = ρ 2 = 0 in (5). To simplify our discussion, we rewrite (4) as Δy t = I t [ρ 1 y t-1 + γ 1 q t ] + (1-I t )[ρ 2 y t-1 + γ 2 q t ] + e t, (9) where q t = [1, t, Δy t-1,.., Δy t-p ], γ 1 = [a 0, a 1, δ 11,.., δ 1p ] and γ 2 = [b 0, b 1, δ 21,.., δ 2p ]. The F-statistic to test the joint hypothesis ρ 1 = ρ 2 = 0 in (5) is given as F(τ) = ˆ( ρ τ )' ˆ ( ( )) 1 V ρτ ˆ( ) ρ τ /2, (10) where ˆ( ρ τ ) = ( ˆ ρ1( τ ), ˆ ρ2( τ )) is the OLS estimator from regression (9) with variance V ( ˆ ρ( τ )). The coefficient ˆ( ρ τ ) can be obtained from the regression using the detrended series y% t 1 after controlling for the effects of q t, Δy t = I t ρ 1 y% t 1 + (1-I t )ρ 2 y% t 1+ e t. Since I t and (1-I t ) are orthogonal, F(τ) is the sum of the two quadratic forms F(τ) = [ ˆ ρ1( τ )' 1 V ( ˆ1 ρ ( τ)) 1 ˆ ρ ( τ ) + ˆ ρ2( τ )' 1 V ( ˆ2 ρ ( τ)) 2 ˆ ρ ( τ )]/2, (11) where ˆ ρi ( τ ) and V ( ˆ ρ ( τ )), i=1,2, are the corresponding OLS estimate and error variance, i respectively. Note that these estimates and their corresponding test statistics are given as a function of τ to denote the dependency on the percentile threshold parameter, which is assumed to be known a priori. The asymptotic distribution of our test statistic is given as follows: Theorem 1. Let M ˆ (, r τ ) be the residuals from the projection of a standard Wiener process W(r) into the subspace generated by the function (1, u(r)) for the drift model, or (1, r, d(r)) for the trend model, and is defined over the interval r [0, 1]. Under the null hypothesis, ρ 1 = ρ 2 = 0 in (5), and F(τ) follows as T, 7

9 1 k k k 2 / 2, (12) k = 1 F(τ) ( M I dw) '( M I M ') ( M I dw) where M = M ˆ (, r τ ), I1 = I(W * (r) > τ) or I(U(r) > τ), and I 2 = 1 I 1. Note that the above asymptotic distribution is simplified by using the expression of the residual process M ˆ (, r τ ) from the projection from an integrating process on the deterministic terms in each regime, and it is given as the sum of two stochastic terms which are orthogonal to each other. More specific expressions of M ˆ (, r τ ) can be given explicitly as in the usual DF type tests for each of the model specifications with drift or trend terms, but it is a matter of more complicated expressions. It is evident that the distribution of the F-statistic is given as functions of stochastic terms and it is clear that F(τ) depends on the percentile parameter τ. The critical values of the F-test in (11) are reported in Table 1 for the TAR and M-TAR models with different specifications regarding a drift and trend. Note that while the threshold parameter is known a priori, the deterministic term parameters will not be nuisance parameters. Thus, regardless of the values of a 0, b 0, a 1 and b 1 in equation (4) the same critical values can be adopted at each value of τ since the test depends only on the threshold percentile parameter. In Table 1 (a) and (c), we report the threshold unit root test critical values at different threshold parameters, τ = 0.1, 0.2,.., 0.5, when the threshold parameter is known a priori. The critical values were obtained using a DGP under the null of a symmetric unit root (ρ 1 = ρ 2 = 0) with i.i.d. errors. All critical values were generated using 50,000 replications. Since the critical values are symmetric for τ = 0.6, 0.7,.., 0.9, they correspond to the critical values for τ = 0.4, 0.3,.., 0.1, respectively. Note that while the critical values do 8

10 not vary much, and even less so in the MTAR model, they do change somewhat at different values of τ. As noted, the critical values in Table 1 (a) and (c) can be utilized whenever the threshold parameter is known or can be consistently estimated a priori. However, if the threshold parameter is unknown, then it must be estimated from the data. There are two possible cases in this regard. The first case applies to the situation where sample separation is not possible under the null of a unit root. In the second case, sample separation is possible even when the null is true. We define these two sources of sample separation as follows: Condition 1: γ 1 γ 2 in (9) Condition 2: ρ 1 ρ 2 in (9). Obviously, Condition 2 will not hold under the null of ρ 1 = ρ 2 = 0 as described in (5). The important question that we wish to address is whether Condition 1 can hold under the null of a unit root. Note that the EG specification implies γ 1 =γ 2 such that Condition 1 does not hold. We first consider the case where Condition 1 does not hold, implying that the deterministic terms and short-run dynamics are symmetric. In this case, unless we assume that the threshold parameter is fixed, the so-called Davies (1987) problem will occur, since the threshold parameter is identified only under the alternative and cannot be identified under the null, ρ 1 =ρ 2 = 0. As noted, if the threshold parameter is assumed to 9

11 be unknown, one common practice is to utilize a supreme type test. In this case, we consider the following endogenous supreme type test: SupF = Max F( τ ), (13) τ where the threshold percentile parameter is estimated as the value that minimizes the sum of squared residuals, which amounts to maximizing the unit root test F-statistic. Note that the supreme type (SupF) test is free of any nuisance parameters under the null. The SupF test statistic can be estimated over the trimmed interval τ [τ 1, τ 2 ] to maximize the probability of identifying two regimes. As is customary, we choose a trimming proportion of 15% so that τ 1 = 0.15 and τ 2 = Critical values for the SupF test were simulated and are provided in Table 1 (b) and (d) for different threshold models with drift and trend. All SupF test critical values were derived using 10,000 iterations in each case, and are provided for different sample sizes ranging from T = 50 to an approximate asymptotic size T = Assume that Condition 1 holds, so that the deterministic terms and/or short-run dynamics (lagged augmented terms) differ in each regime. If so, we know that a threshold parameter exists, regardless of whether the null ρ 1 = ρ 2 = 0 is true. If we know that the deterministic terms and/or short-run dynamics differ in two regimes, then we want to utilize this information in our testing procedure and should assume a priori that two regimes exist. As previously noted, in such cases, there will be confusion on which critical values to use. In each case, the critical values of the endogenous SupF test are significantly greater than the corresponding critical values of the exogenous F-test in Table 1 (a) and (c). Simply put, by ignoring information about Condition 1 researchers can potentially arrive at much different conclusions regarding their research findings. 10

12 Given the above, it is important to recognize that a threshold parameter can be identified when sample separations are evident. As a result, we recommend that practitioners test Condition 1 when utilizing our threshold unit root tests. In the setup of our models, there are three potential sources of differences in γ 1 -γ 2 : the difference in the drift coefficients a 0 b 0, the difference in the trend coefficients a 1 b 1, and the difference in the coefficients of the short-run dynamics δ 1 δ 2. As long as any of these differences occur, then a threshold parameter can be consistently estimated. For example, the value of the threshold parameter could be estimated by a search procedure to identify the value where the sum of squared residuals is minimized. The estimate of the threshold parameter would be expected to approach the true threshold parameter as γ 1 -γ 2 increases. Since the identification of the threshold parameter will not depend on the null or alternative hypotheses if Condition 1 holds, the threshold unit root tests will be valid and the Davies problem (1987) will not occur. Note that testing for Condition 1 is standard, since the testing hypothesis does not involve a long-run persistence parameter restricted to be zero so the usual inference tests can be applied. 1 As such, if the null hypothesis γ 1 = γ 2 is rejected with the usual t-test or F-test, then we can identify the percentile parameter a priori and utilize the critical values in Table 1. What if ρ 1 = ρ 2 < 0? Then the time series y t is stationary and the persistent effects are symmetric. However, in such cases it is still possible that Condition 1 will be satisfied if the deterministic terms and/or short-run dynamics will differ. Then, the 1 It is possible to test the null hypothesis that γ 1 = γ 2 without knowing the order of integration of the time series under investigation, and such tests can be more precise. However, we do not pursue such tests in this paper. 11

13 threshold parameter can be identified. When the long-run persistence parameters differ, we have ρ 1 ρ 2, which holds only under the alternative hypothesis. However, it is not necessary to satisfy both of the conditions ρ 1 ρ 2 and γ 1 γ 2 in order to have stationarity. The stationarity condition is applied only to ρ 1 < 0 or ρ 2 < 0 and does not depend on Condition 1. Fortunately, we can test for a threshold unit root ρ 1 = ρ 2 = 0 in (8), regardless of whether γ 1 =γ 2, because the test statistics are invariant to the parameters γ 1 and γ 2 when the threshold (percentile) parameter is known a priori. Thus, testing the joint restrictions that ρ 1 = ρ 2 = 0 and γ 1 =γ 2 is unnecessary. While adopting such joint tests can be feasible, they will be less powerful and we do not consider them here. 3. Finite Sample Properties In this section, we report Monte Carlo simulations to examine the finite sample power properties of the new EG threshold unit root tests. In each case, the level (TAR) and momentum (MTAR) threshold unit root test results are reported for the models with drift only (subscript D), and with drift and trend (subscript T), respectively. To perform our simulations, pseudo-iid N(0,1) random numbers were generated using the procedure in RATS and all calculations were conducted using the RATS software version 7.0. All of the simulations were performed using 5,000 iterations in each case. To see if the results are sensitive to the threshold parameter, we report results for τ = 0.5 and τ = 0.3, respectively. We begin by examining the results in Table 2A using a sample of size of T = 100. Perhaps most notable among our findings is the significantly higher power of the 12

14 threshold momentum test (MTAR), as compared to the threshold autoregressive test (TAR), in every case. This outcome holds regardless of whether the model includes a drift, or includes a drift and trend, and regardless of the value of the threshold parameter. As expected, the power increases in each test as the value of the persistent parameter (ρ 1 or ρ 1 ) becomes more negative. Also as expected, the power in the models with trend is lower than in the models with drift only. Most notably, the power of the TAR test with trend is the lowest in all cases. Comparing the power at different threshold values, in general, we see somewhat greater power when using the median of the threshold variable as our threshold parameter, τ = 0.5, as compared to τ = 0.3, with the greatest differences displayed in the MTAR model. Moving to results using the larger sample size of T = 250 as displayed in Table 2B, as expected, we see a significant increase in power in every case. Overall, it is clear that the new momentum threshold tests (MTAR) display relatively good power in all cases. While the threshold autoregressive unit root tests (TAR) with trend displays relatively low power in samples of size T = 100, its power increases significantly in the larger sample size T = Empirical Example To provide an empirical example, we utilize our threshold tests to examine the spread in the bank lending minus deposit rate. Asymmetric adjustments in interest rates can have important implications for both banking and monetary policy. For instance, collusive pricing arrangements and/or transactions costs of shopping for a loan might lead banks to raise lending rates more quickly when the spread in lending minus deposit rates is narrowing than widening. Alternatively, banks may be more concerned with limiting adverse customer reaction causing banks to increase lending rates more slowly when 13

15 deposit rates are rising and vice versa (e.g., Hannan and Berger, 1991; Neumark and Sharp, 1992; Rajan, 1992; Thompson, 2006). Moreover, if banks adjust interest rates in an asymmetric manner then monetary policy actions are more likely to have asymmetric effects on the overall economy (e.g., Bruinshoofd and Candelon, 2005). To empirically test the above predictions, we will examine the spread in the prime rate minus 3-month CD rate using U.S. monthly data from 1964: :11. This data set comes from the web site of the Federal Reserve Bank of St. Louis and is the longest consistent time series available at the time of this study. A plot of our data along with summary statistics is provided in Figure 1. While visualization of the times series can be informative, formal testing is required to determine if the series are stationary and if asymmetries are important. To provide a benchmark, we first perform conventional linear unit root tests using the augmented Dickey-Fuller test (ADF) and the Elliot- Rothenberg-Stock DF-GLS test (DF-GLS) for each time series in levels and firstdifferences. The test results are displayed in Table 3. Neither the ADF test nor the more powerful DF-GLS test can reject the unit root in any case in levels (at the 10% significance level). In contrast, the null of a unit root is rejected in all cases except one in the first-differenced series (at the 1% significance level). Thus, according to the conventional linear tests one would likely conclude that these series are non-stationary. We next examine the results of applying our threshold unit root tests. The results are reported in Table 4. We first consider results for the individual interest rate series. For the prime rate, the F-test rejects the null of a unit root in three of four cases (at the 1% or 5% level of significance). The null of symmetric persistent parameters is rejected in two of four cases (at the 1% or 5% significance levels). Rejection of the nulls is 14

16 strongest in the momentum models, which suggest that adjustment speeds depend on the change in the series. The estimated persistent parameters suggest that the prime rate adjusts more quickly to its long-run equilibrium when its change is below the threshold level (approximately equal to zero) and behaves likes a unit root otherwise. Given that F- tests strongly reject the null that the deterministic terms and short-run dynamics are equal in two regimes (at the 1% level of significance), we conclude that the threshold parameter can be consistently estimated and use the more powerful exogenous test critical values in Table 1 (a) and (c). We next examine the results for the 3-month CD rate (in Table 4). The F-test rejects the null of a unit root in all four cases (at the 1% or 5% level of significance). The null of symmetric persistent parameters is rejected in three of four cases (at the 1% or 5% significance levels). Rejection of the nulls is again strongest in the momentum models, which again suggests that adjustment speeds depend on the change in the series. As in the prime rate, the estimated persistent parameters suggest that the deposit rate adjusts more quickly to its long-run equilibrium when its change is below the threshold level (approximately equal to zero) and behaves likes a unit root otherwise. Given that F-tests strongly reject the null that the deterministic terms and short-run dynamics are equal in two regimes (at the 1% level of significance), we again conclude that the threshold parameter can be consistently estimated and use the more powerful exogenous test critical values in Table 1 (a) and (c). We now discuss the results for the spread in the prime interest rate minus the 3- month CD rate, which is the focus of our empirical investigation. The F-test rejects the null of a unit root in three of four cases (at the 1% or 5% level of significance). 15

17 However, the null of symmetric persistent parameters is rejected only in the M-TAR models (at the 1% or 5% significance levels). We therefore focus the remainder of our discussion on the M-TAR models. The estimated persistent parameters suggest that the spread in the lending minus deposit rate adjusts more quickly to its long-run equilibrium when its change is below the threshold level (approximately equal to zero) and behaves likes a unit root otherwise. This difference is most noticeable in the Trend M-TAR model, where the estimated persistent parameter on Y(-) ( = 0.708) is clearly stationary, while the parameter on Y(+) ( = 0.959) is close to a unit root. These findings suggest that the spread in lending minus deposit rates is asymmetric and adjusts more quickly to a long-run equilibrium when the spread is narrowing than widening. This outcome supports the theoretical arguments above that suggest collusive pricing among lenders and/or transactions costs of shopping for a loan, and suggest that monetary policy actions to lower lending rates will take longer to achieve than vice versa. Finally, the null that the deterministic terms and short-run dynamics are equal in two regimes is strongly rejected in each case (at the 1% level of significance), suggesting again that the threshold parameter can be consistently estimated and we use the more powerful exogenous test critical values in Table 1 (a) and (c). Given that a unit root could not be rejected in any case in the linear tests in Table 3, these findings clearly highlight the importance of considering nonlinear tests. 5. Concluding Remarks In this paper, we build on the threshold unit root tests developed by Enders and Granger (1998) and extend their pioneering work in several important directions. Following Enders and Granger, we consider both threshold autoregressive (TAR) and 16

18 momentum (MTAR) threshold models. In particular, we examine a more flexible model specification where the deterministic terms and short-run dynamics can differ in two regimes. Then, using this general model, we consider several important extensions. First, we transform the threshold variable into its percentile value. The resulting threshold parameter is flexible and can take on a range of values between 0 and 1 while using the same set of critical values in each case. Second, we consider a supreme type test to estimate the threshold parameter from the data. Third, we describe how a threshold parameter can be consistently identified under the null if the deterministic terms and/or short-run dynamics differ in two regimes. In such cases, the so-called Davies (1987) problem will not occur. Moreover, if the deterministic terms and/or short-run dynamics differ in each regime, we can utilize this information to adopt more powerful exogenous tests. The asymptotic properties of the threshold tests are derived and finite sample properties are examined in simulations. The power properties of the extended threshold tests are generally good, especially in the momentum (MTAR) model. We conclude by providing an empirical example where we test for asymmetric adjustment paths in the spread of the prime rate minus the 3-month CD rate. Overall, we find strong evidence of asymmetric adjustment speeds and reject the null of a unit root in both regimes. We conclude that the spread in the lending minus deposit rate adjusts more quickly to its long-run equilibrium when the spread is narrowing below a threshold level than when the spread is widening above the threshold. 17

19 References Bruinshoofd, A. and B. Candelon, 2005, Nonlinear Monetary Policy in Europe: Fact or Myth? Economics Letters 86, Davies, R. 1987, Hypothesis Testing When a Nuisance Parameter is Present only Under the Alternative, Biometrika, 74, Enders, Walter, 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 16:3, Hannan, Timothy H., and Allen N. Berger, 1991, The Rigidity of Prices: Evidence from the Banking Industry, American Economic Review 81, Neumark, D. and S. Sharpe, 1992, Market Structure and the Nature of Price Rigidity, Quarterly Journal of Economics 107, Rajan, R., 1992, Insiders and Outsiders: The Choice Between Informed and Arm s Length Debt, Journal of Finance 47, Thompson, Mark A., 2006, Asymmetric Adjustment in the Prime Lending-Deposit Rate Spread, Review of Financial Economics 15, Tong, H., 1983, Threshold Models in Non-Linear Time Series Analysis, New York: Springer-Verlag. 18

20 Table 1. Critical Values of New Threshold and SupF Tests (a) New TR Tests with drift and trend New TAR Tests with threshold variable y t-1 New M-TAR tests with threshold variable Δy t-1 T CV % % % % % % % % % % % % Note: τ is the percentile parameter. (b) New SupF Tests with drift and trend New TAR Tests with threshold variable y t-1 New M-TAR Tests with threshold variable Δy t-1 CV T=50 T=100 T=250 T=1000 T=50 T=100 T=250 T= % % % Note: T denotes the sample size. 19

21 (c) New EG Tests with drift New TAR Tests with threshold variable y t-1 New M-TAR tests with threshold variable Δy t-1 T CV % % % % % % % % % % % % Note: τ is the percentile parameter. (d) New EG SupF Tests with drift New TAR Tests with threshold variable y t-1 New M-TAR Tests with threshold variable Δy t-1 CV T=50 T=100 T=250 T=1000 T=50 T=100 T=250 T= % % % Note: T denotes the sample size. 20

22 Table 2. Power Comparisons of F-statistic Part A. T = 100 ρ 1 ρ 2 TAR D τ = 0.5 MTAR D TAR T MTAR T TAR D τ = 0.3 MTAR D TAR T MTAR T Note: The percentile threshold parameter, τ, is assumed to be 0.5 or 0.3 in the simulations. The subscript of D denotes the case where the regression includes a drift and the subscript T denotes the case where the regression includes a drift and trend. The F-statistic tests the joint null hypothesis that ρ 1 = ρ 2 = 0 when τ is known a priori. 21

23 Table 2. Power Comparisons of F-statistic Part B. T = 250 ρ 1 ρ 2 TAR D τ = 0.5 MTAR D TAR T MTAR T TAR D τ = 0.3 MTAR D TAR T MTAR T Note: The percentile threshold parameter, τ, is assumed to be 0.5 or 0.3 in the simulations. The subscript of D denotes the case where the regression includes a drift and the subscript T denotes the case where the regression includes a drift and trend. The F-statistic tests the joint null hypothesis that ρ 1 = ρ 2 = 0 when τ is known a priori. 22

24 Table 3. Unit Root Test Results in Interest Rates, Prime Rate CD Rate Spread ΔPrime Rate ΔCD Rate ΔSpread ADF *** *** *** Lags DF-GLS *** *** Lags Note: Spread is the monthly U.S. bank prime lending interest rate minus the 3-month CD interest rate. ADF and DF-GLS denote the t-statistics in the ADF and DF-GLS test to test the null hypothesis of a unit root. All tests include an intercept and trend. The number of Lags to correct for serial correlation was determined by a sequential procedure starting with a maximum of 12 lags (a 10% significance level was used to keep the lag). ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively. 23

25 Table 4. Threshold Unit Root Test Results in Interest Rates, Spread Trend TAR Trend M-TAR Drift TAR Drift M-TAR F-test *** *** *** F-test for symmetry *** *** Coefficient of Y(+) Coefficient of Y(-) Threshold value Percentile parameter Selected Lag Drifts 6.476** *** *** Short-run *** *** *** *** Trends *** *** - - Deterministic & SR *** *** *** *** Prime Rate Trend TAR Trend M-TAR Drift TAR Drift M-TAR F-test 5.817* *** 5.277** *** F-test for symmetry 3.441* *** *** Coefficient of Y(+) Coefficient of Y(-) Threshold value Percentile parameter Selected Lag Drifts 4.570** 9.187*** 3.854** *** Short-run 3.197*** *** 3.666*** *** Trends ** - - Deterministic & SR 3.808*** *** 4.154*** *** CD Rate Trend TAR Trend M-TAR Drift TAR Drift M-TAR F-test 6.841** *** 6.299*** *** F-test for symmetry 3.378* *** 4.619** *** Coefficient of Y(+) Coefficient of Y(-) Threshold value Percentile parameter Selected Lag Drifts *** 6.439** *** Short-run 4.898*** *** 4.884*** *** Trends ** - - Deterministic & SR 6.282*** *** 7.518*** *** Note: Spread is the monthly U.S. prime lending rate minus the 3-month CD rate. F-test tests the null of a unit root in two regimes. F-test for symmetry tests the null that the persistence parameters are equal in both regimes. Coefficient of Y(+) and Y(-) are the estimated coefficients of the persistent parameters when the threshold variable is above and below the threshold level, respectively. The Threshold value and the number of lagged augmented terms were jointly determined by minimizing the Schwarz information criteria (SIC). Drifts, Short-run (augmented terms), Trends, and a joint test of these terms indicate F-tests of the null that these terms are symmetric in order to test Condition 1. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively. 24

26 Figure 1. Spread in the Prime Lending Interest Rate minus the 3-Month Certificate of Deposit Interest Rate (%), U.S. monthly data, 1964: :11. The mean spread is and the median spread is 1.870, with a maximum of 6.780, a minimum of , and a standard deviation of