Testing Granger Causality in the presence of threshold effects

International Journal of Forecasting 22 (2006) 771 – 780 www.elsevier.com/locate/ijforecast

Testing Granger Causality in the presence of threshold effects Jing Li * Department of Economics, Finance and Legal Studies, University of Alabama, Tuscaloosa, AL 35487, United States

Abstract This paper proposes a Granger Causality test allowing for threshold effects. The proposed test can be conducted on the basis of the threshold autoregressive distributed lag model or the augmented logistic smooth transition autoregressive model. The proposed test is applied to the U.S. civilian unemployment rate, and it is shown that real investment, real GDP and real interest rate are helpful for improving the in-sample fit of unemployment. D 2006 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. Keywords: Granger Causality; Threshold autoregressive model; Smooth transition autoregressive model; Time series; Unemployment

1. Introduction Recent literature has seen a growing popularity of threshold models motivated largely by asymmetric adjustments of macroeconomic and financial time series. For example, the business cycle asymmetry hypothesis postulates that unemployment rates revert to the equilibrium asymmetrically depending on their previous states. That is, unemployment rates rise more sharply in a recession than they decrease in an expansion. The linear ARIMA models are incapable of capturing nonlinearity of this sort. In contrast, the threshold autoregressive (TAR) model of Tong (1983) generally adequately describes the varying degrees of autoregressive decay for unemployment rates (Caner & Hansen, 2001; Montgomery, Azrnowitz, Tsay, & Tao, 1998; Rothman, 1998). * Tel.: +1 205 348 7591. E-mail address: [email protected].

This paper proposes a threshold Granger Causality test (TGC test hereafter) that relaxes the restrictive linearity of the conventional Granger test. Failing to reject the null hypothesis using the linear Granger test does not necessarily exclude nonlinear forecasting. Thus it is necessary to generalize the linear Granger test for a threshold series. This paper also considers complementary tests to the TGC test, in an attempt to assess predictability in each individual regime specified by the threshold model. The TGC test can be conducted based on the threshold autoregressive distributed lag model (TADL) that augments the TAR model with covariates. The TADL model can be seen as the dynamic switching regression model of Quandt (1958) and is related to the multivariate TAR model of Tsay (1998). See Richard (1980) for a discussion about models with several regimes. The focal point in this paper is testing causality in a single equation, rather than exploiting cross-equation correlation to improve estimation.

0169-2070/$ - see front matter D 2006 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ijforecast.2006.01.003

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J. Li / International Journal of Forecasting 22 (2006) 771–780

Alternatively, the TGC test can be carried out on the basis of augmented smooth transition autoregressive regressions, which may nest the TADL regression as a special case. See van Dijk, Tera¨svirta, and Franses (2002) for a recent survey of STAR models. Other nonlinear Granger Causality tests have been proposed in the literature, such as the causality-in-variance test of Cheung and Ng (1996) and the general bivariate Granger test of Baek and Brock (1992). Unlike those tests, the TGC test proposed in this paper is based on the parametric TADL or augmented LSTAR regression, and hence concentrates on the first moment. The TGC test takes the form of the heteroskedasticity-robust Wald test. Conditional on threshold effects, the TGC test asymptotically follows a standard v 2 distribution for two reasons. First, there are no nuisance parameters (of the threshold value and the delay lag) under the null hypothesis of the TGC test. Second, superconsistent estimates for the threshold value and delay are available. Therefore we can act as if those estimates are known values for the first order asymptotic inference. The TGC test is illustrated by forecasting U.S. civilian unemployment, a series shown to be governed by threshold effects. Macroeconomic theory suggests that real GDP, real investment, real consumption, inflation rates, real interest rates and output gap may have predictive powers for unemployment. That supposition can be evaluated formally by the TGC test. The remainder of the article is organized as follows. Section 2 formulates TGC tests based on the TADL model and augmented LSTAR model. Section 3 applies the TGC test to the U.S. civilian unemployment rate based on those two models. Section 4 concludes.

2. Models and the TGC test 2.1. The basic two-regime TADL model We first consider the two-regime self-exciting threshold autoregressive distributed lag TADL( p, q, s, d) model given by ! p q 2 X X X yt ¼ am þ bmi yti Imt þ cmj V xtj Imt þ et ; m¼1

i¼1

j¼1

ð1Þ

where indicators are defined as I 1t = I( y td N s) and I 2t = 1  I 1t , and the lowercase bold x t = (x 1t ,. . ., x kt )V (excluding constants) is a k  1 covariate vector at time t. The error e t is assumed to be a martingale difference sequence satisfying E(e t | X t1) = 0, where X t1 = ( y t1, . . . , y tp ; x t1, . . . , x tq ). s and d denote the threshold value and delay lag. Sufficient lagged terms of y ti , x tj are added so that no serial correlation is left in the errors. In practice choosing p and q may be guided by the AIC and BIC criteria. The inference space is specified in terms of the lagged dependent variable, though other specifications are possible. No contemporaneous covariates are included (so j starts from 1) to facilitate forecasting. Following Hansen (1997), no explicit distinction is drawn between errors in two regimes, though the TGC test is heteroskedasticity-robust. The random vector ( y t , x t ) is assumed to be jointly stationary and ergodic. Testing for Granger Causality in the threshold cointegration system is beyond the scope of this paper, and might be a future topic for research. Refer to Chan and Tsay (1991) for the geometrical ergodicity of a TAR process. Notice that nonstationary series can be appropriately detrended or differenced to achieve stationarity before applying the TGC test, an approach consistent with the Box– Jenkins methodology. For a difference-stationary process y t , the inference space can be specified based on either Dy td = y td  y td1 or the blong run differenceQ of y t1  y td . We use the former specification. Model (1) does not assume continuity of y t at the threshold, neither does it impose any restriction on shifts of coefficients across regimes. Thanks to the block orthogonality of regressors, fitting regression (1) is equivalent to running separate OLS regressions in each regime. To obtain estimates of the unknown s and d, treat each d a [1, d¯ ] and s a [miny t d , maxy t d ] as a potential delay and threshold, where d¯ is determined by prior knowledge. Then regression (1) specified by corresponding indicators is fitted. After grid-searching, the pair of (d, s) minimizing the residual sum of squares (RSS) is the LS estimate. Chan (1993) shows that (d, s) estimated this way is superconsistent in the TAR model. (d, s) may also be obtained by minimizing BIC or AIC. Note that those approaches lead to identical estimates when p and q are set beforehand. In practice, the lower and upper 15% values of

J. Li / International Journal of Forecasting 22 (2006) 771–780

ordered y td are ignored in estimating (d, s) to ensure adequate observations in each regime (see Andrews, 1993). Finally, it is necessary to plot the RSS against potential thresholds. This step is intended to check for multiple thresholds, which may suggest an augmented LSTAR model as an alternative. Note that when estimating s and d, the general-tospecific strategy of Hendry and Mizon (1999) is adopted since all covariates are included. Alternatively we might grid search s and d in model (1) without covariates, that is, under the null hypothesis of the TGC test. Theoretically there is no guarantee that the two approaches would produce the same estimate. Since the TGC test is based on the Wald statistic (specified later), estimating s and d in the unrestricted model (1) with covariates is preferred. The method of Hansen (1996) can be used to test threshold effects in model (1). As suggested by Hansen (1997), the calculated statistic needs to be compared to the bootstrap critical values owing to issues of nuisance parameters. See Davies (1977) and Andrews and Ploberger (1994) for details. The bootstrap critical values are computed as follows. First we generate a pseudo-random series z t ~iid N(0, 1), regress z t on all regressors in Eq. (1) for a given s, and keep the RSS as RSS(s)u . Next run the restricted regression by imposing b 1i = b 2i , c 1j = c 2j , (8i, j) to get RSSr. Then calculate sups n[RSSr  RSS(s)u ] = RSS(s)u by grid searching s. Those steps can be repeated many times to obtain the bootstrap critical values. See Hansen (1996, 1997) for more details about the bootstrap. Applying the TGC test is warranted once threshold effects are confirmed. 2.2. Extensions of the basic model The first generalization of model (1) allows for more than two regimes. If we have no a priori belief, the number of regimes might be chosen endogenously by the data. Consider the multiple-regime TADL model: ! p q l X X X yt ¼ bmi yti Imt þ cmj Vxtj Imt am þ m¼1

þ et ;

i¼1

ðl z 2Þ;

j¼1

ð2Þ

where I mt = I(s m1 V y td b s m ) with s 0 =  l; s l = l. The point is that the number of regimes l, not predetermined, is selected by the data. Model (2) is

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nevertheless computationally difficult in grid-searching s and d. The second extension is the two-regime augmented LSTAR model (ALSTAR) given as ! p q 2 X X X dmt am þ bmi yti þ cmj V xtj þ et ; yt ¼ m¼1

i¼1

j¼1

ð3Þ 1

where d 1t = 1  d 2t and d 2t = [1 + exp( c( y td  s))] . Note that unlike the TADL model, regime 1 in model (3) is specified by I( y td b s). In the ALSTAR model, parameter c determines the smoothness of transition, and s can be seen as the threshold splitting the two regimes. Note that when 0 b c b l, a smooth transition occurs across two regimes. When c Y l, instantaneous regime-switching takes place so that the ALSTAR model reduces to the TADL model. When c = 0, model (3) is equivalent to the linear ADL model. Model (3) does not require specification of indicators, and s, along with other coefficient estimates, is obtained by nonlinear least squares (NLS). d can be estimated by minimizing RSSs over the range [1, d¯ ]. We do not consider the exponential STAR model because it does not directly nest the two-regime TAR model. The LM-type linearity tests proposed by Luukkonen, Saikkonen, and Tera¨svirta (1988) can be used to test the linearity (or loosely speaking the threshold effects) for model (3). Luukkonen et al. (1988) suggests the first order (or third order) Taylor series approximation for the transition function d 2t around c = 0. The main advantage of the approximation is that identification problems (see Tera¨svirta, 1994) do not arise, and thus the LM-type statistic follows a standard asymptotic v 2 distribution. To implement that test, consider the auxiliary regression ! p q X X y t ¼ a1 þ b1i yti þ c1jV xtj i¼1

þ

p X i¼1

j¼1

b2i yti ytd þ

q X

! c2jV xtj ytd

þ t ;ð4Þ

j¼1

where  t contains the remainder of the Taylor series, and  t = e t under the null hypothesis. Then testing the null H0: c = 0 in model (3) is equivalent to testing H0 : b21 ¼ : : : ¼ b2p ¼ 0 and c21 ¼ : : : ¼ c2q ¼ 0

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in (4). Note that the cross-product term 1 d y td does not appear in the second parentheses in (4) to avoid perfect multicollinearity. See van Dijk et al. (2002) for more details about this test. We emphasize that covariates enter models (1)–(3) additively. This approach is largely practical, due to the difficulty of specifying alternative parametric forms. Consequently the estimations and testing results may not be robust to the misspecification of the function form. We caution the reader in making inferences using this approach. 2.3. The TGC test and complementary tests For models (1) and (3), we consider three null hypotheses given by H00 : H10 :

where the first, second and third subscripts of c are regime, lag and covariate indexes respectively. Likewise, H00 tests predictability in the two regimes while H0i tests predictability in regime i, i = 1, 2. Corresponding null hypotheses for model (2) can be formulated analogously. Note that the linear Granger test is restrictive because it imposes no threshold effect under both the null and the alternative hypotheses. If threshold effects do exist, the linear test is misspecified. All null hypotheses are tested based on the Wald statistic, written as " X 1  X   ˆ zt ztV W ¼ Rh ÞV R eˆ 2t zt zt V 

X

zt zt V

1

#1 RV



 Rhˆ ;

ð5Þ

c11 ¼ c21 ¼ : : : ¼ c1q ¼ c2q ¼ 0; c11

¼ : : : ¼ c1q ¼ 0;

H20 : c21 ¼ : : : ¼ c2q ¼ 0; where the alternative hypotheses state that at least one coefficient is nonzero. Note that H00 implies that none of the covariates has predictive content in the two regimes, so it is related to the standard TGC test. In contrast, H0i implies no predictive content in regime i, i = 1, 2. Strictly speaking, H01 and H02 are not Granger Causality tests but misspecification tests of the dynamic switching regression model. Here they are investigated as complements to the TGC test. For a threshold process, neither H01 nor H02 can be ruled out a priori. It is not hard to imagine a situation in which covariates are binding in a recession and hence have predictive content, while they become non-binding and lose predictive content in an expansion. For unemployment it is meaningful to test H01 and H02. Other null hypotheses are possible. For instance, we might be interested in testing whether a specific covariate (or a sub-vector of covariates) helps forecast y, conditional on the past information of y and other covariates. Those hypotheses can be formulated as H00 : c11i ¼ c21i ¼ : : : ¼ c1qi ¼ c2qi ¼ 0; H10 : c11i ¼ : : : ¼ c1qi ¼ 0;

ia½1; k ;

H20 : c21i ¼ : : : ¼ c2qi ¼ 0;

ia½1; k ;

ia½1; k ;

where R is the selection matrix for the null hypotheses, h are estimates for parameters of interest, z t = Bf(h) / Bh, f = E( y t | X t1) and eˆ t is the OLS or NLS residual. The Wald test is heteroskedasticity-robust and can be simplified if homoskedasticity is imposed. Some remarks are in order regarding the TGC test. First, we emphasize that a test for threshold effects should precede the TGC test, because the TGC test is justified only in the presence of threshold effects. One benefit of the sequential testing procedure is that identification issues can be circumvented since the TGC test is conditional on threshold effects. In comparison, joint testing for threshold effects and Granger Causality might entail a test with nonstandard asymptotic distribution due to nuisance parameters. Second, the fact that W~v 2(m), where m is the number of restrictions, does not rely on whether s and d are known or estimated from the data. Here standard asymptotic results are applicable because (1) there are no parameters unidentified under H00, H01, H02, and (2) superconsistent estimators for s and d are available. The second point can be translated into treating s and d as known values for the first order inference. See Hansen (1997) for more details.

3. An example This section applies the TGC test to the U.S. civilian unemployment rate. The sub-sample of

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Table 1 Data sources and descriptions: The dependent variable and covariates Series 1

Civilian unemployment rate Unemployment rate4

ID

SA2

FRE3

Sources/calculation methods

UNRATE UNM

Y Y

M Q

U.S. Department of Labor: Bureau of Labor Statistics ¼

Real GDP1 Real potential GDP1 Output gap Real personal consumption expenditure1 Real gross private domestic investment1 3-month treasury bill secondary market rate1 Nominal interest rates4

GDPC1 GDPPOT GAP PCECC96 GPDIC1 TB3MS NOMI

Y na na Y Y na na

Q Q Q Q/A/M Q/A M Q

Real interest rates

GDPDEF INF REAL

Y Y na

2 j¼0

 UNRATEtþj =3

U.S. Department of Commerce: Bureau of Economic Analysis U.S. Congress Congressional: Budget Office = (GDPC1  GDPPOT) / 100 U.S. Department of Commerce: Bureau of Economic Analysis U.S. Department of Commerce: Bureau of Economic Analysis Board of Governors of the Federal Reserve System ¼

GDP implicit price deflator1 Inflation rates

P

Q Q

P

2 j¼0

 TB3MS tþj =3

U.S. Department of Commerce: Bureau of Economic Analysis

¼



GDPDEF t GDPDEF t1 GDPDEF t1



 400%

= NOMI  INF

Q

Notes: 1 The series with superscript 1 are downloaded from the Federal Reserve Economic Data database in the Federal Reserve Bank of St Louis at: http://research.stlouisfed.org/fred2/, and other series are calculated. 2 SA: seasonally adjusted; Y: yes; na: not applicable. 3 FRE: data frequency available at the Federal Reserve Economic Data database; Q: quarterly; M: monthly; A: annually. 4 t = Jan, Apr, Jul, Oct.

1948:Q1–1993:Q2 is used in estimation and testing, while the remaining 1993:Q3–2005:Q1 is reserved for the out-of-sample forecast. Remember that our goal is to provide an application of the TGC test. Models in this section are not necessarily the best representations for unemployment. The quarterly unemployment series of unmt is investigated, which is the simple average of the seasonally adjusted monthly data. Readers should be wary of possible impacts of seasonal adjustments on our results. The quarterly frequency is used because covariates like real GDP are at the quarterly frequency. Six macroeconomic variables are considered as covariates: the first-differenced output gap (DGAPt ), real interest rate (DREALt ), inflation rate (DINFt ), log-difference of real consumption (Dlog PCECC96t ), real GDP (Dlog GDPC1t ) and real investment (Dlog GPDIC1t ).1 Here we take differences to ensure stationarity. Data sources and descriptions are reported in Table 1. Plots of all series, with shaded observations of unmt in economic contractions (specified later), are displayed

in Figs. 1 and 2. As shown in Fig. 1, unmt displays asymmetric evolutions. That is, it rises sharply to a peak in a recession and then declines gradually toward mild troughs during an economic recovery. Following Hansen (1997) we work with Dy t = Dunmt . The augmented Dickey–Fuller test and the tests of Enders and Granger (1998) and Shin and Lee (2003) all indicate that Dunmt has no autoregressive unit root. Then sequential least squares are applied to regression (1), in which we set max p = 4 and d¯ = 4 to capture the short-run dynamics of the quarterly data. Table 2 reports the minimum RSSs associated with various values of (d, s, p), and the LS estimate2 is (d = 4, s = 0.2333, p = 4). Fig. 3 plots RSS against possible thresholds for various delays with p = 4, where RSSi , i = 1, 2, 3, 4 is the RSS series related to d = i. From Fig. 3 it is clear that apart from RSS4, the RSSs do not reach the minimum sharply. In addition there is evidence for multiple thresholds (RSS1, RSS2 and RSS3 are W-shaped). This finding implies that 2

1

Actually we use D(100  logPCECC96t ), D(100  logGDPC1t ), D(100  logGPDIC1t ) in estimation and testing.

Those estimates are invariant to the ending period of the regression. We let the end date range from 1988 Q2 to 1995 Q2, and (d, s) remains unchanged.

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J. Li / International Journal of Forecasting 22 (2006) 771–780

Fig. 1. U.S. civilian unemployment rate: 1948:Q1–2005:Q1.

regression (1), the one assuming a unique threshold and instantaneous regime-switching, may not be the best choice for Dunmt . The ALSTAR regression will be fitted shortly as an alternative.

Panel A of Table 3 presents the point estimates of regression (1) with p-values in parentheses. Several remarks are worth noting. First, the Breusch–Godfrey LM test with four lags indicates no autocorrelation in

Fig. 2. Time series of covariates.

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Table 2 Grid-searching d, s with various p in model (1) for Dunmt p = 4a

d =1

d =2

d =3

d = 4a

RSS s

10.33 0.3000

11.12 0.0333

10.86  0.133

p =2

d =1

d =2

d =3

RSS s

11.69 0.6000

11.53 0.0333

11.94 0.5333

p =3

d =1

d =2

d =3

d =4

RSS s

11.14  0.1667

11.32 0.0333

11.47  0.133

10.17 0.2333

d =4

p =1

d =1

d =2

d =3

d =4

10.50 0.2333

RSS s

9.81 0.2333a

12.34 0.5333

12.52 0.4000

12.28 0.5333

11.09 0.2333

Note: RSS is the residual sum of squares. s is the threshold minimizing RSS for given d. a Denotes LS estimate.

residuals at the 5% level (LM(4) = 9.64). Second, the residual variances in the two regimes (r 12 = 0.132, r 22 = 0.110) may imply heteroskedasticity, meaning that robust tests are appropriate. Third, the null hypothesis of no threshold effects in model (1) is rejected at the 1% level by the heteroskedasticity-

robust sub-Wald test proposed by Hansen (1996), justifying the subsequent TGC test. The estimated threshold s = 0.2333 splits the regression function into two regimes. Heuristically, regime 1 specified by I(Dunmt4 N 0.2333) can be thought of as economic contractions since unemployment has been rising.

Fig. 3. RSS against threshold values with p = 4.

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Table 3 Point estimates for Dunmt in models (1) and (3) Panel A: TADL model

Panel B: ALSTAR model

1

Constant Dy t 1 Dy t 2 Dy t 3 Dy t 4 DGAPt4 DGAPt2 DREALt1 DREALt2 DINFt1 DINFt2 Dlog PCECC96t1 Dlog PCECC96t2 Dlog GDPC1t1 Dlog GDPC1t2 Dlog GPDIC1t1 Dlog GPDIC1t2 c s d Threshold test

Regime 1

Regime 2

Regime 11

0.28(0.07) 0.33(0.21)  0.11(0.49)  0.25(0.07)  0.11(0.50) 0.07(0.73)  0.36(0.22)  0.04(0.46) 0.00(0.99) 0.00(0.99) 0.07(0.33)  0.14(0.18)  0.07(0.52)  0.27(0.16) 0.25(0.10) 0.04(0.08)  0.04(0.06) (Not applicable) 0.23333* 4* 37.07(0.00)**

0.26(0.00) 0.36(0.00) 0.18(0.12) 0.19(0.19) 0.07(0.47) 0.28(0.09) 0.21(0.17) 0.06(0.07) 0.09(0.01) 0.02(0.61) 0.04(0.30) 0.02(0.54) 0.02(0.60) 0.22(0.02) 0.06(0.49) 0.00(0.97) 0.01(0.04)

 0.50(0.00) 0.04(0.86) 0.30(0.03)  0.20(0.08)  0.34(0.00)  0.29(0.15)  0.12(0.62) 0.12(0.01) 0.05(0.53) 0.09(0.07)  0.05(0.54) 0.07(0.07) 0.03(0.58) 0.14(0.14) 0.14(0.20) 0.00(0.74)  0.03(0.00) 15.20(0.05)  0.21(0.00) 1* 61.32(0.00)11

  

  

Regime 2

   

   

0.43(0.00) 0.36(0.01) 0.30(0.03) 0.05(0.76) 0.07(0.56) 0.46(0.02) 0.26(0.22) 0.00(0.94) 0.05(0.32) 0.06(0.54) 0.00(0.98) 0.10(0.25) 0.02(0.68) 0.37(0.00) 0.09(0.39) 0.02(0.22) 0.01(0.22)

Note: 1 in the TADL model regime 1 is specified as I(Dunmt4 N 0.23333), while in the ALSTAR model regime 1 is specified as I(Dunm b  0.21). * p-values are not reported since distributions for those estimates are nonstandard. See Hansen (2000) among others. **See Hansen (1997) for details about bootstrapping the threshold test in the TAR model. 11 See van Dijk et al. (2002) for details about the LM-type threshold test in the STAR model.

Table 4 TGC tests for Dunmt based on models (1) and (3) Panel A: TADL model

All x* DGAP DREAL DINF Dlog PCECC96 Dlog GDPC1 Dlog GPDIC1

Panel B: ALSTAR model

H00

H10

H20

H00

H10

H20

107.8 (0.00) 4.73 (0.32) 9.36 (0.05) 2.28 (0.68) 2.61 (0.63) 16.61 (0.00) 10.69 (0.03)

45.51 (0.00) 1.48 (0.48) 0.80 (0.67) 1.08 (0.58) 2.05 (0.36) 4.14 (0.13) 6.04 (0.05)

62.32 (0.00) 3.25 (0.20) 8.56 (0.01) 1.21 (0.55) 0.56 (0.76) 12.48 (0.00) 4.65 (0.10)

198.7 (0.00) 9.03 (0.06) 10.20 (0.04) 3.97 (0.41) 4.46 (0.35) 24.30 (0.00) 17.79 (0.00)

90.99 (0.00) 3.94 (0.14) 6.43 (0.04) 3.77 (0.15) 4.06 (0.13) 9.60 (0.00) 14.79 (0.00)

68.43 (0.00) 5.69 (0.06) 1.15 (0.56) 1.00 (0.61) 1.30 (0.52) 18.22 (0.00) 4.65 (0.10)

Note: H00 tests the null of no Granger Causality in two regimes, while Hi0 tests no Granger Causality in regime i, i = 1, 2. Note that in the TADL model regime 1 is specified as I(Dunmt4 N 0.23333), while in the ALSTAR model regime 1 is specified as I(Dunmt1 b  0.21). *bAll xQ tests no Granger Causality for the six covariates jointly.

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Note that only 40 observations (shaded in Fig. 1) lie in regime 1 among all useful 177 observations. The results of the TGC tests in model (1) are reported in Panel A of Table 4 with p-values in parentheses. H00 is the standard TGC test for the null hypotheses that all covariates (or a single covariate) have no Granger Causality in two regimes, whereas H01 and H02 test no Granger Causality (loosely speaking) in regime 1 and 2 respectively. We have several findings. First, covariates are jointly significant in improving the in-sample fit of Dunmt , whether in two regimes (W = 107.8) or in each individual regime (W = 45.51, 62.32). Second, it is shown that, conditional on lagged values of Dunmt and other covariates, DGAP, DINF and Dlog PCECC96 have no forecasting power either in two regimes (W = 4.73, 2.28, 2.61) or in separate regimes. Interestingly, among the remaining significant covariates, only Dlog log GPDIC1 aids in prediction in regime 1 (W = 6.04) more than in regime 2 (W = 4.65). In other words, only real investment is particularly informative during a recession in the TADL model. By contrast, real interest rate and real GDP are helpful only in expansionary periods (W = 8.56, 12.48). According to the general-to-specific strategy, insignificant covariates and lagged dependent variables may be dropped, and the TGC test can be performed for the remaining covariates. That process may be repeated until all regressors in the final parsimonious model are significant. To conserve space we do not pursue this in model (1). Instead we fit the ALSTAR regression (3) and then carry out the TGC test based on the ALSTAR regression. This is motivated by the finding that multiple thresholds cannot be ruled out, as Fig. 3 indicates. To facilitate comparison, we set3 p = 4, q = 2 when grid searching d a [1, 4] in the ALSTAR model. Then we obtain d = 1 by minimizing RSSs. The final model (3) is specified by d 2t =[1 + exp( c(Dunmt1  s))] 1, and estimation results using NLS are collected in Panel B of Table 3. There, the most noticeable threshold effect is given by the intercepts in regime 1 ( 0.50) and regime 2 (0.43). The LM-type test of Luukkonen et al. (1988) is carried out formally for the null hypothesis of linearity against the ALSTAR

alternative. The computed statistic and p-value are LM =61.32 and p = 0.00, respectively. Hence, applying the TGC test in the ALSTAR model is justified. Note that c = 15.20 and s =  0.21 are both significant. Moreover, the smoothness parameter c is far from infinity, supporting a smooth transition across regimes. We emphasize that the specification for model (3) determines that regime 1 in model (3) corresponds to I(Dunmt1 b  0.21), the expansionary period (since unmt is declining) rather than the contractionary period. Panel B of Table 4 presents testing results in model (3). First, the covariates are shown to jointly improve the in-sample fit (W = 198.7, 90.99, 68.43). Second, DINF and Dlog PCECC96 are still not predictive (W = 3.97, 4.46), but DGAP now suggests Granger Causality in regime 2 (W = 5.69). Furthermore, DGAP and Dlog GDPC1 are the only two covariates having more forecasting information in a recession than in an expansion. When we look at the expansionary period, DREAL, Dlog GDPC1 and Dlog GPDIC1 are helpful (W = 6.43, 9.60, 14.79). A final note is that real GDP is almost equally informative during the contractionary period (W = 18.22) and the expansionary period (W = 9.60). We emphasize that the TGC test, like the linear Granger test, does not necessarily convey information about the out-of-sample forecast. To illustrate this point, we implement a one-step forecast in the TADL model. During 1993:Q3–2005:Q1, the mean squared prediction errors (MSPEs) in regime 1 (having only 4 observations) and regime 2 are (0.07633, 0.03319) for the restricted TAR model and (0.05758, 0.03572) for the unrestricted TADL model. Recall that in regime 2 the covariates are shown to aid in the in-sample fit (W = 62.32), though the TADL model does not significantly outperform the TAR model in regime 2 in terms of the out-of-sample forecast. For regime 2, the DM statistic of Diebold and Mariano (1995), based on mean squared errors and four autocovariances, equals 0.4137 with a p-value of 0.6812, not rejecting the null of equal forecast accuracy.

3 Alternatively we can choose p and q using the information criteria.

This paper has developed new methods of inference for Granger Causality allowing for threshold

4. Conclusion

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J. Li / International Journal of Forecasting 22 (2006) 771–780

effects. The proposed threshold Granger Causality test is based on the TADL model and the augmented LSTAR model. An application of those tests demonstrates that real GDP, real investment and real interest rate are helpful for improving the in-sample fit of U.S. civilian unemployment.

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