Unit root test with partial information on the break date

Introduction

Partial information on the location of break date can help improve the power of the test for unit root under break. It is this observation that informs the unit root testing under a local break-in trend by Harvey, Leybourne, and Taylor (2013) where the authors employ partial information on the break date. It builds on the insight first projected by Andrews (1993), who observed that prior information about the location of the break will help even if the analysts do not have full information about the precise break date.

The setback of most of the unit root tests that allow for breaks is that they lose their power if there is no such break compared to the tests that do not account for such break in the first place. Yet another issue with the procedure for detecting break dates is that when the break dates are not large enough there is tendency that they will not be detected. Undetected breaks when they are present can lead to loss of power in this case even for the tests that do not allow for such break. It then means that testing for unit root under breaks must be carefully sorted out.

In short the idea is to employ restricted search and this involves searching within the domain of the break where it is more likely for the break to have occurred. The benefit of this restricted search is that uncertainty around the break point reduces should there be a break point there indeed. The improvement in power as a result therefore means that the test should not fail to reject when it should.

HLT Approach

Harvey, Leybourne and Taylor (2013), in pursing this objective, adopted the Perron-Rodríguez (2003) approach employing the GLS-based infimum test because of its superior power. It is found that the GLS-based infmum tests perform better among those tests that do not allow for break detection and has greater robustness among those that allow for it. To robustify this approach, they proposed the union of rejections strategy. The union of rejections strategy attempts to derive power from the two discreet worlds, in which case the strategy pools the power inherent in the restricted-range with-trend break and the without-trend break unit roots. Using this strategy, the null of unit root is rejected 

if either the restricted-range with-trend break infimum unit root test or the without-trend break unit root test rejects.

In this way, there is no need for prior break date detection, which in itself can compromise the power of the test. 

Model

We can begin our review of this method by stating the following DGP:\[\begin{align*}y_t=&\mu+\beta t+\gamma_T DT_t (\tau_0)+u_t, \;\; t=1,\dots,T\\ u_t=&\rho_T u_{t-1}+\epsilon_t  \;\; t=2,\dots,T\end{align*}\]where \(DT_t(\tau)=1(t>[\tau T])(t-[\tau T])\). The null hypothesis is \(H_0:\rho_T=1\) against the alternative that \(H_c:\rho_T=1-c/T\) where \(c>0\). A crucial assumption which features in the computation of the asymptotic critical value is that the trend break magnitude is local-to-zero so that uncertainty can be captured, that is, \(\gamma_T=\kappa\omega_\epsilon T^{-1/2}\) with \(\kappa\) is a constant. 

The Procedure to Computing the Union of Rejections

To construct the union of rejections decision rule, the steps are involved have been broken down to a couple of blocks.

• STEP 1: The following sub-steps are involved: 

1. Assume there is a known break date at \(\tau T\), where \(\tau\in(0,1)\). The data are first transformed as:\[Z_{\bar{\rho}}=[y_1,y_2-\rho y_1,\dots,y_T-\rho y_{T-1}]^\prime\] and \[Z_{\bar{\rho},\tau}=[z_1,z_2-\rho z_1,\dots,z_T-\rho z_{T-1}]^\prime\]where \(z_t=[1,t,DT_t(\tau)]^\prime\) and \(\bar{\rho}=1-\bar{c}/T\) with \(\bar{c}=17.6\);
2. Apply the LS regression on the transformed data in Step 1 and obtain the residuals \(\tilde{u}_t=y_t-\tilde{\mu}-\tilde{\beta} t-\tilde{\gamma} DT_t (\tau)\). The GLS estimates for \(\theta\), \(\tilde{\theta}=(\tilde{\mu},\tilde{\beta},\tilde{\gamma})\), are\[\tilde{\theta}=\underset{\theta}{\text{argmin}}\; u_t^\prime u_t;\]
3. The ADF is applied to the residuals obtained in Step 2:\[\Delta \tilde{u}_t=\hat{\pi} \tilde{u}_{t-1}+\sum_{j=1}^k\hat{\psi}_j\Delta\tilde{u}_{t-j}+\hat{e}_t.\]

• STEP 2: Instead of assuming a known break date, HLT make use of the infimum GLS detrended Dickey-Fuller statistic as follows:

1. Define the window mid-point parameter \(\tau_m\) and the window width parameter \(\delta\);
2. Define the search window as\[\Lambda(\tau_m,\delta):=[\tau_m-\delta/2,\tau_m+\delta/2]\]and, if \(\tau_m-\delta/2<0\) or \(\tau_m+\delta/2>1\), then define the following respectively \(\Lambda(\tau_m,\delta):=[\epsilon,\tau_m+\delta/2]\) or \(\Lambda(\tau_m,\delta):=[\tau_m-\delta/2,1-\epsilon]\), where \(\epsilon\) is a small number set to 0.001.
3. Then compute \[MDF(\tau_m,\delta):=\underset{\tau\in\Lambda(\tau_m,\delta)}{\text{inf}}DF^{GLS}(\tau)\]which amounts to repeating the sub-steps Step 1 for every observation corresponding to the fraction defined in the restricted window \(\Lambda(\tau_m,\delta)\) and finding the least DF statistic 

• STEP 3: The Elliot et al (1996) DF-GLS is carried out as follows:

1. The data are first transformed as in Step 1 in the procedure above without including the break date and with \(\bar{c}=13.5\);
2. Apply the LS regression on the transformed data in Step 1 and obtain the residuals \(\tilde{u}_t^e=y_t-\tilde{\mu}-\tilde{\beta} t\). The GLS estimates for \(\theta\), \(\tilde{\theta}=(\tilde{\mu},\tilde{\beta})\), are\[\tilde{\theta}=\underset{\theta}{\text{argmin}}\; u_t^{e\prime} u_t^e;\]
3. The ADF is applied to the residuals obtained in Step 2:\[\Delta \tilde{u}^e_t=\hat{\pi} \tilde{u}_{t-1}^e+\sum_{j=1}^k\hat{\psi}_j\Delta\tilde{u}_{t-j}^e+\hat{e}_t.\]
4. The DF-GLS statistic is the t-value associated with \(\hat{\pi}\) and is denoted \(DF^{GLS}\)

• STEP 4: The union of rejections strategy involves the rejection of the null of unit root, as stated early, 

if either the restricted-range with-trend break infimum unit root test or the without-trend break unit root test rejects.

The decision rule is therefore given by\[U(\tau_m,\delta):=\text{Reject} \;H_0 \;\text{if}\;\left\{DF^{GLS}_U(\tau_m,\delta):=\text{min}[DF^{GLS},\frac{cv_{DF}}{cv_{MDF}}MDF(\tau_m,\delta)]<\lambda cv_{DF}\right\}\]where \(cv_{DF}\) and \(cv_{MDF}\) are the associated critical values and \(\lambda\) is a scaling factor. The critical values \(cv_{DF}\) are reported in Elliot et al (1993) and those of \(cv_{MDF}\) are reported in HLT (2013). The critical values for the scaling factor are also reported in Table 2 of HLT (2013).

Eviews addin

For the purpose of implementing this test seamlessly, I have developed an addin in Eviews. As usual, the philosophy of simplicity has been emphasized. Like the inbuilt unit root tests, the addin has been latched on the series object. This means it is a menu in the time series object's add-ins. To have it listed as seen in Figure 1, you must install the addin. 

In Figure 1, I subject LINV series to HLT unit root. 

Figure 1

The following dialog box presents you with options to choose from. The lag selection criteria include the popular ones such as the Akaike, Schwarz and Hanna-Quinn as well as their modified versions. Additionally, there is the t-statistic for optimal lag length selection. Significance levels indicates the choices for the level of significance for the lag length. The window width and the widow mid-point are also presented and you can also choose them as appropriate. The trimming can become too extreme sometimes. Under this circumstance, you are likely to express errors, whereby the addin will issue error message to inform you appropriately. This is likely going to be the case if the number of observations is too small.

The prior break date edit box can be left empty if there is no such date to be considered. Yet, The window width  and mid-point can be adjusted. A diffuse prior can be expressed with large value of window width. Lower values of window width suggests that the analyst expresses more certainty about the mid-point. For example, if the window width \(\delta=0.050\) is combined with \(\tau_m=0.50\) it means the analyst expresses more conviction that the break date happens around the mid point of the data than when he combines the width \(\delta=0.200\) with the same mid-point.

Figure 2

The output is presented in Figure 3. According to Equation 4 in HLT, one has to compare DF-GLS-U with the corresponding \(\lambda\)-scaled critical value denoted as Lam-sc'd c.v. In case the DF-GLS-U values are less than the Lam-sc'd c.v., then the null hypothesis of unit root is rejected. In this example, the DF-GLS-U are higher than those for Lam-sc'd c.v. Thus, the null hypothesis near unit root is not rejected.

Figure 3

Lastly, if there are reasons to choose a break date around which there is a doubt, this can be entered as a prior date break. In Figure 4, I enter 1976Q1 as the break date and then express the extent of my doubt around this date by selecting the window width as 0.05. 

Figure 4

Compared to window width of 0.200, this is a lot more precisely expressed. Thus, it is no surprise that the break date is found in the neighborhood of the putative break date. This can be seen in Figure 5. 

Figure 5

Happy New Year everyone. Let moderation be your guiding principle as you go out to celebrate the new year. 💥💥💥

Cointegration Tests Under Structural Breaks (Part I)

Introduction

Cointegration testing remains one of the most applied testing procedures in econometrics. Introduced in 1983 by Sir W Granger (Granger, 1983) and Engle and Granger (1987), many testing procedures have been developed. There are the residual-based tests of the null of no cointegration, the most popular being the Engle-Granger (1987) approach and the Phillips-Ouliaris (1990) approach. Both the Engle-Granger and the Phillips-Ouliaris approaches are inbuilt in Eviews as a group object. The difference between the two is that while autocorrelation is corrected by allowing for the lags of the dependent variable in the Engle-Granger approach (which is the ADF approach applied on the residuals from the single-equation bivariate model), in the Phillips-Ouliaris approach, autocorrelation is dealt with non-parametrically with the test based on bias-corrected autocorrelation coefficient and standard error.

One of the problems of testing for cointegration is the presence of structural breaks in the data generating process. The fact is that structural break can seriously mess up the result as the test statistics will often fail to reject unit root because they have low power when structural break is present. To illustrate, the Engle-Granger framework for testing for cointegration is actually the testing for unit root under null using the ADF test statistic. This test will fail to reject the null of unit root, while the PP, assuming the null of stationarity, will fail to reject the alternative hypothesis of unit root. The problem is that the introduction of a structural break into the process gets the ADF test statistic confused. Let \(y_t\) and \(x_t\) be two unit-root processes with a one-time break in \(y_t\):\[\begin{align*} y_t=&y_{t-1}+\epsilon_{1t}+DUM_t\\ x_t=&x_{t-1}+\epsilon_{2t}\end{align*}\]In the first process, we have both the stochastic and the deterministic trends, the latter being a result of the break. We can integrate the first expression to have\[y_t=y_0+\sum_{j=1}^t\epsilon_{1j}+\sum_{j=1}^tDUM_{j}=y_0+\sum_{j=1}^t\epsilon_{1j}+DUM\cdot t\]The impulse has cumulated into a trend, more like something perpetual riding on time factor itself. The second process is made up of the initial value as well as the stochastic trend, i.e.,
\[x_t=x_0+\sum_{j=1}^t\epsilon_{2j}\]Suppose we regress \(y_t\) on \(x_t\). More specifically, the following residuals are generated \(\hat{u}_t=y_t-\hat{\beta} x_t\). Is \(\hat{u}_t\) stationary so we can ascertain the cointegration between \(y_t\) and \(x_t\)? The answer is no. To see why, we rewrite \(\hat{u}_t\) as\[\hat{u}_t=\left(y_0+\sum_{j=1}^t\epsilon_{1j}+DUM\cdot t\right)-\hat{\beta} \left(x_0+\sum_{j=1}^t\epsilon_{2j}\right)\]Rearranged and taxonomized, this expression becomes: \(\hat{u}_t\) as\[\hat{u}_t=\underset{I(0)}{\underbrace{y_0- \hat{\beta}x_0}}+\underset{I(1)}{\underbrace{DUM\cdot t}}+\underset{I(0)}{\underbrace{\left(\underset{I(1)}{\underbrace{\sum_{j=1}^t\epsilon_{1j}}}-\underset{I(1)}{\underbrace{\hat{\beta} \sum_{j=1}^t\epsilon_{2j}}}\right)}}\]Each of the cumulated terms in the bracket is I(1) and their linear combination is I(0). Thus, for \(u_t\) to be I(0) and guarantee stationarity of the residuals and, by so doing, establish cointegration between \(y_t\) and \(x_t\), the term \(DUM\cdot t\) must be set to zero. Indeed, this term is I(1). Thus, without accounting for breaks in the process, both ADF and PP statistics will wrongly accept unit root hypothesis. We therefore failed to establish cointegration between two I(1) variables, whose linear combination would be stationary but for the presence of structural breaks. In sum, structural breaks induces non-stationarity.

The question now is: how do we detect structural breaks in cointegrated relation? A number of test statistics have been proposed to formally integrate structural breaks into cointegration. A retest of many macroeconomic variables previously found to have unit roots has confirmed that indeed those variables are stationary after accounting for breaks. This suggests persistence or permanence in these variables is a result of breaks and not inherent.

Four approaches to accommodating structural breaks in cointegration will be discussed: 

• the Carrion-i-Silvestre-Sansó (Carrion-i-Silvestre and Sansó, 2006) approach. 
• the Gregory-Hansen approach (Gregory and Hansen, 1996a,b), 
• the Hatemi-J (2008) approach, and
• the Arai-Kurozumi (2007) approach

The Carrion-i-Silvestre-Sansó (Carrion-i-Silvestre and Sansó, 2006) approach

In this post, we'll focus on the Carrion-i-Silvestre-Sansó (Carrion-i-Silvestre and Sansó, 2006) approach. There are basically two variants of the the Carrion-i-Silvestre-Sansó test, which depends on whether the regressors in the model are strictly exogenous or not. In each case, there are six specifications.

1. Strict Exogeneity of the Regressors

Six model specifications are investigated. They are termed \(i=A_n, A, B, C, D, E\) and are given by\[y_t=\begin{cases}\Gamma_i(t)+x_t^\prime\beta+\epsilon_t,&{i=A_n, A, B, C}\\\Gamma_i(t)+x_t^\prime\beta_0+x_t^\prime\beta_1DU_t+\epsilon_t,&{i=D, E} \end{cases}\]where\[\begin{cases}\Gamma_{A_n}(t)=\alpha+\theta DU_t\\\Gamma_A(t)=\alpha+\zeta t+\theta DU_t\\\Gamma_B(t)=\alpha+\zeta t+\theta DT_t\\\Gamma_C(t)=\alpha+\zeta t+\theta DU_t+\gamma DT^*_t\\\Gamma_D(t)=\alpha+\theta DU_t\\\Gamma_E(t)=\alpha+\zeta t+\theta DU_t+\gamma DT^*_t\end{cases}\]The dummy variables in the model are constructed as\[DU_t=\begin{cases}1,&\forall t>TB\\0,&otherwise\end{cases}\]and\[DT^*_t=\begin{cases}t-TB,&\forall t>TB\\0,&otherwise\end{cases}\]The first dummy \(DU_t\) is a level shift dummy while \(DT_t^*\) cumulates the effect of the one-off break (impulse) in the data after the break at point \(TB\).

The test statistic is based on \(SC_i(\lambda)=T^{-2}\omega_1^{-2}\sum_{t=1}^TS_{it}^2\) where \(S_{it}=\sum_{j=1}^t\hat{\epsilon}_{ij}\) and \(\omega_1=T^{-1}\sum_{t=1}^T\hat{\epsilon}_t^2+2\sum_{j=1}^{lq}\omega_j\sum_{t=j+1}^T\hat{\epsilon}_t^\prime\hat{\epsilon}_{t-j}\) is the Newey-West nonparametric estimator of the long-run variance of \(\hat{\epsilon}_t\) with \(\omega_j=1-j/(lq+1)\) as the weight on all autocovariances and indicating the more distant autocovariances are the less their weights in the computation of long-run variance. This statistic is a ratio of two variances and as such can be referred to F-statistic. However, there is a presence of nuisance parameter, the break point, with which the statistic varies. To overcome this problem, Carrion-I-Silvestre and Sansó (2006) employ a Monte Carlo simulation to construct a set of critical values reported in their paper.

        2. Non-Strict Exogeneity of the Regressors

For the case where the regressors are not strictly exogenous, Carrion-i-Silvestre and Sansó (2006) propose to use one of the approaches suggested by Phillips and Hansen (1990), Saikkonen (1991), and Stock and Watson (1993) to obtain an efficient estimation of the cointegrating vectors. We adopt the DOLS  in this implementation and it's given by\[y_t=\begin{cases}\Gamma_i(t)+x_t^\prime\beta+\sum_{j=-k}^{k}\Delta x_{t-j}^\prime\gamma_j +\epsilon_t,&{i=A_n, A, B, C}\\\Gamma_i(t)+x_t^\prime\beta_0+x_t^\prime\beta_1DU_t+\sum_{j=-k}^{k}\Delta x_{t-j}^\prime\gamma_j+\epsilon_t,&{i=D, E} \end{cases}\] \(SC_i^+(\lambda)=T^{-2}\omega_1^{-2}\sum_{t=1}^T(S_{it})^{2+}\) where \(S_{it}^+=\sum_{j=1}^t\hat{\epsilon}_{ij}\) 

Eviews addin

The following addin is for implementing the method for non-strictly exogenous regressors with unknown break dates. The break date estimated is one and this is in line with the objective of the Carrion-i-Silvestre-Sanso objective (Ensure you go through their paper as well). I may consider the other case for strictly exogenous variables later. The data used for this example can be sourced here.

The addin is straightforward to use. The figure below shows the spool object saving the graph and the table for the results.

Figure 1

In Figure 2, similar results have been presented. The difference is that Model E has been used and the pre-whitened results have also been included.  

Figure 3

Figure 3 shows the dialog box. You can make your options as you deem appropriate. For example, you can choose any of the methods as shown in Figure 4. Also note that the critical values are not reported. Interested person can see the paper by the authors of the method. Meanwhile, for this method, Carrion-i-Silvestre and Sanso only report the critical values for up to 4 exogeneous variables. Nevertheless, the addin allows you to carry out text for model having more than four exogeneous variables.

Figure 4

The criterion in the dialog box is used to select the optimal numbers for leads/lags.

In the next posts, on cointegration with structural breaks, we shall be looking at all the remaining 3 methods.

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From here, a big thank-you to you.