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. 💥💥💥

Bootstrap ARDL: Eviews addin

Introduction

Let's quickly wrap our heads around the idea of bootstrap ARDL by first looking at the concept of weak exogeneity. The idea is better understood within the VECM approach. Suppose there are two variables of interest in that they are both endogenous. It means we can model them jointly. Recall that in the VECM system of equations, each equation has two parts to it: the short-run (in differences) and the long-run (in levels). The long-run component is a linear combination of the lagged endogenous variables plus some deterministic terms, and the effects are the "loading" factors that convey the impact of this linear combination to the changes in each of the endogenous variables. They are also called the speed of adjustments as they reveal how the short-run adjustment takes place due to the disequilibrium in the long-run component. Long-term feedbacks are therefore through the loading factors. If the loading factor for a particular equation in the system is negligible, then long-term feedback may well be set to zero and we say the particular endogenous variable is weakly exogenous. When endogenous variables are weakly exogenous, the system can be simplified into two sub-models: the conditional and the marginal models. We can then focus on the conditional model, that is, the model whose loading factors are significant, and ignore the marginal model. It all means we have less number of parameters to estimates because the number of equations too has been reduced. 

If you understand the preceding, then you already know the make up of ARDL. In this sense, the dynamic regressors in ARDL are considered weakly exogenous. The model analyzed is termed conditional. A telltale of being a conditional model is the first difference at "lag 0" often included for the regressors, (such as \(\varphi_0^\prime \Delta x_t\) in the following model), and should remind the user that the model employed is conditional; otherwise, it's unconditional:\[\Delta y_t = \alpha+\theta t+\rho y_{t-1} +\gamma^\prime x_{t-1}+\sum_{j=1}^{p-1}\phi_j \Delta y_{t-j}+\varphi_0^\prime \Delta x_t+\sum_{j=1}^{q-1}\varphi_j^\prime \Delta x_{t-j}+\epsilon_t\]Thus in models where the users rotate the dependent variables, this assumption of weak exogeneity of the exogenous variables is being violated.

While weak exogeneity assumption is being violated, especially when authors implicitly assume that the variables can be rotated such that the same variable is being used as a dependent variable in one estimation and as an independent variable in another, the degenerate cases are also common (the degenerate cases are discussed here). The first of these degenerate cases arises when the joint test (F statistic) of the lagged dependent and independent variables is significant and the t-statistic on the lagged dependent variable is significant as well while the F statistic  of the lagged independent variable(s) is not significant. The recommended solution to the lagged dependent degenerate case is to formulate the model such that the dependent variable is I(1). Here again, users also violate this assumption by not ensuring that the dependent variable is I(1).

Added to these issues is the inconclusiveness in bounds testing. How do we decide whether cointegration exists or not if the computed F-statistic falls within the lower and the upper bounds? As is well known, the critical values provided by PSS, or Narayan or even by Sam, McNown and Goh do not provide a clear roadmap on what the decision must be. Experience often shows that the case of fractionally integrated process, \(x_t=\sum_{j=1}^t\Delta ^{(d)}_{t-j}\xi_j\), where \(d\in(-0.5,0.5)\cup(0.5,1.5)\) and \(\Delta_t^{(d)}:=\Gamma(t+d)/\Gamma(d)\Gamma(t+1)\) cannot be ruled out. In other words, series occasionally don't fall perfectly into I(0) or I(1).

To proceed, we can bootstrap. This approach works because no parametric assumptions are made about the distribution. Rather data are allowed to speak. Therefore, through bootstrap a data-based distribution emerges that can be used for making decisions. 

The algorithm used...

The bootstrap steps used in this add-in are as follows, where I'm working with the hypothesis that the model is trend-restricted in a bivariate model, that is, \(H_0:\theta_1=\rho_1=\gamma_1=0\) (You can read more about the five model specifications in PSS here): 

1. Imposing the null hypothesis, e.g., \(H_0:\theta_1=\rho_1=\gamma_1=0\), estimate the restricted model: \[\begin{align*}\Delta y_t =& \alpha_1+\theta_1 t+\rho_1 y_{t-1} +\gamma_1 x_{t-1}+\sum_{j=1}^{p_y-1}\phi_{j,1} \Delta y_{t-j}+\sum_{j=0}^{q_y-1}\varphi_{j,1} \Delta x_{t-j}+\epsilon_{1,t}\\\Delta x_t =& \alpha_2+\theta_2 t+\rho_2 y_{t-1} +\gamma_2 x_{t-1}+\sum_{j=1}^{p_x-1}\phi_{j,2} \Delta y_{t-j}+\sum_{j=0}^{q_x-1}\varphi_{j,2} \Delta x_{t-j}+\epsilon_{2,t}\end{align*}\]and obtain the residuals \(\hat{\epsilon}_{1,t}\) and \(\hat{\epsilon}_{2,t}\). Note that this system needs not be balanced as the orders may not necessarily be the same;
2. Obtain the centered residuals \(\tilde{\epsilon}_{i,t}=(\hat{\epsilon}_{i,t}-\bar{\hat{\epsilon}}_{i,t})\);
3. Resample \(\tilde{\epsilon}_{i,t}\) with replacement to obtain \(\epsilon^*_{i,t}\)
4. Using the model in Step 1, evaluating the system at the estimated values, generate pseudo-data (bootstrap data): \(y_t^*\) and \(x_t^*\), which can be recovered as \(y_t^*=y_{t-1}^*+\Delta y_t^*\) and ditto for \(x_t^*\);
5. Estimate unrestricted model using the bootstrap data:\[\Delta y_t^* = \tilde{\alpha}_1+\tilde{\theta}_1 t+\tilde{\rho}_1 y_{t-1}^* +\tilde{\gamma}_1 x_{t-1}^*+\sum_{j=1}^{p_y-1}\tilde{\phi}_{j,1} \Delta y_{t-j}^*+\sum_{j=0}^{q_y-1}\tilde{\varphi}_{j,1} \Delta x_{t-j}^*\]
6. Test the necessary hypothesis: \(H_0:\tilde{\theta}_1=\tilde{\rho}_1=\tilde{\gamma}_1=0\);
7. Repeat the steps in 3 to 6 B times (say, B=1000).

Eviews addin

The implementation has been synced to addin, which I prefer to working through all these steps each time. You can obtain the addin here. To use it, you just need to estimate your ARDL model as usual. All the 5 specifications in Eviews can be bootstrapped. After estimation of the model, click on the Proc tab of the estimated model and hover to Add-ins for ARDL equation object. The Bootstrap ARDL menu should be located provided it has already been installed.

Figure 1 shows the Bootstrap ARDL addin dialog box. Although the details of the choices that can be made are self-explaining, the coefficient uncertainty deserves some comments. Usually, the bootstrap is carried out at the estimated values of the parameters. While this is innocuous, the "right" thing to do, in my opinion, is to sample from the distributions of the parameters thereby incorporating the fact that they have not been precisely estimated. To give the user this choice, I have included the Check option for Coefficient uncertainty. 

Figures 2 to 5 give different results of the same model under different choices. I think this should available for sensitivity study of the results. In this output, I have appended -F or -t to indicate the F or t statistic.

Figure 1: Bootstrap ARDL Dialog Box

Figure 2: Sample output

Figure 3: Sample output

Figure 4: Sample output

Figure 5: Sample output

Suggestions are welcome. 

Conducting Augmented ARDL in Eviews Using Addin

Introduction

The Augmented ARDL is an approach designed to respond to the question of whether or not the dependent variable should be either I(0) or I(1). With I(0) as the dependent variable, it is difficult to infer long-run relationship between the dependent variable and the regressor(s) even if the F-statistic is above upper critical bound in the well used bounds-testing procedure. The reason is that, in the event that the I(0) is used as the dependent variable, the series will necessarily be stationary (Permit my tautological rigmarole)! This means in jointly testing for long-run relationship via F statistic, the fact that the computed F value is above the upper bound might just reflect the I(0)-ness of the dependent variable. What is more? Other exogenous variables may turn out to be insignificant, suggesting without testing the I(0) dependent variable along with them, the resulting t statistic (if there is only one exogenous variable) or F statistic (if there are more than one exogenous variable) becomes insignificant. Thus, the I(0) variable in the joint relationship may dominate whether or not other variables are significantly contributing to the long-run relationship. The result is always wrong inference.

ARDL at a glance

While the PSS-ARDL approach is a workhorse for estimating and testing for long-run relationship under the joint occurrence of I(0) and I(1) variables, there are certain assumptions the applied researchers often take for granted thereby violating the conditions necessary for using the PSS-ARDL in the first place. For a bivariate specification, the PSS-ARDL(p,q), in its most general form, is given by\[\Delta y_t=\alpha+\beta t+\rho y_{t-1}+\gamma x_{t-1}+\sum_{j=1}^{p-1}\delta_j\Delta y_{t-j}+\sum_{j=0}^{q-1}\theta_j\Delta x_{t-j} +z_t^\prime\Phi+\epsilon_t\]where \(z_t\) represents the exogenous variables and could contain other deterministic variables like dummy variables and \(\Phi\) is the vector of the associated parameters. Based on this specification, Pesaran et al., (2001) highlight five different cases for bounds testing, each informing different null hypothesis testing.  Although some of them are less interesting because they have less practical value, it is instructive to be aware of them:

• CASE 1: No intercept and no trend
• CASE 2: Restricted intercept and no trend
• CASE 3: Unrestricted intercept and no trend
• CASE 4: Unrestricted intercept and restricted trend
• CASE 5: Unrestricted intercept and unrestricted trend

Intercept or trend is restricted if it is included in the long-run or levels relationship. For each of these cases, Pesaran et al., (2001) compute the associated t- and F-statistic critical values. These critical values are reported in that paper and readers are invited to consult the paper to obtain the necessary critical values (if you want since these values are reported pro bono). 

The cases above correspond to the following restrictions on the model:

• CASE 1: The estimated model is given by \[\Delta y_t=\rho y_{t-1}+\gamma x_{t-1}+\sum_{j=1}^{p-1}\delta_j\Delta y_{t-j}+\sum_{j=0}^{q-1}\theta_j\Delta x_{t-j} +z_t^\prime\Phi+\epsilon_t\] and the null hypothesis is \(H_0:\rho=\gamma=0\). This model is recommended if the series have been demeaned and/or detrended. Absent these operations, it should not be used for any analysis except the researcher is strongly persuaded that it is the most suitable for the work or simply for pedagogical purposes.
• CASE 2: The estimated model is \[\Delta y_t=\alpha+\rho y_{t-1}+\gamma x_{t-1}+\sum_{j=1}^{p-1}\delta_j\Delta y_{t-j}+\sum_{j=0}^{q-1}\theta_j\Delta x_{t-j} +z^\prime_t\Phi+\epsilon_t\]where in this case \(\beta=0\). The null hypothesis \(H_0:\alpha=\rho=\gamma=0\). The restrictions imply that both the dependent variable and the regressors are moving around their respective mean values. Think of the parameter \(\alpha\) as \(\alpha=-\rho\zeta_y-\gamma\zeta_x\), where \(\zeta_i\) are the respective mean values or the steady state values to  which the variables gravitate in the long run. Substituting this restriction into the model, we have \[\Delta y_t=\rho(y_{t-1}-\zeta_y)+\gamma (x_{t-1}-\zeta_x)+\sum_{j=1}^{p-1}\delta_j\Delta y_{t-j}+\sum_{j=0}^{q-1}\theta_j\Delta x_{t-j} +z^\prime_t\Phi+\epsilon_t\]This model therefore possesses some practical values and is suitable for modelling the behaviour of some variables in the long run. However, because the dependent variable do not possess the trend due to the absence of intercept in the short run, this specification's utility is limited given that most economic variables are I(1).
• CASE 3: The estimated model is the same as in CASE 2 with \(\beta=0\). However, \(H_0:\rho=\gamma=0\). This implies the intercept is pushed into the short-run relationship and it means the dependent variable has a linear trend,  trending upwards or downwards depending on the direction dictated by \(\alpha\). This characteristic is benign if the dependent variable is really having the trend in it. However, this is not a feature of I(0) dependent variable. As most macroeconomic variables are I(1), this specification is often recommended. In Eviews, it's the default setting for model specification.
• CASE 4: The model estimated for CASE 4 is the full model. Here, trend is restricted while the intercept is unrestricted. The null hypothesis is therefore  \(H_0:\beta=\rho=\gamma=0\). This specification suggests that the dependent variable is trending in the long run. If, in the long run, the dependent variable is not trending, it means this specification might just be a wrong choice to model the dependent variable.
• The last case where both the intercept and the trend are unrestricted is a perverse description of the macroeconomic variables. It is a full model but it means the dependent variable is trending quadratically. This does not fit most cases and is rarely used. The null hypothesis is  \(H_0:\rho=\gamma=0\).

The F statistic and the associated t statistic for bounds testing are reported in PSS. 

Getting More Gist about ARDL from ADF

The F statistic for bounds testing referred to above is necessary, but it is not sufficient, to detect whether or not there is long run relationship between the dependent variable and the regressors. The reason for this is the presence of both I(0) and I(1) and their treatment as the dependent variable in the given model. Note that one of the requirements for valid inference about the existence of cointegration between the dependent and the regressors is that the dependent variable must be I(1). We can get the gist of this point by looking more closely at the relationship between the ARDL and the ADF model. You may be wondering why the dependent variable must be I(1) in the ARDL model specification. The first thing to observe is that ARDL is a multivariate formulation of the augmented Dickey Fuller (ADF). Does that sound strange? 

Suppose \(H_0: \gamma=\theta_0=\theta_1=\cdots=\theta_q=0\), that is, the insignificance of other exogenous variables in the model, cannot be rejected. Then the model reduces to the standard ADF. From this, we can see that if \(\rho\) is significantly negative, stationarity is established. If this is the case, variable \(y_t\) will be reckoned as I(0). Thus, the ADF is given by\[\Delta y_t=\alpha+\rho y_{t-1}+\sum_{j=1}^{p-1}\delta_j\Delta y_{t-j}+\epsilon_t\]The fact that \(y_t\) is stationary at levels means that \(\rho\) must be significant whether or not the coefficients on other variables are significant. Therefore, in a test involving this I(0) variable as a dependent variable and possibly I(1) as independent variable(s), and where the coefficient on the latter is found to be insignificant, it's still possible to find cointegration not because there is one between these variables, but because the significance of the (lag of) dependent variable dominates the joint test and because only a subset of the associated alternative hypothesis is being considered. This is what the bounds testing does without separating the significance of \(\rho\) and \(\gamma\). Note that F test for bounds testing is based on the joint significance of these parameters. However, the joint test of \(\rho\) and \(\gamma\) does not tell us about the significance of \(\gamma\).

Degenerate cases

How then can we proceed here? More tests needed. To find out how, we must first realize what the issues are really like in this case. At the center of this are the two cases of degeneracy. They arise because the bounds testing (a joint F test) involves both the coefficient on the lagged dependent variable \(\rho\) in the model above and the coefficients of lagged exogenous variables. Although PSS reported the t statistic for \(\rho\) separately with a view to having robust inference, not only do the researchers often ignore it, the t statistic so reported along with the F statistic is not enough to avoid the pitfall. In short, the null hypothesis for the bounds testing \(H_{0}: \rho=\gamma=0\) can be seen as a compound one involving  \(H_{0,1}: \rho=0\) and \(H_{0,2}: \gamma=0\). So rejection of either is not a proof of cointegration. This is because the alternative is not just \(H_{0}: \rho\neq\gamma\neq 0\) as often assumed in application; the alternative instead involves \(H_{0,1}: \rho\neq0\) and \(H_{0,2}: \gamma\neq0\) as well. In other words, a more comprehensive hypothesis testing procedure must involve the null hypotheses of these alternatives. Thus, we have the following null hypotheses:

1. \(H_{0}: \rho=\gamma= 0\), and \(H_{1,1}: \rho\neq0\), \(H_{1,2}: \gamma\neq0\) 
2. \(H_{0,1}: \rho=0\) and \(H_{1,1}: \rho\neq0\)
3. \(H_{1,2}: \gamma=0\) and \(H_{1,2}: \gamma\neq0\) 

Taxonomies of Augmented Bounds Test 

Therefore, we state the following taxonomy for testing hypothesis:

• if the null hypotheses (1) and (2) are not rejected but (3) is, we have a case of degenerate lagged independent variable. This case implies absence of cointegration;
• if the null hypotheses (1) and (3) are not rejected but (2) is, we have a case of degenerate lagged dependent variable. This case also implies absence of cointegration; and
• if the null hypotheses (1), (2) and (3) are rejected, then there is cointegration 

We now have a clear roadmap to follow. What this implies is that one needs to augment the testing as stated above. Hence the augmented ARDL testing procedure. With this procedure for testing for cointegration, it is no longer an issue whether or not the dependent variable is I(0) or I(1) as long as all the three null hypotheses are rejected.

Now the Eviews addin...

First note that this addin has been written in Eviews 12. Its functionality in lower version is therefore not guaranteed. 

Using Eviews for testing this hypothesis should be straightforward but may be laborious. Eviews can help you here. All that is needed is reporting all the three cases noted above as against the two cases reported in Eviews. The following addin helps you with all the computations you might need to do. 

To use it, just estimate your ARDL model as usual and then use the Proc tab to locate the add ins. In Figure 1, we have the ARDL method environment. Two variables are included. I choose the maximum lag of eight because I have enough quarterly data, 596 observations in total.

Figure 1

Once the model is estimated, use the Proc tab to locate Add-ins as shown in Figure 2 

Figure 2

Click on Augmented ARDL Bound Test and you will have the figure referred to in Figure 3. The tests are reported underneath what you see here. Just scroll down to look them up.

Figure 3

Figure 4

What is shown in Figure 4 should be the same as reported natively by Eviews. The addition that has been appended by this addin is the Exogenous F-Bounds Test shown in Figure 5. For the confirmation of the test, we append the Wald test for exogenous variables in the spool. It comes under the title exogenous_wald_table. You can click to view it.

Figure 5

In this example, we are sure of cointegration because all the three computed statistics are above the upper bound, suggested no case of degeneracy is lurking in our results.

Note the following...

Before working on the ARDL output, be sure to name it. At the moment, if the output is UNTITLED, Error 169 will be generated. The glitch is a really slippery error. It will be corrected later. 

The results have the fill of the existing table for bounds testing in Eviews but have been appended with the tests for the exogenous variables. The F statistic is used for testing the exogenous variables. Thus, we have the section for Overall F-Bounds Test which is the Null Hypothesis (1) above; the section for the t-Bounds Test which is the Null Hypothesis (2); and, the section for the Exogenous F-Bounds Test which is the Null Hypothesis (3). The first two of these sections should be the same as in the native Eviews report. The last is an addition based on the paper by Sam, McNown and Goh (2018).  

From the application point of view, in this case, the Exogenous F-Bounds test for Cases 2 and 3 are the same:

CASE 2: Restricted intercept and no trend

CASE 3: Unrestricted intercept and no trend

 just as Cases 4 and 5 are the same:

CASE 4: Unrestricted intercept and restricted trend

CASE 5: Unrestricted intercept and unrestricted trend

Therefore, the same critical values are reported for them in the literature. Thus, in the Eviews addin the long run for both cases are reported. 

The link to the addin is here. The data used in this example is here.

Thank you for reading a long post😀.

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.

If you find this addin helpful, drop a message or better still follow!!!

From here, a big thank-you to you.

Quantile-on-Quantile Regression Using Eviews

Introduction

Nonlinearity is everywhere. To model it, analysts have conjectured in their wildest imagination all manners of techniques. Quantile-on-quantile is one of the latest in the research community. If you haven't seen it applied like most other techniques, it's because it requires a lot of heavy lifting in terms of coding. I have brought you this to ease the burden of using one of the most widely used platforms -- Eviews.

Conventionally, quantile regression traces out the effects of the conditional distribution of dependent variable on the dependent variable itself through the impact of the independent variable. It is like asking what the impact of interest rate will be like on inflation if inflation has already reached a particular threshold. Of course, such a question cannot be addressed straightforwardly within the received OLS knowledge. This is where quantile regression stands in to fill the gap.

But then, what about the new estimation strategy quantile-on-quantile regression? Here, both the conditional distributions of the dependent and independent variables modulate the impact of the latter on the former. So, in this case, the question we are interested in is, for example, how extreme inflation (e.g., inflation at say 95 percentile) will respond to extreme interest rate (e.g., at 30-40 percent higher than usual). Put simply, we are interested in how different levels of the independent variable will alter the distribution of the dependent variable. This question cannot be addressed using quantile regression. Because of the existence of two extreme scenarios surfacing within the same policy strategy, the quantile-on-quantile regression comes to the rescue.

QQR is proposed by Sim and Zhou (2015). Yes, it is such a recent method. Again, we are not interested so much in the theory behind this. This can be found in their paper, and I hope the summary here will help you understand that aspect of their paper.

 

Summary of the method

Let the relationship between \(x_t\) and \(y_t\) be given by
\[y_t=\beta^\theta (x_t)+\epsilon_t^\theta\]
Now let \(\tau\)-quantile of \(x_t\) be \(x_t^\tau\). Sim and Zhou suggest the relationship above be approximated by first order Taylor expansion of \(\beta^\theta (x_t)\) around \(x_t^\tau\)
\[\beta^\theta (x_t)\approx \beta_1 (\tau,\theta) + \beta_2 (\tau,\theta)(x_t-x_t^\tau).\]
If follows that
\[y_t= \beta_1 (\tau,\theta) + \beta_2 (\tau,\theta)(x_t-x_t^\tau)+\epsilon_t^\theta\]
At a given value of \(\tau\), the preceding equation can be estimated by quantile regression. Basically, we estimate\[\hat{\beta} (\tau,\theta)=\underset{\beta (\tau,\theta)}{\text{argmin}}\sum_{t=1}^T\rho_\theta \left(y_t - \beta_1 (\tau,\theta)-\beta_2 (\tau,\theta)(x_t-x_t^\tau)\right)\]
where \(\rho_\theta(\cdot)\) is the check function. Rather than estimating this model, the authors realize that there is a need to weight the function appropriately. The reason is that the interest is in the effect exerted locally by the \(\tau\)-quantile of \(x_t\) on \(y_t\). This makes sense in that otherwise the effect will not be contained in the neighbourhood of \(\tau\). They choose the normal kernel function to smooth out unwanted effects that could contaminate the results. The weights so generated are inversely related to the distance between \(x_t\) and \(x_t^\tau\) or, equivalently, between the empirical distribution of \(x_t\), \(F(x_t)\), and \(\tau\). I follow suit in developing the code. Now, the model becomes
\[\hat{\beta} (\tau,\theta)=\underset{\beta (\tau,\theta)}{\text{argmin}}\sum_{t=1}^T\rho_\theta \left(y_t - \beta_1 (\tau,\theta) - \beta_2 (\tau,\theta)(x_t-x_t^\tau)\right)K\left(\frac{(x_t-x_t^\tau)}{h}\right)\]
where \(h\) is the bandwidth. As the choice of bandwidth is critical to getting a good result, in this application, I choose the Silverman optimal bandwidth given by
\[h=\alpha\sigma N^{-1/3}\]

where \(\sigma=\text{min}(IQR/1.34, \text{std}(x))\), \(IQR\) is the inter-quantile range, \(N\) is the sample size and \(\alpha=3.49\).

One snag, however, needs to be pointed out. Eviews does not feature surface plot normally used to present the results in this case. To me, this turns to be an advantage because a more revealing graphical technique has been devised for that purpose. It aligns the boxplots to summarize the results in an equally excellent, if not better, way.

In what follows, I will lead you gently into the world of QQR addin in Eviews

Eviews example

Addin Environment

The QQR addin environment is depicted in Figure 1. If you've already installed the addin, you can click on the Add-ins tab to display the QQR dialogue box as seen below.

Figure 1

The dialogue box is self-explanatory. Three edit boxes are featured. The first one asks you to input the dependent variable followed by a list of exogenous variables. You can include both C and @trend among the exogeneous variables here. However, you should not include the quantile exogeneous variable, which you are required to enter in the second edit box. Note that only one quantile exogeneous variable can be entered here. In the third edit box the period of estimation is indicated.

Example is given in Figure 2. Here, we are estimating the quantile-on-quantile effects of oil price (OILPR) on exchange rate (EXR). We include the third variable, interest rate (INTR). This estimation is carried out over the period of May 2004 and January 2020. Although oil price is an exogenous variable, by entering it in the second variable, we make it the variable whose quantile effect we want to study. 

Figure 2

A couple of options are provided. The Coefficient plots category wants you to choose whether to produce the graphs for all the variables in the model or to just produce the graph for the quantile exogenous variable alone. The default is to generate the graphs for all the coefficients in the model. The Graph category wants you to choose to rotate or not the plots. Orientation may count at times. The default is to rotate the plots. Lastly, there is the Plot Label category. How do you want your graphs labelled? On one side or on both sides. It may not matter much. But beauty they say is in the eyes of beholder. I think I love the double-sided label. Hence the default. These categories are boxed with color code in Figure 3.

Figure 3

Graphical Outputs

As noted above, Eviews has yet to develop either the contour or the surface plot usually favored for the quantile-on-quantile result presentations. In the absence of these valuable tools, I opt for boxplot. Boxplot presents the distribution of the data with a couple of details (median, mean, whiskers, outliers and in Eviews confidence interval). But it is a 2-D plot. This means one can only view one side of the object on the x-y plane. To view the other side of the object, one needs to rotate the object. In other words, one needs two 2-D plots to capture some details of the 3-D objects. That is why we have the two plots for one parameter! The graph is named quantileonquantileplot##. The shade indicates 95% confidence interval. 

In Figures 4-6, I present the graphs for the three coefficients. 

Figure 4

 

Figure 5

Figure 6

The same results are presented in Figures 7-9 but this time not rotated!

Figure 7

Figure 8

Figure 9

External resources

If one really wants to report the contour or surface plot, there is still hope. Eviews has provided an opportunity to interact with external computational software like MATLAB and R. Since I have MATLAB installed on my system, I simply run the following code in Figure 10. The inputs to the snippet include the matrix and vector objects generated and quietly dumped by the QQR addin in the workfile. They are a19\(\times\)k  coefmatrix and a 19-vector taus respectively, where k is the number of parameters estimated.  

Figure 10

Figures 11-13 compare the graphs of the estimated coefficients from the QQR addin with those generated using MATLAB. Therefore, you can still estimate your quantile-on-quantile using the Eviews addin as discussed here and have the surface plots for the estimated coefficients done in MATLAB or R. What is more? R is a open source and free.

Figure 11

Figure 12

Figure 13

Requirement

This addin runs fine on Eviews 12. It hasn't been done yet on lower versions. 

How to get the addin...

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