Rtadf : Testing for Bubbles with EViews

This paper presents Rtadf ( right-tail augmented Dickey-Fuller ), an EViews add-in that facilitates the performance of time series based tests that help detect and date-stamp asset price bubbles. The detection strategy is based on a right-tail variation of the standard augmented Dickey-Fuller (ADF) test where the alternative hypothesis is of a mildly explosive process. Rejection of the null in each of these tests may serve as empirical evidence for an asset price bubble. The add-in implements four types of tests: standard ADF, rolling window ADF, supremum ADF (SADF; Phillips, Wu, and Yu 2011) and generalized SADF (GSADF; Phillips, Shi, and Yu 2015). It calculates the test statistics for each of the above four tests, simulates the corresponding exact ﬁnite sample critical values and p values via Monte Carlo methods, under the assumption of Gaussian innovations, and produces a graphical display of the date stamping procedure.


Introduction
Empirical identification of asset price bubbles in real time, and even in retrospect, is surely not an easy task, and it has been the source of academic and professional debate for several decades. 1 One strand of the empirical literature suggests using time series estimation techniques while exploiting predictions made by finance theory in order to test for the existence of bubbles in the data. The main idea, based on asset pricing theory, suggests that the existence of a bubble component in an observed asset price should be manifested in its dynamics and its stochastic properties. More specifically, theory predicts that if a bubble exists, prices should inherit its explosiveness property. This in turn enables formulating statistical tests aimed at detecting evidence of explosiveness in the data. 2 One of the attempts to test for rational bubbles in the context of the stock market is found in Diba and Grossman (1988), where the authors suggest using reduced form stationarity tests with regard to stock prices and observable market fundamentals, and to rule out bubbles if the former is found no more explosive than the latter. Evans (1991), however, questions the power of such stationarity based tests in the presence of a periodically collapsing bubble (i.e., one that spontaneously occurs and bursts), an apparent feature of actual stock prices seen in the data. Using simulation methods, Evans (1991) shows that standard unit root and cointegration tests fail to reject the null of no bubble in the presence of periodically collapsing bubbles. Despite his findings, Evans (1991) leaves open the question of a better identification strategy.
More recently, new bubble detection strategies were developed and presented by Phillips, Wu, and Yu (2011, hereafter PWY) and Phillips, Shi, and Yu (2015, hereafter PSY). These strategies are based on recursive and rolling ADF unit root tests that enable us to detect bubbles in the data and to date-stamp their occurrence. These types of tests use a right-tail variation of the augmented Dickey-Fuller unit root test wherein the null hypothesis is of a unit root and the alternative is of a mildly explosive process. 3 PWY and PSY show that using recursive and rolling tests results in higher power in the detection of bubbles, compared to standard tests on the whole sample. In a Monte Carlo study, Homm and Breitung (2012) compare several time series based tests for the detection of bubbles and find that the PWY strategy performs relatively well in detecting periodically collapsing bubbles and in real time monitoring. Phillips et al. (2015) show through a Monte Carlo study that the PSY strategy outperforms the PWY strategy in the presence of multiple bubbles. This paper introduces Rtadf, an EViews (IHS Inc. 2013) add-in available from http://www. eviews.com/Addins/rtadf.aipz. It allows end users to easily test for the existence of bubbles, by readily applying four variations of the right-tail ADF unit root test, in line with the reduced form approach for bubble detection described above. The four tests consist of the standard ADF test, a rolling window ADF test, the more recent PWY supremum ADF (SADF) test and the PSY generalized SADF (GSADF) test. The add-in performs two main tasks. First, it calculates the relevant test statistic, according to the selected test. Second, the add-in derives the corresponding exact finite sample critical values by performing a Monte Carlo simulation, under the assumption of Gaussian innovations. Moreover, the add-in allows the user to choose between performing a sequential and a parallel simulation. I show that using the latter option results in significant reduction of computational time as the number of recursions needed to complete the simulation rises. (This is most relevant for the GSADF test.) The rest of the paper is organized as follows. Section 2 presents a basic theoretical model of rational bubbles in a standard asset pricing model. Section 3 introduces the details of the econometric strategy used to detect explosive behavior in asset prices. Section 4 provides general instructions on how to use the Rtadf add-in within the EViews environment. Section 5 presents a hands-on illustration of Rtadf. Finally, Section 6 concludes.
2 For recent surveys on econometric tests for bubbles, see Gürkaynak (2008) and Homm and Breitung (2012). 3 Phillips and Magdalinos (2007) define a mildly explosive root using the following data generating process where δn = 1 + c kn , and where (kn) n∈N is a sequence increasing to ∞ such that kn = o(n) as n → ∞. Limit theory for mildly explosive processes is developed in Phillips and Magdalinos (2007).

Asset pricing with rational bubbles
In this section a formal model of asset pricing with a rational bubble is presented. We start by specifying the definition of the single period return on an asset: where R t+1 > 1 is the gross return on holding the asset from period t to t + 1, P t is the price of the asset measured at the end of period t (i.e., the ex-dividend price) and D t+1 is the dividend the asset holder is entitled to for holding the stock from period t to t + 1 Next, following Campbell and Shiller (1988) we write a log-linear approximation of Equation 1 with p − d being the average log price-to-dividend ratio, and Solving Equation 2 by forward iteration and taking expectations yields the following log-linear approximation of the log price-to-dividend ratio: The right hand side of Equation 3 can be decomposed into two components, where is the fundamental component, stated in terms of the expected dividend growth rate and expected returns, and where is commonly referred to as the rational bubble component. The latter is the focus of the bubble tests described below.
Under the transversality condition, which implies no-Ponzi game, lim i→∞ ρ i E t p t+i = 0, and the possibility of a bubble is ruled out. Thus, the observed price equals the fundamental price. In contrast, the existence of a strictly positive bubble component, i.e., the situation where actual price exceeds what is implied by fundamentals, requires that investors expect to be compensated for overpayment (over the fundamental price) by the expected appreciation of the bubble component. In other words, investors are willing to pay a premium over the fundamental price only because they expect this premium to appreciate in the next period.
Note that this behavior is completely consistent with the rational expectations assumption, hence the name 'rational bubble'.
More importantly, note that Equation 6 implies a submartingale property for b t since where 1 + e (p−d) > 0. Thus, when b t = 0, the log bubble component grows at rate g, where This model reveals important insights regarding the stochastic properties of p t − d t , according to which, we can formulate an econometric test designed to rule out the presence of a rational bubble component in an observed asset price. To see this, note that the stochastic properties of p t − d t , implied by Equation 3, are determined by those of f t and b t . In turn, the dynamics of f t are determined by expected ∆d t and r t . If ∆d t and r t are at most I(1) processes, evidence of explosiveness in p t − d t (in this model) can only be the result of the presence of a bubble, i.e., b t = 0. Thus, a test for the presence of a bubble can be formulated as a test for explosive behavior in log price-to-dividend ratio, p t − d t .

Testing for bubbles
Following the conventions of PSY, assume the following random walk process with an asymptotically negligible drift: where d is a constant, η is a localizing coefficient that controls the magnitude of the drift as the sample size, T , approaches infinity and ε t is the error term. 4 Four test strategies implemented by the Rtadf add-in (which includes the ones suggested by PWY and PSY) are all based on some variation of the following reduced form empirical equation: where y t is the variable in question (e.g., the price of a stock), µ is an intercept, p is the maximum number of lags, φ i for i = 1, . . . , p are the differenced lags coefficients and ε t is the error term. Testing for a bubble (explosive behavior) is based on a right-tail variation of the standard ADF unit root test where the null hypothesis is of a unit root and the alternative is of a mildly explosive autoregressive coefficient. Formally, we test for Before proceeding to a description of the tests included in Rtadf, some notation is needed. For simplicity of exposition, we use a sample interval of [0, 1] (i.e., we normalized the original sample by T ). Denote by δ r 1 ,r 2 and by ADF r 1 ,r 2 the coefficient estimated by Equation 9 and 0 1 r 1 r w = 1 r 2 Sample interval Figure 1: Illustration of the ADF procedure. 0 1 its corresponding ADF statistic over the (normalized) sample [r 1 , r 2 ]. In addition, denote by r w the (fractional) window size of the regression, defined by r w = r 2 − r 1 and by r 0 the fixed initial window, set by the user. The difference between the tests relates to the manner of setting r 1 and r 2 .
The first test included in Rtadf is a simple right-tailed version of the standard ADF unit root test. In this case, r 1 and r 2 are fixed to the first and last observations, respectively, of the whole sample, where in this case, r w = r 0 = 1 (see Figure 1). 5 However, the critical values for testing the null hypothesis differ from the ones used in the usual ADF unit root test since we now need the right-tail of the statistic's nonstandard distribution.
The second type of test, the rolling ADF (RADF) test, is a rolling version of the first test in which the ADF statistic is calculated over a rolling window of fixed size specified by the user, i.e., r w = r 0 for all estimations. At each step of the RADF procedure, the window's start and end point (r 1 and r 2 respectively) are incremented one observation at a time (see Figure 2). Each estimation yields an ADF statistic, denoted as ADF r 1 ,r 2 . The RADF statistic is defined as the supremum ADF r 1 ,r 2 statistic among all possible windows. 6 The SADF test, suggested by PWY, is based on recursive calculations of the ADF statistics with a fixed starting point and an expanding window, where the initial size of the window is set by the user. The estimation procedure is as follows (see Figure 3): The first observation in the sample is set as the starting point of the estimation window, r 1 , i.e., r 1 = 0. Next, the end point of the initial estimation window, r 2 , is set according to some choice of minimal window size, r 0 , such that the initial window size is r w = r 2 (again, in fraction terms). Finally, the regression is recursively estimated, while incrementing the window size, r 2 ∈ [r 0 , 1], one observation at a time. Each estimation yields an ADF statistic denoted as ADF r 2 . Note that in the last step, estimation will be based on the whole sample (i.e., r 2 = 1 and the statistic will be ADF 1 ). The SADF statistic is defined as the supremum value of the ADF r 2 sequence for r 2 ∈ [r 0 , 1]: SADF (r 0 ) = sup 5 The t-statistic from this test matches the one reported by EViews. 6 Note that the windows in the RADF procedure are overlapping.

Test
Null hypothesis Alternative hypothesis ADF Unit root Explosive process SADF Unit root Single periodically collapsing bubble period GSADF Unit root Multiple periodically collapsing bubbles Table 1: A summary of the tests' null and alternative hypotheses according to Phillips et al. (2015).
The fourth and last test is the generalized SADF (GSADF), suggested by PSY. This test generalizes the SADF test by allowing more flexible estimation windows, wherein, unlike the SADF procedure, the starting point, r 1 , is also allowed to vary within the range [0, r 2 − r 0 ] (see Figure 4). Formally, the GSADF statistic is defined as

Date-stamping bubble periods
As PWY and PSY show, the SADF and GSADF procedures can also be used, under general regularity conditions, as a date-stamping strategy that consistently estimates the origination and termination of bubbles. In other words, if the null hypothesis of either of these tests is rejected, one can estimate the start and end points of a specific bubble (or bubbles). The date-stamping procedures will now be presented in brief. 7 The first date-stamping strategy is based on the SADF test. PWY propose comparing each element of the estimated ADF r 2 sequence to the corresponding right-tailed critical values of the standard ADF statistic to identify a bubble initiating at time T r 2 . The estimated origination point of a bubble is the first chronological observation, denoted by T re , in which ADF r 2 crosses the corresponding critical value from below, while the estimated termination point is the first chronological observation after T re , denoted by T r f , in which ADF r 2 crosses the critical value from above. Formally, the estimates of the bubble period (as fractions of the sample) are defined byr where cv β T r 2 is the 100(1 − β T )% critical value of the standard ADF statistic based on [T r 2 ] observations. 8,9 Similarly, the estimates of the bubble period based on the GSADF are given bŷ where cv β T r 2 is the 100(1 − β T )% critical value of the sup ADF statistic based on [T r 2 ] observations. BSADF (r 0 ) for r 2 ∈ [r 0 , 1], is the backward sup ADF statistic that relates to the GSADF statistic by the following relation

Installation
In essence, EViews add-ins are EViews programs packed in a way that makes them feel and look like built-in EViews procedures. 10 This relatively new feature enables adding procedures and functionalities that have yet to be implemented in official releases of EViews. By using the add-ins feature and program language, the user is able to augment standard written programs with an interactive user interface, thus making them more suitable to address a general purpose and user friendly. Moreover, unlike regular EViews programs, add-ins have the ability to run directly from EViews objects and/or by commands.
EViews add-ins are available for EViews users with versions 7.1 and above. Installing the Rtadf add-in (or any other add-in for that matter) on an existing copy of EViews can be done manually by downloading the self-extracting installation file from the download section at the EViews website at http://www.eviews.com/Addins/addins.shtml where it is listed 8 In order to asymptotically eliminate type I errors, there is a need to let βT → 0 as T → 0. However, in applied work it is convenient to use a constant βT such as 5%. 9 Phillips and Yu (2011) argue that the dating rule requires that the duration of the bubble be non-negligible. In Phillips et al. (2015) the authors define log(T )/T as a minimal lasting time for a bubble period.
10 For R users, the concept is similar to R packages and also similar to tool-boxes for MATLAB users. under Rtadf * . 11 Alternatively, EViews users with version 8 can download the add-in while inside EViews by clicking Add-ins → Manage Add-ins, selecting the Rtadf add-in from the list presented under the Available tab and clicking the Install button. In general, note that all other add-ins available on the list are written either by the EViews staff or by outside users.
Each add-in published on the EViews website (including Rtadf) has a corresponding support thread in the EViews add-in Support forum, which can be found at http://forums.eviews. com/viewforum.php?f=2.

Using the add-in
The Rtadf add-in can only be run from a series object. Initiating the add-in's dialog box is done by opening a series object and then clicking Proc → Add-ins → Right Tail ADF tests. The test dialog box, presented in Figure 5, enables the user to set the sample period, type of test, initial window size (as a fraction or number of observations), deterministic terms in the test equation and the number of lags in the ADF equation (p in Equation 9), where it can be either fixed by the user, or automatically selected according to some information criterion. In addition, it allows the user to choose the option of simulating critical values for the test (thus prompting the simulation dialog box described below) and whether to view a graph that includes the sequence of ADF statistics, the corresponding critical value sequence and the actual series.
Toggling the "simulate critical values" option will later prompt the simulation dialog box (see Figure 6). If the "simulate critical values" option is checked, there is also a need to choose the simulation tool. Two options are currently available: EViews and MATLAB. Simulation via EViews uses a single core, while choosing MATLAB will use the multicore capabilities of your machine. Note that simulation via MATLAB is only available for RADF, SADF and GSADF and requires having MATLAB as well as the Parallel Computing Toolbox.
In each case, critical values simulations for all four tests are performed according to the 11 The asterisk next to the add-in's name indicates the add-in was developed by an EViews user rather than by IHS EViews.
following steps: 1. Draw one realization at length T based on the null model (given by Equation 8).
2. Estimate Equation 9 by OLS (with or without recursion, depending on the test).

Repeat
5. Calculate the 90%, 95% and 99% quantiles of the distribution of the relevant statistic.

Quantiles calculated in
Step 4 can then be used for testing the null of unit root against the alternative of an explosive process. In addition, the simulation output includes the p value of the test statistic, defined as the probability of observing a statistic as extreme as we did under the null, calculated as whereτ is the estimated test statistic (ADF, RADF, SADF or GSADF), N is the number of replications, I(·) denotes the indicator function which is equal to 1 if the argument is true and 0 otherwise and τ j are the simulated test statistics (j = 1, . . . , N ). The sequences of critical values, which are necessary for the date-stamping procedure, are also derived by the simulation.
The simulation setup in Rtadf is very flexible, allowing the user to specify the type of test, appropriate sample size, number of replications, deterministic terms in the test equation, initial window size, significance level for the critical value sequence (i.e., β), specification of the parameters of the data generating process for the null hypothesis (i.e., set values for d, η and θ in Equation 8). In addition the user can choose whether to use T or T rw in the null model when calculating the simulated critical value sequence (the latter is more accurate but can be very time consuming for large samples while the former is less accurate but faster). 12 As previously mentioned, the output of the simulation procedure is the 90, 95 and 99 percent quantiles of the finite sample distribution of the statistic in question, the relevant p value of the test statistic and the relevant critical values sequence for the date-stamping process.

Run-time comparisons
The results of the running time simulations are presented in Table 2. All tests were run using EViews 9 and MATLAB 2014a (with the Parallel Computing Toolbox installed). The hardware used includes a core i7-4702MQ CPU with 16 GB RAM.
The table includes the running time (in seconds) for all four tests and for two sample sizes -100 and 200 observations. As we can see, for 100 observations, in the RADF and the SADF tests, there is not much of a difference in running times between EViews and MATLAB.
The reason for this similarity is that for a small number of replications, the communication time between EViews and MATLAB and the time it takes to set up a multi-core session, outweighs the benefits of the parallelized simulation. When the sample size is doubled to   200 observations, the efficiency of multi-core calculations kicks in, and enables MATLAB to outperform EViews by being roughly 50% faster than the single core simulation in EViews. As for the GSADF test, which involves many more estimations per replication, using the parallel computing option improves running time by more than 90%(!).

Usage via the command line
The Rtadf add-in can also be called via the command line. This feature enables using the add-in's capabilities as an integrated part of other EViews programs. The syntax is given by where the available options are detailed in Table 3

win = number
Initial window size (in fraction terms or in number of (default = T (0.01 + 1.8/ √ T ) ) observations).

Simulation options sim
Simulate critical values.

Trw
Use T rw instead of T (default) for calculating the sequence of critical values.

Other options graph
Create a graph of the results.
print Print output from the test.  (trend, model = 4, info = aic, lag = 4, win = 0.02, sim, matlab, rep = 2000, graph, print) is similar to the above command, only now the simulation is run in parallel mode using MATLAB.

Illustration
We now demonstrate in detail how to use Rtadf by replicating the results of the SADF test, reported in Phillips et al. (2015), Table 8 (p. 33) and Figure 8 (p. 35). The analysis is based on monthly data of the S&P 500 price-dividend ratio (the series object name in this example is SNP) over the period of 1871:M1 to 2010:M12 that includes 1680 observations, see Figure 7. 13 In order to start the bubble detection process with the price-dividend ratio (SNP) series, first open the SNP series object and then click Proc → Add-ins → Right tail ADF tests (see Figure 8).
Next, specify the test parameters as in PSY -see Figure 9(a) below -and then click the OK button. Note that just like in PSY, the initial window size is set to 36 observations (which constitutes approximately 2% of the whole sample), the lag length of the ADF test (p in Equation 9) is set to zero. Make sure that the "simulating critical values" option is checked so that the output includes the necessary critical values for testing the null hypothesis.
The simulation parameters are also specified in accordance with PSY by adjusting the parameters in the simulation dialog box, which prompts right after clicking the OK button in the main dialog box, see Figure 9(b). Clicking OK results in finite sample critical values for conducting the SADF test. 14 Summary output of the SADF test is displayed in Figure 10      graph includes the SNP series (in green), the ADF r 2 statistic sequence (in blue) and the corresponding 95% critical values sequence (in red). The data used to plot the graph are now available as series objects within the workfile under the names 'sadf' (the blue line) and 'sadf_cv' (the red line), see Figure 10(b). 17 The add-in successfully identifies two bubble periods, just like in PSY, though with minor differences in the start and end point (see Table 4). In addition, the add-in identifies one bubble period lasting four months in the beginning of the sample and a couple of 'blips' of bubbles lasting for one observation (i.e., one month). The source of discrepancy might be differences in the random number generator used by each software. However, if we ignore "too-short-lasting" bubble periods (PSY recommend to restrict identification to ones lasting more than log(T ) units of time measures, which in this case equals log(1680) ≈ 7 months), the results are very similar.
Note that the whole procedure described in this section can be accomplished via the execution of a single command line: snp.rtadf (model = 3, win = 36, sim, rep = 2000, Trw, graph, print) Concluding the illustration, the SADF test results point to the presence of at least one bubble in the S&P 500 price index at the 1% significance level (since 3.443 > 2.141). However, since 17 The names of these series will change according to the test we use. For instance, if we apply the GSADF test, the output will include two new series objects named 'gsadf' (the sequence of the BSADF statistics values) and 'gsadf_cv' (the sequence of the corresponding critical values). Note that each run of the test runs over the previous output. Hence, if there is a need to save these series, copy them under different names. there seems to be evidence for at least two bubble periods (1879-1880 and 1997-2002), as a second step (which is not pursued here), there is justification to use the GSADF test.

Concluding remarks
This paper presents a new EViews add-in, Rtadf, that implements newly developed asset price bubble detection strategies, all based on right-tail versions of the standard reduced form ADF unit root test, where the null of unit root is tested against the alternative of a mildly explosive process. In this case, rejection of the null for a specific time series may serve as evidence of an asset price bubble. This paper began with a short background on the methodological developments of reduced form econometric approaches for bubble detection alongside a theoretical asset pricing model which helps to clarify the rationale behind the reduced form approach. Next, we gave a brief technical discussion on the bubble detection tests included in Rtadf, and finally, a simple illustration of using the add-in in the context of the S&P 500 stocks index was presented.