Unit-root tests for explosive behavior

Abstract

We present a new command, radf, that tests for explosive behavior in time series. The command computes the right-tail augmented Dickey and Fuller (1979, Journal of the American Statistical Association 74: 427–431) unitroot test and its further developments based on supremum statistics derived from augmented Dickey–Fuller-type regressions estimated using recursive windows (Phillips, Wu, and Yu, 2011, International Economic Review 52: 201–226) and recursive flexible windows (Phillips, Shi, and Yu, 2015, International Economic Review 56: 1043–1078). It allows for the lag length in the test regression and the width of rolling windows to be either specified by the user or determined using data-dependent procedures, and it performs the date-stamping procedures advocated by Phillips, Wu, and Yu (2011) and Phillips, Shi, and Yu (2015) to identify episodes of explosive behavior. It also implements the wild bootstrap proposed by Phillips and Shi (2020, Handbook of Statistics: Financial, Macro and Micro Econometrics Using R, Vol. 42, 61–80) to lessen the potential effects of unconditional heteroskedasticity and account for the multiplicity issue in recursive testing. The use of radf is illustrated with an empirical example.

1 Introduction

The study of the dynamic properties of economic and financial variables occupies a central position in the econometric modeling of time series. One specific type of behavior in which there has always been a great deal of interest, particularly in times of crises or distress, is when the series under consideration exhibits what appears to be explosive behavior. Indeed, analysts have identified several instances in the literature in which variables such as prices appear to increase well beyond the level that could be explained by their fundamentals; see, for example, Garber (2000) for an analysis of famous early bubbles and exuberant behavior.

During the last decade or so, there has been a renewed interest in the application of statistical tests for explosive behavior, mainly because of the appearance of novel theoretical findings by Phillips, Wu, and Yu (2011) and Phillips, Shi, and Yu (2015). These authors, through the further development of unit-root tests, provide a framework of analysis suitable for testing and date-stamping episodes where explosive behavior might have occurred. Empirical implementation of these new testing strategies is possible thanks to available computer codes in MATLAB; ¹ the EViews add-in Rtadf, developed by Caspi (2017); the R Core Team (2020) package exuber, developed by Vasilopoulos, Pavlidis, and Martínez-García (2020a) for the econometric analysis of explosive time series; and psymonitor, developed by Phillips, Shi, and Caspi (2019) for real-time monitoring of asset markets (bubbles and crises).

In this article, we present the community-contributed command radf, which tests for explosive behavior in time series. The radf command implements the right-tail augmented Dickey and Fuller (1979; ADF) unit-root test and its further developments based on supremum statistics derived from ADF-type regressions estimated using recursive windows (Phillips, Wu, and Yu 2011) and recursive flexible windows (Phillips, Shi, and Yu 2015). Similarly to the other software that is currently available, radf supports the implementation of the date-stamping procedures that have been advocated to identify episodes of explosive behavior. In addition, it implements the wild bootstrap proposed by Phillips and Shi (2020) to lessen the potential effects of unconditional heteroskedasticity and multiplicity involved in recursive testing. We believe that the tests computed with the radf command constitute a worthwhile addition to the battery of time-series tests currently available to Stata users.

The article is organized as follows. Section 2 provides an overview of the tests for explosive behavior supported by the radf command. Section 3 describes the command, while section 4 illustrates the use of the command with an empirical example. Section 5 concludes.

2 Tests for explosive behavior

In this section, we offer an overview of the tests for explosive behavior that are provided by the radf command. The reader interested in a thorough and rigorous presentation is referred to Fuller (1976); Dickey and Fuller (1979); Phillips, Wu, and Yu (2011); and Phillips, Shi, and Yu (2015). Following the notation employed by Phillips, Shi, and Yu (2015), radf calculates three tests that are based on the ADF regression with a constant,

$\Delta y_t={\alpha }_{r_1,r_2}+{\beta }_{r_1,r_2}{y}_{t-1}+{\displaystyle \sum _{i=1}^k{\delta }_{r_1,r_2}^i}\Delta y_{t-i}+{\epsilon }_t$

1

where Δ is the first-difference operator, y_t is the realization of the time series of interest at time t, k is a scalar that denotes the number of lags of the dependent variable that are included to account for residual serial correlation, and r ₁ and r ₂, respectively, denote the starting and ending points used for estimation. With T as the total number of time periods in the sample, r ₁ and r ₂ are expressed as fractions of T such that r ₂ = r ₁ + r_w , where r_w is the window size of the regression, also expressed as a fraction of T . The number of observations used to estimate (1) is denoted T_w = ⌊Tr_w⌋, where ⌊·⌋ is the floor function that gives the integer part of the argument. The error term is ε_t .

The unit-root null hypothesis is given by H ₀ : β_{r 1 ,r 2} = 0, while the alternative is that the series of interest exhibits explosive behavior; that is, H ₁ : β_{r 1 ,r 2} > 0. The ADF t statistic required to test H ₀ : β_{r 1 ,r 2} = 0 in (1) is denoted ${\text{ADF}}_{r1}^{r2}$ . In this setting, the command radf calculates the two statistics that were studied by Phillips, Wu, and Yu (2011). The first is the right-tailed ADF statistic based on the full range of observations, that is, when r ₁ = 0 and r ₂ = 1 (that is, r_w = 1). The resulting statistic is thus denoted ${\text{ADF}}_0^1$ . Monte Carlo simulation results reported by Phillips, Wu, and Yu (2011) confirm earlier findings by Diba and Grossman (1988) and Evans (1991) in the sense that standard unit-root tests based on the full sample can detect explosive behavior as long as the episodes of bubbles (or exuberance) do not collapse periodically. ² To deal with the cases where bubbles collapse periodically, Phillips, Wu, and Yu (2011) further consider a second statistic that is based on the supremum t statistic that results from a forward recursive estimation of (1),

$\mathrm{SADF}\left(r_0\right)=\underset{r_2\in \left[r_0,1\right]}{\sup }{\text{ADF}}_0^{r_2}$

2

where the window size r_w expands from the smallest sample window width r ₀, which provides the first t statistic of the recursion, to the last available observation. The supremum ADF (SADF) t statistic given in (2) is the second test statistic computed by the radf command.

In their empirical illustration, Phillips, Wu, and Yu (2011) select the optimal number of lags of the test regression (k) following the general-to-specific methodology, which involves setting k = k _max and testing the statistical significance of ${\delta }_{r_1,r_2}^{\max }$ based on the standard normal distribution (see Campbell and Perron [1991]; Hall [1994]; and Ng and Perron [1995]). The radf command supports the choice of the optimal number of lags with this data-dependent procedure, based on significance levels of 5% and 10%, as well as with the statistical Akaike information criteria (AIC) of Akaike (1974) and the Schwarz information criteria (SIC) of Schwarz (1978).

However, Phillips, Shi, and Yu (2015) indicate that when the sample period is characterized by successive episodes of bubbles, one potential limitation of the recursive approach suggested by Phillips, Wu, and Yu (2011) is that it provides consistent estimates of the origination and ending dates of the first bubble but not subsequent ones. To overcome this limitation, Phillips, Shi, and Yu (2015) put forward the generalized supremum ADF (GSADF) test, which is the third statistic produced by radf. As the name implies, the GSADF test involves a much more extensive set of regressions, in which the first observation used for estimation varies from 0 to r ₂ − r ₀, while the last observation varies from r ₀ to 1. More formally,

$\mathrm{GSADF}\left(r_0\right)=\underset{\underset{r_1\in \left[0,r_2–r_0\right]}{r_2\in \left[r_0,1\right]}}{\sup }{\text{ADF}}_{r_1}^{r_2}$

For practical purposes, Phillips, Shi, and Yu (2015) recommend implementing the GSADF test by setting a low value of k, say, 0 or 1, and determining the minimum window size using the rule $r_0=(0.01+1.8/\sqrt{T})$ . The radf command permits the user to set the number of lags k, while the minimum window size is determined automatically based on the rule already described. Although the command also permits the user to set the minimum window size according to personal preferences, it warns that the critical values have been derived for the window size set by the rule.

The sample sequences described before are summarized graphically in figure 1 (from Phillips, Shi, and Yu [2015]).

Figure 1.

Sample sequences and window widths supported by the radf command

Inference for the right-tail ADF, SADF, and GSADF statistics requires critical values computed using Monte Carlo simulations. In the case of the ${\text{ADF}}_0^1$ and SADF(r₀) statistics, the underlying data-generating process is the well known random walk without a drift,

$y_t=y_{t-}{ }_1+{∊}_t$

where ∊_t ∼ NID(0, σ ²) (NID denotes normally and independently distributed). As for the GSADF(r ₀) statistic, the corresponding critical values are based on a random walk where the drift component is asymptotically negligible,

$y_t=T^{-}{ }^1+y_{t-}{ }_1+{∊}_t$

where ∊_t ∼ NID(0, σ ²). For computational convenience, the radf command takes advantage of a large set of critical values already available in the R Core Team (2020) package exuber; see Vasilopoulos, Pavlidis, and Martínez-García (2020a) and Vasilopoulos et al. (2020b). ³ More specifically, the critical values that we incorporated in radf are the 90%, 95%, and 99% critical values provided by exuber, which were obtained using 2,000 replications, seed equal to 123, initial window size given by $r_0=0.01+1.8/\sqrt{T}$ , and T = 6, 7, 8,…, 600, 700, 800,…, 2000 observations. For 600 < T ≤ 2000, the sample size is used to interpolate between the critical values. For T > 2000, the critical values that are included in the summary results table correspond to those for T = 2000.

The testing strategies based on recursive window and recursive flexible window estimation of (1) provide useful guidance to date-stamp in real time the episodes of explosive behavior if the null hypothesis is rejected. Following the discussion in Phillips, Shi, and Yu (2015), let us suppose that one is interested in assessing whether any particular observation, say, r ₂, belongs to a phase of explosive behavior. These authors recommend performing a SADF test on a sample sequence where the endpoint is fixed at the observation of interest r ₂, and expands backwards to the starting point, r ₁, which varies between 0 and (r ₂ − r ₀). In this frame, the backward SADF (BSADF) statistic is defined as the supremum of the resulting sequence of ADF statistics, that is,

${\mathrm{BSADF}}_{r_2}\left(r_0\right)=\underset{r_1\in \left[0,r_2-r_0\right]}{\sup }{\text{ADF}}_{r_1}^{r_2}$

3

The statistic BSADF _{r 2}(r ₀) is then compared with the corresponding critical values of the SADF(r ₀) for ⌊r ₂ T⌋ observations. Phillips, Shi, and Yu (2015) indicate that this identification procedure is more general than the earlier suggestion in Phillips, Wu, and Yu (2011), which sets r ₁ = 0 in (3) and therefore is more effective at identifying episodes of multiple bubbles. The sample sequences and window widths for date-stamping recommended by Phillips, Wu, and Yu (2011) and Phillips, Shi, and Yu (2015) are illustrated in figure 2, top and bottom, respectively, (from Phillips, Shi, and Yu [2015]).

Figure 2.

Sample sequences and window widths for date-stamping strategies

The radf command permits the user the options to generate and graph the variables that contain the sequences of t ratios and their associated critical values for datestamping the episodes of explosive behavior. Because the critical values for the BSADF statistic are only available for T ≤ 600, these series are not available for larger selected samples.

As a further development, in recent work Phillips and Shi (2020) recommend a wild bootstrap procedure to lessen the potential effects of unconditional heteroskedasticity and to account for the multiplicity issue in recursive testing. This bootstrap scheme can be implemented as an option with the radf command, which provides 90%, 95%, and 99% bootstrap critical values for the three tests described before.

3 The radf command

The command radf calculates the right-tail ADF test statistics for explosive behavior proposed by Phillips, Wu, and Yu (2011) and Phillips, Shi, and Yu (2015). The command permits the user to compute the statistics in which the number of lags of the dependent variable in the test regressions can be determined either by the user or endogenously by using a data-dependent procedure. Options to generate and graph the resulting sequences of recursive and flexible recursive statistics allow the identification of episodes of explosive behavior.

3.1 Syntax

Before using the command radf, and similar to many other Stata time-series commands, it is necessary to tsset the data. radf varname [if] [in] [, prefix(prefix ) maxlag(integer) criterion(string) window(integer) bs seed(integer) boot(integer) print graph] varname may not contain gaps but can contain time-series operators. radf does not support the by: prefix.

3.2 Options

The command radf supports the following options:

prefix( prefix ) can be used to provide a “stub” with which variables created in radf will be named if no more than 600 observations are in the specified sample. If this option is given, four Stata variables will be created for the appropriate range of dates: prefix_SADF, prefix_BSADF, prefix_BSADF_95, and prefix_Exceeding. These variables record the SADF and BSADF statistics, with the third variable displaying the 95% critical values for the BSADF statistic, which vary over the estimation period. The fourth variable is an indicator, set to 1 in each period when the BSADF statistic exceeds its 95% critical value. The prefix() option must be specified to enable the graph option.

maxlag( integer ) sets the number of lags to be included in the test regression to account for residual serial correlation. If not specified, radf sets the number of lags following Schwert (1989), with the formula maxlag = int{4(T/100)^{0 . 25}}, where T is the total number of observations. In either case, the number of lags is reported in the output. If maxlag() is given and the window width is set by the data-dependent procedure, they may conflict. In this case, radf reduces maxlag() so that each ADF regression has positive degrees of freedom.

criterion( string ) can be used to specify AIC, SIC, GTS05, or GTS10 as alternatives to the default value of FIX. By default, radf computes the ADF regressions based on a fixed maxlag(), either one determined by a data-dependent procedure or one specified by the user. Alternatively, the command can determine the optimal number of lags according to the Akaike or Schwarz information criteria, denoted AIC and SIC, respectively, or by following the general-to-specific (GTS) algorithm advocated by Hall (1994) and Ng and Perron (1995). The idea of the GTS algorithm is to start by setting an upper bound on p (denoted p _max), estimating (1) with p = p _max, and testing the statistical significance of δ_{p max}. If this coefficient is statistically significant, using, for example, significance levels of 5% (referred to as GTS_{0 . 05}) or 10% (referred to as GTS_{0 . 10}), one selects p = p _max. Otherwise, the order of the estimated autoregression is reduced by 1 until the coefficient on the last included lag is found to be statistically different from 0.

window( integer ) can be used to select a different window width. The initial window width used to compute both the SADF and the GSADF statistics takes the default value of $r_0=0.01+1.8/\sqrt{T}$ recommended by Phillips, Shi, and Yu (2015). However, because the critical values have been developed for the default window width, a warning is provided if the window width is set by the user, showing the default width and the selected width.

bs computes right-tail Monte Carlo critical values for the 90th, 95th, and 99th percentiles based on the wild bootstrap advocated by Phillips and Shi (2020), using 199 replications. To set a different number of replications, use the boot() option. Also, notice that the bootstrap critical values cannot be replicated unless bs is used along with option seed().

seed (integer ) sets the initial seed for random-number generation.

boot (integer ) sets the number of replications to perform the wild bootstrap advocated by Phillips and Shi (2020).

print specifies that detailed results are to be printed showing the ADF statistics and lag lengths for each of the regressions being estimated.

graph specifies that the time series of the SADF and BSADF statistics, which can be saved as variables with the prefix() option, should be graphed along with their 90% and 95% critical values. The graphs will be saved with names specified by the prefix() option as prefix_SADF.gph and prefix_BSADF.gph. The graph option is not available if more than 600 observations are included in the specified sample, and it requires the use of the prefix() option.

3.3 Stored results

radf stores the following in r():

4 Empirical application

Housing is certainly the most important asset in the portfolio of many individuals and families. For this reason, there is considerable interest in following the dynamic path of property prices so that knowledge can be gained regarding the specific periods of time when they might be viewed as reaching levels that compromise affordability. Indeed, many economists and financial analysts alike feel that property prices are prone to suffering bubble-like phenomena. Before the advent of these tests for explosive behavior, the researcher applying unit-root tests to property price series would have undoubtedly favored the unit-root hypothesis. Then attention would turn to testing the existence of long-run equilibrium relationships with real income (for example, Holly, Pesaran, and Yamagata [2010]) or with other property prices (for example, Holly, Pesaran, and Yamagata [2011]). The findings here are suggestive of episodes of bubbles when applied to certain intervals over sup criteria.

We use data from the International House Price Database of the Federal Reserve Bank of Dallas, which contains quarterly price information on 23 countries that dates back to the first quarter of 1975; see Mack and Martínez-García (2011) for methodological details on the database. To carry out our empirical illustration, we downloaded the data release for the first quarter of 2015 directly from the R console, following the steps described in section 5.2 of Vasilopoulos, Pavlidis, and Martínez-García (2020a), and we created a Stata version of the data that was placed with the Boston College Economics Stata datasets.

We begin by loading the dataset using the community-contributed command bcuse (Baum 2017) and declaring it as a time series:

We would like to test whether the price index series of the United Kingdom and the United States, respectively named uk and us, each contain a unit root, against the alternative that they are explosive processes.

Using the default specifications for criterion() and window() and setting a maxlag() of p = 1, the results of applying the command radf to the two variables of interest over the full range of observations are as follows:

The output indicates that the first observations of the recursive and recursive flexible windows run from the third quarter of 1975 to the second quarter of 2009. Both SADF and GSADF tests provide evidence against the unit-root null hypothesis at the 1% significance level. In the case of ADF, we observe rejection of the null in the case of U.K. prices only at the 10% significance level.

The results reported above are identical to those obtained using the R package exuber and the EViews add-in Rtadf. In terms of computation time, for the variable uk, radf runs in 0.24 seconds on Stata version 17 using a laptop computer equipped with processor Intel Core™ i7 CPU @ 2.80 GHz and installed memory (RAM) of 16.0 GB. Using the R package exuber, the same results can be obtained in only 0.021 seconds. By contrast, with the EViews add-in Rtadf, it takes 0.106, 0.536, and 21.756 seconds to obtain the ADF, SADF, and GSADF statistics, respectively. ⁴ The speed gain achieved by the R package exuber most likely is due to the fact that Vasilopoulos, Pavlidis, and Martínez-García (2020a) use partitioned inversion techniques that avoid the need to invert sequences of matrices when computing the ordinary least-squares estimators, while radf is based on Stata’s Mata language. All in all, we consider that the speed of radf is satisfactory for applied work and also if the user requires the computation of critical values for different values of the number of observations, initial window lengths, and the number of lags.

Given that the results presented above reject the unit-root null hypothesis, one might proceed to plot the sequences of t statistics and corresponding critical values to identify the time periods during which episodes of explosive behavior might have taken place. To this end, we run the previous command using the prefix() and graph options:

The output is omitted to avoid repetition, while the date-stamping plots for the price series in the United Kingdom and the United States are presented in figures 3 and 4, respectively. To facilitate the verification of these results, the variables used to construct the figures are reported in appendix B at the end of the article and are available with the print option. Looking at the more powerful BSADF test and using a 5% significance level, the results for the United Kingdom reveal three episodes of explosive behavior in house prices, which can be date-stamped by the indicator variable prefix_Exceeding: 1987q2–1989q3, 1999q1–2005q3, and 2006q2–2008q1. For house prices in the United States, the identified periods are 1987q1 and 1998q1–2007q2. In terms of the duration of the episodes, Phillips, Shi, and Yu (2015) recommend focusing on those that last more than ln(T ) units of time, which in this illustration is ln(161) ≈ 5 quarters. This further refinement, however, is left at the discretion of the user.

Figure 3.

Date-stamping analysis for the U.K. house price series

Figure 4.

Date-stamping analysis for the U.S. house price series

Finally, we now illustrate the use of the command radf using the options boot(499) and seed(123). To save space, we shall present only the results for the United Kingdom.

As can be seen, in all cases the simulated bootstrap critical values (columns labeled RTMC90, RTMC95, and RTMC99) are much larger than the corresponding tabulated critical values from Vasilopoulos, Pavlidis, and Martínez-García (2020a) (columns labeled Tab90, Tab95, and Tab99). Using the simulated bootstrap critical values, the unit-root null is rejected at the 10% and 5% levels by the SADF and GSADF tests, respectively.

5 Concluding remarks

In this article, we presented the command radf, which calculates the right-tail ADF test statistics for explosive behavior proposed by Phillips, Wu, and Yu (2011) and Phillips, Shi, and Yu (2015). radf permits the user to compute the statistics in which the number of lags of the dependent variable in the test regressions can be determined either by the user or endogenously by using a data-dependent procedure. The options to generate and graph the resulting sequences of recursive and flexible recursive statistics allow the identification of episodes of explosive behavior. Finally, the command radf implements a wild bootstrap scheme recently advocated by Phillips and Shi (2020), which aims to lessen the potential effects of unconditional heteroskedasticity and to account for the multiplicity issue in recursive testing.

7 Programs and supplemental materials

Supplemental Material, sj-zip-1-stj-10.1177_1536867X211063405 - Unit-root tests for explosive behavior

Supplemental Material, sj-zip-1-stj-10.1177_1536867X211063405 for Unit-root tests for explosive behavior by Christopher F. Baum and Jesús Otero in The Stata Journal