Multivariate tests for asset price bubbles Jörg Breitung and Robinson Kruse University of Cologne and Leibniz University Hannover & CREATES January 2015 Preliminary version Abstract Speculative bubbles have played an important role ever since in financial economics. Prices and dividends are typically well approximated by random walks in absence of bubbles. During periods of bubbles, prices follow explosive paths, while dividends do not undergo a structural change. Recently proposed bubble tests focus mainly on the prices, while neglecting a potential link to their fundamental value. In this work, we suggest two different type of tests to deal with the joint dynamics of prices and dividends. The first test builds upon an error correction term in a univariate test regression, while the second one uses a vector autoregressive framework. It is argued that these tests have some advantages over existing ones and the simulation study reveals that large power gains can be achieved. Importantly, the detection of small bubbles is especially improved by applying tests accounting for prices and dividends simultaneously. JEL codes: C12, C22, G10 Keywords: Asset price bubbles, stock prices, dividends, cointegration, explosive processes.
1 Introduction Speculative bubbles have a long history in stock, commodity and real estate markets. They are often considered to be responsible for severe economic crises such as the speculative excess on stock prices prior to the great depression from 1930 1933 or the recent financial crisis of 2007 2008 that was preceded by the US housing bubble. All bubbles are characterised by an explosive path of the underlying market prices, whereas in normal times, prices are well approximated by a random walk process. A standard model to rationalise the occurrence and persistence of speculative bubbles is the framework of rational bubbles (see, e.g., Blanchard and Watson 1982). In such models it is economically rational to invest in an obviously overpriced asset as long as the investor asserts that the price continues to rise at an exponential rate. An alternative approach is adapted by Froot and Obstfeld (1991). It is based on the concept of intrinsic bubbles. This framework directly links the evolution of the bubble to fundamentals like dividend payments. In a recent contribution, Phillips, Wu and Yu (2011) suggest a recursive righttailed Dickey-Fuller test against the explosive alternative. The test is univariate and separately applied to stock prices and dividend payments. A rejection of the unit root hypothesis in favor of explosiveness for stock prices and a corresponding non-rejection for dividends are commonly interpreted as evidence for a bubble. If both tests do not reject, then one typically concludes that bubbles are non-existent. Some inconclusive situations may occur when both tests reject, thereby hinting at possible co-explosiveness, or when only the test for the dividends provides evidence against the null hypothesis. Under the validity of the null hypotheses, stock prices and dividends are free of any explosive component, and according to basic financial theory, both variables shall be cointegrated. An important limitation of the PWY test is its univariate nature which does not allow to test for cointegration directly. Beside these methodological issues, potential gains in power to detect explosive patterns are our main motivation to consider a multivariate framework. It is argued that stock prices and dividends are linked and that tests for bubbles may perform better when taking the joint dynamics of prices and dividends simultaneously into account. In this work, we consider two extensions of the PWY test. First, we propose a test based on an estimated error correction term. By 2
doing so, the test statistic for cointegration, which is a simple t-statistic, exhibits a standard normal limiting distribution. Under the alternative, the variance of the error correction term explodes which leads to a considerable power gain as our simulation study indicates. Second, we adopt an approach by Ahlgren and Nyblom (2008) which allows testing several different hypotheses within a unified framework. These hypotheses are the existence of one or more explosive roots, the presence of unit roots and even cointegration. This test allows for a richer analysis in a vector autoregressive framework by means of OLS estimation and also has higher power against explosive alternatives. In our simulation study, we consider size and power of the two newly proposed testing procedures in comparison to the PWY test which serves as the benchmark. Under the alternative, we focus on the three different popular bubble processes which include a component model, bubbles with a random start over time and periodically collapsing bubbles. It turns out that achievements can be made in terms of power in general. But, especially for small bubbles (which characterized by a relatively low growth rate and a relatively small signal-to-noise ratio) large gains can be made from using the newly suggested tests. Large bubbles are relatively easy to detect and in such situations all tests perform well. Small bubbles, however, are more likely to occur in practice. They are more difficult to detect and thus pose a more challenging problem. The simulation results suggest multivariate tests have a greater chance in detecting these type bubbles. Interestingly, this conclusion holds irrespective of the particular bubble process under study. The rest of the paper is organized as follows: Section 2 covers different bubble processes and the related tests. The simulation study is located in Section 3. In Section 4 we consider an empirical application to a long annual sample for the Standard and Poors 500 index. Main conclusions and an outlook are given in Section 5. 2 Bubble processes and tests 2.1 Bubble processes We start by presenting some of the most widely studied bubble processes. Amongst these are periodically collapsing bubbles, the component model which contains 3
a fundamental value and a bubbles process and some others. The component model is given by the set of equations b t = (1 + g)b t 1 + ε b t (1) f t = µ f + f t 1 + ε f t (2) p t = f t + b t (3) The bubble b t is initialized in the first period at level b 0. The bubble growth rate is given by g 0. Typical values for g, given that data is recorded at a monthly frequency, lie in the range [0, 0.05]. In most studies, one finds typical estimates around 0.025. The fundamental value f t is given by random walk with drift µ f, whereas the stock price p t is constructed by the sum of the fundamental value and the bubble component. Importantly, the size of the bubble component is determined by both, g and σ b (relative to σ f ). 1 In the following, we consider different values for g and σ b, while keeping σ f fixed. Moreover, the initial value of the bubble (b 0 ) also has an impact on the success of bubble detection. Hence, we consider a range of different values for b 0 as well. The component model contains one unit root and one explosive root. When considering randomly starting bubbles, we first note that are conceptually similar to the previous bubble model with the important difference that the bubble does not start in the first period. In fact, the timing of the bubble start is random and determined by a Bernouilli random variable with a certain success rate of 0 < π < 1. Again, the bubbles starts at level b 0. The bubble process is given by the following set of equations: b t = (1 + π 1 gθ t )b t 1 b t 1 = b 0 b t = (1 + g)b t 1 b t 1 > b 0 with θ t B(π). 2 The third bubble process under study is the periodically collapsing bubble 1 Parameter values are set as follows: µ f = 0.00227, σ b = 0.0324 and σ f = 0.05403. 2 Parameters are set as follows: b 0 = 1, f 1 = 1.3, µ f = 0.0373, σ f = 0.1574. 4
process by Evans (1991). It is defined by the following set of equations. b t = (1 + g)b t 1 ε b t b t 1 c b t = [ζ + π 1 (1 + g)θ t (b t 1 (1 + g) 1 ζ)]ε b t b t 1 > c with ε b t = exp(z t σ 2 b /2), z t N(0, σ 2 b ) and θ t B(π). Here, the bubble grows with rate g until it collapses. The probability for such a collapse is given by 1 π in each period. After a collapse, a new bubble starts to grow. 3 In absence of bubbles, stock prices and dividends are likely to be cointegrated and thus having a long-run relationship. In this case, we consider the following data generating process: p t = µ p + βf t 1 + ε p t (4) f t = µ f + f t 1 + ε f t (5) where ε p t N(0, σ 2 p) and ε f t = φε f t 1 + v t with v t N(0, σ 2 f ).4 Another possibility is the absence of bubbles and cointegration. In this case we have that stock prices and dividends are possibly correlated, but not cointegrated: p t = µ p + p t 1 + ε p t (6) f t = µ f + f t 1 + ε f t (7) where ε p t N(0, σp) 2 and ε f t N(0, σf 2 ) are correlated with strength ρ. 2.2 Tests 2.2.1 Univariate tests We consider the following variety of tests. First, we present the univariate PWY test and proceed with the Ahlgren and Nyblom (2008) test. The PWY test is univariate in nature and hence applied to the stock prices and dividends individually. The test statistic is a right-tailed Dickey-Fuller statistic obtained from a 3 Parameters are set as follows: µ f = 0.0373, σ f = 0.1574, f 1 = 1.3, σ b = 0.05, ζ = 0.5, b 0 = 0.5, c = 1. 4 Parameters are set as follows: µ f = 1.906481, µ p = 0.01636848, σ f = 0.1662617, σ p = 0.1783107, β = 0.6721554, φ = 0.6689062. 5
conventional auxiliary OLS test regression k 1 x t = µ + ρx t 1 + β j x t j + ε t where x t = {P t, D t }. The null hypothesis is given by H 0 : ρ = 1, whereas the alternative hypothesis is H 1 : ρ > 1. The limiting distribution is known and simulated critical values for the right-tailed test are reported in PWY. One concludes in favor of a bubble, if one finds a rejection for stock prices, but not for dividends. If both tests do not reject, one concludes that there are no bubbles due to the lack of explosiveness. Note that one cannot directly test for cointegration between stock prices and dividends in the univariate setup. Such a test for cointegration would be a natural way to proceed as both series appear to be characterized as I(1) and finance theory suggests a long-run equilibrium relationship. There two other possible outcomes: (i) both tests reject and (ii) the test statistic for dividends is significant, while the statistic for stock prices is not. Both cases maybe unlikely in practice, but still pose possible outcomes of the testing procedure by PWY. Case (i) would not allow to conclude in favor of a bubble as the surge in stock prices can be validated by a similarly explosive path in dividends. The question of coexplosiveness arisrs. This question amounts to a possible common explosive root and maybe called co-explosiveness. The second case might be seen as unrealistic as it would suggest that the fundamental value would follow an explosive path, while the prices would not. This situation might not be important for practical purposes, while the first one is more likely to be of relevance. The PWY test can be applied to the full sample and to a recursively growing sample. In the latter case, a supremum statistic is considered and critical values are consequently different from the ones for the full sample. Details are provided in PWY. For simplicity of exposition, we currently focus on the full sample case. Extensions to the recursive setup are considered in future. We consider the following extension of the PWY test. The idea builds upon the standard present value model for the fundamental value of a stock (e.g. Camp- j=1 6
bell and Shiller 1987) which implies or P f t = ρ i+1 E t (D t+i ), i=0 = ρ [ E t (D t ) + ϱ E t (D t + D t+1 ) + ϱ 2 E t (D t + D t+1 + D t+2 ) + ] = ϱ [ Et (D t ) + ϱ E t ( D t+1 ) + ϱ 2 E t ( D t+2 ) + ] 1 ϱ S t = P f t 1 r D t = E t (St ) with St = ϱ i D t+i. i=1 If D t is stationary and ρ < 1, then St is stationary as well. Since the (rational expectations) forecast of a stationary variable must also be stationary, it follows that under the assumptions of the present value model for stock prices S t = P f t r 1 D t is a stationary linear combination of stock prices and dividends. As shown by Campbell and Shiller (1987) the model implies that also P f t is a stationary process and, thus, the vector (P f t, D t ) is cointegrated with CI(1,1) in the terminology of Engle and Granger (1987), see also Diba and Grossman (1987). If the stock price P t = P f t + B t entails a bubble component B t subject to the condition E t (B t+1 ) = (1 + r)b t (cf. Blanchard and Watson (1994)), then S t = P t r 1 D t = E t (P t ) + B t is characterized by an explosive pattern. The extended PWY test is similar to the error correction test proposed by Banerjee et al. (1992). The test statistic is based on the t-statistic of φ in the regression P t = φp t 1 + βd t 1 + p γ i P t i + i=1 q δ i D t i + u t. i=1 Since P t is typically found to be (approximately) unpredictable, the lagged differences may be dropped in many empirical applications. Under the null hypothesis the φ(p t 1 + r 1 D t 1 ) is stationary and, therefore, the OLS estimator φ has the usual normal limiting distribution for coefficients attached to stationary variables (cf. Sims, Stock and Watson 1990). It should be noted, however, that the co- 7
efficient φ may be zero even in cases where P t is cointegrated with D t. In this case the error correction term S t 1 helps to predict D t but not P t. This is indeed a plausible scenario in empirical practice. We therefore test the hypothesis of the absence of a speculative bubble based on the hypothesis φ = 0 against the alternative φ > 0. If the error correction term S t 1 = P t 1 r 1 D t helps to predict the price changes P t, the coefficient φ is negative and our test becomes conservative. It is interesting to compare the resulting test with the testing strategy of PWY (2011). The main difference is that our test includes the lagged dividends as an additional regressor. Accordingly, our test runs a regression of P t on the estimated equilibrium errors Ŝt 1 = P t 1 θd t 1 instead of the P t 1 which serves as a regressor in the PWY test. This has two important advantages. First, by using the (asymptotically) stationary variable Ŝt 1 as a regressor we are able to apply the usual critical values based on a standard normal limiting distribution. Second, whereas under the null hypothesis the variance of S t 1 converges to a constant, the sample variance of S t 1 explodes under the alternative. This difference between the statistical properties of S t under the null hypothesis and the alternative leads to a considerable gain in power. 2.2.2 Multivariate tests As argued above, if the stock price P t = P f t + B t entails a bubble component, then S t = P t r 1 D t = E t (Pt ) + B t is characterized by an explosive pattern. Accordingly, the autoregressive representation of the vector y t = [ P t D t ] = A(L)y t 1 + ε t or y t = Πy t 1 + Γ(L) y t 1 + ε t, (8) possesses an explosive root z 0 = ϱ < 1 with A(z 0 ) = 0, A(L) = I A 1 L A p L p, Γ(L) = Γ 0 + Γ 1 L +... + Γ p 1 L p 1 and Π = A(1) I and Γ(L)(1 L) = A(L) A(1). 5 Furthermore, the largest eigenvalue of the matrix Π is equal to r, whereas the other eigenvalue is zero provided that D t I(1). Under the null 5 As usual we suppress possible deterministic terms like vector of constants. In empirical practice the test equations typically include an additional constant. 8
hypothesis that there is no speculative bubble, the real part of one eigenvalue (associated with the linear combination β = (1, r 1 )) is negative, whereas the other eigenvalue is equal to zero. Our second testing strategy invokes a system cointegration test. It is important to note that Johansen s likelihood ratio test does not provide a suitable test procedure for our testing problem. The reason is that this test is essentially two-sided and, thus, it cannot distinguish between stationary and explosive alternatives. In particular, we consider the following bivariate setup for x t = (P t, D t ) which is given by k x t = µ + A j x t j + ε t j=1 where ε t is assumed to be a martingale difference sequence. In order to get rid of the vector of constants µ we work with demeaned data in the following. The demeaning is accounted for when considered the asymptotic properties. 6 The companion form of the model with demeaned data is given by X t = ΨX t 1 + u t We are interested in the determinant equation Ψ λi = 0 which has q 0 explosive roots and s 1 unit roots while the remaining r = 2k q s roots are stationary. The matrix Ψ is estimated by OLS. The ordered eigenvalues of Ψ are labeled as λ 1... λ q λ q+1... λ q+s... λ 2k. In accordance with our bubble processes we consider the following situations: q = {0, 1}, s = {2, 1} and r = {0, 1}. These situations correspond to the case of cointegration (q = 0, s = 1, r = 1), a single explosive and a single unit root which relates to a bubble process (q = 1, s = 1, r = 0) and the absence of cointegration and bubbles (q = 0, s = 2, r = 0). The testing framework of Ahlgren and Nyblom (2008) allows us to consider 6 Prices and dividends are de-meaned individually by subtracting their sample mean. Further improvements in terms of power are expected when other types of demeaning are applied. Promising approaches are recursive and GLS demeaning which work well when testing against stationary alternatives. Their use in the context of explosive alternatives, is however, unexplored yet. 9
the following sequence of pairs of hypotheses. We start by testing for the presence of two unit roots (s = 2), given that there are no explosive roots (q = 0). Under the alternative, there is a unit root and an explosive root in the system. Thereby, we consider testing H 0 : s = 2, q = 0 against H 1 : s < 2, q > 0. The relevant test statistic builds on the largest estimated eigenvalue λ 1 and is given by λ max(s = 2) = (T k)( λ 1 1). This statistic tests the restriction that the largest eigenvalue of Ψ is equal to unity. Therefore, any explosive roots are ruled out under H 0. The limiting distribution of λ max(s = 2) is derived in Ahlgren and Nyblom (2008) where also critical values for various significance levels are reported. The null hypothesis has to be rejected for large values of λ max(s = 2). If the test statistic is insignificant at conventional levels, one concludes that there are no explosive roots in the system which corresponds to the non-existence of bubbles. One natural way to proceed is to test for cointegration. One can simply build the cointegration test on a statistic which uses the second largest eigenvalue: λ min(s = 2) = (T k)( λ q+2 1) where q = 0 given that the statistic λ max(s = 2) is not significant. The limiting distribution of λ min(s = 2) is obtained from Ahlgren and Nyblom (2008). The null hypothesis is rejected in favor of the alternative for small values of λ min(s = 2). Suppose, the initially calculated statistic based on the largest eigenvalue is in fact significant. In this case, one can continue by testing H 0 : s = 1, q = 1 against H 1 : s < 1, q > 1. The bubble processes defined above share the common feature that they exhibit a single unit root and a single explosive root which is in line with H 0. Thus, one would expect a non-rejection. Under the alternative, however, there is no unit root, but two explosive roots which leads to a situation discussed above in the context of the PWY test. The test statistic for H 0 uses the second largest eigenvalue λ max(s = 1) = (T k)( λ 2 1) and rejects for large values. 10
3 Simulation results In our simulation study, we consider empirically relevant parameter settings and a typical sample size of T = 100. Most parameter settings are taken from previous studies, i.e. Homm and Breitung (2012) and PWY (2011), which enables some comparison to related results. The nominal significance level is set equal to five percent. The number of Monte Carlo replications is 5,000 per experiment. The unknown lag length k is selected via BIC with a maximal length of four which corresponds to Schwert s (1989) rule of setting k max = [4(T/100) 0.25 ]. For all statistics small-sample critical values are simulated in order to ensure a fair comparison which is not affected by size distortions stemming from an approximation error to the limiting distribution. 3.1 Unit roots and cointegration When the true data generating process is characterized by either cointegration or simply two unit roots without cointegration, we are considering the size of various methods. Simulation results are reported in Table 1. The different test statistics are the multivariate Ahlgren and Nyblom statistic λ max (s = 2); PWY S and PWY PD refer to the error correction based test. PWY S uses the estimated spread, while PWY PD uses a test regression with prices and dividends. PWY P and PWY D denote the Phillips et al. (2011) test applied to prices (P ) and dividends (D) separately. Table 1: Size of various tests under cointegration and two unit roots λ max (s = 2) PWY S PWY PD PWY P PWY D Cointegration 0.028 0.053 0.055 0.052 0.003 Two unit roots 0.052 0.004 0.004 0.048 0.050 The results indicate that the Ahlgren and Nyblom (2008) test is undersized in the case of cointegration, while the error correction based test statistics are correctly sized. The PWY test applied to the price series displays virtually no size distortions, while the test for the dividends appears to be conservative. Turning to the case of two unit roots we find that the λ max (s = 2) statistic controls size well. Unsurprisingly, the error correction tests are conservative as a different 11
limiting distribution applies in this case. The individual PWY tests are both working well. 3.2 Bubble processes Three different types of bubble processes are studied. Results for the component model are reported in Table 2. The initial value of the bubble b 0 is varied from 0 to 0.1 in steps of 0.02. The growth rate of the bubble g ranges from 0.01 to 0.05. Remaining parameters are kept fixed. Empirical powers are increasing along g for fixed values of b 0. The initial value of the bubble itself has a positive, but negligible impact on the power. A clear result emerges from the simulations: the multivariate test statistic λ max (s = 2) offers considerable power gains over its competing approaches. The PWY P test performs somewhat better than the error correction based tests. For the relatively small value of g = 0.01, all tests have low power in detecting the alternative. When g takes values between 0.02 and 0.04, a clearly better performance can be obtained from the multivariate test. For the largest considered value of g = 0.05, all tests perform nearly similarly well and have very high power. There are small differences between the PWY S and the PWY PD test, whereby the latter is preferred. We finally note that the PWY test applied to dividends shows good performance as only minor upward distortions are observed in some cases. Empirical powers against randomly starting bubbles are given in Table 3. As to be expected, power increases for larger values of g. The probability of a bubble start is given by π in each period. The more likely a start of a bubble is, the more likely is its detection. Hence, π has a positive impact on the tests s power. We start comparing the tests for the smallest value of π = 0.01: for extremely small bubbles, the error correction tests are performing very well in comparison to the other ones. For bubbles with a growth rate of 0.02 and 0.03, the multivariate tests dominates. For large bubbles we observe s similar performance of all tests. For larger values of π we see that the multivariate test dominates. The error correction based tests are performing better than the PWY P test when g is reasonably small. For g 0.03, the univariate test by Phillips et al. (2011) has higher power than the error correction tests. We now turn to the results for the case of periodically collapsing bubbles which are presented in Table 4. A similar picture arises: For small values of 12
Table 2: Component model (bubble plus fundamental) b 0 g λ max (s = 2) PWY S PWY PD PWY P PWY D 0 0.01 0.116 0.112 0.113 0.091 0.066 0.02 0.432 0.221 0.233 0.285 0.063 0.03 0.788 0.499 0.535 0.657 0.054 0.04 0.925 0.779 0.804 0.872 0.059 0.05 0.966 0.908 0.919 0.938 0.053 0.02 0.01 0.115 0.109 0.112 0.088 0.060 0.02 0.435 0.229 0.241 0.297 0.048 0.03 0.780 0.509 0.524 0.652 0.052 0.04 0.917 0.764 0.785 0.853 0.056 0.05 0.970 0.924 0.932 0.948 0.061 0.04 0.01 0.103 0.119 0.124 0.083 0.056 0.02 0.441 0.249 0.264 0.301 0.054 0.03 0.788 0.519 0.547 0.682 0.060 0.04 0.918 0.779 0.805 0.871 0.061 0.05 0.976 0.930 0.939 0.952 0.065 0.06 0.01 0.119 0.117 0.119 0.092 0.056 0.02 0.454 0.263 0.276 0.316 0.058 0.03 0.794 0.561 0.587 0.694 0.061 0.04 0.923 0.814 0.836 0.876 0.053 0.05 0.974 0.923 0.934 0.951 0.060 0.08 0.01 0.115 0.119 0.121 0.088 0.058 0.02 0.446 0.267 0.279 0.320 0.053 0.03 0.807 0.583 0.608 0.714 0.054 0.04 0.929 0.802 0.824 0.886 0.057 0.05 0.978 0.939 0.945 0.958 0.066 0.1 0.01 0.095 0.106 0.106 0.080 0.050 0.02 0.462 0.249 0.269 0.311 0.062 0.03 0.844 0.601 0.636 0.756 0.054 0.04 0.943 0.847 0.870 0.906 0.058 0.05 0.980 0.943 0.951 0.964 0.055 13
Table 3: Randomly starting bubbles π g λ max (s = 2) PWY S PWY PD PWY P PWY D 0.01 0.01 0.183 0.348 0.347 0.094 0.084 0.02 0.582 0.400 0.404 0.241 0.101 0.03 0.625 0.525 0.541 0.576 0.083 0.04 0.644 0.589 0.606 0.643 0.093 0.05 0.651 0.632 0.643 0.656 0.094 0.02 0.01 0.440 0.476 0.477 0.110 0.098 0.02 0.805 0.500 0.507 0.230 0.096 0.03 0.855 0.659 0.684 0.703 0.073 0.04 0.859 0.780 0.795 0.818 0.097 0.05 0.875 0.851 0.860 0.861 0.088 0.05 0.01 0.851 0.575 0.576 0.102 0.089 0.02 0.974 0.580 0.585 0.223 0.083 0.03 0.992 0.792 0.822 0.851 0.080 0.04 0.994 0.944 0.958 0.962 0.096 0.05 0.994 0.980 0.985 0.986 0.081 0.1 0.01 0.771 0.607 0.606 0.104 0.092 0.02 0.968 0.603 0.611 0.224 0.084 0.03 1.000 0.804 0.835 0.900 0.097 0.04 1.000 0.983 0.991 0.998 0.090 0.05 1.000 0.995 0.997 1.000 0.091 0.25 0.01 0.647 0.602 0.603 0.104 0.097 0.02 0.999 0.602 0.590 0.228 0.080 0.03 1.000 0.752 0.776 0.919 0.101 0.04 1.000 0.937 0.935 1.000 0.077 0.05 1.000 0.933 0.929 1.000 0.087 0.5 0.01 0.724 0.593 0.588 0.086 0.078 0.02 1.000 0.604 0.597 0.223 0.095 0.03 1.000 0.778 0.783 0.928 0.086 0.04 1.000 0.936 0.925 1.000 0.090 0.05 1.000 0.933 0.925 1.000 0.093 0.99 0.01 0.758 0.581 0.582 0.086 0.077 0.02 1.000 0.588 0.590 0.212 0.082 0.03 1.000 0.795 0.811 0.933 0.081 0.04 1.000 0.974 0.977 1.000 0.095 0.05 1.000 0.977 0.976 1.000 0.091 14
Table 4: Periodically collapsing bubbles 1 π g λ max (s = 2) PWY S PWY PD PWY P PWY D 0.01 0.01 0.351 0.374 0.375 0.092 0.088 0.02 0.575 0.385 0.392 0.161 0.084 0.03 0.578 0.413 0.441 0.433 0.094 0.04 0.580 0.495 0.526 0.553 0.081 0.05 0.604 0.536 0.558 0.602 0.100 0.02 0.01 0.344 0.333 0.336 0.103 0.096 0.02 0.403 0.291 0.299 0.173 0.092 0.03 0.394 0.294 0.309 0.297 0.093 0.04 0.382 0.310 0.326 0.347 0.081 0.05 0.420 0.368 0.383 0.412 0.082 0.05 0.01 0.194 0.243 0.246 0.100 0.091 0.02 0.193 0.175 0.182 0.120 0.086 0.03 0.189 0.129 0.137 0.141 0.085 0.04 0.189 0.128 0.134 0.160 0.089 0.05 0.192 0.135 0.144 0.164 0.085 0.1 0.01 0.137 0.224 0.227 0.084 0.081 0.02 0.127 0.116 0.123 0.094 0.095 0.03 0.113 0.058 0.063 0.071 0.077 0.04 0.104 0.055 0.059 0.075 0.082 0.05 0.116 0.056 0.058 0.077 0.091 0.25 0.01 0.087 0.199 0.204 0.080 0.084 0.02 0.080 0.090 0.092 0.072 0.086 0.03 0.073 0.034 0.034 0.057 0.085 0.04 0.073 0.019 0.018 0.046 0.093 0.05 0.071 0.017 0.019 0.043 0.089 0.5 0.01 0.081 0.198 0.202 0.098 0.102 0.02 0.065 0.067 0.067 0.067 0.081 0.03 0.062 0.027 0.028 0.061 0.086 0.04 0.058 0.007 0.010 0.040 0.080 0.05 0.070 0.010 0.012 0.036 0.092 0.99 0.01 0.060 0.233 0.235 0.083 0.085 0.02 0.049 0.149 0.147 0.092 0.095 0.03 0.053 0.067 0.067 0.082 0.088 0.04 0.060 0.036 0.035 0.085 0.103 0.05 0.052 0.010 0.010 0.071 0.096 15
0 500 1000 1500 2000 Real S&P500 prices Real S&P500 dividends 1880 1900 1920 1940 1960 1980 2000 Figure 1: S&P 500 annual real prices and dividends (scaled with factor 20). g, the multivariate test and the error correction based tests are able to provide improved power over the PWY P test. For very large values of g, we observe a similar performance of tests, but preference is given to the multivariate test as it offers the highest potential to detect periodically collapsing bubbles. 4 Empirical application In our empirical application, we consider an annual data set of S&P500 real stock prices and dividends. The data is obtained from Robert Shiller s website and covers the period from 1871-2012. The number of observations is therefore T = 142. The two series are shown in Figure 1, where we present real stock prices P t together with scaled dividends, i.e. 20D t which allows a nicer visual comparison. For a long period, prices and dividends move closely together. During the late Nineties, however, prices seem to decouple from their fundamental value. After the peak in the year 2000, prices adjusted towards dividends with some erratic movements in the aftermath. 16
In our subsequent analysis, we consider (i) the full sample including T = 142 annual observations and (ii) a rolling window analysis with a window size of w = 100 years. This choice is made in accordance with our Monte Carlo study where we use small-sample adjusted critical values. The rolling window analysis is carried out in order to uncover periods of explosive behaviour which are probably undetected in a full sample analysis. Due to the facts that bubbles are typically short-lived and that our sample covers more than a century of data points, it appears to be useful to consider a rolling window which can uncover some possible time-variation in the results. Our full sample analysis does not reveal the existence of bubbles, nor the existence of cointegration between prices and dividends. Results are reported in Table 5. The multivariate test statistic using the largest eigenvalue is not significant, neither is the cointegration test statistic using the second largest eigenvalue. The extended PWY tests using an error correction term are not significant as well. The only slightly significant statistic at the 5% level is the PWY test applied to the dividends which is an unusual and somewhat unrealistic result, especially as the PWY statistic for the prices is far from being significant. Overall, the general finding of the non-existence of bubbles and cointegration may not be too surprising, given the very long sample period. Table 5: Empirical results for the full sample 1871-2012 λ max (s = 2) λ max (s = 1) PWY S PWY PD PWY P PWY D 0.171-16.198-1.706-1.709-0.657 0.096 5% Critical values 0.271-22.999 0.961 0.973-0.036-0.036 We now turn to our rolling window analysis which may shed light on possible time-dependence in our results. Results are presented in Figure 2. The upperleft cell contains the results for the multivariate test using the largest eigenvalue. Until the year 1997, the statistic is insignificant hinting at the non-existence of explosiveness until this point in time. In 1998, the statistic exceeds the critical value (displayed as a vertical line) and stays significant until 2002. This suggests a period of explosive prices from 1998 until 2002. The statistic for the second largest eigenvalue can either be used as a cointegration statistic (when compared 17
to the lower critical value), given that the statistic for the largest eigenvalue is not significant, or it can be used to test against the existence of two explosive roots in the system (when comparted to the upper critical value), given that the statistic for the largest eigenvalue is in fact significant. The results obtained from the rolling window scheme suggest that until the start of a bubble in 1998, prices and dividends where cointegrated. During the bubble period, cointegration is lost. Moreover, there is no indication for a second explosive root in the system. After the burst of the bubble, prices and dividends are cointegrated again. For the two year 2011 and 2012, however, the cointegration statistic is no longer significant. The test based on error correction tells us a similar story. Turning to the benchmark test of Philiips et al. (2011), we find also evidence for explosive prices around the Millennium. During this period, the statistic for the dividends does not lead to a rejection of the null hypothesis. But, during the more recent period, the statistic turns out to be significant which is a difficult result to interpret. 5 Conclusions Explosive price paths are a typical feature of speculative bubbles. In a recent contribution, Phillips, Wu and Yu (2011) suggest a recursive right-tailed Dickey- Fuller test against the explosive alternative. Their testing framework is univariate and focusses on the behaviour of prices. In this work, we propose to alternative ways to test for bubbles. The first one is strongly connected to the univariate test, but incorporates dividends as well. A main advantage is that the resulting t-statistic has a standard normal limiting distribution when prices and dividends are cointegrated. According to standard financial theory this should be the case in absence of bubbles. The second approach considers the joint dynamics of prices and dividends within a vector autoregressive framework. Simulation results suggest that the detection of bubbles with a relatively small growth rate, i.e. mildly explosive processes (see e.g. Phillips and Magdalinos 2007), can be improved a lot by using alternative tests. Another result is that the multivariate test statistic performs quite well in many situations and has often higher power than the univariate test by Phillips et al. (2011). 18
5 0 5 10 15 20 Largest EV statistic 30 20 10 5 0 2nd largest EV statistic 1970 1980 1990 2000 2010 Error correction statistic 1970 1980 1990 2000 2010 PWY statistic 2 0 2 4 2 0 2 4 Prices Dividends 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 Figure 2: Results from the rolling window analysis. 19
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