AN EMPIRICAL ANALYSIS OF MEAN REVERSION. OF THE S&P 500 s P/E RATIOS. Abstract

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1 AN EMPIRICAL ANALYSIS OF MEAN REVERSION OF THE S&P 5 s P/E RATIOS Abstract Numerous authors have suggested that the price-earnings (P/E) ratio can be used to predict the future movement of stock prices. Such arguments are based on the belief that P/E ratios are mean-reverting. But are the S&P P/E ratios really mean reverting? A review of the literature finds arguments on both sides, but the issue of mean reversion has not been tested adequately. Using unit roots and multiple structural breaks, we explicitly show that the P/E ratio is stationary around multiple breaks, which means that it will eventually revert to some long-run means. This result supports evidence that high P/E ratios relative to the current long-run mean will be followed by slow growth in stock prices and/or high earnings growth. JEL classification: G14, C12 Key words: P/E Ratios, S&P Ratios, Mean Reversion

2 AN EMPIRICAL ANALYSIS OF MEAN REVERSION OF THE S&P 5 s P/E RATIOS 1. INTRODUCTION Numerous authors have suggested that the price-earnings (P/E) ratio can be used to predict the future movement of stock prices. For example, White (2), Campbell and Shiller (1998, 21), and Siegel (2) argued that the S&P 5 was overvalued in the late 199s and stock prices would decline. Shen (2) found that high P/E ratios generally have been followed by slow growth in stock prices in the next decade. Such arguments are based on the belief that P/E ratios are mean-reverting. According to Campbell and Shiller (21), the mean-reverting property of P/E ratios implies that stock prices will not drift far from their normal nominal price relative to earnings. This implies that P/E ratios tend to remain stable around an historical mean. Given mean-reversion, stock prices will fall if current prices are high as compared to earnings, and vice versa. There are two sources of change contributing to the stability of P/E ratios. One is the adjustment process in stock prices (numerator), and the other is the adjustment in earnings (denominator). Campbell and Shiller (21) state that adjustments through earnings will be weak. Thus, adjustments to high P/E ratios will come from declining stock prices. Therefore the mean-reverting theory has important implications on the long-run stock market outlook. On the other hand, some authors argue that economic fundamentals and economic structures change continuously, and these changes can have permanent persistent effects on P/E ratios. Accordingly, there is no reason to believe that higher or lower P/E ratios 1

3 should revert to some historical level. Following this line of reasoning, P/E ratios should be characterized as a stochastic process that will fluctuate without bounds. Thus, P/E ratios are mean averting, and it will be difficult to predict the movement of stock prices from P/E ratios. For example, when the Dow Jones Industrial Average was about 8,7 in the late 199s, Glassman and Hassett (1999), Elias (1999), and Kadlec (1999) wrote books with titles stating that the Dow would go to 36,, 4,, and 1, respectively. At the end of 24, the Dow was 1,673. Are P/E ratios mean reverting? To our surprise, the literature has neglected this fundamental question. Instead, many authors assume that price earning ratios are mean reverting. These authors then focus on finding evidence of the relationship between P/E ratios and stock prices. Other authors having different views tend to refute the notion of the mean reverting theory simply due to the reason that the relationship is not supported from the data. However, there is a potential problem with both views, since finding or refuting the relationship between P/E ratios and stock prices involves a certain econometric problem. Most readers may agree that stock prices and earnings data are non-stationary. When variables are nonstationary, standard statistical inference can be spurious. Ferson, Sarkissian and Simin (23) point out that there is a spurious regression bias in predictive regressions for stock returns. Pontiff and Schall (1998), Goyal and Welch (22) and Lee, Myers and Swaminathan (1999) make similar arguments in the case of the book-to-market ratio and dividend yield of the Dow Jones Index. It is clear that spurious regressions can lead to incorrect inferences when dealing with highly persistent data. This point cannot be over-emphasized. Ferson, Sarkissian 2

4 and Simin (23) note that the unit root hypothesis is not rejected for 361 of the 5 stock return series they considered. If stock returns are highly persistent, it is obvious that stock prices are nonstationary. Whether a time series is mean reverting or not depends whether it is stationary or nonstationary. If it is nonstationary, there is no tendency to revert to a long-run mean. If P/E ratios are nonstationary, they will not contain predictive information about future stock prices. Stated differently, there will be no statistical evidence to support the belief that high P/E ratios should be followed by slow growth in earnings in order to return to a long-run stable mean. Thus, the question of whether P/E ratios are mean-reverting is of crucial importance to analysts, investors, and academics alike. Thus, it is surprising that a careful analysis of whether P/E ratios are mean-reverting has not been undertaken in the literature. Unit root tests are the appropriate statistical tool to investigate whether a series is stationary or not. The null hypothesis of nonstationarity will only be rejected in favor of the alternative hypothesis of stationarity, if the data provide sufficient statistical evidence for such a conclusion. Other related hypotheses such as a random walk hypothesis or a martingale hypothesis do not necessarily imply mean reversion since they typically assume no serial correlation. It is possible that these hypotheses can be rejected, but the series is still non-stationary with serial correlation. A unit root hypothesis allows for serial correlation, and it is directly related to the underlying property of non-stationarity under general conditions. One important implication is that when P/E ratios are stationary, implying mean reversion of P/E ratios, we can say that there is a long-run 3

5 cointegration relationship between price and earnings. Cointegration holds when one series is a rational forecast of the other, as in the relationship between spot and forward rates in a foreign exchange market. In the case of the P/E ratios used here, it should be noted that the denominator, E, is measured by past earnings, whereas the numerator, P, represents the present value of expected dividends. 1 Then, the implied concept of cointegration relates to our main question of whether P/E ratios would reveal valuable information about the movement of future stock prices. In the presence of cointegration, the relationship between the P/E ratio and stock prices is not spurious. If stock prices and earning are cointegrated, then the P/E ratio will be stationary. This article deals empirically with the stationarity of the S&P 5 P/E ratio time series that spans over 1 years. We use empirical techniques that incorporate recent methodological developments in the literature. Our statistical results provide an explanation for the apparent non-stationarity of P/E ratios, while at the same time arguing that the P/E ratio will eventually revert to some long-run mean(s). First, we show that the time-series of monthly P/E ratios is statistically nonstationary (difference-stationary). However, this result is reversed after allowing for multiple structural breaks. When structural changes are controlled for in the testing regression, the unit root null hypothesis is rejected. Overall, we find that the S&P 5 P/E ratio is stationary around multiple structural changes. Simply stated, the P/E ratios fluctuate around different means. This result has an important implication that without controlling for known structural breaks, 1 Although not discussed here, the P/E ratio can be derived from the Gordon (1962) basic stock 4

6 predictive regressions for stock prices or stock returns can be spurious. Therefore, the price-earning ratio can be used as variable to forecast other variables in the stock market only when structural changes are included in the model. Our results are in line with the recent results in Rapach and Wohar (25) and Lee, List and Starazicich (25), who find that accurate forecasting and empirical verification of theories can depend critically on understanding the appropriate nature of time-series involving structural changes. 2 Improper modeling of structural changes can lead to incorrect forecasting performance and incorrect statistical inference. Our findings also support the arguments for mean reversion in Campbell and Shiller (21), but only when structural changes are allowed. The remainder of this paper is organized as follows. In Section 2, we discuss recent developments in the methods and present our testing methodologies. Section 3 reveals our empirical findings. In Section 4, we examine the issue using regime switching models. Our concluding remarks are in Section METHODOLOGY The P/E ratio of the Standard & Poor s 5 Index is a widely used stock valuation metric. The time series of P/E ratios used here dates back to 1871, providing a very longterm perspective. The S&P 5 dates back to 1923, when it covered 223 companies. It valuation model where the price of a stock reflects present value of expected dividends. 2 Our results are consistent with the finding of Carlson, Pelz, and Wohar (22) who examined the pattern of structural changes in price earning ratios. However, our analysis differs from their analysis. The mean reverting property is assumed in their analysis, while we examine the basic underlying question of whether price earning ratios are mean-reverting or not. 5

7 was expanded to 5 companies in The S&P 5 focuses on the large-cap sector of the stock market. Because it accounts for over 8% coverage of U.S. equities, it serves as an ideal a proxy for the total market (S&P, 24). During the period, P/E ratios ranged from 5 to 44, and the average was about 15 times earnings. The S&P 5 P/E ratios are calculated by dividing the current stock price of a firm by its trailing twelve months reported earnings. 3 Siegel (24) argues that reported earnings (or GAAP) often understate true earnings because they do not take into account write-offs of purchased assets. Siegel (24) stated that Some write-offs, such as $1 billion in the AOL/Time Warner merger, raised the P/E of the S&P 5 by more than five points. In addition, the failure to expense options and pension expenses tends to overstate operating earnings. These are relatively new accounting factors that did not influence earnings throughout most of the period under review. Nevertheless, the fact that P/E ratios reached record high levels in recent years brought into question its usefulness as a valuation metric. As such, we wish to examine whether P/E ratios are stationary or not. Stationarity is a necessary condition for a mean reverting process. Once stationarity is established, it is reasonable, as done in Carlson et al. (22), to identify, potentially time-varying, reversion levels. We begin with conventional unit root tests using the augmented Dickey-Fuller (ADF) tests. 4 The ADF tests have been the most popular amongst a range of tests 3 GAAP Net Income excluding discontinued operations and extraordinary items. 4 See the Appendix. 6

8 designed to differentiate between stationary and non-stationary time series. However, one critical limitation of the ADF tests (and other usual unit root tests) is that they fail to control for structural changes. According to Perron (1989) and Amsler and Lee (1995), conventional unit root tests lose power when existing structural changes are not accounted for. This implies that non-stationarity may be due to the presence of infrequent breaks in the series rather than the absence of mean reversion. An examination of the P/E ratios shown in Figure 1 reveals frequent and large ups and downs. This certainly leads one to wonder if assuming a fixed unconditional mean is rather too restrictive to yield robust results. FIGURE 1 GOES HERE With this in mind, we use a different testing strategy to account for frequent regime changes. Our strategy is to look at the changes in the deterministic term in the model, and to take into account different means for different regimes. This technique may yield different results because testing methods that do not allow for different means can be mis-specified. Thus, we allow for different intercepts for different regimes of priceearning ratios in testing for a unit root. Then, one may use dummy variables to capture structural changes. However, one issue with the dummy variable approach is that we should know a priori how many structural changes have occurred in the data, and where the breaks occurred. Unfortunately, applied researchers rarely know this information, although a few events or major changes in financial markets can be easily identified. Also, there is a technical problem in testing for a unit root with multiple breaks since it is difficult to determine how many breaks have occurred in the data. We note that Lee and 7

9 Strazicich (23) used only two structural breaks. However, the plot of the P/E ratios in Figure 1 reveals that the series cannot be approximated with just one or two structural changes. In addition, the series does not appear to follow step functions or abrupt changes in levels. 5 Instead, through continuous changes, sometimes spontaneous or gradual, the series moves around different levels, while taking long cycles. To deal with these issues, we use a new technique based on a non-linear approximation of the deterministic term in testing for a unit root with structural changes. We use the unit root tests with a Fourier function, as suggested by Enders and Lee (25). Their unit-root test allows for an unknown number of structural breaks with unknown functional forms. Specifically, we approximate unknown functional forms of nonlinearity by using a Fourier function, which is a linear combination of sine and cosine functions. p y t = d t + ρ y t-1 + δi y t -i + u t, (1) i=1 d t = c + a 1 cos(2πkt/t) + a 2 sin(2πkt/t). (2) Here, k is the underlying frequency parameter to be determined from the data. The unit root null hypothesis implies that ρ =, against the alternative hypothesis that ρ <. The Fourier unit root test can detect sharp changes in level, but it is designed to work best when breaks are gradual in the sense of moving toward new levels taking long cycles, which appears to be our case. If changes are instantaneous or abrupt, the traditional 5 Carlson, Pels and Wohar (22) provide strong evidence of structural change by using the tests that assume stationarity of the series. 8

10 approach using dummy variables to capture structural changes may be more appropriate. However, smooth structural breaks can be better approximated by means of a Fourier series. One important feature of this approximation method is that we do not need to assume that the dates and the number of structural changes are known a priori. Thus, the goal is to control for (not model) the effect of unknown forms of nonlinear deterministic terms in testing for a unit root. Next we utilize a regime switching model that can be applied only to stationary data. As a stationary process implies mean reversion to its unconditional mean(s), we will investigate the changing equilibrium levels in more detail. Regime switching models allow a data generating process to switch between two different processes. The processes can differ in their unconditional means, the variance of their residual terms and/or the persistence of the time-series process. As such, regime switching models are sufficiently flexible to allow for a wide variety of complex processes. One important advantage of this method is that we do not need to specify when the breaks occurred as they are determined endogenously from the data. At any point in time, it is unknown which of the processes the system resides in, and therefore the state variable, s t, is modeled as a latent variable. The sample of data will then allow inference on that state variable. In order to complete the model the transition between one state and another has to be modeled. As is common practice, we assumed that the state variable follows a time 9

11 invariant Markov process, which implies constant transition probabilities between the two different states 6. In particular, the following specification for a regime switching model is applied (Hamilton, 1994, pp. 677) in the context of this paper. 2 ( ) 2 ( ) p yt = µ 1 + ρ 1 1i yt i t, t ~, 1 t 1 i + ε ε N σ if s = = p yt = µ 2 + ρ 1 2i yt i εt, εt ~ N, σ 2 if st 2 i= + = (3) The variable s t is a latent variable determining the dynamic process followed at any time t. In its general form, this specification allows for state dependent means, persistence p parameters and innovation variances. If the sum of the persistence parameters ρ ji, j = 1, 2, is smaller than 1, then there is evidence for mean reversion towards the mean of the respective regime, (µ j /(1 - ρ j1 - ρ j2 )). This allows us to investigate the mean reversion properties in the different regimes, thereby adding to the analysis of Carlson et al. (22). i= 1 3. DATA AND TEST RESULTS Following the work of Shiller and others, we use the ten year average as a measure of earnings. All data are from Robert Shiller s webpage. 7 The data are available in monthly observations from January 1871, giving the first ten year average for the 6 More complex regime-switching models have been proposed in the literature. However, it is not the purpose of this paper to find the best fitting nonlinear model for the P/E ratio, but rather to reconcile the statistical evidence with our theoretical prior of mean reversion. As it turns out, this rather simple regime-switching model is sufficient for this purpose

12 previous ten years of earnings for January It should be noted, however, that the monthly earnings data are interpolated from annual data before 1926 and from quarterly data thereafter. The price data are monthly averages of daily closing stock prices. Prices and earnings are expressed in real dollars. The last observation of the data set used in this paper is December 23. FIGURE 2 GOES HERE Calculating a ratio of observations, sampled monthly, and of a moving ten year average will induce a number of characteristics for the time-series of P/E ratios. As shown in Figure 2, the earnings measure is extremely smooth, whereas the share price index, P, contains a lot of short term variation. This implies that in the short-run, the P/E ratio will inherit the statistical properties of P. As it is well known that P is a nonstationary time-series, it is not surprising that the monthly P/E ratio (as will be seen below) will be difficult to distinguish from a nonstationary time-series. It will be interesting to see whether the statistical properties of the time-series of P/E ratio data will change, as the sampling frequency is changed. Sampling the data at lower frequencies (e.g. quarterly, half yearly or annually) reduces the amount of noise content solely due to variations in the nonstationary price series. This might be a sensible strategy because the earnings data are interpolated from at quarterly data (or even annual data before 1926). As shown in Figure 1, P/E ratios have gone through several regimes over the past 1 years. Therefore, we used an alternative method to approximate the unknown form of structural changes with a Fourier function based on equation (2). To obtain the best 11

13 fitting model, we applied a grid search in such a way that the frequency (k) is determined when the corresponding sum of squared residuals in equation (1) is minimized. The results are given in Table 1. TABLE 1 GOES HERE The estimated frequency (k), which is used to define the Fourier function, is computed as 3.7. The t-statistic for a unit root is given as When k = 4 (close to 3.7), the critical values of the Fourier unit root test are -3.34, and at the 1, 5 and 1% significance levels. Thus, it is clear that the null of a unit root is rejected from the unit root test with a Fourier function. Therefore, P/E ratios are seen as stationary from this test. Then, the series has a tendency to return to a long-run equilibrium level, implying a mean reverting process. The coefficient of the cosine function is significant at the 5% level, implying the existence of non-linearity. The plot of the estimated Fourier function along with the time plot of price-earning ratios is shown in Figure 3. It shows that there has been about 3.7 cycles during the period under review. This amounts to a cycle that takes 33 years, on average, to repeat itself. FIGURE 3 GOES HERE Two additional test results based on the ADF tests and the threshold autoregressive (TAR) unit root tests advocated by Enders and Granger (1998) are reported in the Appendix. The essence of the TAR model is to take into account the asymmetric adjustment in testing for a unit root. The results indicate that the null of a unit root is not rejected by these tests. Which method should we rely on? The null is not rejected using the ADF or Enders-Granger tests, but it is rejected from the unit root test 12

14 with a Fourier function. The answer is clear; we rely on the results from more general models. Thus, we favor the results from the Fourier unit root test since the ADF and Enders-Granger TAR tests can be biased when the existing regime shifts are not accounted for. It is well known in the literature that usual unit root tests tend to be biased against rejecting a false unit root null hypothesis if structural changes are not accounted for. Given the apparent structural changes in the P/E ratios, this phenomenon is occurring in the analysis at hand. Therefore, although the P/E ratio time-series may appear to be nonstationary, this result is caused by the apparent bias of the unit root tests ignoring multiple breaks in the P/E ratio series. 4. FURTHER ANALYSIS Given that the null of a unit root hypothesis is rejected for the monthly P/E data, we will further our analysis and use the regime change switching model. This will support the previous finding that, allowing for some structural shifts, the P/E ratio can be considered stationary. It is the ability of the regime switching model to allow for such changes, and furthermore, to infer the time path of such changes from the data, which will supplement and support the results from the analysis of the trigonometric intercept term. This analysis, while it is flexible, is feasible only when the unit root null is rejected. Thus, from this analysis, we may obtain further insight about the properties of P/E ratios. The assumptions of the two models are different. The trigonometric analysis assumes structural change to be of deterministic nature. In contrast, regime switching models, with the regime transition being governed by a Markov process, assumes that the 13

15 structural changes follow a stochastic process. Despite these fundamental differences, it will be demonstrated below that both approaches deliver similar results. This finding provides robust evidence for the stationarity of the P/E ratio conditional on structural changes. For regime switching models to sensibly model the P/E ratio time-series, at least one of the identified regimes should be characterized as a stationary process. As noted before, a stationary process implies mean reversion to its unconditional mean. It seems reasonable to expect this unconditional mean to be around 14 to 15, the long-run average of the P/E ratio. The second regime could potentially allow for some nonstationary behavior, although, to accord with the empirical behavior of the P/E ratio, it would be expected that this regime would not prevail for as long as the stationary regime. TABLE 2 GOES HERE Two autoregressive terms (p = 2) using monthly data are used as the length of the lag in the regime switching model in equation (3) 8. As shown in Table 2, the dynamics in both estimated regimes are similar. Moreover, the autoregressive processes in both regimes are close to a unit root. Different regimes are identified through different residual variances; the more persistent regimes have smaller residual variances. Because the dynamics are close to a unit root, the estimated unconditional means do not have any reasonable interpretation. 8 We used p = 2 for all regime switching models estimated in this paper. It also appeared reasonable to restrict the analysis to two regimes. As it is not the purpose of this paper to find the best regime switching models no further discussion on these choices is provided. 14

16 FIGURE 4 GOES HERE The estimated regime switching model is used to obtain state probabilities, being the probability that at a particular time the system was governed by one of the two states. The probabilities that the P/E ratio process was governed by the second state are displayed in Figure 4 alongside the time series of monthly P/E ratios. It is apparent that the second regime, apart from two episodes, is transitory and does not prevail long. This does not come as a surprise when taking into consideration that the differences between the regimes are confined to the residual variance. These estimation results confirm the results of the unit root testing as both regimes display nonstationary behavior. As previously mentioned, the monthly P/E ratio is likely to inherit the statistical behavior of the nonstationary price series, and the large variations in this series dominate the smooth earnings time series. A lower sampling frequency will potentially reveal features of the P/E ratio time series that are not apparent from the monthly data. For this reason, regime switching models were estimated for P/E ratio data sampled quarterly, half-yearly and annually. This allows for increased variation in the earnings measure and helps to identify the relevant features of the P/E ratios. Two sampling schemes for each of these series were used. First, the first observation in each period was picked and second, all observations in one period were averaged. TABLE 3 GOES HERE 15

17 The estimation results for the six regime switching models are shown in Table 3. For each regime the table displays the unconditional mean (unc mean), the sum of the autoregressive parameters (ρ 1 + ρ 2 ), the residual standard deviation (σ) and the transition probability indicating the likelihood to remain in the respective regime (p 11 or p 22 ). The following conclusions can be drawn from the results displayed in Table 3. The first regime, across the different models with exception of the model for the nonaveraged 3 months data, displays three persistent features. 1) In all these models the unconditional mean is between approximately 13.5 and 15.5, a level frequently referred to as a reasonable long run mean for the P/E ratio. 2) The lower the sampling frequency the smaller is the sum of the autoregressive coefficients, which indicates the stationarity of the P/E ratios in Regime 1. 3) When sampled annually (12m and 12m-avg), the persistence of the process decreases to.768 (when sampling the first month of each period) and.94 respectively (when averaging across the year). The probability to remain in Regime 1 (p 11 ) is consistently estimated between.96 and.98. Regime 1 is therefore estimated to be very stable, although it should be noted that, while these parameters are fairly constant, they imply different expected time spans one would expect the P/E ratio process to remain in Regime 1 9. Regime 1 estimated from the 3 months data is still, as for the monthly data, estimated to be nonstationary and the unconditional mean therefore has no economic meaning. 9 As the sampling frequency decreases, the same transition probability indicates a longer time for which the process is expected to stay in Regime 1. 16

18 The results for Regime 2 are less clear cut. The second regime for all averaged data models and for the 6 months sampled data displays a significantly higher unconditional mean at a value around 2. All these regimes are estimated to be stationary and also differ from their respective first regimes by having a larger residual variance. The results from the 3 months and 12 months sampled data differ slightly from these results. For the 3 months sampled data the unconditional mean is significantly smaller at For the annual, non-averaged data the second regime is estimated to be near nonstationary, consequently the unconditional mean in excess of 1 has no economic interpretation. It should also be noted that for this model, very few observations belong to Regime 2. This can clearly be seen from Figure 5, which displays the P/E ratio time series and the respective probabilities of being in Regime 2. FIGURE 5 GOES HERE Based on these results, it is reasonable to call Regime 1 the default or normal regime because its features coincide with those previously proposed for a theoretically sound P/E ratio process, stationarity and an unconditional mean of about 14 to 15. This interpretation is supported by the results shown in Figure 5. Regime 1 is predominant in all 6 regime switching models. 1 Further insights can be drawn from Figure 5. Regime 2 appears to be more easily identified from the averaged data. As Regime 2 has some rather short episodes, it is likely that by sampling the data only once per period a significant amount of important information might be lost. From the Regime 2 17

19 probabilities across the different regime switching models, at most 4 episodes can be identified. The first is around 19, the second in the 193s, the third in the 196s and the last around 2. All of these have been identified previously. The first and third episodes are characterized by persistent P/E ratios around the unconditional mean of Regime 2, 2. The second and fourth episodes are mainly characterized by one big P/E ratio spike. The fourth however, also displays some persistently high values around 2, whereas the second, apart from the high spike, is mainly characterized by its high variation in the P/E ratio. Most clearly these episodes are identified in the regime switching model with the averaged 6 monthly data. There is a striking similarity between these four episodes identified by the regime switching models and the time-varying intercept identified from the trigonometric expansion and displayed in Figure 3. Although both methodologies are fundamentally different, they deliver the same conclusion. This strengthens the conclusion that it is some structural break causing the apparent nonstationarity of the time-series of P/E ratios. In this case, the structural break appears to be recurrring at regular intervals. The regularity is fortuitous as it facilitates the identification of the break by means of the trigonometric expansion. It remains to be seen whether the regularity in the past structural breaks such high P/E ratio regimes reappear at regular intervals in the future. But, the past data reveal that the structural breaks appear to be recurrring at regular intervals. 1 Note that the probability of being in Regime 1 is 1 minus the probability of being in Regime 2, 18

20 5. CONCLUDING REMARKS This paper demonstrates how to reconcile the apparent statistical nonstationarity of the P/E ratio time-series with the strong theoretical prior that this time-series should be stationary with an unconditional mean reflective of reasonable rates of return. After demonstrating that the time-series of monthly P/E ratios appears to be statistically nonstationary, it was shown that allowing for multiple breaks in the level of the P/E ratio series is sufficient to render the statistical representation of the P/E ratio into a stationary process. This conclusion was arrived at via two very different routes, both of which provide nearly identical answers. First, the change in the level of the series was captured by a deterministic, trigonometric series. Then regime switching models were estimated to the P/E ratio series. Both approaches indicate the presence of periodically recurring high P/E ratio episodes. The similarities between the identified periods are remarkable. Further, both methodologies conclude that, conditional on these structural breaks, the time-series process of P/E ratios is stationary, and thus mean-reverting. As such, we have provided statistical evidence that P/E ratios are mean reverting. This implies that high P/E ratios relative to the current long-run mean will be followed by slow growth in stock prices and/or high earnings growth. Our analysis also suggests that improper modeling of structural changes can lead to incorrect statistical inference. This is an important with the latter being displayed in Figure 5. 19

21 empirical finding for analysts, investors, and academics alike who use the P/E ratio to predict future movements in stock prices or earnings. 2

22 REFERENCES Amsler, C. and Lee, J. "An LM Test for A Unit Root in the Presence of a Structural Change," Econometric Theory, 1995, 11, Campbell, J. Y. and Shiller R.J. "Stock Prices, Earnings, and Expected Dividends," Journal of Finance, 43, 1988, Campbell, J. Y. and Shiller R.J. "Valuation Ratios and The Long-run Stock Market Outlook: An Update," NBER Working Paper No. 8221, April 21. Carlson, J.B., Pelz, E. A., and Wohar, M. E. Will Valuation Ratios Revert to Their Historical Means? Journal of Portfolio Management, 22, Elias, D. Dow 4,: Strategies for Profiting From the Greatest Bull Market in History, New York, McGraw-Hill, Enders, W. and Granger, C. "Unit-Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates," Journal of Business and Economic Statistics 16, 1998, Enders, W. and Lee, J. Testing for a Unit Root with a Nonlinear Fourier Function, working paper, 25, University of Alabama. Ferson, W., S. Sarkissian and T. T. Simin, Spurious Regressions in Financial Economics? Journal of Finance, 58 (4), 23, Glassman, J. and Hassett, K. Dow 36,: The New Strategy for Profiting from the Coming Rise in the Stock Market, New York: Times Business, Gordon, J. The Investment Financing and Valuation of the Corporation, Homewood, IL., Irwin, Goyal, A and I. Welch, Predicting the Equity Premium with Dividend Ratios, Management Science, May 23. Graham B. and Dodd D.L. Security Analysis, 1st ed., McGraw-Hill: New York, Kadlec, C. W. Dow 1,: Fact or Fiction, Old Tappan, NJ, Prentice-Hall Press,

23 Lee, J., J. List, and M. Strazicich, Nonrenewable Resource Prices: Deterministic or Stochastic Trends? Journal of Environmental Economics and Management, 25, forthcoming. Lee, C, J. Myers, and B. Swaminathan, What is the intrinsic value of the Dow? Journal of Finance, 54, 1999, Perron, P. The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis, Econometrica, 57, 1989, Rapach, D., and M. Wohar, Valuation Ratios and Long-Horizon Stock Price Predictability, Journal of Applied Econometrics 2, 25, Shen, P. The P/E Ratio and Stock Market Performance, Economic Review, Federal Reserve Bank of Kansas City, Fourth Quarter, 2, Shiller, R. Webpage - Source of P/E ratio data: Siegel, J.J. Big-Cap Stocks are a Sucker Bet, The Wall Street Journal, March 13, 2, p. A3. Siegel, J. J. The Long-Run Equity Risk Premium, CFA Institute Conference Proceedings; Points of Inflection: New Directions for Portfolio Management, 24, Standard & Poor s. S&P U.S. Indices Methodology, September 28.24, Stambuagh, R. Predictive Regressions, Journal of Financial Economics, 54, 1999, White, C. B. What P/E Will The U.S. Stock Market Support? Financial Analysts Journal, November/December, 2,

24 Table 1 Fourier Unit Root Tests Coefficient t-statistic ρ cosine function sine function Constant augmented terms (omitted) Note: 1. The estimated frequency (k) is When k = 4, the critical values of the Fourier unit root test are -3.34, and at the 1, 5 and 1% significance levels. 3. In the testing regression, 11 augmented terms are used. 23

25 Table 2 Regime Switching Models (Monthly Data) (Restriction) Regime Regime1 Regime 2 µ (1.398) (1.365) unc mean ρ (4.886) (24.913) ρ (-8.596) (-5.274) ρ 1 + ρ σ.391 (34.38) (21.997) Transition probabilities p i p i Note: t-statistics are reported in the parentheses 24

26 Table 3 Regime Switching Models (Monthly Data) Sampling scheme 3months 3m-avg 6m 6m-avg 12m 12m-avg Regime 1 Unc mean ρ 1 + ρ σ p Regime 2 Unc mean ρ 1 + ρ σ p Note: No t-statistics are reported. All autoregressive parameters are estimated to be significantly different from zero at conventional significance levels. For each regime the table displays the unconditional mean (unc mean), the sum of the autoregressive parameters (ρ 1 + ρ 2 ), the residual standard deviation (σ) and the transition probability indicating the likelihood to remain in the respective regime (p 11 or p 22 ). 25

27 4 5 P E Figure 1. Plot of P/E Ratios S&P 5 (left scale) Avg. Earnings (right scale) Figure 2. Plot of S&P 5 Share Price Index and Average Earnings Series 26

28 P E P E F O U R IE R Figure 3. Plot of P/E Ratios and the Fourier Term 27

29 PE monthly Figure 4. Plot of P/E ratios (solid line - left scale) and posterior probabilities of being in the high-mean State (dashed line - right scale). Monthly P/E ratios. 28

30 6 PE quarterly 1 6 avg PE quarterly PE 6 monthly 1 6 avg PE 6 monthly PE annualy 1 6 avg PE annualy Figure 5. Plot of P/E ratios (solid line - left scale) and posterior probabilities of being in the high-mean State (dashed line - right scale). Top left panel: P/E ratio data sampled in the first month of each quarter. Top right panel: P/E ratios averaged over quarters. Middle left panel: P/E ratio data sampled in the first month of each half year. Middle right panel: P/E ratios averaged over half years. Bottom left panel: P/E ratio data sampled in the first month of each year. Bottom right panel: P/E ratios averaged over each year. 29

31 Appendix Appendix Table 1 ADF and Enders-Granger Threshold AR Unit Root Tests ADF TAR M-TAR Lags ρ 1 (t-stat) -.5 (-1.95) -.69 (-2.25) -.57 (-1.66) -.73 (-2.18) -.14 (-.374) -.92 (-2.39) ρ 2 (t-stat) -.49 (-1.4) -.62 (-2.43) -.98 (-2.43) -.45 (-1.14) y t-1 (t-stat).251 (9.9).258 (9.9).256 (9.81) F-stat for TAR and MTAR models ρ 1 = ρ 2 = F-stat for symmetry ρ 1 = ρ 2 (p-value) Q(4) (p-value) 96.1 (.) 7.71 (.1) (.849) 91. (.).36 (.849) 7.76 (.11) (.133) 9.3 (.).699 (.43) 7.75 (.11) ADF t-statistic: Critical values (T 1) at the 1%, 5% and 1% levels are -2.57, and F-test for the linear attractor Threshold (TAR) model: Critical values (T 1) at the 1%, 5% and 1% levels are 3.74, 4.56, and F-test for the momentum Threshold (M-TAR) model: Critical values (T 1) at the 1%, 5% and 1% levels are 4.5, 4.95, and Note: The ADF tests are based on equation (1), when d t is give as d t = a + b t. We used the values of p = and 1. We rely on the results with p = 1 since the coefficient of y t-1 is significant in all cases. It is possible to allow for more augmentation terms, but the reported values of Q(4) statistic and p-values indicates that no further augmentations may be necessary. The so-called threshold autoregressive (TAR) model replaces ρy t-1 in the ADF type equation with ρy t-1 = I t ρ 1 y t-1 + (1-I t )ρ 2 y t-1, where I t is the Heavyside indicator function such that I t = 1 if y t-1 and I t = if y t-1 <. The null hypothesis 3

32 implies the joint restriction ρ 1 = ρ 2 =. The usual F-statistic for ρ 1 = ρ 2 = is used for testing for a unit root. Αn asymmetric adjustment occurs when ρ 1 ρ 2, where each of these parameters measures the degree of autoregressive decay. An alternative scenario is that autoregressive decay is fast when the series is increasing and slow when it is decreasing. In this case, the indicator function is modified as I t = 1 if y t-1 and I t = if y t-1 <. The resulting model is referred to as the momentum threshold autoregressive (M-TAR) model. One limitation of both threshold models is that they do not account for a shift in level. 31

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