JSTOR

I.  Introduction

Research in asset markets has led to a prolific amount of studies related to the time series structure of asset prices, and after years of studies we still are uncertain of their actual behavior, in particular the behavior of stock prices. Fama (1970) surveyed the then state of understanding of asset prices and sent researchers down a path to show that no agent could consistently outperform the market. Using the efficient market hypothesis (EMH) as the explanatory tool to explain the underlying relationships of market and market agents’ interactions, researchers have sought to show that no agent can consistently outperform the market.

The EMH relies on the efficient exploitation of information by economic agents, which implies that it would not be possible for a market agent to earn abnormal profits. EMH assumes that agents, in forming their expectations in the period, are rational in the sense that they do not make systematic forecasting errors, and they know expected market equilibrium prices or expected equilibrium returns. Malkiel (2003) reviews the current evidence and understanding of the market efficiency since Fama’s (1970) paper and concludes that the evidence does support the claim of market efficiency. However, this conclusion is not definitive, and researchers still are trying to account for studies on anomalous, to the efficient market hypothesis, market behavior (for anomalous studies, see, e.g., DeBondt and Thaler [1985]; Jegadeesh and Titman [1993]; Lakonishok, Shleifer, and Vishny [1994]; and Loughran and Ritter [1995]).

We are still on that path set by Fama (1970) but are becoming more open to other possible explanations for market behavior (see, e.g., Shiller [2003] on behavioral finance). The problems with the EMH assumptions are that they apply to all agents at all times and assume that all agents have similar time constraints in their ability to hold assets. This study will assume EMH with a relaxation of the requirement that all agents use all available information at all times.1 Each agent will use only that amount of information that is rational for that particular investor to use at the point of transaction. “Rational to use” is defined as the right amount of information, obtained by traders, to form an informed decision, tailored to their required returns and liquidity constraints. The required returns to motivate a trade can be based on liquidity only, novice traders following perceived market trends, or sentimentally motivated traders. Since there are opportunity costs involved with collecting and interpreting information, it is not assumed here that all agents have the same information nor is it rational for them to have the same information in a usable form.2

Research on behavioral finance, along with the studies on anomalous behavior of markets, suggests that there are various periods when the market behaves in a manner that is uncharacteristic for an efficient market strictly following the EMH assumptions and suggests that there is no reason why expectations of agents cannot vary through time. It is more logical that, as new information is added to the information set or becomes dated, different groups of investors could have diverging expectations, which would lead to multiple expected equilibria returns. This would still maintain relative EMH but now with profit opportunities emerging in the market. This appears to be a reasonable explanation for the inconsistent interpretations and results represented in past studies.

The true underlying market structure of asset prices is still unknown. However, we do know that, for a period of time, it behaves according to the classical definition of an efficient market; then, for a period, it behaves in such a way that researchers are able to systematically find anomalies to the behavior expected of an efficient market. To answer the question as to what is the underlying economic mechanism that causes these asymmetries, we need to identify them in a data stream. It is during these periods that the market becomes ripe for speculation. For the technical trader, these periods may provide an opportunity to identify short‐lived predictable patterns and reap above‐normal returns. For example, Neely, Weller, and Dittmar (1997) find evidence that genetic programming can produce trading rules that identify greater than normal returns for certain time periods in the foreign exchange market (see also, e.g., Lee and Mathur 1996; and Neely and Weller 2003).

It is not the objective of this article to provide a theoretical reason as to why these periods may exist; rather, it is to develop a procedure that can be used to identify them. The procedure developed here incorporates the Enders and Granger (1998) asymmetric stationarity test, henceforth called the E‐G stationarity test. This test provides a tool not previously available to identify stationarity in series that are characterized by an asymmetric adjustment process, such as might be the case for asset markets. In this article, we develop a recursive procedure using the Enders and Granger (1998) stationarity test to identify periods of divergences of stock prices from their normal nonstationary paths.3 Using this procedure, we are able to identify periods of asymmetric stationarity in each of the stock indices of the Group of Seven industrialized countries (henceforth called G7 and consisting of Canada, France, Germany, Italy, Japan, the United Kingdom, and the United States).

The article is organized as follows. Section II describes the data and presents the asymmetric stationary identification procedure. Section III discusses the empirical results, and Section IV provides a moving average technical trading rule scheme to see if technical traders obtain different results in the asymmetric stationary portions of the indices. Section V offers some concluding remarks.

II.  Data and Methodology

All data are from the Dow Jones Global Indices Database.4 Data consist of the aggregate broad‐based national indices of daily closing prices for each of the G7 countries. All indices are calculated using local currency and include trading days from January 1, 1992, to June 12, 2003. Nontrading days are omitted from the data. The use of the G7 national indices provides a convenient grouping of industrial nations to apply the procedure that is developed in this study to identify stationary asymmetric areas in a data stream. It should not be construed here that we are investigating the G7 national indices as a panel data set. The procedure developed below is applied to each national index independently of the others, and no relation between the national indices is claimed here or investigated.

To identify stationarity in times series, researchers generally use conventional unit roots tests such as the Dickey and Fuller (1979) and the Phillips and Perron (1988) tests and their many extensions. These tests are useful for finding unit roots when the tested series exhibits a symmetric adjustment process, which is an assumption of these tests. However, if the series has an asymmetric adjustment process, then conventional unit root tests are not reliable due to the violation of the symmetric adjustment assumption. Enders and Granger (1998) address this shortcoming and provide a stationarity test that allows for an asymmetric adjustment process. The E‐G stationarity test is built on the threshold autoregression (TAR) model. TAR has the property that it splits the series into two groups based on a threshold variable. The model otherwise behaves in an autoregressive manner.5 The TAR model is very useful for explaining why certain data series do not follow an expected economic theory for periods. Using a threshold allows a series to have regimes with different statistical properties. A series for a time period may behave in accordance with economic theory in one regime, and the same series at different times and in another regime may behave in a manner not suggested by economic theory.6

Asset price series have a momentum property to them. Enders and Granger (1998) addressed this issue by looking at a momentum TAR (MTAR) model. MTAR varies from the TAR in the calculation of the function used to determine the threshold variable. Enders and Granger (1998) suggest the MTAR as an alternative to the TAR to allow for decay in the sequence under investigation to be dependent on some previous period’s change instead of the current period, as is the case for the TAR specification. Use of the MTAR specification captures the momentum observed in stock prices, which past studies have observed to move in one direction with a higher magnitude than would be expected of a symmetric adjustment process.7 We found the MTAR to better fit our data and will limit our discussion to the MTAR model. The model used in this article is defined in equation (1): where Δ refers to the first difference, α₀ is the mean of the series of daily closing prices, P_t, is the number of observations in the series. Trend is accounted for by ( ), and if the trend is determined to be insignificant, α₁ is set to 0. I_t is typically called a Heaviside indicator function8 and is equal to one if or equal to zero if , with for this study. One interpretation of the lag length is the relevant historical price that influences market participants’ behavior. By altering the lag length, different adjustment paths are followed.

Using the above model, we perform the E‐G stationarity tests on each index by looking at the data points in a recursive manner. First, we look at the first 50 data points in the index and test them to see if they exhibit stationarity with an asymmetric adjustment process (which we call asymmetric stationarity), stationarity with a symmetric adjustment process (which we call symmetric stationarity), or nonstationarity properties. Then, by adding the next observation from the index, we test that set of observations to see if they exhibit asymmetric stationarity, symmetric stationarity, or nonstationarity properties and continue in this recursive manner until the entire index is exhausted. For example, the first set of observations in the index that is tested for stationarity is data points 1–50 from the price index. The second set of observations tested is data points 1–51 of the price index. Subsequent tests are constructed by including the next data point from the price index in a sequential manner until all points are exhausted in the price index. For simplicity, we will call these sets of observations subseries of the overall index. We use the E‐G stationarity test to determine asymmetric stationarity and the augmented Dickey‐Fuller test to determine symmetric stationarity. In particular, if a subseries is found to be asymmetric stationary, a marker is recorded in a holding vector corresponding to the trading date of the last added data point to the tested portion of the index. A positive test for asymmetric stationarity, determined by both a rejection of the null hypothesis of nonstationarity and the rejection of null hypothesis of a symmetric adjustment process within the E‐G test, is described in detail below. If an E‐G test reveals a rejection of the null hypothesis for nonstationarity but does not reject the null hypothesis for a symmetric adjustment process, then the subseries is subjected to an augmented Dickey‐Fuller test to determine if the last data point in the subseries should be marked with a symmetric stationarity marker or a nonstationarity marker. The use of the augmented Dickey‐Fuller test addresses a weakness of the E‐G stationarity test when the adjustment process is actually symmetric for a stationary series. The following steps describe the details of the E‐G stationarity test and the recursive testing process used in this article.9 The end result of this procedure will produce an index that has asymmetric stationary, symmetric stationary, and nonstationary areas identified for each index.

Step 1. De‐trend and de‐mean the tested subseries by regressing the tested subseries on a constant or a constant plus trend (when a trend is present) and use the residual series, C_t, , with z equal to the number of data points in the tested subseries. Substitute C_t into equation (1) and set the Heaviside indicator function I_t equal to one if or equal to zero if . The lag length, n, in the Heaviside indicator function is determined by the Akaike Information Criterion using lag lengths from 1 to 25. The model specification is now given by equation (2):

Step 2. The sufficient condition for stationarity is given by . Tong (1983) shows that the least squares estimates of ρ₁ and ρ₂ have an asymptotic multivariate distribution provided the series is stationary.10 Check for this condition by estimating equation (2) by a least squares regression. Then test the null hypothesis by comparing the F‐statistic from the regression with the appropriate critical value from Enders and Granger (1998).11 If the null hypothesis is rejected (i.e., the test rejects nonstationarity), then proceed to step 3. If the null hypothesis is not rejected, then stop and consider the subseries as not being stationary.

Step 3. Test for symmetric versus asymmetric adjustment process by using the Wald coefficient test to see if . If the null hypothesis is rejected (indicating and implying that the test is not accepting a symmetric adjustment process) then proceed. If the null hypothesis is not rejected, then stop and perform an augmented Dickey‐Fuller test. If the null hypothesis of the augmented Dickey‐Fuller test is not rejected, then consider the set observations as nonstationary. If the null hypothesis of the augmented Dickey‐Fuller test is rejected, then consider the subseries as symmetric stationary.

Step 4. Perform diagnostic checking of the residuals to determine if a white noise process can reasonably well characterize the sequence. If the residuals are correlated, return to step 2 and reestimate the model in the form . The lag length, k, can be determined by performing diagnostic checks of the residuals by using a correlogram of the residuals and Ljung‐Box tests or the Akaike Information Criterion or Schwarz criterion (see Tong 1983). We find the number of lags, k, by relying on the Akaike Information Criterion.

Step 5. Keep track of the asymmetric stationarity, symmetric stationarity, and nonstationarity results by recording a marker corresponding to the last observation date of the tested subseries to indicate the particular result.

III.  Empirical Results

Table 1 reports descriptive statistics on each price index. It can be seen that all G7 indices are slightly skewed in one direction, with leptokurtic or platykurtic properties, and none of the series has a normal distribution. Additionally, conventional unit root tests are reported and indicate that the series are nonstationary. The indices have descriptive statistics similar to those seen in past studies on these or similar indices.

Table 1 Descriptive Sstatistics for the Closing Prices of the National Stock Indices for the Time Period from January 1, 1992, to June 12, 2003

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The procedure from Section II is applied to each of the G7 national indices. Figures 1–7 show periods of asymmetric stationarity, symmetric stationarity, and nonstationarity, as identified by the markers in step 5 from Section II, for each of the G7 national indices. The asymmetric stationary markers are plotted as the data points associated with actual closing prices in the particular index and are shown by the shaded areas under the closing price series. The indices have been arranged as a sequence of subseries. The subseries are named by a letter followed by a number. The letter identifies the subseries stationarity property with “A” indicating nonstationarity, “B” indicating asymmetric stationarity, and “C” indicating symmetric stationarity. The number that follows is a sequential number of that type of subseries found in the index ordered from the oldest trading days to the most recent. The first 49 observations for each index are always considered nonstationary by assumption since the identification process does not check the first 49 data points and is called A1. A1 is followed by an asymmetric stationary subseries, B1, or a symmetric stationary subseries called C1. Thus, each national index is represented as a sequence of subseries labeled A1, B1 or C1, A2 or B2 or C2, … , Am or Bn or Cs; m = the number of nonstationary subseries, n = the number of asymmetric stationary subseries, and s = the number of symmetric stationary subseries in the index.

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Fig. 1.— Canada’s National Stock Index

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Fig. 2.— France’s National Stock Index

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Fig. 3.— Germany’s National Stock Index

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Fig. 4.— Italy’s National Stock Index

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Fig. 5.— Japan’s National Stock Index

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Fig. 6.— United Kingdom’s National Stock Index

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Fig. 7.— United States’ National Stock Index

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We are particularly interested in the differences in volatility (defined below) and standard deviation of the closing prices between the asymmetric stationary areas and its adjacent areas. For analytical purposes each subseries is checked to ensure that it contains at least 10 data points. If a subseries does not, the data points are removed from consideration. By removing subseries that have fewer then 10 data points it is possible to have two subseries with the same stationarity property adjacent to each other. As such this sequence of subseries is then checked to see if there are two adjacent subseries that have the same stationarity property. If there are, then these two subseries are combined and if any of the previously removed data points were of the same stationarity type they are put back into the adjusted subseries. This could happen if, for example, we originally had a sequence of subseries such as A1, B1, A2, B2, A3, B3 and A2, B2, and B3 all had only two observations; the procedure here would still not consider the observations in A2 and A3 but would consider B1, B2, and B3 observations as an adjusted B1 subseries.12

Figures 1–7 provide a visual representation of where the asymmetric stationary areas are in each of the indices. They show that Japan has the most asymmetric trading days with the United Kingdom, Canada, Italy, United States, and Germany, in order of most to least asymmetric trading days, in the middle. France represents the index with the least asymmetric stationary trading days. It should also be noted that only Italy and the United States have areas in their indices that exhibit symmetric stationarity properties and account for a relatively few number of observations in these indices. Our focus will be on the difference between the asymmetric stationary subseries and their adjacent subseries.

Each of the subseries is compared for dissimilarities in standard deviation, median, mean, and volatility (defined below). Each subseries behaves in the expected manner with respect to being slightly skewed in one direction, with leptokurtic or platykurtic properties. Statistical testing for mean and median equality between asymmetric stationary subseries and the adjacent subseries is conducted but not reported since the expected results of nonequality are found and are not important considering the series being analyzed.13 Of greater interest are the equality comparisons of the measures of deviation.

Equality analyses of standard deviation and volatility are reported in table 2. Volatility is calculated by using the absolute value of the log differences in daily closing prices as the base measure.14 Equality tests are reported as the least significant result from the Bartlett, Levene, or Brown‐Forsythe variance tests. To facilitate our discussion, we define an area under consideration as the asymmetric stationary subseries Bn and its adjacent subseries and name it for the value of n. For example, area 1 in Canada’s index comprises its subseries A1, B1, and A2. Area 1 for Italy’s index comprises its subseries A1, B1, and C1. From table 2 it is seen that, for all but two of the tested areas, the asymmetric stationary subseries and their adjacent nonstationary subseries have statistically different standard deviations mostly at the 1% significance level. The United Kingdom’s area 1 and area 8 are the two exceptions, which are found to have equal standard deviations.

Table 2 Results for Inequality Tests of Standard Deviations and Volatility

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The volatility analyses show results similar to those for the standard deviations of the subseries in the indices. However, there are more mixed results that show equality in volatility between asymmetric stationary subseries and its adjacent subseries. For Canada’s index, the volatility is found to be different, in their respective areas, in areas 4, 5, 8, 9, 11, and 13 at the 1% significance level. Canada’s area 12 has different volatility at the 10% significance level and if the Brown‐Forsythe test is not considered, Canada’s area 7 has different volatility at the 10% significance level. Canada’s areas 1, 2, 3, 6, and 1015 do not have different volatility in their respective areas. France and Germany’s indices are both found to have different volatility in all their areas at the 1% significance level. Italy’s index areas 1, 2, 7, 8, and 9 have different volatility in their respective areas at the 1% significance level, with area 6 at the 10% level of significance. If the Levene test is not included, then Italy’s area 5 has different volatility at the 10% significance level. However, Italy’s areas 3, 4,16 and 10 do not exhibit different volatility in their respective areas. Japan’s index has volatility that is different in its respective areas for areas 1, 3, and 4 at the 1% significance level. If we do not consider the Brown‐Forsythe test, then Japan’s area 5 has different volatility at the 5% significance level. Japan’s area 2 reports equal volatility.17 The United Kingdom’s index has volatility that is different in its respective areas for areas 3 and 10 at the 1% significance level and areas 4 and 9 at the 10% and 5% significance levels, respectively. If we do not consider the Brown‐Forsythe test, then Japan’s areas 2 and 7 have different volatility in their respective areas at the 10% significance level. The United Kingdom’s areas 1, 5, and 818 have equal volatility in their respective areas. The United States’ index has volatility that is different in its respective areas for areas 5, 6, and 7 at the 10%, 1%, and 5% significance levels, respectively. The United States’ areas 1, 2, 3, and 4 have equal volatility in their respective areas.19 The results of the equality tests discussed in this section provide fairly strong evidence that the identified asymmetric stationary areas and its adjacent areas are consistently statistically different.

IV.  Technical Trading Rule Returns in the Asymmetric Stationary Subseries

To provide a practical feel for the differences in the subseries within a particular index, moving average technical trading rules are used to demonstrate potential differences in the subseries’ returns. We do not claim that this is statistical proof but rather complementary evidence that is of interest to money managers, traders, and researchers. In this exercise, we use moving average trading rules, which are one of the simplest and most widely used technical trading schemes. We assume that a position is held every trading day and that there are no transaction costs. Of interest is to see if the identified areas of asymmetric stationary periods generate significantly different results when a technical trading rule is applied and not necessarily if an actual profit can be made.

The moving average rule scheme relies on a signal being generated from past price histories. That signal is then used to forecast which position should be taken. The signal to indicate a long (or buy) position is found as: if the mean of of or zero otherwise, where P = closing price, S = number of days in the short‐term moving average, and L = number of days in the long‐term moving average. This decision rule states that a long position is maintained if the short‐term moving average is greater than or equal to the long‐term moving average. If this condition is not met, then a short position is held. The moving average returns are calculated for each trading day as (see Lee and Mathur 1996; Szakmary and Mathur 1997). This calculation accounts for both the long and the short positions.

We select some of the more common moving average rules used by practicing technical traders. This is similar to the rule selection process used by Brock, Lakonishok, and LeBaron (1992). In this example, we arbitrarily use 2–9 days for the short moving average and 10, 15, 25, 50, and 75 days for the long moving average. To forecast the out‐of‐sample returns for each index we use the initial 100 data points to find the best values to use for the number of days in the long‐term and short‐term moving average by calculating MA_Returns for each combination of short‐term and long‐term moving averages to see which one gives the highest average return. We use this information and apply the trading rule to find our long position and short position signals, which are used to forecast our position held for the next 250 data points. Then the last 100 data points of the previous forecasted points are used to find the best values to use for the number of days in the long‐term and short‐term moving average and are used with the trading rule to find our long position and short position signals, which are used to forecast our position held for the next 250 data points. The process is continued until all the data are exhausted. For example, suppose we are forecasting which position to take for trading day 200. We would be using the short‐term and long‐term moving average lengths determined by the first 100 data points. Then our signal for the 200th trading day would be determined by the rule that for the 199th trading day a long position will be signaled for the 200th trading day if the short‐term moving average is greater than or equal to the long‐term moving average. If this condition is not met, then a short position is held for the 200th day.

For analyzing the returns, table 3 reports the excess daily returns that are calculated as MA_Returns, the average daily return for the period January 1, 1992, through June 12, 2003, using a buy and hold strategy, e^{f(BP, LP, n)}, where n = the number of trading days in the series, BP = the beginning day’s closing price, LP = the last day’s closing price, and f(BP, LP, n) = (natural log of LP)/N − (natural log of BP)/n. This calculation accounts for both the long and the short positions. We also calculate just the long position trading days excess returns as and the short position trading days’ excess returns as . Annualized returns equal daily means multiplied by 250 in percentage form. Results for inequality tests of standard deviations are reported as the least significant result using Bartlett, Levene, Brown‐Forsythe, F‐test, and Siegel‐Tukey tests, unless otherwise noted. Note that excess returns are calculated after the forecasting procedure, and prior knowledge of the final data point in each index is not used in the forecasting procedure.

Table 3 Daily and Annualized Daily Excess Returns from the Best Moving Average Trading Rule

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All discussion of difference is with reference to the combined results of the asymmetric stationary subseries and the combined results of the nonstationary subseries of a particular index. Each index is identified by its country’s name. For the combined long and short positions, the Canadian, German, Italian, and United Kingdom indices generate statistically different excess daily return means at the 10% level of significance or better. For the short position only, Canadian, German, and United Kingdom indices generate statistically different excess daily return means at the 10% level of significance or better. Tests for differences in standard deviations show differences for all indices for the combined position, long position only, and short position only at the 5% level of significance or better with the exception of Japan, which did not reveal any differences in standard deviation.

At first glance these results may not seem too impressive. However, here we are not using the earlier finding that showed that we have different regimes in the data stream. In particular, we find nonstationary periods, asymmetric stationary periods, and for Italy and the United States we also find symmetric stationary periods. In order to use this information, a money manager, technical trader, or researcher would devise a way to capture the information of the regime changes in the trading rule. Here, we are not proposing a trading rule to do this, but we do note that one possible way would be to recognize that the best number of test days used to determine the signals in the trading rule and the number of days in which those signals are applied to may not be the same in all regimes. If one knew, for example, that one was in an asymmetric stationary area of the data stream for the day prior or some number of days prior, one could incorporate into the trading rule the use of a different number of test days and/or forecast days to use, in accordance with the current regime. For example, if we look at the United States and test to see which set of rules would produce the greatest excess profits over the range of 70–120 test days and 230–70 forecast days, we find that the nonstationary areas perform best with 89 test days and 270 forecast days. The asymmetric stationary areas perform best with 86 test days used to establish the rules for the next 253 forecast days. The symmetric stationary areas perform best with 74 test days to establish the rules for the next 232 forecast days. The mean excess daily returns for these areas, given these alternate rules, are 0.001891 for the nonstationary areas, 0.001311 for the asymmetric stationary areas, and 0.001175 for the symmetric stationary areas. This gives a combined improvement of 22.3% in excess daily returns over the rules used to produce table 3.20 Given the speed of today’s computers, it seems reasonable that a moving average rule could be developed to identify a sizable gain for the money manager or trader who incorporates this regime information into the decision process.

The purpose of this exercise is to show how different results may be observed when an index is behaving in an asymmetric stationary manner as opposed to a symmetric stationary or a nonstationary manner and how this information may be of use to a manager. Results that are more robust would be expected if a more sophisticated trading rule was established that uses the regime information found in the series. However, for purposes of this article, we have been able to obtain results generally supporting the proposition that differences exist for the three types of areas found in each of the indices. We further sketch out a feasible way to incorporate this information into a usable rule‐generating function. Given the results here, further research on how to better use this information is promising.

V.  Conclusion

In this article, we develop a procedure using an MTAR model and E‐G stationary tests to find asymmetric stationary periods in an otherwise nonstationary path of each of the G7 national stock indices. Each G7 index reveals the existence of asymmetric stationary periods. Each of the asymmetric subseries and their adjacent nonstationary subseries are shown to have statistically different properties. We further provide anecdotal evidence of the ability of simple technical trading rules to generate significantly different results between the asymmetric stationary trading days and the nonstationary trading days of a particular index. The existence of these asymmetric stationary periods suggests that market information inefficiencies, or asymmetric motives of groups of investors, or management manipulation could exist and are being seen in the data as periods of asymmetric stationary paths.

By using this procedure, future research could focus on these asymmetric stationary periods and explore the possibilities of detecting periods when a market is diverging from an efficient state. This may be a signal that the motivations of corporate managers and efficient production are diverging and that a company’s stock prices are reacting to noneconomic forces.21 In particular, when corporate managers allocate resources to accommodate the market’s short‐term objectives, the most likely outcome for economic efficiency will be poor resource allocation decisions, which will lead to lower productivity and lower economic growth. One vehicle used by corporate management to manipulate the market is the earnings statement. It is found that earnings statements are widely used as indicators for investors to invest in a company22 and for corporate management to be evaluated.23 The importance of earnings statements makes it highly lucrative for corporate management to tailor the timing of expenditures on research and development, capital investment, and other resource decisions according to how the earnings statements will meet the corporate manager’s earnings threshold levels. Degeorge, Patel, and Zeckhauser (1999) provide strong empirical evidence that corporate managers do manipulate earnings statements to meet expectations of three thresholds that show the corporation first has positive profits, has sustained recent performance, and has met analysts’ expectations.24 In doing so, a corporation may alter its production efficiency.

The economic importance is illustrated by the way corporate executives may alter resources by manipulating timings of expenditures and receipts in order to report earnings according to thresholds beneficial to corporate management, a practice referred to as “earnings management.” Lev (2003) provides a good discussion of earnings management and reaches a conclusion that this practice is often difficult to detect, is a concern, and may have substantial adverse social consequences. If earnings management is significant, it is highly unlikely that this practice would reflect characteristic traits in the stock market as described by the textbook explanation of the efficient market. By using the procedure developed in this article, future research work could incorporate these asymmetric stationary days as dummy variables in productivity models to see if there are any direct effects on the economy at large and then investigate the relationship between these asymmetric stationary days and the known cases of corporate management manipulations.

Another use of this procedure is to provide avenues for new forecasting procedures. For example, if a money manager or trader uses asymmetric stationarity information in formulating a trading rule, it is reasonable to expect a superior rule when the series is in a period of asymmetric stationarity. Of interest would be to see if sustained higher excess returns are possible when information on which regimes the market may be currently operating is incorporated in the manager’s decision process. This technique need not be restricted to stock indices but could be applied to any asset price series.