Does beta react to market conditions? Estimates of ‘bull’ and ‘bear’ betas using a nonlinear market model with an endogenous threshold parameter

The authors use a logistic smooth transition market (LSTM) model to investigate whether ‘bull’ and ‘bear’ market betas for Australian industry portfolios returns differ. The LSTM model allows the data to determine a threshold parameter that differentiates between ‘bull’ and ‘bear’ states, and it also allows for smooth transition between these two states. Their results indicate that ‘bull’ and ‘bear’ betas are significantly different for most industries, and that up-market risk is not always lower than down-market risk. LSTM models indicate that the transition between ‘bull’ and ‘bear’ states is abrupt, supporting a dual-beta market modelling framework.


Introduction
The simple linear market model has long been used in tests of the Capital Asset Pricing Model (CAPM), for the measurement of abnormal returns in event studies, and as a benchmark for the performance of mutual funds. See Sharpe (1966), Fama et al. (1969) and Fama and French (1992) for some examples. In such studies, the stability of the beta coefficient in the market model over 'bull' and 'bear' market conditions is assumed, but if beta does in fact vary with market conditions then inferences based on single beta models are likely to be misleading.y Theory that allows for beta to vary over market conditions is well advanced, with work by Bawa and Lindenberg (1977) and Harlow and Rao (1989) providing some important early developments. Direct evidence of the importance of the beta/market condition relationship issue is given by the fact that investment houses regularly publish separate alphas and betas over 'bull' and 'bear' markets, for a range of securities, to offer differing levels of upside potential and downside risk.
Many studies have investigated the relationship between beta risk and stock market conditions. These include studies of individual securities (Fabozzi and Francis 1977, Kim and Zumwalt 1979, and Clinebell et al. 1993, mutual funds Francis 1979, andKao et al. 1998), size based portfolios (Wiggins 1992, Bhardwaj and Brooks 1993, and Howton and Peterson 1998, risk based portfolios (Spiceland andTrapnell 1983, andWiggins 1992) and past performance based portfolios (DeBondt andThaler 1987, andWiggins 1992). While most of these studies have found evidence that beta varies with market conditions, this evidence is mixed and quite weak. Furthermore, most of these studies have used the dual-beta market (DBM) model and simple t-and F-testing methods in conjunction with crude 'up'-and 'down'-market definitions of 'bull' and 'bear' markets to investigate this phenomenon, and they have not considered more general forms of nonlinear behaviour.
There has been substantial divergence in the definition of 'bull' and 'bear' markets in the literature, but only a few studies of beta non-stationarity over 'bull' and 'bear' markets have used a continuously changing timevarying parameter model. An example of a continuously changing beta model is provided by Chen (1982). However, even with considerable refinements in the definition of 'bull' and 'bear' markets, especially in work that uses trend-based indicators of market states, almost all of the existing definitions model the transition from down-market to up-market states (or vice versa) as a discrete jump. Even the latest Markov-switching model by Maheu and McCurdy (2000) assumes an abrupt switch between regimes.
Complementing the existing evidence of two-regime market models is evidence of other sorts of nonlinearities in stock prices. These nonlinearities have been related to various behavioural dynamics of investors, with prominent work along these lines including (i) papers by Peterson (1994) and Guillaume et al. (1995) on investor heterogeneity arising from different risk profiles and different investment horizons; (ii) work on herd behaviour by Lux (1995); and (iii) papers on heterogeneous beliefs regarding market conditions by Brock and Hommes (1998) and Brock and LeBaron (1998). While homogeneous beliefs among investors will imply that they share the same information and that they collectively switch from assuming one market condition (say a 'bull' market) to another (say a 'bear' market), this is quite hard to accept unless we believe in a strong form of efficient market theory. Heterogeneous beliefs among investors are likely to blur the distinction between 'bull' and 'bear' markets, because different investors will interpret their information in different ways.
In this paper we investigate these phenomena with three main aims in mind. First, like many other studies, we wish to determine whether 'bull' and 'bear' market betas differ. Second, in contrast to other studies, we want to investigate the possibility that the transition between market regimes might be gradual, and thereby address the heterogeneous beliefs theory. Third, we want to let the data determine a benchmark criterion for differentiating between market states, allowing for the possibility raised in Harlow and Rao (1989) that such a benchmark might depend on 'investor targets', and might not be the same for all stocks. With these aims in mind, we apply a logistic smooth transition market model (LSTM) to a sample of returns on Australian industry portfolios over the period .y While the dual-beta market (DBM) model used in other studies implies a discrete jump between market states, the LSTM model replaces the indicator function with a logistic smooth function that allows for smooth and continuous transition between these two states. LSTM models are an adaptation of LSTAR (logistic smooth transition autoregressive) models popularized by Tera¨svirta (1994), and in this context smooth transition between states seems particularly appropriate for modelling stock markets with many participants, each switching at different times due to heterogeneous beliefs, different investment targets and differing investment horizons. The LSTM formulation allows the transition between market states to depend on a 'transition variable' that can be chosen by the researcher, and it parametrizes the threshold between states so that this threshold can be estimated. The LSTM model also allows for both the DBM and constant risk models as special cases.
The choice of an appropriate transition or indicator variable is an important consideration when modelling the transition between different market states, because ideally this variable will reflect broad swings in the market, rather than temporary market aberrations. Previous work on 'bull' and 'bear' betas has often failed to recognize this consideration, and has simply used the return on the market index as the market indicator. As shown below, the return on the market index is highly erratic, and does not reflect typical notions of 'bull' and 'bear' states. To overcome this problem, our study uses a rolling twelve-month moving average of market returns to model movement between 'bull' and 'bear' markets. The rolling average series is much smoother than the return on the market portfolio series itself, and it abstracts from noisy and unsystematic movements in the stock market to better capture long-run dependencies and trends in the data.
Our estimates of LSTM models exhibit persuasive evidence that Australian industry portfolio betas vary between 'bull' and 'bear' phases. This result seems to be partly attributable to our use of a trend-based market indicator, but it is also due to the fact that we allow the data to define 'bull' and 'bear' states. We find that although industries tend to spend more time in 'bull' than 'bear' states, the risk associated with 'bull' states is not always smaller than the risk in 'bear' market states. Our LSTM estimates indicate that the transition between 'bull' and 'bear' states is abrupt for most industries, which can be viewed as support for homogeneous investors.
The plan of the paper is as follows. In Section 2 we review the literature on definitions of 'bull' and 'bear' markets and describe a market indicator that will be used in this study to define 'bull' and 'bear' states. In Section 3 we develop our model and describe the methodologies employed in the study. Section 4 discusses the data and the results of our analysis, and Section 5 finishes with some concluding remarks.

Phases of the market
The studies reviewed in Section 1 either compare the market index to a critical threshold value to separate 'up'from 'down'-market periods, or they use a trend-based yWe choose to analyse industry portfolios because the existence of industry-specific risk is recognized, and because one can be more confident of the response of a portfolio beta to changes in market conditions than in the case of a single security beta. scheme to classify markets as 'bull' or 'bear'. The 'up'and 'down'-market scheme dichotomizes the market by comparing the market index to a critical threshold value. Wiggins (1992), for example, defines up (down) months as months when the (excess) market return is greater (less) than zero, while Bhardwaj and Brooks (1993) use the median return on the market portfolio as the demarcating value. Francis (1977, 1979) define substantial up (down) months as months in which the return on the market portfolio is greater (less) than 1.5 times its standard deviation, and thereby separate the market into periods when the market is substantially up, substantially down, or neither. Another way of classifying the market is offered by Granger and Silvapulle (2001), who separate the market into 'bullish', 'bearish' and 'usual' states using quantiles of return distributions. Taleb (2004, pp. 95-97) notes that 'bullish' and 'bearish' characterizations of the market typically incorporate expectations of returns and actions such as whether investors choose to go long or short in a stock, but he also points out that the terms 'bullish' and 'bearish' are commonly used without taking into account all of the relevant features of return distributions.
Several economists (e.g. Neftci 1984 andSkalin andTera¨svirta 2002) have suggested that monthly observations on changes in economic time series are noisy and therefore do not reveal cyclical characteristics. Recognizing this, some authors have used a trend-based approach in their analysis of stock market conditions. Francis (1977, 1979), for example, use 'bull' and 'bear' market dates published in Cohen et al. (1987) to classify their data into 'bull' and 'bear' categories, and, in a similar vein, Gooding and O'Malley (1977) define non-overlapping 'bull' and 'bear' phases based on major peaks and troughs found in the S&P425 Industrial Index. Dukes et al. (1987) used the S&P500 Index to define 'bull' ('bear') markets as periods in which the index increased (decreased) by at least 20% from a trough (peak) to a peak (trough). More noteworthy are the recent studies by Pagan and Sossounov (2003) and Lunde and Timmermann (2004), who each developed sophisticated trend-based definitions of 'bull' and 'bear' markets that focus on systematic movements in the market while ignoring the short-term noise effects. Both papers define 'bull' and 'bear' markets in terms of movements between peaks and troughs, and use pattern recognition dating algorithms to classify 'bull' and 'bear' markets. Both papers find that 'bull' markets tend to last longer than 'bear' markets.
We also use a trend-based definition of 'bull' and 'bear' markets in our analysis, and capture the underlying cyclical movement in the stock market by using the 12-month moving average of the (logarithmic) returns associated with the All Ordinaries Accumulation index as a stock market indicator. Like Pagan and Sossounov (2003) and Lunde and Timmermann (2004), we intend to capture sustained periods of growth or contraction that are normally associated with the concepts of 'bull' and 'bear' markets. Figures 1 and 2 illustrate the rationale behind our approach. The plot of returns for the market index (R mt Þ in figure 1 reveals that, by using this noisy series as an indicator of 'bull' and 'bear' states, most researchers have implicitly assumed that the market jumps in and out of market phases very rapidly, and very frequently. On the other hand, our use of R Ã t , the smoother 12-month moving average of this variable illustrated in figure 2 ensures that our market indicator follows a much smoother path.

The Logistic Smooth Transition Market Model (LSTM)
An unconditional beta for any asset or portfolio can be estimated using the constant risk market model (CRM) regression given by where R it is the return on portfolio i for period t, R mt is the return on the market index for period t, i ¼ covðR it , R mt Þ= 2 mt and " it is the disturbance term which has zero mean and is assumed to be serially independent and homoskedastic. Under this specification i and i are constant with respect to time.  A dual-beta market model (DBM) can be specified as where D t is a dummy variable defining up and down markets (i.e. 'bull' and 'bear' markets) by taking the value one if a market state indicator denoted by M t exceeds some critical value c, and zero otherwise. The market return R mt is often used as the market indicator M t , and the parameter c is often set equal to zero, R mt or the median of R mt : It is also common to omit the U i Â D t term in (2), thereby assuming that the intercept does not vary with the market state.
In this paper we set M t ¼ R Ã t , for reasons discussed above. Since there is no theory with which to specify c and no reason to believe that c will be the same for all industries, we estimate a separate value (c i ) for each industry. Thus, for our DBM models, the market for stocks in industry i will be classified as being in a 'bear' state when R Ã t 5c i and in a 'bull' state when R Ã t 4c i : Notice that in this specification the value of U i measures the difference between the up-and down-market values of the slope coefficient, so that the up-market value of beta is given by i þ U i : Now consider the logistic smooth transition regression model, henceforth called the LSTM model, which is given by and which has (1) and (2) as special limiting cases. The superscript U signifies up-market differential values of the parameters and , F is the logistic smooth transition function with transition variable M t and threshold value c i , and " it $ i:i:d. N(0, 2 i Þ. In our case M t ¼ R Ã t , the 12-month moving average of the return on the market index, because this variable seems to capture the cyclical upturns and downturns that are typically associated with concepts of 'bull' and 'bear' markets. As for our DBM models, we allow for (and estimate) a separate value of c for each industry.
Equation (3) with (4) and (4) is a smooth and continuously increasing function of R Ã t : The transition function FðR Ã t Þ takes a value between 0 and 1, depending on the magnitude of ðR Ã t À c i Þ. When ðR Ã t À c i Þ is large and negative, FðR Ã t Þ % 0 and R it is effectively generated by the linear model R it ¼ i þ i R mt þ " it . In such cases the market for stocks in industry i is very 'bearish'. On the other hand, when ðR Ã t À c i Þ is large and positive, R it is effectively generated by it , and the market for stocks in industry i is very 'bullish'. Intermediate values of ðR Ã t À c i Þ give a convex combination of the two extreme regimes, and as FðR Ã t À c i Þ increases from 0 to 1, the market for industry i moves through a series of market states that range from very 'bearish' to very 'bullish'. The i parameters determine the smoothness of transition between 'bearish' and 'bullish' market states. Note that the DBM is a special case of the LSTM, because FðR Ã t Þ becomes an indicator function when i approaches infinity in (4), with FðR Ã t Þ ¼ 0 for all values of R Ã t smaller than c i and FðR Ã t Þ ¼ 0 otherwise. Also notice that the constant risk market (CRM) model is a special case of the LSTM, because (3) approaches (1) as the smoothness parameter i in (4) approaches zero.
Our LSTM model is a simple adaptation of the logistic smooth transition autoregressive (LSTAR) models promoted by Tera¨svirta (1994), and tests for smooth transition in our framework are very closely related to tests for LSTAR behaviour. To our knowledge, LSTM models have not been used to study 'bull' and 'bear' markets before, although Coutts et al. (1997) used an adaptation of an LSTAR model with a polynomial trend as a transition variable, in an attempt to capture the timing of changes in beta in response to major events.

Tests of linearity against LSTM
As mentioned in Section 3.1, (3) becomes the CRM model when i approaches zero, implying that the constant risk market model is nested within the LSTM model. Thus a natural first step in specifying the model is to test for linearity against the LSTM form. If the null of linearity cannot be rejected then we can conclude that the constant risk market model adequately represents the data generating process. On the other hand, if linearity is rejected we can go on to estimate the nonlinear LSTM model using nonlinear least squares (NLS).
Tests of H 0 : i ¼ 0 are non-standard, since the parameters of (3) are only identified under the alternative H A : i 6 ¼ 0: Following Luukkonen et al. (1988) we replace FðM t Þ in (3) by either a third-or first-order Taylor series linear approximation to FðR Ã t Þ, and expand this to form an auxiliary model in which the alternative H A : i 6 ¼ 0 is equivalent to H A : U i 6 ¼ 0 in equation (3). When a thirdorder Taylor series approximation is used, the expanded and reparametrized equation is with the last six variables in this equation acting as proxies for the nonlinearity. Within this reparametrized setting, the null hypothesis of linearity is H 0 : j ¼ 0 ( j ¼ 3, . . . , 8Þ: A standard Wald test can be used to test this hypothesis (for each i), and the test statistic (called S 3 in what follows) has an F 6, TÀ8 distribution under the null. When a first-order Taylor series is used the expanded and reparametrized equation is The null hypothesis is then H 0 : j ¼ 0 ð j ¼ 3, 4Þ, and the Wald test statistic (denoted by S 1 Þ has an F 2, TÀ4 under the null. When the intercept is not time varying, an augmented first-order test statistic, S Ã 1 can be used. This test includes a fifth regressor, 5 R mt ðR Ã t Þ 3 in (6), and tests the three tests S 3 , S Ã 1 and S 1 as Lagrange Multiplier statistics that have asymptotic 2 distributions, but they point out that the F versions (presented here) perform better in small samples. All of these tests can be adjusted to account for potential heteroskedasticity, and we use White's (1980) adjustments in all that follows.
Simulations undertaken by Luukkonen et al. (1988) and Petruccelli (1990) show that the S 3 , S Ã 1 and S 1 tests are quite powerful in small samples when the true alternative is either a smooth transition or an abrupt regime switching regression. They also show that these tests outperform Tsay's (1989) test for threshold behaviour, even when the data are generated by a threshold process. Thus we expect that the S 3 , S Ã 1 and S 1 tests will have power against our DBM and LSTM models.y Generally, the tests with more regressors (i.e. S 3 tests) are more powerful because they search for non-linearity in more directions, but S 1 tests can become more powerful, when fitting in directions where the nonlinearity is weak interferes with efficiency. The S Ã 1 test is more efficient than the S 3 test when the intercept does not vary, because the latter includes ðR Ã t Þ 2 and ðR Ã t Þ 3 terms, which only appear in the Taylor's series expansion (4) when U i 6 ¼ 0. We perform the S 3 and S Ã 1 tests in the analysis that follows, omitting the S 1 tests because they are similar to the S Ã 1 tests, but not as powerful. Finding evidence of nonlinearity before embarking on the estimation of DBM and LSTM models is important, not only because such evidence justifies the nonlinear form, but also because a failure to reject H 0 : i ¼ 0 provides a warning that the separate identification of parameters for two regimes might not be possible.
Nonlinear least squares (NLS) will provide consistent estimates of the parameters in LSTM and DBM models when the errors follow a martingale difference process, and our assumptions on " it satisfy this requirement. Given normality, NLS is equivalent to MLE in the LSTM case, but this does not hold in the special case of the DBM model, because an abrupt threshold violates the continuity and differentiability conditions needed for MLE. Estimation of the LSTM models is difficult, because the error sum of squares (or likelihood) is often quite flat with respect to i and c i (see Tera¨svirta 1994 for a discussion), but all parameter estimates will have normal distributions, provided that i 6 ¼ 0. The standard NLS technique for estimating a threshold model is to use sequential conditional least squares (SCLS). This involves estimating i , U i , i and U i conditionally for each value of c i as OLS estimates given by where i and zero otherwise. A grid search over the set of potential values for c i (i.e. C i ) is conducted to determine the value ofĉ i which minimizes the residual variance b 2 ðc i Þ, and the final estimates of the parameters are then given by Chan (1993) has demonstrated that the estimator

Results
The data used in this study is adjusted price data ( Figure 1 provides a graph of returns for the market index, and table 1 provides descriptive statistics for the market returns and the returns for each of the 24 industries' indices. The most noticeable feature of the illustrated market index is the sharp dip in October 1987, which was due to the crash in the United States stock market at that time. Separate graphs of individual industry returns show the same feature, although it is more pronounced in some industries (for instance 'banks' and 'insurance' and 'investment & financial services') than it is in others (such as 'gold index'). Turning to the summary statistics, the 'media' industry offered the highest return over this period and 'miscellaneous industrials' the lowest. The standard deviation was highest for 'diversified industrials' and lowest for the 'property trust' industry. In keeping with other studies of financial time series, all return series are leptokurtotic and exhibit negative skewness. Jarque-Bera tests indicate that none of the return series is normally distributed. The estimate of the constant risk market model beta is highest for the 'gold' industry and lowest for the 'property trust' industry, and all betas are statistically significant at the 1% significance level. Fifteen of the betas are statistically different from unity at the 5% level of yOur own simulation experiments, based on 'realistic' parameter values that had been calibrated to reflect our data set, support the published results. zNote that since we have only a finite number of observations on R Ã t , we will only be able to estimate interval in which c i lies with respect to the set of R Ã t (ordered by size). Superconsistency will, however, ensure that the probability of choosing the correct interval approaches 1 very quickly as the sample becomes larger, so that we can expect an accurate choice, given our sample of 253.
significance. Summary statistics for our proposed transition variable are provided on the last line of table 1. The most notable feature here is that the standard deviation of R Ã t is much lower than that for any of the individual indices, and in particular, it is only about one fifth of that for the market index. The small variation in R Ã t relative to that in the market index R t becomes particularly clear, once one compares figures 1 and 2 and notes that these two graphs are on different scales. The cyclical properties of R Ã t have been noted above. We provide evidence of nonlinearity in table 2, using R Ã t as the transition variable in the tests. Since R Ã t exhibits cyclical properties, evidence of nonlinearity due to movements in R Ã t provides support for a type of time variation in beta that is consistent with 'bear' and 'bull' betas. Given that the 1987 observation associated with the stock market crash can potentially influence the test results, we conduct two sets of tests. The first set consists of (heteroskedasticity adjusted) S 3 , S Ã 1 and S 1 tests as described in Section 2 above, while the second set of tests (also heteroskedasticity corrected) includes a 'stock market crash' dummy variable (SMC t ) in all of the test regressions, so as to remove the effect of this outlying observation. The dummy is statistically significant in 13 out of the 24 test regressions. While the two sets of p-values for the tests show some differences, the qualitative conclusions are quite similar. Nevertheless, the results that explicitly account for the outlier are likely to be more reliable, so we place more weight on these. For both sets of tests, the null of linearity is rejected by the S 3 (or S S 3 ) tests for 11 industries at the 5% level and for 12 industries at the 10% level. These tests suggest time-varying betas for approximately half of the series. Joint tests based on systems of index returns (see the last two lines of table 2) provide overall support for differences between 'bull' and 'bear' betas,y and justify our subsequent estimation of nonlinear models. Analogous tests that use R t rather than R Ã t as the transition find stronger evidence of nonlinearity when SMC t is excluded from the test regressions, but much less evidence of nonlinearity once SMC t has accounted for the 1987 stock market crash. These test results are not reported here, but are available on request.
Given the evidence of nonlinearity in table 2, we started to model it by assuming an LSTM specification and a gradual transition between the 'bear' and 'bull' regimes. We imposed the simplifying restriction that alpha was constant over the two market phases, but allowed for the possible effects of the stock market crash by including the SMC t dummy in the lower ('bear') regime. As suggested by Tera¨svirta (1994) we standardized the transition variable to simplify the joint estimation of and c, but The symbols N i ; i ; i ; s i ; i and b i represent the sample size, mean, standard deviation, skewness, kurtosis and estimated beta for industry i. The first eleven observations of each index were trimmed to allow for the construction of R Ã t that was used as the transition variable in the subsequent analysis. The (heteroskedasicity consistent) p-value associated with each beta is less than 0.0000, and a star superscript indicates that i 6 ¼ 1 at the 5% level of significance.
yOur joint tests are based on a single regression of the (vertically stacked T Â k) vector of returns for k industries on I k Z, where I k is an identity matrix of dimension k, and Z contains the set of test regressors. despite this standardization the error sum of squares (ESS) was very flat with respect to , leading to very large and imprecise estimates of . We found that i 510 for only two industries ('miscellaneous industrials' and 'oil and gas'), which suggested that most industries experience rather abrupt changes between 'bear' and 'bull' states. In the two cases in which i 510, we needed to restrict the c i to be no smaller than À0:015, to ensure enough observations in the lower regime to identify the lower regime parameters. For the other industries, our NLS algorithm found i to take values as high as 12 000, implying that these LSTM models were effectively DBM models. Given that f1 þ exp½ÀðR Ã t À cÞg À1 is effectively constant with respect to once becomes large, it is probably more accurate to view a large b as one that has satisfied a particular set of convergence criteria rather than an estimate of itself. This situation is discussed by Tera¨svirta, who notes that for large , the estimation of the other parameters is unaffected by conditioning on : In his work, he simply conditions on ¼ 50 or ¼ 100: Here, to provide readers with an idea of how soon the ESS minimization algorithm starts to treat the model as if it were a DBM, we successively estimate our LSTM models by conditioning on a grid of integer values for each i , and then we report the lowest i that minimizes the error sum of squares measured to four decimal places.
The estimated LSTM models are reported in table 3. The transition parameters are high, providing support for a homogeneous beliefs story, in which investors share the same information and collectively differentiate between 'bull' and 'bear' market states. Accompanying these high values of i are huge t-statistics that are associated with the estimation of the centrality parameter c i . These high t-statistics reflect the 'super-consistency' property associated with estimating the threshold in DBM models, because the reported LSTM models are effectively DBM models and therefore have the same properties. The reported t-statistics for c i are not useful for inference, but they reflect accurate estimation of this parameter. Consistent with Harlow and Rao's (1989) suggestion that investor targets do not just depend on parameters associated with the distribution of market or risk-free returns, we observe that there is no apparent common value for this parameter across different industries. Our empirical evidence shows that 'blanket' decisions to set c i equal to zero or c i equal to the mean of the transition variable for all industries are not necessarily appropriate, and perhaps it explains why most previous research on DBMs has failed to find systematic support for such models. It is noteworthy that more than a third of the estimated centrality parameters are negative, so that for these industries R Ã t needs to be very low before the relevant stock shows evidence of 'bearish' behaviour. Such industries include 'gold', 'building' and 'oil and gas', while other industries such as 'alcohol and tobacco' or 'chemicals' only become 'bullish' when R Ã t has become relatively high.
The up-market differentials in industry-specific betas are recorded in the U column, and 15 of these are statistically significant at the 5% level and another six are are versions of the S 3 and S Ã 1 tests that have been corrected for possible effects of the stock market crash in October 1987 by including a stock market crash dummy (SMC t ) in the test regressions. An SMC superscript on the industry code indicates those industries for which this dummy is statistically significant (at the 5% level). The last two lines contain (joint) tests of the null of linearity in the 19 (3) industries for which 253 (191) observations are available. statistically significant at the 10% level. Our LSTAR models therefore provide widespread support for timevarying betas, with variation linked to movements in the business cycle. Compared with Faff's (2001) work on Australian industry stock portfolios, we provide much stronger evidence of differences between 'bull' and 'bear' betas, and we believe that this is largely attributable to our use of a different regime switching mechanism. Of the statistically significant b U s, 11 are negative, consistent with literature that claims that risk in up markets is lower than risk in down markets. However, we also observe ten industries in which up-market betas are higher, so that overall we cannot conclude that down-market betas are systematically larger than up-market betas (or vice versa).y Industries for which down-market betas are bigger than up-market betas include all five in the 'resource & mining' sectors (i.e. industries in table 1 with codes XGO, XOM, XDR, XSF and XOG). Faff (2001) found that out of all 24 sectors, these five sectors provided the weakest support for a dual-beta CAPM, but here the support is quite clear. Industries in which upmarket betas are bigger than down-market betas include 'transport', 'insurance' and 'investment & financial services'. We note that such industries can offer upside potential with minimal downside risk.
Nearly all of the LSTM models in table 3 are essentially DBM models, so we next consider the direct estimation of DBM specifications. This is relatively straightforward and simply involves using a grid of potential thresholds. For each of these potential thresholds we apply OLS to the two subsamples to obtain a composite ESS, and then we choose the threshold and associated model by minimizing over these composite ESSs. We determine our grid of potential thresholds by considering the ordered values of R Ã t in our sample, using successively larger values of R Ã t to subdivide the sample into two.z Figure 3 shows how the recursive ESS changes with the choice of different thresholds. We illustrate the case for the We do not report t-statistics for , since this parameter was determined via a grid-search. ESS is the minimized error sum of squares. (B) under an estimate for c indicates that c was restricted to be no less than À0.015. yOver all industries, the average down-market beta is approximately 1 (as might be expected), and the average up-market differential is about zero (as might also be expected). zWe considered only those R Ã t that implied at least 15 observations in each regime.
'developers & contractors' industry, which had not shown evidence of nonlinearity in table 2. The plot shows that despite the lack of evidence provided by the nonlinearity tests, the point where ESS is minimized is nevertheless well defined. Analogous plots for other industries that were characterized by stronger evidence of nonlinearity in table 2 typically showed sharper and more dramatic downward spikes at the point where ESS was minimized. We present our estimated DBM models in table 4, and not surprisingly they are very similar to the LSTM models in table 3. The parameter estimates and their standard errors are almost identical, and we observe only slight increases in the ESS for DBM models in the few cases where industry models had experienced smoother transition (i.e. smaller ) in the LSTM framework. As for the LSTM models, the estimation procedure is affected by numerical accuracy limitations, but in this case the limitations arise because our grid of possible values for c can be no finer than the set of observed R Ã t in the sample. We show these limitations by reporting the interval (defined by two values of the ordered R Ã t ) in which b c lies. The most noticeable difference between the b c i reported in the LSTM and DBM tables occur for the 'engineering', 'miscellaneous industrials', 'oil and gas' and 'tourism' industries, and it is in these industries where we see changes in the other estimated parameters. Such changes do not affect our overall conclusion that the betas for most industries are time varying. Further, of those industries in which changes in beta are statistically significant, about half have up-market betas that are lower than down-market betas, and about half have up-market betas that are higher than down-market betas. It is interesting to note that more than half of the industries have less than half of their observations in the down-market regime (T L 5126). This result concurs with those of Pagan and Sossounov (2003) and Lunde and Timmermann (2004), both of whom used trend-based definitions of 'bull' and 'bear' markets to analyse market phase durations and amplitudes.
The R 2 measures reported in table 4 indicate that the DBM models fit the data quite well, and the reported p-values for Ramsey (1969) reset tests find evidence of mispecification in only one of the 24 industries ('retail').
Thus, it appears that the DBM models have captured the time variation in the beta quite well. There is a little evidence of serial correlation in the model residuals for six of the industries (see the LM 1 and LM 12 columns in table 4), but since this was not widespread over all industries and it did not imply any bias in the parameter estimates, we did not explicitly correct for this. However, as noted above, we used White's heteroskedasticity corrections for all of our t-statistics, because heteroskedasticity characterized each of the industry series that we were modelling.
Our estimated DBM models provide consistent evidence that beta varies according to values of R Ã t , and it is interesting to see whether our use of data-determined thresholds is driving this result, whether our use of R Ã t is responsible, or whether each of these factors plays a part. We believe that the use of data-determined thresholds is important, because if we re-estimate table 4 using R Ã t ? 0 to define up and down-market betas, then we find statistically significant differences between up-and down-market betas for only one industry. Defining R Ã median to be the median of R Ã t (0.0096) and repeating the experiment using R Ã t ? R Ã median leads to only four industries in which up-and down-market betas are statistically different. The decision to use R Ã t is also influential because if we re-estimate table 4 using R t as the transition variable, we find statistically significant differences between up-and down-market betas in only eight cases at the 5% level of significance and in only 12 cases at the 10% level. This last finding confirms our earlier intuition regarding our choice of transition variable, because R Ã t reflects broad swings in the market rather than temporary market movements, and this allows the resulting beta to take on 'bull' and 'bear' market interpretations.
We re-estimated table 4 using 6-month and 18-month moving averages of R t to explore the effects of varying the extent of lagged information contained in our market indicator. Our new estimates were very similar to those in table 4 (especially when using the 18-month moving average), with the industry-specific signs of U remaining the same in almost all cases. However, the number of statistically significant U was smaller in both cases, down to 15 (at the 5% level of significance) when the moving average was taken over 18 months, and 14 when the moving average was taken over six months. Further, the R 2 for the regressions involving the six-month moving average were consistently lower. This evidence suggests that a classification of the current state of the market can usefully incorporate returns that go back for at least six months, but not much further than a year. A justification of why lagged information might be relevant when classifying a current market state is that it helps investors to distinguish between temporary and more persistent stock market movements.
Given the work in Siegel (1998) and Resnick and Shoesmith (2002) that looks at the relationship between The two values for c indicate the upper and lower bounds for the threshold R Ã t , and T L counts the number of observations in the down-market regime. We also report p-values for one fitted term Ramsey (1969) reset tests (RR 1 ) and first-and twelfth-order LM serial correlation tests ( LM 1 and LM 12 Þ R 2 and ESS provide measures of fit. stock market performance and business cycle indicators, we also re-estimated table 4 using (i) a coincident business cycle indicator, and (ii) a lagged interest rate spread as a transition variable.y As in the moving average experiments, the resulting estimates were quite similar to those that used R Ã t , although again the number of instances for which U differed from zero at the 5% level of significance was smaller. Thus it appears that although business cycle indicators contain information that is relevant for assessing the state of the stock market, they do not discriminate between 'bull' and 'bear' phases as well as our market-specific indicator.

Conclusion
Previous research on the relationship between beta and market phase offers rather weak evidence that security and portfolio betas are influenced by the alternating forces of 'bull' and 'bear' markets. However, most previous studies have arrived at their conclusions by using the simple threshold DBM model in conjunction with crude 'up'-and 'down'-market definitions that involve a comparison of the market return to an arbitrarily chosen threshold value. In this paper we reinvestigate this phenomenon, testing for differential 'bull' and 'bear' market effects using a trend-based definition of 'bull' and 'bear' markets, and allowing the data to determine an appropriate value for the threshold parameter for each separate industry. We use logistic smooth transition market models for our study, which incorporate smooth transition between the two extreme regimes, and allow an additional investigation of the extent to which investors exhibit heterogeneous beliefs.
Our results differ from previous work in several important respects. Most importantly, we find strong and consistent evidence that beta varies between 'bull' and 'bear' phases. This result seems to be partly attributable to our use of a trend-based market indicator, but it is also due to the fact that we allowed the data to define 'bull' and 'bear' states rather than working with an arbitrarily defined definition. Our LSTM estimates indicate that transition is abrupt for most industries, and we view this as support for homogeneous beliefs among investors due to information symmetry. Specifically, investors collectively differentiate between 'bull' and 'bear' market states for each industry, and once the market indicator crosses each industry-specific threshold value, the beta for that industry reflects a corresponding change in state.
We estimate DBM models because the estimated value of the smoothness parameter in the LSTM models is very large for most industries, and we find that up-market and down-market betas are significantly different in 21 out of 24 industries. The former is lower than the latter in only 11 cases, and is not consistent with literature that claims that risk in up markets is lower than down-market risk. A possible explanation for the lack of a clear relationship between risk and beta is offered by Ang et al. (2006), who use a 'disappointment utility function' and downside correlations (not downside beta) to show that the compensation for bearing downside risk is not simply compensation for market beta.
Our analysis finds very strong evidence that betas in 'bull' and 'bear' markets differ because we are particularly careful about how we define these states. We are careful to ensure that our market indicator captures persistent movements in market returns and that it abstracts from market noise. Our analysis indicates that a noisy indicator such as the market return series itself does not help to identify market states, but that smoother indicators (such as our moving average indicators) do. Our results also show that definitions of 'bull' and 'bear' markets depend on which industry is under consideration. When our market indicator R Ã is declining, some industries go into the 'bear' state well before other industries. This supports Harlow and Rao's (1989) theoretical observation that investor thresholds need not just depend on parameters associated with the distributions of the market or risk-free returns.
Consistent with Maheu and McCurdy (2000), Pagan and Sossounov (2003), and Lunde and Timmermann (2004), we find that, for many industries, stocks spend more time in 'bull'-market than 'bear'-market states. We also find that the threshold between states is typically lower than the mean and/or median of market returns, suggesting that many of the standard 'symmetrical' definitions of 'bull' and 'bear' markets are inappropriate. Taleb (2004, Chapter 6) emphasizes the ways in which asymmetries in the distributions of stock market returns can confound seemingly straightforward notions of 'bull' and 'bear' market classifications, and the alternative 'bull'/'bear' classification mechanism outlined in this paper offers a practical way of accounting for and studying stock market asymmetries.