For our empirical study we use the S&P 500 and FTSE 100 indices. Data for the two indices were obtained from Google Finance and Bloomberg. The setting in Section 2 suggests that the states are triggered by an exogenous variable. We use the University of Michigan’s Index of Consumer Sentiment as our exogenous trigger (MCSI henceforth). We also employ VIX as an additional exogenous trigger and compare the results. The lower bound on the MCSI is 51 and the upper bound is set at 112 so that the regression in eq. (22) can be estimated; for the VIX the lower bound is 10 and the upper bound is 70. We use a 200 point grid on both exogenous triggers.

The MCSI is a monthly survey collected by the University of Michigan. It asks questions on personal finance and economic trends of individuals and households through telephonic interviews. Survey respondents are representative of the American population and each month more than 500 interviews are conducted. More information on the Survey and sample design is available on the MCSI website. The VIX on the other hand is a measure of implied volatility for the S&P500 derived from a variety of different options available on the S&P500. It seeks to estimate future volatility in the index and is therefore representative of investors’ view on the future direction of the stock market index.

Data for the S&P500 index and the MCSI were obtained from January 1978 to June 2015. The underlying assumption required for the MCSI and VIX to be valid trigger variables in this setting is that contemporaneous and one-period lagged values of the two indices do not impact either the MCSI or the VIX. For the VIX, data start from 1st February 1990, but are available at a daily frequency. Thus, for VIX regressions we use daily data from the 1st February 1990 up to 30 June 2015 for both the S&P500 and the FTSE100 Indices.

While data for the MCSI are available from 1964, the survey was initially collected twice a year and the index only becomes a monthly index in 1978. The FTSE 100 index on the other hand is used from its inception in 1984. For the MCSI regressions we use monthly data. The use of MCSI and VIX as trigger variables for FTSE 100 is justified by the strong correlation between the S&P 500 and FTSE 100 indices. The MCSI and VIX do not appear to be independently distributed (first order autocorrelation > 0.9 for MCSI and 0.98 for VIX); thus, they are closer to being Markovian trigger variables.

Although we have not explicitly calculated moments for the case when the trigger variable is Markovian, we refer the interested reader to Knight and Satchell (2011), which discusses these results for the two-state case with a constant drift. We did test with a variety of other potential trigger variables but they failed basic Granger causality tests and thus, cannot be considered exogenous to the indices. Selecting a Markovian variable does not impact our estimators or our estimation strategy; however, we will not be able to use the formulae derived in Section 2 to calculate the mean and variance of our assets. Nevertheless, the formulae offer valuable insights as we will note in our discussion. We first discuss the results using MCSI as the exogenous trigger.

High consumer sentiment, i. e. a positive outlook towards personal finance and general business environment in the country is reflected through a high value of the index. On the other hand low consumer sentiment is reflected as a lower value. We posit that high consumer sentiment that persists for long periods is indicative of explosiveness or bubbles, i. e. if consumers have a very positive outlook they are likely to invest in assets and if a large number of consumers enter the asset markets or in this case the stock market the increase in demand could lead to a switch from efficiency to explosiveness.

Similarly when lower values persist we posit that the market is correcting itself and we get mean reverting behaviour. Figure 1 shows the MCSI and the log of the S&P500 index indicating how the MCSI varies with the log of S&P500 index. We see a spike in consumer sentiment in the run up to the dot com bubble. A similar increase is seen near the 2008–2009 financial crisis. Mean reverting behaviour is observed after the 1979 oil crisis as well as in the aftermath of the financial crisis. While the MCSI may not always respond contemporaneously to movements in the S&P500 index, it nevertheless acts as a valuable trigger variable for our illustrative example.

Figure 1: S&P 500 index and the Michigan consumer sentiment index.

We estimate the autoregressive and drift parameters in eq. (22) using 3 states for the S&P500 & FTSE100 indices respectively and use the MCSI as the trigger variable. The dependant variable in the regression is asset returns instead of log prices in order to ensure consistency of standard errors. We could have used log levels instead of returns but returns are more intuitive; secondly, using levels we would find some non-stationary states. Using the return formulation also ensures that the criterion for the existence of a stationary distribution, specified in Section 2, is satisfied.

We estimate the model with and without the switching drift term$\phantom{\rule{thinmathspace}{0ex}}{\psi}_{t-1}$. When we do use a drift term, we report results with both a switching drift term, i. e. the drift changes in each state and a constant drift term, i. e. the drift does not change across states. The most commonly employed specification in related literature is that with a constant drift. Our aim is to find thresholds ${c}_{1}$ and ${c}_{2}$ for the MCSI that minimize the residual sum of squares for the threshold auto-regression which in turn also yield the parameter estimates for the 2 inefficient states.

Note, that we do not impose any restrictions on the parameters of the other two states; both states may be mean reverting or explosive. The only restriction imposed is$\phantom{\rule{thinmathspace}{0ex}}{c}_{1}>{c}_{2}$. We use the procedure outlined above to estimate the threshold, ${c}_{1}\phantom{\rule{thinmathspace}{0ex}}$ and ${c}_{2}$. 51 and 112 represent the extreme values taken by the trigger variable, MCSI in this case. For our assets these values are reported in below and include results for both stock market indices with and without a (switching/constant) drift term. The table also notes the time the stock-market index is in each of the 3-states; this will allow us to comment on the proportion of time each of the stock-market indices is efficient or inefficient.

Table 7: Non-linear least squares regression results (with standard errors) – with MCSI as the exogenous trigger.

We postulate that$\phantom{\rule{thinmathspace}{0ex}}{\beta}_{1}<0$ if MCSI is low and ${\beta}_{3}>0$ when MCSI is high; the postulated relationship will vary based on our choice of trigger variables). Columns (8) and (9) in report the thresholds corresponding to the minimum sum of squared residuals. Note that for the specification with drift, the intercept is also switching and as illustrated in Section 2, this changes the mean and variance of the process (if they exist) significantly.

Since we use different model specifications for this example, it is no surprise that the results in present a mixed picture. When we consider the case of a shifting drift term in addition to a shifting slope coefficient, a lot of the variance in the series is captured by the shifting drift term. Inclusion of a moving drift substantially reduces the impact of the moving slope terms and we find no more than 32 observations in non-efficient regimes (20 for S&P500 and 32 for FTSE100). This amounts to 5 % inefficiency for the S&P500 and 8 % for the FTSE100. For both indices we also note that the first state is not statistically significant so we may not classify that state as inefficient. We also find little evidence of explosive behaviour due to the slopes.

Thus, explosive and mean reverting episodes under a model with a moving drift are primarily caused by the change in drift. Note that the drift terms are larger in magnitude and appear farther apart which implies that they have a higher variance. As per our formulae in Section 2, a higher variance of the drift parameter leads to a higher variance of the series. We find that the drift term in the explosive regime is statistically significant and greater than the drift term in other regimes. Thus, under this specification explosiveness in the S&P500 and FTSE100 indices is due to a temporary increase in average returns. With a constant drift term, both indices appear stationary and do not have explosive regimes.

One way to systematically beat the market in such a situation will be through predicting when the switches will occur provided that investors are aware of what state they are in as soon as the switch has occurred (and thereby becomes a part of the information set). Thus, we are referring to efficiency in a broader sense. In the conventional auto-regressive sense, a market is said to be efficient if the auto-regressive parameter is 1, i. e. the process is a random walk so that the only change in asset returns is due to unpredictable factors and no gains can be made based on the existing information set. In the threshold auto-regressive case, in addition to the restriction on the auto-regressive parameter we would also require the state switches be unpredictable; although once the switch occurs everyone becomes aware of it. Thus, the information set will also include information about what state the exogenous trigger is in. If markets are weak form efficient all rational investors will find out about the switch at the same time although they may not know when the switch may occur.

For the specification without a drift the results are closer to the behaviour observed in the simulations, i. e. we observe three states although the deviation from efficiency is very small. When ${\beta}_{3}>0$, i. e. we are in the explosive or bubble regime, we observe additional annualized gains of only 1.5 % in the S&P500 index and 1.1 % in the FTSE100 index. It may be argued that the additional annualized gains being captured by the parameter are in fact accounting for the missing drift term. For models with a switching drift, the criterion for a steady-state stationary distribution is trivially satisfied as for both FTSE100 and S&P500 we do not find an explosive slope coefficient. Figure 2 shows the areas that fall under the different states under this specification for log prices based on the thresholds estimated by the procedure outlined above.

Figure 2: S&P 500 index: Results for non-linear least squares with a switching drift with MCSI as the exogenous trigger.

We note that the first state corresponds to periods of relative slow down, i. e. in the aftermath of the 2^{nd} oil price crisis, in the immediate aftermath of the financial crisis and in late 2011 when fears of a double dip recession abounded. The other non-efficient state occurs in the run up to the East Asian financial crisis and the dot-com bubble when consumer sentiment was at an all-time high. Our grid-search results do not indicate a deviation from a random walk during the financial crisis. The area under the non-random walk states has been shaded (blue for mean-reverting and red for explosive). Figure 3 shows similar results for the FTSE100 index. Note that apart from the dot-com bubble period in early 2000, 1998 is identified as a period of explosiveness for both indices. Both indices attained historical highs in the 1998 which is reflected in consumer sentiment.

Figure 3: FTSE100 index: Results for non-linear least squares with a switching drift with MCSI as the exogenous trigger.

By regressing log prices on their lags instead of returns (i. e. add 1 to each coefficient estimated in Section 3), we can calculate the value of the criterion function specified in Section 2 which allows us to comment on whether the series has a stationary distribution. For the case without a drift, the value of the criterion for the S&P500 and FTSE100 is 0.0010 and 0.0005 respectively (the criterion in this case is calculated as $\sum _{i=1}^{3}{\hat{\pi}}_{j}({\hat{\hat{B}}}_{j}$+1). Neither of the two indices satisfies the criterion for a steady-state stationary distribution under specifications without a drift. This indicates that any test for explosiveness that either assumes a constant drift term without shifting slope coefficients (not reported) or that drop the drift term are more likely to find the criterion violated for the S&P500 and FTSE100 indices.

As mentioned before, if MCSI was an independent and identically distributed variable we would be able to use our formulae from Section 2 and be able to calculate the mean and variance for both the S&P500 and the FTSE100 series when they are estimated using a threshold auto-regression. This will have allowed us to compute metrics such as Sharpe ratios enabling us to comment further on market efficiency and investor behaviour. Since MCSI is closer to a Markovian variable we are unable to use the formulae derived earlier. However, our results do allow us to compare efficiency across the two markets. In the following discussion when we talk about inefficient states we are referring to the proportion of time spent by each index in a state that is statistically significantly different from the random-walk.

When specifications with a drift are considered, the FTSE100 index appears more inefficient than the S&P500 index. The S&P500 index is inefficient for 2 % of the time with the switching drift specification and 86 % of the time with a constant drift. In contrast the FTSE 100 index is inefficient for 6.7 % of the time under the switching drift specification and 93 % of the time under the constant drift specification. On the other hand when no drift is included, the S&P500 appears mostly inefficient (95.5 %) compared to the FTSE 100 (68.7 %). If we compare similar periods, i. e. from 1981 onwards, the results remain robust. The mixed results do not offer a clear answer as to which market appears more inefficient; nevertheless the methodology is applicable to other assets. If an asset appears to spend more time in inefficient states under all specifications compared to another we will be able to conclude that the market for that particular asset is inefficient more often. Our results using VIX as the exogenous trigger shed further light on the efficiency question. reports the results for when VIX is used as the exogenous trigger.

Table 8: Non-linear least squares regression results (with standard errors) – with VIX as the exogenous trigger.

Before we discuss the regression results using VIX in detail, it is important to highlight the differences in the two exogenous triggers. While data for the MCSI are only available at a monthly frequency, VIX is reported daily; this allows us to estimate the regression using daily returns thereby allowing us to measure market efficiency at a higher frequency. Thus, VIX is measuring short-term market efficiency while with MCSI we were able to measure medium-term market efficiency. However, since the data for VIX are only available from 1990, we are unable to include observations from the oil price crises and large stock market downturns in the 1970s and 1980s.

Figure 4 plots the S&P500 and the VIX. The pattern suggests that periods of high volatility, marked by a high value of the VIX, tend to correspond to low values of the S&P500 index. Thus, we detect high volatility in the market whenever the S&P500 index is facing a downturn. No such pattern is immediately visible for the opposite scenario. High values of the S&P500 do not always correspond to low values of the VIX. It appears that while the VIX may be a good potential trigger for the stationary or mean reverting regime, it may not have the necessary variance for the explosive state. Also, as we pointed out initially, the VIX is a measure of investment sentiment while the MCSI is a measure of consumer sentiment. This is another reason why the two results can be different; the information sets of consumers and investors may be quite different. Since efficiency is defined with respect to an information set, it is no surprise that the results vary, although we do note some common trends. The prevailing market price that we observe results from the different demand functions of consumers and professional investors. Both indices contain valuable information and it is advisable to consider results from both sets of regression results.

Figure 4: S&P 500 index and VIX – Daily Data.

While results from the MCSI were mixed, the ones obtained when VIX is used as the exogenous trigger are largely in favour of efficient markets. As before, we restricted state 2 to be the efficient state; however, our estimates with VIX contain significant evidence that there are only 2 as opposed to 3 states. ${\beta}_{1}$ is always close to 0 (to 3 decimal places or more) or is statistically insignificant, irrespective of the model specification. It is difficult to distinguish this state from the efficient state. ${\beta}_{3}$ on the other hand does indicate the presence of inefficient periods.

Although the proportion of inefficient observations is small (1 % or less), it accounts for around 50 periods/days. Figure 5 indicates that these tend to occur in 2008, in the aftermath of the financial crisis when volatility as recorded by VIX was very high. These periods also coincide with the periods of mean reversion detected when MCSI was employed as the exogenous variable. We note, however, that the procedure does not indicate the presence of explosive states in this case. It neither detected the dotcom bubble, nor the period in the run up to the financial crisis. The MCSI on the other hand, did flag the dotcom bubble period as inefficient. As mentioned before, VIX does not appear to be a good predictor of the explosive regime although it does do much better for indicating the onset of mean reversion. The results for FTSE100 looked similar and are reported in Figure 6.

Figure 5: S&P 500 index: Results for non-linear least squares with a switching drift with VIX as the exogenous trigger.

Figure 6: FTSE100 index: Results for non-linear least squares with a switching drift with VIX as the exogenous trigger.

Taking both the MCSI and VIX results together, our empirical results supplement our findings in Sections 2 and 3. Explosiveness is more likely to be detected in asset price series where the criterion function is violated and the variance of the switching slope parameters is large (i. e. there are many regimes or the regimes are much farther apart). Inclusion of a switching drift term may explain most of the explosiveness and may make the price series appear efficient. Another way of analysing results could be through comparing the different series.

The set of results reported above also depend on the selection of the trigger variable as we have noted. Finding an appropriate trigger variable that may indicate switches in regimes is non-trivial in practice and will require rigorous theoretical, empirical or experimental basis so that regime identification criteria can be appropriately set. We also need to consider the time period for which we are measuring market efficiency. As we have seen above, markets may appear to be more efficient when higher frequency data are considered.

While we primarily relied on Granger causality tests to determine appropriate lags for the trigger variables and for determining exogeneity, it could be argued that further testing on exogeneity, including testing for non-linear relationships, may be warranted. This requires judgement on the part of the researcher, the specific asset price being considered and the relationship between the asset price and the trigger variable. We did consider other suitable triggers including the fear and greed index and the economic policy uncertainty index but these were either not exogenous or did not include sufficient data. Furthermore, it is also possible to consider a combination of exogenous triggers which would combine elements of investor and consumer sentiment. We leave this exercise for future research as our example was aimed as an illustration of the outlined methodology. Additionally, the researcher also needs to consider the number of states to be used. A price series could exhibit multiple explosive or mean reverting states.

Our methodology can thus work in practice for different asset markets. We have shown with our example that the methodology may be used to identify the incidence of market efficiency for different assets while also highlighting the importance of model-specification when testing for market efficiency. Model specification is not important just in terms of estimation but completely changes the theoretical meaning of the results. Once a researcher has identified an appropriate specification for an asset price or return based on either technical analysis or through solving a structural model, our methodology will allow her to comment on market efficiency for that asset. However, irrespective of the specification, the results may still be used to compare different markets and identify which markets are more efficient for the given information set.

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