A METHODOLOGY OF WHETHER BOND PRICES RETURNS CAN WEAK FORM MARKET EFFICIENCY
- Methodology
- Summary Statistics
The calculation of the descriptive statistics summary will be done using Eviews 8 ( or Econometric Views Version 8) which is a statistical package for Windows, used mainly for time econometric analysis(es) that are -series oriented and was is developed by Quantitative Micro Software (QMS). The summary descriptive statistics will include the mean, minimum, maximum, and standard deviation, as well as kurtosis, and skewness. The summary of descriptive statistics will be followed by normality tests will be performed using kurtosis, and skewness.
Kurtosis measures the flatness or peakedness of the distribution of the series. When kurtosis equals 3, the distribution is normal. However, when the kurtosis is larger than 3, it is known as peaked relative to the normal or leptokurtic. Alternatively, when the kurtosis is less than 3, the distribution of series is termed comparatively flat or platykurtic. For univariate data,,…, , then the kurtosis formula is:
Where is the mean, N is the number of data points, and s is the standard deviation.
The other concept involved in normality testing is skewness which deals with symmetric structure of a series around its means. When skewness of a distribution is considered three outcomes are likely to be observed such as: zero, negative skewness and positive skewness. When the skewness of a distribution is zero, this indicates that the distribution series are normally distributed. However, when the skewness is negative it indicates that the distribution has a long left tail, and when the skewness is positive it means that the distribution has a long right tail. For univariate data,,…, , then the skewness formula is:
Where is the mean, N is the number of data points, and s is the standard deviation.
- Tests for randomness
Jones and Netter (2008) states that based on random walk model, market efficiency implies that successive changes in bond prices or returns are distributed independently and identically. Thus, the data collected in this study was used in order to find out whether a weak form efficiency or random walk pattern was followed by the gilts in each sub-period. In time series analysis was particularly done using statistical methods such as Descriptive statistics, Normality test, Serial correlation test (ACF), Variance ratio test, Runs test and Unit root test (Augmented Dickey Fuller test and Phillips-Perron test) with the assistance of Eviews 8. In addition, there was a differentiation of the empirical analyses into parametric tests and non-parametric tests. This is mainly because the application of parametric tests is usually done when there is a normal distribution, whereas non-parametric tests are applied when the distribution is not normal and they are usually considered as distribution free methods. Therefore, the parametric tests considered in this study include serial correlation test, variance ratio test and Augmented Dickey Fuller unit root test. On the other hand, runs test and Phillips-Perron unit root test were considered as the non-parametric tests in this study. Accordingly, the hypothesis of the study is:
= The series follow a random walk pattern or is a weak-form market efficient.
= The series does not follow the random walk pattern or is not a weak-form market efficient
- Serial correlation test
This is the most common test for randomness in random walk model and it measures the relationship between current period returns and the previous period value or returns.
The model shown below is the basis for testing the independence:
While testing for autocorrelation, the hypotheses are demonstrated as shown below:
Null Hypothesis: (independent)
Alternative Hypothesis: (dependent)
- Durbin Watson Statistic (d) Test
Marsaglia and Tsang (2002) argue that Durbin Watson Statistic (d) is a good test for serial correlation or autocorrelation of returns as shown below:
Durbin Watson Statistic provides a test of=0 (No AR (1)) in the error terms specification shown in this equation. Upon the rejection of the test, this is an indication that there is evidence for AR (1) or first-order serial correlation (autocorrelation of order 1). In addition, when the Durbin Watson Statistic =2, it indicates that there is no serial correlation, whereas when the Durbin Watson Statistic is <2, there is positive serial correlation and if Durbin Watson Statistic is >2, there is negative serial correlation. Thus, in order to test the hypotheses considered in the Watson-Dublin test shown below:
The test statistic is therefore as follows:
Where and and are the observed and predicted values respectively, of individual i.d response variable which becomes smaller with increasing serial correlations. Both lower and upper critical values, and respectively have been tabulated for varied k (the total number of explanatory variables) values and n.
Therefore,
If, then is rejected meaning there is statistical evidence to indicate positive autocorrelation of error terms
If, then is not rejected meaning there is no statistical evidence to indicate positive autocorrelation of error terms
If, then the test is inconclusive
- Variance ratio test
This introduction of this test was first done by Lo and Mackinlay (1988) and it usually acts as a more efficient and powerful method for testing the random walk hypothesis. The performance of Lo and Mackinlay (1988) variance ratio test in this paper uses the Eviews 8 for both heteroskedastic and homoskedastic random walks using the asymptotic distribution. However, Lo-Mackinlay test has one major limitation since it usually ignores the joint nature of the random walk hypothesis testing as a result increasing Type-1 error probability. An improvement of the Lo-Mackinlay test was done by Wright (2000) by proposing the non-parametric variance ratio test on basis of signs and ranks which is more powerful, especially when there is normal distribution of returns.
Therefore, in this paper, the Lo and Mackinlay (1988) and Wright (2000) variance ratio tests are considered:
- Lo-MacKinlay Test
The asymptotic distribution of was proposed by Lo and Mackinlay (1988) test through an assumption that is usually fixed when. In their test they appropriately showed that when equals i.i.d., that is, under the homoscedasticity assumption, meaning that under the null hypothesis.
As a result the test statistic is given by the equation shown below:
The above equation asymptotically follows the standard normal distribution and theasymptotic variance, , is given by the equation shown below:
Moreover, in order for Lo and Mackinlay (1988) test to accomodate which exhibit conditional heteroscedasticity, they proposed the heteroscedasticity robust test statistic, , given by the equation shown below:
The above equation asymptotically follows the standard normal distribution under the null hypothesis presented as , where
The applicability of test is possible to that is generated from a martingale difference time series and the usual standard normal distribution rule is applicable to both tests. Furthermore, Lo and Mackinlay (1989) studied the VR test finite-sample properties and found that there is general closeness in size of two-tailed test and the nominal level, as long as there is robustification of the test against any conditional heteroscedasticity.
Moreover, Lo-MacKinlay test is applicable in two main situations, that is, when there is a drift and when there is no drift. For instance, in the case of a drift, represents an estimator that is unbiased of or variance of the q-differences as shown by the equation below:
On the other hand, in the case of no-drift, represents an estimator that is unbiased of or the variance of the first difference that demonstrated by the equation shown below:
However, the standard Z-test statistic in both cases is calculated as shown in the equations shown below:
- The standard normal test-statistics under homoscedasticity is shown by the equation below:
Where
- The development of the second testis done under heteroscedasticity hypothesis and its expression is as shown below:
Where
Therefore, the main hypotheses of variance ratio test are specified as shown by the null and alternative hypothesis shown below:
- Wright (2000) Test
The other variance ratio test method is the Wright test of the year 2000 which is an improvement of the Lo-MacKinlay test. The Wright test proposes an alternative that is non-parametric to VR tests that are conventionally asymptotic using signs and ranks. The Wright test are exact because they are based on ranks under the i.i.d. assumption, while signs-based tests are exact even when the heteroscedatsticity is conditional, even though low-size distortions may be displayed in ranks-based tests under conditional heteroscedasticity.
For instance, when given first differences variable let among , . . . . be rank of . Therefore, under the null hypothesis i.i.d. sequence generates , an is a the numbers of 1 , . . . , T random permutation with equal probability.
Thus, and statistics were suggested in Wright test and are defined as shown by the equations below:
In the above equations the standardized ranks and are given by the equations shown below:
Where φ (k) is the asymptotic variance, and is the function of the standard normal cumulative distribution inverse. The same exact sampling distribution is followed by R1 and R2 statistics. However, the tests critical values can be obtained through simulation of their exact distributions.
The signs-based tests of first differences are given by the equations shown below:
Where is the asymptotic variance; and
The S1 and S2 tests critical values, similar to R1 and R2 tests, can be obtained through simulation of their exact sampling distribution. Also it is important to note that S1 assumes a zero drift value.
- Runs test
The runs test also referred to as the Walt-Wolfowitz tests was first presented by Abraham Walt and Jacob Wolfowitz. It is considered as a non-parametric test where no distribution assumption is made, and can be used for the examination of whether successive price changes (returns) are independent or not. The computation of number of runs is carried out in form of successive price returns (changes) sequence of the same sign (negative, positive, or zero). The mean is used complete the runs test, and each price changes (returns) is classified into either positive, negative or zero on basis of its position with respect to the mean return. Therefore, when return is larger than the mean, the price return records a positive change, while when the return is less than the mean, the price return records a negative change, and zero is observed when the price return equals to the mean.
In runs test the values are coded where those below the mean are classified as negative, those above the mean are classified as positive and those that are equal to mean as zero. This implies that the definition of a run is that, it is a series of consecutive negative (or positive) values. The runs test is defined by the mull and alternative hypothesis as shown below:
| H0: | The sequence was produced in a random manner |
| Ha: | The sequence was not produced in a random manner |
Therefore, the test statistic in runs test is demonstrated by the equation below:
Where R is the number of runs observed, is the number of runs expected; and is the number of runs’ standard deviation. The computation of values of and can be done as shown in the equations demonstrated below:
Where and represent the number of positive and negative values in a sequence respectively.
The significance level in runs test is and the null hypothesis for runs test is rejected if:
In case of runs test where the sample is large (where and), standard normal table is used to compare the test statistic. This implies that at 5% significance level non-randomness is indicated by a test statistic value greater than 1.96. However, for runs test involving small samples tables for the determination of critical values are dependent on values of and as discussed by (Mendenhall and Reinmuth, 1982).
- Unit root test
The unit root test was first introduced by Dickey and Fuller (1979) and often adopts the autoregressive approach. However, in order to illustrate important statistical issues that are related to this test clearly, a simple AR (1) model is considered as shown below
In the above case the hypotheses of interest are as shown below:
In this case the test statistic is shown below:
Where is the estimate of least squares and is the estimate of usual standard error.
- Augmented Dickey-Fuller (ADF) tests
The ADF test tests the null hypothesis that a time series is against the alternative that it is, on the assumption that the data dynamics have an ARMA structure. The basis of ADF test is on the estimation of the test regression.
Where is a vector of deterministic terms (constant, trend etc.). The lagged difference terms,, are used to approximate the ARMA structure of the errors, and the value of is set so that the error is serially uncorrelated. The error term is also assumed to be homoskedastic. The specification of the deterministic terms depends on the assumed behavior of under the alternative hypothesis of trend stationarity as described in the previous section. Under the null hypothesis, is which implies that. The ADF t-statistic and normalized bias statistic are based on the least squares estimates of the above equation and are given by:
Moreover, the alternative formulation of the ADF test regression is shown in the equation below:
Where π = φ − 1. Under the null hypothesis, is which implies that π = 0. The ADF t-statistic is then the usual t-statistic for testing π = 0 and the ADF normalized bias statistic is
- Phillips-Perron (PP) Unit Root Tests
This test was developed by Phillips and Perron (1988) and the test regression for PP test is given by the equation below:
Where is and may be heteroskedastic. The PP tests correct for any heteroskedasticity in the errors of the test regression through direct modification of the test statistics and. These modified statistics, denoted and, are given by the equations shown below;
In the above equations the terms and are variance parameters estimates that are consistent
Where
- Trading rule tests
Two trading rule tests were examined by Brock et al. (1992), whereby the first trading rule test that they considered was the moving average oscillator, which consisted of two moving averages (MAs) of the index level: , a `short’ moving average represented by order n and demonstrated by the equation shown below:
Also the other moving average which is a `long’ moving average is represented by order and demonstrated by the equation shown below:
In its simplest form, there is generation of a buy (sell) signal by the moving average oscillator rule when rises above (falls below) leading to an initiation of a ‘trend’ when this happens. When the moving average oscillator trading rule test is considered, the most popular MA rule is the 1-200, which sets n=1 (so that is presented in terms of the level of the index currently which is denoted by,) and m=200. Also there are other rules which can be considered in this paper including 1-50, 1-50, 5-150 and 2-200 MA rules. The modifications of the MA rules in this trading rule test are often carried out through introduction of a band around the MA, resulting to a reduction of the number of buy and sell signals through elimination of the occurrence of ‘whiplash’ signals when and are in close proximity to each other. However, in trading rule test the examination of the above discussed MA rules occurs both with and without a one per cent band, that is, the generation of a buy (sell) signal occurs when is below (above) by more than one per cent. Moreover, there is no buy or sell signal that is generated when is inside the band. Therefore, through the MA rule, the trading rule test attempts a simulation of a strategy where traders go short as moves below and long when moves above . However, all trading days are classified as either buys or sells with a band of zero.
The second trading rule test used in the analysis is the trading range break-out (TRB) where the generation of a buy signal occurs when there is penetration of the price returns into a resistance level, mostly referred to as a local maximum. On the other hand, the generation of a sell signal occurs when the price penetrates a support level, mostly referred to as a local minimum. Thus, if
and
then there is generation of a buy signals if and generation of a sell signal occurs if. As with the MA rules, m can also be set at 50, 150 and 200, and the implementation of the TRB rules can be done both with and without a one per cent band.
References
Bogle, J. (1994), Bogle on Mutual Funds: New Perspectives for the Intelligent Investor. New York, NY: Dell.
Brock, W., Lakonishok, J. and LeBaron, B. (1992) “Simple technical trading rules and the stochastic properties of stock returns’’ Journal of Finance, Vol.47 Issue 3, 1731-64.
Hebner, M.T. (2007), Index Funds: The 12-Step Program for Active Investors. London: IFA Publishing.
Cowles, A., and Jones, H. (1997), “Some A Posteriori Probabilities in Stock Market Action”. Econometrica, Vol. 5 Issue 3, pp. 280–294.
Jones, S.L., and Netter, J.M. (2008), “Efficient Capital Markets”. In David R. Henderson (ed.). Concise Encyclopedia of Economics (2nd ed.). Indianapolis: Library of Economics and Liberty.
Kendall, M. (2010), “The Analysis of Economic Time Series”. Journal of the Royal Statistical Society, Vol. 96 Issue 3, pp. 11–25.
Khan, A.M. (1986), “Conformity with Large Speculators: A Test of Efficiency in the Grain Futures Market”. Atlantic Economic Journal, Vol.14 Issue 3, pp. 51–55.
Lo, A.W. and MacKinlay, A.C. (1988), “Stock market prices do not follow random walk: Evidence from a simple specification test”. Review of Financial Studies, Vol.1 Issue 3, pp. 41–66.
Lo, A.W. and MacKinlay, A.C. (1989), “The size and power of the variance ratio test in finite samples: A Monte Carlo investigation”. Journal of Econometrics, Vol.40 Issue 2, pp. 203–238.
Lo, A., and MacKinlay, C. (2001), A Non-random Walk Down Wall St. Boston, MA: Princeton Paperbacks.
Luger, R. (2003), “Exact non-parametric tests for a random walk with unknown drift under conditional heteroscedasticity”. Journal of Econometrics, Vol.115 Issue 4, pp. 259–276.
Malkiel, B.G. (1987), “Efficient market hypothesis,” The New Palgrave: a Dictionary of Economics, Vol. 2 Issue 3, pp. 120–23.
Marsaglia, G. and Tsang, W.W. (2002), “Some Difficult-to-pass Tests of Randomness”. Journal of Statistical Software, Vol. 7 Issue 3, pp. 256-270.
Malkiel, B.G. (1996), A Random Walk Down Wall Street. New York, NY: W. W. Norton.
Mendenhall, W. and Reinmuth, J. (1982), Statistics for Management and Economics, 4th edition. London: Duxbury Press.
Samuelson, P. (2002), “Proof That Properly Anticipated Prices Fluctuate Randomly.” Industrial Management Review, Vol. 6, No. 2, pp. 41–49.
Sipper, M., and Tomassini, M. (1996), “Generating parallel random number generators by cellular programming”, International Journal of Modern Physics, Vol. 7 Issue 2, pp. 181–190.
Whang, Y.J. and Kim, J. (2003), “A multiple variance ratio test using subsampling”. Economics Letters, Vol.79 Issue 3, pp. 225–230.
Working, H. (1990), “Note on the Correlation of First Differences of Averages in a Random Chain”. Econometrica, Vol. 28 Issue 4, pp. 916–918.
Wright, J.H. (2000), “Alternative variance-ratio tests using ranks and signs”. Journal of Business and Economic Statistics, Vol.18 Issue 1, pp. 1–9.
Use the order calculator below and get started! Contact our live support team for any assistance or inquiry.
[order_calculator]