Revised Study of Daily Time Series Data Analysis of Market Prices

Jeffrey Jarrett^*
*Correspondence: Jeffrey Jarrett, Department of Business Management, University of Rhode Island, USA, Email:

Keywords

ARIMA • Time series • Daily variation • Pandemic • Capital markets • Capital markets

Introduction

Capital market efficiency is an important research topic since Fama EF [1] explained these principles as a portion of the efficient market hypothesis. Following Fame’s work many studies were devoted to investigating the randomness of stock price movements for the purpose of demonstrating the efficiency of capital markets. In recent years, other studies demonstrated market inefficiencies by identifying systematic and permanent variations in stock returns. Some of these systematic variations or anomalies as they are referred to are small firm effects, investment recommendations and extraordinary returns to the time or the calendar effect.

These Calendar or time effects do contradict the weak form of the efficient market hypothesis. The weak form refers to the notion that the market is efficient in past price and volume information and we do not predict stock price movements accurately using historical information. If no systematic patterns exist, stock prices are time invariant. On the other hand, if variation in the time series of daily prices of securities markets exist, market inefficiency is present and investors should be able to earn abnormal rates of return not in line with the degree of risk they undertook [2]. In addition, a large number of studies in the literature on predicting prices of traded securities confirm to some degree that patterns exist in stock market prices. We know those interest rates, dividend yields and a variety of macroeconomic variables exhibit clear business cycle patterns. The emerging literature concerning studies of United States securities include [3-6] and Pesaran (made significant contributions in time series and panel data) econometrics. He is the developer of the Global Vector Autoregressive (GVAR) approach used Pesaran HM and Timmermann A [7], Granger CWJ [8] provides an up to that time survey of methods and results. Studies in other places (the United Kingdom) include Clare AD, et al. [9,10], Black A and Fraser P [11] and Pesaran HM and Timmermann A [12]. Furthermore, Caporale G and Gil-Alana LA [13] pointed out that for US stock returns their degree of predictability depends on the process followed by the error term.

The expansion of time series analysis as a discipline permits one to analyse stock market prices in ways not heretofore explored. What is the predictability of the error term and is there predictability in daily stock market returns? Peculiar problems arise when daily patterns are present in stock price data. We know that stock prices possess patterns known as daily effects. For example, Kato K [14] results suggested that were our patterns in stock returns in Japanese securities. He observed low Tuesday and high Wednesday returns within weekly prices. If a week did not have trading on a Friday, he would observe effects related to the Monday of the following week. The following Monday would have low returns indicating that transference of the pattern that would occur on the Friday if trading had occurred which it did not. A second study by Kato K [15] found considerable anomalies on the Tokyo Stock Exchange (TSE), which is an organized exchange similar to the ones in North America.

Only a few studies focused on the investigation of time series components of equity prices and the predictability of these prices. Ray B, et al. [16] investigated a sample of 15 firms and found both permanent and temporary systematic components in individual time series of stock market prices of firms over a lengthy period of time. Mookerjee R and Yu Q [17] investigated the seasonality in stock returns on the Shanghai and Shenzhen stock markets. They documented the seasonal patterns existing on these exchanges and the effects these factors have on risk in investing in securities listed on these exchanges. In addition, they showed that risk in investing is related to the predictability of security prices. Rothlein CJ and Jarrett J [18] investigated the existence of seasonality present in Japanese stock prices, which affect the prices of these securities. They documented the evidence of seasonality in the prices of 55 randomly selected Tokyo Stock Exchange firms over a lengthy period of 18 years (1975 through 1992). In addition, they indicated the accuracy of forecasts or predictions of these firms’ prices are seriously decreased if one does not recognize the patterns in the time series. Jarrett JE and Schilling J [19] studied time series characteristics of the Frankfort stock market, one of the largest markets in the European Union. Pan X and Jarrett EJ [20] other properties of time series of corporate earnings. Finally, Jarrett J and Pan X [21] used another method of multivariate exponential weighted average method to analyse time series method to predict data for similar data in another application involving time series. Last, Jarrett J and Kyper E [22] indicated how patterns in monthly stock prices have predictable patterns. This study differs in that we examine the predictable patterns in the closing daily prices of stock prices. In goes further than the study of Caporale GM and Gil- Alana LA [13] noted before because it attempts to determine the patterns in daily prices of listed securities.

Methodology

The ARIMA (p,d,q) model is a linear combination of two linear models and thus is itself still linear. Autoregressive Moving Average Model of order p, q is a time series model

A time series model, is an autoregressive moving average model of order d, ARIMA (p,d,q), where d or differencing. The difference is done to reduce the white noise measured by the variance of the error term. One of the key features of the ARIMA model is that it is parsimonious and redundant in its parameters. That is, an ARIMA model will often require fewer parameters than an AR (p) or MA (q) model alone.

To determine if a daily pattern can be modelled for a sampled time series, we employ the

Dickey DA and Fuller WA [23] tests which we now illustrate. First an autoregressive process, AR (1), YT = μ + ρyt-1 + єt, where μ and ρ are parameters and єt is white noise. Y is a stationary time series if –1< ρ < 1. If ρ=1, y is a nonstationary series (a random walk with a drift), if the process is started at some point, the variance of y increases steadily with time and goes to infinity. If absolute ρ is greater than unity (1), the series is explosive. Thus, the hypothesis of a stationary series can be evaluated by testing whether the absolute value of ρ is strictly less than unity. The Dickey DA and Fuller WA [23] test takes the unit root as the null hypothesis, that is, Ho: ρ= 1. Since explosive series do not make much sense in economic terms, the alternative hypothesis is stated in terms of Ha: ρ<1. The test is usually carried out by estimating an equation with yt-1 subtracted from both sides of the equation:

Δyt = μ +γ yt −1+ε t where γ = p −1

and the null and alternative hypotheses are

Ho: γ = 0 and Ha: γ < 0. [a one-sided test]. Although this appears as a conventional t-test on the estimated γ, the t-statistic under the null hypothesis of a unit root does not have the conventional t-distribution. Dickey DA and Fuller WA [23] showed that the distribution under the null hypothesis is not standard and simulated the critical values for selected sample sizes. MacKinnon JG [24] implemented a larger set of simulations than those tabulated by Dickey and Fuller. He estimates the response surface using the simulation results, permitting the calculation of Dickey DA and Fuller WA [23] critical values for any sample size and for any number of variables on the right-hand side of the equation. We report our results based on these MacKinnon critical values for the unit root test.

We must also recognize that the simple unit root test noted before is valid only for a series, which is AR (1). If the series is Auto correlated at higher order lags, we violate the assumption of white noise disturbances. The augmented Dickey-Fuller approach controls for higher order autocorrelation by adding lagged difference terms of the response variable y to the right-hand side of the regression model:

Δyt = μ +γ yt −1+τ 1Δyt −1+τ 2Δyt − 2 +...+τ p −1Δyt − p +1+ε

This augmented specification is in turn used to test:

Ho: γ = 0 and Ha: γ < 0.

In the regression model, an additional important result obtained by Fuller is that the asymptotic distribution of the t-statistic on γ is independent of the number of lagged first differences included in the augmented DF regression. Further, the parametric assumption that y follows an AR process restricts the use of the DF test, Said SE and Dickey DA [25] demonstrate that the augmented DF test remains valid even when the time series is movingaverage (MA), provided that enough lagged difference terms are augmented to the regression.

Additionally, one may choose a model with or without a constant term and with or without a linear trend. The purpose of this analysis, however, is not to determine the precise model that best generates the time series of daily observation, but to consider whether a model could be built or not. If we find that the model has a unit root which can be expressed by an AR, MA or mixed ARMA process, we have shown that there is pattern in the daily observations and the notion that daily observation are completely random is nullified.

Results and Discussion

The data analytics

The analysis includes monthly closing stock prices for 49 randomly selected firms listed on organized exchanges in the United States (NYSE and NASDAQ). The original sample was larger but due to problems with acquiring objective data for the sample period several firms from the original sample were not included in the final study. Therefore, we used only firms having the complete sets of time series data that covered the study period (April 1992 to September, 2002). The study period was lengthy enough to minimize any effects of short-term economic fluctuations. In particular, 300 observations on daily closing prices is included in each data set and this number far exceeds the necessary sample size for successful application of Time Series Analysis by ARIMA methods (autoregressive integrated moving average methods). Table 1 contains a list of those firms included in the sample. Note that the firms are also well known enough to eliminate any problems with start-up firms and problems associated with mergers and acquisitions. All data account for stock splits, stock dividends, which may dilute the usefulness of unadjusted information.

The analytics

ARIMA methods permitted us to model the underlying mathematical process which gives rise to the time series. Table 2 contains the results of modeling the 300 observation time series data sets for the forty-nine firms under study. Note all of the forty-eight of the forty-nine firms required one degree of differencing to derive series suitable for modeling. This is consistent with earlier studies of time series data of closing prices, which contain nonstationary properties in the mean. This is usually a result of a trend in the data requiring a filter to generate a stationary series. Usually, one degree of differencing is sufficient to find suitable data for modeling. Only Hillenbrand Industries (hb) did not require a degree of differencing. The autoregressive (AR) terms and the moving average (MA) terms in the time series model are in turn indicated in Table 2 for each of the forty-nine-time series. A term of 1 indicates that the model contains a first-order AR or MA term. The value 2 indicates the existence of a second-order AR or MA term. Finally, the value 5 indicates a fifth-order AR or MA term. Since the number of trading days in a week is five, it is conceivable that daily data may contain a fifth-order term. Most models do not contain a constant term in the final model equation. This is consistent with the notion that first-differenced series usually do not contain constant in the underlying theoretical model.

We examined all modeled time series by producing both plots of the correlograms of the autocorrelations of the residual series. All series that violated the notion that autocorrelations up to 36 lags be within approximate two-sigma limits of zero were eliminated and series was modeled again. Two-sigma limits are applied to indicate significance levels of 5%, which is standard for time series modeling of this type.

In addition, we calculated the Portmanteau Q-statistics for each order of the series for all residual data sets up to 36 lags. In Table 3, we report the Q-statistic for order 5 only. The null hypothesis for each test (if not rejected) indicates that no additional time series modeling is necessary. The probability if large leads one not to reject the null hypothesis. With two exceptions, not all the hypotheses are rejected as indicated by the large p-values or probabilities for the Q-statistics. Only the series for Caterpillar (cat) and Hillenbrand Industries (hb) could not be modeled properly. Stated differently, these two series contained too much noise to indicate that predictability was possible. Thus, the daily closing properties could be model by time series methods and thus pattern exist in these forty-seven series.

Last, Table 3 indicates the existence of inverted roots in the sampled time series. In turn, the augmented Dickey DA and Fuller WA [23] test discussed before was completed for the forty-nine time series of 300 observations. The test statistics and p-values for the null hypothesis that the unit root exists indicate the existence of unit roots for all the time series under study. We present the estimates for the parameters of the model equation in this table as well. All of the p-values were large (at least .066) indicating the existence of unit roots. This verifies the notion that these time series are not completely random and do have properties for modeling at least for periods up to 300 trading days.

Conclusion

Documented in this study the conclusion is that daily closing prices for a sample of firms contain properties and one with enough time, patience and understanding of the mathematics of the underlying processes that give rise to a time series can properly model that time series. The result would permit one to note that time series of closing prices are not random and do have daily affects. Hence, in this study, we indicate substantially the existence of time series components in closing prices of a randomly selected set of firms traded on organized market exchanges in the United States. The results corroborate results of previous studies of international markets and previous studies of markets in the United States where time series daily and monthly components are present in closing prices and indexes of prices of securities. When these properties in closing prices exist, it is possible to forecast the patterns and thus investors can benefit from this information. Furthermore, the results indicate that the weak form of the efficient markets hypothesis is in question when one must make decisions concern investing in stock market securities. Daily variation is neither random nor stochastic and it is possible to predict daily patterns with some degree of accuracy. We suggest, for purposes of prediction that forecasters predict systematic time series components of closing prices. In addition, one cannot understate the importance of stock returns and portfolio risk. These factors coupled with recognition of systematic time series components in stock prices can make one a better forecaster for prices of individual securities. This study was given for a time period that did not occur in a pandemic era. During a pandemic and the it was mishandled by the administration of the United States government and its health agencies may yield very different result. A future study would probably lead to entirely different conclusions. Only a study utilizing multivariate ARIMA modeling would provide the comparison among non-pandemic and pandemic episodes. Since the pandemic era did not end before the finish of this study a new study would provide conclusions concerning the effects during a pandemic era should provide us with informative result. Jarrett J and Kyper E [22] could be repeated with an explanatory variable for the size and variation of the pandemic.

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