Stock Market Price Models

Question 1

a. Consider the following models which a researcher suggests might be a reasonable model of stock market prices

i. What classes of model do these examples represent

The above examples of models represent the Random walk class. A random walk represents a succession of steps in a path. The steps are random and are described on a mathematical space. From the examples above, Yt = Yt-1 + Ut and Yt = 0.5Yt-1 + Ut are random walks, as they are made of a succession of integer steps, for example; Yt-1 and then 0.5Yt-1. The first model has a coefficient of 1 hence it is regarded as a non-stationary process. The second model is an autoregressive stationary process, since the coefficient is 0.5 which is less than 1, and then the process is stationary. The third model represents a moving average process; it has a function made from the residual term.

ii. The autocorrelation and partial autocorrelation functions for each of the processes

The autocorrelation function for instance at lag k, is the correlation between values, which are k intervals apart in a series. For partial autocorrelation functions, also at lag k, is the correlation between values that are k intervals apart in a series, but the values between the intervals are taken into consideration (Degerene, 2003).

For the examples above, the random walk is time dependent and the autocorrelation is as shown below;

Therefore, if we consider a long time series just like time series of stock prices with short term lags, we can obtain an almost unity autocorrelation given as;

Hence, we have a very high autocorrelation, which does not decrease proportionally as lag increases.

In the first model, Yt = Yt-1 + Ut, the coefficient will be 1, and they will exhibit a decreasing autocorrelation coefficient. For the second model, Yt = 0.5Yt-1 + Ut the coefficient bars reduced to 0.5, causing the autocorrelation function to decline rapidly. For the third model, Yt = 0.8Ut-1 + Ut is a Moving Average, and the ACF is determined by order of the cycle.

iii. Model likely to predict stock market prices from a theoretical perspective and if the three represented the way stock market move, which one could we use to make money by predicting the future value of the series?

Looking at the models we have, the stock price at time t depends on the amount at t-1. Therefore, the model we are likely to use from a theoretical perspective is Yt = 0.5Yt-1 + Ut. If the three models represented the way stock market move, the model we could use to make money by predicting the future is still Yt = 0.5Yt-1 + Ut. This is because the model is an improvement of the previous model value. Stock values are base on the closing costs, which can be improved by residual acoustics forms.

iv. The extent of the persistence of shocks to the series in each case

By using the lagged term coefficient, the persistence of a process can be determined. For model 1, we can say that the model is ultimate persistent that is, the process at time t is entirely dependent on its value (t-1), and it is considered highly persistent. For the second model, the model is slightly persistent, and the model partially depends on the values of the prior period. For the third model, it has a coefficient of 1; therefore, the persistence is high.

Question 2

a. Explain why it is crucial to test for non-stationarity in time series before attempting to build an empirical model

In a stationary time-series, all the statistical characteristics of the model such as variance, mean, and autocorrelation do not vary over time; while in a non-stationary time series, the statistical features of the time series model such as autocorrelation, variance and mean vary over time (Mcleod, 1981). In building a mathematical model, it is crucial to test for stationarity since the statistical analysis cannot be done on non-stationary data. The non-stationary data then has to be converted to stationary to allow for further statistical analysis to be done on the set of the time-series data. The logic behind this conversion is to enable statisticians to predict the variance and mean of future values since the model’s mean and variance increases at a constant rate.

b. Test whether the following model is stationary

Yt = Yt-1 + Ut

As explained earlier, a stationary time series is a random model with constant covariance, mean and variance. To test for stationarity, we determine whether mean, variance and autocorrelation are constant for that particular time series model. Using the autocorrelation coefficient and partial autocorrelation coefficients will help determine the stationarity of a particular time-series data. Besides, other methods can be used to test for stationarity including the DF, ADF, GARCH, KPSS and also to use statistical analysis tools such as Gretl (GNU Software), Minitab, Microsoft Excel and many others. Let’s now test whether Yt = Yt-1 + Ut is stationary. Firstly, Yt = Yt-1 + Ut is an ARMA model with coefficients equal to 1. So, its stationary since its mean, variance and covariance are constant, and they are equal to 1.

c. Reasons why ARMA models are particularly useful for financial time series

ARMA stands for Autoregressive Moving Average Models and sometimes is referred to as Box-Jenkins, named after its founder (Rojas, 2008). Univariate ARMA estimates the values of the future construct by making a comparison with the latency of the particular data. The model works best when the data portrays a steady and predictable example over some time. When analysing information which is very long and has stable connections among past perceptions, ARMA is preferred over exponential smoothing strategies. Before applying the ARMA approach, the first task is checking for stationarity; stationarity ensures that the data arrangement stays in a steady-state over a given period. Without taking into consideration the stationarity information, most of the figures related to the data can’t be recorded. Also, ARMA models are essential in financial series application due to their adaptability. The ARMA models are assessed easily, and they can deliver favourable forecasts and most importantly, they don’t require too much learning of the essential factors, which are required in the conventional investigation of econometrics. At times when information needs to be accessed most frequently, logical elements such as accounting ratios and macroeconomics variables cannot be applicable for an interval of monthly interims, instead, ARIMA models can be utilised to ensure consistency of the forecast.

Question 3

a. Stylized features of financial data that cannot be explained using linear time series models

Several features of financial data are suggested regarding linear time series. There are various features of financial data, which cannot be taught using linear time series models, for example, frequency, every time a trade occurs or somebody suggests a new quote, the prices of the stock market are measured, and this implies that the rate of the stock market price data is very non-stationary. Another feature not explained by linear time series models is the financial data, for example price of assets. The price of assets has covariance non-stationarity but for the case of returns, the data can be considered stationary. Linear independence is another feature that cannot be explained in linear time series models, time series models have less evidence about the autoregressive of the data more, especially when the frequency is low. Volatility asymmetries and volatility pooling is another feature not explained in the time series models, and there are clustering and leverage effects in the returns (Granger, 1994).

b. Which of these features could be modelled using a GARCH (1,1) process?

GARCH (1,1) models are specifically created to capture volatility pooling and clustering effects in returns. The GARCH (1,1) can model squared residuals or returns’ dependence, as well as capture unconditional leptokurtosis, even when the residuals of a model are given as by the equation of part d below, the squares are leptokurtic which are standardised from the estimation of GARCH model.

c. Why in the recent research, have researchers preferred GARCH (1,1) models to pure ARCH(p) models?

In most cases, GARCH overcomes all the shortcomings imposed by ARCH(p) model. The ARCH(p) drawbacks include the complexity in deciding on q, the tremendous value of p and violation of non-negativity constraints. The value of variance required when estimating ARCH model should be greater than 0, as variance must be positive. GARCH(1,1) ensures that, all the disadvantages are overcome. The GARCH(1,1) model is characterized by only three parameters in the variance equation which is different from q+1 in the ARCH(p) model. As it is possible to write a GARCH(1,1) model in the form of ARCH(∞), the model is capable of capturing all important dependence of the squared returns..

d. Range of values for the coefficients

Yt are returns, their mean values would be expected to be small and positive as well. Let’s assume the frequency of the data is based on annual returns, then the mean value (µ) would be the daily average percentage return for the total years and the value might be 0.05%. The value of α0is expected to be so small, say α0 = 0.0001 or less than that. The unconditional variance is calculated using the formula α0/ (1-(α1+α2)). The values of α1 and α2 are 0.8 and 0.15 respectively. All values of α are positive and their sum should be equal or close to 1.

Question 4

a. Advantages of constructing a panel of data rather than using pooled data

In a panel data, the data contains time series observations on a number of individuals. The data involves two dimensions, cross-sectional dimension represented with subscript i and time dimension represented by subscript t (Pesaran, 2015). Another advantage of panel data is that of collecting the data, it is easier to collect panel data as compared to collecting pooled data. Also, panel data allows for time series and cross-sectional data to be taught as one lesson in a longitudinal panel data study.

b. Relative advantages and disadvantages of fixed versus random effects specifications and how could you choose between them for application to a particular problem?

Fixed and random effects are longitudinal data found commonly in sociology. Their main advantage is that they are used for time-invariant data control, to avoid omission of variables. In implementing these models, analysts face various challenges, among them is the uncertainty whether to use the fixed versus random effects models. In a fixed effect model, the model parameters include non-random quantities which are fixed. In general, grouping of data can be done according to fixed or random effects for the groups. To choose a fixed effect model, each group must include a specified fixed quantity.

c. Application of panel regression model in academic finance literature

Most of the present works are part of panel data study, let’s consider an institution such as a college, the type of data collected in a college bears observation on the various phenomenons, and they are done over numerous periods. Panel data contains two dimensions, cross-sectional and longitudinal. To determine whether a fixed effect or random effect was employed, let’s consider the variation of the error term. For that case, the error term varies non-stochastically implying that a fixed effect model was applied. The panel regression model can be used in data that contains both dimensional and time series characteristics. In particular, a fixed-effect model minimises difficulties involved in the interpretation of the regression model. The panel regression model indicated that, the results obtained were accurate as the model took into consideration as they were reliable. Pooled regression is characterised by constant coefficients which include slopes and intercepts. If the college employed pooled regression model in its analysis, then they had to combine all data types and run an ordinary regression test. Since there are no specific characteristics of data within the measurement set, and time is not considered, the model would produce different results compared to that of panel regression model.

References

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Shyh-Jier Huang, Kuang-Rong Shih, 2003, Short-term load forecasting via ARMA model identification, IEEE Transactions on power systems journal, vol 18, no. 2, p. 674.

Al Mcleod, WK Li, 1981, Distribution of the residual autocorrelations in multivariate ARMA time series models, Royal Statistical Society Journal, vol 43, no. 2, p.232.

Yves Croissant, Giovanni Millo, 2008, Panel Data Econometrics in R; The plm package, Statistical Software package journal, vol 27, no. 2, p. 40.

Anindya Banerje, 2009, Panel data unit roots and cointegration: an overview, Oxford Bulletin of Economics and Statistics, vol 61, no. 1, p. 620.

Serge Degerene, Sophie Lambert-Lacronix, 2003, Characterization of the partial autocorrelation function of nonstationary time series, vol 87, no. 1, p. 58.

Clive Granger, 1994, Using the mutual information coefficient to identify lags in non-linear models, time-series analysis journal, vol 15, no. 4, p. 380.