Question

Consider the following three models that a researcher suggests might be a
reasonable model of stock market prices
yt = yt−1 + ut
yt = 0.5yt−1 + ut
yt = 0.8ut−1 + ut
(a) What classes of models are these examples of?
(b) What would the autocorrelation function for each of these processes look
like? (You do not need to calculate the acf, simply consider what shape it
might have given the class of model from which it is drawn.)
(c) Which model is more likely to represent stock market prices from a
theoretical perspective, and why? If any of the three models truly
represented the way stock market prices move, which could potentially
be used to make money by forecasting future values of the series?
(d) By making a series of successive substitutions or from your knowledge of
the behaviour of these types of processes, consider the extent of
persistence of shocks in the series in each case.

Answer 1

a)

The first equation is a random walk or root process function, i.e. a non-stationary process with a value of 1.

The second equation describes a stationary Auto Regression cycle, since the coefficient is less than 1.

The third equation is a Moving Average model since its a residual term is a number.

b)

In the first equation, the non-stationary cycle will be fitted with a non decreasing ACF, with correlation bars equal to 1.

ACF will decline very quickly during the second stationary phase, and its coefficients will decrease by half the value of the last period.

At the first order on the horizontal axis, the third cycle would be non-zero because the MA's ACF determines the order of the cycle.

c)

While these processes are quite simplistic, the closest to any stock market process will be a self-reliance process, as stocks are based on previous values, which are improved in the form of residual acoustics. To say that a stock is a random march would mean that it is completely unpredictable, except that this is not so in the longer term. The second method of these three can be used to predict potential stock prices. But a real-world ARIMA (p, d, q) model is typically suitable as a combination of the properties of the given processes to forecast future stock market values.

d)

By the lagging time coefficient, we can say the persistence of a cycle. Persistence is supreme in the first process, i.e. the process at a time t has been extremely persistent or relies entirely on its importance across periods (t-1). The second cycle is less stable because it relies partly on the importance of its previous era. In this cycle, persistence is lower. The continuity with its last error term is high in the third MA cycle, because the coefficient is near 1.

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