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I am studying my lecture notes where I saw this:

$Y_t=Y_{t-1} +u_t$

$Y_t=0.5* Y_{t-1} +u_t$

$y_t=0.8* u_{t-1}+u_t$

The first two models are AR(1) and the third one is an MA(1) model.

In the first model where $\beta$ is 1, the difference between today's value and yesterday's value is error.

Well, which model should I use in order to represent stock market prices from a theoretical perspective? Why? I am a maths student so I dont have economic / finance knowledge.

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First of all, you might want to define what it is that your variables mean. What are Y, u and y? –  rocinante Apr 5 at 12:38
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2 Answers 2

up vote 2 down vote accepted

There is a significant literature on this subject. Stat Tistician provides the right answer, but I would strongly recommend pretending for a second that he didn't provide it and to figure it out on your own why he (and many, many others) reach those conclusions. It will help you learn about time series analysis.

Find some data (historical stock price data can be downloaded easily from any number of sources, most notably yahoo finance) and then plot it (I would suggest a large cap, highly liquid stock that has been in existence a long time or maybe the S&P 500 index). Take the log of the prices (there are other questions that explain why to take logs for series like these, I won't go into it), then plot it. What kind of ARMA model do you think would fit the log prices? Follow the Box-Jenkins methodology (plot PACF, etc.). Fit what you think is the best ARMA model, then examine the residuals. Are they stationary? If not, you've specified the wrong ARMA model. What does the plot of squared residuals look like? What does that mean? Are the residuals normally distributed? What does that mean?

If you can perform that sort of analysis on other types of data you've never seen before, then you will have learned a lot more about time series analysis than simply having the answer.

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What do you mean with I did not provide it? –  Stat Tistician Apr 7 at 15:03
    
No, I said you did provide the right answer. Just that he should think for himself why it's right. –  John Apr 7 at 15:09
    
Ah ok, now I got it. –  Stat Tistician Apr 7 at 15:11
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Well, I cannot give you a perfect answer, since your question lacks some informations which would be necessary to know what you are pointing at.

But:

First of all, I think you are talking about stock market returns, because for those models metioned you need stationarity and that will most likely be the case in returns. So your Y represent the stock returns, e.g. for each day. The prices in general won't be modelled by these models.

Furthermore:

Most likely there will be no autocorrleation in stock market returns, because then you could trade on it and make money (simplified). So in most cases the AR(1) term will be non-significant or just weak significant and the value itself very small. If you use a rolling estimation and plot the AR(1) coefficient over time, you will see that it will be very small all the time and in most market situations it will be non-significant. What reasons should there for using a MA(1) model? I don't know it.

Your first AR(1) model is the so-called "random walk" model (without drift): it assumes that, from one period to the next, the original time series merely takes a random "step" away from its last recorded position.

So in case of random walk you do not know anything about the behaviour, maybe that fits it best?

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