ARMA Model

[marys]

Hi, so I'm trying to fit an ARMA model to describe daily stock returns.

1. If I find that I need an MA(6), do I still have to add MA(1) to MA(5)? Some of them become statistically insignificant as I add more MAs - e.g. MA(2) and MA(3) become insignificant when I add MA(6).

2. Is there something wrong when I find higher order AR? For example I'm looking at my correlogram and I still need an AR(21). But my series is already stationary when I began fitting the model. Should I be checking for something else or can this be explained by underlying stock fundamentals?

3. Are ARMA residuals supposed to follow normal distribution?

4. Any particular tests I should conduct to see if my ARMA model is ok? I mean aside from autocorrelation, heteroskedasticity, etc?

Thanks!

[xprimexinverse]

Hi,

I would help if you could please post a copy of the data.

Alternatively, plot for us the following (1) data in levels (2) ACF and PACF of data in levels (3) data in first differences (4) ACF and PACF of data in first differences.

In the meantime, below is a general response to some of your questions.

Questions for you. What method are you using to identify the ARMA structure? Possibly, Box-Jenkins, a specific-to-general search based on retaining significant lags, minimizing some criteria such as AIC, or some other method?

General points. Residuals from an ARMA model should contain no information - you can check this by examining the residual ACF. You can fine tune these by adding MA terms based on spikes in the residual ACF, however, there is a trade-off between including more parameters for estimation and building a parsimonious model. You'd normally check that the residuals are normal, yes. Essentially, investigate the first four moments and ACF of the residuals.

A satisfactory ARMA model is usually deemed "statistically adequate" if the residuals of the model are white noise. Parsimony may also play a role depending on the method you employ. Lastly, if your model contains AR terms you need to check the stability of the roots. Similarly, if your model contains MA terms you need to assess the invertibility of the model.

All of the above concepts are discussed in the main textbooks about ARMA modelling. Among these are the Box and Jenkins classic textbook and I'd also include Alan Pankratz's books on ARMA modelling, too. Many other books (and lecture notes online), in fact, do a pretty poor job. So, dig out the main references for more in depth details.

Cheers,
Graeme

[marys]

Hi, thank you for the reply.

The following is my original correlogram without the ARMA terms. My original regression was CAPM, so it's just excess returns on stock (y) and excess return on market (x) plus a constant.


When I tried fitting the model I checked for stationarity first. Then when I saw that it is I mostly trial-and-error'd my way and checked the Q-stats of the correlogram as I went along. As you can see from the correlogram there's a stubborn autocorrelation in lag 21 so I added it once lags 1-20 stopped showing autocorrelation. What I have so far is my original regression plus the following ARMA terms:

ma(1) ma(5) ar(2) ar(3) ar(5) ar(7) ar(21)

Like I previously said I deleted the ARMA terms that became insignificant as I added more terms. Is this the right way to go about it? In my textbook they added everything.

Meanwhile this is my correl after I added the ARMA terms:
Estimation output:

My concern is even when the lags no longer show autocorrelation (based on the correlogram anyway), when I check for heteroskedasticity I find that it's still there. Also my residuals are still leptokurtic. So I feel like I must be doing something wrong/not considering something.

Not sure if this is exactly what you meant by data but here:

Level explanatory variable:
First difference of explanatory variable:
Correlogram of explanatory variable (level):
Correlogram of explanatory variable (first difference):

Really appreciate the help.

[xprimexinverse]

Might be easier to help if you could provide the data. Otherwise, this could end up going back and forth. Thanks.

By the way, my initial impression is that you have too many parameters in your model. Think from a theoretical standpoint, if stock markets are unpredictable (random walk hypothesis), then the appropriate ARMA model is going to be relatively simple.

[marys]

Hi, thanks for the reply. I've attached a copy of the file.

_workfile1.xlsx

time series

(114.25 KiB) Downloaded 437 times

Edit: I was thinking about it and isn't a random walk model more appropriate for a non-stationary series? I tested for stationarity and concluded that my series is stationary as far as using ADF is concerned. I am thinking maybe the lagged stock returns (lagged y) could be added? I gave it a shot and I think my correlogram gets a liiiitle better, but again there's lag 21 sticking out. And that brings me back to my original question as to whether I have to actually write AR(1) to AR(20) if it looks like I need an AR(21)... Is it even necessary that I show that many lags? In Tsay's textbook it says m=ln(T), but I'm concerned since typically a month has 21 trading days, so I feel like that should be accounted for.



Also I forgot to mention but my data are all log returns. Not sure if that changes anything.

If you can suggest a simpler ARMA model that can address all the weird stock return characteristics, I'll really appreciate that.

Edit 2: A follow-up question to this is actually about using GARCH to model volatility. Will an ARMA-GARCH be something worth considering? Or does ARMA modelling have to necessarily come before GARCH?