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I have a time series that has at least an ARCH(1) process in the error. The model is:

ŷ = β0 + β1*x + ARCH + e

In other words, a constant, 1 independent variable and an ARCH term and the errors. I have the following residual plots. The first in level form: enter image description here

And standardized:

enter image description here

The second portion of the data set seem to be stationary when transformed into standardized errors, however the first portion has clear periodicity. Assuming that there is no specification error in the theory of my model (big assumption right?), but instead postulating between the following:

  • Could my ARCH process be of the wrong order? Mine is simply ARCH(1)
  • Or should I transform my variables? Currently they are in level form. I did try log form, but both residuals plots still came out the same. I did not try first differencing yet.

If there is another explanation, please share. I feel there are many diagnostic approaches and it's a little overwhelming. My hope is that within the resid plots above there is a sign of sorts that I have missed and perhaps those with more experience will notice.

enter image description here

Arash Howaida
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  • What is your goal? To predict the time series or to predict variation? – keiv.fly Mar 07 '17 at 06:58
  • predict the time series without violating too many time series statistical assumptions – Arash Howaida Mar 07 '17 at 07:33
  • Why an additive ARCH term in the equation for *conditional mean*? Is it an ARCH-in-mean model? – Richard Hardy Mar 07 '17 at 07:37
  • I have been considering adding an MA term, but right now only it's variance is handled discretely. Do you suspect its actually not accounting for conditional mean that has resulted in periodicity in the errors? – Arash Howaida Mar 07 '17 at 07:43
  • Sorry, did not notice your message as you did not include @RichardHardy. I do not understand you question. Could you specify the model along the lines $y_t \sim N(\mu_t,\sigma^2_t)$ where $\mu_t=\dots$ and $\sigma^2_t=\dots$ (fill in the dots)? E.g. $\mu=\beta_0+\beta_1 x_t$ and $\sigma^2_t=\omega+\alpha_1 \varepsilon_{t-1}^2$ where $\varepsilon_t:=y_t-\mu_t$. Like in [this answer](http://stats.stackexchange.com/questions/41509/what-is-the-difference-between-garch-and-arma/231512#231512). – Richard Hardy Mar 07 '17 at 10:51
  • @RichardHardy, I'm afraid I'm not adept enough to give you a precise answer for that. I'm a very visual learner, and sometimes the equations go over my head. Let me try to explain what I can. I'm using software to conduct the regression, I doubt the way it calculated my regression is abnormal. After I ran an auxiliary regression on the first lag of the squared resid, I found an ARCH(1) process. Then I estimated a new model. I basically input two things, the mean equation: y c x1. Then I chose the arch order, ARCH(1), and the software took it over from there. – Arash Howaida Mar 07 '17 at 11:01
  • So probably your model was \begin{aligned} y_t &= \beta_0+\beta_1 x_t + \sigma\varepsilon_t \\ \sigma^2_t &= \omega + \varepsilon_{t-1}^2 \\ \varepsilon_t &\sim N(0,1) \end{aligned} which means there is no ARCH term in the first equation but the error terms has time-varying conditional variance as per the second equation. What was your code to run the model? – Richard Hardy Mar 07 '17 at 11:46
  • @RichardHardy I just entered my y c x1 as in the above comment. I provided a screenshot of the regression printout screen. I'm not at my work computer anymore, so I just whipped this up. It's a different dataset, but the procedure is exactly the same. That should explain some of the details of my model. If you're not familiar with eviews I can explain more if anything in the printout is unclear to you. – Arash Howaida Mar 07 '17 at 12:58
  • I will try to look at EViews documentation to figure out what model your code defines. – Richard Hardy Mar 07 '17 at 14:17
  • @RichardHardy Perhaps for simplicity (and the interest of time, the eviews documentation is lengthy) Let's just assume there are no major specification errors on my part, and also just go with our best guess for what equation eviews used. Would that help? – Arash Howaida Apr 16 '17 at 17:10

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