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I am currently working on a simple linear time series regression model that looks like this:

$P_{t}=\beta _{0}+\beta _{1}X_{t}+\varepsilon _{t}$

Yet, I have problems regarding how to deal with the issue of a very heavy tailed regression model.

Plotting the data against each other results in the following graph:

Data plotted against each other

Qqplotting the residuals of the regression results in the following graph:

QQPlot - Regression Model

Now, in essence I have two questions regarding this issue:

1) I would assume, that the first graph implies non-linearity of the regression model. Is this a reasonable assumption, or can one still assume linearity? If no - how do I establish linearity in this specific case? I have tried my best, yet I can not find a solution.

2) In this case (bivariate linear regression), both graphs basically display the same thing. Is this correct?

I am thankful for any comment or even an answer on how to deal with this issue. I tried reading the threads touching this topic, but I did not find them very helpful regarding my questions.

Thank you so much in advance.

EDIT 1: The dependent variable is electricity price and the independent variable is load data. The endogenity problem here is another issue, which I just have to deal with, as I am explicitly supposed to use the load as the independent variable. The nature of the data also implies, that the fat tails are not caused by measurement errors, but by extreme fluctuations in the electricity market.

EDIT 2: As requested, several time series plots.

Independent variable against time:

enter image description here

Dependent variable against time:

enter image description here

ACF (Regression Model as described above):

enter image description here

PACF (Regression Model as described above):

enter image description here

I wanted to account for seasonality with dummy variables. The ACF/PACF charts indicate AR(1), at least to my understanding. I was planning to apply the chochrane-orcutt method in order to eliminate serial correlation in the error terms.

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    $\begingroup$ At most, normal distribution is an assumption (better read as ideal condition) for the errors, not any of the variables that enter a regression. Nothing in your analysis has time series flavour and the most relevant graph is the scatter plot which poses a challenging mixture of mostly linear behaviour and some marked departures from that for low and high values. The most useful next graph is one not here, plotting the two variables against time. The best model will surely be found by considering the processes underlying your system, about which is nothing is said here. $\endgroup$ – Nick Cox Nov 15 '17 at 17:24
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    $\begingroup$ In addition to answering @NickCox points it might help to say what the variables are as substantive knowledge often helps to explain distribution. $\endgroup$ – mdewey Nov 15 '17 at 17:27
  • $\begingroup$ Thank you, I entirely reedited the question. Yet the problem I have is linearity of the regression model - how does plotting the data against time helps with that? Isn´t that a way to go when dealing with autocorrelation? $\endgroup$ – shenflow Nov 15 '17 at 18:42
  • $\begingroup$ Minimally, you've described what you have done as a time series regression model, but as you've explained it, your model ignores time. (Subscripting here is just cosmetic; you're assuming or asserting instantaneous response, so that lags don't bite.) That may be sensible but without seeing the data as time series we have no way to check on or advise on whether it is sensible. Time series people might also reasonably ask for autocorrelation, cross-correlation etc. $\endgroup$ – Nick Cox Nov 15 '17 at 19:34
  • $\begingroup$ Alright, got it - I edited the question and included the time series charts. I also included an ACF as well as a PACF chart. I was planning to account for seasonality by adding dummy variables and to eliminate autocorrelation by applying cochrane-orcutt. I thought the problem of one explanatory variable displaying a non-linear relationship (even if I expanded the model to a multiple regression model by adding dummy variables) would violate the assumption of a linear model. That is why I thought other information would not add any value to the issue I originally formulated. $\endgroup$ – shenflow Nov 15 '17 at 21:05

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