Consider the autoregressive model of order 2 $$X_{t}=\varphi_1X_{t-1}+\varphi_2X_{t-2}+\varepsilon_t,$$ where $\varepsilon_t$ are zero-mean normally distributed random variables with $\sigma^2$ variance such that these random variables are uncorrelated. Suppose that we have a sample for the above model with sample size, let say, somewhere between 100 and 1000.

As a part of a simulation I would like to know something about the probability distribution of the data. Some examples I made using MATLAB suggests that some normal distribution produces an excellent fitting, which - at least in my opinion - makes sense because the white noise process in the model is normally distributed.

However I am not skilled in the topic. Is there any result in the literature which can give me a theoretical base for this suggestion (that a sample of the above AR(2) model is normally distributed if considered as values of some random variable rather than as a time-series)? If there is not any, then how can I support the good fitting of some normal distribution?


At stationarity, the AR(2) model above is centered normal and its covariance structure $c_n=E(X_tX_{t-n})$ is given by $$ c_0=\frac{1-\varphi_2}{\Delta}\sigma^2,\qquad c_1=\frac{\varphi_1}{\Delta}\sigma^2, $$ where $$ \Delta=(1-\varphi_2)(1-\varphi_1^2-\varphi_2^2)-2\varphi_1^2\varphi_2, $$ and, for every $n\geqslant2$, $$ c_n=\varphi_1c_{n-1}+\varphi_2c_{n-2}. $$

  • $\begingroup$ What is the difference between normal and centered normal distributions? A short description will be more than enought, but I have never heard of centered normal distributions. $\endgroup$ – Marci Mar 23 '14 at 17:12
  • $\begingroup$ Centered normal is $N(0,\sigma^2)$. Normal is $N(\mu,\sigma^2)$. $\endgroup$ – Did Mar 23 '14 at 17:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.