# Time Series: Lesson 1

Welcome!

Watch the videos. Then see whether or not you can answer the problems at the end.

Here, for your amusement and edification, are some problems to work on:

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## 7 thoughts on “Time Series: Lesson 1”

1. MS says:

Let me try to see if I understand the concepts correctly.
In a stationary and a strongly stationary time series the pdf of each xj would have to be identical, but this does not have to hold for a weakly stationary time series.
So if we have two distributions with the same mean and variance – called pdf1 and pdf2 – and every second xj follows pdf1 and the others follow pdf2 I suppose it will be weakly stationary but not strongly stationary.
This is also an example of a white noise process which is not i.i.d. if I understand the concepts correctly.
I do not know which physical process would behave like this.
I found the proof of problem 3 straightforward, but I could use a hint to problem 4.
I like the idea of short videos. It is easy to go back and repeat a detail to understand it better.

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2. Hey, I’m in your target market and I really enjoyed these videos! I don’t think I’m clever enough to answer your questions but here’s a try. An IID time series would be something like the results of tossing a coin. The coin toss outcomes don’t depend on each other. It’s the same coin each time so the probability distribution is the same throughout.
I’ve got a feeling (or a dim memory) that brownian motion and the maybe the noise voltage from a resistor are white noise. I don’t know if they also qualify as IID, or how you would work that out.

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3. PJKar says:

I found an interesting random process that can be used for both problem 1 and problem 2. It is from the exercises in Chapter 1 of Brockwell and Davis. This example was interestng becasue it shows what is required when trying to determine stationarity from first principles.

I will try to use Latex to express the process but there is no telling how it will turn out, If it fails I will rewrite it in script in the next post.

$Z_t \: is \:IID \: N(0,1)\\\\ X_t=Z_t \: if\: t \:is\: even\\\\ X_t=\frac{Z_{t-1}^2 -1}{\sqrt{2}} \: if\: t \:is\: odd$

For problem 2 This process has zero mean and variance 1 for all t. Demonstrating this takes advantage of the fact that the third moment of a standard norrmal distribution is equal to 0 and the 4th moment is 3. So this process is weakly stationary.

It can also be shown that the for t even the probability that X<0 is .5 and the that for t odd the probability of X<0 is .6826 verifying that the distrbutions are different for different t so it is not strong stationary.

For problem 1 if it can shown that the covariance is zero for all all lags h then X is a white process but because the distrbutions are different as indicated above then it is not IID. Showing the required property for the autocovariance of X was not that simple (to me). It seems maybe independence has to be assumed for lags greater than 1

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4. PJKar says:

Where the above refrences Problem 2, it should have it should also have noted the time translation invariance of the second moment for weak stationarity.. This was addressed (albeit lamely) in the reference to problem 1

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5. PJKar says:

Here is an alternative solution to Problem 4 that was given in the presentation in video 2-1. It is an algebraic approach from Papoulus Probability Random Variables and Stochastic Process often used in Enigineering cirrricula. The link is to an image captured Word document.

Clicking the image enlargens it.

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1. PJKar says: