Question

What do “non-stationary error term”imply in modeling the relationship between economic variables? Clearly explain with an economic example

Answer 1

A stationary (time) series is one whose statistical properties such as the mean, variance and autocorrelation are all constant over time.

A non-stationary series is one whose statistical properties change over time.

Non-stationary data should be first converted into stationary data (for example by trend removal), so that further statistical analysis can be done on the de-trended stationary data. This is so because for example, if the series is consistently increasing over time, the sample mean and variance will grow with the size of the sample, and they will always underestimate the mean and variance in future periods. The usual mean of “de-trending” a series is to fit a regression line and then subtract it from the original data.

Examples of non-stationary processes are random walk with or without a drift (a slow steady change) and deterministic trends (trends that are constant, positive, or negative, independent of time for the whole life of the series).

Pure Random Walk (Yt = Yt-1 + εt ) Random walk predicts that the value at time "t" will be equal to the last period value plus a stochastic (non-systematic) component that is a white noise, which means εt is independent and identically distributed with mean "0" and variance "σ²." . It is a non-mean-reverting process that can move away from the mean either in a positive or negative direction. Another characteristic of a random walk is that the variance evolves over time and goes to infinity as time goes to infinity; therefore, a random walk cannot be predicted.

For Example:

A non stationary time series's mean and/or variance are not constant over time.
Consider the model -

y(t) = a + bt + c y(t-1) + u(t) ;
where u(t) is white noise : E[u(t)] = 0 and var[u(t)] = σ2σ2

Non stationary pure Random Walk
If you have a =0 and b=0 and c = 1; then your model becomes
y(t) = y(t-1) + u(t)
This is pure random walk and non stationary.

Why non stationary
Substitute for y(t-1) = y(t-2) + u(t-1)
And then y(t-2) = y(t-3) + u(t-1) and so on and you will get

y(t) = y(0) + u(t) + u(t-1) + u(t-2) + ....

So, we have E[y(t)] = y(0).
Var(y(t)) = t * σ2σ2 - Non stationary!

Also,

Many economic and financial variables are non stationary. Nominal GDP is one such. Below is UK's GDP over the years.