Topic 15: Stationarity

Question 1

What does it mean if a process is stationary?

Answer 1

Its mean and variance are constant over time and the value of covariance between two time periods depends only on the distance between the two periods, not that absolute time

Known as weak stationary, coariance stationary, second order stationary

Question 2

What is the random walk model?

Answer 2

Where Y_t = Y_t-1 + u_t

A random walk with drift is

Y_t = delta + Y_t-1 +u_t

Question 3

How can the random walk model be made stationary?

Answer 3

By considering it’s first difference

Y_t - Y_t-1 = delta

Question 4

What is detrending?

Answer 4

When you have a model like

Yt = B₁ + B₂X_t + u_t

Which is non stationary, as the mean is dependent on X_t, which may be non constant.

If we know B₂ however, we can detrend, forcasting Y_t though

Y_t - B₂X_t

Question 5

What is meant when the model is said to be unit root?

Answer 5

For Y_t = ρY_t-1 + u_t

If ρ = 1, the model is said to be unit root, and is a random walk model.

But if ρ < 1 then the Yt can be said to be stationary

Question 6

What can be the result of modeling two random walk time series together in linear regression, with a very large sample size

Answer 6

Even if the sample size is very large, the normal model can show very significant relationships where none exist. Known as a spurious regression.

As a rule, if R² > d, a spurious regression may be afoot

Question 7

How can we test for stationarity?

Answer 7

• Graphical analysis
• Autocorrelation function
• Formal tests (unit root tests)

Question 8

What is an autocorrelation function (ACF)?

Answer 8

A function that takes k, the number of lags, and returns the covariance at lag k.

Question 9

What is a population correlogram?

Answer 9

When you plot ρk against k, from an Autocorrelatin function

Question 10

What kind of lag length should be use to exame Autocorrelation functions?

Answer 10

Rule of thumb is one third or one quarter of sample size

Question 11

How can we do a test on Autocorrelation function values?

Answer 11

Confidense interval of +- 1.96 x sqr(1/n)

But only relevent for single tests

Question 12

How can one test multiple ρ^ values for significance?

Answer 12

By using the Box-Pierce Q Statistic

A joint hypothesis test, if all the ρ_k up to some value are zero

Q _~ Chi(m)

Question 13

What is the Ljung-Box (LB) statistic?

Answer 13

A variant of the Box-Pierce Q stat

x

Still a large sample test, but better the Box-Pierce for small samples, also what Eviews uses for th Q stat

Question 14

With what model can we estimate ρ through a first difference regression?

Answer 14

consider

Y_t - Y_t-1 = (ρ + 1)Y_t-1 + u_t

we consider the form ΔYt = delta x Y_t-1 + u_t

Question 15

What does it mean if a times series has an integrated order of 1

or I(1)

Answer 15

It means that the time series is stationary after the first difference is taken

Question 16

How can we test for stationarity given the model ΔYt = delta x Yt-1 + u_t

Answer 16

The τ (tau) test, also known as the Dickey-Fuller (DF) test

if |τ| > |DF| reject the null of non-stationarity

Question 17

How do we choose critical values for the Dickey-Fuller (DF) test?

Answer 17

Three different forms of stationarity to consider, all with different critical values

Random walk

Random walk with drift

Random walk with drift around a trend (ie, with B1 t )

Question 18

Can we use the Dickey Fuller test if the error term is autocorrelated?

Answer 18

Nope. We use the Augented Dickey-Fuller (ADF) test instead!

Same tests & critical values as the normal Dickey Fuller test

Question 19

How do we test multiple coefficients in the Dickey-Fuller test

Answer 19

Do a restricted - unrestricted test

But use Dickey-Fuller F critical values

Question 20

What is cointegration?

Answer 20

Where two nonstationary series when modelled together are stationary.

So normal F tests, T tests are all fine.

You call the B₂the cointegrating parameter

To find it, try the model and run unit root tests with altered DF values, that eviews doesn’t even have? idk