What does it mean if a process is stationary?

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

What is the random walk model?

Where Y_{t} = Y_{t-1} + u_{t}

A random walk with drift is

Y_{t} = delta + Y_{t-1 }+u_{t}

How can the random walk model be made stationary?

By considering it's first difference

Y_{t} - Y_{t-1} = delta

What is detrending?

When you have a model like

Yt = B_{1} + B_{2}X_{t} + u_{t}

Which is non stationary, as the mean is dependent on X_{t}, which may be non constant.

If we know B_{2} however, we can detrend, forcasting Y_{t} though

Y_{t} - B_{2}X_{t}

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

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

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

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 R^{2} > d, a spurious regression may be afoot

How can we test for stationarity?

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

What is an autocorrelation function (ACF)?

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

What is a population correlogram?

When you plot ρk against k, from an Autocorrelatin function

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

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

How can we do a test on Autocorrelation function values?

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

But only relevent for single tests

How can one test multiple ρ^ values for significance?

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)

What is the Ljung-Box (LB) statistic?

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

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

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}

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

or I(1)

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

How can we test for stationarity given the model ΔYt = delta x Yt-1 + u_{t}

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

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

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

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 )

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

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

Same tests & critical values as the normal Dickey Fuller test

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

Do a restricted - unrestricted test

But use Dickey-Fuller F critical values

What is cointegration?

Where two nonstationary series when modelled together are stationary.

So normal F tests, T tests are all fine.

You call the B_{2 }the cointegrating parameter

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