Seasonality

Summaries

Seasonality: the time series has repeated pattern

If this occurs (by computing against lag k residual covariance), cannot just use simple AR(1) even though it is not random walk

In this case, add the lag term into the AR(1) model (note that the following is till order 1 because we do not use 2 equations)

y__t = b0 + b1 y__t-1 + b2 y__t-k + error where k is the lag

Autoregressive Conditional Heteroskedasticity (ARCH)

-          in a time series, the error variance depends on previous variance (a special case of serial correlation)

ARCH(1) model: regress error(t) against error(t-1) (i.e. the first lag)

Error(t)^2 = a0+a1 Error(t-1)^2 + error

If a series is nonstationary, we cannot just use AR(1) (becasue t-statistic maybe invalid). Therefore do the following regression:

y(t)- y(t-1) = b0 + (b1-1) y(t-1) + error

This has to be stationary (because (b1-1)<1). Then do t-test (not standard t-value) to see if b1-1 is close to zero.

This method is called the Dickty Fuller test (DF test)

Cointegration:

If 2 time series are both covariance non-stationary, but cointegrated, AR(1) is reliable.

Cointegration means that are economically linked (follow the same trend) and expected not to change.

We form: y(t) = a0 + b1 x(t) + error and then perform DF test (but have to use DF-EG adjusted t-value)