Cours 7

Question 1

What sets regime switching models appart from traditional linear models? What can regime switching models do and be used for?

Answer 1

Non-linear time series models that allow the dynamics of a given time series y_t to differ across regimes.

Regime switching models can:

-capture changes in
time series behavior and the phenomenon that the new dynamics persist for several periods after
a change.

The regimes identified by econometric methods can:
-be used for forecasting, ex.: optimal
portfolio choice, and other economic applications.

Question 2

What is a structural break model? How to define a structural break?

Answer 2

Simplest regime-switching model.

Structural break: significant change in the underlying relationship or pattern of a time series data set. This change can affect the behavior of the series, such as its:
-mean
-variance
-autoregressive properties.

In real life: Structural breaks can arise from various factors, including economic events, policy changes, or shifts in market dynamics.

Question 3

What are 2 limits of structural break models?

Answer 3

1. We can estimate the model only when the date of the structural break
   is known or it has
   been previously estimated.
2. Even assuming that a method to identify structural breaks exists, the use of dummy variables precludes forecasting future regime changes.

Question 4

How do we compute a structural break model?

Answer 4

We can estimate the model by partitioning the sample into two subsamples:
(y1; :::; y_tho) if t<tho and (y_tho+1; ::; yT) if t>=tho.

Regime witch in TIME.

For each subsample, we determine the optimal lag structure (p) using standard
methods (e.g., information criteria).

Question 5

What is a TAR?

Answer 5

Threshold autoregressive (TAR) models are among the most popular non-linear time series models.

A TAR model postulates the existence of two (or more) regimes in which y_t
follows different autoregressive processes.

For instance, the unemployment rate rises sharply during a recession, but
does not decrease as sharply during a boom.

Regime switch according to a VARIABLE.

Question 6

How to define a TAR regime? Use an example.

Answer 6

Each regime is determined by the variable q_t relative to a threshold c.

c could be set at 0 if the economy is at full capacity.

If q_t>0 : This indicates that the economy is expanding and operating above its potential, which might be associated with inflationary pressures or overheating.

If q_t<0: This suggests the economy is underperforming or in a recession, potentially leading to lower inflation or deflation.

Question 7

What is the SETAR and what makes it different from a TAR?

Answer 7

A SETAR is a TAR particular case:

The Self Exciting Threshold Autoregressive (SETAR) model assumes q_t=y_(t-d), where d >0. The delay parameter d and the threshold c can be known or unknown. In the latter case, we need to estimate d and c together with the other parameters.

Question 8

How to estimate a SETAR? What are the 2 cases distinguishable cases?

Answer 8

For given values for the delay parameter d, the threshold c, and the lags p_1 and p_2, the SETAR
model can be estimated via OLS (or, equivalently, by MLE). It is helpful to distinguish between two cases:

1. If epsilon_1t=epsilon_2t=epsilon_t, the parameters (delta_1; delta_2; phi_11; ::; phi_1p; phi_21; ::; phi_2p; sigma_epsilon), the model can be estimated with dummy variable D=1 if we are later than the threshold.
2. If the variance of the error term differs across regimes, we construct the sample y^1t:
   (y11; ::; y1T1) , which contains all the yt such that y(t-d) > c, and the sample y2t=y21; ::; y2T2, which contains all the y_t such that y_(t-d) < c. Then we estimate the equations in (1.2) sepa-
   rately.