TAR Procedure

@TAR is a procedure for testing for and estimating a "self-exciting" threshold autoregression. The testing method is from Hansen(1996).

The regression includes a constant and some set of lags of the dependent variable. These can either be consecutive lags (use P option), or a list with skips in it (use LAGLIST option). As a "self-exciting" TAR, the threshold variable must be one of the lags—all the included lags are tested as possible thresholds.

For a more general procedure for estimating and testing threshold models, see @THRESHTEST, which allows for any series to be a threshold in any linear regression. It does, however, require that the threshold series of interest be known, while @TAR tests across lags.

@TAR( options ) y start end

Parameters

y	series to analyze
start, end	range of y to use. By default, the defined range of y

Options

P=number of lags [12]

LAGLIST=||list of lags to use||

R1=starting quantile of data to use as possible thresholds [.15]

R2=ending quantile of data to use as possible thresholds [.85]

NREPS=number of repetitions for computing approximate p-values [0]

ROBUST/[NOROBUST]

Use heteroscedasticity consistent estimates for standard errors.

[PRINT]/NOPRINT

TITLE=title for report ["Threshold Autoregression"]

Variables Defined

%CDSTAT	Maximum test statistic (for the sup statistic) (REAL)
%SIGNIF	Bootstrapped p-value (for the sup statistic) (REAL)
%RSS	Sum of squared residuals with optimal break (REAL)
%LOGL	Log likelihood of regression with optimal break (REAL)
%%BREAKVALUE	Threshold value at the optimal break (REAL)
%%DELAY	Delay which produces largest test statistic (INTEGER)
%NFREE	Total number of estimated parameters, including variance and threshold (INTEGER)

Example

This does an threshold AR(3) which searches across the 3 lags for the best break value. It uses 1000 repetitions of a (fixed regressor) bootstrap to approximate the significance level of the test.

* Enders, Applied Econometric Time Series, 4th edition

* Example from Section 7.10, pp 458-461

* TAR model

open data terror_types.xls

calendar(q) 1968:1

data(format=xls,org=columns) 1968:01 2000:04 date sky hostage assns threats bombings bomb_k bomb_w other $

deaths casualities

@tar(p=3,nreps=1000) casualities

For another example, see the Hansen(1996) replication.

Output

This is the output from the example above. The optimal break (the one which produces the largest LM statistic) is a threshold value of 37.0 on lag 1. The "P-values" are for a test that there is no break, and are computed using bootstrapping, and so will be (somewhat) different when this is re-run. The sample with the threshold at or below 37.0 has 107 observations and the one above it has 22.

Values in (..) are standard errors.

Threshold Autoregression

Threshold is CASUALITIES{1}=37.0000

Tests for Threshold Effect use 1000 draws

SupLM 17.676028 P-value 0.055000

ExpLM 4.319363 P-value 0.126000

AveLM 5.392957 P-value 0.150000

Variable Full Sample <=Thresh >Thresh

Constant 5.909755 1.456890 -5.381993

( 2.091866) ( 2.224712) ( 16.874353)

CASUALITIES{1} 0.260731 0.534202 0.715470

( 0.087453) ( 0.125333) ( 0.351134)

CASUALITIES{2} 0.309872 0.258074 0.204484

( 0.086218) ( 0.091694) ( 0.216291)

CASUALITIES{3} 0.208695 0.238649 -0.094396

( 0.087018) ( 0.091856) ( 0.194754)

Observations 129 107 22

SEESQ 86.168467 73.478666 93.313311