threshold — Threshold regression - Stata

1 Threshold regressionDescriptionQuick startMenuSyntaxOptionsRemarks and examplesStored resultsMethods and formulasReferencesAlso seeDescriptionthresholdextends linear regression to allow coefficients to differ across regions. Those regionsare identified by a Threshold variable being above or below a Threshold value. The model may havemultiple thresholds, and you can either specify a known number of thresholds or letthresholdfindthat number for you through the bayesian information criterion (BIC), Akaike information criterion(AIC), or Hannan Quinn information criterion (HQIC).

2 Quick startThreshold regression model for the dependent variableywith region-dependent intercepts for tworegions ofxthreshold y, threshvar(x)Add the first lag ofxas a region-invariant variablethreshold y , threshvar(x)Add the first lag ofyas a region-dependent variablethreshold y , threshvar(x) regionvars( ) Threshold regression model ofywith region-dependent intercepts for three regions determined bytwo Threshold values ofxthreshold y, threshvar(x) nthresholds(2)UseBICto select the number of thresholds from a maximum of 5 thresholdsthreshold y, threshvar(x) optthresh(5)MenuStatistics>Time series> Threshold regression model12 Threshold Threshold regressionSyntaxthresholddepvar[indepvar s] [if] [in], threshvar(varname)[options]indepvarsis a list of variables with region-invariant threshvar(varname) Threshold variableregionvars(varlist)include region-varying coefficients for specified covariatesconsinvariantreplace region-varying constant with a region-invariant constantnoconstantsuppress region-varying constant termstrim(#)

3 Trimming percentage; default istrim(10)nthresholds(#)number of thresholds; default isnthresholds(1); not allowedwithoptthresh()optthresh(#[,ictyp e])select optimal number of thresholds less than or equal to#; notallowed withnthresholds()SE/Robustvce(vcetype)vc etypemay beoimorrobustReportinglevel(#)set confidence level; default islevel(95)nocnsreportdo not display constraintsdisplayoptionscontrol columns and column formats, row spacing, line width,display of omitted variables and base and empty cells, andfactor-variable labelingnodotssuppress replication dotsdots(#)display dots every#replicationsAdvancedssrs(stub*|new varlist)create variable with sum of squared residuals (SSRs) for each tentativethresholdconstraints(numlist)ap ply specified linear constraints.

4 Not allowed withoptthresh()coeflegenddisplay legend instead of statistics threshvar()is musttssetyour data before usingthreshold; see [TS] contain factor variables; see[U] Factor ,indepvars,varlist, andvarnamemay contain time-series operators; see[U] Time-series ,collect,rolling, andstatsbyare allowed; see[U] Prefix not appear in the dialog [U] 20 estimation and postestimation commandsfor more capabilities of estimation information criterion (BIC); the defaultaicAkaike information criterion (AIC)hqicHannan Quinn information criterion (HQIC) Threshold Threshold regression 3 Options Model threshvar(varname)specifies the variable from which values are to be selected as ()is (varlist)specifies additional variables whose coefficients vary over the regions definedby the estimated thresholds.

5 By default, only the constant term varies over that the constant term should be region invariant instead of region the region-varying constant terms (intercepts) in the (#)specifies thatthresholdtreat the value at the#th percentile of the Threshold variable asthe first possible Threshold and the value at the (100 #)th percentile as the last possible Threshold .#must be an integer between 1 and 49. The default istrim(10).nthresholds(#)specifies the number of thresholds. Specifying the number of thresholds is equivalentto specifying the number of regions because the number of regions is equal to#+1 default isnthresholds(1), equivalent to 2 (#[,ictype]) specifies thatthresholdchoose the optimal number of thresholds, upto a possible#.

6 By default, the optimal number of thresholds is based on theBIC, but you mayspecify the information criterion (ictype) to be bebic(the default),aic, orhqic. SE/Robust vce(vcetype)specifies the type of standard error reported, which includes types that are derivedfrom asymptotic theory (oim) and that are robust to some kinds of misspecification (robust); see[R]vceoption. Reporting level(#),nocnsreport; see [R] estimation :noci,nopvalues,noomitted,vsquish,noempt ycells,baselevels,allbaselevels,nofvlabe l,fvwrap(#),fvwrapon(style),cformat(%fmt ),pformat(%fmt),sformat(%fmt), andnolstretch; see [R] estimation (#)specify whether to display replication dots.

7 By default, one dot character isdisplayed for each successful replication. A red x is displayed ifcommandreturns an error. Youcan also control whether dots are displayed usingset dots; see [R] display of the replication (#)displays dots every# (0)is a synonym fornodots. Advanced ssrs(stub*|newvarlist)creates a variable containing the sum of squared residuals (SSRs) that wascomputed for each tentative Threshold value during the search for thekth Threshold . For observationswhere the value of the Threshold variable specified inthreshvar()is not a tentative Threshold ,the corresponding value of the variable created byssrs()for that observation will be you specifystub*, Stata will createknew variables with the namesstub1.

8 ,stubk, which willcontain theSSRs for the 1st,..,kth thresholds, wherekis the#specified innthresholds()or the optimal number of thresholds ifoptthresh()is Threshold Threshold regressionIf you specify a list of new variable names, you may requestSSRs for up to the#specified innthresholds(). If you specifyoptthresh(#)and the optimal number of thresholds is less than#, any additional variables will contain only missing (numlist)specifies the constraints by number after they have been defined by usingtheconstraintcommand; see [R] ()may not be specified withoptthresh()The following option is available withthresholdbut is not shown in the dialog box:coeflegend; see [R] estimation and linear regression to allow coefficients to differ across regions.

9 Those regionsare identified by a Threshold variable being above or below a Threshold value. The model may havemultiple thresholds, and you can either specify a known number of thresholds or letthresholdfindthat number for you by minimizing an information models are good alternatives to linear models for capturing abrupt breaks or asymmetriesobserved in most macroeconomic time series over the course of a business cycle. Common thresholdregression models include the Threshold autoregression model and self-exciting Threshold model. Inthe Threshold autoregression model, proposed by Tong (1983), the dependent variable is a function ofits own lags; see Tong (1990) for details.

10 In the self-exciting Threshold model, the lagged dependentvariable is used as the Threshold variable. For a survey of Threshold regression models in economics,refer to Hansen (2011).Formally, consider a Threshold regression with two regions defined by a Threshold . This is writtenasyt=xt +zt 1+ tif < wt yt=xt +zt 2+ tif < wt< whereytis the dependent variable,xtis a 1 kvector of covariates possibly containing laggedvalues ofyt, is ak 1 vector of region-invariant parameters, tis anIIDerror with mean 0 andvariance 2,ztis a vector of exogenous variables with region-specific coefficient vectors 1and 2,andwtis a Threshold variable that may also be one of the variables parameters of interest are , 1, and 2.