# Threshold Autoregressive

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23-Feb-2016Category

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Threshold Autoregressive

Threshold Autoregressive

Several tests have been proposed for assessing the need for nonlinear modeling in time series analysisSome of these tests, such as those studied by Keenan (1985)Keenans test is motivated by approximating a nonlinear stationary time series by a second-order Volterra expansion (Wiener,1958)where {t, < t < } is a sequence of independent and identically distributed zero-mean random variables. The process {Yt} is linear if the double sum on the righthandside vanishes

Keenans test can be implemented as follows:

where the s are autoregressive parameters, s are noise standard deviations, r is the threshold parameter, and {et} is a sequence of independent and identically distributed random variables with zero mean and unit varianceThe process switches between two linear mechanisms dependent on the position of the lag 1 value of the process. When the lag 1 value does not exceed the threshold, we say that the process is in the lower (first) regime, and otherwise it is in the upper regime. Note that the error variance need not be identical for the two regimes, so that the TAR model can account for some conditional heteroscedasticity in the data.As a concrete example, we simulate some data from the following first-order TAR model:

Exhibit 15.8 shows the time series plot of the simulated data of size n = 100

Somewhat cyclical, with asymmetrical cycles where the series tends to drop rather sharply but rises relatively slowly time irreversibility (suggesting that the underlying process is nonlinear)Threshold ModelsThe first-order (self-exciting) threshold autoregressive model can be readily extended to higher order and with a general integer delay:

15.5.1

Note that the autoregressive orders p1 and p2 of the two submodels need not be identical, and the delay parameter d may be larger than the maximum autoregressive ordersHowever, by including zero coefficients if necessary, we may and shall henceforth assume that p1 = p2 = p and 1 d p, which simplifies the notation. The TAR model defined by Equation (15.5.1) is denoted as the TAR(2;p1, p2) model with delay d.TAR model is ergodic and hence asymptotically stationary if |1,1|++ |1,p| < 1 and |2,1| ++ |2,p| < 1The extension to the case of m regimes is straightforward and effected by partitioning the real line into < r1 < r2 r and It = 0 if Yt-1 0So that if Yt-1> r, It=1 and 1 - It= 0

if Yt-1 0, It=0 and 1 - It= 1

To estimate a TAR model in the form (*), create the indicator function and form the variables ItYt-i and (1 It)Yt-i. You can then estimate the equation using OLS

From the numerical example using r = 0

t1234567Yt0.50.3-0.20.0-0.50.40.6Yt-1NA0.50.3-0.20.0-0.50.4Yt-2NANA0.50.3-0.20.0-0.5ItNA110001ItYt-1NA0.50.30000.4(1-It)Yt-1NA00-0.20-0.50ItYt-2NANA0.5000-0.5(1-It)Yt-2NANA00.3-0.20.00To estimate the model using one lag in each regime, you estimate the model using the 4 variabel.To estimate the model using two lag in each regime, you estimate the model using the 6 variabelUnknown Threshold (r)r must lie between the maximum and minimum value of the seriesTry a value of r = y1( the first observation in the band) and estimate an equationIf y2 lies outside the band, there is no need to estimate TAR model using r = y2Next estimate TAR model using r = y3. Continue in this fashion for each observation within the bandThe regression containing the smallest residual sum of squares contains the consistent estimate of the threshold.

Selecting the Delay ParameterThe standar procedure is to estimate a TAR model for each potential value of dThe model with the smallest value of the residual sum of squares contains the most appropriate value of the delay parameter.Alternatively, choose the delay parameter that leads to the smallest value of the AIC or the SBCModel DiagnosticsModel diagnostics via residual analysisThe time series plot of the standardized residuals should look random, as they should be approximately independent and identically distributed if the TAR model is the true data mechanism; that is, if the TAR model is correctly specifiedThe independence assumption of the standardized errors can be checked by examining the sample ACF of the standardized residuals. Nonconstant variance may be checked by examining the sample ACF of the squared standardized residuals or that of the absolute standardized residuals