Materi Analisis Deret Waktu Nonlinier

• _{Several tests have been proposed for assessing}

the need for nonlinear modeling in time series analysis

• _{Some of these tests, such as those studied by}

Keenan (1985)

• _{Keenan’s test is motivated by approximating a}

• where {ε_t, −∞ < t < ∞} is a sequence of

independent and identically distributed zero-mean random variables.

• The process {Y_t} is linear if the double sum on

where the φ’s are autoregressive parameters, σ’s are noise standard deviations, r is the threshold

parameter, and {e_t} is a sequence of independent and identically distributed random variables with zero

• _{The 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}

• _{As a concrete example, we simulate some data}

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

Threshold Models

• _{The 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

• _{However, 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 |φ1,1|+…+ |φ1,p| < 1 and |φ2,1| +

• _{The extension to the case of m regimes is}

straightforward and effected by partitioning the real line into −∞ < r₁ < r₂ <…< r_{m − 1} < ∞, and the position of Y_{t − d} relative to these

• _{While Keenan’s test and Tsay’s test for}

nonlinearity are designed for detecting quadratic nonlinearity, they may not be sensitive to threshold nonlinearity

• _{Here, we discuss a likelihood ratio test with}

the threshold model as the specific alternative

• _{The null hypothesis is an AR(p) model}

• _{the alternative hypothesis of a two-regime}

TAR model of order p and with constant noise

• _{The general model can be rewritten as}

where the notation I( ) is an indicator variable that ⋅

• The null hypothesis states that φ_2,0 = φ_2,1 =…=

φ_2,p = 0.

• _{While the delay may be theoretically larger}

than the autoregressive order, this is seldom the case in practice.

• _{The test is carried out with fixed p and d.}

• _{The likelihood ratio test statistic can be shown}

to be equivalent to

• _{Hence, the sampling distribution of the likelihood ratio test under}

H0 is no longer approximately χ2 with p degrees of freedom.

• _{Instead, it has a nonstandard sampling distribution; see Chan}

(1991) and Tong (1990)

• _{Chan (1991) derived an approximation method for computing the}

p-values of the test that is highly accurate for small p-values.

• _{The test depends on the interval over which the threshold}

parameter is searched.

• _{Typically, the interval is defined to be from the a×100th percentile}

to the b×100th percentile of {Yt}, say from the 25th percentile to

the 75th percentile

• _{The choice of a and b must ensure that there are adequate data}

Estimation of a TAR Model

• _{If r is known, the estimation of a TAR model is}

straightforward.

• _{Separate the observation according to}

whether Y_t-1 is above or below the threshold

• _{Each segment of TAR model can then be}

estimated using OLS.

• The lag lenghts p₁ and p₂ can be determined

• _{A simple numerical example might be helpful.}

• _{Suppose that the first seven observations of a}

time series are as follow :

• _{If you know that threshold is zero, sort the}

observation into two groups according to

whether Y_t-1 is greater that or less than zero

t 1 2 3 4 5 6 7

Y_t

• _{The two regime would look like this;}

• _{The two separate AR(p) processes can be}

• If we restrict var (_1t) = var (_2t), we can rewrite

TAR model as (*)

• if Y_t-1 ≤ 0, I_t=0 and 1 - I_t= 1

• _{To estimate a TAR model in the form (*),}

create the indicator function and form the variables I_tY_t-i and (1 – I_t)Y_t-i.

• _{To 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}

Unknown Threshold (r)

• _{r must lie between the maximum and minimum value of}

the series

• _{Continue in this fashion for each observation within the}

band

• _{The regression containing the smallest residual sum of}

Selecting the Delay Parameter

• _{The standar procedure is to estimate a TAR}

model for each potential value of d

• _{The 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}

Model Diagnostics

Model diagnostics via residual analysis

• _{The 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

• _{The 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