The ARIMA Procedure

Details

You can use the OUTLIER statement to detect changes in the level of the response series that are not accounted for by the currently estimated model. The types of changes considered are Additive Outliers (AO), Level Shifts (LS), and Temporary Changes (TC).

Let $\eta_t$ be a regression variable describing some type of change in the mean response. In time series literature $\eta_t$ is called a shock signature. An additive outlier at some time point i corresponds to a shock signature $\eta_t$ such that $\eta_i=1.0$ and $\eta_t$ is 0.0 at all other points. Similarly a permanent level shift originating at time i has a shock signature such that $\eta_t$ is 0.0 for t < i and 1.0 for $t \geq i$ . A temporary level shift of duration d originating at time i will have $\eta_t$ equal to 1.0 between i and i+d and 0.0 otherwise.

Suppose that the preceding ESTIMATE statement has the ARIMA model

$D(B) Y_t=\mu_t + \frac{\theta(B)}{\phi(B)} a_t$

where Y_t is the response series, D(B) is the differencing polynomial in the backward shift operator B (possibly identity), $\mu_t$ is the transfer function input, $\phi(B)$ and $\theta(B)$ are the AR and MA polynomials, and a_t is the Gaussian white noise series.

The problem of detection of level shifts in the OUTLIER statement is formulated as a problem of sequential selection of shock signatures that improve the model in the ESTIMATE statement. This is similar to the forward selection process in the stepwise regression procedure. The selection process starts with considering shock signatures of the type specified in the TYPE= option, originating at each non-missing measurement. This involves testing $H_{0}\colon\beta=0$ versus $H_{a}\colon \beta \neq 0$ in the model

$D(B) ( Y_t - \beta \eta_t )=\mu_t + \frac{\theta(B)}{\phi(B)} a_t$

for each of these shock signatures. The most significant shock signature, if it also satisfies the significance criterion in ALPHA= option, is included in the model. If no significant shock signature is found then the outlier detection process stops, otherwise this augmented model, which incorporates the selected shock signature in its transfer function input, becomes the null model for the subsequent selection process. This iterative process stops if at any stage no more significant shock signatures are found or if the number of iterations exceed the maximum search number resulting due to the MAXNUM= and MAXPCT= settings. In all these iterations the parameters of the ARIMA model in the ESTIMATE statement are held fixed.

The precise details of the testing procedure for a given shock signature $\eta_t$ are as follows:

The preceding testing problem is equivalent to testing $H_{0}\colon\beta=0$ versus $H_{a}\colon \beta \neq 0$ in the following "regression with ARMA errors" model

$N_t=\beta \zeta_t + \frac{\theta(B)}{\phi(B)} a_t$

where $N_t=( D(B) Y_t - \mu_t )$ is the "noise" process and $\zeta_t=D(B)\eta_t$ is the "effective" shock signature.

In this setting, under H₀, N = ( N₁, N₂, ... , N_n )^T is a mean zero Gaussian vector with variance covariance matrix $\sigma^2 \Sigma$ . Here $\sigma^2$ is the variance of the white noise process a_t and $\Sigma$ is the variance covariance matrix associated with the ARMA model. Moreover, under H_a, N has $\beta \zeta$ as the mean vector where $\zeta=(\zeta_1, \zeta_2, ... , \zeta_n )^T$ . Additionally, the generalized least squares estimate of $\beta$ and its variance is given by

$\hat{\beta} &=&\delta / \kappa \ {\rm Var}( \hat{\beta} ) &=& \sigma^2 /\kappa$

where $\delta=\zeta^T \Sigma^{-1} N$ and $\kappa=\zeta^T \Sigma^{-1} \zeta$ . The test statistic $\tau^2=\delta^2/ (\sigma^2 \kappa)$ is used to test the significance of $\beta$ ,which has an approximate chi-squared distribution with 1 degree of freedom under H₀. The type of estimate of $\sigma^2$ used in the calculation of $\tau^2$ can be specified by the SIGMA= option. The default setting is SIGMA=ROBUST that corresponds to a robust estimate suggested in an outlier detection procedure in X-12-ARIMA, the Census Bureau's time series analysis program; refer to Findley et al. (1998) for additional information. The setting SIGMA=MSE corresponds to the usual mean squared error estimate (MSE) computed the same way as in the ESTIMATE statement with the NODF option. The robust estimate of $\sigma^2$ is computed by the formula

$\hat{\sigma}^2=( 1.49 x {\rm Median}( | \hat{a}_t | ) )^2$

where $\hat{a}_t$ are the standardized residuals of the null ARIMA model.

The quantities $\delta$ and $\kappa$ are efficiently computed by a method described in de Jong and Penzer (1998); refer also to Kohn and Ansley (1985).

Displayed Output

Modeling in the Presence of Outliers

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