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Time Series Analysis and Control Examples

Bayesian Seasonal Adjustment

Seasonal phenomena are frequently observed in many economic and business time series. For example, consumption expenditure might have strong seasonal variations because of Christmas spending. The seasonal phenomena are repeatedly observed after a regular period of time. The number of seasons within a period is defined as the smallest time span for this repetitive observation. Monthly consumption expenditure shows a strong increase during the Christmas season, with 12 seasons per period.

There are three major approaches to seasonal time series: the regression model, the moving average model, and the seasonal ARIMA model.

Regression Model

Let the trend component be T_t = \sum_{i=1}^{m_\alpha} \alpha_i U_{it}and the seasonal component be S_t = \sum_{j=1}^{m_\beta} \beta_j V_{jt}.Then the additive time series can be written as the regression model

y_t = \sum_{i=1}^{m_{\alpha}} \alpha_i U_{it} + 
 \sum_{j=1}^{m_{\beta}} \beta_j V_{jt} + \epsilon_t
In practice, the trend component can be written as the m_{\alpha}th order polynomial, such as
T_t = \sum_{i=0}^{m_{\alpha}} \alpha_i t^i
The seasonal component can be approximated by the seasonal dummies (Djt)
S_t = \sum_{j=1}^{L-1} \beta_j D_{jt}
where L is the number of seasons within a period. The least squares method is applied to estimate parameters \alpha_i and \beta_j.

The seasonally adjusted series is obtained by subtracting the estimated seasonal component from the original series. Usually, the error term \epsilon_t is assumed to be white noise, while sometimes the autocorrelation of the regression residuals needs to be allowed. However, the regression method is not robust to the regression function type, especially at the beginning and end of the series.

Moving Average Model

If you assume that the annual sum of a seasonal time series has small seasonal fluctuations, the nonseasonal component N_t = T_t + \epsilon_t can be estimated by using the moving average method.

\hat{N}_t = \sum_{i=-m}^m \lambda_i y_{t-i}
where m is the positive integer and \lambda_i is the symmetric constant such that \lambda_i = 
\lambda_{-i} and \sum_{i=-m}^m \lambda_i = 1.

When the data are not available, either an asymmetric moving average is used, or the forecast data is augmented to use the symmetric weight. The X-11 procedure is a complex modification of this moving average method.

Seasonal ARIMA Model

The regression and moving average approaches assume that the seasonal component is deterministic and independent of other nonseasonal components. The time series approach is used to handle the stochastic trend and seasonal components.

The general ARIMA model can be written

\prod_{j=1}^m \phi_j(B) 
\prod_{i=1}^k (1-B^{s_i})^{d_i} \tilde{y}_t = 
\theta_0 + \prod_{i=1}^q \theta_i(B)\epsilon_t
where B is the backshift operator and
\phi_j(B) & = & 1 - \phi_1 B -  ...  - \phi_j B^{p_j} \\theta_i(B) & = & 1 - \theta_1 B -  ...  - \theta_i B^{q_i}
and \tilde{y}_t = y_t - {\rm E}(Y_t) if di = 0; otherwise, \tilde{y}_t = y_t.The power of B, si, can be considered as a seasonal factor. Specifically, the Box-Jenkins multiplicative seasonal ARIMA(p,d,q)(P,D,Q)s model is written as
\phi_p(B) \Phi_P(B^s)(1-B)^d(1-B^s)^D \tilde{y}_t =
\theta_q(B) \Theta_Q(B^s) \epsilon_t
ARIMA modeling is appropriate for particular time series and requires burdensome computation.

The TSBAYSEA subroutine combines the simple characteristics of the regression approach and time series modeling. The TSBAYSEA and X-11 procedures use the model-based seasonal adjustment. The symmetric weights of the standard X-11 option can be approximated by using the integrated MA form

(1-B)(1-B^{12})y_t = \theta(B)\epsilon_t
With a fixed value \phi, the TSBAYSEA subroutine is approximated as
(1-\phi B)(1-B)(1-B^{12})y_t = \theta(B) \epsilon_t
The subroutine is flexible enough to handle trading day or leap year effects, the shift of the base observation, and missing values. The TSBAYSEA-type modeling approach has some advantages: it clearly defines the statistical model of the time series; modification of the basic model can be an efficient method of choosing a particular procedure for the seasonal adjustment of a given time series; and the use of the concept of the likelihood provides a minimum AIC model selection approach.

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