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 Time Series Analysis and Examples

### Computational Details

#### Least Squares and Householder Transformation

Consider the univariate AR() process
Define the design matrix .
Let . The least squares estimate, , is the approximation to the maximum likelihood estimate of if is assumed to be Gaussian error disturbances. Combining and as
the matrix can be decomposed as
where is an orthogonal matrix and is an upper triangular matrix, , and .

The least squares estimate that uses Householder transformation is computed by solving the linear system

The unbiased residual variance estimate is
and
In practice, least squares estimation does not require the orthogonal matrix . The TIMSAC subroutines compute the upper triangular matrix without computing the matrix .

#### Bayesian Constrained Least Squares

Consider the additive time series model

Practically, it is not possible to estimate parameters , since the number of parameters exceeds the number of available observations. Let denote the seasonal difference operator with seasons and degree of ; that is, . Suppose that . Some constraints on the trend and seasonal components need to be imposed such that the sum of squares of , , and is small. The constrained least squares estimates are obtained by minimizing

Using matrix notation,
where , , and is the initial guess of . The matrix is a control matrix in which structure varies according to the order of differencing in trend and season.
where
The matrix has the same structure as the matrix , and is the identity matrix. The solution of the constrained least squares method is equivalent to that of maximizing the function
Therefore, the PDF of the data is
The prior PDF of the parameter vector is
When the constant is known, the estimate of is the mean of the posterior distribution, where the posterior PDF of the parameter is proportional to the function . It is obvious that is the minimizer of , where
The value of is determined by the minimum ABIC procedure. The ABIC is defined as

#### State Space and Kalman Filter Method

In this section, the mathematical formulas for state space modeling are introduced. The Kalman filter algorithms are derived from the state space model. As an example, the state space model of the TSDECOMP subroutine is formulated.

Define the following state space model:

where and . If the observations, , and the initial conditions, and , are available, the one-step predictor of the state vector and its mean square error (MSE) matrix are written as
Using the current observation, the filtered value of and its variance are updated.
where and . The log-likelihood function is computed as
where is the conditional variance of the one-step prediction error .

Consider the additive time series decomposition

where is a regressor vector and is a time-varying coefficient vector. Each component has the following constraints:
where and . The AR component is assumed to be stationary. The trading-day component represents the number of the th day of the week in time . If , and (monthly data),
The state vector is defined as
The matrix is

where

The matrix can be denoted as

where
Finally, the matrix is time-varying,

where

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