of the Long-Range Financial Status

of the OASDI Program—September 2004

V. APPENDICES

**Time-Series Modeling**

Time-series analysis is a standard projection technique in econometric modeling. The equations used to project the assumptions are fit using these techniques. This appendix provides details about the models in general, presents statistical methods employed to test these models, and underscores the nuances inherent in the method of determining the equations. The reader may wish to consult Box, Jenkins and Reinsel (1994) or Hamilton (1994) for a standard presentation of this material. A more elementary reference is Pindyck and Rubinfeld (1998).

**Stationary Time Series**

A time series *Y _{t}* is

In particular, the variance of *Y _{t}* is always equal to

**White Noise Process**

Suppose that *ε _{t}*, the error term at time

**Random Walk**

The simplest of all time-series models is the random walk. Here, if *Y _{t}* is the series to be estimated, then the random walk process is given by

**Moving Average (MA) Models**

A time series is called a moving average model of order *q*, or simply an MA(*q*) process, if

*Y _{t}* =

In this equation, *θ*_{1},…,*θ _{q}*
are constant parameters and

**Autoregressive (AR) Models**

A time series is called an autoregressive model of order *p*, or simply an AR(*p*) process, if

*Y _{t}* =

In this equation *φ*_{1},…,*φ _{p}* are constant parameters and δ is a drift term. The naming of this model is apt because the coefficients of the time-series equation are obtained by regressing the equation on itself, more precisely, with its own

**Autoregressive Moving Average (ARMA) Models**

As its name indicates, an autoregressive moving average model of order (*p,q*), or simply an ARMA(*p,q*) model, is a natural combination of an autoregressive and moving average model. The equation which specifies an ARMA(*p,q*) model takes the form

*Y _{t}* =

**Deviations Form of ARMA Model Equations**

It is often more convenient to transform an ARMA model equation into deviations form using the equation

*y _{t}* =

where *μ* is defined, as above, to be the mean of the process. The transformed model may be written as

*y _{t}* =

and has a mean of zero.

Adding the original process mean to both sides of the equation produces

*Y _{t}* =

The lagged variables are left in deviations form, and the constant term, *μ*, on the right-hand side is the process mean.

**Cholesky Decomposition**

Suppose _{} = (*Z*_{1},…, *Z _{n}*)

For our applications, a Cholesky decomposition is used to convert a random vector, _{} of independent standard normal variates to a multivariate normal distribution with mean
_{} = _{}
(the zero vector) and a variance-covariance matrix **V** obtained by using historical data.
If **L** is a lower triangular Cholesky matrix associated with **V** then the vector
**L**_{} has the required multivariate normal distribution.

**Vector Autoregressive Models**

Vector autoregressive models allow the joint modeling of time-series processes. For the sake of simplicity, suppose that three variables
*y*_{1,t} , *y*_{2,t} , and *y*_{3,t} depend on time *t*.
Data may indicate that these variables may be related to each other's past values.
The simplest such case is when the relationship is limited to the time-1 lag, i.e., when
*y*_{1,t} , *y*_{2,t} , and *y*_{3,t}
may be modeled in terms of
*y*_{1,t-1}, *y*_{2,t-1}, and *y*_{3,t-1}.
In this case, a three-variable VAR(1) model takes the form

_{}

for some 3×3 matrix *B*.

Alternatively stated, if _{} = (*y*_{1,t}, *y*_{2,t}, *y*_{3,t})**´**
and _{} = (*ε*_{1,t}, *ε*_{2,t}, *ε*_{3,t})**´**
then the model takes the form _{} = _{}+_{}
for some 3×3 matrix *B*.

The *k*-variable VAR(*p*) model, with *p* lags, naturally extends from this.
If _{} = (*y*_{1,t}, *y*_{2,t},…, *y _{k,t}*)

**Modified Autoregressive Moving Average (ARMA) Models**

The equations used to model the assumptions are autoregressive moving average models with the additional requirement that the mean of the variable *Y _{t}* is always equal to its value under the TR04II. The value of the variable under the TR04II is written

*Y _{t}* =

Writing this equation in "deviations form," with *y _{t}* =

*Y _{t}* =

Since *μ _{t}* =

*Y _{t}* =

Equations in Chapter II are all presented in this form, with the letters *Y* and *y* replaced with a more suggestive naming.