SPECIFY Instruction

SPECIFY(TYPE=SYMMETRIC,other options) other's weight

SPECIFY(TYPE=GENERAL,other options)

# < supp. card> weights (if you don't use the MATRIX option)

SPECIFY defines the prior for a vector autoregressive system. It is a sub-command of SYSTEM and must appear in the system definition after the VARIABLES and LAGS instructions.

Wizard

You can also use the Time Series—VAR (Setup/Estimate) Wizard to define and estimate VAR models.

Parameters

others's weight

This is the relative weight on other variables in a SYMMETRIC prior; it is irrelevant for other prior types. If you omit it, its default value is 1.0.

Description

We use the following notation throughout this section:

•variable j refers to the jth variable listed on the VARIABLES instruction.

•equation i refers to the equation whose dependent variable is variable i.

•the standard deviation of the prior distribution for lag l of variable j in equation i: denoted $S(i,j,l{\kern 1pt} {\kern 1pt} )$.

The function $S(i,j,l{\kern 1pt} {\kern 1pt} )$ takes the following form:

$S(i,j,l{\kern 1pt} {\kern 1pt} ) = \frac{{\left\{ {\gamma {\kern 1pt} g(l){\kern 1pt} {\kern 1pt} f(i,j)} \right\}{\kern 1pt} {\kern 1pt} {s_i}}}{{{s_j}}};\,\,\,\,\,\,\,\,\,\,\,f(i,i) = g(l{\kern 1pt} ) = 1.0$

where $s_i$ is the standard error of a univariate autoregression on the dependent variable of equation i. We multiply by ${s_i}/{s_j}$ to correct for the different scales of the variables. The various options control ${f(i,j)}$, $\gamma $, ${g(l)}$.

Options

TYPE=[SYMMETRIC]/GENERAL

MATRIX=RECTANGULAR matrix (for prior TYPE=GENERAL) [unused]

TYPE selects the ${f(i,j)}$ function. SYMMETRIC provides a restricted form of the function, while GENERAL allows complete generality. See the discussions below.

TIGHTNESS=parameter of overall tightness [0.20]

The overall tightness is the $\gamma$ parameter in the formula above.

MEAN=mean of first own lag (same across equations) [1.0]

MVECTOR=VECTOR of means of first own lags [unused]

Use one of these if you want the means of the prior distributions on the first own lag to be something other than 1.0. The MEAN option sets them all to the specified first lag mean value. The MVECTOR option sets the mean for the first own lag in the ith equation to the ith entry of the VECTOR of first lag means.

LAGTYPE=[HARMONIC]/GEOMETRIC

DECAY=parameter of lag decay [no decay]

These two options control the function ${g(l)}$: how the standard deviation changes with increasing lags. With d=lag decay parameter, SPECIFY uses the formulas:

$g(l){\rm{ }} = {l^{ - d}}$ for LAGTYPE=HARMONIC and

$g(l){\rm{ }} = {d^{l - 1}}$ for LAGTYPE=GEOMETRIC

By default there is no decay of standard deviations with increasing lags. Notice that for HARMONIC, $d=0$ is the default and larger values produce a tighter prior, while for GEOMETRIC, $d=1$ is the default and smaller values produce a tighter prior.

FULL=RECTANGULAR of information on full system prior [unused]

SCALE/[NOSCALE]

These allow you to use priors more general than the standard ones. See the "Full Specification of Priors" below.

[PRINT]/NOPRINT

NOPRINT suppresses the printing of a summary of the prior on the END(SYSTEM) instruction.

Variables Defined

%PRIOR

array of prior weights (RECTANGULAR)

The TYPE=SYMMETRIC Option

The ${f(i,j)}$ function takes the form:

$f\left( {i,j} \right) = \left\{ {\begin{array}{*{20}{c}}{1.0} \hfill & {{\rm{ if }\quad}i = j} \hfill \\ w \hfill & {{\rm{otherwise}}} \hfill \\\end{array}} \right.$

where w is the other’s weight parameter. For example:

system(model=sixvar)

variables ip m1 cpr unemp wage cpi

lags 1 to 12

det constant

specify(type=symmetric,tightness=.10,decay=1.0) .5

end(system)

The TYPE=GENERAL Option

TYPE=GENERAL allows complete freedom in selecting the f(i,j) function. f is either

•input using a supplementary card. You should type the values for f by rows (that is, group by equation). The # goes only on the first supplementary card line, and you must use line continuations ($) if all the numbers do not fit on one card.

•provided using the option MATRIX=RECTANGULAR array. You must set this array up in advance. Element i,j of this array is f(i,j).

system(model=sixvar)

variables ip m1 cpr unemp wage cpi

lags 1 to 12

det constant

specify(type=general)

# 1.0 0.5 0.5 0.2 0.2 0.2 $

0.2 1.0 1.0 0.2 0.2 0.5 $

0.2 1.0 1.0 0.2 0.2 0.2 $

0.2 0.5 0.5 1.0 0.2 0.2 $

0.2 0.5 0.2 0.2 1.0 0.5 $

0.0 0.5 0.2 0.2 0.5 1.0

end(system)

Full Specification of Priors

The primary options of SPECIFY put rather stringent restrictions on the type of priors that you can use. The options FULL and SCALE allow some relaxation of these.

FULL=RECTANGULAR of information on full system prior [unused]

SCALE/[NOSCALE]

With the notation

N : number of equations

L : number of lags

D : number of deterministic variables,

you use FULL to input an (NL+D+1) × N matrix of “dummy observations.” The structure of each column is as follows:

First NL rows	the reciprocals of the standard deviations of the priors on the lags. Use a 0 for a flat prior.
Next D rows	the reciprocals of the standard deviations of the priors on the deterministic variables. Use a 0 for a flat prior.
Last row	the prior mean of the first own lag divided by its prior standard deviation.

Dummy Observations

Suppose you impose the prior ${\beta _k}\sim N\left( {b,{\lambda ^2}} \right)$ upon a coefficient ${\beta _k}$. You can represent this as a “dummy observation” with

$\Psi = \frac{\sigma }{\lambda }\,\;\,{\rm{and}}\,\;r = \frac{\sigma }{\lambda }b$

where $\sigma$ is the standard deviation of the equation being estimated.

The first NL+D elements of each column of the FULL array provide the $\Psi$ values for the dummy observations for the coefficients. The last element provides the r for the first own lag. This allows you:

•complete freedom in setting standard deviations on the lags

•the ability to put mean zero priors on any of the deterministic variables.

The Option SCALE

The SCALE option causes ESTIMATE to compute and insert the ${s_i}$ and ${s_j}$ factors into the formulas above. This lets you concentrate on the form of the prior without worrying about relative magnitudes. NOSCALE (the default with FULL) requires that you provide directly the $\Psi$ and $r$ values that you want.