ERRORS( options )

The instruction ERRORS has two very different functions:

 

•It computes the standard error of forecasts from a dynamic (linear) model.

•It computes the decomposition of variance for a Vector Autoregression (VAR).

 

You can only apply ERRORS to linear models.

 

The syntax for ERRORS has changed quite a bit over the years. A pre-version 7 ERRORS instruction will look quite different from the preferred setup now, as it would use parameters and supplementary cards rather than options and MODELS. If you need it, see "Older Syntax" below.

Wizard

You can use the Time Series—VAR (Forecast/Analyze) Wizard to compute variance decompositions.

Options

MODEL=model name

Of the two ways to input the form of the model to be solved (the other is with supplementary cards), this is the more convenient. MODELs are usually created by GROUP or SYSTEM. FRML’s (formulas) can be included if they are of a simple linear form. (ERRORS requires that the model be fully linear.) If the model includes any identities, those should be last in the model. If you use this, omit the “equation” supplementary cards.

 

STEPS=number of steps to compute [required]

This sets the number of steps (periods) for which you wish to compute responses. If you have set a SMPL, this defaults to the number of steps implied by it. Otherwise, you must use the STEPS option to supply a value.

 

CV=SYMMETRIC covariance matrix of residuals [Covariance matrix for the MODEL]

FACTOR=RECTANGULAR decomposition matrix [unused]

Use CV if you want the orthogonalization computed using a Cholesky factorization of the covariance matrix. If you are using the MODEL option and omit this option, ERRORS defaults to using the estimated covariance matrix for the MODEL.       

 

As an alternative, you can use FACTOR to supply a non-standard factorization of the covariance matrix, such as the factor matrix produced by a CVMODEL instruction.  The form of the factorization does not affect the forecast variance, only the decomposition. (This option was called DECOMP in versions before 7. DECOMP is still recognized as a synonym for FACTOR.)
 

[PRINT]/NOPRINT

TITLE="title for output" [none]

WINDOW="title of window" [not used]

If you use the PRINT option, it will add the TITLE string (if used) to the top of the output (just once).

 

If you use the WINDOW option, a (read-only) spreadsheet window is created with the indicated title and displayed on the screen. This will display \(N\) blocks of \(N+1\) columns, in a format similar to the standard output.

 

LABELS=VECTOR[STRINGS] with labels for shocks

Use the LABELS option if you want to supply your own labels for the shocks. Otherwise, ERRORS will use the names of the variables.

 

STDERRORS=VECTOR[SERIES]for standard errors [not used]

This option saves the standard errors of forecast into a VECTOR of SERIES.

 

IMPULSES/[NOIMPULSES]

Use the IMPULSES option to print the impulse responses which go into the decomposition. An IMPULSE instruction of similar form would produce exactly the same output, so if you want the responses in addition to the decomposition, you need only use the ERRORS instruction with IMPULSES. If you need more flexibility, or if you need to be able to save the impulse responses, use the IMPULSE instruction instead. Note: for a big VAR, this produces a lot of output.

 

ACCUMULATE=||vector of positions||

Indicates which variables (by position in the model) need to be accumulated (integrated)

 

RESULTS=RECTANGULAR[SERIES] for the decomposition [unused]

SAVE=VECTOR[RECTANGULAR] array for the decomposition [unused]

RESULTS saves the decomposition into an \(M \times N\) array of SERIES, where \(M\) is the total number of equations, and \(N\) is the number of estimated equations. The series in entry \((i,j)\) of the array contains the fraction of variance of equation \(i\) that is explained by variable \(j\).

 

SAVE saves the decomposition in an \(M\)-vector of \(L \times N\) arrays, where \(L\) is the number of steps computed. If SAVE=DECOMP, DECOMP(i)(k,j) is the fraction of the variance of equation \(i\) at step \(k\) that is explained by variable \(j\).

 

Both options save fractions, not percentages.

Example of Decomposition

This example computes decompositions for a system of four equations using two orderings.
 

system(model=canmodel)

variables cpr m1 ppi gdp

lags 1 to 4

det constant

end(system)

estimate(cvout=v)

*

*  The first decomposition is done in the original order:

*  CPR-M1-PPI-GDP. The second is GDP-PPI-M1-CPR.

*

errors(model=canmodel,steps=24,cv=v)

errors(model=canmodel,factor=%psdfactor(v,||4,3,2,1||,steps=24)

Technical Information

For a static model such as
 

(1) \({y_t} = {X_t}\beta  + {u_t}\); \({\rm{Var}}\left( {{u_t}} \right) = {\sigma ^2}\)
 

the variance of the error in using \({X_{t + 1}}\hat \beta \) to forecast \({y_{t + 1}}\) is

 

(2) \({\sigma ^2} + {\sigma ^2}{X_{t + 1}}{\left( {{\bf{X'X}}} \right)^{ - 1}}{X'_{t + 1}}\)

 

(See, for instance, Greene(2012), p 81). The first term is due to the equation error \({u_{t + 1}}\), and the second is due to sampling error in using \(\hat \beta \) to estimate \(\beta \). For simple projections, you can get this variance by using the STDERR option of PRJ.
 

The situation is much more complicated for a multiple step forecast in a dynamic model. Take the simplest possible case:

 

(3) \({y_t} = \alpha {\kern 1pt} {y_{t - 1}} + {u_t}\) ; \({\rm{Var}}\left( {{u_t}} \right) = {\sigma ^2}\)

 

For the one-step forecast, there is no difference between this and the static model. However, the two step forecast (forecast for \(t+1\) given \(t-1\)) that uses an estimated \(\hat \alpha \) is \({\hat \alpha ^2}{\kern 1pt} {y_{t - 1}}\) while the actual value is

 

(4) \({y_{t + 1}} = {\alpha ^2}{y_{t - 1}} + \alpha {\kern 1pt} {u_t} + {u_{t + 1}}\)

Thus, the error is

 

(5) \(\alpha {\kern 1pt} {\kern 1pt} {u_t} + {u_{t + 1}} + \left( {{\alpha ^2} - {{\hat \alpha }^2}} \right){\kern 1pt} {\kern 1pt} {y_{t - 1}}\)

 

Note that the effect of sampling error (the last term) depends upon the squares of the coefficients. This term becomes extremely complicated as the size of the model and the number of steps increases.

 

ERRORS ignores the sampling error term and concentrates on the others: the ones due to the effects of the innovations (the \(u\)). Note that the two step forecast error depends not only upon the second period’s innovation, but also upon the first period’s innovation as well. More generally, in the moving average representation,

 

(6) \({{\bf{y}}_t} = \sum\limits_{s = 0}^\infty  {{\Psi _s}{{\bf{u}}_{t - s}}}  = \sum\limits_{s = 0}^\infty  {{\Psi _s}{\bf{F}}{{\bf{v}}_{t - s}}} \)

the error in the K-step ahead forecast is

 

(7) \(\sum\limits_{s = 0}^{K - 1} {{\Psi _s}{{\bf{u}}_{t - s}}}  = \sum\limits_{s = 0}^{K - 1} {{\Psi _s}{\bf{F}}{{\bf{v}}_{t - s}}} \)

where \({\bf{F}} {\bf{F'}}\) is a factorization of the covariance matrix of \(\bf{u}\), and the \(\bf{v}\)'s are orthogonalized innovations. The covariance matrix of the K-step ahead forecasts is

 

(8) \(\sum\limits_{s = 0}^{K - 1} {{\Psi _s}{\bf{F}}} {\bf{F'}}{\kern 1pt} {\Psi '_s} = \sum\limits_{s = 0}^{K - 1} {{\Psi _s}\Sigma } {\kern 1pt} {\Psi '_s}\)

 

This does not depend upon which factorization of \(\Sigma\) is chosen. However, the decomposition of variance, which breaks this sum down into the contributions of the component of \(\bf{v}_t\), does. See the "Impulse and Errors" for a more detailed discussion on the decomposition of variance.

Sample Output

This is part of the output from an ERRORS instruction applied to a six variable VAR. There will be one such table for each endogenous variable.

 

Decomposition of Variance for Series CANRGDPS

 

Step  Std Error     USARGDPS  CANUSXSR  CANCD90D   CANM1S   CANRGDPS  CANCPINF

   1  0.004688318    13.062     2.172     2.321     0.604    81.842     0.000

   2  0.007407510    21.727     2.495     1.291     0.943    73.505     0.040

   3  0.008882190    19.386     4.086     0.977     8.445    67.062     0.044

   4  0.010004321    15.284     4.194     3.754    12.355    64.256     0.156

   5  0.010632775    14.403     4.704     6.786    12.090    61.583     0.434

   6  0.011511805    17.409     5.516    13.026    10.328    53.250     0.472

   7  0.013088464    21.646     5.594    21.954     8.939    41.435     0.432

   8  0.014910398    23.338     5.798    29.436     8.936    31.964     0.529

   9  0.016733054    23.104     6.219    35.950     8.549    25.436     0.742

  10  0.018526192    22.059     6.274    41.696     8.106    20.916     0.949

 

The first column in the output is the standard error of forecast for this variable in the model. This is computed using (8). Since the computation assumes the coefficients are known, it is lower than the true uncertainty when the model has estimated coefficients. The remaining columns provide the decomposition. In each row they add up to 100%. For instance, in the sample above, 81.84% of the variance of the one-step forecast error is due to the innovation in CANRGDPS itself.

Notes

If you want to compute the true uncertainty of forecast, you need to apply the technique of Monte Carlo integration to generate draws from the posterior distribution of the coefficients of the model. You then use SIMULATE to draw random shocks for the innovations during the forecast period.

Older Syntax

The MODEL, STEPS, CV and STDERRS options were added with version 7. Before that, the structure of the model was provided using supplementary cards. If you are writing a new program, we would strongly recommend using the newer methods. However, if you are trying to maintain or understand an older program, this describes the older style.

ERRORS( options )  equations steps VCVmatrix

# equation stderrors newstart column (one card per equation)

Parameters

Supplementary Cards

There is one supplementary card for each equation.
 

Example of Computing Forecast Standard Errors

boxjenk(ma=1,diffs=2,constant,define=cpieq) cpitran $

   1947:3 1979:12 resids

smpl 1980:1 1981:12

forecast 1

# cpieq logcpi

errors(noprint) 1

# cpieq fcsterr

 

estimates an ARIMA(0,2,1) model, then computes forecasts (into LOGCPI) and forecast errors (into FCSTERR). Note that the combination of FORECAST and ERRORS can be done with just one UFORECAST instruction.

 

Copyright © 2025 Thomas A. Doan