Predicted Values

The AUTOREG Procedure

Predicted Values

The AUTOREG procedure can produce two kinds of predicted values for the response series and corresponding residuals and confidence limits. The residuals in both cases are computed as the actual value minus the predicted value. In addition, when GARCH models are estimated, the AUTOREG procedure can output predictions of the conditional error variance.

Predicting the Unconditional Mean

The first type of predicted value is obtained from only the structural part of the model, x_t'b. These are useful in predicting values of new response time series, which are assumed to be described by the same model as the current response time series. The predicted values, residuals, and upper and lower confidence limits for the structural predictions are requested by specifying the PREDICTEDM=, RESIDUALM=, UCLM=, or LCLM= options in the OUTPUT statement. The ALPHACLM= option controls the confidence level for UCLM= and LCLM=. These confidence limits are for estimation of the mean of the dependent variable, x_t'b, where x_t is the column vector of independent variables at observation t.

The predicted values are computed as

$\hat{y}_{t}=x_{t}'b$

and the upper and lower confidence limits as

$\hat{u}_{t}= \hat{y}_{t}+ t_{{\alpha}/2 }\rm{v}$

$\hat{l}_{t}= \hat{y}_{t}- t_{{\alpha}/2 }\rm{v}$

where v² is an estimate of the variance of ${ \hat{y}_{t}}$ and ${t_{{\alpha}/2}}$ is the upper ${\alpha}$ /2 percentage point of the t distribution.

$\rm{Prob}(T \gt t_{{\alpha}/2})={\alpha}/2$

where T is an observation from a t distribution with q degrees of freedom. The value of ${\alpha}$ can be set with the ALPHACLM= option. The degrees of freedom parameter, q, is taken to be the number of observations minus the number of free parameters in the regression and autoregression parts of the model. For the YW estimation method, the value of v is calculated as

$\rm{v}=\sqrt{s^2{x}_{t}' (X'V^{-1}X)^{-1}x_{t} }$

where s² is the error sum of squares divided by q. For the ULS and ML methods, it is calculated as

$\rm{v}=\sqrt{ s^2{x}_{t}'{Wx}_{t} }$

where W is the k×k submatrix of (J'J)^-1 that corresponds to the regression parameters. For details, see "Computational Methods" earlier in this chapter.

Predicting Future Series Realizations

The other predicted values use both the structural part of the model and the predicted values of the error process. These conditional mean values are useful in predicting future values of the current response time series. The predicted values, residuals, and upper and lower confidence limits for future observations conditional on past values are requested by the PREDICTED=, RESIDUAL=, UCL=, or LCL= options in the OUTPUT statement. The ALPHACLI= option controls the confidence level for UCL= and LCL=. These confidence limits are for the predicted value,

$\tilde{y}_{t}=x_{t}'b+ {\nu}_{t| t-1}$

where x_t is the vector of independent variables and ${ {\nu}_{t| t-1}}$ is the minimum variance linear predictor of the error term given the available past values of ${{\nu}_{t-j},j=1,2, { ... }, t-1}$ , and the autoregressive

model for ${{\nu}_{t}}$ . If the m previous values of the structural residuals are available, then

${\nu}_{t| t-1}=- \hat{{\varphi}}_{1} {\nu}_{t-1}- { ... } - \hat{{\varphi}}_{m} {\nu}_{t-m}$

where ${ \hat{{\varphi}}_{1}, { ... } , \hat{{\varphi}}_{m}}$ are the estimated AR parameters. The upper and lower confidence limits are computed as

$\tilde{u}_{t}= \tilde{y}_{t}+ t_{{\alpha}/2}\rm{v}$

$\tilde{l}_{t}= \tilde{y}_{t}- t_{{\alpha}/2}\rm{v}$

where v, in this case, is computed as

$\rm{v}=\sqrt{ s^2(x_{t}' ({X'V}^{-1}X)^{-1}x_{t}+r) }$

where the value rs² is the estimate of the variance of ${ {\nu}_{t| t-1}}$ . At the start of the series, and after missing values, r is generally greater than 1. See "Predicting the Conditional Variance" for computational details of r. The plot of residuals and confidence limits in Example 8.4 later in this chapter illustrates this behavior.

Except to adjust the degrees of freedom for the error sum of squares, the preceding formulas do not account for the fact that the autoregressive parameters are estimated. In particular, the confidence limits are likely to be somewhat too narrow. In large samples, this is probably not an important effect, but it may be appreciable in small samples. Refer to Harvey (1981) for some discussion of this problem for AR(1) models.

Note that at the beginning of the series (the first m observations, where m is the value of the NLAG= option) and after missing values, these residuals do not match the residuals obtained by using OLS on the transformed variables. This is because, in these cases, the predicted noise values must be based on less than a complete set of past noise values and, thus, have larger variance. The GLS transformation for these observations includes a scale factor as well as a linear combination of past values. Put another way, the L^-1 matrix defined in the section "Computational Methods" has the value 1 along the diagonal, except for the first m observations and after missing values.

Predicting the Conditional Variance

The GARCH process can be written

${\epsilon}^2_{t} = {\omega} + \sum_{i=1}^n{( {\alpha}_{i}+ {\gamma}_{i}) {\epsilon}^2_{t-i}} - \sum_{j=1}^p{{\gamma}_{j} {\eta}_{t-j}} + {\eta}_{t}$

where ${ {\eta}_{t}= {\epsilon}^2_{t}- h_{t}}$ and n = max(p,q). This representation shows that the squared residual ${\epsilon}^2_{t}$ follows an ARMA(n,p) process. Then for any d > 0, the conditional expectations are as follows:

$E( {\epsilon}^2_{t+d}| {\Psi}_{t}) = {\omega} + \sum_{i=1}^n{( {\alpha}_{i}+ {\... ...-i}| {\Psi}_{t})} - \sum_{j=1}^p{{\gamma}_{j}E ( {\eta}_{t+d-j}| {\Psi}_{t})}$

The d-step-ahead prediction error, ${\xi}_{t+d}$ = y_t+d - y_t+d|t, has the conditional variance

$V( {\xi}_{t+d}| {\Psi}_{t}) = \sum_{j=0}^{d-1}{g^2_{j} {\sigma}^2_{t+d-j| t}}$

where

${\sigma}^2_{t+d-j| t} = E( {\epsilon}^2_{t+d-j}| {\Psi}_{t})$

Coefficients in the conditional d-step prediction error variance are calculated recursively using the following formula:

$g_{j} = - {\varphi}_{1} g_{j-1} - { ... } - {\varphi}_{m} g_{j-m}$

where g₀=1 and g_j=0 if j<0; ${\varphi}_{1}$ , ..., ${\varphi}_{m}$ are autoregressive parameters. Since the parameters are not known, the conditional variance is computed using the estimated autoregressive parameters. The d-step-ahead prediction error variance is simplified when there are no autoregressive terms:

$V( {\xi}_{t+d}| {\Psi}_{t}) = {\sigma}^2_{t+d| t}$

Therefore, the one-step-ahead prediction error variance is equivalent to the conditional error variance defined in the GARCH process:

$h_{t} = E( {\epsilon}^2_{t}| {\Psi}_{t-1}) = {\sigma}^2_{t| t-1}$

Note that the conditional prediction error variance of the EGARCH and GARCH-M models cannot be calculated using the preceding formula. Therefore, the confidence intervals for the predicted values are computed assuming the homoscedastic conditional error variance. That is, the conditional prediction error variance is identical to the unconditional prediction error variance:

$V( {\xi}_{t+d}| {\Psi}_{t}) = V( {\xi}_{t+d}) = {\sigma}^2\sum_{j=0}^{d-1}{g^2_{j}}$

since ${ {\sigma}^2_{t+d-j| t} = {\sigma}^2}$ . You can compute s²r, which is the second term of the variance for the predicted value ${\tilde{y}_{t}}$ explained previously in "Predicting Future Series Realizations," using the formula ${ {\sigma}^2\sum_{j=0}^{d-1}{g^2_{j}}}$ ; r is estimated from ${\sum_{j=0}^{d-1}{g^2_{j}}}$ using the estimated autoregressive parameters.

Consider the following conditional prediction error variance:

$V( {\xi}_{t+d}| {\Psi}_{t}) = {\sigma}^2 \sum_{j=0}^{d-1}{g^2_{j}} + \sum_{j=0}^{d-1}{g^2_{j} ( {\sigma}^2_{t+d-j| t}} - {\sigma}^2)$

The second term in the preceding equation can be interpreted as the noise from using the homoscedastic conditional variance when the errors follow the GARCH process. However, it is expected that if the GARCH process is covariance stationary, the difference between the conditional prediction error variance and the unconditional prediction error variance disappears as the forecast horizon d increases.

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