R2 Statistics and Other Measures of Fit

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The AUTOREG Procedure

R² Statistics and Other Measures of Fit

This section discusses various goodness-of-fit statistics produced by the AUTOREG procedure.

Total R²

The total R² statistic (Total Rsq) is computed as

R²_tot = 1-[SSE/SST]

where SST is the sum of squares for the original response variable corrected for the mean and SSE is the final error sum of squares. The Total Rsq is a measure of how well the next value can be predicted using the structural part of the model and the past values of the residuals. If the NOINT option is specified, SST is the uncorrected sum of squares.

Regression R²

The regression R² (Reg RSQ) is computed as

R²_reg = 1-[TSSE/TSST]

where TSST is the total sum of squares of the transformed response variable corrected for the transformed intercept, and TSSE is the error sum of squares for this transformed regression problem. If the NOINT option is requested, no correction for the transformed intercept is made. The Reg RSQ is a measure of the fit of the structural part of the model after transforming for the autocorrelation and is the R²for the transformed regression.

The regression R² and the total R² should be the same when there is no autocorrelation correction (OLS regression).

Calculation of Recursive Residuals and CUSUM Statistics

The recursive residuals w_t are computed as

$w_{t} = \frac{e_{t}}{\sqrt{v_{t}}}$

$v_{t} = 1 + x^{'}_{t} [ \sum_{i=1}^{t-1} x_{i}x_{i}^{'}]^{-1}x_{t}$

Note that the forecast error variance of e_t is the scalar multiple of v_t such that ${V(e_{t})= {\sigma}^2 v_{t}}$ .

The CUSUM and CUSUMSQ statistics are computed using the preceding recursive residuals.

$\rm{CUSUM}_{t} = \sum_{i=k+1}^t{\frac{w_{i}}{{\sigma}_{w}}}$

$\rm{CUSUMSQ}_{t} = \frac{\sum_{i=k+1}^t{w^2_{i}}}{\sum_{i=k+1}^T{w^2_{i}}}$

where w_i are the recursive residuals,

${\sigma}_{w} = \sqrt{\frac{\sum_{i=k+1}^T{(w_{i}-\hat{w})^2}}{(T-k-1)}}$

$\hat{w} = \frac{1}{T-k} \sum_{i=k+1}^T{w_{i}}$

and k is the number of regressors.

The CUSUM statistics can be used to test for misspecification of the model. The upper and lower critical values for CUSUM_t are

${+-} a [ \sqrt{T-k} + 2\frac{(t-k)}{(T-k)^{\frac{1}2}}]$

where a = 1.143 for a significance level .01, 0.948 for .05, and 0.850 for .10. These critical values are output by the CUSUMLB= and CUSUMUB= options for the significance level specified by the ALPHACSM= option.

The upper and lower critical values of CUSUMSQ_t are given by

${+-} a + \frac{(t-k)}{T-k}$

where the value of a is obtained from the table by Durbin (1969) if the ${\frac{1}2(T-k)-1{\le}60}$ . Edgerton and Wells (1994) provided the method of obtaining the value of a for large samples.

These critical values are output by the CUSUMSQLB= and CUSUMSQUB= options for the significance level specified by the ALPHACSM= option.

Information Criteria AIC and SBC

The Akaike's information criterion (AIC) and the Schwarz's Bayesian information criterion (SBC) are computed as follows:

AIC = -2ln(L) + 2 k

SBC = -2ln(L) + ln(N) k

In these formulas, L is the value of the likelihood function evaluated at the parameter estimates, N is the number of observations, and k is the number of estimated parameters. Refer to Judge et al. (1985) and Schwarz (1978) for additional details.

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