|_* |_* Tests of autocorrelation |_* |_READ (DATA3-6) YEAR C Y 3 VARIABLES AND 36 OBSERVATIONS STARTING AT OBS 1 |_* |_* Estimate regression of C on Y, saving residuals in U. |_* The RSTAT option reports Residual STATistics. |_* The EXACTDW option reports and exact p-value for the Durbin-Watson statistic. |_* |_OLS C Y / RESID=U RSTAT EXACTDW REQUIRED MEMORY IS PAR= 13 CURRENT PAR= 500 OLS ESTIMATION 36 OBSERVATIONS DEPENDENT VARIABLE= C ...NOTE..SAMPLE RANGE SET TO: 1, 36 DURBIN-WATSON STATISTIC = 0.51370 DURBIN-WATSON P-VALUE = 0.000000 R-SQUARE = 0.9955 R-SQUARE ADJUSTED = 0.9954 VARIANCE OF THE ESTIMATE-SIGMA**2 = 39610. STANDARD ERROR OF THE ESTIMATE-SIGMA = 199.02 SUM OF SQUARED ERRORS-SSE= 0.13468E+07 MEAN OF DEPENDENT VARIABLE = 12491. LOG OF THE LIKELIHOOD FUNCTION = -240.616 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 34 DF P-VALUE CORR. COEFFICIENT AT MEANS Y 0.93274 0.1070E-01 87.20 0.000 0.998 0.9978 1.0308 CONSTANT -384.11 151.3 -2.538 0.016-0.399 0.0000 -0.0308 DURBIN-WATSON = 0.5137 VON NEUMANN RATIO = 0.5284 RHO = 0.76830 RESIDUAL SUM = 0.39790E-12 RESIDUAL VARIANCE = 39610. SUM OF ABSOLUTE ERRORS= 5651.4 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9955 RUNS TEST: 7 RUNS, 19 POS, 0 ZERO, 17 NEG NORMAL STATISTIC = -4.0525 |_* |_* Check autocorrelation diagnostics |_* |_DIAG /ACF REQUIRED MEMORY IS PAR= 5 CURRENT PAR= 500 DEPENDENT VARIABLE = C 36 OBSERVATIONS REGRESSION COEFFICIENTS 0.932737795839 -384.105457205 RESIDUAL CORRELOGRAM LM-TEST FOR HJ:RHO(J)=0, STATISTIC IS STANDARD NORMAL LAG RHO STD ERR T-STAT LM-STAT DW-TEST BOX-PIERCE-LJUNG 1 0.6536 0.1667 3.9217 4.2848 0.5137 16.6984 2 0.4193 0.1667 2.5158 2.9684 0.8589 23.7724 3 0.3304 0.1667 1.9822 2.3747 1.0207 28.2968 4 0.2372 0.1667 1.4231 1.7373 1.1891 30.7017 5 0.1199 0.1667 0.7193 0.9208 1.3785 31.3360 6 0.1095 0.1667 0.6570 0.9024 1.3641 31.8827 7 0.0326 0.1667 0.1959 0.2829 1.4977 31.9330 8 -0.0751 0.1667 -0.4503 0.7050 1.6889 32.2082 9 -0.1232 0.1667 -0.7393 1.1477 1.7791 32.9776 10 -0.1236 0.1667 -0.7419 1.1392 1.7646 33.7820 11 -0.0748 0.1667 -0.4488 0.7141 1.5684 34.0882 LM CHI-SQUARE STATISTIC WITH 11 D.F. IS 30.152 |_* |_* Perform a Breusch-Godfrey LM test of an AR(1) process. |_* Regress U on Y and U lagged. |_GENR ULAG = LAG(U) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO -99999. |_* It is not clear how to treat the first observation, |_* since ULAG is not defined for this observation. |_* Some authors (e.g., Ramanathan) recommend that the first observation be dropped. |_* This can be done by inserting the command |_* SAMPLE 2 36 |_* However, it appears to be better to keep this observation |_* while setting ULAG=0 (see Davidson & MacKinnon (1993) p. 358. |_IF (TIME(0).EQ.1) ULAG=0 |_* Run auxiliary regression. |_* It can be useful to invoke the RSTAT option to detect any serial correlation |_* not captured by the AR(1) specification. |_* The Durbin-Watson statistic is not valid because of the presence of ULAG, |_* but the Durbin h-statistic is asymptotically normally distributed. |_OLS U ULAG Y /RSTAT DLAG REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 500 OLS ESTIMATION 36 OBSERVATIONS DEPENDENT VARIABLE= U ...NOTE..SAMPLE RANGE SET TO: 1, 36 R-SQUARE = 0.5100 R-SQUARE ADJUSTED = 0.4803 VARIANCE OF THE ESTIMATE-SIGMA**2 = 19997. STANDARD ERROR OF THE ESTIMATE-SIGMA = 141.41 SUM OF SQUARED ERRORS-SSE= 0.65991E+06 MEAN OF DEPENDENT VARIABLE = 0.11053E-13 LOG OF THE LIKELIHOOD FUNCTION = -227.776 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 33 DF P-VALUE CORR. COEFFICIENT AT MEANS ULAG 0.78026 0.1331 5.861 0.000 0.714 0.7179*********** Y 0.45892E-02 0.7641E-02 0.6006 0.552 0.104 0.0736*********** CONSTANT -53.629 107.9 -0.4970 0.623-0.086 0.0000*********** DURBIN-WATSON = 1.9591 VON NEUMANN RATIO = 2.0150 RHO = -0.03419 RESIDUAL SUM = -0.11369E-12 RESIDUAL VARIANCE = 19997. SUM OF ABSOLUTE ERRORS= 3857.6 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.5100 RUNS TEST: 17 RUNS, 19 POS, 0 ZERO, 17 NEG NORMAL STATISTIC = -0.6597 DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = -0.34096 |_* One version of the test is an ordinary t-test on ULAG. |_TEST ULAG TEST VALUE = 0.78026 STD. ERROR OF TEST VALUE 0.13314 T STATISTIC = 5.8605827 WITH 33 D.F. P-VALUE= 0.00000 F STATISTIC = 34.346429 WITH 1 AND 33 D.F. P-VALUE= 0.00000 WALD CHI-SQUARE STATISTIC = 34.346429 WITH 1 D.F. P-VALUE= 0.00000 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.02912 |_* Another is an LM test on NR2, which is asymptotically chi-square. |_GEN1 LM = $N*$R2 ..NOTE..CURRENT VALUE OF $N = 36.000 ..NOTE..CURRENT VALUE OF $R2 = 0.51000 |_DISTRIB LM /TYPE=CHI DF=1 CHI-SQUARE PARAMETERS- DF= 1.0000 MEAN= 1.0000 VARIANCE= 2.0000 MODE= 0.0000 DATA PDF CDF 1-CDF LM ROW 1 18.360 0.95980E-05 0.99998 0.18287E-04 ..INPUT FILE COMPLETED..TYPE A NEW COMMAND OR TYPE: STOP TYPE COMMAND