B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:19 DATA STEP Durbin Spirits Data PAGE 1 Variable Label # Cases Mean Std. Dev. Variance Maximum Minimum YEAR 1 YEARS 1870-1938 69 1904.00 20.0624 402.500 1938.00 1870.00 CONSUMP 2 CONSUMPTION OF SPIRITS 69 1.77038 0.271166 0.735312E-01 2.06780 1.25270 INCOME 3 INCOME 69 1.96216 0.964065E-01 0.929421E-02 2.12050 1.76690 PRICE 4 PRICE OF SPIRITS 69 2.11838 0.208861 0.436228E-01 2.50480 1.87270 YEARSQ 5 YEAR**2 69 0.362561E+07 76398.5 0.583673E+10 0.375584E+07 0.349690E+07 AYEAR 6 YEAR - 1870 69 34.0000 20.0624 402.500 68.0000 0.00000 AYEARSQ 7 (YEAR-1980-35)**2 69 397.667 359.522 129256. 1225.00 0.00000 CONSTANT 8 69 1.00000 0.00000 0.00000 1.00000 1.00000 Number of observations in data file 69 Current missing variable code 1.000000000000000E+31 Data begins on (D:M:Y) 1: 1:1870 ends 1: 1:1938. Frequency is 1 B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:19 REGRESSION STEP Durbin Spirits Data PAGE 2 ******************************************************************** Problem Number 1 Subproblem Number 1 F to enter 9.999999776482582E-03 F to remove 4.999999888241291E-03 Tolerance (1.-R**2) for including a variable 1.000000000000000E-05 Maximum Number of Variables Allowed 4 Internal Number of dependent variable 2 Dependent Variable CONSUMP Standard Error of Y 0.2711664355312739 Degrees of Freedom 68 ............ - At step No. 4, Variable No. 1 deleted for tolerance. Diagonal element = 192643.49 . Max step currently 5. NOTE: Variable number 1 was currently in the regression at the prior step. ............. Step Number 5 Analysis of Variance for increase in SS due to variable leaving Variable leaving 1 Source DF SS MS F F Sig. Multiple R 0.977656 Due Regression 2 4.7792 2.3896 713.79 1.000000 Std Error of Y.X 0.578598E-01 Dev. from Reg. 66 0.22095 0.33478E-02 R Square 0.955811 Total 68 5.0001 0.73531E-01 Multiple Regression Equation Variable Coefficient Std. Error T Val. T Sig. P. Cor. Elasticity Partial Cor. for Var. not in equation CONSUMP = Variable Coefficient F for selection INCOME X- 3 -0.1201411 0.1084358 -1.108 0.72809 -0.1351 -0.1332 PRICE X- 4 -1.227648 0.5005206E-01 -24.53 1.00000 -0.9493 -1.469 CONSTANT X- 8 4.606734 0.1520348 30.30 1.00000 Adjusted R Square 0.9544716767978285 -2 * ln(Maximum of Likelihood Function) -200.5167835719893 Akaike Information Criterion (AIC) -192.5167835719893 Scwartz Information Criterion (SIC) -183.5803575536002 Akaike (1970) Finite Prediction Error 3.493308383038779E-03 Generalized Cross Validation 3.499924497400595E-03 Hannan & Quinn (1979) HQ 3.630357458250973E-03 Shibata (1981) 3.480651468607479E-03 Rice (1984) 3.507170717892108E-03 Residual Variance 3.347753867078830E-03 Order of entrance (or deletion) of the variables = 8 4 1 3 -1 Estimate of Computational Error in Coefficients 1 2 3 -0.212595E-10 0.797905E-11 -0.539548E-10 Covariance Matrix of Regression Coefficients Row 1 Variable X- 3 INCOME 0.11758328E-01 Row 2 Variable X- 4 PRICE -0.40232954E-02 0.25052087E-02 Row 3 Variable X- 8 CONSTANT -0.14548831E-01 0.25873630E-02 0.23114595E-01 Program terminated. All variables put in except those for which A(I,I) less than toll. They are .... 1 Residual Statistics for Original data Von Neumann Ratio 1 ... 0.25243 Durbin-Watson TEST..... 0.24878 Von Neumann Ratio 2 ... 0.25243 For D. F. 66 t(.9999)= 4.1419, t(.999)= 3.4441, t(.99)= 2.6524, t(.95)= 1.9966, t(.90)= 1.6683, t(.80)= 1.2945 Skewness test (Alpha 3) = -.220867 , Peakedness test (Alpha 4)= 2.37525 Normality Test -- Extended grid cell size = 6.90 t Stat Infin 1.668 1.295 1.045 0.847 0.678 0.527 0.387 0.254 0.126 Cell No. 5 10 9 3 4 9 5 10 9 5 Interval 1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 Act Per 1.000 0.928 0.783 0.652 0.609 0.551 0.420 0.348 0.203 0.072 Normality Test -- Small sample grid cell size = 13.80 Cell No. 15 12 13 15 14 Interval 1.000 0.800 0.600 0.400 0.200 Act Per 1.000 0.783 0.609 0.420 0.203 Extended grid normality test - Prob of rejecting normality assumption Chi= 9.696 Chi Prob= 0.7130 F(8, 66)= 1.21196 F Prob =0.694069 Small sample normality test - Large grid Chi= 0.4928 Chi Prob= 0.7952E-01 F(3, 66)= 0.164251 F Prob =0.799275E-01 Autocorrelation function of residuals 1 2 3 4 0.863360 0.759578 0.657376 0.531220 F( 23, 23) = 0.8010 1/F = 1.248 Heteroskedasticity at 0.7005 level Sum of squared residuals 0.2209517552274537 Mean squared residual 3.202199351122518E-03 B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:19 REGRESSION STEP Durbin Spirits Data PAGE 3 Doing Gen. Least Squares using residual Dif. Eq. of order 1 Lag Coefficients 1 0.866207 Standard Error of Y 0.4140293971616056 Degrees of Freedom 67 ............. Step Number 4 Analysis of Variance for reduction in SS due to variable entering Variable Entering 3 Source DF SS MS F F Sig. Multiple R 0.934267 Due Regression 3 10.025 3.3416 146.45 1.000000 Std Error of Y.X 0.151053 Dev. from Reg. 64 1.4603 0.22817E-01 R Square 0.872855 Total 67 11.485 0.17142 Multiple Regression Equation Variable Coefficient Std. Error T Val. T Sig. P. Cor. Elasticity Partial Cor. for Var. not in equation CONSUMP = Variable Coefficient F for selection YEAR X- 1 -0.9238011E-02 0.1367314E-02 -6.756 1.00000 -0.6452 -9.935 INCOME X- 3 0.7278257 0.1435748 5.069 1.00000 0.5353 0.8067 PRICE X- 4 -0.8517519 0.7256565E-01 -11.74 1.00000 -0.8263 -1.019 CONSTANT X- 8 19.75221 2.396543 8.242 1.00000 Adjusted R Square 0.8668947126129325 -2 * ln(Maximum of Likelihood Function) -68.20393097575609 Akaike Information Criterion (AIC) -58.20393097575609 Scwartz Information Criterion (SIC) -47.10639244987556 Akaike (1970) Finite Prediction Error 2.415912760358642E-02 Generalized Cross Validation 2.424301346332110E-02 Hannan & Quinn (1979) HQ 2.543834981037451E-02 Shibata (1981) 2.400122480879174E-02 Rice (1984) 2.433808410435373E-02 Residual Variance 2.281695384783162E-02 Order of entrance (or deletion) of the variables = 8 4 1 3 Covariance Matrix of Regression Coefficients Row 1 Variable X- 1 YEAR 0.18695476E-05 Row 2 Variable X- 3 INCOME -0.93091583E-04 0.20613728E-01 Row 3 Variable X- 4 PRICE -0.58694059E-04 0.81633830E-03 0.52657738E-02 Row 4 Variable X- 8 CONSTANT -0.32590869E-02 0.13492587 0.99090040E-01 5.7434198 Program terminated. All variables put in. Residual Statistics for Smoothed Original data For GLS Y and Y estimate scaled by 0.1337931870652028 Von Neumann Ratio 1 ... 2.24041 Durbin-Watson TEST..... 2.20746 Von Neumann Ratio 2 ... 2.24041 For D. F. 64 t(.9999)= 4.1502, t(.999)= 3.4491, t(.99)= 2.6549, t(.95)= 1.9977, t(.90)= 1.6690, t(.80)= 1.2949 Skewness test (Alpha 3) = -.809329 , Peakedness test (Alpha 4)= 8.19174 Normality Test -- Extended grid cell size = 6.80 t Stat Infin 1.669 1.295 1.045 0.847 0.678 0.527 0.387 0.254 0.126 Cell No. 4 4 1 7 8 7 9 9 10 9 Interval 1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 Act Per 1.000 0.941 0.882 0.868 0.765 0.647 0.544 0.412 0.279 0.132 Normality Test -- Small sample grid cell size = 13.60 Cell No. 8 8 15 18 19 Interval 1.000 0.800 0.600 0.400 0.200 Act Per 1.000 0.882 0.765 0.544 0.279 Extended grid normality test - Prob of rejecting normality assumption Chi= 11.12 Chi Prob= 0.8049 F(8, 64)= 1.38971 F Prob =0.781875 Small sample normality test - Large grid Chi= 8.324 Chi Prob= 0.9602 F(3, 64)= 2.77451 F Prob =0.951554 Autocorrelation function of residuals 1 2 3 4 -0.110369 0.128983 0.427210E-01 0.961270E-01 F( 23, 23) = 0.3225 1/F = 3.101 Heteroskedasticity at 0.9956 level Sum of squared residuals 1.460285046254511 Mean squared residual 2.147478009197811E-02 Gen. Least Squares ended by satisfying tolerance. B34S(r) Matrix Command. d/m/y 16/ 9/05. h:m:s 13: 8:19. => CALL LOADDATA$ => CALL LOAD(GLS :STAGING)$ => CALL PRINT(GLS,GLSMOD)$ GLS = Program PROGRAM GLS$ CALL ECHOOFF$ %BASEX=%X$ %BASEY=%Y$ IF(KIND(%MAXGLS).NE.-4)%MAXGLS=1$ I=VFAM(INTEGERS(%MAXGLS+1,NOROWS(%Y)))$ %OLSRES =%RES$ %OLSCOEF=%COEF$ %OLSSE =%SE$ %OLST =%T$ IF(%MAXGLS.GT.0)THEN$ DO KK=1,%MAXGLS$ II=NOROWS(%OLSCOEF)+KK$ %NAMES(II)=OBJECT(RHO_,EVAL('kk'))$ ENDDO$ ENDIF$ PARM=VECTOR(NOROWS(%COEF)+%MAXGLS:)+.1$ GLSY=%Y(I)$ XP=TRANSPOSE(%X)$ MAXIT%=%MAXIT$ FLAM%=%FLAM$ FLU%=%FLU$ EPS2%=%EPS2$ CALL NLLSQ(GLSY YHAT :NAME GLSMOD :PARMS PARM :MAXIT MAXIT% :FLAM FLAM% :FLU FLU% :EPS2 EPS2% :PRINT RESULT )$ CALL TABULATE(%NAMES %LAG %OLSCOEF %OLSSE %OLST %COEF %SE %T :TITLE 'OLS and GLS Estimates using NLLSQ')$ IF(%PLOT.EQ.1)CALL GRAPH(%RES :HEADING 'GLS Residual NLLSQ')$ IF(%NL2SOL.EQ.1)THEN$ PARM=VECTOR(NOROWS(%OLSCOEF)+%MAXGLS:)+.1$ CALL NL2SOL(GLSRES :NAME GLSMOD :PARMS PARM :MAXIT MAXIT% :IVALUE PARM :PRINT )$ CALL TABULATE(%NAMES %LAG %OLSCOEF %OLSSE %OLST %COEF %SE %T :TITLE 'OLS and GLS Estimates using NL2SOL')$ IF(%PLOT.EQ.1)THEN$ CALL GRAPH(%RES :HEADING 'GLS Residual using NL2SOL')$ ENDIF$ ENDIF$ RETURN$ END$ GLSMOD = Program PROGRAM GLSMOD$ IF(%MAXGLS.EQ.0)THEN$ YHAT =PARM*XP$ ENDIF$ IF(%MAXGLS.GT.0)THEN$ P1=PARM(INTEGERS(1,NOROWS(PARM)-%MAXGLS) )$ P2=VECTOR(:PARM(INTEGERS( NOROWS(PARM)-%MAXGLS+1,NOROWS(PARM))))$ YHAT=P1*XP(,I)$ DO J=1,%MAXGLS$ YHAT=YHAT+(P2(J)* %Y(I-J))$ PP=P2(J)*P1$ YHAT=YHAT-(PP*XP(,I-J))$ ENDDO$ ENDIF$ GLSRES=GLSY-YHAT$ RETURN$ END$ => CALL OLSQ(CONSUMP INCOME PRICE YEAR => :PRINT :SAVEX)$ Ordinary Least Squares Estimation Dependent variable CONSUMP Centered R**2 0.9773924575557461 Adjusted R**2 0.9763490325198575 Residual Sum of Squares 0.1130405162708000 Residual Variance 1.739084865704615E-03 Total Sum of Squares 5.000124031594203 Mean of the Dependent Variable 1.770379710144927 Std. Error of Dependent Variable 0.2711664355312771 Sum Absolute Residuals 2.282427644213505 1/Condition XPX 2.096266616089903E-12 Maximum Absolute Residual 8.911539009341696E-02 Number of Observations 69 Variable Lag Coefficient SE t INCOME 0 1.0620261 0.16920535 6.2765515 PRICE 0 -0.85999429 0.58989575E-01 -14.578750 YEAR 0 -0.91158099E-02 0.11572370E-02 -7.8772196 CONSTANT 0 18.864811 1.8133532 10.403275 => CALL GLSSET$ => %MAXGLS=1$ => %NL2SOL=0$ => %PLOT=0$ => %MAXIT=2000$ => %FLAM = 1.0$ => %FLU = 10.$ => %EPS2 = .1E-12$ => RETURN$ => %MAXGLS=1$ => %NL2SOL=1$ => %PLOT=1$ => CALL GLS$ => CALL ECHOOFF$ Nonlinear Estimation using NLLSQ # of observations 68 # parameters 5 Max iterations 2000 Starting Lamda (FLAM) 1.000000000000000 Starting FLU 10.00000000000000 Maximum relative change in sum squares (eps1) 0.000000000000000E+00 Maximum relative change in each parm. (eps2) 1.000000000000000E-13 Initial Parameter Values (TH) 1 2 3 4 5 0.1000 0.1000 0.1000 0.1000 0.1000 Proportions used in calculating difference quotients 1 2 3 4 5 0.1000E-01 0.1000E-01 0.1000E-01 0.1000E-01 0.1000E-01 Sign restriction vector (GT 0.0 means restricted 1 2 3 4 5 0.000 0.000 0.000 0.000 0.000 Initial sum of squares 1972025.518144778 Number of observations = 68 Iteration stops - % change in each parm. LE 1.000000000000000E-13 Correlation Matrix of Estimated Parameters. 1 2 3 4 5 1 1.0000 2 0.0915 1.0000 3 -0.5095 -0.5656 1.0000 4 0.4338 0.5428 -0.9951 1.0000 5 0.1077 0.0183 -0.4364 0.4633 1.0000 Normalizing Elements 1 2 3 4 5 7.155 3.550 0.7054E-01 123.8 2.922 Variance of residuals 4.141801490448914E-04 Sum of squared residuals 2.609334938982816E-02 Standard Error of Estimate 2.035141638915806E-02 Adjusted R Square 0.9944108325955119 Degrees of freedom 63 Number of Iterations 71 1/Conditition of Hessian 1.111405089251697E-11 Durbin Watson 2.166938361313933 # Name Coefficient Standard Error T Value 1 BETA___1 0.73073733 0.14560807 5.0185223 2 BETA___2 -0.85177823 0.72256684E-01 -11.788228 3 BETA___3 -0.90393178E-02 0.14356844E-02 -6.2961733 4 BETA___4 19.364847 2.5203041 7.6835359 5 BETA___5 0.84633254 0.59456817E-01 14.234407 Note: Confidence limits for each parameter on linear hypothesis. OLS and GLS Estimates using NLLSQ Obs %NAMES %LAG %OLSCOEF %OLSSE %OLST %COEF %SE %T 1 INCOME 0 1.062 0.1692 6.277 0.7307 0.1456 5.019 2 PRICE 0 -0.8600 0.5899E-01 -14.58 -0.8518 0.7226E-01 -11.79 3 YEAR 0 -0.9116E-02 0.1157E-02 -7.877 -0.9039E-02 0.1436E-02 -6.296 4 CONSTANT 0 18.86 1.813 10.40 19.36 2.520 7.684 5 RHO_1 NA NA NA NA 0.8463 0.5946E-01 14.23 Note: Graphics not available with this version of B34S Nonlinear Estimation using NL2SNO - Analytic Jacobian Sum of squared Residuals 2.609334938990657E-02 Residual Variance 4.141801490461361E-04 Residual Standard Error 2.035141638918864E-02 # of parameters 5 # of residuals 68 # of iterations 13 # of function evaluations 35 # of gradiant evaluations 14 # of Covariance evaluations 0 Relative Function Tolerance 1.000000000000000E-10 Finite-Difference factor 1.489370874967368E-08 Absolute Function Tolerance 9.999999999999999E-21 False Convergence Tolerance 2.220446049250313E-14 X-Convergence Tolerance 1.489370874967368E-08 2-norm of scaled gradiant 1.360991440505461E-08 2-norm of scaled step size 1.120345842194107E-04 1. / Condition of Hessian 0.4998742805493407 *** relative function convergence *** # Name Coefficient Standard Error T Value 1 BETA___1 0.73073912 0.0000000 0.17976931+309 2 BETA___2 -0.85177785 0.11666154E-03 -7301.2740 3 BETA___3 -0.90393280E-02 0.60129760E-06 -15033.035 4 BETA___4 19.364863 0.27359507E-01 707.79281 5 BETA___5 0.84633267 0.11422204 7.4095390 SE calculated as sqrt |diagonal(Covariance Matrix)| Finite-Difference Hessian indefinite Gradiant Vector 0.302084E-07 0.828733E-08 -0.365165E-05 -0.215027E-08 0.259178E-07 Scale Vector 2.53375 2.77223 2422.29 1.26814 4.81266 OLS and GLS Estimates using NL2SOL Obs %NAMES %LAG %OLSCOEF %OLSSE %OLST %COEF %SE %T 1 INCOME 0 1.062 0.1692 6.277 0.7307 0.000 0.1798+309 2 PRICE 0 -0.8600 0.5899E-01 -14.58 -0.8518 0.1167E-03 -7301. 3 YEAR 0 -0.9116E-02 0.1157E-02 -7.877 -0.9039E-02 0.6013E-06 -0.1503E+05 4 CONSTANT 0 18.86 1.813 10.40 19.36 0.2736E-01 707.8 5 RHO_1 NA NA NA NA 0.8463 0.1142 7.410 Note: Graphics not available with this version of B34S B34S Matrix Command Ending. Last Command reached. Space available in allocator 6856896, peak space used 10036 Number variables used 101, peak number used 102 Number temp variables used 20059, # user temp clean 0 Output from SAS The SAS System 13:08 Friday, September 16, 2005 1 The MEANS Procedure Variable Label N Mean Std Dev Minimum Maximum ャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャ YEAR YEARS 1870-1938 69 1904.00 20.0624026 1870.00 1938.00 CONSUMP CONSUMPTION OF SPIRITS 69 1.7703797 0.2711664 1.2527000 2.0678000 INCOME INCOME 69 1.9621565 0.0964065 1.7669000 2.1205000 PRICE PRICE OF SPIRITS 69 2.1183754 0.2088608 1.8727000 2.5048000 YEARSQ YEAR**2 69 3625612.67 76398.46 3496900.00 3755844.00 AYEAR YEAR - 1870 69 34.0000000 20.0624026 0 68.0000000 AYEARSQ (YEAR-1980-35)**2 69 397.6666667 359.5221365 0 1225.00 CONSTANT 69 1.0000000 0 1.0000000 1.0000000 ャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャャ The SAS System 13:08 Friday, September 16, 2005 2 The AUTOREG Procedure Dependent Variable CONSUMP CONSUMPTION OF SPIRITS Ordinary Least Squares Estimates SSE 0.11304052 DFE 65 MSE 0.00174 Root MSE 0.04170 SBC -229.82402 AIC -238.76045 Regress R-Square 0.9774 Total R-Square 0.9774 Durbin-Watson 0.2852 Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 18.8648 1.8134 10.40 <.0001 INCOME 1 1.0620 0.1692 6.28 <.0001 INCOME PRICE 1 -0.8600 0.0590 -14.58 <.0001 PRICE OF SPIRITS YEAR 1 -0.009116 0.001157 -7.88 <.0001 YEARS 1870-1938 Estimates of Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0 0.00164 1.000000 | |********************| 1 0.00132 0.805266 | |**************** | Preliminary MSE 0.000576 Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value 1 -0.805266 0.074114 -10.87 Algorithm converged. The SAS System 13:08 Friday, September 16, 2005 3 The AUTOREG Procedure Unconditional Least Squares Estimates SSE 0.02918799 DFE 64 MSE 0.0004561 Root MSE 0.02136 SBC -317.1579 AIC -328.32844 Regress R-Square 0.8139 Total R-Square 0.9942 Durbin-Watson 2.0203 Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 15.9067 2.7014 5.89 <.0001 INCOME 1 0.7646 0.1496 5.11 <.0001 INCOME PRICE 1 -0.8848 0.0769 -11.51 <.0001 PRICE OF SPIRITS YEAR 1 -0.007238 0.001535 -4.72 <.0001 YEARS 1870-1938 AR1 1 -0.9186 0.0562 -16.34 <.0001 Autoregressive parameters assumed given. Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 15.9067 2.5501 6.24 <.0001 INCOME 1 0.7646 0.1493 5.12 <.0001 INCOME PRICE 1 -0.8848 0.0767 -11.53 <.0001 PRICE OF SPIRITS YEAR 1 -0.007238 0.001456 -4.97 <.0001 YEARS 1870-1938 The SAS System 13:08 Friday, September 16, 2005 4 The AUTOREG Procedure Dependent Variable CONSUMP CONSUMPTION OF SPIRITS Ordinary Least Squares Estimates SSE 0.11304052 DFE 65 MSE 0.00174 Root MSE 0.04170 SBC -229.82402 AIC -238.76045 Regress R-Square 0.9774 Total R-Square 0.9774 Durbin-Watson 0.2852 Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 18.8648 1.8134 10.40 <.0001 INCOME 1 1.0620 0.1692 6.28 <.0001 INCOME PRICE 1 -0.8600 0.0590 -14.58 <.0001 PRICE OF SPIRITS YEAR 1 -0.009116 0.001157 -7.88 <.0001 YEARS 1870-1938 Estimates of Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0 0.00164 1.000000 | |********************| 1 0.00132 0.805266 | |**************** | Preliminary MSE 0.000576 Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value 1 -0.805266 0.074114 -10.87 Algorithm converged. The SAS System 13:08 Friday, September 16, 2005 5 The AUTOREG Procedure Maximum Likelihood Estimates SSE 0.02922692 DFE 64 MSE 0.0004567 Root MSE 0.02137 SBC -317.25714 AIC -328.42767 Regress R-Square 0.8333 Total R-Square 0.9942 Durbin-Watson 1.9793 Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 15.9728 2.4692 6.47 <.0001 INCOME 1 0.7704 0.1501 5.13 <.0001 INCOME PRICE 1 -0.8877 0.0765 -11.60 <.0001 PRICE OF SPIRITS YEAR 1 -0.007274 0.001422 -5.11 <.0001 YEARS 1870-1938 AR1 1 -0.9006 0.0571 -15.77 <.0001 Autoregressive parameters assumed given. Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 15.9728 2.3715 6.74 <.0001 INCOME 1 0.7704 0.1499 5.14 <.0001 INCOME PRICE 1 -0.8877 0.0764 -11.62 <.0001 PRICE OF SPIRITS YEAR 1 -0.007274 0.001372 -5.30 <.0001 YEARS 1870-1938 The SAS System 13:08 Friday, September 16, 2005 6 The AUTOREG Procedure Dependent Variable CONSUMP CONSUMPTION OF SPIRITS Ordinary Least Squares Estimates SSE 0.11304052 DFE 65 MSE 0.00174 Root MSE 0.04170 SBC -229.82402 AIC -238.76045 Regress R-Square 0.9774 Total R-Square 0.9774 Durbin-Watson 0.2852 Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 18.8648 1.8134 10.40 <.0001 INCOME 1 1.0620 0.1692 6.28 <.0001 INCOME PRICE 1 -0.8600 0.0590 -14.58 <.0001 PRICE OF SPIRITS YEAR 1 -0.009116 0.001157 -7.88 <.0001 YEARS 1870-1938 Estimates of Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0 0.00164 1.000000 | |********************| 1 0.00132 0.805266 | |**************** | Preliminary MSE 0.000576 Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value 1 -0.805266 0.074114 -10.87 Algorithm converged. The SAS System 13:08 Friday, September 16, 2005 7 The AUTOREG Procedure Yule-Walker Estimates SSE 0.02981037 DFE 64 MSE 0.0004658 Root MSE 0.02158 SBC -316.32079 AIC -327.49133 Regress R-Square 0.8832 Total R-Square 0.9940 Durbin-Watson 1.8440 Standard Approx Variable DF Estimate Error t Value Pr > |t| Variable Label Intercept 1 16.1673 2.0592 7.85 <.0001 INCOME 1 0.7928 0.1520 5.22 <.0001 INCOME PRICE 1 -0.8962 0.0748 -11.98 <.0001 PRICE OF SPIRITS YEAR 1 -0.007387 0.001229 -6.01 <.0001 YEARS 1870-1938 B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:37 PGMCALL STEP Durbin Spirits Data PAGE 4 Output from RATS * * Data passed from B34S(r) system to RATS * CALENDAR 1870 ALLOCATE 69 OPEN DATA rats.dat DATA(FORMAT=FREE,ORG=OBS, $ MISSING= 0.1000000000000000E+32 ) / $ YEAR $ CONSUMP $ INCOME $ PRICE $ YEARSQ $ AYEAR $ AYEARSQ $ CONSTANT SET TREND = T TABLE Series Obs Mean Std Error Minimum Maximum YEAR 69 1904.000000 20.062403 1870.000000 1938.000000 CONSUMP 69 1.770380 0.271166 1.252700 2.067800 INCOME 69 1.962157 0.096406 1.766900 2.120500 PRICE 69 2.118375 0.208861 1.872700 2.504800 YEARSQ 69 3625612.666667 76398.464685 3496900.000000 3755844.000000 AYEAR 69 34.000000 20.062403 0.000000 68.000000 AYEARSQ 69 397.666667 359.522137 0.000000 1225.000000 TREND 69 35.000000 20.062403 1.000000 69.000000 * * Various options of RATS AR1 command are tested * linreg consump # constant income price year Linear Regression - Estimation by Least Squares Dependent Variable CONSUMP Annual Data From 1870:01 To 1938:01 Usable Observations 69 Degrees of Freedom 65 Centered R**2 0.977392 R Bar **2 0.976349 Uncentered R**2 0.999489 T x R**2 68.965 Mean of Dependent Variable 1.7703797101 Std Error of Dependent Variable 0.2711664355 Standard Error of Estimate 0.0417023365 Sum of Squared Residuals 0.1130405163 Regression F(3,65) 936.7156 Significance Level of F 0.00000000 Log Likelihood 123.38023 Durbin-Watson Statistic 0.285221 Variable Coeff Std Error T-Stat Signif ******************************************************************************* 1. Constant 18.86481099 1.81335318 10.40327 0.00000000 B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:37 PGMCALL STEP Durbin Spirits Data PAGE 5 2. INCOME 1.06202609 0.16920535 6.27655 0.00000003 3. PRICE -0.85999429 0.05898958 -14.57875 0.00000000 4. YEAR -0.00911581 0.00115724 -7.87722 0.00000000 * ar1(method=corc) consump # constant income price year Regression with AR1 - Estimation by Cochrane-Orcutt Dependent Variable CONSUMP Annual Data From 1871:01 To 1938:01 Usable Observations 68 Degrees of Freedom 63 Centered R**2 0.994745 R Bar **2 0.994411 Uncentered R**2 0.999880 T x R**2 67.992 Mean of Dependent Variable 1.7676426471 Std Error of Dependent Variable 0.2722206591 Standard Error of Estimate 0.0203514164 Sum of Squared Residuals 0.0260933494 Regression F(4,63) 2981.1185 Significance Level of F 0.00000000 Log Likelihood 170.94199 Durbin-Watson Statistic 2.166931 Q(17-1) 37.051543 Significance Level of Q 0.00206232 Variable Coeff Std Error T-Stat Signif ******************************************************************************* 1. Constant 19.36479172 2.23346579 8.67029 0.00000000 2. INCOME 0.73073790 0.14476189 5.04786 0.00000407 3. PRICE -0.85177826 0.07224446 -11.79022 0.00000000 4. YEAR -0.00903929 0.00129171 -6.99790 0.00000000 ******************************************************************************* 5. RHO 0.84632923 0.05945683 14.23435 0.00000000 * ar1(method=hilu) consump # constant income price year Regression with AR1 - Estimation by Hildreth-Lu Search Dependent Variable CONSUMP Annual Data From 1871:01 To 1938:01 Usable Observations 68 Degrees of Freedom 63 Centered R**2 0.994745 R Bar **2 0.994411 Uncentered R**2 0.999880 T x R**2 67.992 Mean of Dependent Variable 1.7676426471 Std Error of Dependent Variable 0.2722206591 Standard Error of Estimate 0.0203514164 Sum of Squared Residuals 0.0260933494 Regression F(4,63) 2981.1185 Significance Level of F 0.00000000 Log Likelihood 170.94199 Durbin-Watson Statistic 2.166936 Q(17-1) 37.051448 Significance Level of Q 0.00206238 Variable Coeff Std Error T-Stat Signif B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:37 PGMCALL STEP Durbin Spirits Data PAGE 6 ******************************************************************************* 1. Constant 19.36482908 2.23348289 8.67024 0.00000000 2. INCOME 0.73073752 0.14476188 5.04786 0.00000407 3. PRICE -0.85177824 0.07224457 -11.79020 0.00000000 4. YEAR -0.00903931 0.00129172 -6.99787 0.00000000 ******************************************************************************* 5. RHO 0.84633145 0.05945684 14.23438 0.00000000 * ar1(method=maxl) consump # constant income price year Regression with AR1 - Estimation by Beach-MacKinnon Dependent Variable CONSUMP Annual Data From 1870:01 To 1938:01 Usable Observations 69 Degrees of Freedom 64 Centered R**2 0.994155 R Bar **2 0.993790 Uncentered R**2 0.999868 T x R**2 68.991 Mean of Dependent Variable 1.7703797101 Std Error of Dependent Variable 0.2711664355 Standard Error of Estimate 0.0213695109 Sum of Squared Residuals 0.0292259837 Log Likelihood 169.21385 Durbin-Watson Statistic 2.108397 Q(17-1) 30.210889 Significance Level of Q 0.01693894 Variable Coeff Std Error T-Stat Signif ******************************************************************************* 1. Constant 15.96698055 2.37345256 6.72732 0.00000001 2. INCOME 0.77013868 0.14990446 5.13753 0.00000282 3. PRICE -0.88766092 0.07641940 -11.61565 0.00000000 4. YEAR -0.00727094 0.00137246 -5.29776 0.00000154 ******************************************************************************* 5. RHO 0.90078850 0.05705435 15.78825 0.00000000 * ar1(method=search) consump # constant income price year Regression with AR1 - Estimation by Maximum Likelihood Search Dependent Variable CONSUMP Annual Data From 1870:01 To 1938:01 Usable Observations 69 Degrees of Freedom 64 Centered R**2 0.994155 R Bar **2 0.993790 Uncentered R**2 0.999868 T x R**2 68.991 Mean of Dependent Variable 1.7703797101 Std Error of Dependent Variable 0.2711664355 Standard Error of Estimate 0.0213695078 Sum of Squared Residuals 0.0292259752 Log Likelihood 169.21385 Durbin-Watson Statistic 2.108401 Q(17-1) 30.210890 Significance Level of Q 0.01693894 Variable Coeff Std Error T-Stat Signif B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:37 PGMCALL STEP Durbin Spirits Data PAGE 7 ******************************************************************************* 1. Constant 15.96697227 2.37347010 6.72727 0.00000001 2. INCOME 0.77013795 0.14990439 5.13753 0.00000282 3. PRICE -0.88766058 0.07641944 -11.61564 0.00000000 4. YEAR -0.00727093 0.00137246 -5.29772 0.00000154 ******************************************************************************* 5. RHO 0.90079059 0.05705366 15.78848 0.00000000 * B34S 8.10Z (D:M:Y) 16/ 9/05 (H:M:S) 13: 8:38 PGMCALL STEP Durbin Spirits Data PAGE 8 B34S normal exit on Date (D:M:Y) 16/ 9/05 at Time (H:M:S) 13: 8:38