Dear all<br><br><br>I use gretl 1.7.4 to estimate a GARCH model by using default menu function (i.e., \Model\Time series\GARCH ) and by the command in console. But they generate different results. The details of iterations are as follows. Any idea?<br>
<br>==== results by menu=====<br><br>Automatic initialization of parameters<br><br>&nbsp;Regression coefficients:<br>&nbsp; theta[0] = 0.0511825<br>&nbsp; theta[1] = 1.08508<br><br>&nbsp;Variance parameters:<br>&nbsp; alpha[0] = 0.1<br>&nbsp;&nbsp; beta[0] = 0.9<br>
<br>Iteration 1: Log-likelihood = -168.929078042<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.11038&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.0851&nbsp;&nbsp;&nbsp;&nbsp; 0.10000&nbsp;&nbsp;&nbsp;&nbsp; 0.90000<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -26.738&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 17.074&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 481.64&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 14.243<br><br>Iteration 2: Log-likelihood = -139.605272823 (steplength = 0.0016)<br>
Parameters:&nbsp;&nbsp;&nbsp;&nbsp; 0.067597&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1124&nbsp;&nbsp;&nbsp;&nbsp; 0.87063&nbsp;&nbsp;&nbsp;&nbsp; 0.92279<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -4.4061&nbsp;&nbsp;&nbsp;&nbsp; 0.82275&nbsp;&nbsp;&nbsp;&nbsp; -15.175&nbsp;&nbsp;&nbsp;&nbsp; -8.1703<br><br>Iteration 3: Log-likelihood = -138.361721087 (steplength = 0.04)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.13607&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1652&nbsp;&nbsp;&nbsp;&nbsp; 0.87232&nbsp;&nbsp;&nbsp;&nbsp; 0.62335<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 14.823&nbsp;&nbsp;&nbsp; -0.51238&nbsp;&nbsp;&nbsp;&nbsp; -14.088&nbsp;&nbsp;&nbsp;&nbsp; -6.9177<br><br>Iteration 4: Log-likelihood = -137.371894269 (steplength = 0.008)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.12485&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1848&nbsp;&nbsp;&nbsp;&nbsp; 0.87847&nbsp;&nbsp;&nbsp;&nbsp; 0.47302<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 13.892&nbsp;&nbsp;&nbsp;&nbsp; -1.0676&nbsp;&nbsp;&nbsp;&nbsp; -13.148&nbsp;&nbsp;&nbsp;&nbsp; -5.2299<br>
<br>Iteration 5: Log-likelihood = -137.147777193 (steplength = 0.04)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.12168&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1207&nbsp;&nbsp;&nbsp;&nbsp; 0.88177&nbsp;&nbsp;&nbsp;&nbsp; 0.43192<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 13.651&nbsp;&nbsp;&nbsp;&nbsp; 0.40209&nbsp;&nbsp;&nbsp;&nbsp; -12.854&nbsp;&nbsp;&nbsp;&nbsp; -4.5731<br><br>Iteration 6: Log-likelihood = -136.065550003 (steplength = 0.2)<br>
Parameters:&nbsp;&nbsp;&nbsp; -0.055939&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1709&nbsp;&nbsp;&nbsp;&nbsp; 0.88165&nbsp;&nbsp;&nbsp;&nbsp; 0.29403<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 7.6020&nbsp;&nbsp;&nbsp;&nbsp; -1.0743&nbsp;&nbsp;&nbsp;&nbsp; -11.185&nbsp;&nbsp;&nbsp; -0.64410<br><br>Iteration 7: Log-likelihood = -135.589722350 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; 0.054295&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1305&nbsp;&nbsp;&nbsp;&nbsp; 0.86108&nbsp;&nbsp;&nbsp;&nbsp; 0.32088<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -3.2400&nbsp;&nbsp;&nbsp; -0.48930&nbsp;&nbsp;&nbsp;&nbsp; -11.296&nbsp;&nbsp;&nbsp;&nbsp; -1.6096<br><br>Iteration 8: Log-likelihood = -135.332923963 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; 0.044387&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1121&nbsp;&nbsp;&nbsp;&nbsp; 0.84171&nbsp;&nbsp;&nbsp;&nbsp; 0.26964<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -2.1320&nbsp;&nbsp; -0.080313&nbsp;&nbsp;&nbsp;&nbsp; -9.7947&nbsp;&nbsp;&nbsp;&nbsp; 0.99314<br>
<br>Iteration 9: Log-likelihood = -134.612248244 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; 0.014804&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.0933&nbsp;&nbsp;&nbsp;&nbsp; 0.76024&nbsp;&nbsp;&nbsp;&nbsp; 0.27015<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.83895&nbsp;&nbsp;&nbsp;&nbsp; 0.60996&nbsp;&nbsp;&nbsp;&nbsp; -7.3230&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3.3510<br><br>Iteration 10: Log-likelihood = -133.876720666 (steplength = 1)<br>
Parameters:&nbsp;&nbsp;&nbsp; -0.034438&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.0938&nbsp;&nbsp;&nbsp;&nbsp; 0.46226&nbsp;&nbsp;&nbsp;&nbsp; 0.40605<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6.1180&nbsp;&nbsp;&nbsp;&nbsp; 0.99752&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 10.528&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6.9157<br><br>Iteration 11: Log-likelihood = -133.215756120 (steplength = 0.008)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; 0.014505&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1018&nbsp;&nbsp;&nbsp;&nbsp; 0.54648&nbsp;&nbsp;&nbsp;&nbsp; 0.46137<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.55270&nbsp;&nbsp;&nbsp;&nbsp; 0.72059&nbsp;&nbsp;&nbsp;&nbsp; -1.2327&nbsp;&nbsp;&nbsp;&nbsp; 0.43556<br><br>Iteration 12: Log-likelihood = -133.185323523 (steplength = 0.04)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; 0.010238&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1316&nbsp;&nbsp;&nbsp;&nbsp; 0.52856&nbsp;&nbsp;&nbsp;&nbsp; 0.49641<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.25051&nbsp;&nbsp;&nbsp; -0.14061&nbsp;&nbsp;&nbsp; -0.71733&nbsp;&nbsp;&nbsp; -0.28480<br>
<br>Iteration 13: Log-likelihood = -133.185060907 (steplength = 0.008)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; 0.011028&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1308&nbsp;&nbsp;&nbsp;&nbsp; 0.52769&nbsp;&nbsp;&nbsp;&nbsp; 0.49709<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.36195&nbsp;&nbsp;&nbsp; -0.11592&nbsp;&nbsp;&nbsp; -0.65345&nbsp;&nbsp;&nbsp; -0.28715<br><br>Iteration 14: Log-likelihood = -133.185045046 (steplength = 0.04)<br>
Parameters:&nbsp;&nbsp;&nbsp; 0.0087095&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1249&nbsp;&nbsp;&nbsp;&nbsp; 0.52597&nbsp;&nbsp;&nbsp;&nbsp; 0.50187<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp; -0.053499&nbsp;&nbsp;&nbsp; 0.085396&nbsp;&nbsp;&nbsp; -0.65223&nbsp;&nbsp;&nbsp; -0.41383<br><br>Iteration 15: Log-likelihood = -133.181789118 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp; 0.0081963&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1273&nbsp;&nbsp;&nbsp;&nbsp; 0.52065&nbsp;&nbsp;&nbsp;&nbsp; 0.49465<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp; -0.016645&nbsp; -0.0072318&nbsp;&nbsp;&nbsp; 0.043542 -0.00045134<br><br>Iteration 16: Log-likelihood = -133.181774062 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp; 0.0081232&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1271&nbsp;&nbsp;&nbsp;&nbsp; 0.52128&nbsp;&nbsp;&nbsp;&nbsp; 0.49410<br>Gradients:&nbsp;&nbsp; -0.00074809 -0.00035241&nbsp; -0.0012373&nbsp; 0.00028761<br>
<br>Iteration 17: Log-likelihood = -133.181774041 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp; 0.0081164&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1271&nbsp;&nbsp;&nbsp;&nbsp; 0.52126&nbsp;&nbsp;&nbsp;&nbsp; 0.49413<br>Gradients:&nbsp;&nbsp; 6.7268e-006 4.3038e-006-1.6833e-005-1.9424e-005<br><br>Iteration 17: Log-likelihood = -133.181774041 (steplength = 1)<br>
Parameters:&nbsp;&nbsp;&nbsp; 0.0081164&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1271&nbsp;&nbsp;&nbsp;&nbsp; 0.52126&nbsp;&nbsp;&nbsp;&nbsp; 0.49413<br>Gradients:&nbsp;&nbsp; 6.7268e-006 4.3038e-006-1.6833e-005-1.9424e-005<br><br><br>--- FINAL VALUES: <br>Log-likelihood = -133.181774041 (steplength = 6.4e-005)<br>Parameters:&nbsp;&nbsp;&nbsp; 0.0081164&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.1271&nbsp;&nbsp;&nbsp;&nbsp; 0.52126&nbsp;&nbsp;&nbsp;&nbsp; 0.49413<br>
Gradients:&nbsp;&nbsp; 6.7268e-006 4.3038e-006-1.6833e-005-1.9424e-005<br><br><br>theta[0]:&nbsp;&nbsp;&nbsp;&nbsp; 0.00381941 (0.0406031)<br>theta[1]:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.530368 (0.0834569)<br>theta[2]:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.115430 (0.0264921)<br>theta[3]:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.494127 (0.192625)<br>
<br>Function evaluations: 47<br>Evaluations of gradient: 17<br><br>Model 11: GARCH estimates using the 99 observations 1980:02-1988:04<br>Dependent variable: Y<br>Standard errors based on Hessian<br><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; VARIABLE&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; COEFFICIENT&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; STDERROR&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; T STAT&nbsp;&nbsp; P-VALUE<br>
<br>&nbsp; const&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.00381941&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0406031&nbsp;&nbsp;&nbsp;&nbsp; 0.094&nbsp;&nbsp; 0.92506<br>&nbsp; Y_1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.530368&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0834569&nbsp;&nbsp;&nbsp;&nbsp; 6.355&nbsp; &lt;0.00001 ***<br><br>&nbsp; alpha(0)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.115430&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0264921&nbsp;&nbsp;&nbsp;&nbsp; 4.357&nbsp;&nbsp; 0.00001 ***<br>
&nbsp; alpha(1)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.494127&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.192625&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2.565&nbsp;&nbsp; 0.01031 **<br><br>&nbsp; Mean of dependent variable = 0.0519415<br>&nbsp; Standard deviation of dep. var. = 0.544934<br>&nbsp; Unconditional error variance = 0.22818<br>&nbsp; Log-likelihood = -58.5562<br>
&nbsp; Akaike information criterion (AIC) = 127.112<br>&nbsp; Schwarz Bayesian criterion (BIC) = 140.088<br>&nbsp; Hannan-Quinn criterion (HQC) = 132.362<br><br><br>==== results by the command in console=====<br>? garch 0 1; Y Y(-1) --verbose<br>
<br>Automatic initialization of parameters<br><br>&nbsp;Regression coefficients:<br>&nbsp; theta[0] = 0.0511825<br>&nbsp; theta[1] = 1.08508<br><br>&nbsp;Variance parameters:<br>&nbsp; alpha[0] = 0.1<br>&nbsp;&nbsp; beta[0] = 0.9<br><br>Iteration 1: Log-likelihood = -270.911199903<br>
Parameters:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.11038&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.0851&nbsp;&nbsp;&nbsp;&nbsp; 0.10000&nbsp;&nbsp;&nbsp;&nbsp; 0.90000<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -125.16&nbsp;&nbsp;&nbsp;&nbsp; -125.16&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1010.7&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 30.388<br><br>Iteration 2: Log-likelihood = -170.537739660 (steplength = 0.0016)<br>Parameters:&nbsp;&nbsp;&nbsp; -0.089875&nbsp;&nbsp;&nbsp;&nbsp; 0.88483&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.7171&nbsp;&nbsp;&nbsp;&nbsp; 0.94862<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -25.749&nbsp;&nbsp;&nbsp;&nbsp; -25.749&nbsp;&nbsp;&nbsp;&nbsp; -8.7587&nbsp;&nbsp;&nbsp;&nbsp; -8.4495<br><br>Iteration 3: Log-likelihood = -161.761522666 (steplength = 0.008)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.29926&nbsp;&nbsp;&nbsp;&nbsp; 0.67545&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.6788&nbsp;&nbsp;&nbsp;&nbsp; 0.88214<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -11.183&nbsp;&nbsp;&nbsp;&nbsp; -11.183&nbsp;&nbsp;&nbsp;&nbsp; -10.840&nbsp;&nbsp;&nbsp;&nbsp; -8.8715<br>
<br>Iteration 4: Log-likelihood = -158.391619816 (steplength = 0.04)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.31167&nbsp;&nbsp;&nbsp;&nbsp; 0.66303&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.6896&nbsp;&nbsp;&nbsp;&nbsp; 0.51071<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -10.407&nbsp;&nbsp;&nbsp;&nbsp; -10.407&nbsp;&nbsp;&nbsp;&nbsp; -10.879&nbsp;&nbsp;&nbsp;&nbsp; -8.1069<br><br>Iteration 5: Log-likelihood = -157.154680098 (steplength = 0.04)<br>
Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.31888&nbsp;&nbsp;&nbsp;&nbsp; 0.65583&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.6928&nbsp;&nbsp;&nbsp;&nbsp; 0.36062<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -10.000&nbsp;&nbsp;&nbsp;&nbsp; -10.000&nbsp;&nbsp;&nbsp;&nbsp; -10.621&nbsp;&nbsp;&nbsp;&nbsp; -6.6638<br><br>Iteration 6: Log-likelihood = -155.742679661 (steplength = 0.2)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.35479&nbsp;&nbsp;&nbsp;&nbsp; 0.61992&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.6858&nbsp;&nbsp;&nbsp;&nbsp; 0.22820<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -7.1755&nbsp;&nbsp;&nbsp;&nbsp; -7.1755&nbsp;&nbsp;&nbsp;&nbsp; -10.191&nbsp;&nbsp;&nbsp;&nbsp; -3.8497<br><br>Iteration 7: Log-likelihood = -154.844791246 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.48452&nbsp;&nbsp;&nbsp;&nbsp; 0.49018&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.6370&nbsp;&nbsp;&nbsp;&nbsp; 0.10641<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5.3640&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5.3640&nbsp;&nbsp;&nbsp;&nbsp; -8.7104&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3.1207<br>
<br>Iteration 8: Log-likelihood = -154.573226547 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45621&nbsp;&nbsp;&nbsp;&nbsp; 0.51850&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.6312&nbsp;&nbsp;&nbsp;&nbsp; 0.19907<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2.1978&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2.1978&nbsp;&nbsp;&nbsp;&nbsp; -10.062&nbsp;&nbsp;&nbsp;&nbsp; -2.3733<br><br>Iteration 9: Log-likelihood = -154.393792855 (steplength = 1)<br>
Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.41196&nbsp;&nbsp;&nbsp;&nbsp; 0.56275&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.6201&nbsp;&nbsp;&nbsp;&nbsp; 0.14415<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -2.0434&nbsp;&nbsp;&nbsp;&nbsp; -2.0434&nbsp;&nbsp;&nbsp;&nbsp; -9.3522&nbsp;&nbsp;&nbsp;&nbsp; 0.38397<br><br>Iteration 10: Log-likelihood = -153.968693027 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.42290&nbsp;&nbsp;&nbsp;&nbsp; 0.55181&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.5786&nbsp;&nbsp;&nbsp;&nbsp; 0.15683<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.97944&nbsp;&nbsp;&nbsp; -0.97944&nbsp;&nbsp;&nbsp;&nbsp; -9.3680&nbsp;&nbsp;&nbsp;&nbsp; 0.21026<br><br>Iteration 11: Log-likelihood = -150.992406742 (steplength = 0.04)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.46207&nbsp;&nbsp;&nbsp;&nbsp; 0.51263&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.2039&nbsp;&nbsp;&nbsp;&nbsp; 0.16524<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3.4088&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3.4088&nbsp;&nbsp;&nbsp;&nbsp; -6.1548&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 7.6249<br>
<br>Iteration 12: Log-likelihood = -148.359103161 (steplength = 0.008)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.43062&nbsp;&nbsp;&nbsp;&nbsp; 0.54409&nbsp;&nbsp;&nbsp;&nbsp; 0.75562&nbsp;&nbsp;&nbsp;&nbsp; 0.32000<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -2.3822&nbsp;&nbsp;&nbsp;&nbsp; -2.3822&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.9263&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6.8635<br><br>Iteration 13: Log-likelihood = -148.097032678 (steplength = 0.008)<br>
Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45911&nbsp;&nbsp;&nbsp;&nbsp; 0.51560&nbsp;&nbsp;&nbsp;&nbsp; 0.71943&nbsp;&nbsp;&nbsp;&nbsp; 0.36973<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.75946&nbsp;&nbsp;&nbsp;&nbsp; 0.75946&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1.8664&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4.7097<br><br>Iteration 14: Log-likelihood = -147.949161098 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45122&nbsp;&nbsp;&nbsp;&nbsp; 0.52348&nbsp;&nbsp;&nbsp;&nbsp; 0.75142&nbsp;&nbsp;&nbsp;&nbsp; 0.44913<br>
Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.66911&nbsp;&nbsp;&nbsp; -0.66911&nbsp;&nbsp;&nbsp;&nbsp; -1.6394&nbsp;&nbsp;&nbsp; -0.55287<br><br>Iteration 15: Log-likelihood = -147.921370789 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45617&nbsp;&nbsp;&nbsp;&nbsp; 0.51854&nbsp;&nbsp;&nbsp;&nbsp; 0.69072&nbsp;&nbsp;&nbsp;&nbsp; 0.47194<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.69320&nbsp;&nbsp;&nbsp; -0.69320&nbsp;&nbsp;&nbsp;&nbsp; 0.78234&nbsp;&nbsp;&nbsp; -0.15366<br>
<br>Iteration 16: Log-likelihood = -147.911096499 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.46083&nbsp;&nbsp;&nbsp;&nbsp; 0.51388&nbsp;&nbsp;&nbsp;&nbsp; 0.71098&nbsp;&nbsp;&nbsp;&nbsp; 0.45901<br>Gradients:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.15242&nbsp;&nbsp;&nbsp;&nbsp; 0.15242&nbsp;&nbsp; -0.086460&nbsp;&nbsp;&nbsp; 0.020146<br><br>Iteration 17: Log-likelihood = -147.910849840 (steplength = 1)<br>
Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45975&nbsp;&nbsp;&nbsp;&nbsp; 0.51495&nbsp;&nbsp;&nbsp;&nbsp; 0.70935&nbsp;&nbsp;&nbsp;&nbsp; 0.46012<br>Gradients:&nbsp;&nbsp;&nbsp; -0.0033672&nbsp; -0.0033672&nbsp;&nbsp; -0.010763&nbsp; -0.0083438<br><br>Iteration 18: Log-likelihood = -147.910848410 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45979&nbsp;&nbsp;&nbsp;&nbsp; 0.51491&nbsp;&nbsp;&nbsp;&nbsp; 0.70914&nbsp;&nbsp;&nbsp;&nbsp; 0.46007<br>
Gradients:&nbsp;&nbsp;&nbsp; 0.00068169&nbsp; 0.00068169&nbsp; 0.00046165 -0.00074704<br><br>Iteration 19: Log-likelihood = -147.910848385 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45979&nbsp;&nbsp;&nbsp;&nbsp; 0.51492&nbsp;&nbsp;&nbsp;&nbsp; 0.70917&nbsp;&nbsp;&nbsp;&nbsp; 0.46004<br>Gradients:&nbsp;&nbsp; 8.9567e-005 8.9567e-005 8.2642e-006 -0.00017120<br>
<br>Iteration 20: Log-likelihood = -147.910848385 (steplength = 0.008)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45979&nbsp;&nbsp;&nbsp;&nbsp; 0.51492&nbsp;&nbsp;&nbsp;&nbsp; 0.70917&nbsp;&nbsp;&nbsp;&nbsp; 0.46004<br>Gradients:&nbsp;&nbsp; 1.1075e-005 1.1075e-005 4.6432e-005 -0.00012421<br><br>Iteration 21: Log-likelihood = -147.910848385 (steplength = 0.008)<br>
Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45979&nbsp;&nbsp;&nbsp;&nbsp; 0.51492&nbsp;&nbsp;&nbsp;&nbsp; 0.70917&nbsp;&nbsp;&nbsp;&nbsp; 0.46004<br>Gradients:&nbsp;&nbsp; 5.2761e-005 5.2761e-005 3.4521e-005-8.0059e-005<br><br>Iteration 22: Log-likelihood = -147.910848385 (steplength = 0.008)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45979&nbsp;&nbsp;&nbsp;&nbsp; 0.51492&nbsp;&nbsp;&nbsp;&nbsp; 0.70917&nbsp;&nbsp;&nbsp;&nbsp; 0.46004<br>
Gradients:&nbsp;&nbsp; 4.1306e-005 4.1306e-005 5.3314e-006-9.2102e-005<br><br>Iteration 23: Log-likelihood = -147.910848385 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45978&nbsp;&nbsp;&nbsp;&nbsp; 0.51492&nbsp;&nbsp;&nbsp;&nbsp; 0.70917&nbsp;&nbsp;&nbsp;&nbsp; 0.46003<br>Gradients:&nbsp; -2.4463e-006-2.4463e-006 2.1778e-006 7.0214e-006<br>
<br>Iteration 23: Log-likelihood = -147.910848385 (steplength = 1)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45978&nbsp;&nbsp;&nbsp;&nbsp; 0.51492&nbsp;&nbsp;&nbsp;&nbsp; 0.70917&nbsp;&nbsp;&nbsp;&nbsp; 0.46003<br>Gradients:&nbsp; -2.4463e-006-2.4463e-006 2.1778e-006 7.0214e-006<br><br><br>--- FINAL VALUES: <br>
Log-likelihood = -147.910848385 (steplength = 0.0016)<br>Parameters:&nbsp;&nbsp;&nbsp;&nbsp; -0.45978&nbsp;&nbsp;&nbsp;&nbsp; 0.51492&nbsp;&nbsp;&nbsp;&nbsp; 0.70917&nbsp;&nbsp;&nbsp;&nbsp; 0.46003<br>Gradients:&nbsp; -2.4463e-006-2.4463e-006 2.1778e-006 7.0214e-006<br><br><br>theta[0]:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.216364 (0.171258)<br>
theta[1]:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.242311 (0.0635179)<br>theta[2]:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.157041 (0.0421745)<br>theta[3]:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.460033 (0.177473)<br><br>Function evaluations: 58<br>Evaluations of gradient: 23<br><br>Model 10: GARCH estimates using the 99 observations 1980:02-1988:04<br>
Dependent variable: Y<br>Standard errors based on Hessian<br><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; VARIABLE&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; COEFFICIENT&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; STDERROR&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; T STAT&nbsp;&nbsp; P-VALUE<br><br>&nbsp; const&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -0.216364&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.171258&nbsp;&nbsp;&nbsp;&nbsp; -1.263&nbsp;&nbsp; 0.20645<br>&nbsp; Y_1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.242311&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0635179&nbsp;&nbsp;&nbsp;&nbsp; 3.815&nbsp;&nbsp; 0.00014 ***<br>
<br>&nbsp; alpha(0)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.157041&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.0421745&nbsp;&nbsp;&nbsp;&nbsp; 3.724&nbsp;&nbsp; 0.00020 ***<br>&nbsp; alpha(1)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.460033&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.177473&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2.592&nbsp;&nbsp; 0.00954 ***<br><br>&nbsp; Mean of dependent variable = 0.0519415<br>&nbsp; Standard deviation of dep. var. = 0.544934<br>
&nbsp; Unconditional error variance = 0.290835<br>&nbsp; Log-likelihood = -73.2853<br>&nbsp; Akaike information criterion (AIC) = 156.571<br>&nbsp; Schwarz Bayesian criterion (BIC) = 169.546<br>&nbsp; Hannan-Quinn criterion (HQC) = 161.82<br><br>
<br>