Hello @Gigiwen    In your version 1 PROC MODEL code, you specified a pure GJR-GARCH model without autoregressive errors; in your version 2 PROC AUTOREG code, you specified a GJR-GARCH model with autoregressive errors(with the NLAG = option in your code). If you intended to specify a pure GJR-GARCH model without autoregressive errors, then you may want to remove the NLAG = option in your version 2 PROC AUTOREG code.   Once you have removed the NLAG = option in your version 2 PROC AUTOREG code, then both versions specify a GJR-GARCH model without AR errors, but they are equivalent with some re-parameterization as discussed below:   In your version 1 PROC MODEL code, it specifies that:   if zlag(resid.y) > 0 then h.y = arch0 + arch1*xlag(resid.y**2,mse.y) + garch1*xlag(h.y,mse.y) ; else h.y = arch0 + arch1*xlag(resid.y**2,mse.y) + garch1*xlag(h.y,mse.y) + phi*xlag(resid.y**2,mse.y) ;   *Note: the resid.y is an internal variable in PROC MODEL defined as: resid.y = pred.y - actual.y   SAS Help Center: Equation Translations   this means that resid.y in PROC MODEL is the negative of the residual usually defined in the literature(which is actual - predicted).  So the variance equation in the version 1 PROC MODEL code is defined as    if epsilon_t-1  < 0 , then  h_t = arch0 + arch1*(epsilon_t-1)**2 + garch1*h_t-1 ;   else if epsilon_t-1 >=0 then h_t = arch0 + arch1*(epsilon_t-1)**2 + garch1*h_t-1 + phi*(epsilon_t-1)**2      While in version 2 PROC AUTOREG code(after removing NLAG = 1 option from your code), the GJR-GARCH model specifies the variance equation as discussed in the following section:   SAS Help Center: GARCH Models   h_t = omega + (alpha1 + I_{epsilon_t-1 < 0}*psi1)*(epsilon_t-1)**2 + gamma1*h_t-1    where I_{epsilon_t-1<0} is the indicator that takes the value 1 when epsilon_t-1 <0, and takes the value of 0 when epsilon_t-1 > = 0. The above variance equation is thus translated as:   if epsilon_t-1 < 0, then h_t = omega + (alpha1 + psi1)*(epsilon_t-1)**2 + gamm1*h_t-1 ;   else if epsilon_t-1 >=0 then  h_t = omega + alpha1*(epsilon_t-1)**2 + gamma1*h_t-1 ;   Corresponding to the parameter names in PROC AUTOREG parameter estimates table,   omega = TARCHA0 alpha1 = TARCHA1 psi1     = TARCHB1  gamma1 = TGARCH1   Comparing the variance equations in the two versions, they are equivalent with the following reparameterization:   Version 1  PROC MODEL                                        Version 2 PROC AUTOREG(after removing NLAG = option) arch0                                                                             TARCHA0 arch1                                                                              TARCHA1 + TARCHB1 arch1 + phi                                                                    TARCHA1 garch1                                                                            TGARCH1   From row 2 and row 3 above, you can see that phi and TARCHB1 is negative of each other:   phi = -TARCHB1   ****************************************************** Or alternatively, you may simply modify your version 1 PROC MODEL code by reversing the variance equations between the cases zlag(resid.y) >0 and  zlag(resid.y)<=0, i.e., specify the following variance equations instead:   /* variance model */ if zlag(resid.y) <= 0 then h.y = arch0 + arch1*xlag(resid.y**2,mse.y) + garch1*xlag(h.y,mse.y) ; else h.y = arch0 + arch1*xlag(resid.y**2,mse.y) + garch1*xlag(h.y,mse.y) + phi*xlag(resid.y**2,mse.y) ;   then the two versions will have one on one matching on all the parameters.   I hope this helps.   Wen   ... View more

Hello @j_elisio ,   For Yule Walker estimation, the AR parameters are estimated from sample autocorrelation functions of first step OLS residuals using Yule Walker equations. They are not estimated in the final step using GLS. The GLS step only obtain regression estimates 'beta'. So the AR parameters are only printed out in the first step of OLS results, but not in the final step parameter estimates table. The following Details section discusses Yule Walker estimation method:   https://go.documentation.sas.com/doc/en/pgmsascdc/v_045/etsug/etsug_autoreg_details02.htm#etsug_autoreg003664   The following section, specifically in item #9, discusses output layout for different estimation methods , including METHOD = YW,    https://go.documentation.sas.com/doc/en/pgmsascdc/v_045/etsug/etsug_autoreg_details57.htm   9. If the NLAG= option is specified with METHOD=ULS or METHOD=ML, the regression parameter estimates are printed again, assuming that the autoregressive parameter estimates are known. In this case, the Standard Error and related statistics for the regression estimates will, in general, be different from the case when they are estimated. Note that from a standpoint of estimation, Yule-Walker and iterated Yule-Walker methods (NLAG= with METHOD=YW, ITYW) generate only one table, assuming AR parameters are given.   I hope this helps.  ... View more

Hello @ZMX ,   You can obtain unconditional expected value and conditional expected values using the EXPECTED and CONDITOINAL options in the OUTPUT statement directly in PROC QLIM.   For the sample selection model, the PREDICTED value is equal to the EXPECTED value, which is also equal to the XBETA value for the continuous dependent variable in the main equation(lwage in the example). You can verify this by specifying the OUTPUT statement like below:    output out = outdata expected xbeta predicted conditional mills ;   and look at the outdata data set.   The conditional expected value differs from the  expected/predicted/xbeta values by adding to it another term,  rho*sigma_lwage*Mills_inlf, in other words,   Predicted_lwage = Expected_lwage = xbeta_lwage  ; Conditional_lwage = xbeta_lwage + rho*sigma_lwage*Mills_inlf ;   I hope this helps. ... View more

hello @srikanthyadav44 ,   The EGARCH equation used in PROC AUTOREG documentation follows that of the original Nelson(1991) paper 'Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica 59:347-70', and Hamilton(1994) 'Time Series Analysis' text book, page 668~669. Specifically, the g(z_t) equation in PROC AUTOREG documentation is the same as in equation (2.2) in Nelson(1991) paper on page 351, (with slight different parameter names in Hamilton(1994) equation). The equation for log(h_t) in PROC AUTOREG documentation is the same as equation (21.2.7) in Hamilton(again with slight different parameter names). For this specification, the parameter theta determines the asymmetric feature of the effect. When theta = 0, positive innovation(z_{t-1} > 0) has the same effect on volatility as a negative innovation(z_{t-1}<0) of the same magnitude. When theta not equal to zero, the effect is different for positive and negative innovation(z_{t-1} term). You may check Hamilton(1994) or Nelson(1991) for more detailed discussions.   I am not familiar with Eviews notation conventions, however, if the following equation you quoted: LOG(GARCH) = C(2) + C(3)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(4)*RESID(-1)/@SQRT(GARCH(-1)) + C(5)*LOG(GARCH(-1))   means the following equation for the volatility:   log(h_t) = C(2) + C(3)*abs(residual_{t-1})/sqrt(h_{t-1})  + C(4)*(residual_{t-1}/sqrt(h_{t-1})) + C(5)*log(h_{t-1})                  (1)   then it seems that this equation (1) is a slight reparameterization of the equation used in PROC AUTOREG documentation and the Nelson(1991) and Hamilton(1994) discussed above, noting that z_t = residual_t/sqrt(h_t). For comparison purpose, I am writing the equation for log(h_t) in PROC AUTOREG documentation (using the same parameter names as the PROC AUTOREG output) below:   log(h_t) = EARCH0 + EARCH1*g(z_{t-1}  + EGARCH1*log(h_{t-1})             (2)   note that g(z_{t-1}) = theta*z_{t-1} + gamma*(abs(z_{t-1}) - E(abs(z_{t-1}))), where gamma = 1, E(abs(z_{t-1})) = sqrt(2/pi) for z_t~N(0,1), so the above equation (2) can be written as:     log(h_t) = EARCH0 + EARCH1*(theta*z_{t-1} + abs(z_{t-1}) - sqrt(2/pi)) + EGARCH1*log(h_{t-1})             rearrange terms you can rewrite it as below in order to make it easier to compare to your Eviews equation (1) above:   log(h_t) = EARCH0 - EARCH1*sqrt(2/pi)                 + EARCH1*theta*z_{t-1}                  + EARCH1*abs(z_{t-1})                         + EGARCH1*log(h_{t-1}                                                          (3)   Comparing equation (3) with equation (1) above, you can see the relationship between the two different parameterizations:   C(2) = EARCH0 - EARCH1*sqrt(2/pi)  C(3) = EARCH1  C(4) = EARCH1*theta C(5) = EGARCH1     You may want to check your Eviews documentation to verify the above relations.      ... View more

Hello @LynnChen    At this time, there is no procedure that provides canned routine for Hansen(2000)'s threshold regression model directly. We may consider this functionality in the future, but at this time, you may consider following options:   For two regimes threshold regression like the following:   you may perform the estimation by looping over all possible gammas(or a grid of gammas) as discussed below: Get distinct w_t and sort them. assume they are {w_(i), i=1,…,T}, where w_(i) is the i-th smallest w. (that’s, all w_t are different, and T is the sample size). OLS regression y on (x, z), and record SSR as SSR_0. For i = 1 to T, OLS regress y on (x, u1, u2), where u1_t=z_t and u2_t=0 if w_t<=w_(i); otherwise, u1_t=0 and u2_t=z_t, and record SSR as SSR_i. Find the smallest SSR among SSR_i, i=0, …, T and the corresponding i; that’s, SSR_iStar<=SSR_i for any i. If i=0, no regime is needed; otherwise, gamma_hat=w_(iStar)   If T is big, take a grid of w_i; e.g., i might be in the range of 10%*T and 90%*T.   For multiple regimes, the combination of gammas should be looped over---but in that case, maybe the new procedure PROC BART is a better choice,   https://go.documentation.sas.com/doc/en/pgmsascdc/v_027/casstat/casstat_bart_details02.htm   Another alternative is similar to the following PROC MODEL switching regression example:    SAS Help Center: Switching Regression Example   which may provide an easier way to estimate this model, if you can accept that the solution is achieved by approximating the step function as the continuous function ProbNorm(.)   --Note that this is a nonlinear optimization problem and the initial values or optimization method might matter.   Following is an example that estimates the threshold model using the PROC MODEL approach:   title1 'Threshold Regression Example'; data switch; call streaminit('pcg', 12345); do t = 1 to 1000; e = rand('normal'); x = rand('normal'); z = rand('normal'); w = rand('uniform'); if(w<0.345) then y = 1 + 1.5*x + 2 * z + 0.5*e; else y = 1 + 1.5*x - 2 * z + 0.5*e; mylogL = (1/2)*( (log(2*3.1415)) + log( 0.5**2 ) + ((0.5*e)*( 1/0.5**2 )*(0.5*e)) ) ; output; end; run; proc means data=switch; var mylogL; run; proc model data=switch; parms sig1=1 int1 b1 b21 b22 gamma=0.5; bounds 0.0001 < sig1; a = (w - gamma)*100; /* Upper bound of integral */ d = probnorm(a); /* Normal CDF as an approx of switch */ /* Regime 1 */ y1 = int1 + x * b1 + z * b21 ; /* Regime 2 */ y2 = int1 + x * b1 + z * b22 ; /* Composite regression equation */ y = (1 - d)*y1 + d*y2; /* Resulting log-likelihood function */ logL = (1/2)*( (log(2*3.1415)) + log( sig1**2 ) + (resid.y*( 1/sig1**2 )*resid.y) ) ; errormodel y ~ general(logL); fit y / method=marquardt converge=1.0e-6; run; quit;   I hope this helps!     ... View more

Because you specified method = ML, but the backward elimination is done only once on the Yule Walker estimates after the OLS step, the significance level for the AR parameters in the Yule Walker stage can be different than when the AR parameters are estimated during the ML estimation. The following section of documentation discusses how the backward elimination is done:   https://go.documentation.sas.com/doc/en/pgmsascdc/v_030/etsug/etsug_autoreg_syntax06.htm#etsug_autoreg003127   For method = ML, the significance level of the selected AR parameters should be consistent with the SLSTAY = level in the 'Backward Elimination of Autoregressive Terms' output table, but may not the case in the Maximum Likelihood Estimates 'Parameter Estimates' output.     I hope this helps.  ... View more

Hello @amanjot_42    In PROC MODEL, you can first assign the h.y variable to a temporary variable, say, h_y = h.y, then use the OUTVARS statement with the OUT = dataset option in the FIT statement to output the values of the h.y into the output data set. For example:   /* Estimate GJR-GARCH Model */ proc model data = gjrgarch ; parms arch0 .1 arch1 .2 garch1 .75 phi .1; /* mean model */ y = intercept ; /* variance model */ if zlag(resid.y) > 0 then h.y = arch0 + arch1*xlag(resid.y**2,mse.y) + garch1*xlag(h.y,mse.y) ; else h.y = arch0 + arch1*xlag(resid.y**2,mse.y) + garch1*xlag(h.y,mse.y) + phi*xlag(resid.y**2,mse.y) ; h_y = h.y ; /* fit the model */ fit y / method = marquardt fiml out = outdata; outvars h_y ; run ; quit ; proc print data = outdata; run;   Please note that the GJR-GARCH model can also be specified directly in PROC AUTOREG with option   TYPE = THRES | THRESHOLD | TGARCH | GJR | GJRGARCH   in the MODEL statement. And you can use CEV = option in the OUTPUT statement in PROC AUTOREG to output the conditional error variance in the output data set, for example:   proc autoreg data =gjrgarch ; model y = /garch =(p=1,q=1, type = gjr); output out = out_auto cev = cev ; run;   proc print data = out_auto; run;   Please also note that the GJR-GARCH model implemented in PROC AUTOREG has slightly different parameterizations on the indicator function as that specified in the above PROC MODEL step, see the equation for h_t in PROC AUTOREG documentation:   https://go.documentation.sas.com/doc/en/pgmsascdc/v_030/etsug/etsug_autoreg_details12.htm#etsug_autoreg005090   where the indicator is equal to 1 when epsilon_t < 0, and equal to 0 otherwise. In the PROC MODEL code above, the indicator is for epsilon_t > 0 instead.  They are equivalent model with this slightly different parameterizations on the indicator function, and you can convert from one to the other as you wish.   I hope this helps. ... View more