Hello @sasalex2024  If you wanted to fit a ARMA(1,2)  model, then your PROC ARIMA code is correct. However, PROC AUTOREG does not support MA terms, and you cannot specify ARMA models in PROC AUTOREG. Your PROC AUTOREG is specifying a regression of consump on its own lag, and the error term in this regression follows AR(2) process. Please see the equations for regressions with AR errors specified in PROC AUTOREG:   https://go.documentation.sas.com/doc/en/pgmsascdc/v_055/etsug/etsug_autoreg_gettingstarted01.htm     I hope this helps.     ... View more

Hello @kky6196  Your PROC AUTOREG code is fitting a GARCH(1,1) model for the  daily_return variable for each country, and saves the conditional error variance from the GARCH(1,1) model into the output data set. The subsequent data step is transforming the conditional error variance to the standard deviation by taking the square root. If this is what you intended to do, then it looks good.   I hope this helps.     ... View more

Hello @sasalex2024    I am not aware of an absolute answer to the question of which estimation method is preferred always, but you may find the following information and/or references helpful to you to decide which method you want to use:   (1). In PROC ARIMA documentation you are referencing, the following statement follows:   SAS Help Center: Estimation Details   ...Maximum likelihood estimates are more expensive to compute than the conditional least squares estimates; however, they may be preferable in some cases (Ansley and Newbold 1980; Davidson 1981).   You may want to check the two references mentioned above to see the cases discussed where ML is preferable.      (2). SAS for Forecasting Time Series, third edition, by Brocklebank, Dickey, and Choi: "Although CLS, ULS, and ML should give similar results for reasonably large data sets, studies comparing the three methods indicate that ML is the most accurate."   (3). Forecasting with Univariate Box-Jenkins Models, by Pankratz: "Box and Jenkins (ref.) favor estimates chosen according to the maximum likelihood criterion. Mathematical statisticians frequently prefer the ML approach to estimation problems because the resulting estimates often have attractive statistical properties. ....." "However, finding exact ML estimates of ARIMA models can be cumbersome and may require relatively large amounts of computer time. For this reason, Box and Jenkins suggest using the least-squares(LS) criterion. ....."       I hope this helps.   Wen   ... View more

Hello @sasalex2024  I assume the example you are looking at is from this section of the PROC ARIMA documentation: SAS Help Center: Input Variables and Regression with ARMA Errors   which illustrates the steps usually taken to identify a regression model with ARMA errors. You usually first run the regression with inputs but without ARMA specification: proc arima data=a;    identify var=sales crosscorr=(price income) noprint;    estimate input=(price income) plot; run;   You obtain the ACF, PACF, IACF plots for the residuals in this regression model, i.e., y - mu - beta*input, which help you identify proper ARMA specifications. If you decide an ARMA(1,1) process, then you can specify the regression with ARMA error by:   estimate p=1 q=1 input=(price income); run;   The ACF, PACF, IACF plots for the residuals in this second step with ARMA specifications are the residuals obtained in the model with ARMA error already taken into account, i.e., residual = actual - forecast, where the ARMA parameters played a role in computing the forecast, so the ACF, PACF, IACF plots in this step will give you indication of whether the ARMA process are properly specified. In this second step, ACF, PACF, and IACF plots for y-mu - beta*input are usually not of interest, since you already get the ACF plots of the residuals in the simple regression model in step one and know that the residuals from the simple regression model show some ARMA patterns and is not white noise.    If by saying "If we ignore the entire noise part (the ARMA(1,1) process), and want to compute Nt as "sales - constant - coeff1*price - coeff2*income", so that to build a SACF of the generated Nt series, can you please tell how this can be done? Or, perhaps there is an easier way to do it? I've read about the "plot" option, but that one produces ACF for the residuals obtained before the full regression with ARMA disturbance is run.",  you mean that although you have already identified an ARMA(1,1) process in the first step regression residual ACF plots, and have estimated the model with ARMA specification in the second step, but you still wanted to get ACF's of y - mu - beta*x only, but using the estimates of mu and beta's in the second step estimation(the model with ARMA specification), rather than ACF's of y - mu - beta*x using parameters in the first step regression model, if that is indeed what you intended to obtain, then you may do that by using OUTEST = option in the ESTIMATE statement to save the parameters estimates for mu and beta obtained in the second step ARMA model estimation, then run a subsequent PROC ARIMA step of a simple regression model(without ARMA specification) but by specifying initial values for mu and beta's to be equal to the saved parameter estimates from the second step(ARMA model), with INITVAL=(initializer-spec variable …) option and MU = option, together with NOEST option in the ESTIMATE statement   SAS Help Center: ESTIMATE Statement   so that the final estimates for both mu and beta's are fixed at the initial values supplied.   For example, suppose in your second step ARMA model estimation, estimated mu = 1, parameter estimate for price = 2, parameter estimate for income = 3, then you can specify a subsequent PROC ARIMA step to specify a simple regression model only, and fix the mu and price, income variable parameters as:     proc arima data=a; identify var=sales crosscorr=(price income); estimate input=(price income) mu = 1 initval = (2 price 3 income) noest ; run;   then the ACF, PACF, IACF's produced in the above step are obtained for residual = y - mu - beta*input, while the mu and beta's are fixed at the parameters values obtained from the previous step regression with ARMA error specification.   I hope this helps. If I misunderstood what you intended to get, then please clarify what you want and I will further look into it.   Thanks,   Wen ... View more

Hello @santisouknee  You may take a look at the following thread discussing similar issue in generating future forecast in PROC ARIMA with input variables:   https://communities.sas.com/t5/SAS-Forecasting-and-Econometrics/PROC-ARIMA-Forecast-quot-The-value-for-option-LEAD-has-been/m-p/844239   Please let me know if you have a different situation than discussed there, and if so, I will take a further look.     ... View more

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