Hi @DavyJones ,   @Ksharp 's explanation and your understanding are correct.  Based on the table you provided, the values in the "Autocorrelations" portion of the "Autocorrelation Check for White Noise" table contain the autocorrelation coefficients at the individual lags.  For example, the first row in the table contains the autocorrelation coefficients at lag1, lag2, ...lag6.  In other words, the fifth value in that row, 0.427, is the autocorrelation coefficient at lag 5.  The second row contains the autocorrelation coefficients for lags 7, 8, 9,...12.  The third value in the second row, 0.375, is the autocorrelation coefficient at lag 9.  If you add the OUTCOV= option to the IDENTIFY statement, you can create a data set with the autocorrelations in a format that you might prefer.  For example:   proc arima data=mydata plot(only)=(series(corr)) ; identify var=tsvar nlag=20 outcov=mycorr; run; proc print data=mycorr; run; Regarding your second question, the Chi-Square statistic that is printed on each row is the Ljung-Box statistic.  It is used to test the null hypothesis that the series is white noise (ie. no autocorrelation).  These test statistics are computed by using sets of autocorrelation coefficients, therefore, the first Chi-Square statistic and associated DF and Pr>ChiSq values are based on the set of the first 6 autocorrelations.  The Chi-Square statistic and its associated DF and p-value in the second row are based on the set of the first 12 autocorrelations, etc.   If you want to either obtain a test of significance on each individual autocorrelation coefficient or compute the Ljung-Box test statistic using autocorrelations up through lag "p", then you can use the TIMESERIES procedure.  Please see the following code for an example:   proc timeseries data=mydata outcorr=corr plots=corr; var tsvar; corr lag n acf acfprob wn wnprob /nlag=30; run; proc print data=corr; run; For additional details on the calculations performed by the CORR statement, please see the following documentation link:   https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_timeseries_details08.htm&docsetVersion=15.1&locale=en    I hope this helps! DW ... View more

Hi @somebody    PROCs ARIMA and TIMESERIES should return the same crosscorrelations when the same values of Y and X are passed to both procedures.  For example, the following code returns identical results between the two procedures--both when Y and X have all nonmissing values, and when Y and/or X have one or more embedded missing values in the DATA= data set passed to the procedures: proc arima data=test; identify var=y crosscorr=(x) nlag=10 outcov=arima_ccf (where=(crossvar='x')) noprint ; run; quit; proc timeseries data=test outcrosscorr=ts_ccf out=_null_; var y; crossvar x; crosscorr lag n ccov ccf/ nlag=10; run; (Note that the above code is slightly different than the code you are currently running for these procedures.)   If you observe differences in the results between PROC ARIMA and PROC TIMESERIES, then it could be due to the inclusion of the ID statement in your PROC TIMESERIES step.  If the DATA= data set you are using has gaps in it (such as omitted observations for holidays), then the ID statement in PROC TIMESERIES will first fill in any gaps in the data with missing values prior to computing the crosscorrelations.  The PROC ARIMA code does not do that.  You might want to try either omitting the ID statement in your PROC TIMESERIES step or creating an OUT= data set from your PROC TIMESERIES step and using that data set as input into your PROC ARIMA analysis to see if that resolves the differences you observed.  The following section of the PROC TIMESERIES documentation includes details on how the cross-covariances and crosscorrelations are computed in the TIMESERIES and ARIMA procedures:   https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_timeseries_details14.htm&docsetVersion=15.1&locale=en    The crosscorrelations computed by PROCs ARIMA and TIMESERIES differ from the "manual computation" of the correlations using PROC CORR. When you use PROC CORR to compute the correlation between an analysis variable and the lags of another variable, PROC CORR uses the mean of the analysis variable and the mean of each individual lag variable in the calculations.  On the other hand, PROCs ARIMA and TIMESERIES use the mean of the VAR variable and the mean of the CROSSVAR variable in the calculations--not the mean of each lagged variable.  For example, if you compute Corr(y_t,x_t-4), then PROC CORR uses the mean of Y and the mean of the Lag4(X) variable.  If you examine the crosscorrelation coefficient between Y and X at lag 4 computed by PROC TIMESERIES or PROC ARIMA, then its computation is based on the mean of Y and the mean of X.     The reason crosscovariances and crosscorrelations are computed by time series methods using the overall mean of each variable is because of the assumption of stationarity in both the mean and variance of each series.  This is also noted in the following Wikipedia page under the "Time series analysis" section:   https://en.wikipedia.org/wiki/Covariance_and_correlation    There is no option to change the behavior in either PROC CORR or PROCs TIMESERIES or ARIMA to resolve this discrepancy.  For more details on the computations used in PROC CORR, please see the following documentation link:   https://go.documentation.sas.com/?docsetId=procstat&docsetTarget=procstat_corr_details01.htm&docsetVersion=9.4&locale=en    I hope this helps! DW   ... View more

Hi @somebody    PROC ARIMA assumes the data are equally-spaced and sorted sequentially over time, therefore, a time ID variable is not required to compute the cross-correlations.  If you have gaps in the data, then these gaps should be filled in with missing values (or some other value of your choosing) prior to running the procedure in order to preserve the correct spacing between observations.     PROC ARIMA supports a BY statement so you can perform a separate analysis for each BY group.  Within each BY group, PROC ARIMA assumes the data is equally-spaced and sorted sequentially over time.   I hope this helps! DW ... View more

Hi @braam ,    By default, PROC EXPAND fits a cubic spline function through the nonmissing values of your analysis variable prior to performing any transformations specified in the TRANSFORM= option.  This can result in slight precision differences from your original data for some observations, as @Reeza noted.  If you do not need to do any missing value interpolation or frequency conversion in PROC EXPAND prior to computing your transformations, then you are correct to specify the METHOD=NONE option.  For more details on the METHOD=NONE option, please see the following documentation link:   https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_expand_details13.htm&docsetVersion=15.1&locale=en#etsug_expand000779      Another alternative is to use the ROUND function to round the results from PROC EXPAND prior to making your comparisons.  For example, you could add the statement:   lagvar=round(lagvar,1e-8);  in your TEMP3 data step prior to your IF statement.   I hope this helps! DW ... View more

Hi @ubshams ,    The default Yule-Walker method is a two-step estimator, where the AR coefficient(s) are estimated from the OLS residuals.  These coefficients are used to compute S , which is then used in a subsequent GLS estimation to obtain the final Yule-Walker parameter estimates.  In the output generated by PROC AUTOREG when the METHOD=YW option is specified, you should see a table called "Estimates of Autoregressive Parameters".  This table is displayed at the end of the OLS estimation results and before the "Yule-Walker Estimates" table of fit statistics.  The updated coefficients for the independent variables in your model are in the final "Parameter Estimates" table near the end of the output.   For more information on the computational details for the YW method, please see the following section of the documentation:   https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_autoreg_details02.htm&docsetVersion=15.1&locale=en#etsug_autoreg003659    I hope this helps! DW ... View more

Hi @JacAder,    PROC ARIMA can be used to fit a regression model with ARMA errors, as shown in the link below: https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_arima_gettingstarted25.htm&docsetVersion=15.1&locale=en#etsug_arima001029    However, if you want to fit the model:   "change in sale = f(price, income, lag1 of change in sale, lag1 error, lag2 error, error term)"   then you can use other SAS procedures, such as PROC UCM or PROC VARMAX.  For your desired model, PROC VARMAX provides the most straightforward approach.  Your model, residual plot, and saved output can be obtained with the following code.  Note:  Although PROC VARMAX supports an OUTEST= data set, the code below illustrates how to output the ParameterEstimates table via an ODS OUTPUT statement.  It saves the parameter estimates table in a different format than the OUTEST= data set.   proc varmax data=have plots=(model residual); ods output ParameterEstimates=arma_est; id date interval=month; model sales = price income / p=1 q=2 dify(1);* difx(1); output out=out lead=0; run; proc print data=arma_est; run; data all_varmax; merge have out; by date; run; proc print data=all_varmax; var date sales price income for1 res1; run; When the response variable is differenced, it is common to apply the same order of differencing to the input variables as well.  I included the DIFX option in the MODEL statement to illustrate how to difference the input variables, but commented it out, since that was not used in your model description.   For more details on the mathematical models fit by the ARIMA and VARMAX procedures, please see the following links: ARIMA:  https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_arima_gettingstarted13.htm&docsetVersion=15.1&locale=en    VARMAX:  https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_varmax_details02.htm&docsetVersion=15.1&locale=en   I hope this helps! DW ... View more

Hi @Golf ,    Please see the following sections of the PROC VARMAX documentation, which describe the tests supported when fitting a VECM:   General Tests and Restrictions on Parameters: https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_varmax_details49.htm&docsetVersion=15.1&locale=en#etsug_varmax017856    TEST Statement: https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_varmax_syntax46.htm&docsetVersion=15.1&locale=en    In the latter link, please see the paragraph beginning, "For the vector error correction models, you can test the hypothesis..."   I hope this helps! DW ... View more

Hi @sivakoya ,   Method 2 in your example is an ARIMA model with input variables.  To forecast a response series using an ARIMA model with inputs, PROC ARIMA needs values of each of the input series for the LEAD= forecast horizon.  You can either provide future values of the input variables in the DATA= data set (setting the response variable to missing for those future observations) or you can use PROC ARIMA to model and forecast the input variables in addition to the response series.   For more details, please see the Forecasting with Input Variables section of the PROC ARIMA documentation at the following link:  https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_arima_gettingstarted40.htm&docsetVersion=15.1&locale=en   I hope this helps! DW   ... View more