| Library: | times |
| See also: | ar1res armacls armalik arimacls arima11 arimaf |
| Quantlet: | ariols | |
| Description: | Estimation of the ARI(p,d) process by OLS and diagnostics. Residuals diagnostics include their timeplot with two-standard error bounds, correlograms and Ljung-Box statistics with p-values. Parameter diagnostics include their t-statistics with p-values and variance-covariance matrix. Computed model selection criteria are AIC and BIC. |
| Usage: | A = ariols(y,p,d,const{,rcheck{,pcheck{,msc}}}) | |
| Input: | ||
| y | input series length n | |
| p | integer, order of autorregresive process | |
| d | integer, order of differences | |
| const | string, a constant is calculated if const = "constant" | |
| rcheck | optional string, residual diagnostic checking, if rcheck = "rcheck", residual diagnostics are calculated and plots of the residuals and the corresponding correlograms are displayed, if rcheck = "prcheck", the plots are omitted | |
| pcheck | optional string, parameter diagnostic statistics are computed if pcheck = "pcheck" | |
| msc | optional string, if msc = "msc" model selection criteria are computed | |
| Output: | ||
| A.b | estimated coefficients. If const = "constant", the first element represents the constant term. | |
| A.bst | asymptotic standard deviations of estimated parameters. | |
| A.wnv | estimated variance of the innovations. | |
| A.checkr | if rcheck="prcheck", list containing the residual diagnostics: 1. res = n-p x 1 vector of residuals (i.e., one step ahead prediction errors computed with estimated parameters), 2. stat = scalar with statistic for testing H0: zero mean innovations (asymptotically normal under H0) and 3. acfQ = 30 x 4 matrix with the residuals acf, Ljung-Box statistic with p-values and pacf. The p-values are computed for M>p, and the first p values are filled with zeros. If rcheck="rcheck", the output contains all of the above mentioned residual diagnostics and plots additionally the residuals and their acf and pacf (correlograms). Otherwise, checkr=list containing the residuals. | |
| A.checkp | if pcheck="pcheck", list containing the parameter diagnostics: 1. est = string that informs if the necessary stationary condition (phi1+phi2+...+phip) holds or not. If the estimated values are stationary, it takes value 0. 2. bt = p x 2 or 1+p x 2 matrix of t-statistics and p-values (asymp. normal) for testing parameter significance and 3. bvar = variance covariance matrix of parameter estimates. Otherwise, checkp=0. | |
| A.ic | if msc="msc", 2 x 1 vector containing two model selection criteria: the Akaike Information Criterion, AIC, and the Schwarz Information Criterion, SIC. Otherwise, ic = 0. | |
; Example 1: zero mean process; estimation and parameter diagnostics
library("times") ; loads the quantlets from times library
randomize(33) ; sets the seed
b = #(0.5,-0.3) ; AR coefficients
y = genarma(b,0,normal(300)) ; generates AR(2) series
ar2 = ariols(y,2,0,"noconstant","0","pcheck")
ar2.checkp.est ; informs if the process is stationary
ar2.b ; estimated AR coefficients
ar2.checkp.bvar ; covariance matrix for estimated coefficients
ar2.checkp.bt ; t- and p-values for estimated coefficients
ar2.wnv ; estimated variance of the innovations
ar2.checkr.res[1] ; first residual
Contents of est [1,] "0" Contents of b [1,] 0.51878 [2,] -0.23496 Contents of bvar [1,] 0.0031933 -0.0013434 [2,] -0.0013434 0.0031972 Contents of bt [1,] 9.1804 5.421e-20 [2,] -4.1554 3.2468e-05 Contents of wnv [1,] 1.0518 Contents of _tmp [1,] -0.065255 ; first residual
; Example 2: nonzero mean process; estimation, residual diagnostics and model selection criteria
library("times") ; loads the quantlets from times library
randomize(33) ; sets the seed
b = #(0.5,-0.3) ; AR coefficients
y = genarma(b,0,normal(300)) + 0.5 ; generates AR(2) series
ar2 = ariols(y,2,0,"constant","prcheck","0","msc")
ar2.b ; estimated AR coefficients
ar2.wnv ; estimated variance of the innovations
ar2.checkr.acfQ[15,2] ; Ljung-Box with M=15
ar2.ic ; AIC and BIC
Contents of b [1,] 0.32307 [2,] 0.51796 [3,] -0.23599 Contents of wnv [1,] 1.0541 Contents of _tmp [1,] 7.3197 Contents of ic [1,] 0.055948 [2,] 0.080761