SAS Introduction to Time Series Forecasting-libre

8/19/2019 SAS Introduction to Time Series Forecasting-libre

1/34

Quick Review about How to Use SAS to

Analyze Time Series Data

1. Get to know SAS

How to Start SAS?

)f you use computer in this laboratory, please start SAS from Desktop or Start/programs.

You can use the SAS software at the laboratory of the Computer center of our university, or even

by the server of our university if you have the permission.

You can get a temporary license of the SAS software by contacting our computer assistant.

Five main windows

Program Editor -- Edit SAS programs

Log – Records the running messages of SAS session, which is very helpful for program

debugging.

Output – Display output from SAS procedures

Explorer – Manage SAS datasets or Create new libraries

Result – Show a tree-like summary of your Output window

Several important shortcuts

Open a new Program Editor window

Open SAS program which is composed before

Save your program as external files

Create a new library

Open Explorer window to manage SAS datasets

Submit the whole program or just submit a few lines SAS programs to SAS System

2. How to use SAS

Two important concepts

SAS library – A folder in which the SAS data set is. You can create a new library by libname or

shortcut .

SAS data set – Temporary and Permanent SAS data set.


2/34

Structure of SAS program

DATA step – Deal with SAS dataset, or change raw data into a SAS data set, which can be

identified by SAS System and dealt with by PROC step

=====================================

DATA dataset name;

INPUT variable;

CARDS;

…………………..data line

;

=====================================

The dataset name must contain no more than 8 characters alphabet a, b…, digit , … or

underscore (_)), and begin with alphabet or underscore.

PROC step – Deal with SAS data set, and output results of analysis

=====================================

PROC procedure name DATA= dataset name;

RUN;

=====================================


3/34

The procedure name is the name of SAS Command, and includes PRINT, PLOT, GPLOT , and

INSIGHT etc.

3. Change raw data into SAS datasetCreate a new library

Library Name Physical Path

Using SAS program.

Using shortcut.

SAS data set name

For example, lib1.blood means that data set blood is saved in the library lib1.

The library_name can be sashelp, sasuser , maps, work or lib1. The dataset_name is due to you,

such as blood .

When library_name is equal to work , the data set work.dataset_name is temporary SAS data set,

which will be deleted automatically when you shut down the SAS software. At this time, the

work can be ignored. For example, you use blood or work.blood as the name of the data set.

Three methods to deal with data through DATA Step

The size of raw data is small.

Lib1 D:\example

Libname lib D:\example

library_name.dataset_name

DATA dataset name;

INPUT variable ;CARDS;

………………. data line)

;


4/34


5/34

OUTLIER options;

FORECAST options;

RUN;

QUIT;BY

A BY statement can be used in the ARIMA procedure to process a data set in groups of

observations defined by the BY variables. Note that all IDENTIFY, ESTIMATE, and FORECAST

statements specified are applied to all BY groups.

IDENTIFY

ALPHA= significance-level: The ALPHA= option specifies the significance level for tests in the

IDENTIFY statement. The default is 0.05.

ESACF: computes the extended sample autocorrelation function and uses these estimates to

tentatively identify the autoregressive and moving average orders of mixed models.The ESACF option generates two tables. The first table displays extended sample

autocorrelation estimates, and the second table displays probability values that can be used to

test the significance of these estimates. The P= (pmin: pmax) and Q= (qmin: qmax) options

determine the size of the table.

NLAG= number: indicates the number of lags to consider in computing the autocorrelations and

cross-correlations.

STATIONARITY=(ADF= AR orders DLAG= s) or STATIONARITY=(DICKEY= AR orders DLAG= s):

performs augmented Dickey-Fuller tests. If the DLAG=s option specified with s is greater than

one, seasonal Dickey-Fuller tests are performed. The maximum allowable value of s is 12. The

default value of s is one.

VAR= variable ( d1, d2, ..., dk ) : names the variable containing the time series to analyze. The

VAR= option is required. A list of differencing lags can be placed in parentheses after the

variable name to request that the series be differenced at these lags. For example, VAR=X(1)

takes the first differences of X. VAR=X(1,1) requests that X be differenced twice, both times with

lag 1, producing a second difference series, which is (Xt-Xt-1)-(Xt-1-Xt-2)=Xt-2Xt-1+Xt-2 .

VAR=X(2) differences X once at lag two (Xt-Xt-2) . If differencing is specified, it is the

differenced series that is processed by any subsequent ESTIMATE statement.

ESTIMATE

METHOD=ML/ULS /CLS: specifies the estimation method to use. METHOD=ML specifies the

maximum likelihood method. METHOD=ULS specifies the unconditional least-squares method.

METHOD=CLS specifies the conditional least-squares method. METHOD=CLS is the default.

P= order: specifies the autoregressive part of the model. By default, no autoregressive

parameters are fit. P=(l1, l2, ..., lk) defines a model with autoregressive parameters at the


6/34

specified lags. P= order is equivalent to P=(1, 2, ..., order). A concatenation of parenthesized lists

specifies a factored model. For example, P=(1,2,5)(6,12) specifies the autoregressive model

Q= order: specifies the moving average part of the model.

NOCONSTANT/NOINT: suppresses the fitting of a constant (or intercept) parameter in the

model. (That is, the parameter is omitted.)

PLOT: plots the residual autocorrelation functions. The sample autocorrelation, the sample

inverse autocorrelation, and the sample partial autocorrelation functions of the model residuals

are plotted.

FORECAST

ALPHA= n: sets the size of the forecast confidence limits. The ALPHA= value must be between 0

and 1. When you specify ALPHA=, the upper and lower confidence limits will have a confidence

level. The default is ALPHA=.05, which produces 95% confidence intervals. ALPHA values are

rounded to the nearest hundredth.

ID= variable: names a variable in the input data set that identifies the time periods associated

with the observations.

INTERVAL= interval /n: specifies the time interval between observations.

LEAD= n: specifies the number of multistep forecast values to compute.

OUT= SAS-data-set: writes the forecast (and other values) to an output data set.


7/34

Fitting the ARIMA Model to a Simulated Time Series

0. Simulate an AR(2) time series data

The model: Z(t)=0.5*Z(t-1)+0.4Z(t-2)+a(t)

The SAS program:

Simulate an MA(2):

/* Create a new library */

libname ts 'D:/TimeSeries';

/* Simulate an AR(2) process */

data ts.ar;

z1=0; z2=0;

do t = -50 to 200;

a = rannor( 32565 );z = z1*0.5 + z2*0.4 + a;

if t > 0 then output;

z2=z1; z1=z;

end;

keep z t;

run;

/* Simulate an MA(2) process */

data ts.ma;

a1=0; a2=0;

do t = -50 to 200;

a = rannor( 32565 );

z = a + a1*0.2+a2*0.5;


a2=a1; a1=a;

end;

keep z t;

run;


8/34

Simulate an ARMA(1,1):

1. Draw the time plot

The SAS program:

The result:

Simulated AR(2) Time Series

/* Draw the time plot */

symbol i=join v=none;

proc gplot data=ts.ar;

plot z*t;

run;

quit;

/* Simulate an ARMA(1,1) process */

data ts.arma;

z1=0; a1=0;

do t = -50 to 200;a = rannor( 32565 );

z = z1*0.5 + a + a1*0.3;


a1=a; z1=z;

end;

keep z t;

run;


9/34

2. Identify some suitable models

The SAS program:

The summary of the output:

/* Identify some suitable models with minimum requirement */ proc arima data=ts.ar;

identify alpha=0.05 var=z nlag=20;

run;

/* Use EACF to identify the orders of ARMA models */

identify alpha=0.05 var=z nlag=20 esacf p=(0:6) q=(0:8);

run;

/* Use Dickey-Fuller unit root tests to check the stationarity */

identify alpha=0.05 var=z nlag=20 stationarity=(dickey=(1, 2, 4));

run;

/* Take differencing on the data and analyze again */

identify alpha=0.05 var=z(1) nlag=20 stationarity=(dickey=5);run;

quit;


10/34

The detailed output without differencing:


11/34

Series Correlation Panel

3 deterministic trends

different values of k

3 different tests


12/34

The detailed output after first differencing:

Series Correlation Panel


13/34

We may reach three possible models:

ARIMA(3,0,0); ARIMA(0,1,1); and ARIMA(2,1,0).

3. Estimate the models

Candidate models: AR(3), ARMA(3,1) with AR coefficient at lag 2 suppressed and ARIMA(2,1,0)

without intercept.

The SAS program:

/* Identify some suitable models with minimum requirement */

proc arima data=ts.ar;

identify alpha=0.05 var=z nlag=20;

run;

/* Use EACF to identify the orders of ARMA models */

identify alpha=0.05 var=z nlag=20 esacf p=(0:6) q=(0:8);

run;

/* Use Dickey-Fuller unit root tests to check the stationarity */

identify alpha=0.05 var=z nlag=20 stationarity=(dickey=(1, 2, 4));

run;

/* Take diffferencing on the data and analyze again */

identify alpha=0.05 var=z(1) nlag=20 stationarity=(dickey=5);

run;

/* Use CLS method to estimate the AR(3) model */

identify var=z;

run;

estimate method=cls p=3 plot;

run;/* Use ULS method to estimate the ARMA(3,1) model */

/* with the second coefficient is suppressed */

estimate method=uls p=(1,3) q=1 plot;

run;

/* Use ML method to estimate the ARIMA(2,1,0) model without

intercept */

identify var=z(1);

run;

estimate method=ml p=2 noint plot;

run;

quit;


14/34


The estimated AR(3) model:


15/34

The important outputs for the fitted AR(3) model:

Mean

Estimated

parameters

P values of

significanceIntercep

Variance of the

white noise

Standard deviation

of the white noise


16/34

Outputs for ARMA(3,1) with AR coefficient at lag 2 suppressed:


17/34

Outputs for ARIMA(2,1,0) without intercept:


18/34

4. Diagnostic checking for the fitted ARIMA(2,1,0)

The SAS program:


/* Diagnostic checking for the fitted ARIMA(2,1,0) */


identify var=z(1);run;


run;

forecast out=ts.dc lead=0 id=t;

run;

quit;



proc gplot data=ts.dc;

plot residual*t;run;

quit;

/* Perform the normality test */

proc univariate data=ts.dc normal plot;

var residual;

run;


19/34

The time plot:

A normality test:


20/34

Distribution plot and Q-Q plot for normality:

Sample autocorrelation function (ACF) of the residuals and Sample partial ACF of the residuals:


21/34

Ljung-Box test:

Analysis of over-parameterized models:o The SAS program:

o

The first over-parameterized model based on the sample partial ACF:

/* Analysis of over-parameterized models */


identify var=z(1) nlag=20;

run;


run;

estimate method=ml p=(1,2)(6) noint plot;

run;

estimate method=ml p=2 q=(6) noint plot;run;

quit;

Test statistic

Degree of

freedomP-values


22/34

o The second over-parameterized model based on the sample ACF:

o

Three fitted models:


23/34

o

Conclusion is that the fitted ARIMA(2,1,0) is not adequate!


24/34

5. Do forecasting with the fitted ARIMA(2,1,0) model

The SAS program:

The results:

/* Do forecasting by using the fitted ARIMA(2,1,0) model */


identify var=z(1) nlag=20;

run;


run;

forecast out=ts.out lead=50 id=t;

run;

quit;

/* Draw the time plot */ symbol i=join v=none;

proc gplot data=ts.out;

plot z*t=1 forecast*t=2 l95*t=3 u95*t=3/overlay;

run;

quit;


25/34

1

Fitting the Seasonal ARIMA Model to

The Airline Passenger Data

0. The data

The airline passenger data records the number of passengers traveling by air per month from

January, 1949 to December, 1960.

It is given as Series G in Box and Jenkins (1976), and has been used in time series analysis

literature as a standard example of a non-stationary seasonal time series.

1. Draw the time plot

The SAS program:

The time plot:

/* Create a new library */

libname ts 'D:/TimeSeries';



proc gplot data=sashelp.air;

plot air*date;

run;

quit;


26/34

2

Taking log transformation and drawing the time plot again.

The time plot:

2. Identify some suitable models

The SAS program:

/* Take log transformation*/

data ts.lair;

set sashelp.air;

lair=log(air);run;



proc gplot data=ts.lair;

plot lair*date;

run;

quit;

/* Identify some suitable models*/

proc arima data=ts.lair;

identify alpha=0.05 var=lair;

run;

/* Take differencing since the sample ACF decays slowly */

identify alpha=0.05 var=lair(1);

run;

/* Take seasonal differencing since the sample ACF decays slowly

especially after periods */

identify alpha=0.05 var=lair(1,12);

run;


27/34

3

The sample ACF of original sequence:

The sample ACF of the sequence after common differencing:

The sample ACF of the sequence after both common differencing and seasonal differencing:


28/34

4

3. Estimate the seasonal ARIMA(0,1,1)X(0,1,1)12 model

The SAS program:

The estimated model:



run;

/* Estimate the ARIMA(0,1,1)X(0,1,1)12 model to the data */

estimate method=ml q=(1)(12) plot;

run;


29/34

5

4. Diagnostic checking the fitted seasonal ARIMA(0,1,1)X(0,1,1)12 model

The SAS program:

The sample ACF of residuals:

proc arima data=ts.lair;identify alpha=0.05 var=lair(1,12);

run;

/* Estimate the ARIMA(0,1,1)X(0,1,1)12 model to the data */


run;

/* Diagnostic checking by overfit AR part */

estimate method=ml p=(9) q=(1)(12) plot;

run;

/* Diagnostic checking by overfit MA part */

estimate method=ml q=(1)(12)(23) plot;

run;

/* Export the data to do further diagnostic checking*/

forecast out=ts.out lead=0 id=date;

run;

quit;




plot residual*date;

run;

quit;

/* Perform the normality test */

proc univariate data=ts.out normal plot;var residual;

run;


30/34

6

The sample PACF of residuals:

Ljung-Box test:

Diagnostic checking by overfitting the AR part and the MA part:


31/34

7

Compare the

estimated

coefficients

Compare

the model

criteria


32/34

8

The time plot of the residuals:

Normality tests:

Distribution plot and Q-Q plot for normality:


33/34

9

5. Do forecasting with the fitted seasonal ARIMA(0,1,1)X(0,1,1)12 model

The SAS program:

/* Do forecasting with the fitted seasonal ARIMA(0,1,1)X(0,1,1)12 model */



run;


run;

forecast out=ts.out lead=24 id=date interval=month;

run;

quit;




plot lair*date=1 forecast*date=2 l95*date=3 u95*date=3/overlay;

run;

quit;


34/34

10

The result:

Documents

SAS Introduction to Time Series Forecasting-libre