PROBLEM 12: CORRECTING FOR AUTOCORRELATION

(Kmenta 1986, 2nd edition, pp. 311-323 and P&R pp. 154-158).

OBJECT OF THE PROBLEM

To correct for autocorrelation using the Hildreth-Lu method. To

use the Cochrane-Orcutt procedure for correcting for

autocorrelation as found in PROC AUTOREG. To gain some experience

using the Econometric Software Package (ESP).

INTRODUCTION

One of several methods available for correcting for

autocorrelation is the Hildreth-Lu method. This method uses a

grid search procedure to find the value of rho which minimizes

the sum of squared residuals. Rho is used to transform all of the

variables in the model by subtracting from them the value of rho

multiplied by the value of the previous observation of the

variable.

Because SAS does not have a procedure for doing the

Hildreth-Lu method we use ESP to do the analysis.

On pages 316 to 317 Kmenta presents an example of the use of

the Hildreth-Lu procedure involving the estimation of a

consumption function.

The Cochrane-Orcutt procedure for correcting for

autocorrelation uses the OLS residuals to calculate rho. Rho is

then used to transform the observations of the dependent and

independent variables, and these new variables are used in a

reestimation of the model. The residuals from the new model are

then used to transform the transformed variables. The process

continues until convergence is reached in the values of the

betas.

PROC AUTOREG in SAS uses the Cochrane-Orcutt method to

correct for autocorrelation. The order (iteration) of the error

process must be specified in AUTOREG.

PROBLEM PROGRAM FOR HILDRETH-LU PROCEDURE

The following program can be used to complete Kmenta's problem.

Note: Before submitting this problem for batch processing, the

dataset must be unnumbered. ESP will not run on numbered

datasets. Unnumbering can be done while in EDIT mode by typing in

UNNUM. Be sure that verify is on.

//PROB12A JOB (AF,A592),yourname,REGION=4096K

/*OPENBIN

//S1 EXEC ESP

//FT24F001 DD DUMMY

//FT34F001 DD DUMMY

//SYSIN DD *

$$NAME, HILDLU$

LOAD$

SMPL 1 20$

IDGENR ID 1 1$

OLSQ CONS,C,MONEY$

HILU -.95,.9,.05,CONS,C,MONEY$

CORC CONS,C,MONEY$

END$

SMPL 1 20$

LOAD CONS$

214.6 217.7 219.6 227.2 230.9 233.3 234.1 232.3 233.7 236.5 238.7

243.2 249.4 254.3 260.9 263.3 265.6 268.2 270.4 275.6$

SMPL 1 20$

LOAD MONEY$

159.3 161.2 162.8 164.6 165.9 167.9 168.3 169.7 170.5 171.6 173.9

176.1 178.0 179.1 180.2 181.2 181.6 182.5 183.3 184.3$

END$

//

DISCUSSION OF THE HILDRETH-LU PROGRAM

Because this is the first problem in which you use ESP this

section will carefully discuss the program. The JCL for the

program is similar to that found in SAS. The ESP portion of the

program begins with a NAME card. Note that in ESP a line ends

with a "$," not a ";" as in SAS. The NAME card simply identifies

the program. The NAME card is followed by the LOAD card which

helps read in our data to the program. The next card, the SMPL

card, controls the number of observations to be used in the

analysis. In this case we are using observations one to twenty.

The next part of the program begins our estimation

procedures. First, we just run ordinary least squares (OLS) for

comparison purposes. Second we run the Hildreth-Lu procedure,

specified as "HILU." Within the HILU statement we must specify

the variables to be used in the analysis as well as the range of

iterations and the steps of iteration. The first number following

HILU specifies the beginning value for searching for rho. The

second number specifies the last value for iteration, and the

next number the amount to increment between iterations. The next

value on the line specifies the dependent variable. The last

observation(s) specifies the independent variable(s). The C is

used to include the intercept term in the analysis.

In addition to running the Hildreth-Lu correction we also do

the Cochrane-Orcutt procedure (CORC) so that you can compare

results. For all these models the first variable specified is the

dependent variable, CONS, and is followed by the intercept

term,C, and the explanatory variable, MONEY.

The section of the program following the first "END"

statement reads in the data. It uses a SMPL card to control the

number of variables. The procedure chosen here for loading the

data is to first load the consumption data in an array and then

the money data. They could have been loaded simultaneously but

only with one observation per line (i.e. LOAD CONS MONEY$). With

the technique used we can put many observations on one line. This

part of the program is completed with an END statement. the data

for the problem are those supplied by Kmenta on pp. 316-317.

INSTRUCTIONS FOR WRITE-UP

Give an intuitive and econometrics notation explanation of each

step of the program, and an intuitive explanation of what the

problem is trying to accomplish. Write out each of the estimated

models and analyze them (in terms of t and F-statistics, R-

Square, etc.). Give an intuitive explanation of each model. Label

all output. Draw a Durbin-Watson distribution, for each model

complete with critical points to help in the analysis of the

Durbin-Watson statistic supplied. How many iterations does it

take for convergence? What are the properties of the OLS

estimators (before and after correction for autocorrelation)?

The Cochrane-Orcutt procedure for correcting for

autocorrelation uses the OLS residuals to calculate rho. Rho is

then used to transform the observations of the dependent and

independent variables, and these new variables are used in a

reestimation of the model. The residuals from the new model are

then used to transform the transformed variables. The process

continues until convergence is reached in the values of the

betas.

PROC AUTOREG in SAS uses the Cochrane-Orcutt method to

correct for autocorrelation. The order (iteration) of the error

process must be specified in AUTOREG.

PROBLEM PROGRAM FOR COCHRANE-ORCUTT ITERATIVE PROCEDURE

The following program is an example of the use of PROC AUTOREG:

//PROB12B JOB (AF,A592),yourname,REGION=4096K

/*OPENBIN

//S1 EXEC SAS

//ESPDATA DD DSN=ACAD.ECONOMIC.QRTRLY,DISP=SHR

//SYSIN DD *

DATA MACRO; INFILE ESPDATA;

INPUT GNP 28-36 #2 MS 55-63 #4 UNE 1-9 BEH 28-36 #10;

N+1; IF N>100 THEN STOP;

LABEL GNP = GROSS NATIONAL PRODUCT

MS = M1 DEFINITION OF MONEY SUPPLY

UNE = UNEMPLOYMENT RATE

BEH = BUDGET EXPENDITURES HIGH EMPLOYMENT;

PROC REG; MODEL UNE=GNP MS BEH/DW;

PROC AUTOREG; MODEL UNE=GNP MS BEH/NLAG=1;

MODEL UNE=GNP MS BEH/NLAG=3;

MODEL UNE=GNP MS BEH/NLAG=5;

MODEL UNE=GNP MS BEH/NLAG=7;

//

DISCUSSION OF THE COCHRANE-ORCUTT PROGRAM

The procedure for calling the data should be familiar to you by

now. Once we have the data we first run PROC REG to check the

Durbin-Watson statisitcs. Then we use PROC AUTOREG and specify

lags of 1, 3, 5, and 7 for the autoregressive process. By

comparing the AUTOREG results we can determine which process is

best. This is done by looking at the t-statistics, Rsquare, and

MSE in the AUTOREG output.

INSTRUCTIONS FOR WRITE-UP

Give an intuitive and econometrics notation explanation of each

step of both programs, and an intuitive explanation of what each

problem is trying to accomplish. Write out each of the estimated

models and analyze them (in terms of t and F-statistics, R-

Square, etc.). Give an intuitive explanation of each model. Is

there autocorrelation in the primary model? How can you tell?

Which model corrects for the autocorrelation best (assuming that

there is some)? Why? Label all output. Draw a Durbin-Watson

distribution, for each model complete with critical points to

help in the analysis of the Durbin-Watson statistic supplied. How

many iterations does it take for convergence, in the Hildreth-Lu

method? What are the properties of the OLS estimators (before and

after correction for autocorrelation)? In general, what are the

properties of the OLS estimators (before and after correction for

autocorrelation)?