d. = (1 + Xi(X. 1X, 1) X.) 2)

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1 INTERFACING SAS SOFTWARE WITH THE B34S SYSTEM RECURSIVE RESIDUAL OPTION: A BRIEF LOOK AT THEORY AND AN EXAMPLE Houston H. Stokes University of Illinois at Chicago Abstract The recursive residual technique, developed by Brown-Durbin-Evans (1975) has proved to be a powerful tool to test for dynamic problems in OLS regression models. Code was initially developed to implement these procedures in the B34S system in the late 70s. The PROC CB34S allows the B34S system to run under the SAS software system. The use of the recursive residual procedure is illustrated by means of a sample dataset from the SAS/ETS manual. I. Introduction The B34S data analyses program,' developed by Houston H. Stokes, contains a number of statistical procedures, one of which is the recursive residual procedure that implements the equation specification suggestions of Brown Durbin-Evans (1975) and Dufour (1979).2 The purpose of the recursive residual procedure is to test whether the population density function underlying an OLS regression model is stable over time, or in the case of cross section models, stable for all values of the right-hand variables. The technique involves estimating a model for a subset of the dataset and testing how the coefficients change as more observations are added. 2. Theory In a time series model the usual procedure for a model with N observations and K right-hand variables is to start with a regression on the first K observations (if the X'X matrix is full rank) and add an observation at a time. A shift in the underlying population density function for the coefficients will be reflected in a visual movement of the coefficients. An alternative variant is to start with the most recent data and work backwards, adding an observation at a time until the coefficients begin to change. At this point one must truncate the dataset. This procedure will allow one to answer the question: "How much of the available dataset should I use?" While at first it might appear that all observations should be used to maximize the degrees of freedom, such an approach would not be appropriate if the cost of the added degrees of freedom was data coming from a different underlying model, which would bias the estimated coefficients. It is often difficult to determine what interaction terms to try when estimating a cross section model. The potential risk of not finding interaction terms is that estimated coefficients may be biased. PROC RSQUARE in SAS will only search among models where the independent variables have been built. With th,s procedure a1 i possible lnteractlon terms are not considered. As an example of the use of the recursive residual procedures, consider a cross section OLS regression of the form MODEL CON SUMP = INCOME AGE;. If the dataset were sorted with respect to INCOME and regressions were calculated, adding a higher INCOME observation each time, a significant movement in the INCOME coefficient would suggest that the effect of income on consumption, holding age constant, was not stable. If the coefficient of AGE were to change, it would indicate that there was an implicit AGE-INCOME interaction that was not specified in the MODEL statement and that the appropriate model to test was CONSUMPTION = f(age, INCOME, AGE*INCOME). Brown-Durbin-Evans (1975) provide formulas for the updating of the coefficient on K variables, provided that X'X for the first K observations is full rank. It is not possible to fall into degeneracy as new observations are added. The N+K recursive first step ahead residuals are of the LUS class whose sum of squares equals the OLS residuals. The one step ahead forecast error vector for observation i, vi uses the coefficient vector for the first i-i observations b i _ 1 to predict the dependent variable y. ana, using ith observation on the independent variables Xi' is defined over the range i=k+l to i=t as i=k+l,... T. 1) Brown-Durbin-Evans (1975) show that under the assumption of constancy of regression parameters and homoskedasticity, Vi has mean zero and variance a2*(d; **2) where for i=k+l... T I I -1 5 d. = (1 + Xi(X. 1X, 1) X.) 2) Dividing Vi by di gives the standardized onestep-ahead prediction error Wi of the recursive residual. Dufour (1979) has shown that the vectors of standardized differences of individual coefficients constitute K sets of LUS residuals, which have the same absolute values as the recursive residuals. The RR procedure will allow these and other tests to be performed for any step ahead recursive residual. Other tests that can be performed include the CUSUM test, which is defined as i CUSUM. = ('. k 1 wj)/ a 1 L J = + 3) a CUSUMSQ test defined as 76

2 and the HarveY-Collier (1977) test' on the recursive residual wfo If it were known that for a time series with N observations there was a change in the underlying structure at the ni th observation, then the most appropriate test would be a Chow test (distributed as F) on a regression run for observation 1 to ni and a regression run from observation ni + 1 to N. The problem is that the position "i' where the structure changed, it is not known. The Quandt Log-Likelihood Ratio test (Li) will provide some information on where ni occurs and is defined as: i = (.5 * i * In(01)2) + (.5 * CT-i) * In(02)2) - (.5 * T * 1n(02» 5) where ai, 02 and a are the standard errors of the regression fitted for the first i observations, the last T-i observations and the whole T observations. The minimum point of L; is valuable in selecting the "break u in the regression. 3. EXAMPLE To illustrate some of the features of the B34S recursive residual procedure, a model for interest rates developed by Pindyck and Rubinfeld, and discussed in the SAS/ETS Guide (5th edition) on page 48, was estimated and selected results displayed in Figure 1-7. The interest rate model can be written in the SAS SYSNLlN fannat as: R = R1 + R2 * Y + R3 * DIFY + R4 * DIFM + R5 * PLRS; where, using SAS notation, DIFY = DIF(Y); DIFM = DIF(M); PLRS = (LAG1(R) + LAG2(R»;. OLS estimates for the coefficients (with t scores in brackets) were R1 = (-1.57), R2 = (2.96), R3 = (.2133), R4 = (-1.731) and R5 =.3266 (6.23). The difficulty with just interpreting final OLS results is that the maintained, but not tested, assumption is that the population density function underlying the coefficients is stable. To test this assumption, recursive OLS coefficients are calculated and reported in Figure 1. The final values agree with what was reported earlier but, starting from the 6th observation, as additional observations were added, the values of the coefficients shifted markedly. For example, R1 moved steadily up from to , R2 decreased fr~n to R3, which was found to be insignificant for the total sample, fluctuated about zero as might be expected. R4, which was found to be significant for the total.. LISTING OF 8 COEFFICIENTS CBS #, " " ". 3l ". 37. as. " l s. ". 50. 'l. Ii: Ii: ~: 59. ". ". 44. ". 47. ". 40. 'l. 70. ". 7l " so..l. 82. sa. 84. ". so. OIFH ::~gt~5 ::tl~~ :Jn61 ::ii:g~ :Jiili :J~fZ~ ::iaf~i ::~Jn :Jgj~ ::~n2 ::~~~~~ :JA~Af ::nii~ ::~~~~ :J~~~ :J~~~! ::~~~~~ < , ' lS227D-<l1 -, [ < :ll6t~8:g~ :~4~~8:g~ ::U~j~ [HJ Figure 1 PLRS CONSTANT :~7:~~ :i5~4lb ( D-Ol : li~~ ::*l~~~ :f~~l :1:~n : ltl~i :1; ~~~~ :~~~~ :~:~~f :~~~~A :l:~~h ') sample, started positive (.32574), reached a low point of for the first 17 observations, then began slowly increasing to its final value of R5, which was highly significant (t=6.23) for the complete sample period, started negative ( ), then gradually increased over the sample period. Granted that for the first few observations it is hard to obtain a good estimate of the population from such a small sample, in the case of R5 half way through the sample (at observation 43) its value was approximately two thirds (.20544) of what was finally estimated. Such movement for a "highly significant" coefficient tends to give one caution concerning blind interpretation of the t scores. 77

3 B COEF '. PLOT OF B COEFFICIENT FOR Y.30971E--tl2 """''''*.. *****,j,,j,,j,,j,,j,*,j,*,j, _*.,j,,j,,j,,,,j,"**_*_.,j,,,,j,,j,,j,.,j,,j,,j,,j,,,,j,,j,...,j, _,j,,j,,j,,j,...,j,,j, TIME Fieure 2 B COEF.46181E-ol.. PLOT OF B COEFFICIENT FOR DIFY ".., * * **. *-****_... _*_... _-_......,,**... _... _....-._* Figure J TIME 78

4 B COEF.32!i74 " PLOT OF B COEFFICIENT FOR DIFH " , "'-***--"'''''''*-''''''''''''--'''''''''**'''-''''''**'''**'''''''''-''''''''''''''''''-'''''''''''''''.'''-''''''''''''-''' _-"''''''''''-''' ''''''.'''.. TJME Figure 4 B COEF.3~43 PLOT OF B COEFFICIENT FOR PLRS '" " "''''''''''''''''''''''''''''''''''-**'''''''''''''''-''''''-'''****-*'''*'''-'''''''''''''''''''''-'''''''''''''''-**''''''-''''''***'''''''''-"''''''''''''''''-''''''''''''.''' Figure.5 TJME 79

5 PLOT OF THE QUANDT LIKELIHOOD RATIO MINIMUM IS AN INDICATION OF A SHIFT IN STRUCTURE LANDA '. ***-***************--*******-***--*****-*************-****---***-****************-*** TIME Fieure 6 CUSUMSQ PLOT OF CUSUMSQ OF lth STEP AHEAD RECURSIVE RESIDUAL CO(.95)" OF... _ *...' " **... - * **** **- *- **** *** ***** * _ ****-.. * *********-***************-*******-********-*****************-******-*****************-** TESTS ON CUSUHSQ OF lth STEP AHEAD RECURSIVE RESIDUAL MAXIMUM DISTANCE CUSUMSQ FROM DIAGONAL.3818 OF 39 PROBABILITV FROM DURBIN TABLE DH.1445 MOD VN.1631 SIEGEL SIGN (#... ) 0 HARVEY-COLLIER PSI..( / ).. T OF Figure 7 TIME 80

6 Space prevents illustration of all test output available such as lists and plots and further tests on the recursive residual. Figure 2-5 show plots of the estimated coefficients listed in Figure 1 which are often easier to interpret than lists of the coefficients. The plot of the coefficient for the significant variable Y shows stability in the latter part of the sample (see Figure 2). In Figure 3 the plot of the insignificant coefficient for DIFY moves about zero, as is expected. In Figure 4 the plot of the coefficient for DIFM is shown to gradually increase throughout the sample period. In Figure 5 the plot of the coefficient for PLRS, which for the complete sample was highly significant (t=6.23), was shown in the early part of the sample to be negative. In the latter part of the sample this coefficient was quite stable. The Quandt Log-Likelihood test has been reported in Figure 6 to test where the structure might have changed. This statistic suggests a change in the underlying population density function approximately 3/4 of the way through the sample. One possibil i ty waul d be to truncate the sampl e. If the latest period were of most interest, due to potential forecasting applications, perhaps only latter period observations could be used. If it is assumed that these should include the last 1/2 of the sample, for observation 45 to 86 the estimated values become R1 =.632 (.31), R2 =.0022) (.98), R3 = (-.99), R4 = (-.30) and R5 =.3098 (3.32). The latter period results suggest that only R5 is significant and that any other variables found significant for the whole period obtain this Significance frrnn the first half of the sample. Figure 7 reports the CUSUMSQ test and presents a summary of the perfomance of the regression. The Harvey-Collier Psi test significantly indicates problems (a t statistic of -3.2), confirmed by the CUSUMSQ test, which was found to differ from the diagonal by a maximum of When this value is entered into the appropriate tables, it indicates that estimated population density function is highly unstable. 4. CONCLUS IONS The above research suggests that in regression analysis, it is important to detennine whether the sample has been taken frrnn a stable population or whether the population has been shifting over time. The latter problem can be observed if the population density function has in fact shifted or if important variables have been excluded in the sample. (If the shift in the population function could be predicted, these variables could possibly be incorporated in the model.) In some cases the nature of these exclude variables can be determined from the pattern of the recursive coefficients. When data is not available, often it is possible to deduce the direction of the estimated bias of the coefficients. The above arguments suggest that recursive residual analysis is an important regression diagnostic tool that should not be ignored. FOOTNOTES 1. The 834S basic reference is "The 834S Data Analysis Program: A Short Writeup," by Houston H. Stokes. Uni vers i ty of III i noi s College of Business Working Paper Series, Report FY 77-1 revised Additional documentation is conta; ned in chapters 1, 2 and 9 of "Econometric Analysis with the B34S Program," by Houston H. Stokes. Uni vers i ty of III i noi s Call ege of Business Working Paper Series, Report FY S is approximately 78,000 FORTRAN statements and contains options in the area of time series, PROBIT, L08IT, Error Component, simultaneous equations, Markov model estimation, optimal control and BLUS analysis. An attempt has been made to strap together the developmental code in these areas so that the various procedures can be run as a system. In 1983 the PROC CB34S was written so that the whole B34S system would be seen by SAS as a single procedure, although B34S is usually run in the next job step. The intention of this paper is to illustrate the use of the recursive residual procedure, not discuss the B34S system in detail. 2. The basic reference for the recursive residual is litechniques for Testing the Consistency of RegreSSion Relationships over Time," by R. J. Brown, J. Durbin and J. Evans,,Journal of the Royal Statistical Society Series B Vol. 1 (1975) pp Jean-Marl e Dufour, "Methods for Specification Errors Analysis with Macroeconomic Applications,1I unpublished Ph.D Dissertation, U of Chicago 1979, extended the pioneering work of Brown-Durbin-Evans by proposing non-parametric tests. 3. Harvey, Andrew and Patrick Collier., "A Comparison of'the Power of Some Tests for Heteroskedasticity in the General Linear Modell! Journal of Econometrics Vol. 6 (1977) pp. ~119. The equatlon for the test is not given due to space limitations. Dr. Houston H. Stokes Head Department of Economics University of Illinois Box 4348 Chicago, IL