... t t t k kt t. We assume that there are observations covering n periods of time. Autocorrelation (also called

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1 Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Modeling Auocorrelaion Because auocorrelaion is primarily a phenomenon of ime series daa, i is convenien o represen he linear regression model using as a subscrip o represen ime: (13B.1) Y 0 1X1 X... X where = 1,,, n k k We assume ha here are observaions covering n periods of ime. Auocorrelaion (also called serial correlaion) exiss when he error erms 1,,, n are no independen of one anoher. There are many ways we migh envision non-independence among he errors. The firs-order auoregressive model (someimes called he AR1 model) is a common way of hinking abou correlaed errors: (13B.) 1 u where 1 +1 where is he auocorrelaion parameer and u is a well-behaved (i.e., normally disribued, homoscedasic, non-auocorrelaed) random disurbance wih mean zero and consan variance. As you can see from equaion (13B.), if = 0, hen is also well-behaved because = u. If = 0 hen is unaffeced by error -1. In oher words, if = 0 here is no carry-over from period 1 o period. However, if is no zero, hen error is affeced by error -1. In fac, i is fairly easy o show ha is affeced by all of he prior errors. Because he same relaionship holds beween every error and is predecessor, we can subsiue 1 u 1 ino equaion (13B.) o obain u ( u ) u u u Coninuing wih subsiuions in his fashion, we can show ha u u u... u 1 0 This says ha he error in period is affeced by all he prior random errors. Only if = 0 will auocorrelaion vanish so ha = u. Serial correlaion is a violaion of he assumpion of independence, creaing problems wih he -ess and confidence inervals ha are repored in regression sofware. More specifically, if > 0 (a common occurrence in ime series daa) he repored MSE ends o underesimae he variance of he errors. The ANOVA es saisic for overall significance may hus be inflaed, as well as he -es saisics for individual predicor significance. While he leas-squares esimaes remain unbiased, hey are no longer minimum variance unbiased esimaors (see MVUE in Chaper 8). Therefore, a es for auocorrelaion is desirable when auocorrelaion is suspeced. Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Page 1

2 Durbin-Wason Tes Because he rue errors 1,,, n are unobservable, we use he regression residuals e 1, e,, e n in our es for auocorrelaion. The Durbin-Wason saisic is calculaed as (13.xx) DW n ( e e ) n 1 e 1 The range of DW is 0 DW 4. An approximae relaionship exiss beween DW and : Thus: DW (1 ) = +1 DW (1 1) = 0 Perfec posiive auocorrelaion = 0 DW (1 0) = No auocorrelaion = 1 DW [1 ( 1)] =4 Perfec negaive auocorrelaion When DW is much less han, we suspec posiive serial correlaion (a common condiion in ime series daa). Conversely, when DW is much greaer han, we suspec negaive serial correlaion (a less common condiion in ime series daa). When DW = we have no sample evidence of auocorrelaion. Some praciioners (e.g., hp://help.sap.com) sugges a rule of humb ha wihin he range 1.5 o.5 here is lile cause for alarm. However, more precise ess are desirable. Criical Values for Durbin Wason Tes To es for auocorrelaion, he es saisic is compared o lower and upper criical values (d L and d U ) for a specified level of significance α. The criical values depend on he sample size (n) and he number of predicors (k). Criical values for =.05 are shown in Table 13B.1 for various sample sizes and numbers of predicors. This able only goes up o k = 5 predicors and only shows seleced sample sizes. This suffices o illusrae he DW es, and should cover he ypes of problems you will encouner as an inroducory saisics suden. However, you can easily find more complee ables if you need hem (see able foonoe, chaper references, or Google). Wihin Table 13B.1 you can inerpolae beween sample sizes if you find i necessary. Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Page

3 Table 13B.1 Durbin-Wason Criical 5% Values k = 1 k = k = 3 k = 4 k = 5 n d L d U d L d U d L d U d L d U d L d U Source: Excerps from N. E. Savin and Kenneh J. Whie, The Durbin-Wason Tes for Serial Correlaion wih Exreme Sample Sizes or Many Regressors, Economerica, Vol. 45, No. 8 (Nov., 1977), pp Used wih permission of he Economeric Sociey. To es for posiive auocorrelaion, he hypoheses are: H 0 : = 0 H 1 : > 0 (errors are no auocorrelaed) (errors are posiively auocorrelaed) The inerpreaion of he es for posiive auocorrelaion is shown in words and visually: If DW < d L conclude H 1 (errors are posiively auocorrelaed) If DW > d U conclude H 0 (errors are no posiively auocorrelaed) If d L DW d U he es is inconclusive Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Page 3

4 Example: Changes in Consumer Prices Are changes in consumer prices over ime relaed o changes in manufacuring capaciy uilizaion, changes in he money supply, and unemploymen raes? Table 13B. shows daa for 40 recen years. The variables o be invesigaed are: CPI ChCPI = change in he Consumer Price Index (all iems) CapUil = change in he manufacuring capaciy uilizaion rae ChgM1 = change in he M1 componen of he money supply ChgM = change in he M componen of he money supply Unem = unemploymen rae (percen) Table 13B. Seleced U.S. Economic Variables, CPI Year ChCPI CapUil ChgM1 ChgM Unem Source: Economic repor of he Presiden, February, 011. Only he firs hree and las hree observaions are shown. Regression Analysis R² Adjused R² 0.31 n 40 R k 4 Sd. Error.751 Dep. Var. ChCPI ANOVA able Source SS df MS F p-value Regression Residual Toal Regression oupu Variables Coefficiens Sd. error (df=35) p-value VIF Inercep CapUil ChgM ChgM Unem Durbin-Wason = 0.83 The fied regression is shown here. Overall, he regression is significan a α =.01. The bes predicor is CapUil (p =.0007) followed by Unem (p =.0188). The money supply predicor ChgM is weak (p =.1389) and ChgM1 is no significan. The inercep differs significanly from zero, bu is no of ineres here. Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Page 4

5 Residual (gridlines = sd. error) Some of he predicor signs are in line wih a priori expecaions, bu his naïve model is of lile economic ineres. We will no analyze i in deail, as our objecive here is o only examine he paern of residuals. To es for posiive auocorrelaion, he hypoheses are: H 0 : = 0 (errors are no auocorrelaed) H 1 : >0 (errors are posiively auocorrelaed) The Durbin-Wason es saisic (shown above in he compuer prinou) is DW = There are k = 4 predicors and n = 40, so d L = 1.85 and d U = The decision rule is: If DW < d L conclude H 1 (errors are posiively auocorrelaed) If DW > d U conclude H 0 (errors are no posiively auocorrelaed) If d L DW d U he es is inconclusive Because DW < d L, we conclude ha posiive auocorrelaion exiss in he errors. This paern can be seen in Figure 13B.1 as a cyclic paern, i.e., a series of runs of residuals of he same sign ( ec). The residuals have a zero mean (as he mus) bu he series of 40 residuals only crosses he zero axis line 1 imes. Chance alone would sugges more sign changes (i.e., more cenerline crossings). FIGURE 13B.1 Residual Plo Over Time 8.5 Residuals Observaion Negaive Auocorrelaion In our example (and in mos economic ime series models) we would wan o es for posiive auocorrelaion. However, o es for negaive auocorrelaion, he hypoheses would be: H 0 : = 0 H 1 : < 0 (errors are no auocorrelaed) (errors are negaively auocorrelaed) The es is similar o posiive auocorrelaion, excep ha he es saisic is 4 DW. Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Page 5

6 For negaive auocorrelaion, he es is: If 4 DW < d L conclude H 1 (errors are negaively auocorrelaed) If 4 DW > d U conclude H 0 (errors are no negaively auocorrelaed) If d L 4 DW d U he es is inconclusive Cavea for Durbin-Wason Tes The D-W ables do no apply if you have lagged values of he response variable (e.g., Y 1 or Y ) among he lis of predicors. A his sage of your raining, i is bes o avoid such predicors. Exercise Noe: * indicaes opional porions for hose who wan a greaer challenge. 13B.1 Below are daa on several economic variables ha migh help predic per capia consumer spending (annual daa covering ). The response variable in he proposed model is ConsCap and he hree predicors are YdCap, Unem, and r3-mo, where: ConsCap = YdCap = Unem = r3-mo = per capia consumpion expendiures (curren dollars) per capia disposable personal income (curren dollars) unemploymen rae (percen) hree-monh U.S. Treasury bill rae (percen) TABLE 13B.3 U.S. Economic Daa, Consumpion Year ConsCap YdCap Unem r3-mo Source: Economic Repor of he Presiden, February, 011. Insrucions: Esimae he regression. Include a able of residuals and he Durbin-Wason es (in Miniab, look under Opions, while in MegaSa you mus check a box if you wan he DW es saisic). (b) Discuss he overall significance of he model, and ell which predicors are significan a α =.05 (c) Find he 5% criical values in Table 13B.1 and sae he decision rule for he DW es for posiive auocorrelaion. Hin: To be conservaive, use he nex lower sample size if your n is no in he able (or inerpolae he able values). (d) Wha is your conclusion abou auocorrelaion? (e) Plo he residuals in ime order. Describe he paern. (f) How many imes do he residuals sign change (coun he crossings of he zero axis). Wha does his ell you? (g*) Based on wha you know abou economics, are he signs of he predicors logical a priori (ignoring hose ha are insignifican)? (h*) Re-esimae he model using variables ha have been ransformed using firs differences (see he daa spreadshee). Does his reduce auocorrelaion? Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Page 6

7 RELATED READINGS Durbin, J.; and Wason, G. S. Tesing for Serial Correlaion in Leas Squares Regression, I. Biomerika 37, 1950, Durbin, J., and Wason, G. S. Tesing for Serial Correlaion in Leas Squares Regression, II. Biomerika 38, 1951, Gujarai, Damodar; and Dawn Porer. Basic Economerics. 5 h ed. McGraw-Hill, 009. Kuner, Michael H.; Chrisopher J. Nachsheim; John Neer;, and William Li. Applied Linear Saisical Models. 5 h ed. McGraw-Hill, 005. Savin, N. E. and Kenneh J. Whie, The Durbin-Wason Tes for Serial Correlaion wih Exreme Sample Sizes or Many Regressors, Economerica, 45, 1977, Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Page 7