Presentation on theme: "Lecture #9 Autocorrelation Serial Correlation"— Presentation transcript:
1Lecture #9 Autocorrelation Serial Correlation Studenmund (2006) Chapter 9AutocorrelationSerial CorrelationObjectivesThe nature of autocorrelationThe consequences of autocorrelationTesting the existence of autocorrelationCorrecting autocorrelation
2Time Series Data Time series process of economic variables e.g., GDP, M1, interest rate, exchange rate,imports, exports, inflation rate, etc.RealizationAn observed time series data set generated from a time series processRemark:Age is not a realization of time series process.Time trend is not a time series process too.
3Cyclical or seasonal random Decomposition of time seriesXt = Trend + seasonal + randomTrendXttimeCyclical orseasonalrandom
4Example: Static Phillips curve model inflatt = 0 + 1unemployt + t Static ModelsCt = 0 + 1Ydt + tSubscript “t” indicates time. The regression is a contemporaneous relationship, i.e., how does current consumption (C) be affected by current Yd?Example: Static Phillips curve modelinflatt = 0 + 1unemployt + tinflat: inflation rateunemploy: unemployment rateContemporaneous relation:a. Equilibrium relationb. No dynamic effects, i.e., immediate effect only
5Ct+1=0+0Ydt+1+1Ydt+tCt=0 +0Ydt+1Ydt-1+t Finite Distributed Lag ModelsEconomic actionat time tEffectCt =0+0Ydt+tEffectat time t+1Ct+1=0+0Ydt+1+1Ydt+tCt=0 +0Ydt+1Ydt-1+tEffectat time t+2Effectat time t+q….Forward Distributed Lag Effect (with order q)….Ct+q=0+1Ydt+q+…+1Ydt+tCt=0+1Ydt+…+1Ydt-q+t
6Backward Distributed Lag Effect Economic actionat time tEffectat time t-1Backward Distributed Lag EffectEffectat time t-2Effectat time t-3Effectat time t-q….Yt= 0+0Zt+1Zt-1+2Zt-2+…+2Zt-q+tInitial state: zt = zt-1 = zt-2 = c
7C = 0 + 0Ydt + 1Ydt-1 + 2Ydt-2 + t Long-run propensity (LRP) = (0 + 1 + 2)Permanent unit change in C for 1 unit permanent (long-run) change in Yd.Distributed Lag model in general:Ct = 0 + 0Ydt + 1Ydt-1 +…+ qYdt-q+ other factors + tLRP (or long run multiplier) = 0 + q
8Time Trends Linear time trend Yt = 0 + 1t + t Constant absolute changeExponential time trendln(Yt) = 0 + 1t + t Constant growth rateQuadratic time trendYt = 0 + 1t + 2t2 + t Accelerate changeFor advances on time series analysis and modeling , welcome to take ECON 3670
9Definition: First-order of Autocorrelation, AR(1) If Cov (t, s) = E (t s) where t sYt = 0 + 1 X1t + t t = 1,……,Tand ift = t-1 + utwhere < < ( : RHO)and ut ~ iid (0, u2) (white noise)This scheme is called first-order autocorrelation and denotes as AR(1)Autoregressive : The regression of t can be explained by itself lagged one period.(RHO) : the first-order autocorrelation coefficientor ‘coefficient of autocovariance’
10Example of serial correlation: u1990… … … ….… … ….u2002u2003u2004u2005u2006u2007Year Consumptiont = 0 + 1 Incomet + errortExample of serial correlation:TaxPay2006TaxPay2007Error termrepresentsother factorsthat affectconsumptionThe current year Tax Pay may be determined by previous year rateTaxPay2007 = TaxPay u2007 t = t-1 + utut ~ iid(0, u2)
12timei^xPositive autocorrelationtimei^xPositive autocorrelationtimei^Cyclical: Positive autocorrelationxThe current errorterm tends to havethe same sign asthe previous one.
13Negative autocorrelation timei^xThe current error term tends to have the opposite sign from the previous.No autocorrelationxtimei^The current error term tends to be randomly appeared from the previous.
14The meaning of : The error term t at time t is a linear The meaning of : The error term t at time t is a linear combination of the current and past disturbance.0 < < 1-1 < < 0The further the period is in the past, the smaller is the weight of that error term (t-1) in determining t = 1The past is equal importance to the current. > 1The past is more importance than the current.
15The consequences of serial correlation: The estimated coefficients are still unbiased.E(k) = k^BLUE^2. The variances of the k is no longer the smallest3. The standard error of the estimated coefficient, Se(k)becomes large^Therefore, when AR(1) is existing in the regression,The estimation will not be “BLUE”
16The AR(1) variance is not the smallest Example:Two variable regression model: Yt = 0 + 1X1t + tThe OLS estimator of 1,^ x y xt2If E(t t-1) = 0then Var (1) =2===> 1 =If E(tt-1) 0, and t = t-1 + ut , thenVar (1)AR1= 2^ xt xt xt xt+2 xt2 xt xt xt2-1 < < 1+ ….If = 0, zero autocorrelation, than Var(1)AR1 = Var(1)^If 0, autocorrelation, than Var(1)AR1 > Var(1)The AR(1) variance is not the smallest
17Autoregressive scheme: t = t-1 + ut ==> t = [ t-2 + ut-1] + ut==> t-2 = t-3 + ut => t = 2 [ t-3 + ut-2] + ut-1 + ut==> t-1 = t-2 + ut t = 2 t-2 + ut-1 + utt = 3 t-3 + 2 ut-2 + ut-1 + utE(t t-1) =21 - 2E(t t-3) = 2 2E(t t-2) = 2…………….E(t t-k) = k-1 2It means the more periods in the past,the less effect on current periodk-1 becomes smaller and smaller
23Durbin-Watson test(Cont.) II. H0 : ≥ no negative autocorrelationH1 : < yes, negative autocorrelationwe use (4-d) (when d is greater than 2)if (4 - d) < dLor dL < d < ==> reject H0if dL (4 - d) duor du d 4 - dL ==> inconclusiveif dL (4 - d) duor du > d > ==> not reject H01.271.452dLdu4(4 - dL)(4-dU)2.732.55
24Durbin-Watson test(Cont.) II. H0 : = No autocorrelationH1 : two-tailed test for auto correlationeither positive or negative AR(1)If d < dLor d > 4 - dL==> reject H0If du < d < 4 - du ==> not reject H0If dL d duor 4 - du d 4 - dL==> inconclusive
25For example :UMt = CAPt CAPt Tt^(15.6) (2.0) (3.7) (10.3)R2 = F = = SSR = DW = n = 68_(i) K = 3 (number of independent variable)Observed(ii) n = 68 , = significance level0.05(iii) dL = , du =dL = , du =Reject H0, positive autocorrelation exists(excluding intercept)
27The assumptions underlying the d(DW) statistics : 1. Intercept term must be included.2. X’s are nonstochastic3. Only test AR(1) : t = t-1 + ut where ut ~ iid (0, u2)4. Not include the lagged dependent variable,Yt = 0+ 1 Xt1 + 2 Xt2 + …… + kXtk + Yt-1 + t(autoregressive model)5. No missing observationN.A N.A.N.A N.A.95...YXmissing
28Lagrange Multiplier (LM) Test or called Durbin’s m test Or Breusch-Godfrey (BG) test of higher-order autocorrelation^Test Procedures:(1) Run OLS and obtain the residuals t.^(2) Run t against all the regressors in the modelplus the additional regressors, t-1, t-2, t-3,…, t-p.t = 0 + 1 Xt + t-1 + t-2 + t-3 + … + t-p + uObtain the R2 value from this regression.(3) compute the BG-statistic: (n-p)R2(4) compare the BG-statistic to the 2p (p is # of degree-order)(5) If BG > 2p, reject Ho,it means there is a higher-order autocorrelationIf BG < 2p, not reject Ho,it means there is a no higher-order autocorrelation
314. Cochrane-Orcutt Iterative Procedure (5). If DW test shows that the autocorrelation still existing, thanit needs to iterate the procedures from (4). Obtains the t*(6). Run OLSt* = t-1* + ut’^ ( )DW22and obtains which is the second-round estimated Xt** = Xt - Xt Yt-1 = 0 + 1 Xt-1 + t-1(7). Use the to transform the variable^Yt** = Yt - Yt Yt = 0 + 1 Xt + t
32Cochrane-Orcutt Iterative procedure(Cont.) (8). Run OLS onYt** = 0** + 1** Xt** + t**Where is^(Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + (t - t-1)(9). Check on the DW3 -statistic, if the autocorrelation is still existing, than go into third-round procedures and so on.Until the estimated ’s differs a little^( - < 0.01).
33Example: Studenmund (2006) Exercise 14 and Table 9.1, pp.342-344 (1)Low DW statisticObtain theResiduals(Usually after you run regression, the residuals will be immediately stored in this icon
34(2)Give a new namefor the residual seriesRun regression of the current residual on the lagged residualObtain the estimated ρ(“rho”)^
35(3)Transform the Y* and X*New series are created,but each first observationis lost.
36(4)Run thetransformedregressionObtain the estimated resultwhich is improved
37The is the EVIEWS’ Command to run the iterative procedure (5)~(9)The Cochrane-Orcutt Iterative procedure in the EVIEWSThe is the EVIEWS’ Command to run the iterative procedure
38The result of the Iterative procedure This is theestimated ρEachvariableistransformedThe DWis improved
42Including thislagged term of YObtain the estimatedρ(“rho”)^
43Limitation of Durbin-Watson Test: Lagged Dependent Variable and AutocorrelationYt = 0 + 1 X1t + 2 X2t + …… + k Xk.t + 1 Yt-1 +tDW statistic will often be closed to 2 orDW does not converge to 2 (1 - )^DW is not reliableDurbin-h Test:Compute h* = ^1 - n*Var (1)nCompare h* to Z where Zc ~ N (0,1) normal distributionIf |h*| > Zc => reject H0 : = 0 (no autocorrelation)
44Durbin-h Test: n ^ Compute h* = 1 - n*Var (1) h* = 4.458 > Z Therefore rejectH0 : = 0 (no autocorrelation)