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# Autocorrelation

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### Autocorrelation

1. 1. What is autocorrelation? The term autocorrelation may be defined as correlation between members of series of observation ordered in time (as in time series data) or space (as in cross sectional data). Let us consider the general linear regression model isY=Xβ+ut One of the basic assumption of this model is that the error termut’s are mutually independent or uncorrelated, i.e. cov(ut+ut+s)≠0, for all t and t+s. but this assumption of uncorrelated error is not valid for certain cases such as in time series data where the successive error tends to the highly correlated i.e. cov(ut+ut+s)≠0, for all t≠s. There is a correlation between successive values of ut’s. these types of correlation is known as autocorrelation. Serial correlation Serial correlation is defined as ‘lag correlation’ between two different series. For example- the correlation between time series such as u1, u2,…..,u10 and v2,v3,…v11, where u and v are two different time series is called serial correlation. Distinguish between serial correlation and autocorrelation Auto Correlation Serial Correlation When the correlation occurs in same When the correlation occurs in series then the correlation is called different series then the correlation auto correlation is called serial correlation Auto correlation of a series with In serial correlation we find itself lagged by time unit correlation lagged between two different series Let us consider two time series Let us consider two time series datadatau1, u2,…..,u10 u1, u2,…..,u10 u2, u3,…..,u11 v2,v3,…v11 What is auto regression and what is auto regression model? Write down its assumptions. Auto regression: a regression is known as auto regression if one of the explanatory variables is the lagged value of the dependent variable.
2. 2. Auto regression model: Let Yt=β1+β2Xt+ut where t denotestheobservation at time t. one can assume that disturbance term are generalised as usersut=ρut-1+vt whereρ is known as coefficient of auto covariance and vt is the disturbance term. This model is known as AR(1) model. Because it can be interpreted as the regression of ut on itself lagged one period i.e. its immediate first value are involved. Assumption: 1. E[ut]=0 2. V[ut]=σu2 3. Cov[ut,ut-1]≠0 If you depend on the value of two successive periods then the linear relationship contain first and second order auto correlation coefficient. Hence the second order auto regressive model is given byut=ρut-1+vt ut-1=ρut-2+vt-1 Question: For an auto regressive model i.e. ut=ρut-1+vt 2 where, │ρ│<1 and v[vt]=σv Show that, i) ii) E[ut]=0 Var[ut]= σu2= iii) Cov[ut,ut-1]= i) = σu2 We are given, ut=ρut-1+vt………….(i) and v[vt]=σv2 now taking expectation in the equation (i) E[ut] =E[ρut-1+vt] =ρE[ut-1]+E[vt] =ρ.0+0 =0 ∴E[ut]=0
3. 3. ii) Var[ut]= V[ρut-1+vt] =ρ2var(ut-1)+var(vt) =ρ2var(ut-1)+σv2……………(ii) Var(ut)=1 and var(ut-1)=σu2 From equation (ii), Var[ut]=ρ2σu2+σv2 ⇒σu2= ρ2σu2+σv2 ⇒σu2- ρ2σu2=σv2 ⇒σu2(1-ρ2)=σv2 ⇒σu2= ∴Var[ut]= Question: find the first order auto regressive scheme or structure Or, derive the consequence of autocorreltation. Or, show that, first order autoregressive schemeu t= ρrvt-r ⇒to find the consiqence of autocorrelation let us consider a sample regression model with time t. Yt=β0+ β1Xt+ut……………………..(i) Where ut follows the first order auto regressive scheme, ut=ρut-1+vt whereρ is the coefficient of auto covariance. │ρ│≤1 i. e. -1≤ρ≤1 Vt is a random term; which fulfills all usual assumption of r. v i. e.E[vt]=0 var[vt]=E[vt,vt-r]=σv2, when r=0 and var[vt]==0 when r≠0 Now we can write, Ut-1=ρut-2+vt-1 Ut-2=ρut-3+vt-2 . . Ut-r=ρut-(r+1)+vt-r Now we perform continuous substitutions of lagged values of u in equation (i) as follows substitute ut-1 and obtain-
4. 4. Ut= ρ[ρut-2+vt-1] +vt =ρ2ut-2+ρvt-1+vt Again substitute ut-2, ut= ρ[ρut-2+vt-1] +vt What happens if the disturbance term are correlated? CLRM⇒cov(ui,uj│xi,xj) = E(ui,uj)=0 The Classical Linear Regression Model assumes that the disturbance term in any observation is not influenced by the disturbance term in any other observation. For example- if we are dealing with quarterly time series data involving the regression of output on labor and capital inputs. If say there is a labor strike affecting output in one quarter there is no reason to believe that this demonstration will be carried over to the next part i.e. if output is lower to the first quarter there is no reason to believe that it will be continue to lower next quarter. If we are dealing with cross sectional data involving the regression of family consumption expenditure on family income, the affect of an increase of one family’s income on its consumption expenditure is not expected to affect the consumption expenditure of another family.
5. 5. Consequence of autocorrelation: (1) When the disturbance terms (µ’s) are seriously correlated then the least square (OLS) estimate are unbiased but optimality property (M.V property) is not satisfy (2) If the disturbance term µ’s are autocorrelated then the OLS variance is greater than the variance of estimate calculated by other method then the usual t and F test of significance are no longer misleading conclusion about the estimate regression. (3) If the disturbance term are autocorrelated then the OLS estimate are non-asymptotic. (4) The variance of random term is may be seriously under estimated if the µi’s are autocorrelated. What remedial measures can be taken to alleviate autocorrelation problem? (1) Try to find out if the autocorrelation is pure correlation and not the result of mis-specification of the model. (2) If it is pure correlation one can use appropriate transformation of the original model so that in the transformation model we don’t have the problem of (Pure) autocorrelation as in the case of heteroscedasticity we will have to use some type of generalised least square model or GLS model. (3) In large samples we can use Newey-West method to obtain standard error of OLS estimators that are correlated from autocorrelation. This method is actually an extension of whites’ heteroscedasticity consistent