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**Lecture #9 Autocorrelation Serial Correlation**

Studenmund (2006) Chapter 9 Autocorrelation Serial Correlation Objectives The nature of autocorrelation The consequences of autocorrelation Testing the existence of autocorrelation Correcting autocorrelation

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**Time Series Data Time series process of economic variables**

e.g., GDP, M1, interest rate, exchange rate, imports, exports, inflation rate, etc. Realization An observed time series data set generated from a time series process Remark: Age is not a realization of time series process. Time trend is not a time series process too.

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**Cyclical or seasonal random**

Decomposition of time series Xt = Trend + seasonal + random Trend Xt time Cyclical or seasonal random

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**Example: Static Phillips curve model inflatt = 0 + 1unemployt + t**

Static Models Ct = 0 + 1Ydt + t Subscript “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 model inflatt = 0 + 1unemployt + t inflat: inflation rate unemploy: unemployment rate Contemporaneous relation: a. Equilibrium relation b. No dynamic effects, i.e., immediate effect only

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**Ct+1=0+0Ydt+1+1Ydt+tCt=0 +0Ydt+1Ydt-1+t**

Finite Distributed Lag Models Economic action at time t Effect Ct =0+0Ydt+t Effect at time t+1 Ct+1=0+0Ydt+1+1Ydt+tCt=0 +0Ydt+1Ydt-1+t Effect at time t+2 Effect at time t+q …. Forward Distributed Lag Effect (with order q) …. Ct+q=0+1Ydt+q+…+1Ydt+tCt=0+1Ydt+…+1Ydt-q+t

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**Backward Distributed Lag Effect**

Economic action at time t Effect at time t-1 Backward Distributed Lag Effect Effect at time t-2 Effect at time t-3 Effect at time t-q …. Yt= 0+0Zt+1Zt-1+2Zt-2+…+2Zt-q+t Initial state: zt = zt-1 = zt-2 = c

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**C = 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 + t LRP (or long run multiplier) = 0 + q

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**Time Trends Linear time trend**

Yt = 0 + 1t + t Constant absolute change Exponential time trend ln(Yt) = 0 + 1t + t Constant growth rate Quadratic time trend Yt = 0 + 1t + 2t2 + t Accelerate change For advances on time series analysis and modeling , welcome to take ECON 3670

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**Definition: First-order of Autocorrelation, AR(1)**

If Cov (t, s) = E (t s) where t s Yt = 0 + 1 X1t + t t = 1,……,T and if t = t-1 + ut where < < ( : 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 coefficient or ‘coefficient of autocovariance’

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**Example of serial correlation:**

u1990 … … … …. … … …. u2002 u2003 u2004 u2005 u2006 u2007 Year Consumptiont = 0 + 1 Incomet + errort Example of serial correlation: TaxPay2006 TaxPay2007 Error term represents other factors that affect consumption The current year Tax Pay may be determined by previous year rate TaxPay2007 = TaxPay u2007 t = t-1 + ut ut ~ iid(0, u2)

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If t = 1 t-1 + ut it is AR(1), first-order autoregressive If t = 1 t-1 + 2 t-2 + ut it is AR(2), second-order autoregressive High order autocorrelation If t = 1 t-1 + 2 t-2 + 3 t-3 + ut it is AR(2), third-order autoregressive If t = 1 t-1 + 2 t-2 + …… + n t-n + ut it is AR(n), nth-order autoregressive ………………………………………………. Autocorrelation AR(1) : Cov (t t-1) > => 0 < < positive AR(1) Cov (t t-1) < => -1 < < negative AR(1) -1 < < 1

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time i ^ x Positive autocorrelation time i ^ x Positive autocorrelation time i ^ Cyclical: Positive autocorrelation x The current error term tends to have the same sign as the previous one.

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**Negative autocorrelation**

time i ^ x The current error term tends to have the opposite sign from the previous. No autocorrelation x time i ^ The current error term tends to be randomly appeared from the previous.

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**The 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 < < 0 The further the period is in the past, the smaller is the weight of that error term (t-1) in determining t = 1 The past is equal importance to the current. > 1 The past is more importance than the current.

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**The consequences of serial correlation:**

The estimated coefficients are still unbiased. E(k) = k ^ BLUE ^ 2. The variances of the k is no longer the smallest 3. 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”

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**The AR(1) variance is not the smallest**

Example: Two variable regression model: Yt = 0 + 1X1t + t The OLS estimator of 1, ^ x y xt2 If E(t t-1) = 0 then Var (1) = 2 ===> 1 = If E(tt-1) 0, and t = t-1 + ut , then Var (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

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**Autoregressive 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 + ut t = 3 t-3 + 2 ut-2 + ut-1 + ut E(t t-1) = 2 1 - 2 E(t t-3) = 2 2 E(t t-2) = 2 ……………. E(t t-k) = k-1 2 It means the more periods in the past, the less effect on current period k-1 becomes smaller and smaller

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**How to detect autocorrelation ?**

DW* or d*

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**5% level of significance,**

k = 1, n=24 k is the number of independent variables (excluding the intercept) dL = 1.27 du = 1.45 DW* = DW* < dL

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**Durbin-Watson Autocorrelation test**

From OLS regression result: where d or DW* = Check DW Statistic Table (At 5% level of significance, k’ = 1, n=24) dL = 1.27 du = 1.45 1.27 1.45 2 dL du DW* 0.9107 Reject H0 region H0 : no autocorrelation = 0 H1 : yes, autocorrelation exists. or > 0 positive autocorrelation

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**Durbin-Watson test OLS : Y = 0 + 1 X2 + …… + k Xk + t**

obtain t , DW-statistic(d) ^ Assuming AR(1) process: t = t-1 + ut I H0 : ≤ no positive autocorrelation H1 : > yes, positive autocorrelation -1 < < 1 Compare d* and dL, du (critical values) DW* if d* < dL ==> reject H0 if d* > du ==> not reject H0 if dL d* du ==> this test is inconclusive

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**^ d 2(1-r) Durbin-Watson test(Cont.) DW = 2 (1 - ) (t - t-1)2**

==> ≈ 1 - ==> ≈ 1- ^ d 2 Since -1 1 ^ implies 0 d 4 1.27 1.45 2 dL du 4 (4-dL) (4-dU) 2.73 2.55

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**Durbin-Watson test(Cont.)**

II. H0 : ≥ no negative autocorrelation H1 : < yes, negative autocorrelation we use (4-d) (when d is greater than 2) if (4 - d) < dL or dL < d < ==> reject H0 if dL (4 - d) du or du d 4 - dL ==> inconclusive if dL (4 - d) du or du > d > ==> not reject H0 1.27 1.45 2 dL du 4 (4 - dL) (4-dU) 2.73 2.55

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**Durbin-Watson test(Cont.)**

II. H0 : = No autocorrelation H1 : two-tailed test for auto correlation either positive or negative AR(1) If d < dL or d > 4 - dL ==> reject H0 If du < d < 4 - du ==> not reject H0 If dL d du or 4 - du d 4 - dL ==> inconclusive

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For 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 level 0.05 (iii) dL = , du = dL = , du = Reject H0, positive autocorrelation exists (excluding intercept)

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**negative autocorrelation**

H0 : = 0 positive autocorrelation H1 : > 0 dL du 2 DW (d) 4-du 4-dL 4 inconclusive reject H0 H0 : = 0 negative autocorrelation H1 : < 0 not reject reject H0 not reject inconclusive 1% & 5% Critical values 0.23

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**The assumptions underlying the d(DW) statistics :**

1. Intercept term must be included. 2. X’s are nonstochastic 3. 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 observation N.A N.A. N.A N.A. 95 ... Y X missing

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**Lagrange 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 model plus the additional regressors, t-1, t-2, t-3,…, t-p. t = 0 + 1 Xt + t-1 + t-2 + t-3 + … + t-p + u Obtain 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 autocorrelation If BG < 2p, not reject Ho, it means there is a no higher-order autocorrelation

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**D D Remedy: 1. First-difference transformation Yt = 0 + 1 Xt + t**

Yt-1 = 0 + 1 Xt-1 + t assume = 1 ==> Yt - Yt-1 = 0 - 0 + 1 (Xt - Xt-1) + (t - t-1) ==> Yt = 1 DXt + t D no intercept 2. Add a trend (T) Yt = 0 + 1 Xt + 2 T + t Yt-1 = 0 + 1 Xt-1 + 2 (T -1) + t-1 ==> (Yt - Yt-1) = (0 - 0) + 1 (Xt - Xt-1) + 2 [T- (T -1)] + (t - t-1) ==> DYt = 1 DXt + 2*1 + ’t ==> Yt = 2* + 1 DXt + ’t D If 1* > 0 => an upward trend in Y ^ (2 > 0)

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**3. Cochrane-Orcutt Two-step procedure (CORC)**

(1). Run OLS on Yt = 0 + 1 Xt + t and obtains t ^ Generalized Least Squares (GLS) method (2). Run OLS on t = t-1 + ut ^ and obtains Where u~(0, ) (3). Use the to transform the variables : ^ Yt* = Yt - Yt-1 Xt* = Xt - Xt-1 Yt = 0 + 1 Xt + t -) Yt-1 = 0 + 1 Xt-1 + t-1 ^ (Yt - Yt-1)= 0(1-) +1(Xt - Xt-1) + (t -t-1) ^ (4). Run OLS on Yt* = 0* + 1* Xt* + ut

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**4. Cochrane-Orcutt Iterative Procedure**

(5). If DW test shows that the autocorrelation still existing, than it needs to iterate the procedures from (4). Obtains the t* (6). Run OLS t* = t-1* + ut’ ^ ( ) DW2 2 and 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

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**Cochrane-Orcutt Iterative procedure(Cont.)**

(8). Run OLS on Yt** = 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).

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**Example: Studenmund (2006) Exercise 14 and Table 9.1, pp.342-344**

(1) Low DW statistic Obtain the Residuals (Usually after you run regression, the residuals will be immediately stored in this icon

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(2) Give a new name for the residual series Run regression of the current residual on the lagged residual Obtain the estimated ρ(“rho”) ^

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(3) Transform the Y* and X* New series are created, but each first observation is lost.

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(4) Run the transformed regression Obtain the estimated result which is improved

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**The is the EVIEWS’ Command to run the iterative procedure**

(5)~(9) The Cochrane-Orcutt Iterative procedure in the EVIEWS The is the EVIEWS’ Command to run the iterative procedure

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**The result of the Iterative procedure**

This is the estimated ρ Each variable is transformed The DW is improved

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**Generalized least Squares (GLS) 5. Prais-Winsten transformation**

Yt = 0 + 1 Xt + t t = 1,……,T (1) Assume AR(1) : t = t-1 + ut < < 1 Yt-1 = 0 + 1 Xt-1 + t (2) (1) - (2) => (Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + (t - t-1) GLS => Yt* = 0* + 1* Xt* + ut

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To avoid the loss of the first observation, the first observation of Y1* and X1* should be transformed as : Edit the figure here To restore the first observation Y1* = 2 (Y1) ^ X1* = 2 (X1) ^ but Y2* = Y2 - Y1 ; X2* = X2 - X1 ^ Y3* = Y3 - Y2 ; X3* = X3 - X2 …... Yt* = Yt - Yt-1 ; Xt* = Xt - Xt-1

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**6. Durbin’s Two-step method :**

Since (Yt - Yt-1) = 0 (1 - ) + 1 (Xt - Xt-1) + ut => Yt = 0* + 1 Xt - 1 Xt-1 + Yt-1 + ut Yt = 0 + 1 Xt + t 6. Durbin’s Two-step method : I. Run OLS => this specification Yt = 0* + 1* Xt - 2* Xt-1 + 3* Yt-1 + ut Obtain 3* as an estimated (RHO) ^ II. Transforming the variables : Yt* = Yt - 3* Yt as Yt* = Yt - Yt-1 and Xt* = Xt - 3* Xt as Xt* = Xt - Xt-1 ^ III. Run OLS on model : Yt* = 0 + 1 Xt* + ’t and 1 = 1 ^ where 0 = 0 (1 - )

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Including this lagged term of Y Obtain the estimated ρ(“rho”) ^

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**Limitation of Durbin-Watson Test:**

Lagged Dependent Variable and Autocorrelation Yt = 0 + 1 X1t + 2 X2t + …… + k Xk.t + 1 Yt-1 +t DW statistic will often be closed to 2 or DW does not converge to 2 (1 - ) ^ DW is not reliable Durbin-h Test: Compute h* = ^ 1 - n*Var (1) n Compare h* to Z where Zc ~ N (0,1) normal distribution If |h*| > Zc => reject H0 : = 0 (no autocorrelation)

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**Durbin-h Test: n ^ Compute h* = 1 - n*Var (1) h* = 4.458 > Z**

Therefore reject H0 : = 0 (no autocorrelation)

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