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9.1 Lecture #9 Studenmund (2006) Chapter 9 Objectives The nature of autocorrelation The consequences of autocorrelation Testing the existence of autocorrelation Correcting autocorrelation

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9.2 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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9.3 Decomposition of time series Trend random Cyclical or seasonal XtXt time X t = Trend + seasonal + random

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9.4 Static Models C t = 0 + 1 Yd t + 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 inflat t = 0 + 1 unemploy t + t inflat: inflation rate unemploy: unemployment rate

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9.5 Lag Finite Distributed Lag Models Forward Distributed Lag Effect (with order q) Effect at time t+2 Economic action at time t Effect at time t C t = 0 + 0 Yd t + t Effect at time t+1 C t+1 = 0 + 0 Yd t+1 + 1 Yd t + t C t = 0 + 0 Yd t + 1 Yd t-1 + t Effect at time t+q …. C t+q = 0 + 1 Y d t+q +…+ 1 Y d t + t C t = 0 + 1 Y d t +…+ 1 Y d t-q + t

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9.6 Economic action at time t Effect at time t-1 Backward Distributed Lag Effect Y t = 0 + 0 Z t + 1 Z t-1 + 2 Z t-2 +…+ 2 Z t-q + t Initial state: z t = z t-1 = z t-2 = c Effect at time t-q …. Effect at time t-3 Effect at time t-2

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9.7 0 1 2 C = 0 + 0 Yd t + 1 Yd t-1 + 2 Yd t-2 + t 0 + 1 + 2 Long-run propensity (LRP) = ( 0 + 1 + 2 ) permanent Permanent unit change in C for 1 unit permanent (long-run) change in Yd. Distributed Lag model in general: C t = 0 + 0 Yd t + 1 Yd t-1 +…+ q Yd t-q + other factors + t LRP (or long run multiplier) = 0 + q

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

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9.9 Definition: First-order of Autocorrelation, AR(1) If Cov ( t, s ) = E ( t s ) 0 where t s Y t = 0 + 1 X 1t + t t = 1,……,T and if t = t-1 + u t RHO where -1 < < 1 ( : RHO) and u t ~ iid (0, u 2 ) (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 (RHO) : the first-order autocorrelation coefficient or ‘coefficient of autocovariance’

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u 1990 … … … ….... … … … u u u u u u 2007 Year Consumption t = 0 + 1 Income t + error t Example of serial correlation: TaxPay 2006 TaxPay 2007 Error term represents other factors that affect consumption u t ~ iid(0, u 2 ) u TaxPay 2007 = TaxPay u 2007 u t = t-1 + u t The current year Tax Pay may be determined by previous year rate

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

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9.12 time 0 ii ^ x x x x x x x x x Positive autocorrelation time 0 ii ^ x x x x x x x x Positive autocorrelation time0 ii ^ Cyclical: Positive autocorrelation x x x x x x x x x x x x x x x x The current error term tends to have the same sign as the previous one.

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9.13 Negative autocorrelation time ii ^ x x x x x x x x x x x x x x No autocorrelation x x x x x x x x x x x x x x x x x x x x x x x 0 time x x x ii ^ The current error term tends to have the opposite sign from the previous. The current error term tends to be randomly appeared from the previous.

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9.14 The meaning of 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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9.15 The consequences of serial correlation: 3. The standard error of the estimated coefficient, Se( k ) becomes large ^ ^ no longer the smallest 2. The variances of the k is no longer the smallest Therefore, when AR(1) is existing in the regression, The estimation will not be “BLUE” BLUE still unbiased 1.The estimated coefficients are still unbiased. E( k ) = k ^

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9.16 If E( t t-1 ) 0, and t = t-1 + u t, then If = 0, zero autocorrelation, than Var( 1 ) AR1 = Var( 1 ) ^ ^ > If 0, autocorrelation, than Var( 1 ) AR1 > Var( 1 ) ^ ^ Two variable regression model: Y t = 0 + 1 X 1t + t The OLS estimator of 1, ^ x y xt2 xt2 If E( t t-1 ) = 0 then Var ( 1 ) = ^ 22 xt2 xt2 ===> 1 = Var ( 1 ) AR1 = + + 2 ^ 2 2 2 x t x t+1 x t x t+2 x t 2 x t 2 -1 < < 1 + …. The AR(1) variance is not the smallest Example:

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9.17 t = t-1 + u t ==> t = [ t-2 + u t-1 ] + u t ==> t-2 = t-3 + u t-2 => t = 2 [ t-3 + u t-2 ] + u t-1 + u t ==> t-1 = t-2 + u t-1 t = 2 t-2 + u t-1 + u t t = 3 t-3 + 2 u t-2 + u t-1 + u t E( t t-1 ) = 22 1 - 2 E( t t-3 ) = 2 2 E( t t-2 ) = 2 ……………. E( t t-k ) = k-1 2 Autoregressive scheme: It means the more periods in the past, the less effect on current period k-1 becomes smaller and smaller

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9.18 How to detect autocorrelation ? DW* or d*

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9.195% level of significance, k = 1 k = 1,n=24 DW* = d L = 1.27 d u = 1.45 k k is the number of independent variables (excluding the intercept) < d L DW* < d L

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9.20 From OLS regression result: where d or DW * = Check DW Statistic Table (At 5% level of significance, k’ = 1, n=24) d L = 1.27 d u = dLdL dudu DW DW * Durbin-Watson Autocorrelation test Reject H 0 region H 0 : no autocorrelation = 0 H 1 : yes, autocorrelation exists. or > 0 positive autocorrelation

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9.21 Durbin-Watson test OLS : Y = 0 + 1 X 2 + …… + k X k + t obtain t, DW-statistic(d) ^ Assuming AR(1) process: t = t-1 + u t I. H 0 : ≤ 0 no positive autocorrelation H 1 : > 0 yes, positive autocorrelation -1 < < 1 dd L d u Compare d * and d L, d u (critical values) DW * dd L if d * reject H 0 d u if d * > d u ==> not reject H 0 d L dd u if d L d * d u ==> this test is inconclusive

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9.22 Durbin-Watson test(Cont.) -1 Since -1 1 ^ d implies 0 d 4 DW = 2 (1 - ) ( t - t-1 ) 2 t=2 T ^^ t2 t2 t=1 T ^ ^ d(d)d(d) d d 2(1 ) ^ d d ≈ 2 (1- ) ==> ≈ 1 - ==> ≈ 1- ^ d 2 d 2 ^ ^ dLdL dudu 4 (4-d L ) (4-d L ) (4-d U )

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9.23 Durbin-Watson test(Cont.) II. H 0 : ≥0 no negative autocorrelation H 1 : < 0 yes, negative autocorrelation d we use (4-d) (when d is greater than 2) if (4 - d) < d L or 4 - d L reject H 0 if d L (4 - d) d u or 4 - d u > d > 2 ==> not reject H 0 if d L (4 - d) d u or 4 - d u d 4 - d L ==> inconclusive dLdL dudu 4 (4 - d L ) (4-d U )

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9.24 Durbin-Watson test(Cont.) II. H 0 : =0 No autocorrelation H 1 : 0 two-tailed test for auto correlation either positive or negative AR(1) If d < d L or d > 4 - d L ==> reject H 0 If d u not reject H 0 If d L d d u or 4 - d u d 4 - d L ==> inconclusive

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9.25 For example : UM t = CAP t CAP t T t ^ (15.6) (2.0) (3.7) (10.3) DW = 0.23 R 2 = 0.78 F = 78.9 = SSR = 29.3 DW = 0.23 n = 68 _ ^ (i) K = 3 (number of independent variable) Observed (ii) n = 68, = 0.01 significance level 0.05 (iii) d L = 1.525, d u = d L = 1.372, d u = Reject H 0, positive autocorrelation exists (excluding intercept)

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9.26 H 0 : = 0 positive autocorrelation H 1 : > 0 0 dLdL dudu 2 reject H 0 not reject inconclusive DW (d) 4-d u 4-d L 4 inconclusive reject H 0 H 0 : = 0 negative autocorrelation H 1 : < 0 not reject % & 5% Critical values 0.23

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9.27 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 + u t where u t ~ iid (0, u 2 ) 4. Not include the lagged dependent variable, Y t = 0 + 1 X t 1 + 2 X t 2 + …… + k X t k + Y t-1 + t (autoregressive model) 5. No missing observation N.A. N.A. 82 N.A. N.A YX missing

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9.28 Lagrange Multiplier (LM) Test 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. (n-p)R 2 (3) compute the BG-statistic: (n-p)R 2 p (4) compare the BG-statistic to the 2 p (p is # of degree-order) (5) If BG > 2 p, reject Ho, it means there is a higher-order autocorrelation If BG < 2 p, not reject Ho, it means there is a no higher-order autocorrelation ^ ^^ ^ ^ ^^ ^ ^ (2) Run t against all the regressors in the model plus the additional regressors, t-1, t-2, t-3,…, t-p. p t = 0 + 1 X t + t-1 + t-2 + t-3 + … + t-p + u R 2 Obtain the R 2 value from this regression. ^

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9.29 Remedy: 1. First-difference transformation Y t = 0 + 1 X t + t Y t-1 = 0 + 1 X t-1 + t-1 assume = 1 ==> Y t - Y t-1 = 0 - 0 + 1 (X t - X t-1 ) + ( t - t-1 ) ==> Y t = 1 X t + t no intercept 2. Add a trend (T) Y t = 0 + 1 X t + 2 T + t Y t-1 = 0 + 1 X t-1 + 2 (T -1) + t-1 ==> (Y t - Y t-1 ) = ( 0 - 0 ) + 1 (X t - X t-1 ) + 2 [T- (T -1)] + ( t - t-1 ) ==> Y t = 1 X t + 2 *1 + ’ t ==> Y t = 2 * + 1 X t + ’ t If 1 * > 0 => an upward trend in Y ^ ( 2 > 0) ^

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9.30 Cochrane-Orcutt Two-step procedure 3. Cochrane-Orcutt Two-step procedure (CORC) (1). Run OLS on Y t = 0 + 1 X t + t and obtains t ^ (3). Use the to transform the variables : ^ Y t * = Y t - Y t-1 ^ ^ X t * = X t - X t-1 -) Y t-1 = 0 + 1 X t-1 + t-1 ^ ^ ^^ Y t = 0 + 1 X t + t (4). Run OLS on Y t * = 0 * + 1 * X t * + u t (2). Run OLS on t = t-1 + u t ^ and obtains ^ ^ Where u~(0, ) Generalized Least Squares Least Squares(GLS)method (Y t - Y t-1 )= 0 (1- ) + 1 (X t - X t-1 ) + ( t - t-1 ) ^ ^ ^ ^

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9.31 Cochrane-Orcutt 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 * + u t ’ ^ ^ (1 - ) DW 2 2 ^ ^ and obtains which is the second-round estimated ^ ^ X t ** = X t - X t-1 Y t-1 = 0 + 1 X t-1 + t-1 (7). Use the to transform the variable ^ ^ Y t ** = Y t - Y t-1 Y t = 0 + 1 X t + t ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

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9.32 Cochrane-Orcutt Iterative procedure(Cont.) (8). Run OLS on Y t ** = 0 ** + 1 ** X t ** + t ** Where is ^ ^ (Y t - Y t-1 ) = 0 (1 - ) + 1 (X t - X t-1 ) + ( t - t-1 ) ^ ^ ^ ^ ^ ^ (9). Check on the DW 3 -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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9.33 Example: Studenmund (2006) Exercise 14 and Table 9.1, pp (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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9.34(2) Give a new name for the residual series Run regression of the current residual on the lagged residual ρ Obtain the estimated ρ(“rho”) ^

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9.35(3) Transform the Y* and X* New series are created, but each first observation is lost.

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9.36(4) Obtain the estimated result which is improved Run the transformed regression

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

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9.38 The result of the Iterative procedure The DW is improved This is the ρ estimated ρ Each variable is transformed

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9.39 Generalized least Squares (GLS) Y t = 0 + 1 X t + t t = 1,……,T (1) Assume AR(1) : t = t-1 + u t -1 < < 1 Y t-1 = 0 + 1 X t-1 + t-1 (2) (1) - (2) => (Y t - Y t-1 ) = 0 (1 - ) + 1 (X t - X t-1 ) + ( t - t-1 ) GLS => Y t * = 0 * + 1 * X t * + u t 5. Prais-Winsten transformation

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9.40 To avoid the loss of the first observation, the first observation of Y 1 * and X 1 * should be transformed as : X 1 * = 1 - 2 (X 1 ) ^ Y 1 * = 1 - 2 (Y 1 ) ^ Edit the figure here To restore the first observation but Y 2 * = Y 2 - Y 1 ; X 2 * = X 2 - X 1 ^^ Y 3 * = Y 3 - Y 2 ; X 3 * = X 3 - X 2 ^^ …... Y t * = Y t - Y t-1 ; X t * = X t - X t-1 ^ ^

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

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

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9.43 Lagged Dependent Variable Lagged Dependent Variable and Autocorrelation Compare h * to Z where Z c ~ N (0,1) normal distribution If |h * | > Z c => reject H 0 : = 0 (no autocorrelation) Y t = 0 + 1 X 1 t + 2 X 2 t + …… + k X k.t + 1 Y t-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 ^ Limitation of Durbin-Watson Test:

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9.44 Durbin-h Test: Compute h * = ^ 1 - n*Var ( 1 ) n ^ Therefore reject H0 : = 0 (no autocorrelation) h* = > Z

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