7. Autocorrelation (Violation of Assumption #B3)


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1 7. Autocorrelation (Violation of Assumption #B3) Assumption #B3: The error term u i is not autocorrelated, i.e. Cov(u i, u j ) = 0 for all i = 1,..., N and j = 1,..., N where i j Where do we typically find autocorrelation: Timeseries data sets (less frequently in crosssectional data sets) 169
2 Outlook: Strong similarities between heteroskedasticity and autocorrelation with respect to consequences and estimation procedures (GLS, FGLS estimators) Example: [I] Estimation of a pricerevenue function (monthly data) Variables: y i = monthly revenue quantity (in 1000 pieces) x i = selling price (in euros) 170
3 Month Obs. Price Revenue Month Obs. Price Revenue 01: : : : : : : : : : : : : : : : : : : : : : : : Dependent Variable: REVENUE Method: Least Squares Date: 11/17/04 Time: 13:50 Sample: 2002: :12 Included observations: 24 Variable Coefficient Std. Error tstatistic Prob. C PRICE Rsquared Mean dependent var Adjusted Rsquared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Fstatistic DurbinWatson stat Prob(Fstatistic)
4 Example: [II] Linear model with a single regressor: y i = α + β x i + u i (i = 1,..., 24) Price increase of 1 euro decreases the monthly revenue by pieces 172
5 Revenue Price
6 Evidently: Line connecting the residuals rarely crosses the regression line we frequently find that positive u i 1 values are followed by positive u i values negative u i 1 values are followed by negative u i values Cov(u i 1, u i ) 0 (violation of Assumption #B3) Question: Impact on estimation and testing procedures 174
7 7.1 Consequences Note: We assume an explicit pattern of autocorrelation (alternative patterns are not considered here) Definition 7.1: (AR(1) process) Let u 1,..., u N be the error terms of the linear regression model. Furthermore, let ρ R be a (constant) parameter and let e 1,..., e N denote additional error terms that satisfy all Bassumptions (#B1 #B4). If u i = ρu i 1 + e i, (i = 2,..., N) we say that the error term u i follows a firstorder autoregressive process (in symbols: u i AR(1)). 175
8 Remarks: An AR(1) process regresses u i on its predecessor value u i 1 plus the new random shock e i For ρ = 1 or ρ = 1 we have socalled random walks (important stochastic processes) For ρ > 1 processes become explosive in this lecture: 1 < ρ < 1 Now: Expected values, (co)variances, correlation coefficients of an AR(1) process 176
9 Theorem 7.2: (Moments of an AR(1) process) Let the error term u i (i = 1,..., N) follow an AR(1) process according to Definition 7.1 where 1 < ρ < 1. Furthermore, let Var(e i ) σ 2 e denote the constant variance of all e i. We then have for all (admissible) i = 1,..., N: (Proof: class) E(u i ) = 0, Var(u i ) = σ2 e 1 ρ 2 σ2, Cov(u i, u i τ ) = ρ τ σ 2 e 1 ρ 2 = ρτ σ 2 0, Corr(u i, u i τ ) = ρ τ. 177
10 Obviously: If the error term u i follows an AR(1) process with 1 < ρ < 1, then the Assumptions #B1, #B2 are satisfied whereas #B3 is violated Now: Autocorrelation in matrix notation (u i s follow an AR(1) process) Notation: u = u 1 u 2. u N, u 1 = u 0 u 1. u N 1, e = e 1 e 2. e N 178
11 Matrix representation: [I] Linear regression model y = Xβ + u with AR(1) error terms ( 1 < ρ < 1) u = ρu 1 + e Theorem 7.2 yields Cov(u) 179
12 Matrix representation: [II] Due to σ 2 σ 2 e /(1 ρ 2 ) we obtain Cov(u) = σ 2 Cov(u 1, u 2 ) Cov(u 1, u N ) Cov(u 2, u 1 ) σ 2 Cov(u 2, u N )... Cov(u N, u 1 ) Cov(u N, u 2 ) σ 2 = σ 2 ρσ 2 ρ N 1 σ 2 ρσ 2 σ 2 ρ N 2 σ 2... ρ N 1 σ 2 ρ N 2 σ 2 σ 2 = σ 2 Ω 180
13 Matrix representation: [II] where Ω = 1 ρ ρ N 1 ρ 1 ρ N 2... ρ N 1 ρ N
14 Question: Is there any transformation of the autocorrelated model so that the parameter vector β remains unchanged autocorrelation vanishes the transformed model y = X β + u satisfies all #A, #B, #C assumptions? (cf. Section 6, Slide 111) 182
15 Hope: If yes, then the OLS estimator of the transformed model (the GLS estimator) would be BLUE (cf. Section 6, Slides ) Result: In analogy to the line of argument given on Slides under heteroskedasticity the following result obtains: there exists a regular matrix P so that the transformed model Py = PXβ + Pu satisfies all #A, #B, #Cassumptions 183
16 Form of P in the autocorrelated model: [I] P has to satisfy the following equations: P P = Ω 1 and PΩP = I N (see Slides 117, 120) First, the inverse of Ω from Slide 181 is given by Ω 1 = 1 1 ρ 2 (check it) 1 ρ ρ 1 + ρ 2 ρ ρ. 1 + ρ ρ 2 ρ ρ 1 184
17 Form of P in the autocorrelated model: [II] The form of P is given by P = 1 1 ρ 2 1 ρ ρ ρ ρ 1 185
18 Form of P in the autocorrelated model: [III] transformed model: y = X β + u where y = Py = X = PX = 1 ρ 2 u 1 e 2. 1 ρ 2 y 1 y 2 ρy 1, u = Pu =. y N ρy N 1 e N 1 ρ 2 1 ρ 2 x 11 1 ρ 2 x K1 1 ρ. x 12 ρx 11.. x K2 ρx K1. 1 ρ x 1N ρx 1(N 1) x KN ρx K(N 1) 186
19 Remarks: The transformed model y = X β + u satisfies all #A, #B, #Cassumptions The parameter vector β remains unchanged consequences of autocorrelation parallel those of heteroskedasticity 187
20 Consequences of autocorrelation: [I] The OLS estimator β = ( X X ) 1 X y is still unbiased, but no longer BLUE (cf. Theorem 6.1, Slide 109) The covariance matrix of the OLS estimator is given by Cov ( β ) = σ 2 ( X X ) 1 X ΩX ( X X ) 1 The GLS estimator is BLUE β GLS = [ X X ] 1 X y = [ X Ω 1 X ] 1 X Ω 1 y 188
21 Consequences of autocorrelation: [II] Its covariance matrix is given by ( Cov β GLS) = σ 2 [ X Ω 1 X ] 1 (cf. Theorem 6.3, Slide 123) Unbiased estimator of σ 2 : ˆσ 2 = û û N K 1 = (Pû) Pû N K 1 189
22 Impact of neglecting autocorrelation: [I] OLS estimator of β is unbiased, but inefficient β = ( X X ) 1 X y The estimator ˆσ 2 (X X) 1 of the covariance matrix Cov ( β ) is biased The estimator ˆσ 2 û û = N K 1 of the errorterm variance is biased 190
23 Impact of neglecting autocorrelation: [II] test statistics are based on biased estimators hypothesis tests are likely to be unreliable (t, F tests) 191
24 7.2 Diagnostics Graphical analysis: [I] First, estimation of the model y = Xβ + u by OLS, i.e. β = ( X X ) 1 X y calculation of the residuals û = y X β 192
25 Graphical analysis: [II] Plot of the residuals versus time slow swings around zero positive autocorrelation fast swings around zero negative autocorrelation Scatterplot of û i 1 versus û i positive slope positive autocorrelation negative slope negative autocorrelation Example: [I] Pricerevenue function on Slides
26 Dependent Variable: Revenue Method: Least Squares Date: 11/17/04 Time: 13:50 Sample: 2002: :12 Included observations: 24 Variable Coefficient Std. Error tstatistic Prob. C PRICE Rsquared Mean dependent var Adjusted Rsquared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Fstatistic DurbinWatson stat Prob(Fstatistic) OBS RESID(1) RESID
27 Residuals :01 02:04 02:07 02:10 03:01 03:04 03:07 03:10 i = 1,..., u i u i 1
28 Obviously: Positive dependence between û i 1 and û i indication of positive autocorrelation a conceivable specification of an AR(1) errorterm process could be with 0 < ρ < 1 u i = ρu i 1 + e i 196
29 Now: Use the pair of residuals (û i 1, û i ) to estimate ρ Model û i = ρû i 1 + e i (i = 2,..., N) OLS estimator of ρ: ˆρ = N i=2 (ûi 1 û ) ( û i û ) N (ûi 1 û ) 2 = N i=2 N û i 1 û i i=2 û 2 i 1 i=2 197
30 Remarks: For the sum of the residuals computed via OLS we always have N i=1 û i = 0 and thus û = 1 N (cf. Von Auer, 2007, p. 57) N i=1 û i = 0 Since for i = 1 there is no residual û i 1 = û 0, we only have i = 2,..., N observations 198
31 OLS estimate of ρ in the pricerevenue example: ˆρ = Dependent Variable: RESID Method: Least Squares Date: 11/20/04 Time: 19:15 Sample(adjusted): 2002: :12 Included observations: 23 after adjusting endpoints Variable Coefficient Std. Error tstatistic Prob. RESID(1) Rsquared Mean dependent var Adjusted Rsquared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood DurbinWatson stat Question: Is ρ significantly different from zero? DurbinWatson test for autocorrelation 199
32 DurbinWatson test: [I] Most popular test for autocorrelation (due to Durbin & Watson, 1950, 1951) Tests for both, positive (ρ > 0) and negative autocorrelation (ρ < 0) Test statistic: calculate the residuals û i via OLS test statistic: DW = N i=2 (ûi û i 1 ) 2 N / û 2 i i=1 200
33 DurbinWatson test: [II] Relation to the OLS estimator ˆρ from Slide 197: DW 2(1 ˆρ) Properties: Since 1 < ρ < 1 it follows that 0 < DW < 4 no autocorrelation: ˆρ 0 DW 2 positive autocorrelation: ˆρ 1 DW 0 negative autocorrelation: ˆρ 1 DW 4 201
34 DurbinWatson test: [III] Test for positive autocorrelation: hypotheses: H 0 : ρ 0 versus H 1 : ρ > 0 distribution of DW under H 0 depends on sampling size (N) number of exogenous regressors (K) specific values of the regressors x 1i,..., x Ki exact calculation by econometric software 202
35 DurbinWatson test: [IV] distribution under H 0 of DW has lower and upper bounds exact critical values at the αlevel (d α ) have lower and upper bounds (i.e. d L α d α d U α) (for α = 0.05 see Von Auer, 2007, p. 402) explicit decision rule: reject H 0 : ρ 0 if DW < d L α do not reject H 0 : ρ 0 if DW > d U α no decision if d L α DW d U α 203
36 DurbinWatson test: [V] Test for negative autocorrelation: hypotheses: H 0 : ρ 0 versus H 1 : ρ < 0 explicit decision rule: Reject H 0 : ρ 0 if DW > 4 d L α Do not reject H 0 : ρ 0 if DW < 4 d U α No decision if 4 d U α DW 4 dl α 204
37 Example: Estimation of the pricerevenue function (Slides 170, 171) Test for positive autocorrelation at the 5% level: H 0 : ρ 0 versus H 1 : ρ > 0 We have N = 24, K = 1, DW = , d0.05 L = 1.27, du 0.05 = 1.45 and thus DW = < 1.27 = d0.05 L reject H 0 at the 5%level 205
38 Drawbacks of the DurbinWatson test: Frequently there is no decision (e.g. if DW [d L α, d U α]) when testing for pos. autocorrelation) DWTest is unreliable if predecessor values like y i 1, y i 2,... are used as regressors (socalled lagmodels) DWtest only tests for AR(1)autocorrelation 206
39 7.3 Feasible Estimation Procedures Now: Estimation of the autocorrelated model y = Xβ + u with AR(1) errorterms u = ρu 1 + e ( 1 < ρ < 1) 207
40 Problem: From the data set (X, y) we do not have direct knowledge about the autocorrelation parameter ρ FGLS estimation Two feasible estimation procedures: GLS approach (Hildreth & Lu) FGLS approach (Cochrane & Orcutt) 208
41 1. Method by Hildreth & Lu: [I] Search algorithm Consider the GLS estimator β GLS = [ X Ω 1 X ] 1 X Ω 1 y where Ω 1 = 1 1 ρ 2 (cf. Slides 184, 188) 1 ρ ρ 1 + ρ 2 ρ ρ. 1 + ρ ρ 2 ρ ρ 1 209
42 1. Method by Hildreth & Lu: [II] Perform GLS estimation for distinct ρvalues ( 1 < ρ < 1) compute the sum of squared residuals û û for each estimation Find the ρvalue with the minimal sum of squared residuals GLS estimator of β associated with this ρvalue is called HildrethLu estimator 210
43 Example: Data on pricerevenue function ρ u * ' u * α β HildrethLu estimates: ˆα HL = , ˆβ HL =
44 2. Method by Cochrane & Orcutt: [I] Iterative multistep procedure Procedure: 1. OLS estimation of the model y = Xβ + u 2. Save the residuals û = y X β 212
45 2. Method by Cochrane & Orcutt: [II] 3. Consider the resgression û = ρû 1 + e and estimate ρ by the OLS estimator ˆρ = N û i 1 û i i=2 N û 2 i 1 i=2 4. Use ˆρ to apply the FGLS estimator [ β FGLS = X Ω 1 1 X] X Ω 1 y 213
46 2. Method by Cochrane & Orcutt: [III] 5. Improvement due to iteration: Compute the new residuals û (2) FGLS = y X β Reestimate ρ as in Step #3 Find the new FGLS estimator β FGLS(2) Repeat Step #5 until the FGLS estimator of β does not exhibit any further (substantial) change 214
47 Example: [I] Consider the pricerevenue example Estimate OLS estimate Iteration #1 Iteration #2 Iteration #3 ρ α β CochraneOrcutt estimates: ˆα CO = , ˆβ CO = (possibly further iterations) 215
48 Example: [II] Contrasting both estimation results: Parameter OLS estimate HildrethLu CochraneOrcutt
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