Reading Assignment. Serial Correlation and Heteroskedasticity. Chapters 12 and 11. Kennedy: Chapter 8. AREC-ECON 535 Lec F1 1

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1 Reading Assignment Serial Correlation and Heteroskedasticity Chapters 1 and 11. Kennedy: Chapter 8. AREC-ECON 535 Lec F1 1

2 Serial Correlation or Autocorrelation y t = β 0 + β 1 x 1t + β x t β k x kt + e t where E(e t e s ) 0 for t s. Common problem with samples over time. Positive Serial Correlation Negative Serial Correlation e t e t t t AREC-ECON 535 Lec F1

3 Suppose y t = β 0 + β 1 x 1t + β x t β k x kt + e t e t = ρ e t-1 + u t 0 ρ < 1 where E(e t ) = 0 E(u t ) = 0 E(e t ) = σ E(u t ) = σ u E(e t e t-s ) 0 E(u t u t-s ) = 0. This is a first-order autoregressive error. We are assuming that we can find a process that explains the serial correlation. (Also need the errors to be stationary ) AREC-ECON 535 Lec F1 3

4 e t = ρ e t-1 + u t so V(e t ) = σ = V(ρ e t-1 ) + V(u t ) = ρ V(e t-1 ) + V(u t ) = ρ σ + σ u σ = σ u / (1 - ρ ) Constant and function of σ u and ρ. Further, what does the magnitude of ρ imply about V(e t )? Cov(e t, e t-s ) = ρ s σ = ρ s σ u / (1 - ρ ) Non-zero. This violates an OLS assumption. Further, what does the magnitude of ρ imply about Cov(e t,e t-s )? AREC-ECON 535 Lec F1 4

5 Why does serial correlation occur? Dependent variable is a dynamic process. Inertia, lagged adjustments, lagged expectations. Explicit dynamic models are improving... Specification Bias: missing variable. If missing variable follows dynamic process... Specification Bias: Incorrect functional form. Consequences of Serial Correlation ˆ are unbiased but not efficient. Variances of OLS estimators are biased and inconsistent. Positive serial correlation: ˆ too small, t and F-tests too large. Negative serial correlation: ˆ too large, t and F-tests too small. AREC-ECON 535 Lec F1 5

6 Detection of Serial Correlation 1) Graph the residuals against time. ) Durbin-Watson d test H 0 : ρ = 0 DW-d = (e t - e t-1 ) / e t DW-d = (1 - ρ) or ρ = 1 - (d/) If ρ Then DW-d 0 4 AREC-ECON 535 Lec F1 6

7 The exact distribution of DW-d under H 0 is unknown because DW-d depends on e t which depends on the X's which are different for every model. Upper and lower critical values. 0 d l d u 4 Table values of d l and d u depend on α, k, and T. This is only important from the standpoint of econometric history AREC-ECON 535 Lec F1 7

8 Econometric research says use d u you have serial correlation unless you fail to reject the H 0. Further, some computer programs will calculate exact critical values. Comments on DW-d statistic: Assumes first-order autoregressive error. Invalid without intercept and with lagged dependent variable. If DW-d < R then regression results are spurious. Spurious?: If you have y t = y t-1 + u 1t and x t = x t-1 + u t then run y t = β 0 + β 1 x t + e t you would think the slope would be close to zero. That s not the case. The t-statistics have a 60-80% chance of being significant and the R approaches 1. This is a digression but that s trouble. However, if you run y t = β 0 + β 1 x t + u t then you get the anticipated result. The slope is close to zero, the t-statistic is small, and the R is low. AREC-ECON 535 Lec F1 8

9 3) Breusch-Godfrey (LM) test (addresses all DW-d limitations) e t = ρ 1 e t-1 + ρ e t ρ p e t-p + u t H 0 : ρ 1 = ρ =... = ρ p = 0 Steps of test: a) Run OLS regression and obtain ê t b) Regress ê t on X s in model and eˆ t 1,..., eˆ t p c) Obtain R of this artificial regression. If sample size is large (T - p)r ~ χ p ex) (3-5) = 3.38 > = χ 5 (5%) AREC-ECON 535 Lec F1 9

10 d) Econometric research shows an F-test on the auxiliary regression has better small sample properties. H 0 : ρ 1 = ρ =... = ρ p = 0 Comments: Examine t-statistics on auxiliary regression. Use several lags lengths of autoregressive error. AREC-ECON 535 Lec F1 10

11 Correcting for Serial Correlation Procedure: transform the error term into a random variable that meets OLS assumption. i.e., E (e t e s ) = 0 for t s. Generalized Differencing: when ρ is known. y t = β 0 + β 1 x t + e t and e t = ρ e t-1 + u t y t = β 0 + β 1 x t + ρ e t-1 + u t y t = β 0 + β 1 x t + ρ (y t-1 - β 0 - β 1 x t-1 ) + u t (y t - ρy t-1 ) = (β 0 - ρβ 0 ) + β 1 (x t - ρ x t-1 ) + u t y t = β 0 + β 1 x t + e t Run OLS on this model. ˆ are BLUE. So, if we knew ρ we could fix the problem. AREC-ECON 535 Lec F1 11

12 Cochrane-Orcutt procedure y t = β 0 + β 1 x t + e t and e t = ρ e t-1 + u t a) Estimate ˆ 0 and ˆ 1 via OLS using y t and x t. b) Calculate ê t using ˆ 0 and ˆ 1. c) Estimate ˆ via OLS using ê t and ê t. d) Calculate y t and x t using ˆ y t = (y t - ˆ y t-1 ) and x t = (x t - ˆ x t-1 ). e) Estimate ˆ 0 and ˆ 1 via OLS using y t and x t. Repeat until estimate of ρ converges. We are replacing an unknown parameter with a consistent estimate of that unknown parameter and then iterating the estimates should improve. EViews: y t = β 0 + β 1 x t + ρ e t-1 + u t AREC-ECON 535 Lec F1 1

13 Comment on generalized differencing. Lose one observation. This is not efficient. Prais-Winsten transformation for first observation. y 1 = y 1 (1 - ρ ) x 1 = x 1 (1 - ρ ) e 1 = e 1 (1 - ρ ) Most programs will do this. Use with caution first observation could be influential. If P-W improves efficiency then how should the results β s and V(β) s change? What would they do if the observation is influential? AREC-ECON 535 Lec F1 13

14 Maximum Likelihood Approach The Cochrane-Orcutt procedure is an Estimated Generalized Least Squares (EGLS) method and relies on a consistency argument. We can consistently estimate β and then ρ and σ u. Alternatively, we can use maximum likelihood where all parameters are estimated simultaneously. The likelihood function looks like (1 )( y x) ln L ln ln ln(1 ) T 1 1 [( yt yt 1) ( xt xt 1) ] ln lnu t u u u and unlike the ML example earlier there is no closed for expressions for the set of parameters β, σ u, and ρ. Changing one parameter will change others. So we have to use nonlinear methods. Does your software do this? There are also several ways to simplify the Hessian in the optimization problem. Which does your software do? AREC-ECON 535 Lec F1 14

15 Example OLS: NYSE Index t = CPI t + e t (Std err) (.39) (0.1969) (p-value) (0.0001) (0.0001) R = 0.913, F-test = , P-Value = , σ = DW-d = where d l = and d u = EGLS: NYSE Index t = CPI t + e t (Std err) (5.06) (0.4136) (P-value) (0.0001) (0.0001) e t = e t-1 + u t (0.0305) (0.0001) R = , F = , P-Value = , σ = AREC-ECON 535 Lec F1 15