Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time

Introductory Econometrics Based on the textbook by Wooldridge: Introductory Econometrics: A Modern Approach Robert M. Kunst [email protected] University of Vienna and Institute for Advanced Studies Vienna

January 14, 2013

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time

Outline

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time-series regression Basic issues Testing for autocorrelation Generalized least squares OLS with corrected standard errors

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time Basic issues

OLS with serially correlated errors Presume that assumptions on uncorrelated errors are violated, i.e. corr(ut , us ) 6= 0 for some s, t: ◮

◮

◮

As long as assumptions TS.1–TS.3 are fulfilled, OLS remains unbiased (Remember that TS.3 fails to hold in dynamic regressions); If TS.1′ –TS.3′ hold, OLS remains consistent (Remember that TS.3′ holds in a well-specified dynamic regression); Assumption TS.5 is violated, typically OLS is inefficient, and indicated variance estimates are incorrect.

Thus, in static regressions, ‘autocorrelation’ (correlated errors) has consequences similar to heteroskedasticity. In a dynamic regression model, however, autocorrelation usually makes OLS inconsistent. Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time Basic issues

An important element: AR(1) processes A simple model for autocorrelation is provided by the first-order autoregressive or AR(1) process yt = φ0 + φ1 yt−1 + εt , with εt uncorrelated and homoskedastic (‘white noise’) and |φ1 | < 1. This process is stationary, and its autocovariances C (h) = φh1 σy2 decrease fast with increasing h. For φ1 = 0, the AR(1) becomes uncorrelated white noise. The excluded case φ1 = 1 is the non-stationary random walk. The model is easily generalized to higher orders AR(p). Autocorrelation may of course follow more complex patterns. Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time Basic issues

The prototypical example: autoregression with AR errors Consider the dynamic regression model yt = β0 + β1 yt−1 + ut , with |β1 | < 1, and ut = ρut−1 + εt with |ρ| < 1 and εt serially uncorrelated. With these assumptions, ut and yt are stationary. It is easily shown that Pn (yt − y¯+ )(yt−1 − y¯ ) β1 + ρ cov(yt , yt−1 ) = βˆ1 = t=2Pn−1 → 6= β1 2) 2 1 + β ρ var(y (y − y ¯ ) 1 t t t=1

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time Basic issues

Remarks on dynamic regression with AR errors ◮

◮

These models violate TS.3′ due to dynamic misspecification. They are simply the wrong models for the data. In the example, inserting the regressor yt−2 leads to correct dynamic specification, to valid TS.3′ and TS.5, and to consistent OLS; Dynamic misspecification often but not always implies inconsistency. For example, yt = β0 + β1 yt−1 + ut with ut = ρut−2 + εt fulfils TS.3′ , and OLS becomes consistent.

It makes sense to aim at correct dynamic specification. This aim requires good tests for autocorrelation.

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time Testing for autocorrelation

The Durbin-Watson test Durbin and Watson (1950,1951) analyzed the small-sample distribution of the statistic Pn (ˆ ut − uˆt−1 )2 DW = t=2Pn ˆt2 t=1 u for OLS residuals uˆt under the null of TS.1–TS.5 with normal ut . This statistic is sensitive to deviations from the null ρ = 0 in the sense of yt = Xt′ β + ut ,

ut = ρut−1 + εt ,

|ρ| < 1,

with TS.1–TS.4 valid also under the alternative ρ 6= 0.

Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time Testing for autocorrelation

Some properties of the Durbin-Watson statistic DW It generally holds that 0 ≤ DW ≤ 4, such that the distribution always has a bounded support on [0, 4]. Because of DW ≈ 2(1 − ρˆ), with ρˆ an estimate of ρ, ◮

DW ∈ (0, 2) indicates positive autocorrelation ρ > 0;

◮

The ideal value DW = 2 indicates absence of autocorrelation;

◮

DW ∈ (2, 4) indicates negative autocorrelation ρ < 0.

As n → ∞, DW will converge to 2 under the null ρ = 0. With normality, one might expect tables of significance points depending on n and an exact test. Introductory Econometrics

University of Vienna and Institute for Advanced Studies Vienna

Heteroskedasticity Regressions with time-series observations Asymptotics of OLS in time-series regression Serial correlation in time Testing for autocorrelation

The distribution of DW under the null Unfortunately, the DW test is not similar, its null distribution depends on more than just n, changes with the regressor matrix X . It is customary to consider tabulated upper and lower bounds dU and dL for the significance points. ρ>0

ρ