Econometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Transcription

1 Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall / 23

2 Recommended Reading For the today Serial correlation and heteroskedasticity in time series regressions. Chapter 12 (pp ). For the next week Pooling cross sections across time. Simple panel data methods. Chapter 13 (pp ). 2 / 23

3 Properties of OLS with Serially Correlated Errors: Unbiasedness and Consistency OLS is unbiased under the first 3 Gauss-Markov assumptions for time series regression. But we do not assume anything about serial correlation present often in the economic data. As long as explanatory variables are strictly exogenous, ˆβ OLS are unbiased, regardless the degree of serial correlation in the errors. Last lecture, we also relaxed the strict exogeneity to E(u t x t ) = 0 and by assuming weak dependence of the data, we have shown that ˆβ OLS are still consistent (although not necessarily unbiased). But what about assumption on serial correlation? 3 / 23

4 Properties of OLS with Serially Correlated Errors: Efficiency and Inference Gaus-Markov theorem requires both homoskedasticity and serially uncorrelated errors. Thus, OLS is no longer BLUE in the presence of serial correlation....and standard errors and test statistics are not valid. Let s assume model with AR(1) errors: y t = β 0 + β 1 x t + u t, u t = ρu t 1 + ɛ t, for t = 1, 2,..., n, where ρ < 1 and ɛ t uncorrelated random variables with zero mean and variance σ 2 ɛ. 4 / 23

5 Properties of OLS with Serially Correlated Errors: Efficiency and Inference cont. The OLS estimator is then: where SST x = n t=1 x2 t ˆβ 1 = β 1 + SST 1 x n x t u t, t=1 5 / 23

6 Properties of OLS with Serially Correlated Errors: Efficiency and Inference cont. Variance of ˆβ 1 conditional on X is: ( n ) V ar( ˆβ 1 ) = SSTx 2 V ar x t u t = SST 2 x t=1 n n 1 n t x 2 t V ar(u t ) + 2 x t x t+j E(u t u t+j ) t=1 = σ 2 /SST x }{{} variance of ˆβ 1 t=1 j=1 n 1 n t + 2(σ 2 /SSTx 2 ) ρ j x t x t+j t=1 j=1 } {{ } bias where σ 2 = V ar(u t ) and we used the fact from last lecture E(u t u t+j ) = Cov(u t, u t+j ) = ρ j σ 2 If we ignore the serial correlation and estimate the variance in the usual way, variance estimator will be biased (as ρ 0) 6 / 23

7 Properties of OLS with Serially Correlated Errors: Efficiency and Inference cont. Consequences: In most economic applications, ρ > 0 and usual OLS variance underestimates the true variance of the OLS We tend to think that OLS slope estimator is more precise than it actually is. Main consequence is that standard errors are invalid t statistics for testing single hypotheses are invalid statistical inference is invalid. 7 / 23

8 Testing for AR(1) Serial Correlation We need to be able to test for serial correlation in the error terms in the multiple linear regression model: y t = β 0 + β 1 x t1 +,... + β k x tk + u t, with u t = ρu t 1 + ɛ t, t = 1, 2,... n. The null hypothesis is that there is no serial correlation. H 0 : ρ = 0 With strictly exogenous regressors, the test is very straightforward - simply regress the OLS residuals û t on lagged residuals û t 1 t statistics of ˆρ coefficient can be used to test H 0 : ρ = 0 against H A : ρ 0 (or sometimes even H A : ρ > 0) 8 / 23

9 Testing for AR(1) Serial Correlation cont. An alternative is the Durbin-Watson (DW) statistic: DW = n t=2 (û t û t 1 ) 2 n. t=1 û2 t DW 2(1 ˆρ). ˆρ 0 DW 2. ˆρ > 0 DW < 2. The DW is little problematic, we have 2 sets of critical values, d L (lower) and d U (upper): DW < d L reject the H 0 : ρ = 0 in favor of H A : ρ > 0. DW > d U fail to reject the H 0 : ρ = 0. d L DW d U the test is inconclusive. 9 / 23

10 Testing for AR(1) Serial Correlation cont. In case we do not have strictly exogenous regressors (one or more x tj is correlated with u t 1 ), t test nor DW test does not work. In this case, we can regress û t on x t1, x t2,... x tk, û t 1 for all t = 2,..., n. t statistics of ˆρ coefficient of û t 1 can be used to test the null of no serial correlation. The inclusion of x t1, x t2,... x tk explicitly allows each x tj to be correlated with u t 1 no need for strict exogeneity. 10 / 23

11 Testing for Higher Order Serial Correlation We can easily extend the test for second order (AR(2)) serial correlation. In the model y t = ρ 1 u t 1 + ρ 2 u t 2 + ɛ t, we test the H 0 : ρ 1 = 0, ρ 2 = 0 We regress û t on x t1, x t2,... x tk, û t 1, û t 2 for all t = 3,..., n...and obtain F test for joint significance of û t 1 and û t 2. If they are jointly significant, we reject the null errors are serially correlated of order two. 11 / 23

12 Testing for Higher Order Serial Correlation cont. We can include q lags to test high order serial correlation. Regress û t on x t1, x t2,... x tk, û t 1, û t 2,..., û t q for all t = (q + 1),..., n. Use F test to test joint significance of û t 1, û t 2,..., û t q Or use LM version of test Breusch-Godfrey test: LM = (n q)r 2 û, where R 2 û is usual R2 from the regression above. Under the null hypothesis, LM a χ 2 q. 12 / 23

13 Correcting for Serial Correlation When correlation is detected, we need to treat it. We know that OLS may be inefficient. So how do we obtain BLUE estimator in the AR(1) setting? We assume all 4 Gauss-Markov Assumptions, but we relax Assumption 5 and assume errors to follow AR(1): u t = ρu t 1 + ɛ t, t = 1, 2,... V ar(u t ) = σ 2 ɛ /(1 ρ 2 ) We need to transform the regression equation so we have no serial correlation in the errors. 13 / 23

14 Correcting for Serial Correlation cont. Consider following regression: For t 2, we can write: y t = β 0 + β 1 x t + u t, u t = ρu t 1 + ɛ t y t 1 = β 0 + β 1 x t 1 + u t 1, y t = β 0 + β 1 x t + u t By multiplying first equation by ρ and subtracting it from second equation, we get: ỹ t = (1 ρ)β 0 + β 1 x t + ɛ t where ỹ t = y t ρy t 1 and x t = x t ρx t 1 This is called quasi-differencing. BUT we never know the value of ρ 14 / 23

15 Feasible GLS Estimation with AR(1) Errors The problem with this GLS estimator is that we never know the value of ρ. But we already know how to obtain the ρ estimator: Simply regress the OLS residuals on their lagged values and get ˆρ. Feasible GLS (FGLS) Estimation with AR(1) Errors Run the OLS regression of y t on x t1,..., x tk and obtain residuals û t, t = 1, 2,..., n. Run the regression of û t on û t 1 to obtain estimate ˆρ. Run OLS equation: ỹ t = β 0 x t0 + β 1 x t β k x t + error t, where x t0 = (1 ˆρ), x t1 = x t ρx t 1 for t 2, and x t0 = (1 ρ 2 ) 1/2, x t1 = (1 ρ 2 ) 1/2 x 1 for t = 1 15 / 23

16 Feasible GLS Estimation with AR(1) Errors GLS is BLUE under Assumptions 1 5 and we can use t and F tests from the transformed equation for the inference. These tests are asymptotically valid if Ass.1 5 hold in transformed model (along with stationary and weak dependence in the original variables) Distributions conditional on X are exact (with minimum variance) if Ass 6. holds fro ɛ t. FGLS estimator is called Prais-Winsten estimator If we just omit first equation (t = 1), it is called Cochrane-Orcutt estimator. FGLS estimators are not unbiased, but are consistent. Asymptotically, both procedures are the same and FGLS is more efficient than OLS. This method can be extended for higher order serial correlation, AR(q) in the error term. 16 / 23

17 Serial Correlation-Robust Standard Errors Problem: If the regressors are not strictly exogenous, FGLS is no longer consistent. If strict exogeneity does not hold, it s possible to calculate serial correlation (and heteroskedasticity) robust standard errors of OLS estimate. We know that OLS will be inefficient. The idea is to scale OLS standard errors to take into account serial correlation. 17 / 23

18 Serial Correlation-Robust Standard Errors cont. Estimate the model with OLS to obtain residuals û t, ˆσ and the usual standard errors se( ˆβ 1 ), which are incorrect. Run the auxiliary regression of x t1 on x t2, x t3,..., x tk (with constant) and get residuals ˆr t. For a chosen integer g > 0 (typically integer part of n 1/4 ): ( n g n ) ˆν = â 2 t + 2 [1 h/(g + 1)] â t â t h, t=1 h=1 where â t = ˆr t û t, t = 1, 2,..., n. Serial Correlation-Robust Standard Error se( ˆβ 1 ) = [ se( ˆβ 1 ) /ˆσ] 2 ˆν t=h+1 Similarly for ˆβ j. SC robust standard errors can poorly behave in small samples in presence of large serial correlation. 18 / 23

19 Heteroskedasticity in Time Series Regressions OLS estimators are unbiased (with Ass. 1-3) and consistent (Ass. 1A-3A). OLS inference is invalid, if Ass.4 (homoskedasticity) fail. Heteroskedasticity-robust statistics can be easily derived in the same manner as in cross-sectional data (if Ass. 1A,2A,3A and 5A hold). However, in small samples we know that these robust standard errors may be large. we want to test for heteroskedasticity. We can use the same tests as for the cross-sectional case, but we need to have no serial correlation in the errors. Also for the Breusch-Pagan test where we specify u 2 t = δ 0 + δ 1 x t δ k x tk + ν t and test H 0 : δ 1 = δ 2 =... = δ k = 0, we need ν to be homoskedastic and serially uncorrelated. If we find heteroskedasticity, we can use heteroskedasticity robust statistics. 19 / 23

20 Autoregressive Conditional Heteroskedasticity Many times, we find dynamic form of the heteroskedasticity in economic data. We can have E[u 2 t X] = V ar(u t X) = V ar(u t ) = σ 2, but still: E[u 2 t X, u t 1, u t 2,...] = E[u 2 t X, u t 1 ] = α 0 + α 1 u 2 t 1. Thus u 2 t = α 0 + α 1 u 2 t 1 + ν t, where E[ν X, u t 1, u t 1,...] = 0. Engle (1982) suggested looking at the conditional variance of u t given past errors - autoregressive conditional heteroskedasticity (ARCH) model. So even when the errors are not correlated (Ass. 5 holds), its squares can be correlated. OLS is still BLUE with ARCH errors and inference is valid if Ass. 6 (normality) holds. Even if Normality does not hold, we know that asymptotically OLS inference is valid under Ass 1A 5A and we can have ARCH effects. 20 / 23

21 Autoregressive Conditional Heteroskedasticity cont. So why do we need to care about ARCH errors? Because we can obtain asymptotically more efficient estimators than OLS. Details will be provided at Mgr. courses, not Bc. level. ARCH model have become important for empirical finance as it captures time-varying volatility in the stock markets. Rob Engle received a Nobel Prize in 2003 for it. Example of stock market returns on the next slide. 21 / 23

22 Autoregressive Conditional Heteroskedasticity cont. Prices of DJI stock market index Returns of DJI stock market index / 23

23 Thank you Thank you very much for your attention! 23 / 23