Heteroskedasticity and Autocorrelation

Transcription

1 Heteroskedasticity and Autocorrelation Fall 2008 Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

2 Heteroskedasticity We now relax the assumption of homoskedasticity, while all other assumptions remain to hold. Heteroskedasticity is said to occur when the variance of the unobservable error u, conditional on independent variables, is not constant. Var (u i jx i ) = σ 2 i. In particular, the variance of the error may be a function of independent variables: Var (u i jx i ) = σ 2 h (X i ). Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

3 Consequences of Heteroskedasticity First, note that we do not need the homoskedasticity asssumption to show the unbiasedness of OLS. Thus, OLS is still unbiased. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

4 Consequences of Heteroskedasticity First, note that we do not need the homoskedasticity asssumption to show the unbiasedness of OLS. Thus, OLS is still unbiased. However, the homoskedasticity assumption is needed to show the e ciency of OLS. Hence, OLS is not BLUE any longer. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

5 Consequences of Heteroskedasticity First, note that we do not need the homoskedasticity asssumption to show the unbiasedness of OLS. Thus, OLS is still unbiased. However, the homoskedasticity assumption is needed to show the e ciency of OLS. Hence, OLS is not BLUE any longer. The variances of the OLS estimators are biased in this case. Thus, the usual OLS t statistic and con dence intervals are no longer valid for inference problem. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

6 Consequences of Heteroskedasticity First, note that we do not need the homoskedasticity asssumption to show the unbiasedness of OLS. Thus, OLS is still unbiased. However, the homoskedasticity assumption is needed to show the e ciency of OLS. Hence, OLS is not BLUE any longer. The variances of the OLS estimators are biased in this case. Thus, the usual OLS t statistic and con dence intervals are no longer valid for inference problem. We can still use the OLS estimators by nding heteroskedasticity-robust estimators of the variances. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

7 Consequences of Heteroskedasticity First, note that we do not need the homoskedasticity asssumption to show the unbiasedness of OLS. Thus, OLS is still unbiased. However, the homoskedasticity assumption is needed to show the e ciency of OLS. Hence, OLS is not BLUE any longer. The variances of the OLS estimators are biased in this case. Thus, the usual OLS t statistic and con dence intervals are no longer valid for inference problem. We can still use the OLS estimators by nding heteroskedasticity-robust estimators of the variances. Alternatively, we can devise an e cient estimator by re-weighting the data appropriately to take into account of heteroskedasticity. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

8 How Does the Heteroskedasticity Look? Consider the following true regression model with heteroskedastic errors: Y i = 1 + 2X i + u i, where u i N 0, Xi 2 Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

9 How Does the Heteroskedasticity Look? Consider the following true regression model with heteroskedastic errors: Y i = 1 + 2X i + u i, where u i N 0, Xi 2 y/linear prediction x Environmental Econometrics (GR03) yhetero - Autocorr Linear prediction Fall / 17

10 How Does the Heteroskedasticity Look? Alternatively, we can graph the residuals bu i with X i. Is there a constant spread across values of X? Residuals x Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

11 Testing for Heteroskedasticity: Breusch-Pagan Test Assume that heteroskedasticity is of the linear form of independent variables: σ 2 i = δ 0 + δ 1 X i1 + + δ k X ik. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

12 Testing for Heteroskedasticity: Breusch-Pagan Test Assume that heteroskedasticity is of the linear form of independent variables: σ 2 i = δ 0 + δ 1 X i1 + + δ k X ik. The hypotheses are H 0 : Var (u i jx i ) = σ 2 and H 1 : not H 0. The null can be written H 0 : δ 1 = = δ k = 0. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

13 Testing for Heteroskedasticity: Breusch-Pagan Test Assume that heteroskedasticity is of the linear form of independent variables: σ 2 i = δ 0 + δ 1 X i1 + + δ k X ik. The hypotheses are H 0 : Var (u i jx i ) = σ 2 and H 1 : not H 0. The null can be written H 0 : δ 1 = = δ k = 0. Since we never know the actual errors in the population model, we use their estimates, bu i, which is the OLS residual: bu 2 i = δ 0 + δ 1 X i1 + + δ k X ik + v i. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

14 Testing for Heteroskedasticity: Breusch-Pagan Test Assume that heteroskedasticity is of the linear form of independent variables: σ 2 i = δ 0 + δ 1 X i1 + + δ k X ik. The hypotheses are H 0 : Var (u i jx i ) = σ 2 and H 1 : not H 0. The null can be written H 0 : δ 1 = = δ k = 0. Since we never know the actual errors in the population model, we use their estimates, bu i, which is the OLS residual: bu 2 i = δ 0 + δ 1 X i1 + + δ k X ik + v i. A form of the Breusch-Pagan test is constructed as BP test: N R 2 bu 2 a X 2 k. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

15 Testing for Heteroskedasticity: White Test The White test is explicitly intended to test for forms of heteroskedasticity: the relation of u 2 with all independent variables (X i ), the squares of th independent variables Xi 2, and all the cross products (X i X j for i 6= j). Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

16 Testing for Heteroskedasticity: White Test The White test is explicitly intended to test for forms of heteroskedasticity: the relation of u 2 with all independent variables (X i ), the squares of th independent variables Xi 2, and all the cross products (X i X j for i 6= j). Just as we did in the Breusch-Pagan test, we regress bu i on all the above variables and compute the Rbu 2 and construct the statistic of 2 same form. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

17 Testing for Heteroskedasticity: White Test The White test is explicitly intended to test for forms of heteroskedasticity: the relation of u 2 with all independent variables (X i ), the squares of th independent variables Xi 2, and all the cross products (X i X j for i 6= j). Just as we did in the Breusch-Pagan test, we regress bu i on all the above variables and compute the Rbu 2 and construct the statistic of 2 same form. The abundance of independent variables is a weakness in the pure form of the White test. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

18 Heteroskedasticity-Robust Standard Errors Consider the simple regression model, Y i = β 0 + β 1 X i + u i, and allow heteroskedasticity. Then, note that the variance of bβ 1 is Var b β 1 jx = N i=1 X i X 2 σ 2 i n N i=1 X i X o 2 2. White (1980) suggested the following: Get the OLS residual bu i. Get a valid estimator of Var b β 1 jx : \ Var b β 1 jx = N i =1 X i X 2 bu 2 i n N i =1 X i X 2 o 2. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

19 Generalized Least Squares Estimation If we correctly specify the form of the variance, then there exists a more e cient estimator (Generalized Least Squares, GLS) than OLS. Suppose the true model is: Y i = β 0 + β 1 X i + u i, Var (u i jx ) = σ 2 i. Suppose we know exactly the form of heteroskedasticity. Then we divide each term of the equation by σ i : Y i /σ i = β 0 /σ i + β 1 X i /σ i + u i /σ i Yi = β 0 + β 1 X i + ui, Var (ui jx ) = 1 Perform the OLS regression of Yi bβ GLS 1 = N i=1 Xi N i=1 on X i : X Y i X i X 2 Y In GLS, less weight is given the observations with a higher error variance. Obviously, GLS is unbiased and, indeed, is BLUE. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

20 Feasible GLS The problem is we usually do not know the form of variance, σ i. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

21 Feasible GLS The problem is we usually do not know the form of variance, σ i. Instead of σ i, we can use bσ i in the GLS estimation, called the Feasible GLS (FGLS) estimator. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

22 Feasible GLS The problem is we usually do not know the form of variance, σ i. Instead of σ i, we can use bσ i in the GLS estimation, called the Feasible GLS (FGLS) estimator. Run the OLS regression to get the residuals, bu i. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

23 Feasible GLS The problem is we usually do not know the form of variance, σ i. Instead of σ i, we can use bσ i in the GLS estimation, called the Feasible GLS (FGLS) estimator. Run the OLS regression to get the residuals, bu i. Model the relation of errors with independent variables: σ 2 i = f (X i ) Estimate bσ i using the following OLS regression: bu 2 i = f (X i ) + v i Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

24 Feasible GLS The problem is we usually do not know the form of variance, σ i. Instead of σ i, we can use bσ i in the GLS estimation, called the Feasible GLS (FGLS) estimator. Run the OLS regression to get the residuals, bu i. Model the relation of errors with independent variables: σ 2 i = f (X i ) Estimate bσ i using the following OLS regression: bu 2 i = f (X i ) + v i The feasible GLS estimator is where X i bβ FGLS 1 = N i =1 X i N i =1 = X i /bσ i and Y i = Y i /bσ i. X Yi Xi X 2 Y Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17,

25 Autocorrelation The error terms are said to be autocorrelated if and only if Cov (u i, u j ) 6= 0, for i 6= j. (Time Series Data) The error term at one date can be correlated with the error terms in the previous periods: Autoregressive process of order k = 1, 2,..., AR (k) : u t = ρ 1 u t 1 + ρ 2 u t ρ k u t k + v t. Moving average process of order k = 1, 2,..., MA (k) : u t = v t + λ 1 v t λ k v t k. (Cross-section Data) The error terms may be correlated with each other in terms of socio and geographical distance such as the distance between towns and neighborhood e ects. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

26 Consequences of Autocorrelation Assuming all other assumptions remain to hold, under the condition of autocorrelation, the OLS estimator is still unbiased. the OLS is not BLUE any more. The usual OLS standard errors and test statitics are no longer valid. We can nd an autocorrelation-robust estimator of the variance after we perform the OLS regression. Alternatively, we can devise an e cient estimator by re-weighting the data appropriately to take into account of autocorrelation. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

27 Autocorrelation-robust Standard Errors In order to correct the unknown form of autocorrelation in the error terms, Newey and West (1987) suggested \ Var b β 1 jx = 1 n N t=1 X t X 2 o 2 ( N bu t 2 X t X 2 t=1 + L N l=1 t=l+1 w l bu t bu t l X t X X t l X ), where w l = 1 l L + 1. The correlation between u t and u t l is approximated with l 1 L+1 but bu t 1. The above standard error is also robust to arbitrary heteroskedasticity. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

28 Testing for Autocorrelation: AR(1) We start to test the presence of AR(1) serial correlation: u t = ρu t 1 + e t. Then, the nulll hypothesis that the errors are serially uncorrelated is In order to test the null, H 0 : ρ = 0. run the OLS regression of Y t on X t1,..., X tk to obtain the OLS residuals, bu t run the regression of bu t on bu t 1 for all t = 2,..., N to estimate bρ construct the t statistic. The t test is not valid if Cov (u t 1, X tj ) 6= 0, for some j. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

29 Testing for Autocorrelation: AR(1) Another test for AR(1) serial correlation is the Durbin-Watson test: based on the OLS residuals, d = N t=2 (bu t bu t 1 ) 2 N t=1 bu 2 t = 2 (1 r) + bu2 1 + bu2 N N t=1 bu 2 t ' 2 (1 bρ), where r = N t=2 bu t bu t 1 N. t=1 bu t 2 The DW test works as follows with two critical values, d L and d U : d 2 [d U, 4 d U ] =) not reject the null; either d d L or d 4 d L =) reject the null; d 2 (d L, d U ) or d 2 (4 d U, 4 d L ) =) inconclusive The DW test is also not valid if Cov (u t 1, X tj ) 6= 0, for some j. An important case is the regression with a lagged dependent variable: Y t = β 0 + β 1 X t + β 2 Y t 1 + u t, u t = ρu t 1 + e t Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

30 Testing for Autocorrelation: Breusch-Godfrey Test The Breusch-Godfrey(BG) test is more general and test for higher order serial correlation, AR (q): u t = ρ 1 u t 1 + ρ 2 u t ρ q u t q + v t. Then, the null hypothesis is In order to construct the BG test, H 0 : ρ 1 = = ρ q = 0. run the OLS regression of Y t on X t1,..., X tk to obtain the OLS residuals, bu t regress bu t on X t1,..., X tk, bu t 1,..., bu t k compute the F statistic Alternatively, compute (N degrees of freedom. q) Rbu 2, which follows the chi-square with q Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17

31 GLS under Autocorrelation Consider the following model: Y t = β 0 + β 1 X t + u t, u t = ρu t 1 + v t Assuming we know this relation, we can rewrite Y t ρy t 1 = β 0 (1 ρ) + β 1 (X t ρx t 1 ) + v t and run the OLS regression. If ρ is not known, then we can do the feasible GLS in the following way: run the OLS with the original equation to get bu t run the regression: bu t = ρbu t 1 + w t, to get bρ transform the model using bρ and run OLS. Environmental Econometrics (GR03) Hetero - Autocorr Fall / 17