Serial Correlation and Heteroscedasticity in Time series Regressions Econometric (EC3090) - Week 11 Agustín Bénétrix 1
Properties of OLS with serially correlated errors OLS still unbiased and consistent if errors are serially correlated Correctness of R-squared also does not depend on serial correlation OLS standard errors and tests will be invalid if there is serial correlation OLS will not be efficient anymore if there is serial correlation 2
Serial correlation and the presence of lagged dependent variables Is OLS inconsistent if there are serially correlated and lagged dependent variables? No: Including enough lags so that weak dependence holds guarantees consistency Including too few lags will cause an omitted variable problem and serial correlation because some lagged dependent variable end up in the error term 3
Efficiency and Inference In the presence of serial correlation the standard errors and test statistics are not valid (even asymptotically) 4
Efficiency and Inference Let s compute the variance of the OLS estimator in the presence of serial correlation.. Assume that the error term follows an AR(1) process : are uncorrelated random variables with mean zero and variance 5
Efficiency and Inference Consider the variance of the OLS slope estimator in the simple regression model The OLS estimator of can be written as where 6
Efficiency and Inference The variance of conditional on X is given by is the variance of under the Gauss- Markov assumptions when 7
Serial correlation in the presence of lagged dependent variables Suppose we have and assume that for stability The expression above can always be written as an error term as This satisfies the key assumption of no perfect collinearity so and are consistent 8
Serial correlation in the presence of lagged dependent variables However, the error term can be serially correlated The condition ensures that is uncorrelated with but it does not ensure that and are uncorrelated The covariance between and is which is not necessarily zero 9
Serial correlation in the presence of lagged dependent variables Thus, the errors exhibit serial correlation and the model contains a lagged dependent variable, but OLS consistently estimates and because these are the parameters in the conditional expectation The serial correlation in the errors will cause the usual OLS statistics to be invalid for testing purposes, but it will not affect consistency 10
Testing for serial correlation Testing for AR(1) serial correlation with strictly exogenous regressors AR(1) model for serial correlation (with an i.i.d. series e t ) Test in Replace true unobserved errors by estimated residuals 11
Example: Static Phillips curve Reject null hypothesis of no serial correlation 12
Durbin-Watson test under classical assumptions Under the 6 time series assumptions the Durbin- Watson test is an exact test (previous t-test is only valid asymptotically) vs. Reject if, "Accept" if Unfortunately, the Durbin-Watson test works with a lower and and an upper bound for the critical value. In the area between the bounds the test result is inconclusive. 13
Testing for serial correlation Testing for AR(1) serial correlation with general regressors The t-test for autocorrelation can be easily generalized to allow for the possibility that the explanatory variables are not strictly exogenous: Test for The test now allows for the possibility that the strict exogeneity assumption is violated The test may be carried out in a heteroscedasticity robust way 14
Testing for serial correlation Breusch-Godfrey test for AR(q) serial correlation 15
Correcting for serial correlation with strictly exogenous regressors Under the assumption of AR(1) errors, one can transform the model so that it satisfies all Gauss- Markov assumptions. For this model, OLS is BLUE Simple case of regression with only one explanatory variable. The general case works analogously Lag and multiply by 16
Correcting for serial correlation with strictly exogenous regressors The transformed error satisfies the GMassumptions Problem: The AR(1)-coefficient is not known and has to be estimated 17
Correcting for serial correlation with strictly exogenous regressors Replacing the unknown by leads to a FGLSestimator There are two variants: Cochrane-Orcutt estimation omits the first observation Prais-Winsten estimation adds a transformed first observation In smaller samples, Prais-Winsten estimation should be more efficient 18
Comparing OLS and FGLS with autocorrelation For consistency of FGLS more than the strict exogeneity assumption is needed because the transformed regressors include variables from different periods If OLS and FGLS differ dramatically this might indicate violation of strict exogeneity 19
Serial correlation-robust inference after OLS In the presence of serial correlation, OLS standard errors overstate statistical significance because there is less independent variation One can compute serial correlation-robust standard errors after OLS This is useful because FGLS requires strict exogeneity and assumes a very specific form of serial correlation (AR(1) or, generally, AR(q)) 20
Serial correlation-robust inference after OLS Serial correlation-robust standard errors: The usual OLS standard errors are normalized and then inflated by a correction factor Serial correlation-robust F- and t-tests are also available 21
Correction factor for serial correlation (Newey-West formula) : this term is the product of the residuals and the residuals of a regression of x tj on all other explanatory variables 22
Correction factor for serial correlation (Newey-West formula) The integer g controls how much serial correlation is allowed g=2: g=3: The weight of higher order autocorrelations is declining 23
Discussion of serial correlation-robust standard errors The formulas are also robust to heteroscedasticity, they are therefore called "heteroscedasticity and autocorrelation consistent" (HAC) For the integer g, values such as g=2 or g=3 are normally sufficient (there are more involved rules of thumb for how to choose g) Serial correlation-robust standard errors are only valid asymptotically; they may be severely biased if the sample size is not large enough 24
Discussion of serial correlation-robust standard errors The bias is the higher the more autocorrelation there is if the series are highly correlated, it might be a good idea to difference them first Serial correlation-robust errors should be used if there is serial correlation and strict exogeneity fails (e.g. in the presence of lagged depependent variables) 25
Heteroscedasticity in time series regressions Heteroscedasticity usually receives less attention than serial correlation Heteroscedasticity-robust standard errors also work for time series Heteroscedasticity is automatically corrected for if one uses the serial correlation-robust formulas for standard errors and test statistics 26
Testing for heteroscedasticity The usual heteroscedasticity tests assume absence of serial correlation Before testing for heteroscedasticity one should therefore test for serial correlation first, using a heteroscedasticity-robust test if necessary After serial correlation has been corrected for, test for heteroscedasticity 27
Example: Serial correlation and homoscedasticity in the EMH Test equation for the EMH Test for serial correlation: No evidence for serial correlation 28
Example: Serial correlation and homoscedasticity in the EMH Test for heteroscedasticity: Strong evidence for heteroscedasticity Note: Volatility is higher if returns are low 29
Autoregressive Conditional Heteroscedasticity (ARCH) Even if there is no heteroscedasticity in the usual sense (the error variance depends on the explanatory variables), there may be heteroscedasticity in the sense that the variance depends on how volatile the time series was in previous periods: ARCH(1) model 30
Consequences of ARCH in static and distributed lag models If there are no lagged dependent variables among the regressors, i.e. in static or distributed lag models, OLS remains BLUE under TS.1-TS.5 Also, OLS is consistent etc. for this case under assumptions TS.1 -TS.5 In this case, the assumption of homokedasticity still holds under ARCH 31
Consequences of ARCH in dynamic models In dynamic models, i.e. models including lagged dependent variables, the homoscedasticity assumption will necessarily be violated: because 32
Consequences of ARCH in dynamic models This means that the error variance indirectly depends on explanatory variables In this case, heteroscedasticity-robust standard error and test statistics should be computed, or a FGLS/WLS-procedure should be applied Using a FGLS/WLS-procedure will also increase efficiency 33
Example: Testing for ARCH-effects in stock returns Are there ARCH-effects in these errors? 34
Example: Testing for ARCH-effects in stock returns Estimating equation for ARCH(1) model There are statistically significant ARCH-effects: If returns were particularly high or low (squared returns were high) they tend to be particularly high or low again, i.e. high volatility is followed by high volatility 35
A FGLS procedure for serial correlation and heteroscedasticity Given or estimated model for heteroscedasticity: Model for serial correlation: 36
A FGLS procedure for serial correlation and heteroscedasticity Estimate transformed model by Cochrane- Orcutt or Prais-Winsten techniques (because of serial correlation in transformed error term) 37