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Autocorrelation Walter Sosa-Escudero Econ 471. Econometric Analysis. Spring 2009 April 23, 2009 Walter Sosa-Escudero Autocorrelation

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Time-Series Observations Consider the following model ··· Kt , ,t =1 ,...,T Here denotes periods, ,...,T . This is a model for time series observations. Example: consumption and income for a given country, in several periods. In time-series analysis the way observations are sorted is very important. Walter Sosa-Escudero Autocorrelation

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The no-serial correlation assumption is Cov ,u )=0 ,, t s, meaning that the error terms of two diﬀerent periods must be linearly unrelated. In the time-series context this assumption is known as no autocorrelation Remember that serial correlation invalidates the Gauss-Markov Theorem: OLS is still unbiasded (why?) but is not the best linear unbiased estimator. Walter Sosa-Escudero Autocorrelation

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A simple model for autocorrelation Consider, WLOG, the simple model for time series: , t =1 ,...,T and let the error term be speciﬁed by φu This structure is known as the linear model with ﬁrst-order autorregresive serial correlation. We will assume and that )=0 )= and Cov , )=0 , t =0 Walter Sosa-Escudero Autocorrelation

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A closer look at φu It is easy to see that the error term is linked over time, impliying the presence of serial correlation. is explicitely linked to its immediate past when =0 No autocorrelation in this setup: =0 φu is known as a ﬁrst-order autorregresive (AR(1)) process. Intuition: suppose φ> , then tends to be ‘close’ to its previous value . Example: wheather? The is a requisite for stability. What would happen is, say, φ> Walter Sosa-Escudero Autocorrelation

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Testing for AR(1) autocorrelation The Durbin-Watson test It is a test for =0 (no ﬁrst order autocorrelation) vs. =0 The Durbin-Watson test is based on the following statistic: DW =2 =1 where are OLS residuals. Walter Sosa-Escudero Autocorrelation

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Intuition behind DW DW =2 =1 DW =2 =1 =2 =1 =2 =1 Suppose that the number of observations is large. Then the ﬁrst term should be very close to one. Second term: same thing Third term: it can be shown that in the AR(1) structure Cor ,u )= Cov ,u Then the third term is nothing but an estimate of having replaced by Walter Sosa-Escudero Autocorrelation

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Then, the following approximation for the Durbin-Watson holds: DW =1+1 =2(1 Consider testing =0 vs φ> When =0 is true (no autocorrelation), DW When φ> DW < Then we should accept (no autocorrelation) when DW and reject (positive serial correlation) when DW is signiﬁcantly smaller than 2 Walter Sosa-Escudero Autocorrelation

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A problem with DW Consider =0 vs φ> (a test for positive autocorrelation). According to the previous intuition, we should reject if DW is signiﬁcantly smaller than 2. If we proceed as usual, we would reject if DW where is a critical value from the distribution of DVW Problem: the distribution of DW depends on the data used, it cannot be tabulated in general: we cannot get Solution: even though we cannot get , we can get bounds and such that Walter Sosa-Escudero Autocorrelation

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Then the procedure for a test for positive serial correlation is as follows If DW >d , then DW >d : accept If DW , then DW : reject If we cannot tell if DW or DW >d the test is inconclusive Walter Sosa-Escudero Autocorrelation

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Comments on the DW test It is a rather old-fashioned test. It requires a very special table. We need to assume normal errors. The model must include an intercept. It is crucial that the is non-stochastic. The Durbin-Watson procedure is a test for a particular form of serial correlation, the AR(1) process. It is not informative about more general patterns of autocorrelation. Walter Sosa-Escudero Autocorrelation

Page 12

More general autocorrelation: the Breusch-Pagan test Now consider the following model: , t =1 ,...,T (1) ··· (2) were satisﬁed the same assumptions as before This is the two-variable model with autocorrelation of order AR No autocorrelation ··· =0 The alternative hypothesis is =0 =0 ∨··· =0 Walter Sosa-Escudero Autocorrelation

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The Breusch-Pagan test for AR autocorrelation It is very similar to our test for heteroskedasticity Estimate by OLS, save residuals Regress on ,e ,...,e and , get of this auxiliar regression. Test statistic . Under , asymptotically, it has a distribution with degrees of freedom. Intuition: like regressing on ,...,u , but replacing the ’s by ’s. Under the of this auxiliar regression should be zero, and diﬀerent from zero under the alternative. Walter Sosa-Escudero Autocorrelation

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Comments: The Breusch-Pagan test does not require to be non-random It explores a more general pattern of serial correlation than the DW test, which also explores the AR(1) case. The choice of is problematic: intuition suggests to go for a large . But for each lag we are loseone observations, a large reduces the number of observations and the power of the test. Walter Sosa-Escudero Autocorrelation

Page 15

Estimation under Autocorrelaton: a modern view We will see in the next lecture that the presence of autocorrelation can be handled by using a dynamic regression model Nevertheless, we will explore one possible strategy. Walter Sosa-Escudero Autocorrelation

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Consider the simple linear model with AR (1) autocorrelation , t =1 ,...,T φu Since the model is valid for every period, the two following statements hold: Walter Sosa-Escudero Autocorrelation

Page 17

Now multiply both sides by , and substract, and given that φu we get: φY φX φu φY φX This is a non-linear (in parameters) regression model with no autocorrelation: we have been able to get rid of serial correlation, but now we need to estimate a non-linear model. We will not cover the technical details, but estimation of such model can be easily handled in any standard computer package. Walter Sosa-Escudero Autocorrelation

Econometric Analysis Spring 2009 April 23 2009 Walter SosaEscudero Autocorrelation brPage 2br TimeSeries Observations Consider the following model Kt t 1 T Here denotes periods T This is a model for time series observations Example ID: 22324

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Page 1

Autocorrelation Walter Sosa-Escudero Econ 471. Econometric Analysis. Spring 2009 April 23, 2009 Walter Sosa-Escudero Autocorrelation

Page 2

Time-Series Observations Consider the following model ··· Kt , ,t =1 ,...,T Here denotes periods, ,...,T . This is a model for time series observations. Example: consumption and income for a given country, in several periods. In time-series analysis the way observations are sorted is very important. Walter Sosa-Escudero Autocorrelation

Page 3

The no-serial correlation assumption is Cov ,u )=0 ,, t s, meaning that the error terms of two diﬀerent periods must be linearly unrelated. In the time-series context this assumption is known as no autocorrelation Remember that serial correlation invalidates the Gauss-Markov Theorem: OLS is still unbiasded (why?) but is not the best linear unbiased estimator. Walter Sosa-Escudero Autocorrelation

Page 4

A simple model for autocorrelation Consider, WLOG, the simple model for time series: , t =1 ,...,T and let the error term be speciﬁed by φu This structure is known as the linear model with ﬁrst-order autorregresive serial correlation. We will assume and that )=0 )= and Cov , )=0 , t =0 Walter Sosa-Escudero Autocorrelation

Page 5

A closer look at φu It is easy to see that the error term is linked over time, impliying the presence of serial correlation. is explicitely linked to its immediate past when =0 No autocorrelation in this setup: =0 φu is known as a ﬁrst-order autorregresive (AR(1)) process. Intuition: suppose φ> , then tends to be ‘close’ to its previous value . Example: wheather? The is a requisite for stability. What would happen is, say, φ> Walter Sosa-Escudero Autocorrelation

Page 6

Testing for AR(1) autocorrelation The Durbin-Watson test It is a test for =0 (no ﬁrst order autocorrelation) vs. =0 The Durbin-Watson test is based on the following statistic: DW =2 =1 where are OLS residuals. Walter Sosa-Escudero Autocorrelation

Page 7

Intuition behind DW DW =2 =1 DW =2 =1 =2 =1 =2 =1 Suppose that the number of observations is large. Then the ﬁrst term should be very close to one. Second term: same thing Third term: it can be shown that in the AR(1) structure Cor ,u )= Cov ,u Then the third term is nothing but an estimate of having replaced by Walter Sosa-Escudero Autocorrelation

Page 8

Then, the following approximation for the Durbin-Watson holds: DW =1+1 =2(1 Consider testing =0 vs φ> When =0 is true (no autocorrelation), DW When φ> DW < Then we should accept (no autocorrelation) when DW and reject (positive serial correlation) when DW is signiﬁcantly smaller than 2 Walter Sosa-Escudero Autocorrelation

Page 9

A problem with DW Consider =0 vs φ> (a test for positive autocorrelation). According to the previous intuition, we should reject if DW is signiﬁcantly smaller than 2. If we proceed as usual, we would reject if DW where is a critical value from the distribution of DVW Problem: the distribution of DW depends on the data used, it cannot be tabulated in general: we cannot get Solution: even though we cannot get , we can get bounds and such that Walter Sosa-Escudero Autocorrelation

Page 10

Then the procedure for a test for positive serial correlation is as follows If DW >d , then DW >d : accept If DW , then DW : reject If we cannot tell if DW or DW >d the test is inconclusive Walter Sosa-Escudero Autocorrelation

Page 11

Comments on the DW test It is a rather old-fashioned test. It requires a very special table. We need to assume normal errors. The model must include an intercept. It is crucial that the is non-stochastic. The Durbin-Watson procedure is a test for a particular form of serial correlation, the AR(1) process. It is not informative about more general patterns of autocorrelation. Walter Sosa-Escudero Autocorrelation

Page 12

More general autocorrelation: the Breusch-Pagan test Now consider the following model: , t =1 ,...,T (1) ··· (2) were satisﬁed the same assumptions as before This is the two-variable model with autocorrelation of order AR No autocorrelation ··· =0 The alternative hypothesis is =0 =0 ∨··· =0 Walter Sosa-Escudero Autocorrelation

Page 13

The Breusch-Pagan test for AR autocorrelation It is very similar to our test for heteroskedasticity Estimate by OLS, save residuals Regress on ,e ,...,e and , get of this auxiliar regression. Test statistic . Under , asymptotically, it has a distribution with degrees of freedom. Intuition: like regressing on ,...,u , but replacing the ’s by ’s. Under the of this auxiliar regression should be zero, and diﬀerent from zero under the alternative. Walter Sosa-Escudero Autocorrelation

Page 14

Comments: The Breusch-Pagan test does not require to be non-random It explores a more general pattern of serial correlation than the DW test, which also explores the AR(1) case. The choice of is problematic: intuition suggests to go for a large . But for each lag we are loseone observations, a large reduces the number of observations and the power of the test. Walter Sosa-Escudero Autocorrelation

Page 15

Estimation under Autocorrelaton: a modern view We will see in the next lecture that the presence of autocorrelation can be handled by using a dynamic regression model Nevertheless, we will explore one possible strategy. Walter Sosa-Escudero Autocorrelation

Page 16

Consider the simple linear model with AR (1) autocorrelation , t =1 ,...,T φu Since the model is valid for every period, the two following statements hold: Walter Sosa-Escudero Autocorrelation

Page 17

Now multiply both sides by , and substract, and given that φu we get: φY φX φu φY φX This is a non-linear (in parameters) regression model with no autocorrelation: we have been able to get rid of serial correlation, but now we need to estimate a non-linear model. We will not cover the technical details, but estimation of such model can be easily handled in any standard computer package. Walter Sosa-Escudero Autocorrelation

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