Christopher Dougherty EC220 - Introduction to econometrics (revision lectures 2011) Slideshow: autocorrelation Original citation: Dougherty, C. (2011) EC220 - Introduction to econometrics (revision lectures 2011). [Teaching Resource] © 2011 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

AUTOCORRELATION 1 Assumption B.5 states that the values of the disturbance term in the observations in the sample are generated independently of each other. Y =  1 +  2 X Y X

AUTOCORRELATION 2 In the graph above, it is clear that this assumption is not valid. Positive values tend to be followed by positive ones, and negative values by negative ones. Successive values tend to have the same sign. This is described as positive autocorrelation. Y X Y =  1 +  2 X

AUTOCORRELATION 3 In this graph, positive values tend to be followed by negative ones, and negative values by positive ones. This is an example of negative autocorrelation. Y X Y =  1 +  2 X

AUTOCORRELATION 4 Intuition: The error term captures everything that isn’t included as a regressor. So, for example, if we regress expenditure on clothing on total income and the price of clothing, the error term is capturing other things that might explain expenditure on clothing. Examples: higher wages in China, the cost of cotton, the weather. Any of these could cause autocorrelation, if they are correlated from one year to the next.

First-order autoregressive autocorrelation: AR(1) AUTOCORRELATION 4 A particularly common type of autocorrelation, at least as an approximation, is first-order autoregressive autocorrelation, usually denoted AR(1) autocorrelation. It is autoregressive, because u t depends on lagged values of itself, and first-order, because it depends only on its previous value. u t also depends on  t, an injection of fresh randomness at time t, often described as the innovation at time t.

CONSEQUENCES OF AUTOCORRELATION 1 The consequences of autocorrelation for OLS are similar to those of heteroscedasticity. In general, the regression coefficients remain unbiased, but OLS is inefficient. Also, the standard errors will be estimated incorrectly: usually the residuals will make the OLS coefficient’s variance appear smaller than it really is. Intuition: imagine the extreme case where, if the first error is positive, they are all positive (because they’re autocorrelated). The OLS fitted line will be above the true line, making the residuals much smaller than the actual errors.

CONSEQUENCES OF AUTOCORRELATION 4 Showing the OLS estimate is unbiased: you should now know how to do this in a variety of contexts. The method is always the same. Decompose the estimator, and take expectations. In the case of autocorrelation, you need to show E(u t ) = 0.

CONSEQUENCES OF AUTOCORRELATION 7 For example, suppose you know the error is AR(1). Lagging and substitute, we can express u t in terms of current and lagged values of  t with diminishing weights. Since, by definition, the expected value of each innovation is zero, the expected value of u t is zero.

CONSEQUENCES OF AUTOCORRELATION 13 Now we come to the special case where OLS yields inconsistent estimators if the disturbance term is subject to autocorrelation: if the model specification includes a lagged dependent variable. This is because both u t and Y t-1 depend on u t-1, so they will be correlated. We’ve seen that when a regressor is correlated with the error term, OLS will be inconsistent. Note: in Ch 13 we will see that there is another special case where OLS is inconsistent: if the autocorrelation is so severe that the error term is non- stationary.

ELIMINATING AR(1) AUTOCORRELATION 4 Luckily, if we know what type of autocorrelation we’re dealing with, it’s often easy to get rid of by simply manipulating the specification by adding or subtracting terms from both sides of the regression equation. For example, if we know the error term is AR(1). It’s useful to remember this technique, as we very often assume errors are AR(1). Note that the new specification involves a restriction on the parameters. To fit it, we cannot use OLS, we need to use a non-linear estimation technique (i.e. eviews will do this for you).

Copyright Christopher Dougherty 2000–2011. This slideshow may be freely copied for personal use