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Unformatted text preview: Introduction to Econometrics Lecture 15: Multiple regression and the analysis of time series Multiple regression using time series data This lecture is an introduction to the problems which arise when we try to explain the relationship between economic time series using simple or multiple regression. It turns out that the procedures which we should adopt depend crucially on whether the time series which enter our model are stationary – that is I(0) – or nonstationary – typically I(1). This means that before using regression to relate two or more time series we should use the tests described in the last handout to test the order of integration for each series separately. The first part of the lecture describes how to formulate regression models for stationary or nonstationary time series data: the second part describes an additional diagnostic test (for Serial Correlation) which is useful with time series data. Modelling I(0) series Suppose that we find that all the series which we wish to include in our model are stationary (this implies that the Dickey-Fuller test, or equivalent, rejects the null of nonstationarity). Then we can use conventional regression techniques to estimate models of the form y t = α + β x t + u t (the t subscript is used to indicate that y and x are time series). Under suitable assumptions OLS estimators of the parameters of this model will be unbiased and efficient, even though y t and x t are not the outcome of a random sampling procedure. The equation given above is a static regression equation, because all the variables are observed at the same date t. In many contexts we want to allow some of the x variables to affect y with a lag: thus consumption might be a function of both current and last period’s income. This implies that we want to specify and estimate a dynamic regression equation. The simplest case is where the explanatory variables include current and lagged exogenous variables, but not the lagged value of the dependent variable y which we are trying to explain. Thus the equation has the form y t = α + β 1 x t + β 2 z t + β 3 z t-1 + u t An equation like this is sometimes described by saying that y is explained by a distributed lag (a sort of moving average) of z. Again, under suitable assumptions we can show that there is a sense in which OLS estimators of this equation are BLUE. If we want to include lagged values of y as a regressor, so that the equation becomes y t = α + β 1 x t + β 2 z t + β 3 z t-1 + γ y t-1 + u t things become a bit more complicated in statistical terms (although it’s still straightforward to calculate the regression coefficients). This is because the fact that y t-1 is included in the regressors means that we cannot prove OLS estimators are unbiased, although we can show that (again with suitable assumptions) they have desirable properties in ‘large’ samples. Proving all of this is complicated, and beyond the syllabus for this course, so you don’t need to know how to do it. This last sort of equation is sometimes syllabus for this course, so you don’t need to know how to do it....
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