Lecture 22 – Spurious and Cointegrating Regressions The time series regression theory and application developed in Econ 672 and 674 have assumed that the time series we are working with are stationary (or trend stationary). If we are working with difference stationary time series (with or without drift), regressions in the levels of those series are problematic: • spurious regressions • cointegrating regressionsSpurious Regressions Suppose yt and xt are mutually independent random walks. For example, for t = 1,2…,T yt = yt-1 + ε1t , y0 = 0, ε1t ~ iid N(0,1) xt = xt-1 + ε2t, x0 = 0, ε2t ~ iid N(0,1) E(ε1t,ε2s) = 0 for t,s = 1,…,T What happens if we regress y on 1,x ? Granger and Newbold (1974) conducted Monte Carlo experiments and concluded that such regressions tend to be characterized by • t-tests with actual rejection rates of H0:β1=0 that are much too large relative to the nominal size of the test. • High R2’s • small DW statisticsThe high rejection rates and R2s mean that the researcher will be frequently misled into concluding that there is a significant relationship between y and x. Granger and Newbold concluded that time series regressions that yield high R2 and low DW values should be viewed with caution, because this combination could reflect a spurious regression. Phillips (1986) developed the econometric theory that explains the Granger-Newbold results. Phillips showed that if yt and xt are mutually independent I(1) process then in the regression of y on 1,x : • 0ˆ1P→β• ∞→Ptβˆ• 0pDW →• , where S is a random variable SRD→2So, the OLS estimator of β1 is consistent, converging in probability to 0. However the t-ratio diverges and so for any test size the probability of rejecting the null converges to 1! The p-value of the test of H0:β1 =0 goes to 0. The DW statistic converges to zero, but the R2 does not. Notes: The same problem arises • when regressing an I(1) process on a trending regressor (e.g., a linear trend). • in finite samples when y and x are persistent I(0) processes.Cointegrating Regressions – Suppose yt and xt are I(1) process where yt = β0 + β1xt + εt , where εt ~ I(0) Note that β1 must be non-zero. (Why?) In this case we say that y and x are cointegrated and we call the regression of y on x (or x on y) a cointegrating regression. In this case (as in the spurious regression case) the OLS estimator is (super-) consistent. This holds even if y and x are jointly determined! (The regression of x on y is a consistent estimator of 1/β1). However, standard asymptotics do not apply with regard to the asymptotic distribution of the OLS estimator. More on this
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