Lecture 14

Lecture 14 - Introduction to Econometrics Lecture 14:...

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Introduction to Econometrics Lecture 14: Testing time series for nonstationarity Testing stationarity - non-trended data To test the null hypothesis that x t is nonstationary against the alternative of a stationary process, first consider the simplest possible case where our null hypothesis is that x t is a random walk with zero mean. We can estimate the autoregression x t = ρ x t-1 + e t and conduct the one-sided hypothesis test H 0 : ρ = 1 H 1 : ρ < 1 using a conventional t-test. However there are two reasons why this is not sensible. The first is trivial: it is easier to transform the equation by subtracting x t-1 from both sides, giving the modified regression (x t - x t-1 ) = ( ρ - 1)x t-1 + e t = β x t-1 + e t and the modified null and alternative hypotheses H’ 0 : β = 0 H’ 1 : β < 0 This transformation makes no difference to the properties of the test. The second reason is more sophisticated. It turns out that if x t is a nonstationary process, it has an infinite variance, and the statistical theory by which we can establish that the statistic β /SE ) has a t-distribution no longer works: instead we need to use a different, and non-standard, distribution for this statistic. The necessary distribution has been tabulated (originally by Dickey and Fuller). The Dickey-Fuller procedure for testing the null hypothesis that a zero-mean series x t is I(1) is a: compute the regression x t - x t-1 = β x t-1 + e t β = ρ - 1 (note there is no constant because we assume x t has zero mean) b: compute the t-statistic for β and perform a 1-sided test of H’ 0 : β = 0 against H’ 1 : β < 0 comparing the t-statistic with the special critical values tabulated in Dickey-Fuller Table 1 (for τ ). These critical values are higher than those which would be used if the distribution was a standard normal (or t), partly because the distribution of β is not symmetric around zero ( β e < 0 in 68% of cases when the true value of β = 0). Using the standard normal distribution would lead to too many rejections of H 0 , the nonstationarity hypothesis. Example 1 is based on the deviations of log bond prices from their sample mean. Computing the regression gives x t - x t-1 = -0.0384x t-1 + e t Introduction to Econometrics 14-1 Technical Notes
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with a t-ratio on the lagged value of x t of -1.381. In this case the relevant critical value from the tables for the t-statistic at the 5% level is -1.95, so we cannot reject the hypothesis that the log bond price is a random walk. In this case we would have reached the same conclusion if we had incorrectly used the standard t-distribution, since the 1-sided critical value for t is -1.67. Note that the fact that the Dickey- Fuller critical value, -1.95, is very close to the familiar 1.96 critical value for 2-sided tests using the Normal distribution is pure coincidence: the tests are completely different. Suppose we have a series x
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This note was uploaded on 03/07/2012 for the course ECON 201 taught by Professor Cowell during the Spring '10 term at LSE.

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Lecture 14 - Introduction to Econometrics Lecture 14:...

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