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Exercise 2

# Exercise 2 - University of California Department of...

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University of California D. Steigerwald Department of Economics Economics 241B Exercise 2 1. Consider the bivariate regression model , t t t U X Y in which the regressor is exogenous and all observations are measured as deviations from means. You wish to test a hypothesis about the value of β with the OLS estimator B . Notation: There are n observations and n I is the identity matrix of dimension n . a. A colleague tells you: “Always form the null hypothesis in such a way that rejecting the null hypothesis is of interest.” Do you think this advice is sensible? b. To test hypotheses about β , the quantity of interest is the statistic . / 2 t t t X U X B If we assume   n I N X U 2 , 0 ~ | , then what is the distribution of (B- β )|X ? Does the distribution depend on X ? Is the distribution exact, or does it hold only asymptotically? c. If the variance of (B- β )|X depends on parameters, construct a statistic (a function of B- β ) whose variance does not depend on parameters (and so is a pivotal statistic). For the pivotal statistic you construct, give both the distribution conditional on X and the unconditional distribution. If σ ² is unknown, we must use an estimator of σ ² to form the statistic. Call the resulting statistic the estimated statistic. Does the distribution of the estimated statistic depend on X ? For the estimated statistic, give both the conditional (on X ) and unconditional distributions. d. The Durbin-Watson statistic is   . ˆ ˆ ˆ 1 2 2 2 1 n t t n t t t U U U DW

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From your results in the preceding parts, what can you infer about the distribution, conditional on X , of DW ? If the distribution of DW conditional on X depends on X , how does this affect the resultant critical values from the
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Exercise 2 - University of California Department of...

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