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The Holly Seven Steps of Developing a Statistical Test

The Holly Seven Steps of Developing a Statistical Test. Tugrul Temel Center for World Food Studies Free University, Amsterdam February 2000. Step 1. A Data-Generating Process. Given a random sample of n i.i.d. random variables (r.v.) Independence of obs. on a r.v

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The Holly Seven Steps of Developing a Statistical Test

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  1. The Holly Seven Steps of Developing a Statistical Test Tugrul Temel Center for World Food Studies Free University, Amsterdam February 2000

  2. Step 1. A Data-Generating Process • Given a random sample of n i.i.d. random variables (r.v.) • Independence of obs. on a r.v • Identical (or constant) prob. dist. for r.v. • Stationary dist. for r.v. • The Borel field • The history does not forget • sample space, events, prob. of a set of events (or test size) • A Borel measurable function • Integration over events

  3. The Holy Seven Stepsfor a Statistical Test • A data-generating process • A model • The hypotheses • Asymptotic distributions of distance functions implied by the hypotheses • A test statistic • A critical region • A decision rule

  4. Step 2. A Model • Model specification • Parametric, semi-parametric, and non-parametric • Misspecification in • degree of polynomial, choice of variables, or both • Measurement error in • dependent, independent, or both • Nonlinearity in • policy variables, coefficients, or both

  5. Step 3. The Hypotheses • The null and alternative hypotheses • The underlying distance function (d) • The underlying loss and risk functions • Subjectivity of risk functions • Form of objective functions, i.e., Mean Squared Error • A transformation of d, T=T(d)

  6. Step 4. The Asymptotic Distribution of Tne • Tn is an estimator of T • Tne is the estimated Tn • A value of Tne far from zero is evidence against the null. To tell how far Tne must be from zero to reject the null, we find its asymptotic distribution. • Apply a CLT and a LLN to obtain a normal dist. for Tne • Estimate the variance of Tne

  7. Step 5. A Test Statistic (TS) • Define TS=f(Tne) • Using different random samples (i), find a distribution of TS(i) • Is the TS sufficient? • In reality, we have only one random sample, i.e., i=1. Obtain TS(i) by applying • The classical approach • The Bayesian approach • The bootstrap

  8. Step 6. A Critical Region • Construct a critical region given the distribution of TS(i) and the test size (alpha) • The link between the critical region and the Borel field

  9. Step 7. A Decision Rule • Reject the null if TS falls in the critical region or accept the null if it falls in the confidence region • Asymptotic versus the bootstraped values of critical region

  10. The Bootstrap • The bootstrap to detect • violations of parametric distributional assumptions, like a non-normal error structure assumed in OLS regression • characteristics of TS

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