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143

Appendix 2 Statistical Inference

The fundamental goal of statistics is to estimate the values of model parameters and to associate a level of confidence with the results. Only when a model is properly specified and parameter estimates are robust, unbiased and efficient can the model predict well. The questions answered in this appendix concern the accuracy of parameter estimates: With what degree of confidence can we say that a model parameter takes some value, such as zero?6 It may also be of interest to test the equality of certain parameters, or that all parameters are zero, or more general linear restrictions on several model parameters.7

A.2.1 Hypothesis Testing and Confidence Intervals

There are two methods of inference on the true values of model parameters: confidence intervals and hypothesis tests. The framework for hypothesis tests is conceptually very simple:

1. Set up the null and alternative hypotheses, H0 and Hq. The null hypothesis of no change or no effect consists of one or more restrictions on the true model parameters. The alternatives may be either one-sided (< or >) or two-sided ( ).

2. State the test statistic, X. A variety of test statistics are often available. The most common parametric test statistics fall into one of three categories: Wald tests, Lagrange multiplier (LM) tests or likelihood ratio (LR) tests. These tests have different power properties and different distributions in small samples, although they are asymptotically equivalent (§A.2.5).

3. Choose a significance level, a. The level of significance with which one can state results determines the size of the test, which is the probability of a type I error (the rejection of a true null hypothesis). Increasing the significance level reduces the size, but also increases the probability of a type II error (the failure to reject a false null hypothesis) and so reduces its power. The power of a test is the probability that it will reject a false null hypothesis. When more than one test is available it is preferable to choose the most powerful test of any given size.

4. Determine the critical region CRa. The critical values of the test statistic depend on the chosen significance level a. For a fixed a, and for a two-sided alternative, upper and lower critical values CVtj and CVL give the critical region CRa - {(-oo, CVL] and [CVtj, oo)}. A one-sided (<) alternative has the critical region (-00, CVL] and a one-sided (>) alternative has the

6The hypothesis pi, = 0 is a particularly interesting inference in a regression model, because it implies that the associated explanatory variable Xt has no effect on the dependent variable.

Non-linear hypotheses require statistical methods that are beyond the scope of this book, but the> are well covered in other econometrics texts, such as Campbell el al. (1997).



(b) Lower critical region Upper critical region

Figure A.6 Critical region of (a) a one-sided (>) statistical test; (b) a two-sided statistical test.

critical region [CVtj, oo). In each case the size of the test is the area above the critical region under the density function of the test statistic as shown in Figure A.6. In the majority of cases the critical values of a test statistic are tabulated. In fact many test statistics have one of four standard sampling distributions under the null hypothesis: the normal distribution, the (-distribution, the -distribution or the chi-squared (/2) distribution. Critical values for these distributions are tabulated at the end of the book.

5. Evaluate the test statistic, X*. This value will be based on sample data, and it may or may not assume the values of model parameters that are given under the null and alternative hypothesis. In Wald tests the test statistic is calculated without the restrictions of the null hypothesis, but LM tests employ restricted estimators and LR tests use both unrestricted and restricted estimators.

6. Apply the decision rule. The decision is reject H0 in favour of Hj at 100a% if X* e CRa.

Note that, for the same value of a. the upper and lower critical values will not be the same for the one-sided test as for the two-sided test: in the latter case the critical region will be split between the tails, while in the former case it will fall entirely in one tail.



As its name suggests, a confidence interval is a range within which one is, to some degree, confident the true parameter lies. So (A, B) is a 95% confidence interval for 9 if ( < 9 < ) = 0.95. The range of the confidence interval depends on the degree of confidence and the distribution of the parameter estimator.

Confidence intervals allow error bounds to be placed on the true values of model parameters using the estimated standard errors of model parameter estimates. Consider, for example, an OLS estimator b of a parameter (3 in a linear regression model. The simple t-ratio will have a /-distribution if the OLS estimators are normally distributed:

/-ratio - (b- P)/(est. s.e. b) ~ /v.

Here "est. s.e. stands for estimated standard error, the square root of the estimated variance given in the estimated covariance matrix (A. 1.17).9 The degrees of freedom v are T - k, where T is the sample size and is the number of variables in the regression (including the constant).10

This /-statistic may be used for hypotheses with only one linear restriction on parameters, as described below. It also forms the basis of confidence intervals for p. Denote by /v0.o25tne 2.5% critical value of the /-distribution on v degrees of freedom. Since

Prob((7> - P)/(est. s.e. b) < -tvfim5) = 0.025

Prob((7> - P)/(est. s.e. b) > -tvfim5) = 0.025,

we have

Prob(7> - /v 0025(est. s.e. b) < P < b + tv0 025(est. s.e. b)) = 0.95.

Thus a two-sided 95% confidence interval for p is

- v,o.o25(est. s.e. b), b + tvQm5(cst. s.e. bj).

The size a of the critical region determines the degree of confidence, just as it does the significance level of a hypothesis test. In general, a two-sided 100(1 - a)% confidence interval for P is

(b - ?v,a/2(est- s-e- b), b + /va/2(est. s.e. b))

Confidence intervals may also be one-sided, just as hypothesis tests. A onesided upper 100(1 - a)% confidence interval for p is

(b - /v,a(est- s-e- b), oo).

9The terminology standard error, rather than standard deviation, is used because we are referring to an estimate rather than an arbitrary random variable.

"One degree of freedom is lost for every constraint on the data that is necessary to calculate the estimates.



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