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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECON771 Testing Against One-Sided Alternatives

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Testing Against One-Sided Alternatives

In order to determine a rule for rejecting $\mathrm{H}_0$, we need to decide on the relevant alternative hypothesis. First consider a one-sided alternative of the form
$$\mathrm{H}_1: \beta_j>0 .$$

This means that we do not care about alternatives to $\mathrm{H}0$ of the form $\mathrm{H}_1: \beta_j<0$; for some reason, perhaps on the basis of introspection or economic theory, we are ruling out population values of $\beta_j$ less than zero. (Another way to think about this is that the null hypothesis is actually $\mathrm{H}_0: \beta_j \leq 0$; in either case, the statistic $t{\hat{\beta}j}$ is used as the test statistic.) How should we choose a rejection rule? We must first decide on a significance level or the probability of rejecting $\mathrm{H}_0$ when it is in fact true. For concreteness, suppose we have decided on a $5 \%$ significance level, as this is the most popular choice. Thus, we are willing to mistakenly reject $\mathrm{H}_0$ when it is true $5 \%$ of the time. Now, while $t{\hat{\beta}j}$ has a $t$ distribution under $\mathrm{H}_0$ – so that it has zero mean-under the alternative $\beta_j>0$, the expected value of $t{\hat{\beta}j}$ is positive. Thus, we are looking for a “sufficiently large” positive value of $t{\hat{\beta}j}$ in order to reject $\mathrm{H}_0: \beta_j=0$ in favor of $\mathrm{H}_1: \beta_j>0$. Negative values of $t{\hat{\beta}_j}$ provide no evidence in favor of $\mathrm{H}_1$.

The definition of “sufficiently large,” with a $5 \%$ significance level, is the $95^{\text {th }}$ percentile in a $t$ distribution with $n-k-1$ degrees of freedom; denote this by $c$. In other words, the rejection rule is that $\mathrm{H}0$ is rejected in favor of $\mathrm{H}_1$ at the $5 \%$ significance level if $$t{\hat{\beta}_j}>c .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Two-Sided Alternatives

In applications, it is common to test the null hypothesis $\mathrm{H}_0: \beta_j=0$ against a two-sided alternative, that is,
$$\mathrm{H}_1: \beta_j \neq 0 .$$
Under this alternative, $x_j$ has a ceteris paribus effect on $y$ without specifying whether the effect is positive or negative. This is the relevant alternative when the sign of $\beta_j$ is not well-determined by theory (or common sense). Even when we know whether $\beta_j$ is positive or negative under the alternative, a two-sided test is often prudent. At a minimum, using a two-sided alternative prevents us from looking at the estimated equation and then basing the alternative on whether $\hat{\beta}j$ is positive or negative. Using the regression estimates to help us formulate the null or alternative hypotheses is not allowed because classical statistical inference presumes that we state the null and alternative about the population before looking at the data. For example, we should not first estimate the equation relating math performance to enrollment, note that the estimated effect is negative, and then decide the relevant alternative is $\mathrm{H}_1: \beta{\text {enroll }}<0$.

When the alternative is two-sided, we are interested in the absolute value of the $t$ statistic. The rejection rule for $\mathrm{H}0: \beta_j=0$ against (4.10) is $$\left|t{\hat{\beta}_j}\right|>c, \quad \text { (4.11) }$$
where $|\cdot|$ denotes absolute value and $c$ is an appropriately chosen critical value. To find $c$, we again specify a significance level, say $5 \%$. For a two-tailed test, $c$ is chosen to make the area in each tail of the $t$ distribution equal $2.5 \%$. In other words, $c$ is the $97.5^{\text {th }}$ percentile in the $t$ distribution with $n-k-1$ degrees of freedom. When $n-k-1=$ 25 , the $5 \%$ critical value for a two-sided test is $c=2.060$. Figure $4.4$ provides an illustration of this distribution.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Testing Against One-Sided Alternatives

$$\mathrm{H}_1: \beta_j>0 .$$

$$t \hat{\beta}_j>c .$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|TwoSided Alternatives

$$\mathrm{H}_1: \beta_j \neq 0 .$$

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