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经济代写|计量经济学代写Introduction to Econometrics代考|ECO400 Computing p-values for F Tests

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经济代写|计量经济学代写Introduction to Econometrics代考|Computing p-values for F Tests

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

For reporting the outcomes of $F$ tests, $p$-values are especially useful. Since the $F$ distribution depends on the numerator and denominator $d f$, it is difficult to get a feel for how strong or weak the evidence is against the null hypothesis simply by looking at the value of the $F$ statistic and one or two critical values.

In the $F$ testing context, the $p$-value is defined as
$$p \text {-value }=\mathrm{P}(\mathscr{F}>F), \quad \text { (4.43) }$$
where, for emphasis, we let $\mathscr{F}$ denote an $F$ random variable with $(q, n-k-1)$ degrees of freedom, and $F$ is the actual value of the test statistic. The $p$-value still has the same interpretation as it did for $t$ statistics: it is the probability of observing a value of the $F$ at least as large as we did, given that the null hypothesis is true. A small $p$-value is evidence against $\mathrm{H}_0$. For example, $p$-value $=.016$ means that the chance of observing a value of $F$ as large as we did when the null hypothesis was true is only $1.6 \%$; we usually reject $\mathrm{H}_0$ in such cases. If the $p$-value $=.314$, then the chance of observing a value of the $F$ statistic as large as we did under the null hypothesis is $31.4 \%$. Most would find this to be pretty weak evidence against $\mathrm{H}_0$.

As with $t$ testing, once the $p$-value has been computed, the $F$ test can be carried out at any significance level. For example, if the $p$-value $=.024$, we reject $\mathrm{H}_0$ at the $5 \%$ significance level but not at the $1 \%$ level.

The $p$-value for the $F$ test in Example $4.9$ is $.238$, and so the null hypothesis that $\beta_{\text {motheduc }}$ and $\beta_{\text {fatheduc }}$ are both zero is not rejected at even the $20 \%$ significance level.
Many econometrics packages have a built-in feature for testing multiple exclusion restrictions. These packages have several advantages over calculating the statistics by hand: we will less likely make a mistake, $p$-values are computed automatically, and the problem of missing data, as in Example 4.9, is handled without any additional work on our part.

经济代写|计量经济学代写Introduction to Econometrics代考|Computing pvalues for $F$ Tests

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

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

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

$$p \text {-value }=\mathrm{P}(\mathscr{F}>F),$$

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