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经济代写|计量经济学代写Introduction to Econometrics代考|BEA242 Other Large Sample Tests: The Lagrange Multiplier Statistic

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经济代写|计量经济学代写Introduction to Econometrics代考|Other Large Sample Tests: The Lagrange Multiplier Statistic

Once we enter the realm of asymptotic analysis, there are other test statistics that can be used for hypothesis testing. For most purposes, there is little reason to go beyond the usual $t$ and $F$ statistics: as we just saw, these statistics have large sample justification without the normality assumption. Nevertheless, sometimes it is useful to have other ways to test multiple exclusion restrictions, and we now cover the Lagrange multiplier LM statistic, which has achieved some popularity in modern econometrics.

The name “Lagrange multiplier statistic” comes from constrained optimization, a topic beyond the scope of this text. [See Davidson and MacKinnon (1993).] The name score statistic – which also comes from optimization using calculus-is used as well. Fortunately, in the linear regression framework, it is simple to motivate the $L M$ statistic without delving into complicated mathematics.

The form of the $L M$ statistic we derive here relies on the Gauss-Markov assumptions, the same assumptions that justify the $F$ statistic in large samples. We do not need the normality assumption.

To derive the $L M$ statistic, consider the usual multiple regression model with $k$ independent variables:
$$y=\beta_0+\beta_1 x_1+\ldots+\beta_k x_k+u .$$

经济代写|计量经济学代写Introduction to Econometrics代考|THE LAGRANGE MULTIPLIER STATISTIC FOR q EXCLUSION RESTRICTIONS:

(i) Regress $y$ on the restricted set of independent variables and save the residuals, $\tilde{u}$.
(ii) Regress $\tilde{u}$ on all of the independent variables and obtain the $R$-squared, say $R_u^2$ (to distinguish it from the $R$-squareds obtained with $y$ as the dependent variable).
(iii) Compute $L M=n R_u^2$ [the sample size times the $R$-squared obtained from step (ii)].
(iv) Compare $L M$ to the appropriate critical value, $c$, in a $\chi_q^2$ distribution; if $L M>$ $c$, the null hypothesis is rejected. Even better, obtain the $p$-value as the probability that a $\chi_q^2$ random variable exceeds the value of the test statistic. If the $p$-value is less than the desired significance level, then $\mathrm{H}_0$ is rejected. If not, we fail to reject $\mathrm{H}_0$. The rejection rule is essentially the same as for $F$ testing.
Because of its form, the $L M$ statistic is sometimes referred to as the $\mathbf{n}-\boldsymbol{R}$-squared statistic. Unlike with the $F$ statistic, the degrees of freedom in the unrestricted model plays no role in carrying out the $L M$ test. All that matters is the number of restrictions being tested $(q)$, the size of the auxiliary $R$-squared $\left(R_u^2\right)$, and the sample size $(n)$. The $d f$ in the unrestricted model plays no role because of the asymptotic nature of the $L M$ statistic. But we must be sure to multiply $R_u^2$ by the sample size to obtain $L M$; a seemingly low value of the $R$-squared can still lead to joint significance if $n$ is large.

经济代写|计量经济学代写Introduction to Econometrics代考|Other Large Sample Tests: The Lagrange Multiplier Statistic

“Lagrange multiplier statistic”这个名称来自约束优化，这个主题超出了本文的范围。[参见 Davidson 和 MacKinnon (1993)]。也使用名称得分统计一一它也来自使用微积分的优化。幸运的是，在线性回归框架中， 很容易激发 $L M$ 无需深入研究筫杂的数学即可进行㧤计。

$$y=\beta_0+\beta_1 x_1+\ldots+\beta_k x_k+u .$$

经济代写|计量经济学代写Introduction to Econometrics代考|THE LAGRANGE MULTIPLIER STATISTIC FOR q EXCLUSION RESTRICTIONS:

(i) 回归 $y$ 在受限制的自变量集上并保存残差， $\tilde{u}$.
(ii) 回归 $\tilde{u}$ 在所有自变量上并获得 $R$-平方，说 $R_u^2$ (为了区别于 $R$-平方获得 $y$ 作为因变量)。
(iii) 计算 $L M=n R_u^2$ [样本量乘以 $R$-从步骤 (ii)] 获得的平方。
(iv) 比较 $L M$ 到适当的临界值， $c$ ，在一个 $\chi_q^2$ 分配; 如果 $L M>c$ ，原假设被拒绝。更好的是，获得 $p$-value 作为 $\mathrm{a}$ 的概率 $\chi_q^2$ 随机变量超过了检验统计量的值。如果 $p$ – 值小于所需的显着性水平，则 $\mathrm{H}_0$ 被拒绝了。如果没 有，我们拒绝 $\mathrm{H}_0$. 拒绝规则与 for 基本相同 $F$ 测试。

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