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# 统计代写|线性回归代写Linear Regression代考|Hypotheses Concerning One of the Terms

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## 统计代写|线性回归代写Linear Regression代考|Hypotheses Concerning One of the Terms

Obtaining information on one of the terms may be of interest. Can we do as well, understanding the mean function for Fuel if we delete the Tax variable? This amounts to the following hypothesis test of
\begin{aligned} & \mathrm{NH}: \quad \beta_1=0, \quad \beta_0, \beta_2, \beta_3, \beta_4 \text { arbitrary } \ & \mathrm{AH}: \quad \beta_1 \neq 0, \quad \beta_0, \beta_2, \beta_3, \beta_4 \text { arbitrary } \ & \end{aligned}
The following procedure can be used. First, fit the mean function that excludes the term for Tax and get the residual sum of squares for this smaller mean function.

Then fit again, this time including Tax, and once again get the residual sum of squares. Subtracting the residual sum of squares for the larger mean function from the residual sum of squares for the smaller mean function will give the sum of squares for regression on Tax after adjusting for the terms that are in both mean functions, Dlic, Income and $\log$ (Miles). Here is a summary of the computations that are needed:
\begin{tabular}{lcccccc}
& Df & SS & MS & F & Pr $(>F)$ \
Excluding Tax & 47 & 211964 & & & & \
Including Tax & 46 & 193700 & & & & \
\hline Difference & 1 & 18264 & 18264 & 4.34 & 0.043
\end{tabular}
The row marked “Excluding Tax” gives the df and $R S S$ for the mean function without $\operatorname{Tax}$, and the next line gives these values for the larger mean function including Tax. The difference between these two given on the next line is the sum of squares explained by Tax after adjusting for the other terms in the mean function. The $F$-test is given by $F=(18,264 / 1) / \hat{\sigma}^2=4.34$, which, when compared to the $F$ distribution with $(1,46)$ df gives a significance level of about 0.04 . We thus have modest evidence that the coefficient for Tax is different from zero. This is called a partial $F$-test. Partial $F$-tests can be generalized to testing several coefficients to be zero, but we delay that generalization to Section 5.4 .

## 统计代写|线性回归代写Linear Regression代考|Relationship to the t-Statistic

Another reasonable procedure for testing the importance of Tax is simply to compare the estimate of the coefficient divided by its standard error to the $t\left(n-p^{\prime}\right)$ distribution. One can show that the square of this $t$-statistic is the same number of the $F$-statistic just computed, so these two procedures are identical. Therefore, the $t$-statistic tests hypothesis (3.22) concerning the importance of terms adjusted for all the other terms, not ignoring them.

From Table 3.3, the $t$-statistic for Tax is $t=-2.083$, and $t^2=(-2.083)^2=$ 4.34 , the same as the $F$-statistic we just computed. The significance level for $\operatorname{Tax}$ given in Table 3.3 also agrees with the significance level we just obtained for the $F$-test, and so the significance level reported is for the two-sided test. To test the hypothesis that $\beta_1=0$ against the one-sided alternative that $\beta_1<0$, we could again use the same $t$-value, but the significance level would be one-half of the value for the two-sided test.

A $t$-test that $\beta_j$ has a specific value versus a two-sided or one-sided alternative (with all other coefficients arbitrary) can be carried out as described in Section 2.8.

In Section 3.1 , we discussed adding a term to a simple regression mean function. The same general procedure can be used to add a term to any linear regression mean function. For the added-variable plot for a term, say $X_1$, plot the residuals from the regression of $Y$ on all the other $X$ ‘s versus the residuals for the regression of $X_1$ on all the other $X \mathrm{~s}$. One can show (Problem 3.2) that (1) the slope of the regression in the added-variable plot is the estimated coefficient for $X_1$ in the regression with all the terms, and (2) the $t$-test for testing the slope to the zero in the added-variable plot is essentially the same as the $t$-test for testing $\beta_1=0$ in the fit of the larger mean function, the only difference being a correction for degrees of freedom.

## 统计代写|线性回归代写Linear Regression代考|Hypotheses Concerning One of the Terms

\begin{aligned} & \mathrm{NH}: \quad \beta_1=0, \quad \beta_0, \beta_2, \beta_3, \beta_4 \text { arbitrary } \ & \mathrm{AH}: \quad \beta_1 \neq 0, \quad \beta_0, \beta_2, \beta_3, \beta_4 \text { arbitrary } \ & \end{aligned}

\begin{tabular}{lcccccc}
& Df & SS & MS & F & Pr $(>F)$ \Excluding Tax & 47 & 211964 & & & &\Including Tax & 46 & 193700 & & & &\hline Difference & 1 & 18264 & 18264 & 4.34 & 0.043
\end{tabular}

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