The following model was fitted to data on 90 German chemical...

The following model was fitted to data on 90 German chemical companies: $$ \begin{aligned} \hat{y}= & 0.819+\underset{(1.79)}{2.11 x_1}+\underset{(1.94)}{0.96 x_2}-\underset{(0.144)}{0.059 x_3}+\underset{(4.08)}{5.87 x_4} \\ & +\underset{\left(0.00226 x_5\right.}{0.0025)} \quad \bar{R}^2=.410 \end{aligned} $$ where the numbers in parentheses are estimated coefficient standard errors and $$ \begin{aligned} y & =\text { share price } \\ x_1 & =\text { earnings per share } \\ x_2 & =\text { funds flow per share } \\ x_3 & =\text { dividends per share } \\ x_4 & =\text { book value per share } \\ x_5 & =\text { a measure of growth } \end{aligned} $$ a. Test at the $10 \%$ level the null hypothesis that the coefficient on $x_1$ is 0 in the population regression against the alternative that the true coefficient is positive. b. Test at the $10 \%$ level the null hypothesis that the coefficient on $x_2$ is 0 in the population regression against the alternative that the true coefficient is positive. c. The variable $X_2$ was dropped from the original model, and the regression of $Y$ on $\left(X_1, X_3, X_4, X_5\right)$ was estimated. The estimated coefficient on $X_1$ was 2.95 with standard error 0.63 . How can this result be reconciled with the conclusion of part a?

The following model was fitted to data on 90 German chemical companies:
$$
\begin{aligned}
\hat{y}= & 0.819+\underset{(1.79)}{2.11 x_1}+\underset{(1.94)}{0.96 x_2}-\underset{(0.144)}{0.059 x_3}+\underset{(4.08)}{5.87 x_4} \\
& +\underset{\left(0.00226 x_5\right.}{0.0025)} \quad \bar{R}^2=.410
\end{aligned}
$$
where the numbers in parentheses are estimated coefficient standard errors and
$$
\begin{aligned}
y & =\text { share price } \\
x_1 & =\text { earnings per share } \\
x_2 & =\text { funds flow per share } \\
x_3 & =\text { dividends per share } \\
x_4 & =\text { book value per share } \\
x_5 & =\text { a measure of growth }
\end{aligned}
$$
a. Test at the $10 \%$ level the null hypothesis that the coefficient on $x_1$ is 0 in the population regression against the alternative that the true coefficient is positive.
b. Test at the $10 \%$ level the null hypothesis that the coefficient on $x_2$ is 0 in the population regression against the alternative that the true coefficient is positive.
c. The variable $X_2$ was dropped from the original model, and the regression of $Y$ on $\left(X_1, X_3, X_4, X_5\right)$ was estimated. The estimated coefficient on $X_1$ was 2.95 with standard error 0.63 . How can this result be reconciled with the conclusion of part a?

Key Concepts

Multiple Regression Analysis

This concept involves modeling a dependent variable using multiple independent variables. In the context of regression modeling, it allows us to understand the impact of several predictors simultaneously while controlling for other influences, thus providing a more accurate picture of the relationships among variables.

Hypothesis Testing for Regression Coefficients

This is the process of evaluating whether a predictor has a statistically significant relationship with the dependent variable. It involves formulating a null hypothesis that a given coefficient is zero (indicating no effect) and using sample data to determine if there is enough evidence to reject that hypothesis in favor of an alternative, such as a positive effect.

t-Statistic

The t-statistic is a key component in testing regression coefficients. It is calculated by dividing the estimated coefficient by its standard error, and it measures how many standard errors the estimate is away from the hypothesized value under the null hypothesis. A larger absolute t-value indicates stronger evidence against the null hypothesis.

Significance Level

The significance level, often denoted by alpha, is the probability threshold for rejecting the null hypothesis. In hypothesis testing, choosing a 10% significance level means that there is a 10% risk of incorrectly rejecting a true null hypothesis, thereby controlling the likelihood of a Type I error.

One-Sided Test

A one-sided test is used when the research hypothesis specifies a direction of the effect, such as testing whether a coefficient is greater than zero. This approach focuses on detecting an effect in one predetermined direction, which can provide more power when the direction of the effect is known in advance.

Omitted Variable Bias

Omitted variable bias occurs when a relevant variable is left out of the regression model, causing the estimated coefficients of the included variables to be biased if the omitted variable is correlated with them. This concept is important as it explains potential discrepancies in coefficient estimates when the model changes, such as when a variable is dropped from the analysis.

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