Hypothesis Testing

Hypothesis testing is used to test whether the estimated regression coefficients are statistically significant. Hypothesis testing can be performed using the confidence interval approach or the t-test approach.

In the previous learning objective, we discussed the confidence interval approach. In this section, we will explore the t-test approach.

T-test Approach

The t-test of significance is performed following the steps below:

1. Set the significance level for the test.
2. Formulate the null and the alternative hypothesis.

     iii. Calculate the t-statistic using the formula:

$$\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}$$

Where:

• \(b_{1}\) = True slope coefficient
• \(\widehat{b_{1}}\) = Point estimator for \(b_{1}\)
• \(\widehat{S_{b_{1}}}\) = Standard error of the regression coefficient

1. Compare the absolute value of the t-statistic to the critical t value \(t_{c}\).

• Reject the null hypothesis if the absolute value of the t-statistic is greater than the critical t-value, i.e., \(t>+t_{\text{critical}}\) or \(<-t_{\text{critical}}\).

Example: Hypothesis Testing of the Significance of Regression Coefficients

Shah generates the following output from the regression analysis of inflation on unemployment.

$$\small{\begin{array}{llll}\hline{}& \textbf{Regression Statistics} &{}&{}\\ \hline{}& \text{Multiple R} & 0.8766 &{} \\ {}& \text{R Square} & 0.7684 &{} \\ {}& \text{Adjusted R Square} & 0.7394 & {}\\ {}& \text{Standard Error} & 0.0063 &{}\\ {}& \text{Observations} & 10 &{}\\ \hline {}& & & \\ \hline{} & \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t Stat}\\ \hline \text{Intercept} & 0.0710 & 0.0094 & 7.5160 \\\text{Forecast (Slope)} & -0.9041 & 0.1755 & -5.1516\\ \hline\end{array}}$$

At the 5% significant level, test the null hypothesis that the slope coefficient is significantly different from one, i.e.,

 $$ H_{0}: b_{1}= 1 vs. H_{a}: b_{1}≠1 $$

Solution

The calculated t-statistic, \(\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}\) is equal to:

$$\begin{align*}\text{t}&= \frac{-0.9041-1}{0.1755}\\&=-10.85\end{align*}$$

The critical two-tail t-values from the table with \(n-2=8\) degrees of freedom are:

$$\text{t}_{c}=±2.306$$

Notice that \(|t|>t_{c}\) i.e., (\(10.85>2.306\))

Thus, we reject the null hypothesis and conclude that the estimated slope coefficient is statistically different from one.

Question

Neeth Shinu, CFA, is forecasting price elasticity of supply for certain a product. Shinu uses the quantity of the product supplied for the past 5months as the dependent variable and price per unit of the product as the independent variable. The regression results are shown below.

$$\small{\begin{array}{lccccc}\hline \textbf{Regression Statistics} & & & & & \\ \hline \text{Multiple R} & 0.9971 & {}& {}&{}\\ \text{R Square} & 0.9941 & & & \\ \text{Adjusted R Square} & 0.9922 & & & & \\ \text{Standard Error} & 3.6515 & & & \\ \text{Observations} & 5 & & & \\ \hline {}& \textbf{Coefficients} & \textbf{Standard Error} & \textbf{t Stat} & \textbf{P-value}\\ \hline\text{Intercept} & -159 & 10.520 & (15.114) & 0.001\\ \text{Slope} & 0.26 & 0.012 & 22.517 & 0.000\\ \hline\end{array}}$$

Which of the following most likely reports the correct value of the t-statistic for the slope and most accurately evaluates its statistical significance with 95% confidence?

1. \(t=21.67\); slope is significantly different from zero.
2. \(t= 3.18\); slope is significantly different from zero.
3. \(t=22.57\); slope is not significantly different from zero.

Solution

The correct answer is A.

The t-statistic is calculated using the formula:

$$\text{t}=\frac{\widehat{b_{1}}-b_1}{\widehat{S_{b_{1}}}}$$

Where:

• \(b_{1}\) = True slope coefficient
• \(\widehat{b_{1}}\) = Point estimator for \(b_{1}\)
• \(\widehat{S_{b_{1}}}\) = Standard error of the regression coefficient

$$\begin{align*}\text{t}&=\frac{0.26-0}{0.012}\\&=21.67\end{align*}$$

The critical two-tail t-values from the t-table with \(n-2 = 3\) degrees of freedom are:

$$t_{c}=\pm 3.18$$

Reading 0: Introduction to Linear Regression

LOS 0 (f) Formulate a null and alternative hypothesis about a population value of a regression coefficient, and determine whether the null hypothesis is rejected at a given level of significance