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7.1 Hypothesis Tests and Confidence Intervals for a Single Coefficient

We first discuss how to compute standard errors, how to test hypotheses and how to construct confidence intervals for a single regression coefficient $\beta_j$ in a multiple regression model. The basic idea is summarized in Key Concept 7.1.

Testing the Hypothesis $\beta_j = \beta_{j,0}$ Against the Alternative $\beta_j \neq \beta_{j,0}$

1. Compute the standard error of $\hat{\beta_j}$
2. Compute the $t$-statistic, $t^{act} = \frac{\hat{\beta}_j - \beta_{j,0}} {SE(\hat{\beta_j})}$
3. Compute the $p$-value, $p\text{-value} = 2 \Phi(-|t^{act}|)$
where $t^{act}$ is the value of the $t$-statistic actually computed. Reject the hypothesis at the $5\%$ significance level if the $p$-value is less than $0.05$ or, equivalently, if $|t^{act}| > 1.96$. The standard error and (typically) the $t$-statistic and the corresponding $p$-value for testing $\beta_j = 0$ are computed automatically by suitable R functions, e.g., by summary.

Testing a single hypothesis about the significance of a coefficient in the multiple regression model proceeds as in in the simple regression model.

You can easily see this by inspecting the coefficient summary of the regression model

$TestScore = \beta_0 + \beta_1 \times size \beta_2 \times english + u$

already discussed in Chapter 6. Let us review this:

model <- lm(score ~ size + english, data = CASchools)
coeftest(model, vcov. = vcovHC, type = "HC1")
#>
#> t test of coefficients:
#>
#>               Estimate Std. Error  t value Pr(>|t|)
#> (Intercept) 686.032245   8.728225  78.5993  < 2e-16 ***
#> size         -1.101296   0.432847  -2.5443  0.01131 *
#> english      -0.649777   0.031032 -20.9391  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

You may check that these quantities are computed as in the simple regression model by computing the $t$-statistics or $p$-values by hand using the output above and R as a calculator.

For example, using the definition of the $p$-value for a two-sided test as given in Key Concept 7.1, we can confirm the $p$-value for a test of the hypothesis that the coefficient $\beta_1$, the coefficient on size, to be approximately zero.

# compute two-sided p-value
2 * (1 - pt(abs(coeftest(model, vcov. = vcovHC, type = "HC1")[2, 3]),
df = model\$df.residual))
#>  0.01130921

Confidence Intervals for a Single Coefficient in Multiple Regression

A $95\%$ two-sided confidence interval for the coefficient $\beta_j$ is an interval that contains the true value of $\beta_j$ with a $95 \%$ probability; that is, it contains the true value of $\beta_j$ in $95 \%$ of all repeated samples. Equivalently, it is the set of values of $\beta_j$ that cannot be rejected by a $5 \%$ two-sided hypothesis test. When the sample size is large, the $95 \%$ confidence interval for $\beta_j$ is $\left[\hat{\beta_j}- 1.96 \times SE(\hat{\beta}_j), \hat{\beta_j} + 1.96 \times SE(\hat{\beta_j})\right].$