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## 7.3 Joint Hypothesis Testing using the F-Statistic

The estimated model is

$\widehat{TestScore} = \underset{(15.21)}{649.58} -\underset{(0.48)}{0.29} \times STR - \underset{(0.04)}{0.66} \times english + \underset{(1.41)}{3.87} \times expenditure.$

Now, can we reject the hypothesis that the coefficient on $size$ and the coefficient on $expenditure$ are zero? To answer this, we have to resort to joint hypothesis tests. A joint hypothesis imposes restrictions on multiple regression coefficients. This is different from conducting individual $t$-tests where a restriction is imposed on a single coefficient. Chapter 7.2 of the book explains why testing hypotheses about the model coefficients one at a time is different from testing them jointly.

The homoskedasticity-only $F$-Statistic is given by

$F = \frac{(SSR_{\text{restricted}} - SSR_{\text{unrestricted}})/q}{SSR_{\text{unrestricted}} / (n-k-1)}$

with $SSR_{restricted}$ being the sum of squared residuals from the restricted regression, i.e., the regression where we impose the restriction. $SSR_{unrestricted}$ is the sum of squared residuals from the full model, $q$ is the number of restrictions under the null and $k$ is the number of regressors in the unrestricted regression.

It is fairly easy to conduct $F$-tests in R. We can use the function linearHypothesis()contained in the package car.

# estimate the multiple regression model
model <- lm(score ~ STR + english + expenditure, data = CASchools)

# execute the function on the model object and provide both linear restrictions
# to be tested as strings
linearHypothesis(model, c("STR=0", "expenditure=0"))
#> Linear hypothesis test
#>
#> Hypothesis:
#> STR = 0
#> expenditure = 0
#>
#> Model 1: restricted model
#> Model 2: score ~ STR + english + expenditure
#>
#>   Res.Df   RSS Df Sum of Sq      F   Pr(>F)
#> 1    418 89000
#> 2    416 85700  2    3300.3 8.0101 0.000386 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The output reveals that the $F$-statistic for this joint hypothesis test is about $8.01$ and the corresponding $p$-value is $0.0004$. Thus, we can reject the null hypothesis that both coefficients are zero at any level of significance commonly used in practice.

A heteroskedasticity-robust version of this $F$-test (which leads to the same conclusion) can be conducted as follows:

# heteroskedasticity-robust F-test
linearHypothesis(model, c("STR=0", "expenditure=0"), white.adjust = "hc1")
#> Linear hypothesis test
#>
#> Hypothesis:
#> STR = 0
#> expenditure = 0
#>
#> Model 1: restricted model
#> Model 2: score ~ STR + english + expenditure
#>
#> Note: Coefficient covariance matrix supplied.
#>
#>   Res.Df Df      F   Pr(>F)
#> 1    418
#> 2    416  2 5.4337 0.004682 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The standard output of a model summary also reports an $F$-statistic and the corresponding $p$-value. The null hypothesis belonging to this $F$-test is that all of the population coefficients in the model except for the intercept are zero, so the hypotheses are $H_0: \beta_1=0, \ \beta_2 =0, \ \beta_3 =0 \quad \text{vs.} \quad H_1: \beta_j \neq 0 \ \text{for at least one} \ j=1,2,3.$

This is also called the overall regression $F$-statistic and the null hypothesis is obviously different from testing if only $\beta_1$ and $\beta_3$ are zero.

We now check whether the $F$-statistic belonging to the $p$-value listed in the model’s summary coincides with the result reported by linearHypothesis().

# execute the function on the model object and provide the restrictions
# to be tested as a character vector
linearHypothesis(model, c("STR=0", "english=0", "expenditure=0"))
#> Linear hypothesis test
#>
#> Hypothesis:
#> STR = 0
#> english = 0
#> expenditure = 0
#>
#> Model 1: restricted model
#> Model 2: score ~ STR + english + expenditure
#>
#>   Res.Df    RSS Df Sum of Sq      F    Pr(>F)
#> 1    419 152110
#> 2    416  85700  3     66410 107.45 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# Access the overall F-statistic from the model's summary
summary(model)\$fstatistic
#>    value    numdf    dendf
#> 107.4547   3.0000 416.0000

The entry value is the overall $F$-statistics and it equals the result of linearHypothesis(). The $F$-test rejects the null hypothesis that the model has no power in explaining test scores. It is important to know that the $F$-statistic reported by summary is not robust to heteroskedasticity.