Econ 265: Introduction to Econometrics

Topic 5: Inference

Moshi Alam

Introduction

Population model: \(y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \ldots + \beta_k x_{ki} + u_i\).

• estimate the parameters using OLS
• obtain the standard errors of the OLS estimates
• And imlement them in R

Now, based on sample estimates and standard errors. we will answer questions like:

• Whether the estimated coefficient is statistically significant at any confidence level?
  • Means: Whether the true population parameter is different from zero at any confidence level?

library(wooldridge)
wage_data <- wage1
reg1 <- lm(log(wage) ~ educ + exper + tenure, data = wage_data)
summary(reg1)

Call:
lm(formula = log(wage) ~ educ + exper + tenure, data = wage_data)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.05802 -0.29645 -0.03265  0.28788  1.42809 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.284360   0.104190   2.729  0.00656 ** 
educ        0.092029   0.007330  12.555  < 2e-16 ***
exper       0.004121   0.001723   2.391  0.01714 *  
tenure      0.022067   0.003094   7.133 3.29e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4409 on 522 degrees of freedom
Multiple R-squared:  0.316, Adjusted R-squared:  0.3121 
F-statistic: 80.39 on 3 and 522 DF,  p-value: < 2.2e-16

Refreshers from Econ 160

Normal Distributions

Random variable: \(X \sim N(\mu, \sigma^2)\)

PDF: \(f(X = x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\), where

• \(E(X) = \mu\) is the mean and \(Var(X) = \sigma^2\) is the variance
• The distirbution is symmetric around \(\mu\)

Properties:

• \(Z = \frac{X - \mu}{\sigma}\) is the standard normal random variable with mean 0 and variance 1
• Transformation of X such as \(aX + b \sim N(a\mu + b, a^2\sigma^2)\)
• If X & Y are independent normal random variables, then \(X + Y \sim N(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2)\)
• Any linear combination of normal random variables is normal

\(\chi^2_n\) and \(t_n\)

\(Z_i \sim N(0,1)\) for \(i = 1, 2, \ldots, n\) independent standard normal random variables

• The sum of squared independent standard normal random variables follows a chi-square distribution with \(n\) d.o.f.. Let \(Q = \sum_{i=1}^n Z_i^2\) \[Q \sim \chi^2_n\]
  • \(E(Q) = n\) and \(Var(Q) = 2n\)
• The ratio of a standard normal random variable and a chi-square random variable follows a t-distribution. Let \(T = \frac{Z}{\sqrt{Q/n}}\) \[T \sim t_n\]
  • \(E(T) = 0\) and \(Var(T) = n/(n-2)\) for \(n > 2\)
  • Shape of the t-distribution is similar to the normal distribution but more spread out (heavier tails). Check out Figure B-9 in Math Refresher B
• As sample size increases, the t-distribution approaches the standard normal distribution

sampling distributions

The distribution of the sample statistic (e.g., sample mean, sample variance, regression coefficients) over repeated independent sampling

• Sampling variability: The variability of the sample statistic across different samples
• Standard error: The estimate of the standard deviation of the sampling distribution of a sample statistic
• Central Limit Theorem (CLT): Let \(\{X_1, X_2, \ldots, X_n\}\) be a random sample of size \(n\) from a population with mean \(\mu\) and variance \(\sigma^2\). Then, as \(n\) approaches infinity, \(\bar{X} \sim N(\mu, \sigma^2/n)\) so that \(\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)\)
• Recall from Econ 160:
  • The sampling distribution of the sample proportion \(\hat{p}\) approaches a normal distribution with mean \(p\) and variance \(p(1-p)/n\).
  • The sampling distribution of sample means follows a \(t\)-distribution with \(n-1\) degrees of freedom

Hypothesis testing

• Null hypothesis \(H_0\): A statement about the population parameter that is assumed to be true and test against an alternative hypothesis \(H_1\)
• Type I error: Rejecting the \(H_0\) when it is true
• Type II error: Failing to reject the \(H_0\) when it is false
• Significance level: The probability of committing a Type I error, denoted by \(\alpha\)
• Confidence interval: A range of values where we are \(1-\alpha\) confident that the true population parameter lies within
  • estimate \(\pm\) critical value \(\times\) standard error
  • What happens if the CI contains the hypothesized value?
• Tests can only allow us to reject the \(H_0\) or fail to reject the \(H_0\) but never accept the \(H_0\). Why?

• p-value: The probability of observing the sample (statistic) if \(H_0\) were true
  • What does it mean when p-value is less than \(\alpha\)?
• Critical value: The value that separates the rejection region from the non-rejection region in hypothesis testing
  • What is critical value for a two-tailed test with \(\alpha = 0.05\)?
• Critical region: The range of values that leads to the rejection of the \(H_0\)
• One-tailed test: A hypothesis test that tests the \(H_0\) in one direction
• Two-tailed test: A hypothesis test that tests the \(H_0\) in both directions

Hypothesis testing in MLR

Big picture

• To test hypotheses about regression coeffcients \(\hat{\beta}_j\), we need to know:
  • the sampling distribution of the \(\hat{\beta}_j\)’s
• Once we have the sampling distribution of the \(\hat{\beta}_j\)
  • We can get the standard errors of the \(\hat{\beta}_j\)’s
    • Estimates of the standard deviation of the sampling distribution of the \(\hat{\beta}_j\)’s
• We can make inferences about the population parameters \(\beta_j\)’s’ based on
  • the estimates, standard errors and the sampling distribution of the \(\hat{\beta}_j\)’s
• Turns out that under some assumptions, the \(\hat{\beta}_j\)’s follow a \(t\) distribution (in practice)
  • It requires an assumption on the distribution of the errors \(u_i\)’s MLR.6

Assumption MLR.6:

Assumption MLR.6: Normality of Errors

The population error \(u_i\) is normally distributed with mean 0 and constant variance \(\sigma^2\) and independent of the regressors for all \(i = 1, 2, \ldots, n\) \[u_i \sim N(0, \sigma^2)\]

• This directly implies MLR.4 and MLR.5

• The full set of assumptions MLR.1-MLR.6 are called the classical linear model (CLM) assumptions. Together they imply: \[y \mid x_1, \ldots x_k \sim N(\beta_0 + \beta_1 x_{1i} + \ldots + \beta_k x_{ki}, \sigma^2)\]

• Hard to defend this very strong assumption

  • Thankfully not required for large samples (Ch-5)

Theorem 4.1: Normality of OLS Estimators

Under the CLM assumptions MLR. 1 through MLR.6, conditional on the sample values of the independent variables,

\[ \hat{\beta}_j \sim \operatorname{Normal}\left(\beta_j , \operatorname{Var}\left(\hat{\beta}_j\right)\right), \]

where \(\operatorname{Var}\left(\hat{\beta}_{j}\right)\)was given in Chapter 3 [equation (3.51)]. Therefore,

\[ \frac{\hat{\beta}_j-\beta_j}{\operatorname{sd}\left(\hat{\beta}_j\right)} \sim \operatorname{Normal}(0,1) \quad \text { for } j=0,1, \ldots, k \]

Theorem 4.2: t-distribution of OLS Estimates

Under the CLM assumptions MLR. 1 through MLR.6, conditional on the sample values of the independent variables,

\[ \frac{\hat{\beta}_j-\beta_j}{\operatorname{se}\left(\hat{\beta}_j\right)} \sim t_{n-k-1} \quad \text { for } j=0,1, \ldots, k \]

where \(k+1\) is the number of unknown parameters in the population model and \(n-k-1\) is the d.o.f.

Observe the differences!

Single hypothesis testing

Single hypothesis testing

Inference on the \(j\)th population parameter \(\beta_j\):

• Null hypothesis: \(H_0: \beta_j = b\), where \(b\) is a hypothesized value typically 0
• Note that this is a test on the \(j\)th population parameter while holding all other parameters constant
• One-sided alternative: \(H_1: \beta_j > b\) or \(H_1: \beta_j < b\)
• Two-sided alternative: \(H_1: \beta_j \neq b\)
• set b= 0. Test statistic: \(t_{\hat{\beta}_j} = \frac{\hat{\beta}_j - b}{\operatorname{se}(\hat{\beta}_j)} = \frac{\hat{\beta}_j}{\operatorname{se}(\hat{\beta}_j)}\)
• We know that the sampling distribution of \(t_{\hat{\beta}_j}\) follows…

One-sided test

Fix a level of significance \(\alpha\) (e.g., 0.05)

• alternative: \(H_1: \beta_j > 0\) (or \(H_1: \beta_j < 0)\)
  • Reject \(H_0\) if \(t_{\hat{\beta}_j} > t_{\alpha, n-k-1}^*\)
  • p-value = \(P(T > t_{\hat{\beta}_j})\) where \(T \sim t_{n-k-1}\)
  • Confidence interval: \(\hat{\beta}_j \pm t_{\alpha, n-k-1}^* \times \operatorname{se}(\hat{\beta}_j)\)
• For large samples, t-distribution \(\rightarrow\) standard normal
  • the critical value \(\approx \pm 1.65\) for \(\alpha = 0.05\)

One-sided test (right)

One-sided test (left)

Examples

library(wooldridge)
nrow(meap93) - 4 - 1 # DOF

[1] 403

summary(lm(math10 ~ totcomp + staff + enroll, data=meap93))

Call:
lm(formula = math10 ~ totcomp + staff + enroll, data = meap93)

Residuals:
    Min      1Q  Median      3Q     Max 
-22.235  -7.008  -0.807   6.097  40.689 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.2740209  6.1137938   0.372    0.710    
totcomp      0.0004586  0.0001004   4.570 6.49e-06 ***
staff        0.0479199  0.0398140   1.204    0.229    
enroll      -0.0001976  0.0002152  -0.918    0.359    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 10.24 on 404 degrees of freedom
Multiple R-squared:  0.05406,   Adjusted R-squared:  0.04704 
F-statistic: 7.697 on 3 and 404 DF,  p-value: 5.179e-05

library(wooldridge)
nrow(meap93) - 4 - 1 # DOF

[1] 403

summary(lm(math10 ~ log(totcomp) + log(staff) + log(enroll), data=meap93))

Call:
lm(formula = math10 ~ log(totcomp) + log(staff) + log(enroll), 
    data = meap93)

Residuals:
    Min      1Q  Median      3Q     Max 
-22.735  -6.838  -0.835   6.139  39.718 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -207.6649    48.7031  -4.264 2.50e-05 ***
log(totcomp)   21.1550     4.0555   5.216 2.92e-07 ***
log(staff)      3.9800     4.1897   0.950   0.3427    
log(enroll)    -1.2680     0.6932  -1.829   0.0681 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 10.18 on 404 degrees of freedom
Multiple R-squared:  0.06538,   Adjusted R-squared:  0.05844 
F-statistic:  9.42 on 3 and 404 DF,  p-value: 4.974e-06

Two-sided test

Fix a level of significance \(\alpha\) (e.g., 0.05)

• alternative: \(H_1: \beta_j \neq 0\)
  • Reject \(H_0\) if \(|t_{\hat{\beta}_j}| > t_{\alpha/2, n-k-1}^*\)
  • p-value = \(P(|T| > |t_{\hat{\beta}_j}|)\)
  • Confidence interval: \(\hat{\beta}_j \pm t_{\alpha/2, n-k-1}^* \times \operatorname{se}(\hat{\beta}_j)\)
• For large samples, t-distribution \(\rightarrow\) standard normal
  • the critical value \(\approx \pm 1.96\) for \(\alpha = 0.05\)

Two-sided test

Examples

library(wooldridge)
nrow(meap93) - 4 - 1 # DOF

[1] 403

summary(lm(math10 ~ log(totcomp) + log(staff) + log(enroll), data=meap93))

Call:
lm(formula = math10 ~ log(totcomp) + log(staff) + log(enroll), 
    data = meap93)

Residuals:
    Min      1Q  Median      3Q     Max 
-22.735  -6.838  -0.835   6.139  39.718 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -207.6649    48.7031  -4.264 2.50e-05 ***
log(totcomp)   21.1550     4.0555   5.216 2.92e-07 ***
log(staff)      3.9800     4.1897   0.950   0.3427    
log(enroll)    -1.2680     0.6932  -1.829   0.0681 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 10.18 on 404 degrees of freedom
Multiple R-squared:  0.06538,   Adjusted R-squared:  0.05844 
F-statistic:  9.42 on 3 and 404 DF,  p-value: 4.974e-06

Computing p-values

• p-value is the probability of observing the sample (statistic) if \(H_0\) is true
• Indeed you can do this using a table.
• But R can do this for you

t_value <- 2.5    #  t-value
df <- 100         #  degrees of freedom

# Two-tailed p-value
p_value <- 2 * (1 - pt(abs(t_value), df))
print(p_value)

[1] 0.01404579

# One-tailed p-value (right tail)
p_right <- 1 - pt(t_value, df)
print(p_right)

[1] 0.007022895

# One-tailed p-value (left tail)
p_left <- pt(t_value, df)
print(p_left)

[1] 0.9929771

Testing Linear combinations

\[log(wage_i) = \beta_0 + \beta_1 jc_i + \beta_2 univ_i + \beta_3 exper_i + u_i\]

library(wooldridge)
summary(lm(lwage ~ jc + univ + exper, data = twoyear))

Call:
lm(formula = lwage ~ jc + univ + exper, data = twoyear)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.10362 -0.28132  0.00551  0.28518  1.78167 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.4723256  0.0210602  69.910   <2e-16 ***
jc          0.0666967  0.0068288   9.767   <2e-16 ***
univ        0.0768762  0.0023087  33.298   <2e-16 ***
exper       0.0049442  0.0001575  31.397   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4301 on 6759 degrees of freedom
Multiple R-squared:  0.2224,    Adjusted R-squared:  0.2221 
F-statistic: 644.5 on 3 and 6759 DF,  p-value: < 2.2e-16

• Does attending \(jc_i\) has lower effect on wages as going to university \(univ_i\)?
• What are we testing?

\[log(wage_i) = \beta_0 + \beta_1 jc_i + \beta_2 univ_i + \beta_3 exper_i + u_i\] \(H_0: \beta_1 = \beta_2\) and \(H_1: \beta_1 < \beta_2\)

• Call \(\beta_1 - \beta_2 = \theta\) and so \(H_0: \theta = 0\) and \(H_1: \theta < 0\)
• Test statistic: \(t_{\hat{\theta}} = \frac{\hat{\theta}}{\operatorname{se}(\hat{\theta})} = \frac{\hat{\beta}_1 - \hat{\beta}_2}{\operatorname{se}(\hat{\beta}_1-\hat{\beta}_2)} = \frac{\hat{\beta}_1 - \hat{\beta}_2}{\sqrt{\operatorname{se}(\hat{\beta}_1)^2 + \operatorname{se}(\hat{\beta}_2)^2 + 2\widehat{\operatorname{cov}(\hat{\beta}_1, \hat{\beta}_2)}}}\)
  • Reject \(H_0\)
    • if \(t_{\hat{\theta}} < t_{\alpha, n-k-1}^*\)
    • Or if p-value = \(P(T < t_{\hat{\theta}})\) where \(T \sim t_{n-k-1}\) is less than \(\alpha\)
    • Or if CI: \([\hat{\theta} \pm t_{\alpha, n-k-1}^* \times \operatorname{se}(\hat{\theta})]\) does not contain zero
• Easy to calculate \(t_{\hat{\theta}}\) if \(\widehat{\operatorname{cov}(\hat{\beta}_1, \hat{\beta}_2)}\) is known

Implementation

• \[log(wage_i) = \beta_0 + \beta_1 jc_i + \beta_2 univ_i + \beta_3 exper_i + u_i\]
• \[log(wage_i) = \beta_0 + (\theta + \beta_2) jc_i + \beta_2 univ_i + \beta_3 exper_i + u_i\]
• \[log(wage_i) = \beta_0 + \theta jc_i + \beta_2 (jc_i + univ_i) + \beta_3 exper_i + u_i\]

R code

\[log(wage_i) = \beta_0 + \theta jc_i + \beta_2 (jc_i + univ_i) + \beta_3 exper_i + u_i\]

library(wooldridge)
summary(lm(lwage ~ jc + I(jc + univ) + exper, data = twoyear))

Call:
lm(formula = lwage ~ jc + I(jc + univ) + exper, data = twoyear)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.10362 -0.28132  0.00551  0.28518  1.78167 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)   1.4723256  0.0210602  69.910   <2e-16 ***
jc           -0.0101795  0.0069359  -1.468    0.142    
I(jc + univ)  0.0768762  0.0023087  33.298   <2e-16 ***
exper         0.0049442  0.0001575  31.397   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4301 on 6759 degrees of freedom
Multiple R-squared:  0.2224,    Adjusted R-squared:  0.2221 
F-statistic: 644.5 on 3 and 6759 DF,  p-value: < 2.2e-16

Testing Multiple Linear Restrictions

Testing multiple restrictions

• The \(t\)-test is used to test a hypothesis about a single population parameter \(\beta_j\)
• The \(F\)-test is used to test multiple hypotheses about multiple population parameters jointly
• Example: \(\log (\text { salary }_i )= \beta_0+\beta_1 \text { years }_i +\beta_2 \text { gamesyr }_i +\beta_3 \text { bavg }_i +\beta_4 \text { hrunsyr }_i +\beta_5 \text { rbisyr }_i +u_i\)
  • \(\mathrm{H}_0: \beta_3=0, \beta_4=0, \beta_5=0\)
  • \(\mathrm{H}_1:\) \(H_0\) is not true: At least one of \(\beta_3, \beta_4, \beta_5\) is not zero

library(wooldridge)
summary(lm(log(salary) ~ years + gamesyr + bavg + hrunsyr + rbisyr, data = mlb1))

Call:
lm(formula = log(salary) ~ years + gamesyr + bavg + hrunsyr + 
    rbisyr, data = mlb1)

Residuals:
     Min       1Q   Median       3Q      Max 
-3.02508 -0.45034 -0.04013  0.47014  2.68924 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.119e+01  2.888e-01  38.752  < 2e-16 ***
years       6.886e-02  1.211e-02   5.684 2.79e-08 ***
gamesyr     1.255e-02  2.647e-03   4.742 3.09e-06 ***
bavg        9.786e-04  1.104e-03   0.887    0.376    
hrunsyr     1.443e-02  1.606e-02   0.899    0.369    
rbisyr      1.077e-02  7.175e-03   1.500    0.134    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7266 on 347 degrees of freedom
Multiple R-squared:  0.6278,    Adjusted R-squared:  0.6224 
F-statistic: 117.1 on 5 and 347 DF,  p-value: < 2.2e-16

• Based on what we have learned so far, what do we think about \(\mathrm{H}_0: \beta_3=0, \beta_4=0, \beta_5=0\)?

Let us run both models

Unrestricted model

library(wooldridge)
unrestricted <-  lm(log(salary) ~ years + gamesyr + bavg + hrunsyr + rbisyr, data = mlb1)
summary(unrestricted)

Call:
lm(formula = log(salary) ~ years + gamesyr + bavg + hrunsyr + 
    rbisyr, data = mlb1)

Residuals:
     Min       1Q   Median       3Q      Max 
-3.02508 -0.45034 -0.04013  0.47014  2.68924 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.119e+01  2.888e-01  38.752  < 2e-16 ***
years       6.886e-02  1.211e-02   5.684 2.79e-08 ***
gamesyr     1.255e-02  2.647e-03   4.742 3.09e-06 ***
bavg        9.786e-04  1.104e-03   0.887    0.376    
hrunsyr     1.443e-02  1.606e-02   0.899    0.369    
rbisyr      1.077e-02  7.175e-03   1.500    0.134    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7266 on 347 degrees of freedom
Multiple R-squared:  0.6278,    Adjusted R-squared:  0.6224 
F-statistic: 117.1 on 5 and 347 DF,  p-value: < 2.2e-16

sum(unrestricted$residuals^2)

[1] 183.1863

Restricted model

library(wooldridge)
restricted <-  lm(log(salary) ~ years + gamesyr, data = mlb1)
summary(restricted)

Call:
lm(formula = log(salary) ~ years + gamesyr, data = mlb1)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.66858 -0.46412 -0.01177  0.49219  2.68829 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 11.223804   0.108312 103.625  < 2e-16 ***
years        0.071318   0.012505   5.703  2.5e-08 ***
gamesyr      0.020174   0.001343  15.023  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7527 on 350 degrees of freedom
Multiple R-squared:  0.5971,    Adjusted R-squared:  0.5948 
F-statistic: 259.3 on 2 and 350 DF,  p-value: < 2.2e-16

sum(restricted$residuals^2)

[1] 198.3115

• Is the increase in SSR large enough to reject the null hypothesis?
• But what is the test statistic that we can use?

The F-test

• \(y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \ldots + \beta_k x_{ki} + u_i\)
• Hypotheses Test of \(q\) exclusion restrictions
  • \(H_0: \beta_{k-q} = \beta_{k-q+1} = \ldots = \beta_k = 0\)
  • \(H_1:\) At least one of \(\beta_{k-q}, \beta_{k-q+1}, \ldots, \beta_k\) is not zero
• Test statistic: \(F = \frac{(SSR_r - SSR_{ur})/q}{SSR_{ur}/(n-k-1)}\)
  • Can bee re-written as \(\frac{(R^2_{UR} - R^2_R)/q}{(1 - R^2_{UR})/(n-k-1)}\)
  • Follows an \(F\)-distribution with \(q\) and \(n-k-1\) d.o.f.
  • Note:
    • \(q = dof_r - dof_{ur}\)
    • F > 0

Example F sampling distribution

Going back to the example

\(\log(salary_i)= \beta_0+\beta_1 years_i +\beta_2 gamesyr_i +\beta_3 bavg_i +\beta_4 hrunsyr_i +\beta_5 rbisyr_i +u_i\)

• \(\mathrm{H}_0: \beta_3=0, \beta_4=0, \beta_5=0\)
• \(\mathrm{H}_1:\) \(H_0\) is not true.

library(wooldridge)
unrestricted <-  lm(log(salary) ~ years + gamesyr + bavg + hrunsyr + rbisyr, data = mlb1)
restricted <-  lm(log(salary) ~ years + gamesyr, data = mlb1)
F_stat <- ((sum(restricted$residuals^2) - sum(unrestricted$residuals^2))/3)/(sum(unrestricted$residuals^2)/(nrow(mlb1) - 5-1))
print(F_stat)

[1] 9.550254

critical_F <- qf(0.05, df1 = 3, df2 = nrow(mlb1) - 5-1, lower.tail = F)
print(critical_F)

[1] 2.630641

if (F_stat > critical_F) {
  print("Reject H0")
} else {
  print("Fail to reject H0")
}

[1] "Reject H0"

\(\log(salary_i)= \beta_0+\beta_1 years_i +\beta_2 gamesyr_i +\beta_3 bavg_i +\beta_4 hrunsyr_i +\beta_5 rbisyr_i +u_i\)

• But recall when we ran the unrestricted model, we found that \(\beta_3, \beta_4, \beta_5\) were not statistically significant.
• So, whats going on here with the joint F-test?
• \(hrunsyr\) and \(rbisyr\) are highly correlated.
  • multicollinearity \(\rightarrow\) large standard errors \(\rightarrow\) low t-stats \(\rightarrow\) individual statistical insignificance
• The F-test is a joint test (including bavg) and is not affected by multicollinearity
• Hence F-tests of joint hypotheses can be useful in the presence of multicollinearity

Relationship between t and F tests

• For a single restriction \(q=1\), the F-test is equivalent to the t-test
  • \(F_{1,n-k-1} = t^2_{n-k-1}\)
• For single hypothesis testing, the t-test is more powerful
  • F-tests remain under-powered than t-tests (See Math refresher C)

Reporting Regression Results

• Report Estimated Coefficients
  • Interpret key variables’ estimates in economic or practical terms.
• Include Standard Errors
  • Preferred over just \(t\)-statistics as they help interpret hypothesis tests and confidence intervals.
• Report \(R^2\) and Other Fit Statistics
  • \(R^2\) is essential for goodness-of-fit.
    
  • Reporting F-statistics helps test exclusion restrictions.
• Summarize in Tables for Multiple Models
  • If multiple equations are estimated, use tables instead of inline equations.
    
  • Dependent variable should be clearly indicated.
    
  • Independent variables should be listed in the first column.
    
  • Standard errors in parentheses below estimates.

Salary and Benefits Tradeoff example

• Total compensation (\(totcomp\)) consists of salary and benefits:\[totcomp = salary + benefits = salary \left( 1 + \frac{benefits}{salary} \right)\]
• Taking the log transformation: \(\log(totcomp) = \log(salary) + \log(1 + b/s).\)
• For small \(b/s\), can approximate \(\log(1 + b/s) \approx b/s.\)
• This leads to the econometric model:\[\log(salary) = \beta_0 + \beta_1 (b/s) + \text{other *controls*}. \]
• Hypothesis Test: Testing the salary-benefits tradeoff:
  • \(H_0: \beta_1 = -1\) (full tradeoff)
  • \(H_1: \beta_1 \neq -1\) (partial or no tradeoff)
• Data from MEAP93 controls for enrollment, staff size, dropout, and graduation rates.

library(stargazer, wooldridge)
model1 <- lm(log(salary) ~ I(benefits/salary), data = meap93)
model2 <- lm(log(salary) ~ I(benefits/salary) + I(log(enroll)) + I(log(staff)), data = meap93)
model3 <- lm(log(salary) ~ I(benefits/salary) + I(log(enroll)) + I(log(staff)) + droprate + gradrate, data = meap93)

stargazer(model1, model2, model3, 
          type = "text", 
          title = "Regression Results",
          omit.stat = c("ser"), # Omit some stats if needed
          dep.var.labels = "Log(salary)",
          column.labels = c("Model 1", "Model 2", "Model 3"),
          covariate.labels = c("benefits to salary ratio", "log of enrollment", "log of staff", "Drop out rate", "Graduation rate", "Intercept"))

Regression Results
================================================================================================
                                                   Dependent variable:                          
                         -----------------------------------------------------------------------
                                                       Log(salary)                              
                                 Model 1                 Model 2                 Model 3        
                                   (1)                     (2)                     (3)          
------------------------------------------------------------------------------------------------
benefits to salary ratio        -0.825***               -0.605***               -0.589***       
                                 (0.200)                 (0.165)                 (0.165)        
                                                                                                
log of enrollment                                       0.087***                0.088***        
                                                         (0.007)                 (0.007)        
                                                                                                
log of staff                                            -0.222***               -0.218***       
                                                         (0.050)                 (0.050)        
                                                                                                
Drop out rate                                                                    -0.0003        
                                                                                 (0.002)        
                                                                                                
Graduation rate                                                                   0.001         
                                                                                 (0.001)        
                                                                                                
Intercept                       10.523***               10.844***               10.738***       
                                 (0.042)                 (0.252)                 (0.258)        
                                                                                                
------------------------------------------------------------------------------------------------
Observations                       408                     408                     408          
R2                                0.040                   0.353                   0.361         
Adjusted R2                       0.038                   0.348                   0.353         
F Statistic              17.050*** (df = 1; 406) 73.386*** (df = 3; 404) 45.428*** (df = 5; 402)
================================================================================================
Note:                                                                *p<0.1; **p<0.05; ***p<0.01

modelsummary package

library(modelsummary)
models <- list(
  "Model 1" = model1,
  "Model 2" = model2,
  "Model 3" = model3
)
modelsummary(models, stars = TRUE, output = "markdown")

	Model 1	Model 2	Model 3
+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001
(Intercept)	10.523***	10.844***	10.738***
	(0.042)	(0.252)	(0.258)
I(benefits/salary)	-0.825***	-0.605***	-0.589***
	(0.200)	(0.165)	(0.165)
I(log(enroll))		0.087***	0.088***
		(0.007)	(0.007)
I(log(staff))		-0.222***	-0.218***
		(0.050)	(0.050)
droprate			-0.000
			(0.002)
gradrate			0.001
			(0.001)
Num.Obs.	408	408	408
R2	0.040	0.353	0.361
R2 Adj.	0.038	0.348	0.353
AIC	8070.3	7913.7	7912.4
BIC	8082.4	7933.7	7940.5
Log.Lik.	192.417	272.763	275.397
F	17.050	73.386	45.428
RMSE	0.15	0.12	0.12