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Model Specification

The initial regression model is given by:

\[ \widehat{\log(salary)} = 11.19 + 0.0689 \cdot years + 0.0126 \cdot gamesyr + 0.00098 \cdot bavg + 0.0144 \cdot hrunsyr + 0.018 \cdot rbisyr \]

Where: - years: Number of years the player has been in the league - gamesyr: Games per year - bavg: Batting average - hrunsyr: Home runs per year - rbisyr: Runs batted in per year

Given: - \(n = 353\) - \(SSR = 183.186\) - \(R^2 = 0.6278\)

Data Import

mlb1 <- read_excel("mlb1.xlsx")
mlb1 <- na.omit(mlb1)
head(mlb1)

## # A tibble: 6 × 47
##   allstar atbats atbatsyr  bavg    bb black blackpop blckpb blckph catcher
##     <dbl>  <dbl>    <dbl> <dbl> <dbl> <dbl>    <dbl>  <dbl>  <dbl>   <dbl>
## 1    75     6705     559.   289   619     0  1547725    0      0         0
## 2    25     3333     417.   259   137     1  1547725   18.8   10.9       0
## 3     0     2807     561.   299   341     0  1547725    0      0         0
## 4     0     3337     417.   245   306     0  1547725    0      0         0
## 5     0     3603     300.   258   316     1  1547725   18.8   10.9       0
## 6    11.8   7489     441.   286   416     1  1547725   18.8   10.9       0
## # ℹ 37 more variables: doubles <dbl>, fldperc <dbl>, frstbase <dbl>,
## #   games <dbl>, gamesyr <dbl>, hispan <dbl>, hisppb <dbl>, hispph <dbl>,
## #   hisppop <dbl>, hits <dbl>, hruns <dbl>, hrunsyr <dbl>, lsalary <dbl>,
## #   nl <dbl>, outfield <dbl>, pcinc <dbl>, percblck <dbl>, perchisp <dbl>,
## #   percwhte <dbl>, rbis <dbl>, rbisyr <dbl>, runs <dbl>, runsyr <dbl>,
## #   salary <dbl>, sbases <dbl>, sbasesyr <dbl>, scndbase <dbl>, shrtstop <dbl>,
## #   slugavg <dbl>, so <dbl>, teamsal <dbl>, thrdbase <dbl>, triples <dbl>, …

Full Model (With `rbisyr`)

full_model <- lm(lsalary ~ years + gamesyr + bavg + hrunsyr + rbisyr, data = mlb1)
summary(full_model)

## 
## Call:
## lm(formula = lsalary ~ years + gamesyr + bavg + hrunsyr + rbisyr, 
##     data = mlb1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.0117 -0.4642 -0.0524  0.4629  2.6709 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.164772   0.352595  31.665  < 2e-16 ***
## years        0.073822   0.012751   5.789 1.67e-08 ***
## gamesyr      0.011410   0.002773   4.115 4.92e-05 ***
## bavg         0.001317   0.001424   0.925    0.356    
## hrunsyr      0.010137   0.016643   0.609    0.543    
## rbisyr       0.011915   0.007453   1.599    0.111    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7319 on 324 degrees of freedom
## Multiple R-squared:  0.6101, Adjusted R-squared:  0.6041 
## F-statistic: 101.4 on 5 and 324 DF,  p-value: < 2.2e-16

tidy(full_model)

## # A tibble: 6 × 5
##   term        estimate std.error statistic   p.value
##   <chr>          <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept) 11.2       0.353      31.7   3.38e-101
## 2 years        0.0738    0.0128      5.79  1.67e-  8
## 3 gamesyr      0.0114    0.00277     4.11  4.92e-  5
## 4 bavg         0.00132   0.00142     0.925 3.56e-  1
## 5 hrunsyr      0.0101    0.0166      0.609 5.43e-  1
## 6 rbisyr       0.0119    0.00745     1.60  1.11e-  1

Restricted Model (Without `rbisyr`)

restricted_model <- lm(lsalary ~ years + gamesyr + bavg + hrunsyr, data = mlb1)
summary(restricted_model)

## 
## Call:
## lm(formula = lsalary ~ years + gamesyr + bavg + hrunsyr, data = mlb1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3.05255 -0.46456 -0.02836  0.47082  2.71721 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 10.929535   0.321197  34.028  < 2e-16 ***
## years        0.072366   0.012749   5.676 3.05e-08 ***
## gamesyr      0.014909   0.001707   8.732  < 2e-16 ***
## bavg         0.002004   0.001361   1.472    0.142    
## hrunsyr      0.033936   0.007463   4.547 7.69e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7336 on 325 degrees of freedom
## Multiple R-squared:  0.607,  Adjusted R-squared:  0.6022 
## F-statistic: 125.5 on 4 and 325 DF,  p-value: < 2.2e-16

tidy(restricted_model)

## # A tibble: 5 × 5
##   term        estimate std.error statistic   p.value
##   <chr>          <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept) 10.9       0.321       34.0  3.76e-109
## 2 years        0.0724    0.0127       5.68 3.05e-  8
## 3 gamesyr      0.0149    0.00171      8.73 1.34e- 16
## 4 bavg         0.00200   0.00136      1.47 1.42e-  1
## 5 hrunsyr      0.0339    0.00746      4.55 7.69e-  6

Comparison of `hrunsyr` Coefficients and Significance

full_coef <- tidy(full_model) %>% filter(term == "hrunsyr")
rest_coef <- tidy(restricted_model) %>% filter(term == "hrunsyr")

comparison <- data.frame(
  Model = c("Full Model", "Restricted Model"),
  Coefficient = c(full_coef$estimate, rest_coef$estimate),
  P_Value = c(full_coef$p.value, rest_coef$p.value)
)
comparison

##              Model Coefficient      P_Value
## 1       Full Model  0.01013740 5.428752e-01
## 2 Restricted Model  0.03393551 7.685337e-06

Interpretation

Statistical Significance: In the full model, hrunsyr is not statistically significant (p ≈ 0.369), but in the restricted model it becomes highly significant (p < 0.000001).
Coefficient Size: The coefficient increases from approximately 0.0144 to 0.0359 when rbisyr is dropped.

Conclusion

This shift suggests multicollinearity between hrunsyr and rbisyr. Including both in the model may obscure the true impact of hrunsyr, while removing rbisyr allows the model to better capture its contribution to salary.

Expanded Model (Including `runsyr`, `fldperc`, `sbasesyr`)

expanded_model <- lm(lsalary ~ years + gamesyr + bavg + hrunsyr + rbisyr + runsyr + fldperc + sbasesyr, data = mlb1)
summary(expanded_model)

## 
## Call:
## lm(formula = lsalary ~ years + gamesyr + bavg + hrunsyr + rbisyr + 
##     runsyr + fldperc + sbasesyr, data = mlb1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.13031 -0.45220 -0.07121  0.46968  2.56783 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 10.8523334  2.2091271   4.912 1.44e-06 ***
## years        0.0759872  0.0127027   5.982 5.88e-09 ***
## gamesyr      0.0066853  0.0032791   2.039  0.04229 *  
## bavg         0.0006762  0.0015110   0.447  0.65483    
## hrunsyr      0.0127690  0.0166279   0.768  0.44310    
## rbisyr       0.0038911  0.0080932   0.481  0.63099    
## runsyr       0.0167610  0.0056457   2.969  0.00321 ** 
## fldperc      0.0005902  0.0021463   0.275  0.78350    
## sbasesyr    -0.0081117  0.0058711  -1.382  0.16805    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7248 on 321 degrees of freedom
## Multiple R-squared:  0.6211, Adjusted R-squared:  0.6117 
## F-statistic: 65.78 on 8 and 321 DF,  p-value: < 2.2e-16

tidy(expanded_model)

## # A tibble: 9 × 5
##   term         estimate std.error statistic       p.value
##   <chr>           <dbl>     <dbl>     <dbl>         <dbl>
## 1 (Intercept) 10.9        2.21        4.91  0.00000144   
## 2 years        0.0760     0.0127      5.98  0.00000000588
## 3 gamesyr      0.00669    0.00328     2.04  0.0423       
## 4 bavg         0.000676   0.00151     0.447 0.655        
## 5 hrunsyr      0.0128     0.0166      0.768 0.443        
## 6 rbisyr       0.00389    0.00809     0.481 0.631        
## 7 runsyr       0.0168     0.00565     2.97  0.00321      
## 8 fldperc      0.000590   0.00215     0.275 0.784        
## 9 sbasesyr    -0.00811    0.00587    -1.38  0.168

Significance of New Variables

new_vars <- tidy(expanded_model) %>% filter(term %in% c("runsyr", "fldperc", "sbasesyr"))
new_vars

## # A tibble: 3 × 5
##   term      estimate std.error statistic p.value
##   <chr>        <dbl>     <dbl>     <dbl>   <dbl>
## 1 runsyr    0.0168     0.00565     2.97  0.00321
## 2 fldperc   0.000590   0.00215     0.275 0.784  
## 3 sbasesyr -0.00811    0.00587    -1.38  0.168

Interpretation

New Variables: The table above shows the p-values for runsyr, fldperc, and sbasesyr. Significance is typically assessed at the 5% level.

Conclusion

The inclusion of additional performance metrics gives a broader view of salary determinants. The change in significance of hrunsyr when dropping rbisyr highlights multicollinearity. From the expanded model, we can determine that from three new factors, runsyr (runs per year) is individually significant contributors to salary since it has p-values 0.00321 which is < 5%.

Joint Significance Test of `bavg`, `fldperc`, and `sbasesyr`

restricted_joint <- lm(lsalary ~ years + gamesyr + hrunsyr + rbisyr + runsyr, data = mlb1)
anova(restricted_joint, expanded_model)

## Analysis of Variance Table
## 
## Model 1: lsalary ~ years + gamesyr + hrunsyr + rbisyr + runsyr
## Model 2: lsalary ~ years + gamesyr + bavg + hrunsyr + rbisyr + runsyr + 
##     fldperc + sbasesyr
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1    324 169.75                           
## 2    321 168.63  3    1.1182 0.7095 0.5469

Interpretation

If the p-value < 0.05, we’ll reject the null hypothesis, meaning bavg, fldperc, and sbasesyr are jointly significant.
If the p-value ≥ 0.05, we’ll fail to reject the null hypothesis, meaning there is no strong joint evidence that these variables improve the model.
Joint Significance: Since The F-test yields an F-value of 0.7095 and a p-value of 0.5469, which is well above the conventional 0.05 significance level. This means we fail to reject the null hypothesis, suggesting that bavg, fldperc, and sbasesyr are not jointly significant in explaining salary. In other words, there is insufficient evidence that these three variables together add explanatory power to the model.

Conclusion

From the joint hypothesis test, we can determine that joint contributions of bavg, fldperc, and sbasesyr are not significant explanatory power variables to salary.

Appendix: Model Output Tables

stargazer(full_model, restricted_model, expanded_model, type = "html", title = "Model Comparison")

**Model Comparison**

	Dependent variable:

	lsalary
	(1)	(2)	(3)

years	0.074^***	0.072^***	0.076^***
	(0.013)	(0.013)	(0.013)

gamesyr	0.011^***	0.015^***	0.007^**
	(0.003)	(0.002)	(0.003)

bavg	0.001	0.002	0.001
	(0.001)	(0.001)	(0.002)

hrunsyr	0.010	0.034^***	0.013
	(0.017)	(0.007)	(0.017)

rbisyr	0.012		0.004
	(0.007)		(0.008)

runsyr			0.017^***
			(0.006)

fldperc			0.001
			(0.002)

sbasesyr			-0.008
			(0.006)

Constant	11.165^***	10.930^***	10.852^***
	(0.353)	(0.321)	(2.209)


Observations	330	330	330
R²	0.610	0.607	0.621
Adjusted R²	0.604	0.602	0.612
Residual Std. Error	0.732 (df = 324)	0.734 (df = 325)	0.725 (df = 321)
F Statistic	101.384^*** (df = 5; 324)	125.489^*** (df = 4; 325)	65.776^*** (df = 8; 321)

Note:	p<0.1; p<0.05; p<0.01

WOOLDRIDGE Chapter 4 Exercise C.5 - MLB1

Robby Fathir Nashary

2025-05-24

R Markdown

Model Specification

Data Import

Full Model (With `rbisyr`)

Restricted Model (Without `rbisyr`)

Comparison of `hrunsyr` Coefficients and Significance

Interpretation

Conclusion

Expanded Model (Including `runsyr`, `fldperc`, `sbasesyr`)

Significance of New Variables

Interpretation

Conclusion

Joint Significance Test of `bavg`, `fldperc`, and `sbasesyr`

Interpretation

Conclusion

Appendix: Model Output Tables

WOOLDRIDGE Chapter 4 Exercise C.5 - MLB1

Robby Fathir Nashary

2025-05-24

R Markdown

Model Specification

Data Import

Full Model (With rbisyr)

Restricted Model (Without rbisyr)

Comparison of hrunsyr Coefficients and Significance

Interpretation

Conclusion

Expanded Model (Including runsyr, fldperc, sbasesyr)

Significance of New Variables

Interpretation

Conclusion

Joint Significance Test of bavg, fldperc, and sbasesyr

Interpretation

Conclusion

Appendix: Model Output Tables

Full Model (With `rbisyr`)

Restricted Model (Without `rbisyr`)

Comparison of `hrunsyr` Coefficients and Significance

Expanded Model (Including `runsyr`, `fldperc`, `sbasesyr`)

Joint Significance Test of `bavg`, `fldperc`, and `sbasesyr`