I.

1.

# Load the dataset
data(mtcars)

# Fit a multivariate regression model
model <- lm(mpg ~ wt + hp, data = mtcars)

# Print the estimating equation
summary(model)
## 
## Call:
## lm(formula = mpg ~ wt + hp, data = mtcars)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -3.941 -1.600 -0.182  1.050  5.854 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 37.22727    1.59879  23.285  < 2e-16 ***
## wt          -3.87783    0.63273  -6.129 1.12e-06 ***
## hp          -0.03177    0.00903  -3.519  0.00145 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.593 on 29 degrees of freedom
## Multiple R-squared:  0.8268, Adjusted R-squared:  0.8148 
## F-statistic: 69.21 on 2 and 29 DF,  p-value: 9.109e-12
library(stargazer)
## 
## Please cite as:
##  Hlavac, Marek (2022). stargazer: Well-Formatted Regression and Summary Statistics Tables.
##  R package version 5.2.3. https://CRAN.R-project.org/package=stargazer
stargazer(model, type = "text")
## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                                 mpg            
## -----------------------------------------------
## wt                           -3.878***         
##                               (0.633)          
##                                                
## hp                           -0.032***         
##                               (0.009)          
##                                                
## Constant                     37.227***         
##                               (1.599)          
##                                                
## -----------------------------------------------
## Observations                    32             
## R2                             0.827           
## Adjusted R2                    0.815           
## Residual Std. Error       2.593 (df = 29)      
## F Statistic           69.211*** (df = 2; 29)   
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

2.

The regression output shows that the slope coefficient for the variable “wt” is -3.87783 and for “hp” is -0.03177. Both coefficients are negative, which is expected, indicating that as the weight and horsepower increase, the miles per gallon (mpg) decreases. The magnitude of the coefficients is meaningful as they indicate the change in the dependent variable associated with a one-unit increase in the independent variable. For example, a one-unit increase in weight is associated with a decrease of approximately 3.88 miles per gallon, holding other variables constant. Similarly, a one-unit increase in horsepower is associated with a decrease of approximately 0.032 miles per gallon. The statistical significance is indicated by the p-values, which are all less than 0.05, suggesting that the effects of both variables on mpg are statistically significant.

3.

Residuals represent the differences between the observed values of the dependent variable (mpg) and the predicted values from the regression model. They provide information about the unexplained variation in the data. By examining the residuals, we can assess the goodness of fit of the model and check for violations of assumptions. In this case, the residuals range from -3.941 to 5.854, indicating that there are some observations for which the model’s predictions differ from the actual values by a relatively large amount. The residuals appear to be approximately symmetrically distributed around zero, which is a desirable property. However, there may be some heteroscedasticity present, as the spread of the residuals appears to increase as the predicted values increase.

4.

The Gauss Markov assumptions are a set of conditions that, when satisfied, make the ordinary least squares (OLS) estimator the Best Linear Unbiased Estimator (BLUE). The key assumptions are: a. Linearity: The relationship between the dependent variable and the independent variables is linear. b. Independence: The observations in the dataset are independent of each other. c. Homoscedasticity: The variance of the error term is constant across all levels of the independent variables. d. No endogeneity: The error term is not correlated with the independent variables.

5.

“OLS is BLUE” stands for “Ordinary Least Squares is Best Linear Unbiased Estimator.” It means that under the Gauss Markov assumptions, the OLS estimator is the best among all linear unbiased estimators in terms of having the minimum variance. In other words, among all possible linear combinations of the observed values of the dependent variable, the OLS estimator has the smallest variance, making it the most efficient estimator. The property of being unbiased means that, on average, the OLS estimator is equal to the true population parameter being estimated.

II.

Firstly, it can help address issues related to non-linear relationships between variables. In many cases, the relationship between the dependent variable and an independent variable may not be linear. By taking the logarithm of the variable, you can transform the relationship into a logarithmic scale, which can make it easier to capture non-linear patterns. This transformation allows for a more flexible and accurate representation of the relationship within a linear regression framework. Secondly, taking the logarithm of a variable can help stabilize the variance. In some situations, the variability of the dependent variable may increase or decrease disproportionately as the independent variable changes. This violation of the homoscedasticity assumption can lead to biased and inefficient parameter estimates. By applying a logarithmic transformation, you can often mitigate heteroscedasticity, making the residuals more homoscedastic and improving the reliability of the regression results. Lastly, the logarithmic transformation can enable a more interpretable interpretation of the regression coefficients. When the dependent variable is transformed, the coefficients in the regression model represent percentage changes rather than absolute changes. This percentage change interpretation can be more intuitive and meaningful in certain contexts, particularly when dealing with variables that exhibit exponential growth or decay. It allows for easier comparisons and facilitates a better understanding of the relationships between the variables.