1. State the Gauss-Markov Assumptions.

The Gauss-Markov assumptions refer to the key conditions under which ordinary least squares (OLS) regression estimates are unbiased, efficient, and have minimum variance. These assumptions include:

Linearity: The relationship between the dependent variable and the independent variables is linear in parameters.

Mathematically, Yi=β0+β1​X1i+β2​X2i+…+βk​Xki​+ϵi

Zero Conditional Mean: The expected value of the error term given any value of the independent variables is zero, ensuring that the errors are not systematically related to the explanatory variables.

Mathematically, E(ϵi​∣X)=0 implies Cov(Xi,ϵji)=0

No Perfect Collinearity: The independent variables are not perfectly correlated with each other.

Homoscedasticity: The error term has constant variance across all values of the independent variables. Mathematically, it is expressed as $Var(_i)=$ for all i.

Independence: The independence assumption in linear regression pertains to the independence of errors across observations. It means that the error for one observation is not correlated with the error for another observation.

Mathematically, it is expressed as Cov(ϵi​,ϵj​)=0 for all i≠j

Normality of Errors: The error term follows a normal distribution.

Exogeneity: The independent variables are uncorrelated with the error term, ensuring that they are not influenced by factors not included in the model.

  1. Explain each assumption (what does it mean) and why we need to make it as if the crowd is non-technical i.e. understands only plain English. Avoid mathematical terms like matrix, inverse, rank, linearity, et cetra. This is a good interview question by the way. Try to explain the intuition/logic behind the assumption.

Linearity: This assumption means that the relationship between the dependent variable (the one we want to predict) and the independent variables (the ones we use to make predictions) can be represented by a straight line or a simple mathematical equation. It assumes that the changes in the dependent variable are directly proportional to the changes in the independent variables.

Independence: This assumption states that the errors or residuals (the differences between the predicted values and the actual values) for each observation are not influenced by or related to the errors of other observations. In other words, each observation’s error is independent of the errors of all other observations.

Homoscedasticity: Homoscedasticity means that the spread or variability of the errors is constant across all levels of the independent variables. This assumption implies that the errors are not systematically larger or smaller for certain values of the predictors. It ensures that our model’s predictions are equally reliable across the entire range of the independent variables.

No Perfect Collinearity: This assumption states that the independent variables are not perfectly correlated with each other. It means that there is no exact linear relationship between any two predictors. If perfect collinearity exists, it becomes impossible to estimate the individual effects of each predictor, as they are redundant or interchangeable.

Zero Conditional Mean: This assumption assumes that the average or expected value of the error term is zero for any given set of values of the independent variables. It implies that, on average, our model is not systematically overestimating or underestimating the true values of the dependent variable.

No Endogeneity: Endogeneity means that the independent variables are not influenced by or related to the error term. This assumption ensures that the predictors are not affected by any hidden factors or circumstances that may also influence the dependent variable. It helps us avoid biased and inconsistent estimates.

Normality: This assumption states that the errors follow a normal distribution. It means that the distribution of the errors is symmetric, bell-shaped, and centered around zero. Assuming normality allows us to use certain statistical techniques that rely on this distribution, such as hypothesis testing and constructing confidence intervals.

  1. Explain each assumption and why we need to make it as if the crowd has some technical background (like I assumed some matrix algebra. familiarity in the lecture).

Explanation of each Gauss-Markov assumption assuming some technical background:

Linearity: This assumption assumes that the relationship between the dependent variable and the independent variables can be expressed as a linear combination of the independent variables, multiplied by unknown parameters. It allows us to use linear regression models, which are mathematically tractable and have well-developed statistical properties. It is given by, Y = β0 + β1X1 + β2X2 + … + βp*Xp + ε Y represents the response variable. X1, X2, …, Xp represent the predictor variables. β0, β1, β2, …, βp are the regression coefficients, representing the effect of each predictor on the response. ε represents the random error term.

Independence: The assumption of independence states that the errors or residuals in our model are not correlated with each other. This is important because it enables us to treat each observation as a separate and unrelated data point, allowing for valid statistical inference and reliable estimation of model parameters.It is given by, εi ⊥ εj for all i ≠ j εi represents the residual for the ith observation. εj represents the residual for the jth observation. ⊥ denotes independence.

Homoscedasticity: Homoscedasticity assumes that the errors have constant variance across all levels of the independent variables. It ensures that the spread of the residuals is consistent, regardless of the values of the predictors. It is given by, Var(ε) = σ^2,Var(ε) represents the variance of the residuals,σ^2 represents a constant value. Violations of homoscedasticity, known as heteroscedasticity, can lead to biased standard errors and inefficient parameter estimates.

No Perfect Collinearity: Perfect collinearity refers to the situation where one or more independent variables are perfectly correlated with each other. This assumption is important because it guarantees that the design matrix, which represents the independent variables, has full rank, allowing for the existence of unique solutions to the regression equations.

Zero Conditional Mean: The assumption of zero conditional mean, also known as the exogeneity assumption, states that the expected value of the error term is zero given the values of the independent variables. This assumption is crucial for unbiasedness, as it ensures that the error term oes not have any systematic relationship with the predictors.

No Endogeneity: Endogeneity refers to situations where the independent variables are correlated with the error term. Violations of this assumption can arise due to omitted variables, measurement errors, or simultaneous causality. By assuming no endogeneity, we can establish causal relationships between the independent variables and the dependent variable.

Normality: The assumption of normality states that the errors or residuals in our model follow a normal distribution. This assumption is important for conducting hypothesis tests, constructing confidence intervals, and applying other statistical techniques that rely on the normal distribution. Departures from normality may not severely affect estimators’ consistency but can affect the efficiency and validity of statistical inference.It is given by,ε ~ N(0, σ^2) where ε represents the residual. N denotes the normal distribution. 0 represents the mean of the residuals. σ^2 represents the variance of the residuals.

II)Find a cross-sectional dataset for simplicity with more than 120 rows/observations (traditionally considered a large sample size).

  1. Run a simple linear regression. Of course, I am expecting you to explain the linear regression you are running i.e. be sure to type out the estimating equation, tell me the units of the dependent and the independent variable, and interpret the slope parameter along with the intercept term. Do you find your coefficients are statistically significant (good interview question again), and if so at what level (alpha)? Is the economic magnitude meaningful?
# Load the iris dataset
data(iris)

str(iris)
## 'data.frame':    150 obs. of  5 variables:
##  $ Sepal.Length: num  5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
##  $ Sepal.Width : num  3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
##  $ Petal.Length: num  1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
##  $ Petal.Width : num  0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
##  $ Species     : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...
# Run a simple linear regression
lm_model <- lm(Petal.Length ~ Sepal.Length, data = iris)

# Print the regression results
summary(lm_model)
## 
## Call:
## lm(formula = Petal.Length ~ Sepal.Length, data = iris)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.47747 -0.59072 -0.00668  0.60484  2.49512 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -7.10144    0.50666  -14.02   <2e-16 ***
## Sepal.Length  1.85843    0.08586   21.65   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8678 on 148 degrees of freedom
## Multiple R-squared:   0.76,  Adjusted R-squared:  0.7583 
## F-statistic: 468.6 on 1 and 148 DF,  p-value: < 2.2e-16

The estimating equation of the simple linear regression model is:

Petal.Length = β₀ + β₁ * Sepal.Length + ε

where Petal.Length represents the dependent variable (in centimeters), Sepal.Length represents the independent variable (in centimeters), β₀ is the intercept term, β₁ is the slope parameter, and ε represents the error term.

Interpretation of the coefficients:

Intercept (β₀): It represents the estimated average petal length when the sepal length is zero. In the context of the iris dataset, this interpretation may not have a meaningful real-world interpretation since sepal length cannot be zero in practice.

Slope (β₁): It represents the estimated change in petal length (in centimeters) for a one-unit increase in sepal length. In other words, it tells us the expected increase or decrease in petal length associated with a one-unit change in sepal length. To determine the statistical significance of the coefficients, we can look at the p-values in the regression summary. The p-value represents the probability of observing a coefficient as extreme as the estimated coefficient, assuming the null hypothesis that there is no relationship between the variables.

If the p-value is less than a specified significance level (alpha), typically 0.05, we consider the coefficient statistically significant. The regression summary in R provides the p-values for each coefficient.

The economic magnitude of the coefficients depends on the specific context and measurement units of the variables. In this case, the slope coefficient (β₁) represents the change in petal length (in centimeters) for a one-unit increase in sepal length. The magnitude of the coefficient indicates the strength of the relationship between the variables. It is meaningful if the coefficient’s magnitude is practically significant and aligns with the context of the problem and the specific domain knowledge.

  1. Store the regression results in an object called “my_reg”.
# Load the iris dataset
data(iris)

# Run a simple linear regression and store the results in "my_reg"
my_reg <- lm(Petal.Length ~ Sepal.Length, data = iris)

# Print the regression results
summary(my_reg)
## 
## Call:
## lm(formula = Petal.Length ~ Sepal.Length, data = iris)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.47747 -0.59072 -0.00668  0.60484  2.49512 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -7.10144    0.50666  -14.02   <2e-16 ***
## Sepal.Length  1.85843    0.08586   21.65   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8678 on 148 degrees of freedom
## Multiple R-squared:   0.76,  Adjusted R-squared:  0.7583 
## F-statistic: 468.6 on 1 and 148 DF,  p-value: < 2.2e-16

Now, create the 4 linear regression plots we saw in class using the “plot(my_reg)” command in R.
1. Tell us what each of the 4 charts measures what is the logic behind the chart setup? Also, are there any trends/rules of thumb you should rely on to analyze the model fit from these charts? Please try to make your own notes Download notesfrom the textbooks/internet. 8 sentences min.

# Load the required libraries
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.4.1
library(car)
## Warning: package 'car' was built under R version 4.4.1
## Loading required package: carData
## Warning: package 'carData' was built under R version 4.4.1
# Load the iris dataset
data(iris)

# Run a simple linear regression
my_reg <- lm(Petal.Length ~ Sepal.Length, data = iris)

# Generate the diagnostic charts
# Residuals vs. Fitted Values Plot
plot1 <- ggplot(my_reg, aes(.fitted, .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, color = "red", linetype = "dashed") +
  xlab("Fitted Values") +
  ylab("Residuals") +
  ggtitle("Residuals vs. Fitted Values Plot")

# Normal Q-Q Plot
plot2 <- ggplot(my_reg, aes(sample = .stdresid)) +
  stat_qq() +
  stat_qq_line(color = "red") +
  xlab("Theoretical Quantiles") +
  ylab("Standardized Residuals") +
  ggtitle("Normal Q-Q Plot")

# Scale-Location Plot
plot3 <- ggplot(my_reg, aes(.fitted, sqrt(abs(.stdresid)))) +
  geom_point() +
  geom_smooth() +
  xlab("Fitted Values") +
  ylab("Square Root of Absolute Standardized Residuals") +
  ggtitle("Scale-Location Plot")

# Residuals vs. Leverage Plot
plot4 <- ggplot(my_reg, aes(.hat, .stdresid)) +
  geom_point() +
  geom_hline(yintercept = 0, color = "red", linetype = "dashed") +
  xlab("Leverage") +
  ylab("Standardized Residuals") +
  ggtitle("Residuals vs. Leverage Plot")

# Display the charts
plot1

plot2

plot3
## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

plot4

When analyzing the fit of a linear regression model, there are several diagnostic charts that can provide insights into the model’s performance. Here are four commonly used charts and their purposes:

Residuals vs. Fitted Values Plot: This chart plots the residuals (vertical axis) against the predicted or fitted values (horizontal axis) from the regression model.The red dashed line at y = 0 represents the expected value of residuals when the model is a good fit. The logic behind this chart is to examine whether there is a pattern or relationship between the residuals and the predicted values. Ideally, we want to see a random scatter of points with no discernible pattern. Rules of Thumb: If the points form a horizontal band around zero, it suggests that the model assumptions of constant variance and linearity are reasonable. If there is a U-shaped or inverted U-shaped pattern, it may indicate a non-linear relationship between the dependent variable and the predictors.

Normal Q-Q Plot (Quantile-Quantile Plot): This chart assesses the normality assumption of the residuals.The x-axis represents the quantiles of the theoretical normal distribution, while the y-axis represents the quantiles of the residuals. The logic is to compare the observed quantiles of the residuals against the quantiles expected from a normal distribution. If the residuals follow a normal distribution, the points on the plot should closely follow a straight line. Rules of Thumb: If the points lie approximately along a straight line, it suggests that the residuals follow a normal distribution. Deviations from the straight line indicate departures from normality. For example, if the points curve upwards or downwards, it suggests non-normality.

Scale-Location Plot (Square Root of Residuals vs. Fitted Values): This chart examines the assumption of constant variance (homoscedasticity) of the residuals.The spread of points around a horizontal line represents the variability of the residuals at different levels of the fitted values. The logic is to plot the square root of the absolute values of the residuals (vertical axis) against the predicted or fitted values (horizontal axis). Rules of Thumb: If the points form a horizontal band around zero, it suggests that the assumption of constant variance is reasonable. If the points show a funnel shape or a discernible pattern, it indicates heteroscedasticity, where the spread of the residuals varies across the range of the predicted values.

Residuals vs. Leverage Plot: This chart helps identify influential observations, which can have a significant impact on the regression model. The logic is to plot the standardized residuals (vertical axis) against the leverage values (horizontal axis), which measure the influence of each observation on the regression coefficients. Rules of Thumb: Points that fall outside a certain threshold (e.g., Cook’s distance) may be considered influential observations. High leverage points (extreme values of the predictor variables) can have a disproportionate influence on the regression results.

  1. Now, given the linear regression you ran, what is the chart suggesting? Are Gauss-Markov Assumptions are seriously violated? 4 sentences max.

Based on the diagnostic charts for the linear regression model, the Residuals vs. Fitted Values Plot shows a random scatter of points around zero, suggesting that the assumption of constant variance is reasonable. The Normal Q-Q Plot indicates that the residuals approximately follow a straight line, implying that the assumption of normality is not seriously violated. The Scale-Location Plot exhibits a relatively constant spread of residuals, supporting the assumption of constant variance. The Residuals vs. Leverage Plot does not show any influential observations that seriously violate the Gauss-Markov assumptions. Overall, the charts suggest that the Gauss-Markov assumptions are not seriously violated by the linear regression model.

  1. Sometimes it helps if you transform a variable (log, square root, et cetra) in terms of better model fit (Overcome problems due to nonconstant variance or Overcome problems due to nonlinearity), but it can make the interpretation of coefficients harder. Try to play around by transforming the variables and tell us if the model fit improves or not. 3 sentences max.
# Load the required libraries
library(ggplot2)
library(car)

# Load the iris dataset
data(iris)

# Original model
orig_model <- lm(Petal.Length ~ Sepal.Length, data = iris)

# Function to assess model fit
assess_model_fit <- function(model) {
  plot1 <- plot(model, which = 1)
  plot2 <- plot(model, which = 2)
  plot3 <- plot(model, which = 3)
  plot4 <- plot(model, which = 5)
  
  list(plot1 = plot1, plot2 = plot2, plot3 = plot3, plot4 = plot4)
}

# Assess model fit of the original model
original_fit <- assess_model_fit(orig_model)

# Transformation: Log transformation of Sepal.Length
iris$log_Sepal_Length <- log(iris$Sepal.Length)
log_model <- lm(Petal.Length ~ log_Sepal_Length, data = iris)
log_fit <- assess_model_fit(log_model)

# Transformation: Square root transformation of Sepal.Length
iris$sqrt_Sepal_Length <- sqrt(iris$Sepal.Length)
sqrt_model <- lm(Petal.Length ~ sqrt_Sepal_Length, data = iris)
sqrt_fit <- assess_model_fit(sqrt_model)

# Transformation: Log transformation of Petal.Length
iris$log_Petal_Length <- log(iris$Petal.Length)
log_Petal_Model <- lm(log_Petal_Length ~ Sepal.Length, data = iris)
log_Petal_fit <- assess_model_fit(log_Petal_Model)

# Compare model fit for different transformations
comparison <- list(
  original_fit = original_fit,
  log_fit = log_fit,
  sqrt_fit = sqrt_fit,
  log_Petal_fit = log_Petal_fit
)

comparison
## $original_fit
## $original_fit$plot1
## NULL
## 
## $original_fit$plot2
## NULL
## 
## $original_fit$plot3
## NULL
## 
## $original_fit$plot4
## NULL
## 
## 
## $log_fit
## $log_fit$plot1
## NULL
## 
## $log_fit$plot2
## NULL
## 
## $log_fit$plot3
## NULL
## 
## $log_fit$plot4
## NULL
## 
## 
## $sqrt_fit
## $sqrt_fit$plot1
## NULL
## 
## $sqrt_fit$plot2
## NULL
## 
## $sqrt_fit$plot3
## NULL
## 
## $sqrt_fit$plot4
## NULL
## 
## 
## $log_Petal_fit
## $log_Petal_fit$plot1
## NULL
## 
## $log_Petal_fit$plot2
## NULL
## 
## $log_Petal_fit$plot3
## NULL
## 
## $log_Petal_fit$plot4
## NULL

Based on the comparison of the diagnostic charts, the log transformation of Sepal.Length shows a slight improvement in model fit, particularly in terms of the linearity and normality assumptions. However, the square root transformation of Sepal.Length and log transformation of Petal.Length do not appear to significantly enhance the model fit compared to the original model.

By transforming variables in the linear regression model, it is possible to improve the model fit, especially when dealing with issues related to nonconstant variance or nonlinearity. However, it may make the interpretation of coefficients more challenging as the transformed variables have different scales or units. Experimenting with variable transformations can potentially enhance the model fit, but careful consideration is required when interpreting the coefficients in the transformed model.