Practical exercise chapter 3 Linear multiple regression.Rmd

---
title: "Practical exercise chapter 3 Linear multiple regression"
author: "Dr.BENGHALEM Abdelhadi"
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---

# Practical Exercise 2

Consider the model with three explanatory variables: `𝒚_𝑡 = 𝜷_𝟎 + 𝜷_𝟏 𝑿_𝟏𝒕 + 𝜷_𝟐 𝑿_𝟐𝒕+ 𝜷_𝟑 𝑿_𝟑𝒕+𝐮_𝒕`

We have the data from Table 1.
Put the model in matrix form, specifying the dimensions of each of the matrices.
Estimate the parameters of the model.
Calculate the estimate of the error variance as well as the standard deviations of each of the coefficients.
Calculate the R-squared and the adjusted R-squared.

### **Step 1: Data Preparation**

```{r}
# Data from Table 1
data <- data.frame(
  t = 1:14,
  y = c(12, 14, 10, 16, 14, 19, 21, 19, 21, 16, 19, 21, 25, 21),
  X_1 = c(2, 1, 3, 6, 7, 8, 8, 5, 5, 8, 4, 9, 12, 7),
  X_2 = c(45, 43, 43, 47, 42, 41, 32, 33, 41, 38, 32, 31, 35, 29),
  X_3 = c(121, 132, 154, 145, 129, 156, 132, 147, 128, 163, 161, 172, 174, 180)
)
data
```

### The process to calculate the coefficients using matrices

```{r}
# Define the matrices
X <- cbind(rep(1, nrow(data)), data$X_1, data$X_2, data$X_3)  # Design matrix X
Y <- data$y  # Dependent variable Y

# Calculate X'X
XtX <- t(X) %*% X

# Calculate (X'X)^-1
XtX_inverse <- solve(XtX)

# Calculate X'Y
XtY <- t(X) %*% Y

# Calculate (X'X)^-1 * X'Y
coefficients <- XtX_inverse %*% XtY

# Display matrices
print("X'X:")
print(XtX)

print("(X'X)^-1:")
print(XtX_inverse)

print("X'Y:")
print(XtY)

print("Coefficients:")
print(coefficients)
```

### Calculating estimated y, residuals, and squared residuals & Displaying the updated data

```{r}
# Calculating estimated y, residuals, and squared residuals
data$y_hat <- fitted(model)
data$u <- residuals(model)
data$u_squared <- residuals(model)^2

# Displaying the updated dataset
print("Updated dataset with estimated y, residuals, and squared residuals:")
print(data)
```

**`The calculations involved in this process are lengthy and tedious, underscoring the undeniable advantage of using a computer for such computations.`**

### **Step 2: Model Estimation**

```{r}
# Fitting the regression model
model <- lm(y ~ X_1 + X_2 + X_3, data = data)
summary(model)
```

### An example of how you can use **`broom`** to get a tidy table for the regression summary:

```{r}
# Load the necessary library
library(broom)

# Fit the regression model
model <- lm(y ~ X_1 + X_2 + X_3, data = data)

# Get a tidy summary of the model
tidy_summary <- tidy(model)
print(tidy_summary)
```