$5.$ The basic consumption function model explains that consumption depends on disposable income. This analysis examines how macroeconomic factors$-$monthly real disposable personal income $($DSPIC$96),$ real personal consumption expenditures $($PCEC$96),$ and the $5-$year Treasury interest rate $($GS$5)-$influence consumption behavior. The dataset, covering U$.$S$.$ data from $2002$ to $2021,$ includes summary statistics $($Table $1)$ and regression results $($Table $2).$

Table $2$: Regression Results

Column $(1)$ is simple linear model with dependent consumption, while column $(2)$ use log consumption as dependent variable and log income as independent variable.

Tasks:

a$.$ OLS Methodology: Briefly describe the Ordinary Least Squares $($OLS$)$ method. What are the decision variables and the objective function?

b$.$ Estimated Equation: Using column $(1)$ of Table $2,$ write the regression equation and interpret the coefficient of disposable personal income $($DSPIC$96).$ Is it statistically significant?

c$.$ Log Transformation: Explain how the interpretation of the coefficient changes in column

$(2),$ where both consumption $($PCEC$96)$ and income $($DSPIC$96)$ are logarithmically transformed.

d$.$ Impact of GS$5$: In column $(3),$ the $5-$year Treasury interest rate $($GS$5)$ is added. Write the regression equation and explain how GS$5$ affects consumption.

e$.$ Prediction: Using column $(3)\text{'}$s regression equation, predict the monthly consumption $($PCEC$96)$ when disposable income $($DSPIC$96)$ is $$10,000$ billion and GS$5$ is $2\%.$

f$.$ Model Comparison: Compare the R$-$squared and adjusted R$-$squared values across the three models in Table $2.$ Which model best explains consumption variability? Justify your choice.