The problem you have uploaded involves conducting an Ordinary Least Squares (OLS) regression on two real-life random variables $X$ and $Y$, and interpreting the results. Let's break down the tasks and how to approach them.

Problem Breakdown:

1. Data Collection:

   • You're required to gather data for two real-life variables $X$ and $Y$. The sample size must be at least 100. These could be variables like time spent studying ($X$) and test scores ($Y$), or advertising expenditure ($X$) and sales ($Y$), etc.

2. Linear Model Construction:

   • The linear relationship is given by the equation: $Y_t = \beta_1 + \beta_2 X_t + u_t$ where:
     • $Y_t$ is the dependent variable at time $t$,
     • $X_t$ is the independent variable at time $t$,
     • $\beta_1$ is the intercept,
     • $\beta_2$ is the slope (which shows how much $Y$ changes for each unit increase in $X$),
     • $u_t$ is the error term at time $t$.

3. Estimating the Coefficients via OLS:

   • Use the OLS formula to estimate $\hat{\beta_1}$ and $\hat{\beta_2}$. In OLS, the coefficients are chosen to minimize the sum of the squared differences between the actual and predicted $Y_t$ values.

   The formulas for the OLS estimates of the coefficients are: $\hat{\beta_2} = \frac{\sum (X_t - \bar{X})(Y_t - \bar{Y})}{\sum (X_t - \bar{X})^2}$ $\hat{\beta_1} = \bar{Y} - \hat{\beta_2}\bar{X}$ Where $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively.

4. Variance of Estimates:

   • To compute the variance of $\hat{\beta_1}$ and $\hat{\beta_2}$, use the following formulas: $\text{var}(\hat{\beta_2}) = \frac{\sigma^2}{\sum (X_t - \bar{X})^2}$ $\text{var}(\hat{\beta_1}) = \sigma^2 \left(\frac{1}{n} + \frac{\bar{X}^2}{\sum (X_t - \bar{X})^2}\right)$ Where $\sigma^2$ is the estimated variance of the residuals, calculated as: $\hat{\sigma}^2 = \frac{\sum (Y_t - \hat{Y}_t)^2}{n-2}$

5. Standard Errors:

   • The standard errors of $\hat{\beta_1}$ and $\hat{\beta_2}$ are simply the square roots of their variances: $\text{se}(\hat{\beta_2}) = \sqrt{\text{var}(\hat{\beta_2})}, \quad \text{se}(\hat{\beta_1}) = \sqrt{\text{var}(\hat{\beta_1})}$

Next Steps:

1. Gather the Data: Identify two correlated variables and collect a sample of size 100 or more.

2. OLS Estimation: Using the data, apply the formulas to compute the estimates $\hat{\beta_1}$ and $\hat{\beta_2}$.

3. Variance and SE Computation: After computing the residuals, calculate the variances and standard errors for the estimates.

Would you like assistance with a specific part of the process? For example, if you have collected data, I can help you with the calculations. Let me know if you'd like more detailed explanations or further steps.

Here are 5 related questions:

1. How would you collect a sample of data for the variables $X$ and $Y$?
2. How does the choice of sample size affect the accuracy of the OLS estimates?
3. What assumptions must be satisfied for the OLS estimates to be valid?
4. How would multicollinearity affect the estimates of $\beta_1$ and $\beta_2$?
5. How can the goodness-of-fit of the model be evaluated using $R^2$?

Tip:

Always check for potential violations of OLS assumptions (such as heteroscedasticity or autocorrelation) before interpreting the estimates.