Teaching Notes for Applied Econometrics and Economic Modelling Lab Session - Day 3

**1.1 Recap on the results of exercises.

• Linear regression Model(population):

$$\text{Sales_i} = \beta_1 + \beta_2 \times \text{Advertising_i} + \epsilon_i$$

• $\beta_1 + \beta_2 \times \text{Advertising_i}$ is the population regression line.

• Create a table showing the values of $\hat{\beta}_2$, $s_{\hat{\beta}_2}$, and $t-stat$.

• $R^2$=0.75

• $s$ (the standard error of the regression)= 8.74

• Recap the concepts of hypothesis testing.

1.2 Confidence Intervals

• Objective: Recap the concepts of confidence intervals.
• Key Points:
  • Confidence Interval for the Slope:
    • What is it?: A range of values within which the true slope coefficient is expected to lie with a certain level of confidence (e.g., 95%).
    • Why is it important?: It provides a measure of the uncertainty around the slope estimate.

1.3 Hands-On Calculation of Confidence Interval for the Slope

• Objective: Calculate the confidence interval for the slope manually using Google Sheets.
• Exercise: Use the Advertising-Sales dataset.
  • Step 1: Compute the standard error of the slope coefficient ($s_{\hat{\beta}_2}$).
    • Instruction: Use the formula: $$s_{\hat{\beta}_2} = \sqrt{\frac{s^2}{\sum (x_i - \bar{x})^2}}$$
    • Why?: This measures the variability of the slope estimate.
  • Step 2: Calculate the 95% confidence interval for the slope.
    • Instruction: Use the formula: $$\text{CI} = \hat{\beta}_2 \pm 2 \times s_{\hat{\beta}_2}$$
    • Why?: This gives a range within which the true slope is likely to lie.
  • Discussion: What does the confidence interval tell us about the slope? How does it help in interpreting the regression results?

2.1 Linear Model for Men and Women

• Objective: Fit a linear trend model to the Olympic 100m dataset for men and women.
• Exercise: Use the Olympic 100m dataset.
  • Step 1: Fit a linear trend model to the data.
    • Instruction: Use the formula: $$W_i = \beta_1 + \beta_2 G_i + \epsilon_i$$ where $W_i$ is the winning time (seconds) and $G_i$ is the year for i = 1,…,15 (from 1=1948 to 15=2004).
      • Find $\hat{\beta}_2 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$ and $\hat{\beta}_1 = \bar{y} - \hat{\beta}_2 \bar{x}$.
    • Why?: This provides a baseline forecast based on a linear trend.
  • Step 2: Calculate the outcomes of the linear model.

    • Instruction: Compute $s^2 = \frac{1}{n-2} \sum_{i=1}^{n} e_i^2$ and $\sigma_{\hat{\beta}_2}^2 = \frac{s^2}{\sum (x_i - \bar{x})^2}$

    • Create a table summarizing the results.

    • Construct a 95% CI for the slope coefficients.

    • A 95% two-sided confidence interval for the slope coefficient is an interval that contains the true value of the slope coefficient with a 95% probability; that is, it contains the true value of the slope coefficient in 95% of all possible randomly drawn samples.

    • Women seem to have made most progress.

    • Model assumes fixed gain (in seconds per game).

    • Negative slope means winning times decrease over time (improvement).

2.2 Non-Linear (Exponential) Model for Men and Women

• Objective: Fit a non-linear (exponential) trend model to the data and transform it into a log-linear form.
• Exercise: Use the Olympic 100m dataset.
  • Step 1: Fit a non-linear (exponential) trend model to the data.
    • Instruction: Use the formula: $$W_i = e^{\beta_1 + \beta_2 G_i + \epsilon_i}$$
    • Why?: This model can capture exponential trends in the data and provide a more accurate forecast.
  • Step 2: Transform the non-linear model into a log-linear form.
    • Instruction: Take the natural logarithm of both sides of the non-linear model: $$\ln(W_i) = \beta_1 + \beta_2 G_i + \epsilon_i$$
    • Why?: This allows us to use linear regression techniques on the transformed model.
  • Step 3: Calculate the outcomes of the non-linear model.
    • Instruction: Compute the predicted winning times for men and women using the non-linear model.
    • Why?: This gives the forecasted winning times based on the non-linear trend.

2.3 Forecasting Winning Times for 2008 and 2012

• Objective: Use the fitted models to forecast winning times for men and women in the Olympic games of 2008 and 2012.
• Exercise: Use the Olympic 100m dataset.
  • Step 1: Forecast winning times for 2008 and 2012 using the linear model.
    • Instruction: Plug in the years 2008 and 2012 into the linear equation: $$W_i = \beta_1 + \beta_2 G_i$$
    • Why?: This provides a forecast based on the linear trend.
  • Step 2: Forecast winning times for 2008 and 2012 using the non-linear model.
    • Instruction: Plug in the years 2008 and 2012 into the log-linear equation and then exponentiate the result: $$W_i = e^{\beta_1 + \beta_2 G_i}$$
    • Why?: This provides a forecast based on the non-linear trend.
  • Discussion: How do the forecasts differ between the linear and non-linear models? Which model is more realistic?

2.4 Introduction to GRETL and Verification of Results

• Objective: Introduce GRETL and verify the results of the linear and non-linear models.
• Exercise: Use the Olympic 100m dataset.
  • Step 1: Load the dataset into GRETL.
    • Instruction: Use the GRETL interface to load the dataset.
    • Why?: This allows us to perform regression analysis using GRETL.
  • Step 2: Perform linear regression in GRETL.
    • Instruction: Use GRETL to run the linear regression and compare the results with the manual calculations.
    • Why?: This verifies the accuracy of the manual calculations.
  • Step 3: Perform non-linear regression in GRETL.
    • Instruction: Use GRETL to run the non-linear regression and compare the results with the manual calculations.
    • Why?: This verifies the accuracy of the manual calculations.
  • Discussion: How do the GRETL results compare with the manual calculations? What are the benefits of using GRETL for regression analysis?