User:Minahil Siddiqui/sandbox

Question: 01

a ) - Based on the provided information and the regression results, the estimated regression model can be written as an equation:

Years of schooling = β0 + β1 * motheduc + β2 * fatheduc + ε

Using the estimated coefficients from the table:

Years of schooling = 6.964355 + 0.3041971 * motheduc + 0.1902858 * fatheduc + ε

Here, "years_of_schooling" is the individual's years of schooling (dependent variable), "motheduc" is the mother's years of schooling, and "fatheduc" is the father's years of schooling. β0, β1, and β2 are the coefficients representing the intercept, the effect of mother's education, and the effect of father's education, respectively. ε represents the error term, which accounts for the unexplained variation in the dependent variable.

b-) Fatheduc ; The coefficient for fatheduc is 0.1902858. This means that, on average, a one-year increase in a father's years of schooling is associated with a 0.1902858-year increase in the child's years of schooling, holding the mother's education constant. This positive relationship suggests that higher levels of father's education are associated with higher levels of education in their children.

To Find the overall goodness of fit of the model, we can consider the R-squared, adjusted R-squared, and Root Mean Squared Error (RMSE) values.

R-squared: The R-squared value in the provided table is 0.2493, which means that 24.93% of the variance in the dependent variable (individual's years of schooling) can be explained by the independent variables (mother's and father's years of schooling). While an R-squared of 0.2493 indicates that the model is capturing some of the variation in the dependent variable, it also suggests that there may be other factors not included in the model that could help explain the remaining 75.07% of the variation.

Adjusted R-squared: The adjusted R-squared value in the table is 0.2480. The adjusted R-squared takes into account the number of independent variables and the sample size, penalizing models that include too many predictors. In this case, the adjusted R-squared is very close to the R-squared value, indicating that the model is not overly complex and that the included variables contribute meaningfully to the explanation of the dependent variable's variance.

Root Mean Squared Error (RMSE): The RMSE value in the table is 2.0416. RMSE is a measure of the average difference between the observed values and the predicted values by the model. A lower RMSE value indicates a better fit of the model, as the differences between observed and predicted values are smaller. In this case, the RMSE of 2.0416 provides an idea of the typical prediction error of the model, though it's hard to evaluate the RMSE in isolation without comparing it to other models or knowing the scale of the dependent variable.

The model shows a moderate goodness of fit, as it explains about 24.93% of the variance in the individual's years of schooling. While the model provides some insights into the relationship between parental education and the child's education, there may be other factors not included in the model that could help explain more of the variance in the dependent variable. The adjusted R-squared value and the RMSE both suggest that the model is not overly complex, but it might be worth exploring additional variables to improve the overall goodness of fit.

d- When you include the ability score ('abil') in the second regression (Table 2), the coefficients for parental education (motheduc and fatheduc) change compared to Table 1 because the model is now accounting for the effect of cognitive skills on the dependent variable (individual's years of schooling).

The change in the coefficients of parental education can be attributed to the following reasons:

Omitted variable bias: In the first regression (Table 1), the model did not account for the ability score. If the ability score is correlated with both parental education and the individual's years of schooling, omitting it from the model could lead to biased estimates of the parental education coefficients. By including the ability score in the second regression, the model accounts for its effect and provides a more accurate estimate of the relationship between parental education and the individual's years of schooling.

Multicollinearity: There might be a correlation between parental education and the ability score. If this is the case, including the ability score in the second regression helps to partial out the effect of parental education on the ability score, leading to more precise estimates of the coefficients for motheduc and fatheduc. However, it is important to note that if the multicollinearity is very high, it could still lead to inflated standard errors and less reliable estimates of the coefficients.

E)  In Table 2, the model includes both parental education variables (motheduc and fatheduc) and the ability score (abil) as predictors of the child's level of education. While including the ability score helps to reduce the omitted variable bias present in Table 1, it is not possible to definitively conclude that the impact of parental education on their children's level of education is entirely unbiased.

There are several reasons why the estimates in Table 2 might still be biased:

Unobserved factors: The model may not include all relevant factors affecting the child's level of education. There could still be other unobserved factors that are correlated with parental education and the child's level of education, leading to biased estimates.

Measurement error: If the variables included in the model, such as motheduc, fatheduc, or abil, are measured with error, the estimated coefficients could be biased.

Multicollinearity: If there is high multicollinearity between the independent variables (motheduc, fatheduc, and abil), the estimates of the coefficients may be less precise and reliable.

Model misspecification: The model assumes a linear relationship between the independent and dependent variables, but if the true relationship is non-linear, the estimated coefficients may be biased.

Question: 02

a) Interpretation of the regression coefficients in Table 3:

Under both specifications, the regression coefficients provide an estimate of the relationship between the average group mean and the exam scores. The coefficients are expressed as the expected change in exam scores for a one-unit increase in the average group mean, holding all other variables constant.

The first specification shows that the estimated coefficient on the average group mean is 0.441, with a standard error of 0.20. The asterisk next to the coefficient indicates that it is statistically significant at the 5% level. This suggests that having classmates with higher cognitive scores is associated with higher exam scores, but the magnitude of the effect is small. The standard error of the estimate is relatively large, indicating that the estimate is not very precise.

The second specification shows that the estimated coefficient on the average group mean is 0.628, with a standard error of 0.21. The asterisks next to the coefficient indicate that it is statistically significant at the 1% level. This suggests that having classmates with higher cognitive scores is associated with higher exam scores, and the magnitude of the effect is larger. The standard error of the estimate is still relatively large, indicating that the estimate is not very precise.

It is important to note that these estimates are based on the data and model used in the analysis and may not generalize to other populations or settings. Additionally, the estimates may be biased if there are omitted variables or if the model specification is incorrect.

b) Improving the model to estimate the impact of peer effect:

To improve the model and estimate the impact of peer effect more accurately, several variables could be included in the model. Some suggestions include:

Student characteristics: Including variables such as student background, prior academic performance, and demographic characteristics can help control for the potential confounding effects of these variables on exam scores.

Classroom characteristics: Including variables such as class size, teacher quality, and classroom resources can help control for the potential confounding effects of these variables on exam scores.

School characteristics: Including variables such as school size, location, and resources can help control for the potential confounding effects of these variables on exam scores.

Time-varying variables: Including variables that change over time, such as student motivation or teacher quality, can help control for the potential confounding effects of these variables on exam scores.

Interaction terms: Including interaction terms between the average group mean and other variables can help capture the non-linear relationship between peer effect and exam scores.

In addition to including these variables, it may be useful to consider alternative model specifications, such as fixed-effects models or two-stage least squares models, to address potential endogeneity and omitted variable bias.

c) Measuring the causal impact of peers:

To measure the causal impact of peers, it is important to control for any confounding variables that may affect both peer effect and exam scores. This can be done using a variety of methods, including:

Instrumental variables: Using an instrumental variable, such as a lottery-based assignment to classrooms, can help control for any confounding variables that may affect both peer effect and exam scores.

Differences-in-differences: Comparing the change in exam scores between students in classrooms with higher and lower average group means can help control for any confounding variables that are constant over time.

Fixed-effects models: Including fixed-effects for classrooms or schools can help control for any confounding variables that are constant within these units.

In addition to controlling for confounding variables, it is important to consider the appropriate data and model specifications to estimate the causal impact of peers. This includes using appropriate data, such as longitudinal data or data from randomized experiments, and using appropriate model specifications, such as difference-in-differences models, fixed-effects models, or instrumental variables models, to account for any confounding variables and omitted variable bias.

In conclusion, to measure the causal impact of peers on exam scores, it is important to have appropriate data and to control for any confounding variables that may affect both peer effect and exam scores. This can be done through a combination of appropriate data, model specifications, and control variables, and may require a more complex analysis than a simple regression model.