Dummy Variables:
Dummy variables, also known as indicator variables or binary variables, are used in regression analysis to represent categorical data in a quantitative way. They take the value of 1 if a specific category is present and 0 if the category is absent. Dummy variables allow us to include categorical information in regression models, enabling us to estimate the effect of categorical variables on the dependent variable while considering the effects of other independent variables.
For example, consider a regression model to predict house prices with a categorical variable “neighborhood” having three categories: A, B, and C.
We can create two dummy variables:
“neighborhood_B” and “neighborhood_C.”
If a house belongs to neighborhood B, the “neighborhood_B” dummy variable would take the value 1 and “neighborhood_C” would take the value 0.
Similarly, if the house belongs to neighborhood C, “neighborhood_B” would be 0, and “neighborhood_C” would be 1.
Using dummy variables in regression allows us to estimate separate intercepts for each category and determine how each category affects the dependent variable compared to the reference category (usually the one not represented by any dummy variable).
Truncated Variables:
Truncated variables are those that have a restricted range due to the way they were collected or defined. In statistical terms, a truncated variable is one that has a conditional restriction based on another variable. For example, suppose we are studying the heights of male and female adults in a population, and the heights of females are only measured for those aged 18 and above. In this case, the variable “height” for females would be considered truncated since it is only observed for individuals aged 18 and above.
Truncated variables can impact the analysis, especially if the truncation is not random and is related to the dependent variable. In such cases, ignoring the truncation could lead to biased results. Truncation can occur in various contexts, such as survey data, experimental data, or data with specific inclusion/exclusion criteria.
When analyzing truncated data, specific statistical methods, such as Heckman selection models, can be used to account for the truncation bias and obtain unbiased parameter estimates. These methods take into account the selection process and allow for consistent estimation of the relationships between variables in the presence of truncation.
Diagnostic Checking
Diagnostic checking is a critical step in regression analysis to assess the validity of the model assumptions and identify any potential issues or violations. It involves examining the residuals (the differences between the observed and predicted values) to ensure that they meet the assumptions of the regression model. The diagnostic checking process helps verify if the regression model is appropriate for the data and if any adjustments or transformations are needed to improve the model’s performance.
Common diagnostic checks performed in regression analysis:
- Residual Plots: Residual plots are scatter plots that display the residuals against the predicted values, the independent variables, or the time (in time series analysis). These plots help identify patterns or trends in the residuals, such as heteroskedasticity, nonlinearity, or outliers.
- Normality of Residuals: Checking the normality of residuals is essential, especially when using the Ordinary Least Squares (OLS) method. A histogram or a Q-Q plot of the residuals can help assess if they follow a normal distribution.
- Homoskedasticity (Constant Variance): Plotting the residuals against the predicted values can help detect heteroskedasticity (non-constant variance). If the spread of residuals changes systematically with predicted values, it indicates heteroskedasticity.
- Outliers and Influential Points: Identifying and investigating outliers and influential data points can be crucial. Outliers can significantly affect the regression model’s results, while influential points can heavily influence the estimated coefficients.
- Cook’s Distance and Leverage: Cook’s distance and leverage are measures used to identify influential observations that may have a significant impact on the regression results. High leverage points can lead to substantial changes in the coefficient estimates.
- Collinearity: Examining the correlations between independent variables can help detect multicollinearity issues. High correlations between predictors may indicate potential collinearity problems.
- Durbin-Watson Test: The Durbin-Watson test is used to check for serial correlation (autocorrelation) in the residuals, especially in time series data.
- Variance Inflation Factor (VIF): VIF is used to quantify the extent of multicollinearity in the model. High VIF values suggest potential collinearity problems.
- R-squared and Adjusted R-squared: R-squared and adjusted R-squared are measures of how well the model fits the data. A low R-squared or a significantly lower adjusted R-squared compared to R-squared may indicate overfitting.
- Residual Analysis for Time Series Models: For time series data, additional diagnostic checks, such as autocorrelation function (ACF) plots and partial autocorrelation function (PACF) plots, can help assess the adequacy of the time series model.
Once diagnostic checks are performed, adjustments to the model or data transformations may be needed based on the findings. It is essential to ensure that the regression model’s assumptions are met and that the results are reliable and valid for making accurate interpretations and predictions.