Residual plots

1. Residual Plots

Residual plots are a powerful and fundamental tool in Regression Analysis used to assess the validity of the assumptions underlying linear regression models. They are crucial for determining if a linear model is appropriate for the data, and for identifying potential problems that might need to be addressed before drawing conclusions from the analysis. This article provides a comprehensive introduction to residual plots, covering their creation, interpretation, common patterns, and how to address issues they reveal. It's geared towards beginners with a basic understanding of regression but no prior experience with residual analysis.

1. 1. What are Residuals?

Before diving into plots, we need to understand *residuals*. In a Linear Regression model, we attempt to find the best-fitting line (or hyperplane in multiple regression) that describes the relationship between a dependent variable (Y) and one or more independent variables (X). The actual values of Y will rarely fall perfectly on this line. The difference between the observed (actual) value of Y and the predicted value of Y (based on the regression equation) is called the *residual*.

Mathematically:

Residual = Observed Value - Predicted Value = Y_i - Ŷ_i

Where:

• Y_i is the actual value of the dependent variable for observation *i*.
• Ŷ_i is the predicted value of the dependent variable for observation *i*.

Residuals represent the error in the prediction made by the model. A good model minimizes the sum of squared residuals, meaning it produces predictions that are as close as possible to the actual values. However, simply minimizing the residuals isn't enough. We need to examine the *pattern* of these residuals to ensure the assumptions of the linear regression model are met. This is where residual plots come in.

1. 1. Why Use Residual Plots?

The core purpose of residual plots is to check the assumptions of linear regression. These assumptions are:

1. **Linearity:** The relationship between the independent and dependent variables is linear. 2. **Independence:** The residuals are independent of each other. This means there's no correlation between the errors. 3. **Homoscedasticity:** The residuals have constant variance across all levels of the independent variable(s). (Equal spread of errors). 4. **Normality:** The residuals are normally distributed.

Violations of these assumptions can lead to inaccurate and unreliable regression results. Residual plots help us identify these violations. Specifically, they allow us to detect:

• **Non-linearity:** Indicates the relationship is not linear and a different model (e.g., polynomial regression, Exponential Smoothing or a non-parametric method) might be more appropriate.
• **Heteroscedasticity:** Indicates the variance of the errors is not constant. This can affect the accuracy of standard errors and hypothesis tests.
• **Autocorrelation:** Indicates that residuals are correlated with each other, often occurring in time series data. This violates the independence assumption.
• **Outliers:** Identifies observations with unusually large residuals, which may be errors in the data or represent influential points. Understanding Support and Resistance levels can sometimes help contextualize outliers.

1. 1. Common Types of Residual Plots

There are several types of residual plots, each providing different information.

1. 1. 1. 1. Residuals vs. Fitted Values Plot (also known as the Residual Plot)

This is the most commonly used residual plot. It plots the residuals on the y-axis against the predicted (fitted) values on the x-axis.

• **What to look for:** Ideally, the residuals should be randomly scattered around zero with no discernible pattern.

   * **Random Scatter:** Indicates the assumptions of linearity and homoscedasticity are likely met.
   * **Funnel Shape:**  Indicates heteroscedasticity (non-constant variance). The spread of the residuals increases or decreases as the fitted values increase.
   * **Curvature:** Indicates non-linearity. The residuals form a curved pattern, suggesting a non-linear relationship between the variables.  This could be addressed with a Moving Average or another smoothing technique.
   * **Clusters:**  Suggests non-linearity or the presence of subgroups within the data.

1. 1. 1. 2. Normal Probability Plot (also known as a Q-Q Plot)

This plot assesses the normality of the residuals. It plots the sorted residuals against the theoretical quantiles of a normal distribution.

• **What to look for:** If the residuals are normally distributed, the points on the plot should fall approximately along a straight diagonal line.

   * **Straight Line:** Indicates normality.
   * **S-Shape:** Indicates non-normality.  The residuals deviate systematically from the line.
   * **Deviations at the Ends:**  Suggests the presence of outliers.

1. 1. 1. 3. Residuals vs. Predictor Variables Plots

These plots examine the relationship between the residuals and each individual predictor variable. They are created by plotting the residuals against each independent variable in the model.

• **What to look for:** Similar to the residual vs. fitted values plot, look for random scatter. Any systematic pattern (e.g., curvature, funnel shape) indicates a problem. These plots can reveal that a predictor variable needs to be transformed or that an important variable is missing from the model. Consider using Fibonacci Retracements to explore potential relationships within the predictor variables.

1. 1. 1. 4. Residuals vs. Order (for Time Series Data)

This plot is specifically for time series data. It plots the residuals against the order in which the data was collected.

• **What to look for:** Look for patterns or trends in the residuals over time.

   * **Random Scatter:** Indicates independence.
   * **Autocorrelation:**  The residuals show a pattern (e.g., positive or negative correlation) over time. This suggests that the errors are not independent.  This is a common problem and might require using time series models like ARIMA or GARCH.  Understanding Elliott Wave Theory can sometimes provide context for patterns in time series data.

1. 1. Interpreting Common Patterns in Residual Plots

Let's delve deeper into interpreting some common patterns observed in residual plots:

• **Heteroscedasticity (Funnel Shape):** This indicates that the variance of the errors is not constant. Possible causes include:

   * **Scale Effect:** The variance of the errors increases as the magnitude of the dependent variable increases.
   * **Model Misspecification:** The model is not capturing the true relationship between the variables.
   * **Outliers:**  A few influential outliers can distort the variance.
   * **Remedies:**
       * **Transform the Dependent Variable:** Applying a transformation like a logarithmic transformation can stabilize the variance.  Consider using a Bollinger Bands strategy after transformation.
       * **Weighted Least Squares Regression:**  This method assigns different weights to observations based on their variance.
       * **Robust Regression:** This method is less sensitive to outliers.

• **Non-Linearity (Curvature):** This indicates that the relationship between the variables is not linear.

   * **Possible Causes:**
       * **Incorrect Functional Form:** The model is using a linear equation when a non-linear equation is more appropriate.
       * **Missing Variables:** An important variable is missing from the model.
   * **Remedies:**
       * **Add Polynomial Terms:** Include squared or cubed terms of the independent variable(s) in the model.
       * **Transform the Variables:** Apply a transformation to one or both variables to linearize the relationship.
       * **Use a Non-Linear Model:** Consider using a non-parametric regression model.

• **Autocorrelation (Pattern in Residuals vs. Order):** This indicates that the residuals are correlated with each other, violating the independence assumption. This is especially common in time series data.

   * **Possible Causes:**
       * **Time Dependence:** The value of the dependent variable at one time point is influenced by its value at previous time points.
       * **Model Misspecification:** The model is not capturing the time dependence in the data.
   * **Remedies:**
       * **Include Lagged Variables:** Include lagged values of the dependent variable as predictors in the model.
       * **Use Time Series Models:** Use models specifically designed for time series data, such as ARIMA or GARCH. MACD can sometimes help identify trends in time series data prior to modelling.

• **Outliers (Points Far From the Pattern):** Outliers are observations with unusually large residuals.

   * **Possible Causes:**
       * **Data Errors:**  Errors in data entry or measurement.
       * **Influential Points:** Observations that have a significant impact on the regression results.
       * **Genuine Anomalies:**  Observations that represent unusual but legitimate events.
   * **Remedies:**
       * **Verify Data:** Check the data for errors.
       * **Remove Outliers:**  If the outliers are due to data errors, they should be removed.  Be cautious about removing outliers without a valid reason.
       * **Robust Regression:** Use a robust regression method that is less sensitive to outliers.  Look to Candlestick Patterns for potential signals around outlier events.

1. 1. Tools for Creating Residual Plots

Many statistical software packages can create residual plots. Some popular options include:

• **R:** A powerful statistical programming language with extensive capabilities for regression analysis and residual diagnostics.
• **Python (with libraries like Statsmodels and Scikit-learn):** Another popular programming language with a rich ecosystem of statistical tools.
• **SPSS:** A user-friendly statistical software package.
• **Excel:** While limited, Excel can create basic residual plots.
• **MATLAB:** A numerical computing environment that can be used for regression analysis.

1. 1. Best Practices

• **Always create multiple residual plots:** Don't rely on just one plot to assess the model assumptions.
• **Examine the plots carefully:** Look for any patterns or deviations from the expected random scatter.
• **Consider the context of the data:** The interpretation of residual plots should be informed by your understanding of the data and the research question.
• **Document your findings:** Keep a record of the residual plots and your interpretations.
• **Don’t ignore violations:** If you identify violations of the assumptions, address them before drawing conclusions from the regression analysis. Consider using Ichimoku Cloud analysis for broader contextual understanding.
• **Iterate:** Regression modelling is an iterative process. You may need to refine the model and re-examine the residual plots several times to arrive at a satisfactory solution. Use Trend Lines to help identify potential patterns.

Understanding and utilizing residual plots is paramount for building reliable and insightful regression models. They are an essential part of any sound statistical analysis. Remember to combine this with other technical analysis tools like Relative Strength Index (RSI), Stochastic Oscillator and Average True Range (ATR) for a comprehensive view.

Regression Analysis Linear Regression Exponential Smoothing ARIMA GARCH Support and Resistance Moving Average Fibonacci Retracements Elliott Wave Theory Bollinger Bands MACD Ichimoku Cloud Trend Lines Relative Strength Index (RSI) Stochastic Oscillator Average True Range (ATR) Time Series Analysis Data Analysis Statistical Modelling Model Validation Hypothesis Testing Outlier Detection Homoscedasticity Non-linearity Autocorrelation Normality Tests Weighted Least Squares Robust Regression Data Transformation

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