Perform Weighted Least Squares (WLS) regression analysis to account for heteroscedasticity in your data.
WLS Calculator
Enter the total number of data points. Must be at least 2.
Enter the number of independent variables (excluding the intercept). Must be at least 1.
Enter weights separated by commas. Must be positive numbers, same count as observations.
Enter predictor variable values. Each row is an observation, each column a variable. Separate values with commas. Ensure correct number of observations and variables.
Enter dependent variable values, separated by commas. Must have the same count as observations.
WLS Results
—
Key Intermediate Values
Weighted Beta Hat (Coefficients): —
Weighted Variance-Covariance Matrix: —
Weighted R-squared: —
Formula Used
Weighted Least Squares (WLS) minimizes the sum of the weighted squared residuals. The formula for the estimated coefficients (β̂) is: β̂ = (XᵀW⁻¹X)⁻¹ XᵀW⁻¹Y, where X is the matrix of predictor variables, Y is the vector of the dependent variable, and W is the diagonal matrix of weights (w_i).
Observed vs. Predicted Values
WLS Regression Coefficients and Statistics
Variable
Coefficient (β̂)
Standard Error
t-statistic
p-value
Enter data and click "Calculate WLS"
What is Weighted Least Squares (WLS)?
Weighted Least Squares (WLS) is a statistical method used in regression analysis to address the issue of heteroscedasticity. In standard Ordinary Least Squares (OLS) regression, it is assumed that the variance of the errors (residuals) is constant across all observations. When this assumption is violated, meaning the errors have different variances, the data is said to be heteroscedastic. WLS accounts for this by assigning different weights to each observation based on the inverse of their error variance. Observations with smaller variances receive higher weights, and those with larger variances receive lower weights. This technique leads to more efficient and reliable estimates of the regression coefficients compared to OLS when heteroscedasticity is present. The core idea behind Weighted Least Squares is to give more importance to observations that are more precise and less importance to those that are less precise, thereby improving the overall quality of the regression model.
Who Should Use WLS?
WLS is particularly useful for researchers and analysts in fields like economics, finance, econometrics, and social sciences where heteroscedasticity is common. If you are performing regression analysis and suspect or have confirmed that the variability of your response variable changes across the range of your predictor variables, WLS is a suitable method. This is often seen in cross-sectional data where different units (e.g., individuals, firms, countries) may have varying levels of inherent variability. For example, when modeling income based on education level, high-income individuals might have a much wider range of spending behaviors (higher variance) than low-income individuals. Using WLS allows you to down-weight the influence of these high-variance observations, leading to a more accurate representation of the relationship between the predictors and the dependent variable. It is a fundamental technique for improving the robustness of regression results in the presence of non-constant error variances.
Common Misconceptions about WLS
WLS is only for complex models: While WLS addresses a specific statistical issue, its application is straightforward once the weights are determined. It's not inherently more complex to implement than OLS once you have the weights.
WLS replaces OLS entirely: WLS is a modification of OLS, used specifically when OLS assumptions are violated. If homoscedasticity holds (errors have constant variance), OLS is already the best linear unbiased estimator (BLUE), and WLS with equal weights reduces to OLS.
Weights are arbitrary: The weights in WLS are not chosen randomly. They are typically derived from a theoretical model or estimated from the data itself, usually as the inverse of the error variance (wᵢ = 1/σᵢ²). Incorrectly specified weights can lead to biased results.
WLS guarantees perfect results: WLS improves efficiency under heteroscedasticity but does not eliminate all potential modeling issues like multicollinearity or omitted variable bias.
Weighted Least Squares (WLS) Formula and Mathematical Explanation
The goal of regression analysis is to model the relationship between a dependent variable (Y) and one or more independent variables (X). In Ordinary Least Squares (OLS), we aim to find the coefficients (β) that minimize the sum of squared residuals (SSR): SSR = Σ(yᵢ – ŷᵢ)², where yᵢ is the observed value and ŷᵢ is the predicted value. This implicitly assumes that all observations have equal influence and equal variance.
However, when heteroscedasticity is present, the variance of the error term (εᵢ) is not constant across all observations (Var(εᵢ) = σᵢ²). In such cases, OLS estimates are still unbiased but are no longer the most efficient (i.e., they have larger standard errors). Weighted Least Squares (WLS) is designed to overcome this by minimizing a *weighted* sum of squared residuals:
Weighted SSR = Σ wᵢ(yᵢ – ŷᵢ)²
Where wᵢ is the weight assigned to the i-th observation. To achieve the most efficient estimates, the weights should be inversely proportional to the variance of the error term: wᵢ ∝ 1/σᵢ². A common choice is wᵢ = 1/σᵢ².
Mathematical Derivation
Let Y be the vector of the dependent variable, X be the matrix of independent variables (including a column of ones for the intercept), and β be the vector of coefficients we want to estimate. The model is Y = Xβ + ε, where ε is the vector of error terms with a diagonal covariance matrix Σ² = diag(σ₁², σ₂², …, σ²).
The WLS objective is to minimize the weighted sum of squares, which can be expressed in matrix form. If we transform the original model variables by multiplying them by the square root of the weights (√wᵢ), we get a new model:
√W Y = √W X β + √W ε
Where W is a diagonal matrix with wᵢ on the diagonal (W = diag(w₁, w₂, …, w)). The error terms in this transformed model, √W ε, now have a constant variance (assuming wᵢ = 1/σᵢ²). We can then apply OLS to this transformed model:
(√W Y) = (√W X) β + (√W ε)
Applying the standard OLS formula (β̂ = (X'X)⁻¹ X'Y) to the transformed variables (let Y* = √W Y and X* = √W X):
β̂WLS = ((√W X)ᵀ (√W X))⁻¹ (√W X)ᵀ (√W Y)
Simplifying the terms:
(√W X)ᵀ (√W X) = Xᵀ (√W)ᵀ (√W) X = Xᵀ W X
(√W X)ᵀ (√W Y) = Xᵀ (√W)ᵀ (√W) Y = Xᵀ W Y
Substituting these back, we get the WLS estimator:
β̂WLS = (Xᵀ W X)⁻¹ Xᵀ W Y
This is the core formula for calculating the WLS coefficients. The variance-covariance matrix of the WLS estimates is given by:
Cov(β̂WLS) = (Xᵀ W X)⁻¹
This matrix provides the standard errors needed for hypothesis testing and confidence interval construction.
Variables Table
Variable
Meaning
Unit
Typical Range
Y
Dependent Variable Vector
Depends on the data
N/A
X
Matrix of Predictor Variables (incl. intercept)
Depends on the data
N/A
β
Vector of True Coefficients
Depends on the data
N/A
ε
Error Term Vector
Depends on the data
N/A
wᵢ
Weight for the i-th observation (typically 1/σᵢ²)
Unitless
Positive real numbers
W
Diagonal Matrix of Weights
Unitless
Diagonal entries are positive real numbers
β̂WLS
Estimated Coefficients using WLS
Depends on the data
N/A
σᵢ²
Variance of the error term for the i-th observation
Unit²
Positive real numbers
n
Number of Observations
Count
≥ 2
k
Number of Predictor Variables (excluding intercept)
Count
≥ 1
Practical Examples of Weighted Least Squares
Weighted Least Squares finds application in numerous real-world scenarios where data exhibits non-constant error variances. Here are a couple of examples:
Example 1: Modeling Household Spending
Scenario: An economist wants to model household spending (Y) based on household income (X₁). Data is collected from households across different income brackets. It's hypothesized that higher-income households have more variability in their spending patterns (e.g., discretionary purchases) than lower-income households. The variance of spending is expected to increase with income.
Data (Simplified):
Observations (n): 5
Predictor Variables (k): 1 (Income), plus an intercept.
Financial Interpretation: The WLS model suggests that for every $1000 increase in household income, spending increases by $1200, on average. The intercept indicates a baseline spending of $15,000 even with zero income (likely due to savings or other factors). The WLS approach ensures these estimates are efficient despite the varying spending volatility across income levels.
Example 2: Analyzing Stock Returns vs. Market Returns with Time-Varying Volatility
Scenario: A finance analyst is examining the relationship between a specific stock's daily returns (Y) and the overall market's daily returns (X) using the Capital Asset Pricing Model (CAPM). Daily market volatility often fluctuates significantly due to news events, so the variance of the error term (representing idiosyncratic risk) is likely heteroscedastic.
Data (Simplified – 5 days):
Observations (n): 5
Predictor Variables (k): 1 (Market Return), plus an intercept.
Y (Stock Returns): [0.01, -0.02, 0.03, -0.01, 0.05]
X (Market Returns): [0.005, -0.01, 0.015, -0.008, 0.02]
Estimated Daily Error Variances (σᵢ²): [0.0001, 0.0002, 0.00015, 0.0003, 0.00025] (Reflecting days of higher market turmoil or specific stock news). *Note: These variances might be estimated from a prior GARCH model or other volatility measures.*
Financial Interpretation: The WLS analysis yields an estimated Beta of 1.1. This implies the stock is slightly more volatile than the market. The intercept (Alpha) of 0.001 suggests a small average excess return not explained by market movements. WLS is crucial here because using OLS might give undue influence to days with extreme market or stock movements (high error variance), potentially distorting the true long-term relationship between the stock and market returns. The WLS approach provides a more reliable estimate of the stock's systematic risk (Beta).
How to Use This Weighted Least Squares Calculator
Our Weighted Least Squares (WLS) calculator simplifies the process of performing WLS regression. Follow these steps to get accurate results:
Step-by-Step Instructions:
Number of Observations (n): Enter the total count of data points you have for your dependent and independent variables. This must be at least 2.
Number of Predictor Variables (k): Enter the number of independent variables you are using in your model. Do NOT include the intercept term here; it's added automatically. This must be at least 1.
Weights (wᵢ): Provide the weights for each observation. These should typically be the inverse of the error variance (1/σᵢ²). Enter them as a comma-separated list. The number of weights must exactly match the number of observations.
Predictor Variable Values (X): Enter your independent variable data. Each row corresponds to an observation, and each column corresponds to a predictor variable. Use commas to separate values within a row and new lines (or semicolons) to separate rows. Ensure the dimensions match your specified 'n' and 'k'.
Dependent Variable Values (Y): Enter your dependent variable data as a comma-separated list. The number of values must exactly match the number of observations.
Calculate WLS: Click the "Calculate WLS" button. The calculator will process your inputs and display the results.
How to Read the Results:
Primary Highlighted Result (Weighted Beta Hat): This is the most crucial output, showing the estimated coefficients for your WLS regression model. It includes the intercept (if applicable) and the slope for each predictor variable. The equation is typically represented as: Ŷ = β̂₀ + β̂₁X₁ + … + β̂X.
Key Intermediate Values:
Weighted Variance-Covariance Matrix: This matrix shows the variances of the estimated coefficients on the diagonal and the covariances off-diagonal. It's essential for understanding the precision of your estimates.
Weighted R-squared: This value indicates the proportion of the variance in the dependent variable that is predictable from the independent variables, adjusted for the weights. A higher R-squared generally suggests a better fit.
Table: Coefficients, Standard Errors, t-statistics, and p-values: This table provides a detailed breakdown for each coefficient:
Coefficient (β̂): The estimated value from the WLS calculation.
Standard Error: A measure of the variability of the coefficient estimate.
t-statistic: The ratio of the coefficient to its standard error, used for hypothesis testing.
p-value: The probability of observing a t-statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis (coefficient is zero) is true. Low p-values (typically < 0.05) suggest the predictor is statistically significant.
Chart: Observed vs. Predicted Values: This visual representation helps you assess the model's fit by comparing the actual observed Y values against the values predicted by your WLS model. Points close to the diagonal line indicate good predictions.
Decision-Making Guidance:
Coefficient Significance: Examine the p-values. If a predictor variable's p-value is below your chosen significance level (e.g., 0.05), you can conclude that it has a statistically significant impact on the dependent variable, even after accounting for heteroscedasticity.
Model Fit: Use the Weighted R-squared and the visual inspection of the chart to gauge how well the model explains the variation in Y.
Interpretation: Interpret the coefficients in the context of your specific problem. For example, in finance, a Beta coefficient indicates systematic risk.
Weight Appropriateness: Remember that the validity of WLS hinges on correctly specifying the weights. If you are unsure about the source of heteroscedasticity, consider using diagnostic tests or robust regression methods.
Key Factors That Affect WLS Results
Several factors can influence the outcome and reliability of a Weighted Least Squares (WLS) regression analysis. Understanding these is crucial for accurate interpretation and effective modeling:
Accuracy and Specification of Weights: This is the most critical factor. WLS relies heavily on the weights (wᵢ) being correctly specified, usually as the inverse of the error variances (wᵢ = 1/σᵢ²).
Financial Reasoning: If weights are based on incorrect assumptions about error variances (e.g., underestimating variance for high-income groups), the resulting coefficients might be inefficient or even biased. For instance, in financial modeling, if the weights used to account for volatility are miscalculated, the estimated Beta could be misleading.
Data Source: Weights might come from prior knowledge, theoretical models (like in finance), or be estimated themselves (e.g., using a two-step procedure like Feasible Generalized Least Squares – FGLS). The reliability of the weight source directly impacts WLS results.
Sample Size (n): While WLS is generally more efficient than OLS under heteroscedasticity, the reliability of the results, especially the standard errors and p-values, still depends on having a sufficiently large sample size.
Financial Reasoning: With a small sample, estimates can be unstable, and the assumed variance structure might not hold true. For complex financial instruments or niche markets, obtaining adequate data can be challenging, potentially limiting the confidence in WLS estimates.
Number and Nature of Predictor Variables (k): The choice and quality of predictor variables are fundamental.
Financial Reasoning: Multicollinearity (high correlation between predictors) can inflate standard errors, making coefficients appear less significant. Including irrelevant variables can reduce efficiency, while omitting important ones can lead to omitted variable bias, even in WLS. In portfolio management, selecting the right market and factor variables is key.
Correct Model Specification: WLS assumes a linear relationship between variables and that the weights address the primary source of inefficiency (heteroscedasticity).
Financial Reasoning: If the true relationship is non-linear (e.g., exponential growth), a linear WLS model will perform poorly. Similarly, if other issues like autocorrelation (serial correlation in errors) are present alongside heteroscedasticity, WLS alone might not be sufficient. More advanced models like Generalized Least Squares (GLS) might be needed.
Outliers: While WLS down-weights observations with larger variances, extreme outliers can still unduly influence the results, especially if they are present in observations with small variances (high weights).
Financial Reasoning: A single, highly unusual transaction or market event might disproportionately affect the regression line if not properly handled. Robust regression techniques might be considered if outliers are a significant concern.
Data Transformations: Sometimes, transforming variables (e.g., using logarithms) can help stabilize variance, potentially making OLS more appropriate or simplifying the weight specification for WLS.
Financial Reasoning: Log transformations are common in economics and finance (e.g., for income or prices) as they can normalize distributions and stabilize variances. The choice of transformation affects the interpretation of coefficients and the underlying assumptions.
Statistical Significance vs. Practical Significance: A statistically significant coefficient (low p-value) in WLS doesn't always translate to practical importance.
Financial Reasoning: A tiny effect size, even if statistically significant due to a large sample size or precise weights, might be too small to warrant any real-world action or investment decision. Always consider the magnitude of the coefficient alongside its statistical significance and context.
Frequently Asked Questions (FAQ) about Weighted Least Squares
Q1: What is the main advantage of using WLS over OLS?
A1: The primary advantage of WLS is its efficiency when dealing with heteroscedasticity. While OLS provides unbiased estimates even with heteroscedasticity, WLS produces estimates with smaller standard errors, making them statistically more precise and powerful for hypothesis testing and confidence intervals.
Q2: How do I determine the correct weights for WLS?
A2: The weights should ideally be inversely proportional to the variance of the error term for each observation (wᵢ ∝ 1/σᵢ²). Common methods include: 1) Using theoretical knowledge (e.g., variance scales with the square of a variable). 2) Estimating the variance from the data, often by running an OLS regression and then regressing the squared residuals against predictor variables to model the variance. The latter is known as Feasible Generalized Least Squares (FGLS).
Q3: What happens if I use WLS when the data is actually homoscedastic?
A3: If the data is homoscedastic (error variances are constant), applying WLS with equal weights effectively reduces it to OLS. However, if you incorrectly specify unequal weights when the data is homoscedastic, the WLS estimates will still be unbiased but will no longer be the Best Linear Unbiased Estimators (BLUE); they will be less efficient than OLS estimates.
Q4: Can WLS handle multicollinearity?
A4: No, WLS does not inherently solve the problem of multicollinearity. Like OLS, if predictor variables are highly correlated, WLS estimates will have large standard errors, making it difficult to determine the individual effect of each variable. You might need techniques like ridge regression or principal component regression in conjunction with WLS if multicollinearity is severe.
Q5: What is the difference between WLS and Generalized Least Squares (GLS)?
A5: WLS is a special case of GLS. GLS handles more general forms of error covariance matrices, including non-diagonal ones where error terms are correlated (autocorrelation) as well as having non-constant variances (heteroscedasticity). WLS specifically addresses only heteroscedasticity, assuming the error terms are uncorrelated.
Q6: Is Weighted R-squared interpreted the same as regular R-squared?
A6: The Weighted R-squared measures the proportion of variance explained by the model, but it accounts for the weights. While a higher value is generally better, its direct comparison with OLS R-squared can be nuanced. It's more about assessing the model's fit within the WLS framework.
Q7: Can I use WLS with categorical predictor variables?
A7: Yes, just like OLS, WLS can incorporate categorical predictor variables (typically through dummy coding). The weights are applied to each observation regardless of whether the predictors are continuous or categorical.
Q8: What does a negative weight imply in WLS?
A8: Standard WLS requires positive weights, typically derived from variances. A negative weight is theoretically incorrect and usually indicates an error in the calculation or specification of the weights, or a misunderstanding of the underlying variance structure.
Q9: Does WLS affect the interpretation of coefficients compared to OLS?
A9: The interpretation of the coefficients themselves (e.g., the change in Y for a one-unit change in X) remains the same. However, WLS provides estimates that are statistically more efficient under heteroscedasticity, meaning the standard errors are smaller, leading to potentially different conclusions about statistical significance.
Related Tools and Internal Resources
OLS Regression CalculatorUnderstand the basics of Ordinary Least Squares regression before exploring WLS.
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if(xValid) {
var rows = xValuesInput.value.split('\n').map(function(row) { return row.trim(); }).filter(function(row) { return row !== ''; });
for (var r = 0; r < rows.length; r++) {
X_data.push(rows[r].split(',').map(function(val) { return parseFloat(val.trim()); }));
}
}
} else {
xValuesInput.value = ''; // Clear X values if n or k are invalid
}
var yValuesInput = document.getElementById('yValuesInput');
var yValid = true;
var Y_data = [];
if (nValid) { // Only validate Y if n is valid
yValid = validateArrayInput('yValuesInput', 'yValuesError', parseInt(document.getElementById('observationsInput').value), null, null);
if(yValid) {
Y_data = yValuesInput.value.split(',').map(function(y) { return parseFloat(y.trim()); });
}
} else {
yValuesInput.value = ''; // Clear Y values if n is invalid
}
if (!nValid || !kValid || !wValid || !xValid || !yValid) {
document.getElementById('primaryResult').innerHTML = 'Please correct the errors.';
document.getElementById('intermediateBetaHat').innerHTML = 'Weighted Beta Hat (Coefficients): —';
document.getElementById('intermediateVariance').innerHTML = 'Weighted Variance-Covariance Matrix: —';
document.getElementById('intermediateR2').innerHTML = 'Weighted R-squared: —';
clearTable();
clearChart();
return;
}
var n = parseInt(document.getElementById('observationsInput').value);
var k = parseInt(document.getElementById('variablesInput').value);
// Prepare matrices
// Add intercept column to X
var X_matrix = new Array(n);
for (var i = 0; i < n; i++) {
X_matrix[i] = new Array(k + 1);
X_matrix[i][0] = 1; // Intercept
for (var j = 0; j < k; j++) {
X_matrix[i][j + 1] = X_data[i][j];
}
}
var Y_vector = new Array(n);
for (var i = 0; i < n; i++) {
Y_vector[i] = [Y_data[i]]; // Make it a column vector
}
// Create Weight Matrix W
var W_matrix = new Array(n);
for (var i = 0; i 0) {
W_matrix[i][i] = weights[i];
} else {
// Handle case of non-positive weight, though validation should catch this
document.getElementById('weightsError').textContent = "Weights must be positive.";
document.getElementById('weightsError').style.display = 'block';
return;
}
}
// Calculate WLS coefficients: beta_hat = (X'WX)^(-1) X'WY
var Xt = transposeMatrix(X_matrix); // X transpose
var W = W_matrix;
var Y = Y_vector;
var W_half = new Array(n); // For calculating X'WX, sqrt(W) not needed for X'WX
for(var i=0; i<n; i++) {
W_half[i] = [Math.sqrt(W_matrix[i][i])]; // Diagonal elements are weights
}
// Using W directly
var XWX = multiplyMatrices(multiplyMatrices(Xt, W), X_matrix);
var XWY = multiplyMatrices(multiplyMatrices(Xt, W), Y);
var XWX_inv = inverseMatrix(XWX);
if (!XWX_inv) {
document.getElementById('primaryResult').innerHTML = 'Error: Cannot invert XWX matrix.';
return;
}
var beta_hat = multiplyMatrices(XWX_inv, XWY);
// Calculate residuals and weighted R-squared
var Y_pred = multiplyMatrices(X_matrix, beta_hat);
var residuals = new Array(n);
var weighted_residuals_sq_sum = 0;
var total_weighted_sum_sq = 0;
for (var i = 0; i < n; i++) {
residuals[i] = Y_data[i] – Y_pred[i][0];
weighted_residuals_sq_sum += weights[i] * Math.pow(residuals[i], 2);
// For total sum of squares, use weighted deviations from the weighted mean of Y
// This calculation can be complex. A simpler approach for R-squared in WLS is often:
// R^2 = 1 – (ESS / TSS), where ESS = sum(w_i * (y_i – y_hat_i)^2) and TSS = sum(w_i * (y_i – y_bar_w)^2)
// y_bar_w needs careful calculation (weighted mean)
}
// Calculate weighted mean of Y
var sum_w = weights.reduce(function(a, b) { return a + b; }, 0);
var sum_wy = 0;
for(var i=0; i<n; i++) {
sum_wy += weights[i] * Y_data[i];
}
var weighted_mean_Y = sum_wy / sum_w;
// Calculate Total Sum of Squares (TSS) with weights
var TSS_w = 0;
for(var i=0; i<n; i++) {
TSS_w += weights[i] * Math.pow(Y_data[i] – weighted_mean_Y, 2);
}
// Calculate Explained Sum of Squares (ESS) with weights
var ESS_w = TSS_w – weighted_residuals_sq_sum;
var R2_w = (TSS_w === 0) ? 1 : ESS_w / TSS_w; // Handle division by zero
// Variance-Covariance Matrix calculation: Cov(beta_hat) = (X'WX)^(-1)
var cov_beta_hat = XWX_inv; // Already calculated inverse
// — Display Results —
document.getElementById('primaryResult').innerHTML = 'β̂ = ' + formatVector(beta_hat);
document.getElementById('intermediateBetaHat').innerHTML = 'Weighted Beta Hat (Coefficients): ' + formatVector(beta_hat);
document.getElementById('intermediateVariance').innerHTML = 'Weighted Variance-Covariance Matrix: ' + formatMatrix(cov_beta_hat);
document.getElementById('intermediateR2').innerHTML = 'Weighted R-squared: ' + R2_w.toFixed(4);
// Populate the results table (including standard errors, t-stats, p-values)
populateResultsTable(beta_hat, cov_beta_hat, X_matrix, n, k + 1); // k+1 includes intercept
updateChart(Y_data, Y_pred.map(function(row){ return row[0]; })); // Pass predicted values as a flat array
}
// Helper to format a vector for display
function formatVector(vector) {
var formatted = '[';
for (var i = 0; i < vector.length; i++) {
if (Array.isArray(vector[i])) { // Handle column vector [ [val] ]
formatted += vector[i][0].toFixed(4);
} else { // Handle flat array
formatted += vector[i].toFixed(4);
}
if (i < vector.length – 1) {
formatted += ', ';
}
}
formatted += ']';
return formatted;
}
// Helper to format a matrix for display
function formatMatrix(matrix) {
var formatted = '[';
for (var i = 0; i < matrix.length; i++) {
formatted += '[';
for (var j = 0; j < matrix[i].length; j++) {
formatted += matrix[i][j].toFixed(4);
if (j < matrix[i].length – 1) {
formatted += ', ';
}
}
formatted += ']';
if (i < matrix.length – 1) {
formatted += ', ';
}
}
formatted += ']';
return formatted;
}
// Populate results table with coefficients, SE, t-stat, p-value
function populateResultsTable(beta_hat, cov_matrix, X_matrix, n, num_vars_incl_intercept) {
var tableBody = document.getElementById('resultsTableBody');
tableBody.innerHTML = ''; // Clear previous rows
if (!beta_hat || !cov_matrix) return;
// Standard errors are the square roots of the diagonal elements of the covariance matrix
var std_errors = [];
for (var i = 0; i = 0) {
std_errors.push(Math.sqrt(cov_matrix[i][i]));
} else {
std_errors.push(NaN); // Indicate error or missing value
}
}
// Calculate t-statistics and p-values (requires degrees of freedom)
// Degrees of freedom for WLS is typically n – k – 1 (where k is number of predictors, not including intercept)
var k_predictors = num_vars_incl_intercept – 1;
var df = n – k_predictors – 1;
if (df <= 0) df = 1; // Ensure df is positive
var t_stats = [];
var p_values = [];
for (var i = 0; i < num_vars_incl_intercept; i++) {
var beta = beta_hat[i] ? (Array.isArray(beta_hat[i]) ? beta_hat[i][0] : beta_hat[i]) : 0;
var se = std_errors[i];
if (isNaN(beta) || isNaN(se) || se === 0) {
t_stats.push(NaN);
p_values.push(NaN);
} else {
var t_stat = beta / se;
t_stats.push(t_stat);
// Calculate p-value using a simplified approximation or placeholder if complex CDF is unavailable
// For a true p-value, you'd need a T-distribution CDF function.
// Placeholder: Use a simple comparison or a library function if available.
// Here, we'll use a common approximation or placeholder logic.
p_values.push(calculatePValue(t_stat, df));
}
}
// Variable names
var var_names = ['Intercept'];
var k_input = parseInt(document.getElementById('variablesInput').value);
for(var i=1; i <= k_input; i++) {
var_names.push('X' + i);
}
// Add rows to the table
for (var i = 0; i 3.5) return 0.0005; // Very unlikely
if (abs_t > 3.0) return 0.002;
if (abs_t > 2.5) return 0.01;
if (abs_t > 2.0) return 0.05;
if (abs_t > 1.5) return 0.1;
if (abs_t > 1.0) return 0.3;
return 0.5; // If t_stat is near 0
}
// Function to update the chart
var chartInstance = null; // To keep track of the chart instance
var wlsChartCanvas = document.getElementById('wlsChart');
var chartCtx = wlsChartCanvas ? wlsChartCanvas.getContext('2d') : null;
function updateChart(observedY, predictedY) {
if (!chartCtx) return;
// Destroy previous chart instance if it exists
if (chartInstance) {
chartInstance.destroy();
}
// Ensure observedY and predictedY have the same length
var length = Math.min(observedY.length, predictedY.length);
var labels = Array.apply(null, {length: length}).map(Number.call, Number); // 0, 1, 2, …
// Create a new chart instance
chartInstance = new Chart(chartCtx, {
type: 'scatter', // Use scatter for plotting points
data: {
datasets: [
{
label: 'Observed Y',
data: observedY.slice(0, length).map(function(y, i) { return { x: labels[i], y: y }; }),
backgroundColor: 'rgba(0, 74, 153, 0.6)',
borderColor: 'rgba(0, 74, 153, 1)',
borderWidth: 1,
pointRadius: 5,
showLine: false // Don't connect observed points
},
{
label: 'Predicted Y',
// For predicted Y, we can plot it as a line against the index, or as scatter points
// Plotting as a line might be more intuitive to show the trend
data: predictedY.slice(0, length).map(function(y, i) { return { x: labels[i], y: y }; }),
backgroundColor: 'rgba(40, 167, 69, 0.6)',
borderColor: 'rgba(40, 167, 69, 1)',
borderWidth: 1,
pointRadius: 5,
showLine: true // Connect predicted points to show the regression line trend
}
]
},
options: {
scales: {
x: {
type: 'linear',
position: 'bottom',
title: {
display: true,
text: 'Observation Index'
}
},
y: {
title: {
display: true,
text: 'Dependent Variable (Y)'
}
}
},
plugins: {
legend: {
display: true,
position: 'top'
},
tooltip: {
callbacks: {
label: function(context) {
var label = context.dataset.label || ";
if (label) {
label += ': ';
}
label += context.parsed.y.toFixed(4);
return label;
}
}
}
},
aspectRatio: 1.5 // Adjust aspect ratio for better visualization on different screens
}
});
}
function clearChart() {
if (chartInstance) {
chartInstance.destroy();
chartInstance = null;
}
if(chartCtx) {
chartCtx.clearRect(0, 0, wlsChartCanvas.width, wlsChartCanvas.height);
}
document.getElementById('chartContainer').style.display = 'none'; // Hide if cleared
}
function hideChartContainer() {
document.getElementById('chartContainer').style.display = 'none';
}
function showChartContainer() {
document.getElementById('chartContainer').style.display = 'block';
}
function clearTable() {
var tableBody = document.getElementById('resultsTableBody');
tableBody.innerHTML = '
Enter data and click "Calculate WLS"
';
}
// Function to reset the calculator
function resetCalculator() {
document.getElementById('observationsInput').value = '10';
document.getElementById('variablesInput').value = '2';
document.getElementById('weightsInput').value = '1,1,1,1,1,1,1,1,1,1'; // Default to equal weights
document.getElementById('xValuesInput').value = '1,2\n2,4\n3,6\n4,8\n5,10\n6,12\n7,14\n8,16\n9,18\n10,20'; // Example X values
document.getElementById('yValuesInput').value = '3,5,7,9,11,13,15,17,19,21'; // Example Y values corresponding to X
// Clear errors
document.getElementById('observationsError').textContent = ";
document.getElementById('variablesError').textContent = ";
document.getElementById('weightsError').textContent = ";
document.getElementById('xValuesError').textContent = ";
document.getElementById('yValuesError').textContent = ";
// Clear results
document.getElementById('primaryResult').innerHTML = '—';
document.getElementById('intermediateBetaHat').innerHTML = 'Weighted Beta Hat (Coefficients): —';
document.getElementById('intermediateVariance').innerHTML = 'Weighted Variance-Covariance Matrix: —';
document.getElementById('intermediateR2').innerHTML = 'Weighted R-squared: —';
clearTable();
clearChart();
hideChartContainer(); // Hide chart on reset
// Optionally, re-run calculation with defaults
// calculateWLS();
}
// Function to copy results to clipboard
function copyResults() {
var resultsText = "WLS Calculation Results:\n\n";
resultsText += "Primary Result (Coefficients): " + document.getElementById('primaryResult').textContent.replace('Weighted Beta Hat (Coefficients): ', ") + "\n";
resultsText += "Weighted R-squared: " + document.getElementById('intermediateR2').textContent.replace('Weighted R-squared: ', ") + "\n\n";
resultsText += "Key Intermediate Values:\n";
resultsText += "- " + document.getElementById('intermediateBetaHat').textContent + "\n";
resultsText += "- " + document.getElementById('intermediateVariance').textContent + "\n";
resultsText += "- " + document.getElementById('intermediateR2').textContent + "\n\n";
resultsText += "Regression Table:\n";
var table = document.getElementById('resultsTable');
var rows = table.rows;
for (var i = 0; i < rows.length; i++) {
var cells = rows[i].cells;
var rowText = "";
for (var j = 0; j < cells.length; j++) {
rowText += cells[j].textContent + "\t"; // Use tab for basic formatting
}
resultsText += rowText.trim() + "\n";
}
resultsText += "\nAssumptions/Inputs:\n";
resultsText += "- Observations (n): " + document.getElementById('observationsInput').value + "\n";
resultsText += "- Predictor Variables (k): " + document.getElementById('variablesInput').value + "\n";
resultsText += "- Weights (w_i): " + document.getElementById('weightsInput').value + "\n";
resultsText += "- X Values:\n" + document.getElementById('xValuesInput').value + "\n";
resultsText += "- Y Values: " + document.getElementById('yValuesInput').value + "\n";
// Use Clipboard API
navigator.clipboard.writeText(resultsText).then(function() {
// Optionally provide feedback to user
var copyButton = document.querySelector('.copy');
copyButton.textContent = 'Copied!';
setTimeout(function() {
copyButton.textContent = 'Copy Results';
}, 2000);
}).catch(function(err) {
console.error('Failed to copy: ', err);
// Fallback for older browsers or if permissions are denied
alert('Failed to copy results. Please copy manually.');
});
}
// Initial setup for chart canvas size based on container
function setupChartCanvas() {
var chartContainer = document.getElementById('chartContainer');
if (chartContainer && chartCtx) {
var containerWidth = chartContainer.offsetWidth;
// Maintain a reasonable aspect ratio, e.g., 16:9 or 4:3
var aspectRatio = 16 / 9;
wlsChartCanvas.width = containerWidth;
wlsChartCanvas.height = containerWidth / aspectRatio;
}
}
// Add event listeners for real-time updates (optional, can be triggered by button)
// document.getElementById('observationsInput').addEventListener('input', calculateWLS);
// document.getElementById('variablesInput').addEventListener('input', calculateWLS);
// document.getElementById('weightsInput').addEventListener('input', calculateWLS);
// document.getElementById('xValuesInput').addEventListener('input', calculateWLS);
// document.getElementById('yValuesInput').addEventListener('input', calculateWLS);
// Initial setup and calculation
window.onload = function() {
resetCalculator(); // Set default values
setupChartCanvas(); // Setup canvas size
// calculateWLS(); // Optionally calculate immediately on load
};
// Re-setup canvas size on window resize
window.addEventListener('resize', setupChartCanvas);
// Ensure chart is displayed when results are calculated
document.getElementById('results-container').addEventListener('DOMSubtreeModified', function() {
// Check if primary result has been updated from '—'
if (document.getElementById('primaryResult').textContent !== '—') {
showChartContainer();
}
}, false);