Quick Answer: Generalized Least Squares Regression Result

The verified Python and R analyses use 649 complete observations. The outcome is G3. Numeric predictors are G1, G2, studytime, failures, absences, age and Medu. Categorical predictors are school, sex and address. The baseline OLS residuals show strong heteroscedasticity, so inverse estimated error-variance weights are used for feasible GLS.

In the Python report, OLS has R² = 0.8524 and adjusted R² = 0.8501. Feasible GLS/WLS has R² = 0.8695 and adjusted R² = 0.8674. Its AIC decreases from 2143.3400 to 2037.0053, and BIC decreases from 2192.5698 to 2086.2351. MAE improves from 0.7791 to 0.7721, while unweighted RMSE changes slightly from 1.2403 to 1.2423.

What Is Generalized Least Squares Regression?

Generalized Least Squares Regression is a linear regression estimator designed for errors with a non-spherical covariance structure. Ordinary least squares assumes that, conditional on the predictors, errors have equal variance and are uncorrelated. GLS allows error variances to differ and errors to be correlated according to a specified covariance matrix.

When the covariance matrix is known, GLS transforms the data so that the transformed errors satisfy the ordinary least-squares conditions. When the covariance structure is unknown and estimated from data, the method is called feasible generalized least squares.

What Changes from OLS?

GLS can change coefficient estimates, standard errors, confidence intervals, p-values and fitted values. It does not merely replace conventional standard errors. If the weighting or covariance model is correct, GLS can be more efficient than OLS. If it is misspecified, the apparent precision gain can be misleading.

Why Ordinary Least Squares Can Be Inefficient

Under heteroscedasticity, OLS coefficient estimates can remain unbiased when the conditional-mean model is correct, but ordinary OLS standard errors become unreliable and the estimator is no longer the most efficient linear unbiased estimator. High-variance observations receive the same influence as low-variance observations even though their outcomes are less precisely measured.

Under serial or clustered correlation, OLS also ignores dependence between errors. The effective information is smaller than the raw record count suggests. GLS addresses these problems by modelling a covariance matrix rather than assuming the identity matrix multiplied by one variance.

Generalized Least Squares Matrix Formula

The linear model is:

When Ω is known and positive definite, the GLS coefficient estimator is:

If Ω = σ²I, the formula reduces to OLS. If Ω is diagonal with observation-specific variances, GLS becomes weighted least squares. If Ω contains off-diagonal correlations, GLS also accounts for error dependence.

Whitening Transformation

Suppose a matrix P satisfies P′P = Ω⁻¹. Multiplying the model by P produces transformed errors with identity covariance:

OLS applied to the transformed outcome and predictors produces the GLS estimator. This interpretation helps explain why GLS is still a least-squares method: it minimizes squared residuals in transformed space.

Feasible Generalized Least Squares and the Variance Model

The complete covariance matrix is rarely known. Feasible GLS first fits OLS, uses OLS residuals to estimate the error-variance pattern, constructs an estimated covariance matrix and then refits the regression using that estimate.

Worked Analysis Variance Model

The predicted log variance is exponentiated to obtain a positive estimated variance. The feasible GLS weight is:

Observations with smaller predicted error variance receive larger weights. Observations with larger predicted variance receive smaller weights. The full-data weight distribution spans approximately 1.9 to 5.1 in the supplied histogram, with the greatest concentration around 2.7 to 3.4.

Why Log Residual Squared?

Squared residuals are nonnegative and highly skewed. Modelling their logarithm ensures that exponentiated predictions are positive and can stabilize the variance relationship. A small constant prevents taking the logarithm of zero.

One-Step vs Iterated FGLS

The supplied workflow is a practical one-step FGLS correction: OLS residuals estimate variance, and the weighted model is fitted once. Iterated FGLS repeats variance estimation and weighting until convergence. Iteration may improve internal consistency but can also amplify a misspecified variance model.

GLS vs WLS vs FGLS vs GLSAR

The feasible WLS and diagonal GLS models produce nearly identical results: Python R² = 0.8695, adjusted R² = 0.8674 and MAE = 0.7721 for both. Their AIC values differ only from 2037.0053 to 2037.0021 because the implementations represent the same diagonal covariance correction.

When to Use Generalized Least Squares Regression

The decision should begin with the error structure and research goal, not with a desire to obtain a larger R² or smaller p-value.

Dataset and Variables Used

The Python and R reports use the same 649 complete observations and the same formula. G3 is the continuous outcome. Seven numeric predictors and three categorical predictors enter both OLS and FGLS.

Variable Dictionary

Error-Variance Variables

OLS fitted values, OLS residuals, squared residuals, predicted log variance, estimated variance and inverse estimated variance weights are derived quantities. They are not substantive predictors in the main G3 equation; they define the FGLS covariance correction.

OLS Diagnostic Results That Motivate GLS

Python and R use somewhat different auxiliary-test implementations, so the exact Breusch–Pagan and White statistics differ. Both software packages produce the same decision: heteroscedasticity is statistically clear. The Durbin–Watson statistic and GLSAR estimate show that serial dependence is not the principal problem.

OLS, Feasible GLS, Diagonal GLS and GLSAR Model Comparison

Python Model Comparison

Feasible WLS and diagonal GLS are numerically equivalent for the estimated diagonal covariance matrix. Their weighted R² exceeds OLS by approximately 0.0171. MAE improves by 0.0070, but RMSE increases by 0.0020. The results therefore support the variance correction without claiming that every unweighted error criterion improves.

R Model Comparison

R reproduces OLS R² = 0.8523746, adjusted R² = 0.8500607, RMSE = 1.240328 and MAE = 0.7790877. Feasible GLS/WLS has R² = 0.8694957, adjusted R² = 0.8674502, RMSE = 1.242313 and MAE = 0.7721485.

R reports AIC/BIC as 2145.340/2199.045 for OLS and 2039.003/2092.708 for FGLS. These differ from Python by additive likelihood-convention constants. Compare AIC and BIC within the same software, not across different likelihood definitions.

OLS and Feasible GLS Coefficient Interpretation

Exact OLS Coefficients from the Reports

FGLS Coefficient Changes Shown in Chart 6

The PDF coefficient table is truncated after the FGLS intercept, which is exactly -0.3473, SE = 0.7111, p = 0.6254, 95% CI [-1.7436, 1.0490]. The comparison chart displays the remaining plotted FGLS coefficients graphically. Approximate bar values are school_MS ≈ -0.15, sex_M ≈ -0.16, address_U ≈ 0.08, G1 ≈ 0.14, G2 ≈ 0.86, studytime ≈ 0.05, failures ≈ -0.29 and absences ≈ 0.01.

The substantive pattern remains stable: G2 is the dominant positive coefficient, G1 is positive, and failures is negative. Weighting changes the magnitudes modestly, particularly making the failures coefficient more negative. Exact FGLS coefficient reporting should use the generated CSV table rather than reverse-reading the bar chart.

How Generalized Least Squares Weights Work

The full model assigns larger weights to observations with smaller estimated error variance. A weight twice as large does not mean an observation is twice as important substantively. It means the fitted covariance model treats it as having greater conditional precision.

Weights should be normalized only when needed for interpretation or software stability. Multiplying every weight by the same constant does not change WLS coefficient estimates, although it may affect reported likelihood scales.

Weight Diagnostics

Inspect minimum, maximum, quantiles and relationships with fitted values and predictors. Extremely large weights can cause a few observations to dominate. Extremely small weights can effectively remove observations from coefficient estimation. Trimming weights changes the estimand and requires justification.

Generalized Least Squares Regression Assumptions

GLS does not require raw outcomes to be normally distributed for coefficient consistency. Normality is mainly relevant to exact finite-sample inference and predictive distribution assumptions.

Python Chart-by-Chart Interpretation with Real Values

Python Chart 1: G3 Outcome Distribution

The analysis uses 649 G3 observations. The distribution has mean 11.9060, median 12, SD 3.2307, first quartile 10, third quartile 14, minimum 0 and maximum 19. The highest frequencies occur at G3 = 11 with 104 records, G3 = 10 with 97, G3 = 13 with 82 and G3 = 12 with 72.

The histogram is concentrated around grades 10–15 but includes 15 zero scores. These low outcomes create the long negative residual tail visible in both OLS and FGLS diagnostic plots. GLS changes weighting; it does not transform the bounded integer outcome into a normal variable.

Python Chart 2: Observed vs Fitted Values for OLS and FGLS

Both fitted series track the 45-degree line closely through the middle of the grade distribution. OLS has R² = 0.8524, RMSE = 1.2403 and MAE = 0.7791. FGLS has weighted R² = 0.8695, RMSE = 1.2423 and MAE = 0.7721.

The two point clouds overlap heavily because coefficient changes are modest. Several zero outcomes receive fitted values between approximately 5 and 10, producing large negative residuals. FGLS improves the weighted objective and MAE, but it does not eliminate these unusual cases.

Python Chart 3: OLS Residuals vs Fitted Values

OLS residuals range visually from approximately -9.1 to 5.5. The main residual band is concentrated between about -2 and 2, but the distribution includes large negative points around fitted values 5–10. The diagonal stripes arise because observed G3 is integer valued while fitted values are continuous.

Formal tests confirm that the visible spread is not constant: Breusch–Pagan LM = 34.4086, p = 0.000157, and White LM = 106.8797, p = 0.000348. These values, rather than visual appearance alone, motivate the FGLS correction.

Python Chart 4: Feasible GLS Residuals vs Fitted Values

The FGLS residual plot retains a similar raw-residual range, approximately -9.1 to 5.6, because weighting does not guarantee smaller residuals for every observation. The central point band remains near zero and the same integer-outcome diagonal structure is visible.

FGLS MAE decreases from 0.7791 to 0.7721, but RMSE rises slightly from 1.2403 to 1.2423. The correct interpretation is not that every residual became smaller. The model redistributes influence according to estimated precision and improves the weighted likelihood and average absolute error.

Python Chart 5: OLS vs FGLS Absolute Residual Spread

The two boxplots have very similar central ranges and extreme outliers reaching above 9. OLS MAE is 0.7791, while FGLS MAE is 0.7721, an improvement of about 0.0070. The boxplot median appears slightly lower for FGLS, but the difference is small.

The chart is a useful warning against exaggerated claims. GLS is strongly justified by the variance diagnostics, yet the unweighted absolute-residual distribution changes only modestly. Efficiency correction and dramatic point-prediction improvement are not the same thing.

Python Chart 6: OLS vs Feasible GLS Coefficient Comparison

OLS plotted coefficients are school_MS = -0.1788, sex_M = -0.1917, address_U = 0.1082, G1 = 0.1337, G2 = 0.8814, studytime = 0.0641, failures = -0.2315 and absences = 0.0177. The FGLS bars remain close but shift approximately to -0.15, -0.16, 0.08, 0.14, 0.86, 0.05, -0.29 and 0.01.

G2 remains the dominant effect under both estimators. The largest visible change occurs for failures, which becomes more negative under FGLS. Exact FGLS coefficients should be copied from the output CSV because the PDF coefficient table is truncated.

Python Chart 7: Feasible GLS Weight Distribution

Weights span approximately 1.9 to 5.1. The highest-frequency bin is around 2.8–3.0 with slightly more than 80 observations. Other dense regions occur around 2.5–2.8 and 3.1–3.5. Relatively few records receive weights above 4.5.

The moderate range indicates that the variance model changes influence without creating extremely dominant observations. Higher weights correspond to lower predicted error variance; lower weights correspond to noisier observations.

Python Chart 8: OLS Diagnostic p-Values

Every displayed p-value is far left of the .05 reference line. Python reports Breusch–Pagan LM p = 0.000157, Breusch–Pagan F p near 0.0001, White LM p = 0.000348, White F p near 0.0001 and Jarque–Bera p < 0.0001.

The heteroscedasticity tests support GLS. The Jarque–Bera result supports residual-distribution caution but does not determine the choice between OLS and GLS. Durbin–Watson is excluded because it is a statistic rather than a p-value.

Python Chart 9: Observed and Fitted Means by G1

At G1 = 0, the observed mean is approximately 11.0, while OLS and FGLS means are near 10.1. At G1 = 5, the observed mean is approximately 2.6, whereas both fitted means are near 4.0. From G1 = 8 through 17, the three lines generally track closely.

At G1 = 18, the fitted means rise to about 18.5, compared with an observed mean near 18.0. At G1 = 19, the observed mean is about 18.0 and fitted means are about 17.7–17.8. The chart shows excellent central calibration with larger discrepancies at sparse extreme G1 levels.

R Charts, Tables and Independent Validation

The R report independently reproduces the 649-row formula, heteroscedasticity decision, model comparison and nine chart topics. The supplied list repeats the same chart URLs, so the R section uses those confirmed assets while interpreting the R-specific numerical results.

Open the complete Generalized Least Squares Regression R report PDF

R Chart 1: Outcome Distribution Validation

R uses the same outcome mean 11.9060, median 12, SD 3.2307 and range 0–19. This confirms that OLS and FGLS differences are estimator differences rather than changes in the analyzed sample.

R Chart 2: Observed vs Fitted Validation

R reports OLS RMSE = 1.240328 and MAE = 0.7790877. FGLS reports RMSE = 1.242313 and MAE = 0.7721485. The same conclusion holds: average absolute error improves slightly, while RMSE changes minimally in the opposite direction.

R Chart 3: OLS Residual Diagnostic Validation

R’s Breusch–Pagan LM statistic is 19.0968 with p = 1.24 × 10⁻⁵. The White statistic is 20.0366 with p = 4.46 × 10⁻⁵. Both validate the nonconstant variance decision despite differing from Python’s auxiliary-test statistics.

R Chart 4: FGLS Residual Diagnostic Validation

The weighted fit modifies the coefficient solution and influence structure, but raw residual outliers remain. Post-FGLS residual checks should evaluate whether the variance relationship is reduced in transformed or standardized residuals, not only whether every raw residual shrinks.

R Chart 5: Residual Spread Validation

R MAE decreases by approximately 0.00694. The boxplots remain similar because the strongest correction is to observation influence and likelihood, not a universal compression of raw residual magnitudes.

R Chart 6: Coefficient Comparison Validation

R reproduces exact OLS coefficients G1 = 0.133732, G2 = 0.881441 and failures = -0.231465. The FGLS comparison preserves the dominant G2 effect and negative failures direction.

R Chart 7: Weight Distribution Validation

The distribution concentrates between approximately 2 and 4, with a thinner upper tail to about 5.1. This indicates meaningful but not extreme reweighting.

R Chart 8: Diagnostic p-Value Validation

R’s heteroscedasticity p-values are even smaller than Python’s: 0.0000124 and 0.0000446. Both are far below .05. Jarque–Bera is effectively zero, confirming nonnormal residual tails.

R Chart 9: Observed and Fitted Means by G1 Validation

The observed and fitted mean lines nearly overlap for most G1 values, with the largest discrepancy near G1 = 5. R therefore confirms strong central fit and sparse-level instability at the extremes.

SPSS Output Status and Corrected GLS Workflow

Open the currently supplied SPSS PDF

Required SPSS Export Correction

OUTPUT EXPORT
  /CONTENTS EXPORT=ALL LAYERS=PRINTSETTING MODELVIEWS=PRINTSETTING
  /PDF DOCUMENTFILE=
'D:\DATA ANALYSIS\H Regression Tests and Models\Generalized Least Squares Regression\SPSS_Output\pdf\generalized_least_squares_regression_spss_report.pdf'.

Do not use EMBEDBOOKMARKS or EMBEDFONTS as separate OUTPUT EXPORT subcommands. Regenerate the Viewer output and PDF after removing them.

Practical SPSS FGLS Procedure

SPSS does not provide the same one-command heteroscedastic FGLS workflow as the Python/R scripts. A compatible approach is:

1. Fit baseline REGRESSION and save predicted values and residuals.
2. Compute squared residuals and log squared residuals with a small constant.
3. Regress log residual squared on absolute fitted value or selected variance predictors.
4. Save predicted log variance.
5. Compute estimated variance as EXP(predicted log variance).
6. Compute inverse variance weight.
7. Use WLS or weighted REGRESSION with the estimated weight.
8. Save a valid SAV, SPV and PDF.

Excel Worked Generalized Least Squares Results

The Excel workbook is a separate 10-row formula demonstration. It does not reproduce the 649-row model coefficients. It explains how OLS residuals lead to estimated variances, inverse weights and FGLS fitted values.

Excel Demonstration Summary

Ten-Row Formula Demonstration

For row 1, observed Y = 10, OLS fitted = 10.4 and OLS residual = -0.4. The predicted log variance is 1.15, estimated variance is 3.1582, inverse weight is 0.3166, FGLS fitted = 10.2 and FGLS residual = -0.2.

Row 7 has the smallest weight, approximately 0.1738, because its estimated variance is 5.7546. Row 8 has the largest weight, approximately 0.4066, because its estimated variance is 2.4596. This directly demonstrates inverse-variance weighting.

Excel Formula Patterns

Python, R, SPSS and Excel GLS Workflows

Generalized Least Squares Regression Code

Python Code

import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
from statsmodels.stats.diagnostic import het_breuschpagan, het_white
from statsmodels.stats.stattools import durbin_watson

df = pd.read_csv("dataset.csv")

formula = (
    "G3 ~ G1 + G2 + studytime + failures + absences + age + Medu"
    " + C(school) + C(sex) + C(address)"
)

work = df[
    ["G3", "G1", "G2", "studytime", "failures",
     "absences", "age", "Medu", "school", "sex", "address"]
].dropna().copy()

# OLS baseline
ols = smf.ols(formula, data=work).fit()
work["ols_fitted"] = ols.fittedvalues
work["ols_resid"] = ols.resid

# Heteroscedasticity tests
bp = het_breuschpagan(ols.resid, ols.model.exog)
white = het_white(ols.resid, ols.model.exog)
print("BP LM p:", bp[1])
print("White LM p:", white[1])
print("DW:", durbin_watson(ols.resid))

# Variance model
small = 1e-8
work["log_resid_sq"] = np.log(work["ols_resid"] ** 2 + small)
variance_model = smf.ols(
    "log_resid_sq ~ I(abs(ols_fitted))",
    data=work
).fit()

work["predicted_log_variance"] = variance_model.predict(work)
work["estimated_variance"] = np.exp(work["predicted_log_variance"])
work["gls_weight"] = 1.0 / work["estimated_variance"]

# Feasible GLS as WLS
fgls = smf.wls(
    formula,
    data=work,
    weights=work["gls_weight"]
).fit()

# Equivalent diagonal GLS
sigma = np.diag(work["estimated_variance"].to_numpy())
gls = sm.GLS(
    ols.model.endog,
    ols.model.exog,
    sigma=sigma
).fit()

print(fgls.summary())
print(gls.summary())

R Code

df <- read.csv("dataset.csv", stringsAsFactors = TRUE)

formula_gls <- G3 ~ G1 + G2 + studytime + failures +
  absences + age + Medu + school + sex + address

work <- na.omit(df[
  c("G3", "G1", "G2", "studytime", "failures",
    "absences", "age", "Medu", "school", "sex", "address")
])

ols <- lm(formula_gls, data = work)
work$ols_fitted <- fitted(ols)
work$ols_residual <- residuals(ols)

small <- 1e-8
work$log_resid_sq <- log(work$ols_residual^2 + small)

variance_model <- lm(
  log_resid_sq ~ abs(ols_fitted),
  data = work
)

work$predicted_log_variance <- predict(variance_model)
work$estimated_variance <- exp(work$predicted_log_variance)
work$gls_weight <- 1 / work$estimated_variance

fgls <- lm(
  formula_gls,
  data = work,
  weights = gls_weight
)

summary(ols)
summary(fgls)

library(lmtest)
library(sandwich)
bptest(ols)
dwtest(ols)

SPSS Compatibility-Safe Syntax Pattern

* Baseline OLS.
REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age Medu
    school_MS sex_M address_U
  /STATISTICS COEFF OUTS R ANOVA CI(95) COLLIN TOL
  /SAVE PRED(ols_pred) RESID(ols_resid).

COMPUTE resid_sq = ols_resid**2.
COMPUTE log_resid_sq = LN(resid_sq + .000001).
COMPUTE abs_ols_pred = ABS(ols_pred).
EXECUTE.

* Variance model.
REGRESSION
  /DEPENDENT log_resid_sq
  /METHOD=ENTER abs_ols_pred
  /SAVE PRED(pred_log_variance).

COMPUTE estimated_variance = EXP(pred_log_variance).
COMPUTE gls_weight = 1 / estimated_variance.
EXECUTE.

WEIGHT BY gls_weight.

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age Medu
    school_MS sex_M address_U
  /STATISTICS COEFF OUTS R ANOVA CI(95) COLLIN TOL
  /SAVE PRED(fgls_pred) RESID(fgls_resid).

WEIGHT OFF.

OUTPUT SAVE OUTFILE=
'D:\DATA ANALYSIS\H Regression Tests and Models\Generalized Least Squares Regression\SPSS_Output\spv\generalized_least_squares_regression_spss_output.spv'.

OUTPUT EXPORT
  /CONTENTS EXPORT=ALL LAYERS=PRINTSETTING MODELVIEWS=PRINTSETTING
  /PDF DOCUMENTFILE=
'D:\DATA ANALYSIS\H Regression Tests and Models\Generalized Least Squares Regression\SPSS_Output\pdf\generalized_least_squares_regression_spss_report.pdf'.

Excel Formula Pattern

OLS residual:
=ObservedY-OLSFitted

Residual squared:
=OLSResidual^2

Log residual squared:
=LN(ResidualSquared+0.000001)

Estimated variance:
=EXP(PredictedLogVariance)

GLS inverse variance weight:
=1/EstimatedVariance

Weighted least squares objective:
=SUMPRODUCT(WeightRange,ResidualRange,ResidualRange)

Generalized Least Squares vs Heteroscedasticity-Robust Standard Errors

A useful sensitivity analysis reports OLS with HC3 standard errors alongside FGLS. If substantive conclusions change dramatically, investigate whether a small set of weights or variance-model assumptions drives the difference.

Prediction, Fit and Model Selection in GLS

Weighted R² is calculated using the weighted residual objective and should not automatically be compared with unweighted OLS R² as though they were the same quantity. AIC and BIC also depend on likelihood conventions and weight scaling.

How to Interpret This Model Comparison

For prediction, evaluate cross-validated unweighted error, weighted error and calibration across variance strata. In-sample weighted AIC improvement is not enough to establish superior out-of-sample prediction.

APA-Style Reporting for Generalized Least Squares Regression

Full Python Result

Coefficient Summary

SPSS Qualification

Generalized Least Squares Reporting Checklist

Common Generalized Least Squares Mistakes

Generalized Least Squares Troubleshooting Guide

Weights Are Infinite or Missing

Predicted variance may be zero, negative before exponentiation, missing or numerically extreme. Use a log-variance model, verify finite fitted values and apply reasonable numerical bounds only with documented justification.

FGLS Coefficients Change Dramatically

Inspect whether a small number of observations receive very large weights. Compare with OLS HC3 results and refit after checking data errors. Dramatic changes may indicate variance-model misspecification.

FGLS R Squared Is Higher but RMSE Is Worse

This is possible because weighted R² optimizes a different objective than unweighted RMSE. In this analysis, FGLS adjusted R² rises to .8674 while RMSE changes from 1.2403 to 1.2423.

Python and R AIC Values Differ by a Constant

Different implementations include different likelihood constants or weight normalizations. Compare differences between OLS and FGLS inside Python or inside R, not raw values across languages.

Breusch–Pagan and White Statistics Differ Across Software

The auxiliary regressions and degrees of freedom may differ. Focus on whether the same null decision is reached. Here every implementation strongly rejects constant variance.

Durbin–Watson Is Near 2 but GLSAR Is Included

GLSAR is a comparison model. With ρ = 0.0661 and DW = 1.8723, autocorrelation is weak; the main correction remains heteroscedastic FGLS.

SPSS PDF Contains Only EMBEDBOOKMARKS Error

Remove the unsupported subcommand, rerun the entire syntax, save the Viewer file and export with the safe PDF block. Do not merely rename the failed one-page PDF.

FGLS Residual Plot Still Has Outliers

GLS does not guarantee elimination of unusual outcomes. Check studentized residuals, Cook’s distance, Mahalanobis distance and outlier detection.

Variance Model Produces No Improvement

The chosen variance predictor may not describe heteroscedasticity. Consider multiple variance predictors, group-specific variances, smooth functions or robust OLS instead of forcing FGLS.

Downloads and Resources

Related Salar Cafe Guides

Final Generalized Least Squares Regression Conclusion

The 649-row analysis provides strong evidence that OLS errors are heteroscedastic. Feasible GLS uses inverse estimated error-variance weights and increases the reported weighted adjusted R² from 0.8501 to 0.8674. It lowers MAE from 0.7791 to 0.7721 and substantially improves within-software likelihood criteria, while unweighted RMSE remains nearly unchanged.

G2 remains the dominant academic predictor, G1 remains positive, and failures remains negative. The correction changes coefficient magnitude and influence modestly rather than reversing the substantive model.

The correct final report should state the variance model, explain the weight construction, distinguish weighted from unweighted fit measures, provide sensitivity analysis against robust OLS and regenerate the SPSS output with compatibility-safe PDF syntax before publishing SPSS-specific numerical conclusions.