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Polynomial regression might seem different from linear regression at first glance, but it’s still considered a linear model. Why? It all comes down to how the model parameters are used.

✔️ Linear in Parameters: In polynomial regression, the model remains linear in terms of its coefficients. Even if we include terms like squared or cubic versions of the input variable, the relationship with the parameters stays linear.

✔️ Transformed Features: Instead of treating the data as simple inputs, polynomial regression transforms the original input features (e.g., turning an input into its squared or cubic form). However, the relationship between the coefficients and the target variable remains linear.

✔️ Optimization Stays Linear: The method for estimating the coefficients, such as Ordinary Least Squares, remains the same because the relationship with the parameters does not become non-linear.

❌ Non-Linear Model? If a regression model involves coefficients in non-linear ways, such as multiplying them together or applying complex functions to them, it’s no longer considered linear regression.

Quantile regression is a valuable tool for analyzing the relationship between variables, especially when data is not evenly distributed or has outliers.

Unlike traditional linear regression, which focuses only on the mean, quantile regression allows us to predict different points across the distribution of the target variable.

Challenges:
❌ Compared to linear regression, quantile regression requires more computational power and can be harder to interpret for non-experts.
❌ Larger sample sizes might be needed to achieve stable and reliable quantile estimates, especially for extreme percentiles.
❌ The model's results might be less intuitive if you are accustomed to traditional regression techniques, which could limit ease of communication.

Advantages:
✔️ Quantile regression helps to explore trends at various quantiles, offering a more detailed picture of your data.
✔️ This method is highly effective for non-normal data, particularly when there are outliers or heavy tails.
✔️ It is ideal for situations where extreme values or various percentiles are as important as the central trend.

How to handle quantile regression in practice:
🔹 Stata: qreg var1 var2 var3, quantile(.25)