Econometrics # 16 : Assumptions of Classical Linear Regression Model (CLRM) with English [CC]

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6 chapters6 takeaways10 key terms5 questions

Overview

This video explains the assumptions of the Classical Linear Regression Model (CLRM), focusing on the four assumptions not covered in previous lectures. It details the requirements for linearity in parameters, the zero mean of error terms, the independence of independent variables from error terms, and the normality of error terms. The video also introduces the Gauss-Markov theorem, which states that if six core assumptions are met, Ordinary Least Squares (OLS) estimators are Best Linear Unbiased Estimators (BLUE). Finally, it touches upon the Central Limit Theorem and its relation to the normality assumption for hypothesis testing in large samples.

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Chapters

The regression model must be linear with respect to its parameters (coefficients like alpha and beta).
Non-linearity in variables is permissible (e.g., beta * X^2), as long as the parameters themselves are not raised to powers or involved in non-linear functions.
OLS has the capacity to estimate parameters only when they appear linearly in the model.

This assumption ensures that the OLS method can correctly estimate the relationship between variables by solving linear equations, preventing biased or unachievable estimates.

A model like y = alpha + beta1*x + beta2*x^2 is acceptable because the parameters (beta1, beta2) are linear, even though the variable 'x' is squared.

The expected value (average) of the error term (e) must be zero.
This assumption is crucial for ensuring that the estimated values from the model are unbiased and close to the actual values on average.
The presence of a constant term (intercept) in the regression model automatically ensures that the mean of the error term is zero.

If the error term's mean is not zero, the model's predictions will be systematically off, leading to biased parameter estimates and unreliable conclusions.

If the average error is 10, the model y = alpha + beta*x + 10 will consistently overestimate or underestimate y. Adding a constant term and adjusting the intercept effectively centers the error term around zero.

Independent variables (predictors) should not be correlated with the error term.
If they are correlated, it indicates that omitted variables (part of the error term) are influencing the independent variables, or vice versa.
This correlation leads to endogeneity, causing the estimated coefficients to be overestimated or biased.

This assumption prevents omitted variable bias and ensures that the estimated coefficients solely reflect the impact of the included independent variables, not confounding factors.

If 'x' is positively related to the error term 'e', and 'e' contains unobserved factors affecting 'y', then an increase in 'x' might be wrongly attributed to the coefficient of 'x' when it's partly due to the unobserved factors.

The error terms are assumed to follow a normal distribution.
This assumption is often considered optional for estimation purposes, especially with large sample sizes, due to the Central Limit Theorem.
It is essential for hypothesis testing (e.g., calculating p-values and confidence intervals) when using small sample sizes.

Normality allows for accurate statistical inference, enabling us to test hypotheses about the parameters and determine the significance of the model's findings.

When testing if a coefficient is statistically different from zero, the assumption of normally distributed errors allows us to use standard t-tests and F-tests.

The Gauss-Markov theorem states that if the first six CLRM assumptions (linearity, zero error mean, no correlation between independent variables and errors, no autocorrelation, no heteroscedasticity, and linearity in parameters) hold, then OLS estimators are BLUE.
BLUE stands for Best Linear Unbiased Estimator.
Unbiased means the average of the estimates equals the true parameter value.
Best means the estimator has the minimum variance among all linear unbiased estimators, making it the most precise.

This theorem provides the theoretical foundation for why OLS is a preferred method in econometrics, guaranteeing the optimal properties of the estimates under specific conditions.

If all six assumptions are met, the OLS estimates for alpha and beta will be the most accurate and reliable linear unbiased estimates possible.

The Central Limit Theorem (CLT) states that the distribution of sample means (or sums) approaches a normal distribution as the sample size increases, regardless of the original population distribution.
In econometrics, the CLT implies that with a sufficiently large sample size (often considered n >= 30), the error terms in a regression model will approximate a normal distribution.
This large-sample property makes the normality assumption less critical for estimation and allows for valid hypothesis testing even if the original errors are not perfectly normal.

The CLT justifies using statistical inference methods that rely on normality, even when the underlying error distribution is unknown or non-normal, provided the sample size is adequate.

Even if the errors in a population are skewed, taking a sample of 100 observations for a regression analysis means the distribution of the estimated coefficients will be approximately normal due to the CLT.

Key takeaways

1The Classical Linear Regression Model (CLRM) relies on several key assumptions to ensure reliable and unbiased estimates.
2Linearity in parameters is crucial; the model's coefficients must be linear, though variables can be non-linear.
3A zero mean for the error term is vital for unbiased estimation, and the inclusion of an intercept term ensures this.
4Independent variables must not be correlated with the error term to avoid endogeneity and biased coefficients.
5The Gauss-Markov theorem guarantees that OLS estimators are the 'Best Linear Unbiased Estimators' (BLUE) if the first six CLRM assumptions are met.
6The normality of error terms is primarily needed for hypothesis testing with small samples; the Central Limit Theorem supports its approximation in large samples.

Key terms

Classical Linear Regression Model (CLRM)Linearity in ParametersError TermEndogeneityGauss-Markov TheoremBest Linear Unbiased Estimator (BLUE)Ordinary Least Squares (OLS)Central Limit Theorem (CLT)ParameterEstimate

Test your understanding

1What is the difference between a model being linear in parameters versus linear in variables, and why is the former essential for OLS?
2How does the assumption of a zero mean error term contribute to the unbiasedness of regression estimates?
3Explain the consequences of endogeneity, where independent variables are correlated with the error term, on the estimated coefficients.
4Under what conditions does the Gauss-Markov theorem guarantee that OLS estimators are BLUE?
5How does the Central Limit Theorem relax the strict requirement of normally distributed errors for hypothesis testing in large sample regression analysis?