Appunti schematici Econometria

Goldfeld Organing and White OLS

Based on the Goldfeld approach, we establish a system where we calculate the auxiliary models using White's OLS estimators. This involves analyzing time series and dynamic models to identify patterns and trends.

Linear Model Hypothesis

The linear bivariate model is expressed as:

• 1 + 2 + LINEAR MODEL HYPOTHESIS (|): 1 + (|): ² (,)=0 ~ (1+2,²) + 0 (/).
• ()=0 ℎ, ()=²(,)=0, ()=0 ~ (0,²).

To derive the OLS estimators for parameters 1 and 2, we resolve:

• 1, 2 ∑₌₁(1−1−2)² ∂∑/∂1 −2∑(1−1−2)=0
• ∂∑/∂2 −2∑(1−1−2)
• ∑1=1+2∑∑1=1∑+2∑²1=̄+2̄∑1=̄∑̄+2̄+2∑²=+̄̄=(+)(+)

For the Total Sum of Squares (TSS) we have:

• TSS [ (1−̄) ]² ∑1−2̄1+̄²∑1²−2̄̄+̄²∑1²−̄²̄1−̄²=̄−̄²+ ∑(1−1)=∑₌₁_{}²

Non-linear Relationships

Exploring non-linear models involves analyzing a quadratic function. A percentage change in x results in a percentage change in E(y) as per the transformation 2bx/log. We then consider the exponential and logarithmic transformations to model the relationships, such as exp(log1) aT exp(bx) e−xlog. Logarithmic transformations are applied to further refine the model.

Multiple Regression and Goodness of Fit

Measures used to evaluate the goodness of fit in multiple regression include:

• R² = ESS/TSS = 1 - RSS/TSS
• Adjusted R² = 1 - [(RSS/n-k) / (TSS/n-1)] = 1 - n-1/n-k (1-R²) = 1 - k/n-k n-1/n-k R²
• AIC (Akaike Information Criterion): log e'e/n + 2k/n
• BIC (Schwarz Information Criterion): log e'e/n + k/n log(n)

OLS Sampling Distribution

Under classical assumptions, the Ordinary Least Squares (OLS) estimators have specific characteristics:

• Unbiased: E(b_j) = β_j
• Efficient
• Consistent: b_j → β_j

The variance of the estimator b is given by:

• var(b) = σ² (X'X)^-1

Bivariate Case

For bivariate cases, the matrix X'X and its inverse are:

• X'X = [ x² Σx Σx Σx² ]
• (X'X)^-1 = [ Σx² -Σx -Σx n ]

In terms of variance and covariance:

• var(b₁) = σ² Σx² / D = σ² (1/n + x̄² / Σx²)
• var(b₂) = σ² n / D = σ² / Σx²
• cov(b₁, b₂) = -σ² x̄ / Σx²