Formulario modelli econometrici

It looks like you have a series of statistical and econometric formulas and tests, but they are not properly formatted or labeled. I'll try to interpret them and provide some context where possible. 1. **Hypothesis Testing for Autocorrelation (Durbin-Watson Test):** - The Durbin-Watson statistic is given by: \[ d = 2(1 - r_k) \approx 2 \left(1 - \frac{\sum_{t=2}^T (\hat{u}_t - \hat{u}_{t-1})^2}{\sum_{t=1}^T \hat{u}_t^2}\right) \] - Where \( r_k \) is the sample autocorrelation of the residuals. 2. **Global Significance Test (F-test):** - The F-statistic for testing the overall significance of a regression model: \[ F = \frac{(SSR_R - SSR_U)/k}{SSR_U / (n-k-1)} \] - Where \( SSR_R \) is the regression sum of squares, \( SSR_U \) is the total sum of squares, and \( k \) is the number of independent variables. 3. **Model Selection Criteria:** - Adjusted R-squared: \[ R^2_{adj} = 1 - \frac{SSR / (n-k-1)}{SST / (n-1)} \] - Akaike Information Criterion (AIC): \[ AIC = n \ln\left(\frac{SSR}{n}\right) + 2k \] - Bayesian Information Criterion (BIC): \[ BIC = n \ln\left(\frac{SSR}{n}\right) + k \ln(n) \] 4. **Breusch-Pagan Test for Heteroskedasticity:** - The test statistic: \[ LM = n R^2 \] - Where \( n \) is the number of observations and \( R^2 \) is the coefficient of determination from a regression of squared residuals on independent variables. 5. **Robust Variance Estimation:** - The robust variance estimator: \[ \hat{\sigma}^2 = \frac{1}{n-2} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 \] 6. **Okun's Law:** - The relationship between unemployment rate and GDP growth: \[ \Delta u_t = -\beta (\Delta gdp_t) + \epsilon_t \] - Where \( \Delta u_t \) is the change in the unemployment rate, \( \Delta gdp_t \) is the change in GDP, and \( \epsilon_t \) is an error term. 7. **Coefficient of Autocorrelation:** - The coefficient of autocorrelation: \[ r_k = \frac{\sum_{t=2}^T (u_t - \bar{u})(u_{t-1} - \bar{u})}{\sqrt{\sum_{t=1}^T (u_t - \bar{u})^2 \sum_{t=1}^T (u_{t-1} - \bar{u})^2}} \] 8. **Test for \( r_