ECONOMETRICS TEMA 2

(Linearity) Any estimator that is a weighted average of $y_{i}$ s is called a linear estimator. TRUE. FALSE.

(Unbiasedness) Unbiasedness states that an estimate from any one individual sample is guaranteed to be very close to the true parameter value. TRUE. FALSE.

(Unbiasedness) We can say that a specific numerical estimate is unbiased, but we cannot say that the least squares estimation procedure is unbiased. TRUE. FALSE.

(Variance of an estimator) The variance of the estimator $\hat{\beta}_{2}$ is the average of the squared distances between the possible values of the estimator and its mean given by $\beta_{2}$. TRUE. FALSE.

(Variance of an estimator) The variance of an estimator measures the precision of the estimator and tells us how much the estimates can vary from sample to sample. TRUE. FALSE.

(Variances and covariance of OLS) To find the mathematical expressions for the variances and covariance of $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$, the assumption concerning the normal distribution of the errors is strictly required. TRUE. FALSE.

(Variances and covariance of OLS) The larger the variance term $\sigma^{2}$, the smaller the uncertainty there is in the statistical model, and the smaller the variances of the least squares estimators. TRUE. FALSE.

(Variances and covariance of OLS) The larger the sum of squares, $\sum_{i=1}^{N}(x_{i}-\overline{x})^{2}$, the smaller the variances of the least squares estimators and the more precisely we can estimate the unknown parameters. TRUE. FALSE.

(Variances and covariance of OLS) High variation in the explanatory variable $x$ leads to a lower precision of estimation for the regression parameters. TRUE. FALSE.

(Variances and covariance of OLS) The larger the sample size $N$, the smaller the variances and covariance of the least squares estimators. TRUE. FALSE.

(Variances and covariance of OLS) The larger the term $\sum_{i=1}^{N}x_{i}^{2}$, the smaller the variance of the least squares estimator $\hat{\beta}_{1}$. TRUE. FALSE.

(The Gauss-Markov Theorem) Under the assumptions of the linear regression model (where normality is not needed), the estimators $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$ are the Best Linear Unbiased Estimators (BLUE). TRUE. FALSE.

(The Gauss-Markov Theorem) The Gauss-Markov Theorem states that $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$ are the best of all possible estimators, including all nonlinear and biased estimators. TRUE. FALSE.

(The Gauss-Markov Theorem) When comparing two linear and unbiased estimators, we always want to use the one with the larger variance because it gives a higher probability of being close to the true parameter. TRUE. FALSE.

(The Gauss-Markov Theorem) If any of the underlying regression assumptions are violated (apart from the normality assumption), then $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$ are no longer the best linear unbiased estimators. TRUE. FALSE.

(Probability distributions) If we make the normality assumption about the error term, then the least squares estimators are normally distributed. TRUE. FALSE.

(A Central Limit Theorem) If the errors are not normally distributed, but all remaining assumptions hold, we can often approximate the normal distribution of the estimators if the sample size $N$ is sufficiently large. TRUE. FALSE.

(Estimating the variance of the error term) Because the random errors $e_{i}$ are unobservable, we can estimate their variance using least squares residuals ($\hat{e}_{i}$) instead. TRUE. FALSE.

(Estimating the variance of the error term) If the true random errors $e_{i}$ were observable, we could estimate the variance using the expression $\hat{\sigma}^{2}=\frac{\sum e_{i}^{2}}{N}$. TRUE. FALSE.

(Estimating the variance of the error term) The formula for the unbiased estimator of $\sigma^{2}$ using residuals is $\hat{\sigma}^{2}=\frac{\sum\hat{e}_{i}^{2}}{N-2}$. TRUE. FALSE.

(Estimating the variance of the error term) The number 2 subtracted in the denominator of the unbiased error variance formula represents the number of explanatory variables $x$ in the dataset. TRUE. FALSE.

(Standard errors) Standard errors of $\hat{\beta}_{1}$ and $\hat{\beta}_{2}$ are defined as the square roots of their estimated variances. TRUE. FALSE.

(Variance matrix) In a variance-covariance matrix of the least squares estimators, the estimated variances are arrayed in the "off-diagonal" positions, while the covariances are on the main diagonal. TRUE. FALSE.