Chapter 1-4

Question 1

What is econometrics?

Answer 1

Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.

Econometrics is the mathematical formalization of economic theory for its validation through statistical procedures.

Question 2

What is econometrics looking for?

Answer 2

3 uses:
-> Describing the economic reality;
-> Test hypotheses about economic theory;
-> Predict future economic activity.

Question 3

Temporary Data Series

Answer 3

-> Dependency;
-> Data frequency (days, months, years);
-> Homogeneity.

Question 4

Cross-sectional Data Series

Answer 4

-> The order of the data does not matter
-> We leave aside any small difference in the taking of the data (t)

Question 5

Variables

Answer 5

-> They are the (mathematical) entities representative of the economic phenomena.
-> Quantitative and qualitative variables/dichotomic/categorical.
-> In econometrics, when variables are analyzed together, we are fundamentally interested in the relationship between them.
-> In our regression analysis we are interested in the statistical dependence and not the functional or deterministic dependence. Therefore, we will deal, fundamentally, with random or stochastic variables.

Question 6

Mixed or Combined Data Series

Answer 6

-> Longitudinal or Panel data

Question 7

If the relationship between X and Y is stochastic or random:

Answer 7

A change in the value of the X variable causes a change in the probability distribution of the Y variable.

Question 8

ECONOMETRIC MODEL

Answer 8

It is an economic model that contains the necessary specifications for its empirical application.
Y = β1 + β2X2 + β3X3 + … + βnXn + ɛ

Question 9

If the relationship between X and Y is deterministic

Answer 9

A change in the value of the X variable causes a change in the value of the Y variable.

Question 10

HYPOTHESES OR BASIC ASSUMPTIONS OF THE LINEAR REGRESSION MODEL.
-> Assumptions about the random disturbance

Answer 10

-> 1. EXOGENEITY (or weak assumption of “conditional mean equal to zero”): Every random disturbance has a zero conditional mean. THE MEAN VALUE OF THE DISTRIBUTION OF DISTURBANCES εi GIVEN xi, IS EQUAL TO ZERO;
-> 2. ASSUMPTION OF NO SERIAL CORRELATION OR NO AUTOCORRELATION*: There is no correlation between the random disturbances. The disturbances are independent of each other. It is equivalent to saying that their covariances are zero.
-> 3. ASSUMPTION OF HOMOSCEDASTICITY: All distributions of random disturbances have equal variance, which we will call σ2.
-> 4. The random disturbances are distributed Normally

Question 11

HYPOTHESES OR BASIC ASSUMPTIONS OF THE LINEAR REGRESSION MODEL.
-> Assumptions and conditions on the model coefficients or parameters:

Answer 11

-> 1. CONDITION OF LINEARITY: THE REGRESSION MODEL IS LINEAR IN THE PARAMETERS.

-> 2. CONDITION OF IDENTIFICATION: The sample size (n) must be equal to or greater than the number of coefficients or parameters to be estimated (k).

-> 3. STRUCTURAL STABILITY HYPOTHESIS OF THE PARAMETERS: The structural parameters, β, are constant for all sample units and for all possible samples.

Question 12

HYPOTHESES OR BASIC ASSUMPTIONS OF THE LINEAR REGRESSION MODEL.
-> Assumptions about the model variable(s):

Answer 12

-> 1. If the observations are drawn by simple random sampling from a single large population, (Xi , Yi), for i = 1, 2,…, n, they are independent and identically distributed (I.I.D)
-> 2. High outliers are unlikely

Question 13

HYPOTHESES OR BASIC ASSUMPTIONS OF THE LINEAR REGRESSION MODEL.
-> Assumptions about the Xi explanatory variable(s):
(Only for multivariate models)

Answer 13

NON-MULTICOLLINEARITY: There is no exact linear relationship between the vectors formed by the sample observations of the explanatory variables. In other words, the data matrix X columns are linearly independent.

Question 14

HYPOTHESES OR BASIC ASSUMPTIONS OF THE LINEAR REGRESSION MODEL.
-> Assumptions about the model:

Answer 14

THE REGRESSION MODEL IS CORRECTLY SPECIFIED (THERE ARE NO SPECIFICATION BIASES OR ERRORS).
After accepting these assumptions, in the multivariate case, the objective of the regression analysis is, like in the cases of 2 variables, to estimate the mean of Y conditional on the fixed values of the Xk.

Question 15

β1 interpretation

Answer 15

-> 1. Univariate: is the mean value of Yi when the value of explanatory variable Xi is zero;
-> 2. Multivariate: is the mean value of Yi when the value of ALL explanatory variables is zero;

Question 16

ESTIMATION OF THE MODEL
OLS vs. MLE

Answer 16

-> In a statistical sense, the OLS method is more robust ->for its application, it does not require knowing the probability distribution of the random variable modelling the population. To use the MLE method, we have to know or assume this distribution (for ɛ).
-> The MLE method has good asymptotic properties -> large sample estimation method.

Question 17

(β2) … (βk) interpretation

Answer 17

-> 1. Univariate: is the average variation of the mean value of Yi in the presence of unit variations of explanatory variable Xi.
-> 2. Multivariate: the average variation of the mean value of the endogenous variable in the presence of unit variations of (X2)… (Xk), with all other explanatory variables remaining constant (“ceteris paribus” clause).

Question 18

Why do we use the OLS method?

Answer 18

-> It is easier to apply;
-> The OLS and MLE estimators of the model coefficients (βi) are identical;
-> In large samples, the OLS and MLE estimators of σ2 (variance of the disturbances) do not differ considerably.

Question 19

GAUSS-MARKOV THEOREM

Answer 19

Given the assumptions of the classical linear regression model, the least squares estimators are linear, unbiased and minimum variance estimators (BLUE, Best Linear, Unbiased, Estimator).

Question 20

THE LEAST SQUARES METHOD

Answer 20

The sum of the squares of the differences between the observed (real values) and the calculated quantities (estimated values) must be minimum. (Gauss)

Question 21

β1 estimated, interpretation

Answer 21

-> 1. Univariate: It is the estimated mean value of Yi when the value of explanatory variable Xi is zero.
-> 2. Multivariate: It is the estimated mean value of Yi when all explanatory variables are equal to zero.

Question 22

(β2) estimated … (βk) estimated, interpretation

Answer 22

-> 1. Univariate: Estimated average variation in the estimated mean value of Yi in the presence of unit changes in explanatory variable Xi;
-> 2. Multivariate: Estimated average variation of the estimated mean value of the endogenous variable in the presence of unit variations of (X2)…(Xk), with all other explanatory variables remaining constant (“ceteris paribus” clause).

Question 23

PROPERTIES of the estimated regression line

Answer 23

-> It passes through the sample means of X and Y.
-> The sum of the estimated values is equal to the sum of the observed values, therefore, the mean value of Y (observed) = with the mean value of Y estimated.
-> The mean value of the residuals is 0.
-> The estimated values of Y (also the residuals) are invariant to (non-singular) linear transformations of X; in other words, the estimated values of Y do not change when X is replaced by a linear function g(Xi) = a + bXi, where a and b are known constants.
-> The residuals ei are not correlated with the estimated values of Yi.
-> The residuals ei are not correlated with Xi.

Question 24

MAXIMUM LIKELIHOOD

Answer 24

The aim is to maximise the likelihood function of the sample.