Simple Linear Regression Model; Estimation and Inference

Question 1

Xi

Answer 1

The independent variable, explanatory variable, predictor variable or regressor , exogenous.

Question 2

Yi

Answer 2

The dependent variable,

explained variable, response variable or regressand, endogenous.

Question 3

What is a statistical relationship?

Answer 3

Statistical relationships among variables deal with random or stochastic variables, that is, variables that have probability distributions.

Question 4

What is a deterministic relationship?

Answer 4

Functional or deterministic dependency, we also deal with variables, but these variables are not random or stochastic.

Question 5

Regression versus causation

Answer 5

Although regression analysis deals with the dependence of one variable on other variables, it does not necessarily imply causation. A statistical relationship, however strong and however suggestive, can never establish causal connection.

Question 6

Regression versus correlation

Answer 6

Correlation Analysis: the primary objective is to measure the strength or degree of linear association between two variables (both are assumed to be random)
Regression Analysis: we try to estimate or predict the average value of one variable (dependent, and assumed to be stochastic) on the basis of the fixed values of other variables (independent, and non-stochastic)

Question 7

Population Regression Function

Answer 7

It states merely that the expected value of the distribution of Y given Xi is functionally related to Xi.
If we join conditional mean values, we obtain what is known as the population regression line (PRL), or more generally, the population regression curve. More simply, it is the regression of Y on X.
Geometrically, then, a population regression curve is simply the locus of the conditional means of the dependent variable for the fixed values of the explanatory variable(s).
conditional expectation function (CEF) or population regression function (PRF) or population regression (PR) for short.
• In simple terms, it tells how the mean or average response of Y varies with X.

Question 8

Linearity in variables

Answer 8

The first and perhaps more “natural” meaning of linearity is that the conditional expectation of Y is a linear function of Xi,

Question 9

Linearity in parameters

Answer 9

The second interpretation of linearity is that the conditional expectation of
Y, E(Y | Xi), is a linear function of the parameters, the β’s; it may or may not be linear in the variable X.
Of the two interpretations of linearity, linearity in the parameters is relevant for the development of the regression theory.

Question 10

Properties of Expected Values

Answer 10

•The expected value of a constant is the constant itself. Thus, if b is a constant,
E(b) = b.
• If a and b are constants,
E(aX + b) = aE(X) + b
• If X and Y are independent random variables, then E(XY) = E(X)E(Y)
• If X is a random variable with PDF f(x) and if g(X) is any function of X, then
If discrete?
If continuous?

Question 11

Variance

Answer 11

Let X be a random variable and let
E(X) = μ. The distribution, or spread, of the X values around the expected value can be measured by the variance, which is defined as
Var (X) = σ^2 = E(X − μ)^2
E(X^2) - μ^2
E(X^2) - [E(X)]^2

Question 12

Deterministic component of the function

Answer 12

The part of the variation in Y explained by the changes in X

Question 13

Random component of the function

Answer 13

The variation in Y that is explained by other factors other than X.

Question 14

Point Estimation

Answer 14

The end result is a point estimate, which is a single number that serves as an intelligent guess of the value of the parameter.

Question 15

Interval estimation

Answer 15

The estimate consists of a range of values thought to contain an unknown parameter.

Question 16

Ui

Answer 16

The disturbance term u is a surrogate for all those variables that are omitted from the model but that collectively affect Y.

Question 17

Why not introduce these variables into the model explicitly?

Answer 17

Omission of variables from the function: in economic reality each variable is influenced by a very large number of factors.

Intrinsic randomness in human behavior: Human reactions are to a certain extent unpredictable and may cause deviations from the normal behavior pattern depicted by the line.

Wrong functional form: We may have linearised a possibly nonlinear relationship. Or we may have left out of the model some equations.

Errors in aggregation; we often use aggregate data, in which we add magnitudes referring to individuals whose behavior is dissimilar.

Errors of Measurement: these are inevitable due to the methods of collecting and processing statistical information.