EMF 2026 deel 1

Role of u

Captures all unobserved factors other than x that affect y

Cross-sectional data

Observations on one or more variables for different units at a single point in time

Time-series data

Observations on one or more variables collected at multiple points in time

Panel data

A time series for each cross-sectional unit in the dataset

Combines cross-sectional and time-series dimensions

3 types of random variables

Bernoulli: can take only value of 0 or 1

Discrete: takes only a finite number of values

Continuous: takes infinitely many values

Probability Distribution Function

Shows the probability of each possible score

Probabilities of each score can be defined as:

• p_j = P(X=j)

• 0 < p_j < 1 and total sum equals 1

Probability Density Function

Describes the relative likelihood/density of possible values of a random variable

For a continuous variable, probabilities correspond to areas under the PDF

Cumulative Distribution Function

Used to know the probability that a random variable is below or above a certain value

Gives the cumulative probability up to x:

• F(x) = P (X<x)

Independence

X and Y are independent if knowing the outcome of one does not change the probabilities of the possible outcomes of the other

P(X=x, Y=y) = P(X=x)P(Y=y)

Conditional distribution

Distribution of Y given X:

• f_Y|X(y|x) = f_X,Y(x,y) / f_X(x)

If X,Y are independent, then f_Y|X(y|x) = f_Y(y)

Measures of Central Tendency

E(X) = mu_X, the expected value, the mean of a random variable X

The median of a random variable X

Measures of Dispersion

Var(X) = sigma_X² , the variance, measure of how far a random variable X is from its mean

sd(X) = sigma_X , the standard deviation, positive square root of the variance

Measures of Association

Cov(X,Y) = sigma_XY

Corr(X,Y) = rho_XY

Variance

Var(X) = E[(X-mu)²]

Covariance

Cov(X,Y) = E[(X-mu_X)(Y-mu_Y)]

Measures linear dependence between two random variables

Positive, X and Y tend to move together

Negative, X and Y tend to move in opposite directions.

Covariance depends on the unit of measurement

Correlation

Corr(X,Y) = Cov(X,Y) / sd(X)sd(Y)

Measures how X,Y vary with each other regardless of their units of measurement

Standard normal distribution

Special case of the normal distribution where mu=0, and sigma=1

If X ~ N(mu, sigma²), then Z = X-mu / sigma ~ N(0, 1)

Chi-square distribution

X has a chi-square distribution with n degrees of freedom, denoted X ~ X_n²

If X = SUM Z_i² , where Z_i ~ N(0, 1), i = 1, …, n are independent of each other

t Distribution

T has a t distribution with n degrees of freedom, denoted T ~ t_n

If T = Z / sqrt(X/n), where Z ~ N(0,1), X ~ X_n² and they are independent of each other

F distribution

F has an f distribution with (k₁, k₂) degrees of freedom, denoted F ~ F_k1,k2

If F = (X₁/k₁) / (X₂/k₂), where X₁ ~ X_k1² , X₂ ~ X_k2² and they are independent of each other

Symmetric matrix

Square matrix that is symmetric along the leading diagonal

Diagonal matrix

Square matrix with non-zero elements only on the leading diagonal

Identity matrix

Diagonal matrix with 1 on the leading diagonal

Transpose

The transpose of a matrix A’ is obtained by switching the rows and columns of a matrix

Full rank matrix

Has a rank equal to its dimension

The rank of a matrix is the maximum number of linear independent rows or columns contained in the matrix

Singular matrix has a rank smaller than its dimension

Matrix algebra

Addition and subtraction requires the matrices to have the same number of rows and columns

Multiplication and division by a scalar s implies that all elements of the matrix are multiplied or divided by that scalar

Two matrices can be multiplied when the number of columns of the 1st matrix = number of rows of the 2nd matrix

Multiplication is performed along the rows of the 1st matrix and down the columns of the 2nd matrix

Inverse matrix

Inverse of a matrix A^-1 is the matrix that when pre- or post- multiplied by A results in the identity matrix

A^-1 = 1 / ad - bc ((d, -b) (-c, a))

Bivariate Linear Regression Model

y_i = a + Bx_i + u_i

a is the constant coefficient

B is the slope coefficient

x the explanatory variable

y the dependent variable

u contains all unobserved factors affecting y

Process of finding a and B is called Ordinary Least Squares

OLS

1. Take the vertical distances defined as u_i^{^} between each point in the graph and each candidate fitted line

2. Takes the squares of each distance and sums them: SUM u_i^{^2}

3. Finds the estimated coefficients that minimize the sum of the squared residuals

OLS estimators

a^{^} = y - B^{^} x

B^{^} = SUM(y_i-y^-)(x_i-x^-) / SUM(x_i-x^-)² = Cov(y,x)/Var(x)

Standardized coefficient

B^{^} sigma_x / sigma_y

it expresses the effect in std units

A one std increase in x changes the predicted y by B^{^} sigma_x / sigma_y std

Quadratic model

Using the derivative

The effect of a one-unit increase in x is different at starting values of x

dy/dx = B₁ + 2B₂x_i

Logarithmic transformation

1. Can reduce the influence of outliers

2. Can help to make a positively skewed distribution closer to a normal distribution

3. Allows for nonlinear relationship with a constant percentage increase

4. Permits interpretation through elasticities

Level-Log

dy = (B/100)% dx

log-level

%dy = 100B dx

log-log

%dy = B %dx

Assumption 1

Linear in parameters

The population model is linear in parameters: y_i = a + Bx_i + u_i

OLS cannot be used for a model that is nonlinear in parameters

such as y_i = 1/ (a + Bx_i) + u_i

Assumption 2

Random sample

We have a random sample from the population

Systematically selecting particular observations can make the OLS results unreliable

Assumption 3

Sample variation

There must be sample variation in the explanatory variable

x cannot have a variance of zero in the sample

Assumption 4

Zero conditional mean

The error u has an expected value of zero given any value of x

E(u | x) = E(u) = 0

Thus, the average value of the unobserved factors is the same for every value of x

Assumption 5

Homoscedasticity

The variance of u is constant and finite for any value of x

Var(u | x) = sigma² < inf

If Var(u_i | x_i) = sigma_i² we have heteroscedasticity

BLUE

Best Linear Unbiased Estimator

Under the CLRM assumptions, OLS estimators have the desirable properties of unbiasedness and efficiency

Best means minimum variance among linear unbiased estimators

P1 Unbiasedness

An estimator is unbiased when its expected value equals the true population value

E(a^{^}) = a, E(B^{^}) = B

Requires A1 to A4 to hold

The difference between E(B^{^}) and B is bias

Efficiency

OLS estimators have the minimum variance among the class of linear unbiased estimators

No other linear unbiased estimators have a smaller variance

OLS SE

SE measure the precision of the estimated coefficients

A smaller SE means a more precise estimate

RSS

Residual Sum of Squares

Variation in y not explained by the model

RSS = SUM (y_i - y_i^{^})²

TSS

Total Sum of Squares

Measures the total variation in y around its mean

TSS = SUM (y_i - y^-)²

ESS

Explained Sum of Squares

Measures the variation in y explained by the model

ESS = SUM (y_i^{^} - y^-)²

TSS decomposition

TSS = ESS + RSS

R-squared

Standard goodness of fit measure

R² = ESS/TSS = 1 - RSS/TSS

Measures the proportion of the sample variation in y explained by x

Assumption 6

Normality

The population error u is independent of the explanatory variables x and normally distributed

u ~ N(0, sigma²)

t-Test

Used to test a hypothesis about one population parameter

H₀: B = B₀

t_B = B^{^} - B₀ / se(B^{^})

Interpretation t statistic

Measures how many standard errors B^{^} is away from the hypothesized value B₀

A larger t_B provides stronger evidence against null hypothesis

Significance level

The probability of rejecting H₀ when H₀ is actually true

P(reject H₀ | H₀ true)

Type 1 error

p-Value

The smallest significance level at which H₀ would be rejected

For a two-sided t-test

p-val = P(T_B > t_B | H₀)

It is the probability, assuming H₀ is true, of obtaining a t statistic at least as extreme as the observed one

p-Value decision rule

Compare the p-value with the chosen significance level

p-value < significance value => reject H₀

Confidence Interval

Provides a range of likely values for the true parameter B

A 95% CI is defined by

B^{^} ± 1.96 se(B^{^})

Interpret regression results

Check the statistical significance of each variable

Only if statistically significant, interpret its economic significance

Interpret the overall goodness of fit of the model