Appunti quantitative methods in financial economics

Quantitative methods in financial economics

Prof. Roberto Iannacone
Academic Year: 2020-2021
Lorenzo Guglielmetti
Course started: 28/09/2020

Data types

• Cross-section: Observations are relative to different individuals.
• Time series: Observations referring at a regular time interval; the order of observations is relevant.
• Panel data: Group of cross-sectional units who are observed over time (N units T periods).

Simple linear regression

The probability density function describes the probability of observing different values of Y at a certain level of X.

SLR functions

E(Y|X) = µ = ß₁ + ß₂X | X → ß₁ intercepts, it represents the value of Y when X is equal to 0. ß₂ → slope, it represents how much Y varies for a given change in X.

Hypothesis of the econometric model

• E(Y|X) = µ = ß₁ + ß₂X
• var(Y|X) = Homoskedasticity
• cov (Y_i, Y_j) = 0, the values of y are all statistically independent
• The variable X isn’t random and must take at least 2 distinct values
• Y ~ N(ß₁ + ß₂X; σ²)

Hypothesis simple regression

• Y = ß₁ + ß₂X_i + e_i
• E(e|X) = 0 E(Y|X) = ß₁ + ß₂X
• var(e) = var(Y)
• cov(e_i, e_j) = cov(y_i, y_j) = 0
• The variable X isn’t random and must take at least 2 distinct values
• e ~ N(0; σ²)

Ordinary least squares method

Method that minimizes the sum of the squares of the distances of the points from the line (e_i); the distances are squared to prevent large positive distances from being canceled by large negative distances. b₁ and b₂ are the least squares estimates of ß₁ and ß₂, and e^ are the residuals of the least squares.

b₂ = ∑[(X_i – X̄)(Y_i – Ȳ)] / ∑(X_i – X̄)²
b₁ = Ȳ – b₂X̄

Unbiasedness property

If we took the averages of the estimates from many samples, these averages would approach the values of the real parameters b₁ and b₂.

Gauss-Markov Theorem

Under the SR1-SR5 hypothesis, the estimators b₁ and b₂ have the smallest variance of all linear and correct estimators of ß₁ and ß₂ and they are the best unbiased linear estimators (BLUE) of ß₁ and ß₂.

Prediction

Under the SR1-SR6 hypothesis, we want to predict the value of Y₀ for a given value of X₀. The least squares predictor of Y₀ is:

Ŷ₀ = b₁ + b₂X₀

The least squares estimator b₁ and b₂ follow that the least squares predictor must also be random with forecast error (f):

f = Y₀ - Ŷ₀ = (ß₁ + ß₂X₀ + e₀) - (b₁ + b₂X₀)

E(f) = ß₁ + ß₂X₀ + E(e₀) - [E(b₁) + E(b₂)X₀] = ß₁ + ß₂X₀ + 0 - ß₁ - ß₂X₀ = 0

Ŷ₀ is the best linear unbiased predictor (BLUP) of Y₀ if assuming SR1-SR5 hold. This result is reasonable given that the least squares estimators b₁ and b₂ are BLUE.

R²

∑(Y_i – Ȳ)² = ∑(Ŷ_i – Ȳ)² + ∑e_i²

SST = SSR + SSE
Ȳ = Y barretta

• SST: Measures the total change in Y around the sample mean.
• SSR: Measures the part of the total variation in Y around the sample mean explained by the model.
• SSE: Part of the variation in Y not explained by the model.

R² = SSR/SST = 1 - SSE/SST

• If R² = 1, all sample points are exactly on the least squares line, so SSE = 0, and the model first data "perfectly".
• If R² = 0, the sample data for Y and X are not related and show no linear association, then the estimate of the regression line is horizontal and identical to Ȳ, so SSR = 0.

Example: R² = 0.385 means that 38.5% of the change in Y is explained by our regression model, which uses only one X as an explanatory variable.

Non-linearity

Log-log model: ln(Y_i) = ln(ß₁) + ß₂ ln(X_i) + e_i

Y is a non-linear function of the X_i, but the function is linear in logarithms, so that ordinary least squares may be applied.

Log-linear model: ln(Y_i) = ß₁ + ß₂ X_i + e_i

• If ß₂ > 0, the function increases at an increasing rate; its slope is larger for larger values of Y.
• If ß₂ < 0, the function decreases but at a decreasing rate.

Prediction

If we consider w = e^σ_Y2/2, then Y = ln(w) ~ N(µ, σ²) and w is said to have a log-normal distribution.

E(w) = e^µ+σ2/2

Ŷ_c = [E(Y)]̂_c = Ŷ_ne^σ2/2 where Ŷ_c is corrected predictor and Ŷ_n is natural predictor from the model: Ŷ_n = exp(b₁ + b₂X)

Name function slope and elasticity

Multivariate regression

Y = ß₀ + ß₁X_i1 + ß₂X_i2 + ... + ß_kX_ik + e_i

This model can eliminate the distortion from omitted variables through inclusion as additional regressors and then estimate the effect of a regressor while maintaining the other variables constant.

ß_k measures the effect of a change in the variable X_k on the expected value of Y, keeping all the other variables constant. ß₀ represents the intercept term.

Hypothesis multivariate regression

• Y = ß₀ + ß₁X_i1 + ß₂X_i2 + ... + ß_kX_ik + e_i
• E(e_i) = 0 E(Y_i) = ß₀ + ß₁X_i1 + ß₂X_i2 + ... + ß_kX_ik
• var(Y_i) = var(e_i) = σ²
• cov(e_i, e_j) = cov(Y_i, Y_j) = 0
• The X values are not stochastic and they are not exact linear functions of the other explanatory variables.
• Y ~ N(ß₀ + ß₁X_i1 + ß₂X_i2 + ... + ß_kX_ik, σ²) e ~ N(0; σ²)

Gauss-Markov Theorem

If the MR1-MR5 hypothesis is valid, then the least square estimators are the BLUE of the parameters.

Variable interaction

Y = ß₀ + ß₁X_i1 + ß₂X_i2 + ß₃X_i1X_i2

P-values

Let p be the p-value and t the calculated value of the t-statistic.

• H₀: ß_j = c → |t|
• H₁: ß_j ≠ c → p = 2(1 - pt(abs(t), df)) = sum probabilities to the right of |t| and to the left of |t|
• H₁: ß_j > c → p = 1 - pt(t, df) = probability to the right of t
• H₁: ß_j < c → p = pt(t, df) = probability to the left of t

Joint hypothesis

Imposes 2 or more constraints on regression coefficients considering a multiple conjecture such as:

• H₀: ß₀ = ß₂ = ß₄ = ß₅ = 0 aren’t true
• H₁: One or more of the q constraints under H₀

The test to verify a joint null hypothesis uses the test statistic F calculated using a simple formula based on the sum of the squares of the residuals of 2 regressions:

1. In the first regression, called restricted model (R), the H₀ assumes that the coefficients are zero, thus excluding the relevant regressors from the regression.
2. In the second regression, called unrestricted model (U), the H₁ is valid.

F = [(SSE_R - SSE_U)/q] / [SSE_U/(n - k - 1)] ~ F_q,n-k-1

Rejection region: F > F_c

Multicollinearity

No variable is redundant and in its absence the least squares procedure ceases to be valid.

Perfect multicollinearity occurs if one of the regressors is an exact linear combination of the other regressors.

Nonexact multicollinearity arises when one of the regressors is highly correlated, though not perfectly, with the other regressors. It implies that one or more regression coefficients could be inaccurately estimated.

The effects of these inaccurate results are:

• When the standard errors of the estimator are large, it’s likely that the usual t-test will lead to the conclusion that the parameter estimates are not significantly different from 0 even if the values of R² are high.
• Estimators may be very sensitive to the addition or omission of some observations or to the omission of an apparently insignificant variable.

VIF = 1 / (1 - R_k²) where R_k² is the R² from regressing the variable X_k on all the remaining regressors.

If VIF > 10, a regressor produces collinearity.

Indicator variables

Classic approach to time series analysis: A time series Y can be modeled as the sum or product...