Time-Series Concepts/Linear Regression

Question 1

Stochastic Process

Answer 1

• a sequence of random variables indexed by time
• e.g. X_t = 3_t + ϵ_t

Question 2

Deterministic Process

Answer 2

• no randomness
• e.g. X_t = 3_t

Question 3

Weak Stationarity

Answer 3

• The mean is constant: E[X_t] = μ
• The variance is constant: Var(X_t) = σ²
• The covariance between X_t and X_t+h depends only on h, not t

Question 4

Autocorrelation

Answer 4

How related a time series value, X_t, is to its past values, X_t-k

Question 5

Partial Autocorrelation

Answer 5

Isolates the direct relationship between X_t and X_t-k, removing the influence of intermediate lags

Question 6

White noise process

Answer 6

a sequence of uncorrelated random variables with:
* mean E[ϵ_t] = 0
* constant variance Var(ϵ_t) = σ²
* no autocorrelation Cov(ϵ_t, ϵ_t-k) = 0 ∀ k
!= 0

Question 7

If residuals of a model are white noise ….

Answer 7

the model fits well

Question 8

What does OLS do?

Answer 8

• OLS estimates parameters by minimising the sum of squared residuals
• residual: Y_t - (β₀ + β₁X_t)

Question 9

Key properties of OLS

Answer 9

• Unbiasedness: On average, OLS estimates are correct if assumptions are met
• Efficiency: OLS has the smallest variance among unbiased estimators (Gauss-Markov Theorem)

Question 10

Gauss-Markov Assumptions

Answer 10

• Linearity
• Exogeneity
• Homoscedasticity
• No Autocorrelation of errors
• No perfect multicollinearity

Question 11

Linearity

Answer 11

The relationship between the dependent variable and the independent variable must be linear

Question 12

Exogeneity

Answer 12

The independent variables must not be correlated with the error term

Question 13

Homoscedasticity

Constant Variance of Errors

Answer 13

The variance of the error term must be constant across all levels of X_t
* Var(ϵ_t) = σ² (constant for all t)

Question 14

No autocorrelation of errors

Answer 14

The errors should not be correlated with each other
* Cov(ϵ_t, ϵ_t-k) = 0 for any lag k

Question 14

Why is homoscedasticity important?

Answer 14

If the variance of errors changes with X_t, OLS estimates will still be unbiased but standard errors will be incorrect leading to invalid statistical inferences

Question 14

No perfect multicollinearity

Answer 14

The independent variables must not be perfectly correlated i.e. it cannot be expressed as an exact linear combination of others

Question 14

Errors are normally distributed

Answer 14

The error term should follow a normal distribution
* ϵ_t ∼ N(0, σ²)

Question 15

Stricty exogenous regressors

Answer 15

X_t must not be influenced by past, present or future values of Y_t or ϵ_t
* If this holds, OLS produces unbiased and efficient estimates

Question 16

Weakly exogenous regressors

Answer 16

X_t can be influenced by past values of Y_t, but not future errors
* in this case, OLS still works but inference may require largers samples (asymptotic properties)

Question 17

T-test

Answer 17

Tests the significance of individual coefficients
* Null hypothesis H₀: β₁ = 0 (no relationship)
* Alternative Hypothesis H₁: β₁ != 0 (relationship)

Question 18

F-test

Answer 18

Tests if multiple coefficients are jointly zero
* Null Hypothesis H₀: All tested coefficients are zero
* Alternative Hypothesis H₁: At least one coefficient is not zero