Time Series exam study guide

Stationary Process

A time series process where the probability distributions are stable over time.

Covariance Stationary Process

A stochastic process where mean, variance, and covariance are constant over time.

Weak Dependence

A stationary time series process is weakly dependent if random variables xt and xt+h are almost independent as h increases.

MA(1) Model

A moving average process of order one defined as xt = et + α1et-1.

AR(1) Model

An autoregressive process of order one defined as yt = ρ1yt-1 + et.

Consistency

An estimator is consistent if it approaches the true parameter value as the sample size increases.

Unbiasedness

An estimator is unbiased if its expected value equals the true parameter value.

Random Walk

A time series process where the next period's value is equal to the current value plus an independent error term.

Unit Root Process

A highly persistent time series where the current value equals the last period's value plus a weakly dependent disturbance.

I(1) Process

A time series process that is integrated of order one, meaning its first difference is weakly dependent.

Cointegration

Exists if a linear combination of two I(1) variables is I(0), indicating they do not drift apart.

Engle-Granger Test

A test applied to the residuals to check for cointegration between two I(1) variables.

Infinite Distributed Lag Model

A model that considers the effects of current and all past values of an explanatory variable.

Granger Causality

A limited notion of causality where past values of one series help predict future values of another series.

ARCH Model

A model for dynamic heteroskedasticity where the variance of the error term depends on past squared errors.

Dickey-Fuller Test

A test for the unit root null hypothesis in an AR(1) model.

Heteroskedasticity

The condition where the variance of the errors varies across observations.

Serial Correlation

The situation where errors in a regression model are correlated across observations.

HAC Standard Errors

Standard errors that are robust to the presence of both heteroskedasticity and serial correlation.

Quasi-Differencing

A method used to estimate the correlation of errors and to transform variables to eliminate serial correlation.

Prais-Winsten Procedure

A preferred method for correcting serial correlation in regression analysis.

First-Differencing

A method used to transform an integrated time series process into a weakly dependent process.

Asymptotic Normality

A property of estimators that allows them to follow a normal distribution as the sample size approaches infinity.

Best Linear Unbiased Estimator (BLUE)

The estimator that is linear, unbiased, and has the minimum variance among all linear estimators.

Risk of Spurious Regression

Running a regression that might indicate a relationship when none exists, due to misspecifications or integrated processes.

White Noise Process

A process such that: E(et) = E(es) = 0 for all s not equal to t, gamma_k = gamma_0 for k = 0, and gamma_k = 0 for k not equal to 0.

White Noise Process

A white noise process is a sequence of random variables where each variable has a mean of zero, constant variance, and no autocorrelations (covariances) at any lags other than lag zero. Mathematically, E(et) = E(es) = 0 for all s not equal to t, with gamma_k (autocovariance) equal to gamma_0 for k = 0, and gamma_k = 0 for all other k.

Gaussian (Normal) White Noise

This refers to a type of white noise where the random variables et are normally distributed with a mean of 0. In this case, the autocovariance is zero for all lags except lag zero, meaning that gamma_k = 0 for k not equal to 0.

Strictly Stationary

A process is considered strictly stationary if the joint distribution of any subsequence of observations remains unchanged when compared to the joint distribution of the full sequence of observations. This means that statistical properties do not depend on time.

Microstructure Noise

This term refers to very high-frequency noise in financial time series, typically characterized by fluctuations occurring at intervals of 6 minutes or less.

Assumption 1 of a Time Series

This assumption states that a time series is infinitely long; it has been continuously observed from the past and will continue to be observed into the future without any regard for initial or end conditions.

Stochastic Process

A stochastic process is a collection of random variables indexed according to time. It models systems or phenomena that exhibit inherent randomness.

Strongly (Strictly) Stationary

A process is said to be strongly or strictly stationary if it is nth order stationary for all values of n from 1 to infinity. Essentially, this means that all moments of the distribution are time invariant.

Strict Stationarity Implies Weak Stationarity

The concept that if a process is strictly stationary, it also implies weak stationarity, which means that only the first and second moments of the distribution need to be time invariant.

Best Linear Unbiased Estimator (BLUE)

The BLUE refers to an estimator that is linear, unbiased, and has the minimum variance among all linear unbiased estimators, ensuring the most accurate estimates with the least error.

Stationary Process

A stationary process in time series analysis refers to a stochastic process where the statistical properties such as mean, variance, and autocovariance do not change over time. This means that the probability distributions of the variables involved remain consistent and are invariant to shifts in time, making such processes essential for reliable statistical inference and modeling.