Time Series Analysis: Introduction to Stationarity

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Questions and Answers

In the context of general regression, what condition must be met regarding the design matrix A to ensure that $A'A$ is invertible?

The matrix A must be a square matrix.

The rank of A must be equal to T, and p must be greater than or equal to T.

The number of rows (T) must be less than the number of columns (p).

The rank of A must be equal to p, and T must be greater than or equal to p. (correct)

Polynomial regression is a more general form of regression than general regression.

False (B)

In the context of polynomial regression, if `p = 2`, what specific type of regression does this represent?

linear regression

In the least squares sense, general regression chooses the coefficients that minimize the sum of the ______ of the differences between observed and predicted values.

squares Signup and view all the answers

What does the rank of a matrix signify?

The maximal number of linearly independent columns or rows in the matrix. (A) Signup and view all the answers

The design matrix (A) in general regression is a square matrix.

False (B) Signup and view all the answers

Match the regression type with its corresponding formula.

Polynomial Regression = $mt = β0 + β1 t + β2 t^2 +...+ βp−1 t^{p−1}$ General Regression = $mt = β1 f1(t) + β2 f2(t) +...+ βp fp(t)$ Signup and view all the answers

In linear regression, what does the formula $\alpha = \frac{\sum_{t=1}^{T} (t - t_T)x_t}{\sum_{t=1}^{T} (t - t_T)^2}$ represent?

The optimal slope of the regression line (C) Signup and view all the answers

Given a general regression model predicting a time series $x_t$, which of the following is being minimized in a least squares sense?

The sum of squared errors between observed values $x_t$ and the predicted trend $m_t$. (B) Signup and view all the answers

In the context of linear regression, the term 'residuals' refers to the difference between the observed values and the trend line.

True (A) Signup and view all the answers

In linear regression, what is the interpretation of ( \beta ) in the equation ( \beta = x_T - \alpha t_T )?

The y-intercept Signup and view all the answers

In linear regression, the optimal values for ( \alpha ) and ( \beta ) are found by minimizing the sum of squared ________.

errors Signup and view all the answers

Match the terms with their corresponding descriptions in the context of regression analysis:

$\alpha$ = Slope of the regression line $\beta$ = Y-intercept of the regression line Residuals = Difference between observed and predicted values $t_T$ = Time index Signup and view all the answers

What does a high value of residuals in a regression model indicate?

A poor fit of the model to the data (A) Signup and view all the answers

Polynomial regression is a special case of linear regression.

False (B) Signup and view all the answers

What is the primary goal of regression analysis?

To model the trend between variables Signup and view all the answers

Which of the following conditions is necessary and sufficient for a random variable $X$ to be normally distributed with mean $\mu$ and variance $\sigma^2$?

The characteristic function $\varphi_X$ of $X$ is given by $\varphi_X(u) = e^{iu\mu - \sigma^2 u^2/2}$ for all $u \in \mathbb{R}$. (B) Signup and view all the answers

If a random vector $X = (X_1, ..., X_d)'$ is Gaussian, then all its components must be uncorrelated.

False (B) Signup and view all the answers

What two parameters uniquely determine the distribution of a Gaussian random vector $X = (X_1, ..., X_d)'$?

mean and covariance matrix Signup and view all the answers

A symmetric positive semidefinite matrix $\Sigma$ satisfies $\Sigma' = \Sigma$ and $x'\Sigma x \geq 0$ for all $x \in \mathbb{R}^d$. Given $\mu \in \mathbb{R}^d$ and a symmetric positive semidefinite matrix $\Sigma \in \mathbb{R}^{d \times d}$, there ________ a random vector $X$ with the distribution $N(\mu, \Sigma)$.

exists Signup and view all the answers

Match the following properties of a random vector $X \sim N(\mu, \Sigma)$ with their corresponding descriptions:

$\mu$ = Expectation of X $\Sigma$ = Covariance matrix of X $f_X(x)$ (when $\Sigma$ is invertible) = Probability density function of X $\varphi_X(u)$ = Characteristic function of X Signup and view all the answers

If a fitted model class proves inadequate, what is the recommended course of action?

Begin again with a different model class. (A) Signup and view all the answers

Suppose $X \sim N(\mu, \Sigma)$ where $\Sigma$ is invertible. What is the exponent in the probability density function $f_X(x)$?

$-\frac{1}{2}(x - \mu)' \Sigma^{-1} (x - \mu)$ (D) Signup and view all the answers

A good model cannot be used for forecasting future values.

False (B) Signup and view all the answers

Besides forecasting, what is another application of a well-fitted model mentioned in the text?

hypothesis testing Signup and view all the answers

If $(X_t){t \in \mathbb{Z}}$ is a Gaussian time series, then for any $n \in \mathbb{N}$ and $t_1, ..., t_n \in \mathbb{Z}$, the vector $(X{t_1}, ..., X_{t_n})'$ is normally distributed.

True (A) Signup and view all the answers

A time series $(X_t)_{t \in \mathbb{Z}}$ is said to be a Gaussian time series if all of its ________-dimensional distributions are normally distributed.

finite Signup and view all the answers

Before modeling residuals to a stationary model, one usually estimates or eliminates ______ and seasonality in data first.

trend Signup and view all the answers

What property is typically imposed on the residuals when fitting a model to data?

Stationarity (C) Signup and view all the answers

What are the two types of stationarity?

Strict and weak (A) Signup and view all the answers

Match the action with the scenario of time series analysis:

Model is not good = Fit another model class Model is good = Test certain hypotheses Interested in predictions = Forecast future values Signup and view all the answers

Weak and second-order stationarity are different concepts.

False (B) Signup and view all the answers

For an MA(q) process, which of the following is the correct expression for the autocovariance function $γ_X(h)$ when $h ≥ 0$, given $E[X_t^2] < ∞$, $E[X_t] = 0$, and $θ_0 = 1$?

$γ_X(h) = \sum_{j=0}^{q-|h|} θ_j θ_{|h|+j} σ^2$ (D) Signup and view all the answers

A weakly stationary time series is always strictly stationary.

False (B) Signup and view all the answers

What are the parameters that define a normal distribution?

Mean and variance (C) Signup and view all the answers

If a random variable X is normally distributed with a variance of 0, then X is equal to its ______ almost surely.

mean Signup and view all the answers

What is the standard notation for a standard normal distribution?

N(0, 1) Signup and view all the answers

What is the purpose of the characteristic function $φ_X(u)$ of a random vector X?

To uniquely describe the distribution of X (C) Signup and view all the answers

The Euclidean product in $R^d$ is used in the definition of the characteristic function of a random vector.

True (A) Signup and view all the answers

Match the following terms with their descriptions:

Dirac measure = A probability measure concentrated at a single point. Normal distribution = A probability distribution characterized by a mean and variance. Characteristic function = A function that uniquely describes the distribution of a random vector. Weakly stationary = A time series with constant mean and autocovariance. Signup and view all the answers

When the absolute value of φ1 is less than 1, what mathematical expression defines the autocovariance function (ACVF) γX(h) for h ∈ N0?

$γX(h) = \frac{σ^2}{1 - |φ1|^2} φ1^h$ (C) Signup and view all the answers

If |φ1| > 1, how does the autocovariance function γX(h) differ from the case where |φ1| < 1 for h ∈ N0?

The denominator changes sign, becoming |φ1|^2 - 1. (A) Signup and view all the answers

The sequence $(φ1^j)_{j∈N0}$ represents a linear filter if $|φ1| < 1$.

True (A) Signup and view all the answers

Write the formula for calculating Xt, the weakly stationary solution, using the linear filter involving φ1 and Zt−k.

Xt := ∑(from k=0 to ∞) φ1^k * Zt−k Signup and view all the answers

For a weakly stationary solution to exist, the sum of the absolute values of the filter coefficients must be ______.

finite Signup and view all the answers

Which condition must be met to ensure that the sequence $(φ1^j)_{j∈N0}$ is a linear filter?

$|φ1| < 1$ (C) Signup and view all the answers

Match the following conditions of |φ1| with the corresponding formula for the autocovariance function (ACVF) γX(h) for h ∈ N0:

|φ1| < 1 = $γX(h) = \frac{σ^2}{1 - |φ1|^2} φ1^h$ |φ1| > 1 = $γX(h) = \frac{σ^2}{|φ1|^2 - 1} φ1^{-h}$ Signup and view all the answers

If h < 0, how can you express the autocovariance function γX(h) using γX(−h)?

γX(h) = γX(−h) Signup and view all the answers

Flashcards

Model Class Fitting

The process of selecting a statistical model to represent data.

Forecasting

Predicting future values based on a model.