Lab4_quiz_TimeSeries

What condition must be met for a time series to be considered ergodic?

• The autocovariance function must tend to zero as the lag increases.

• The mean and variance must be constant over time.

• The process must have periodic components.

• The ACF must remain constant at all lags.

Answer explanation

For a time series to be considered ergodic, the autocovariance function must tend to zero as the lag increases. This indicates that the process is stable over time and that its statistical properties can be inferred from a single, sufficiently long realization.

How does ergodicity differ from stationarity?

• Ergodicity refers to memory loss in the process, while stationarity refers to constant statistical properties.

• Stationarity implies no dependence on time, while ergodicity refers to dependence on initial conditions.

• Ergodicity guarantees stationarity, but stationarity does not guarantee ergodicity.

• Ergodicity only applies to random walks, while stationarity applies to all processes.

Answer explanation

The correct choice highlights that ergodicity involves memory loss in a process, meaning past states do not influence future states, while stationarity indicates that statistical properties remain constant over time.

Which of the following properties is required for a time series to be weakly stationary?

• The series has a constant mean, constant variance, and autocovariance depends only on the lag, not time.

• The variance grows over time.

• The autocorrelation increases with time.

• The process has a random trend component.

Answer explanation

A time series is weakly stationary if it has a constant mean, constant variance, and its autocovariance depends only on the lag, not on time. This ensures that the statistical properties do not change over time.

What is the key difference between a random walk and white noise?

• A random walk has a constant mean, while white noise has a variable mean.

• A random walk accumulates past values and has a long memory, while white noise is independent with no memory.

• White noise has a trend, while a random walk does not.

• Both are stationary processes with constant variance.

Answer explanation

The key difference is that a random walk accumulates past values, leading to long-term dependence, while white noise consists of independent values with no memory, making it a purely random process.

If a time series follows a White Noise process, what can be said about its predictability?

• Future values can be predicted based on past observations

• Future values are completely unpredictable from past values.

• The process shows periodic patterns that aid in forecasting.

• The process has predictable trends.

Answer explanation

In a White Noise process, future values are random and do not depend on past values, making them completely unpredictable. This rules out any patterns or trends that could aid in forecasting.

What are the main properties of a White Noise process?

• It has a zero mean, constant variance, and zero autocorrelation for all non-zero lags.

• It has a non-zero mean, increasing variance, and autocorrelation that decreases with time.

• It exhibits a trend, seasonality, and correlation between observations.

• It has a variable mean, constant variance, and non-zero autocorrelation for all lags.

Answer explanation

A White Noise process is characterized by having a zero mean, constant variance, and zero autocorrelation for all non-zero lags, making the correct choice the first option.

What is the difference between weak stationarity and strict stationarity in time series analysis?**

• Weak stationarity requires constant mean and variance, while strict stationarity requires the entire distribution to remain the same over time.

• Weak stationarity implies no autocorrelation, while strict stationarity allows for autocorrelation.

• Strict stationarity only applies to non-linear processes, while weak stationarity applies to linear processes.

• Weak stationarity requires independence between observations, while strict stationarity does not.

Answer explanation

Weak stationarity requires that the mean and variance of the time series remain constant over time, while strict stationarity demands that the entire probability distribution of the series does not change, making the first choice correct.

When seasonal patterns are present in a time series, how do the autocorrelations behave?

• Autocorrelations are large for all lags

• Autocorrelations are larger for lags at multiples of the seasonal frequency than for other lags.

• Autocorrelations are zero for all lags.

• Autocorrelations are consistent regardless of the lag.

Answer explanation

When seasonal patterns are present, autocorrelations peak at lags that are multiples of the seasonal frequency, indicating a strong relationship at those intervals compared to other lags.

In a dataset characterized as random data (white noise), what should the autocorrelations look like?

• Autocorrelations should be significantly positive for all lags.

• Autocorrelations should be significantly negative for all lags.

• Autocorrelations should be near zero for all lags, within confidence intervals.

• Autocorrelations should show a steady increase as lags increase.

Answer explanation

In random data (white noise), autocorrelations should be near zero for all lags, indicating no predictable pattern. This aligns with the correct choice, as significant positive or negative autocorrelations would suggest a structure that doesn't exist.

Looking at the ACF (Auto-Correlation Function) plots, which of the time series shows significant autocorrelations at many lags, indicating seasonality?

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Answer explanation

The ACF plot for Monthly air passengers (C) shows significant autocorrelations at multiple lags, indicating a seasonal pattern. In contrast, the other time series do not exhibit such strong seasonal autocorrelations.