Let {et: t=-1,0,1, …} be a sequence of independent,...

Let $\left\{e_{t}: t=-1,0,1, \ldots\right\}$ be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by $$x_{t}=e_{t}-(1 / 2) e_{t-1}+(1 / 2) e_{t-2}, t=1,2, \ldots$$ i. Find $\mathrm{E}\left(x_{t}\right)$ and $\operatorname{Var}\left(x_{t}\right) .$ Do either of these depend on $t ?$ ii. Show that $\operatorname{Corr}\left(x_{t}, x_{t+1}\right)=-1 / 2$ and $\operatorname{Corr}\left(x_{t}, x_{t+2}\right)=1 / 3 .$ (Hint: It is easiest to use the formula in Problem $1 .$ ) iii. What is $\operatorname{Corr}\left(x_{t}, x_{t+h}\right)$ for $h>2 ?$ iv. Is $\left\{x_{t}\right\}$ an asymptotically uncorrelated process?

Let $\left\{e_{t}: t=-1,0,1, \ldots\right\}$ be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by $$x_{t}=e_{t}-(1 / 2) e_{t-1}+(1 / 2) e_{t-2}, t=1,2, \ldots$$ i. Find $\mathrm{E}\left(x_{t}\right)$ and $\operatorname{Var}\left(x_{t}\right) .$ Do either of these depend on $t ?$
ii. Show that $\operatorname{Corr}\left(x_{t}, x_{t+1}\right)=-1 / 2$ and $\operatorname{Corr}\left(x_{t}, x_{t+2}\right)=1 / 3 .$ (Hint: It is easiest to use the formula
in Problem $1 .$ )
iii. What is $\operatorname{Corr}\left(x_{t}, x_{t+h}\right)$ for $h>2 ?$
iv. Is $\left\{x_{t}\right\}$ an asymptotically uncorrelated process?

Key Concepts

Asymptotically Uncorrelated Process

An asymptotically uncorrelated process is one in which the autocorrelation between observations decreases to zero as the lag between them increases. In the context of finite-order MA processes, correlations typically vanish beyond the order of the synthesis (i.e., beyond lag 2 in an MA(2) model). This property is important because it informs short-term dependency and implies that, for large lags, the series behaves almost like a sequence of independent observations.

Expectation and Variance Calculation

Calculating the expectation and variance of a linear combination of random variables, such as in the MA process, involves using the linearity of expectation and the independence properties of the error terms. These calculations illustrate how the mean remains unchanged by the linear operations when the white noise has a mean of zero, and the variance is computed by summing up the weighted variances of the individual terms, considering the absence of correlation between different error terms.

Autocovariance and Autocorrelation Functions

The autocovariance function measures the degree of linear dependence between observations at different time lags in a time series, while the autocorrelation function normalizes these values to lie between -1 and 1. For MA processes, the autocovariance function will be nonzero for only a finite number of lags due to the finite number of error terms involved. The problem demonstrates how to compute specific lag correlations, which are essential in identifying and estimating the structure of the process.

Moving Average Process

A moving average (MA) process is a time series model in which the current value is expressed as a linear combination of current and past white noise error terms. In the context of the given problem, the process is a finite-order MA model (specifically an MA(2)), which means that only a limited number of past white noise terms contribute to the current observation. This structure is key in understanding the autocovariance and autocorrelation properties of the series.

White Noise Process

A white noise process is a sequence of random variables that are independent and identically distributed, typically with a constant mean (often zero) and a constant variance. This concept is fundamental because it serves as the building block for more complex time series models, where the individual shocks or innovations are assumed to be independent with no autocorrelation.

Stationarity

Stationarity refers to a property of a time series where its statistical characteristics, such as the mean and variance, remain constant over time. In the context of the problem, stationarity is implied because the expectation and variance of x? do not depend on t. This property is important for the analysis of time series since many inferential techniques rely on the assumption of constant means and variances.

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