Question

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?

Short answer

E(x_t) = 0; Var(x_t) = 3/2. Corr(x_t, x_{t+1}) = -1/2, Corr(x_t, x_{t+2}) = 1/3, Corr(x_t, x_{t+h}) = 0 for h > 2. The process is asymptotically uncorrelated.

Step-by-step solution

Calculate E(x_t)

The expectation of the process is found using the linear property of expectation. Since the sequence \( \{e_t\} \) consists of independent, identically distributed random variables with mean zero:\[ \mathrm{E}(x_t) = \mathrm{E}(e_t) - \frac{1}{2}\mathrm{E}(e_{t-1}) + \frac{1}{2}\mathrm{E}(e_{t-2}) \]Each term equals zero: \[ \mathrm{E}(e_t) = \mathrm{E}(e_{t-1}) = \mathrm{E}(e_{t-2}) = 0 \]Thus,\[ \mathrm{E}(x_t) = 0 \]

Calculate Var(x_t)

The variance is found using the property that the variance of a sum of independent variables is the sum of their variances.\[ \operatorname{Var}(x_t) = \operatorname{Var}(e_t) + \left(\frac{1}{2}\right)^2 \operatorname{Var}(e_{t-1}) + \left(\frac{1}{2}\right)^2 \operatorname{Var}(e_{t-2}) \]Given the variance of each term \( e_t \) is 1:\[ \operatorname{Var}(e_t) = \operatorname{Var}(e_{t-1}) = \operatorname{Var}(e_{t-2}) = 1 \]Thus,\[ \operatorname{Var}(x_t) = 1 + \frac{1}{4} + \frac{1}{4} = \frac{3}{2} \]

Does E(x_t) and Var(x_t) Depend on t?

Both \( \mathrm{E}(x_t) \) and \( \operatorname{Var}(x_t) \) are constants, showing they do not depend on \( t \).

Calculate Corr(x_t, x_{t+1})

To find the correlation between \( x_t \) and \( x_{t+1} \):\[ x_{t+1} = e_{t+1} - \frac{1}{2}e_t + \frac{1}{2}e_{t-1} \]Using the fact that only \( e_t \) appears in both expressions, the covariance is:\[ \operatorname{Cov}(x_t, x_{t+1}) = \operatorname{Cov}(e_t, -\frac{1}{2}e_t) = -\frac{1}{2}\operatorname{Var}(e_t) = -\frac{1}{2} \]Thus,\[ \operatorname{Corr}(x_t, x_{t+1}) = \frac{-1/2}{\sqrt{3/2} \cdot \sqrt{3/2}} = -\frac{1}{2} \]

Calculate Corr(x_t, x_{t+2})

To find the correlation between \( x_t \) and \( x_{t+2} \):\[ x_{t+2} = e_{t+2} - \frac{1}{2}e_{t+1} + \frac{1}{2}e_t \]Using the fact that only \( e_t \) appears in both expressions, the covariance is:\[ \operatorname{Cov}(x_t, x_{t+2}) = \operatorname{Cov}(e_t, \frac{1}{2}e_t) = \frac{1}{2}\operatorname{Var}(e_t) = \frac{1}{2} \]Thus,\[ \operatorname{Corr}(x_t, x_{t+2}) = \frac{1/2}{\sqrt{3/2} \cdot \sqrt{3/2}} = \frac{1}{3} \]

Determine Corr(x_t, x_{t+h}) for h > 2

For \( h > 2 \), no common terms appear in both \( x_t \) and \( x_{t+h} \), therefore:\[ \operatorname{Cov}(x_t, x_{t+h}) = 0 \]Leading to:\[ \operatorname{Corr}(x_t, x_{t+h}) = 0 \]

Is the Process Asymptotically Uncorrelated?

Since \( \operatorname{Corr}(x_t, x_{t+h}) \to 0 \) for \( h > 2 \), the process \( \{x_t\} \) is asymptotically uncorrelated.

Key concepts

Independent Random Variables

A fundamental concept in understanding stochastic processes is recognizing the nature of independent random variables. Independent random variables are those whose occurrences do not affect each other. This means that knowing the outcome of one variable provides no additional information about the outcome of another.
For the stochastic process defined in the exercise, we have a sequence of such independent random variables, all with mean zero and variance equal to one. In math terms, this is denoted as:

• \( ext{E}(e_t) = 0 \)
• \( ext{Var}(e_t) = 1 \)

This characteristic is pivotal when calculating expectations and variances. By understanding that each component is independent, calculations simplify significantly as correlations or covariances between different random variables in the sequence are always zero, unless explicitly present in both expressions of the stochastic process (i.e., overlaps occur).

Correlation Analysis

Correlation analysis helps us measure the degree to which two variables are related. In this context, it enables us to understand how different terms in a time series might relate over time.
The correlation between two random variables, say \( x_t \) and \( x_{t+h} \), is given by the formula:\[\text{Corr}(x_t, x_{t+h}) = \frac{\text{Cov}(x_t, x_{t+h})}{\sqrt{\text{Var}(x_t)} \cdot \sqrt{\text{Var}(x_{t+h})}}\]In the provided solution, the correlation between \( x_t \) and \( x_{t+1} \) is found to be \(-1/2\), and between \( x_t \) and \( x_{t+2} \) it is \(1/3\). Understanding these results is crucial to predicting how current terms in a sequence might affect future ones.
Importantly, for \( h > 2 \), the correlation becomes zero because no overlapping terms are present in the expressions of \( x_t \) and \( x_{t+h} \), leading to zero covariance.

Variance and Expectation

In the context of our stochastic process, expectation and variance provide insights into the central tendency and variability of the process, respectively. Both mean and variance are crucial for describing the process's distribution.
For the process given, expectation is calculated as the sum of expectations of the components:\[\text{E}(x_t) = \text{E}(e_t) - \frac{1}{2}\text{E}(e_{t-1}) + \frac{1}{2}\text{E}(e_{t-2}) = 0\]Since each component has a mean of zero, the total expectation equals zero. Similarly, the variance relies on the sum of the variances of these independent components:\[ \text{Var}(x_t) = \text{Var}(e_t) + \left(\frac{1}{2}\right)^2 \text{Var}(e_{t-1}) + \left(\frac{1}{2}\right)^2 \text{Var}(e_{t-2}) = \frac{3}{2}\]The variance reflects the spread of the process away from its mean, providing a measure of the "average" squared deviations from the expected value.
Since both expectation and variance are constants, they do not depend on time \( t \), indicating stationary properties of the linear combinations of the process.

Asymptotically Uncorrelated Processes

The property of being asymptotically uncorrelated emerges when the correlation between consecutive terms in a sequence diminishes as the gap between them increases. In simpler terms, as we examine pairs of terms in the process farther apart in time, their connection weakens or becomes non-existent.
From the calculations earlier, we deduce that \( \text{Corr}(x_t, x_{t+h}) \) approaches zero for \( h > 2 \). This means there is no linear relationship or predictability between such widely spaced terms.
Generally, this property implies long-term independence, which is vital when forecasting or analyzing time series. In practice, such a characteristic is beneficial as it suggests that past influences on the system dissipate with time, ensuring robustness and adaptability of the model in dynamic environments.