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Chapter 11: Problem 1

Let \(\left\\{x_{t}: t=1,2, \ldots\right\\}\) be a covariance stationary process and define \(\gamma_{h}=\operatorname{Cov}\left(x_{t}, x_{t+h}\right)\) for \(h \geq 0\) [Therefore, \(\left.\gamma_{0}=\operatorname{Var}\left(x_{t}\right) .\right]\) Show that \(\operatorname{Corr}\left(x_{t}, x_{t+h}\right)=\gamma_{h} / \gamma_{0}\).

Short Answer

Expert verified
\(\operatorname{Corr}(x_t, x_{t+h}) = \frac{\gamma_h}{\gamma_0}\).

Step by step solution

01

Define Covariance

The covariance of a stationary process \(x_t\) is given by \(\gamma_h = \operatorname{Cov}(x_t, x_{t+h})\). This is the measure of how much two random variables change together, where \(h\) indicates the lag between them.
02

Define Variance

The variance of \(x_t\), denoted \(\gamma_0\), is given as \(\operatorname{Var}(x_t)\). Variance is a measure of the dispersion of the random variable values from the mean.
03

Define Correlation

The correlation coefficient \(\operatorname{Corr}(x_t, x_{t+h})\) measures the strength and direction of a linear relationship between \(x_t\) and \(x_{t+h}\). It is defined as the covariance divided by the product of the standard deviations of \(x_t\) and \(x_{t+h}\).
04

Calculate Standard Deviations

Since \(\gamma_0 = \operatorname{Var}(x_t)\), the standard deviation is \(\sqrt{\gamma_0}\) because variance is the square of the standard deviation.
05

Use Correlation Formula

The correlation coefficient is given by:\[\operatorname{Corr}(x_t, x_{t+h}) = \frac{\operatorname{Cov}(x_t, x_{t+h})}{\sqrt{\operatorname{Var}(x_t)} \cdot \sqrt{\operatorname{Var}(x_{t+h})}}\] Since the process is stationary, \(\operatorname{Var}(x_{t+h}) = \operatorname{Var}(x_t)\), therefore:\[\operatorname{Corr}(x_t, x_{t+h}) = \frac{\gamma_h}{\sqrt{\gamma_0} \cdot \sqrt{\gamma_0}} = \frac{\gamma_h}{\gamma_0}\]
06

Conclusion

Thus, we have shown that \(\operatorname{Corr}(x_t, x_{t+h}) = \frac{\gamma_h}{\gamma_0}\), as required by the problem statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Correlation Coefficient
The correlation coefficient is a crucial concept when studying a covariance stationary process. It measures how strongly two random variables move together. Specifically, for two variables like \(x_t\) and \(x_{t+h}\), it tells us about the linear relationship between them.

Here are key points about the correlation coefficient:
  • The value ranges from -1 to 1.
  • A value of 1 indicates a perfect positive linear relationship.
  • A value of -1 indicates a perfect negative linear relationship.
  • A value of 0 suggests no linear relationship.
To calculate the correlation between \(x_t\) and \(x_{t+h}\) in a covariance stationary process, divide their covariance \(\gamma_h\) by the variance \(\gamma_0\). This is represented as: \[\operatorname{Corr}(x_t, x_{t+h}) = \frac{\gamma_h}{\gamma_0}\]
This formula simplifies because, in a stationary process, variances remain constant over time.
Exploring Variance
Variance is a fundamental measure in statistics that tells us about the spread of a set of values. It helps us understand how much these values differ from the average value.

In the context of your covariance stationary process, variance is represented by \(\gamma_0 = \operatorname{Var}(x_t)\). Here鈥檚 what variance tells us:
  • It quantifies how data points differ from the mean.
  • Larger variance indicates that data points are more spread out.
  • Smaller variance suggests data points are more clustered around the mean.
Understanding variance is key to grasping how the process behaves over time. It sets the foundation for other measurements like the standard deviation, and it determines the stationary characteristic of the process. Since \(\gamma_0\) remains constant in a stationary process, it simplifies calculations such as the correlation coefficient.
Diving into Covariance
Covariance is a statistic that captures how much two random variables change together. In the context of a stationary process, it helps analyze relationships over time.

Key aspects of covariance:
  • Covariance between two variables \(x_t\) and \(x_{t+h}\) is represented as \(\gamma_h = \operatorname{Cov}(x_t, x_{t+h})\).
  • Positive covariance means that as one variable increases, the other tends to increase.
  • Negative covariance means that as one variable increases, the other tends to decrease.
  • A covariance of zero indicates no relationship.
While covariance gives insight into variable relationships, its magnitude is not standardized. This is why we often convert it into a correlation coefficient, which normalizes covariance, making it easier to interpret.
Simplifying with Standard Deviation
Standard deviation is one of the most widely used measures of variability. It provides an intuitive sense of the spread of the data.

In the case of a covariance stationary process, the standard deviation is linked to variance \(\gamma_0\). You find it by taking the square root of variance:\[\text{Standard deviation} = \sqrt{\gamma_0}\]
Key insights about standard deviation include:
  • A larger standard deviation implies more spread in the data values.
  • A smaller standard deviation suggests that values cluster closer to the mean.
  • Standard deviation is useful as it brings variance to the same unit as the data.
In a stationary process, knowing the standard deviation helps in understanding variability at any point in time. It is crucial because it is part of the correlation formula, simplifying the relationship between time series observations.

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Most popular questions from this chapter

Let \(h y 6_{t}\) denote the three-month holding yield (in percent) from buying a six-month T-bill at time \((t\)- 1) and selling it at time \(t\) (three months hence) as a three-month T-bill. Let \(h y_{t-1}\) be the three-month holding yield from buying a three-month T-bill at time \((t-1) .\) At time \((t-1), h y 3_{t-1}\) is known, whereas \(h y 6_{t}\) is unknown because \(p s_{t}\) (the price of three-month T-bills) is unknown at time \((t-1) .\) The expectations hypothesis (EH) says that these two different three-month investments should be the same, on average. Mathematically, we can write this as a conditional expectation: $$\mathrm{E}\left(h y 6_{t} | I_{t-1}\right)=h y 3_{t-1}$$ and testing \(\mathrm{H}_{0}: \beta_{1}=1 .\) (We can also test \(\mathrm{H}_{0}: \beta_{0}=0,\) but we often allow for a term premium for buying assets with different maturities, so that \(\beta_{0} \neq 0 .\) ) i. Estimating the previous equation by OLS using the data in INTQRT (spaced every three months) gives $$\begin{aligned} \widehat{h y 6_{t}} &=-.058+1.104 \mathrm{hy}_{t-1} \\ &\quad\quad(.070)(.039) \\ n &=123, R^{2}=.866 \end{aligned}$$ Do you reject \(\mathrm{H}_{0}: \beta_{1}=1\) against \(\mathrm{H}_{0}: \beta_{1} \neq 1\) at the \(1 \%\) significance level? Does the estimate seem practically different from one? ii. Another implication of the EH is that no other variables dated as \(t-1\) or earlier should help explain \(h y 6_{t},\) once \(h y g_{t-1}\) has been controlled for. Including one lag of the spread between six- month and three-month T-bill rates gives $$\begin{aligned} \widehat{h y 6}_{t}=&-.123+1.053 \mathrm{hy} 3_{t-1}+.480\left(\mathrm{r} 6_{t-1}-r 3_{t-1}\right) \\ &(.067)\quad\space\space(.039) \\ n &=123, R^{2}=.885 \end{aligned}$$Now is the coefficient on \(h y 3_{t-1}\) statistically different from one? Is the lagged spread term significant? According to this equation, if, at time \(t-1, r 6\) is above \(r 3,\) should you invest in six-month or three-month T-bills? iii. The sample correlation between \(h y\) s \(_{t}\) and \(h y s_{t-1}\) is. \(914 .\) Why might this raise some concerns with the previous analysis? iv. How would you test for seasonality in the equation estimated in part (ii)?

Let \(\left\\{y_{t}: t=1,2, \ldots\right\\}\) follow a random walk, as in \((11.20),\) with \(y_{0}=0 .\) Show that \(\operatorname{Corr}\left(y_{t}, y_{t+h}\right)=\sqrt{t /(t+h)}\) for \(t \geq 1, h>0\).

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?
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