Question

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)?

Short answer

i. Reject \(\beta_1 = 1\); practically, small difference. ii. \(\beta_1\) not statistically different; spread term significant; invest in six-month if \(r6>r3.\) iii. High correlation suggests multicollinearity issues. iv. Test for seasonality using quarterly dummies.

Step-by-step solution

Hypothesis Testing for Coefficient

First, determine if the coefficient \(\beta_1\) from the regression in part (i) is statistically different from 1. The null hypothesis \(H_0: \beta_1 = 1\) is tested against the alternative \(H_1: \beta_1 eq 1\). Given the estimated coefficient \(1.104\) and the standard error \(0.039\), you perform a t-test:\[ t = \frac{1.104 - 1}{0.039} \approx 2.67\]With \(n-2 = 121\) degrees of freedom, at a 1% significance level, check if \( |t| \) exceeds the critical value (approximately 2.576 for a two-tailed test). Since \(2.67 > 2.576\), reject the null hypothesis, suggesting \(\beta_1\) is statistically different from 1.

Practical Significance of Coefficient

Even though statistically different from 1, check if \(1.104\) is practically different. A 0.104 increase over 1 might not represent a substantial economic significance, especially in financial contexts where small differences can often be negligible.

Effect of Lagged Spread Term

Analyze the regression results from part (ii) with the term \(1.053\) for \(\mathrm{hy}3_{t-1}\) and 0.480 for the lagged spread. Perform a t-test for \(\beta_1 = 1\) similarity: \[ t = \frac{1.053 - 1}{0.039} \approx 1.36\]At a 1% significance level, the computed t-value does not surpass the critical value of 2.576, thus accept \(H_0: \beta_1 = 1\). For the spread, significance depends on its t-statistic, which is\[ t = \frac{0.480}{0.067} \approx 7.16\]which is significant as it exceeds the critical value.

Investment Decision According to Rate Spread

If \(r6 > r3\), a positive spread exists implying potential excess returns by investing in six-month T-bills, given the positive coefficient on the spread.

Concerns with High Correlation

A high correlation (0.914) between \(h y s_{t}\) and \(h y s_{t-1}\) can indicate multicollinearity issues or possible problems with autocorrelation, which affect the reliability of the regression coefficients' estimates.

Testing for Seasonality

To test for seasonality in the regression, introduce dummy variables for different quarters (if applicable) to capture cyclical patterns in the data. Each dummy except one (as baseline) is included in the regression to see if any seasonal pattern significantly influences the dependent variable.

Key concepts

Expectations Hypothesis

The Expectations Hypothesis (EH) is a fundamental concept in finance and econometrics. This theory proposes that the returns from different investment strategies—specifically strategies with varying time horizons—should be equal, on average. In practical terms, this means that the expected yield from investing in a longer-dated asset should equal the yield from sequentially rolling over shorter-dated assets to match the longer time period.
\(E(hy6_t | I_{t-1}) = hy3_{t-1}\)
captures this idea by indicating that the expected return from holding a six-month T-bill for three months should equal the return from a three-month T-bill, assuming market efficiency and rational expectations.
However, real-world factors such as risk premiums or term premiums might cause deviations. In econometric analysis, these deviations are scrutinized to test if the EH holds true using statistical methods, such as regression analysis.

OLS Regression

Ordinary Least Squares (OLS) Regression is a type of linear regression used to estimate the relationship between a dependent variable and one or more independent variables. In the context of testing the EH, OLS helps in estimating the relationship between different holding yields of T-bills.
The regression equation takes the form:
\[\widehat{hy6_t} = \beta_0 + \beta_1 hy3_{t-1} + \epsilon_t\]where \(\beta_0\) is a constant (intercept), \(\beta_1\) is the coefficient of the independent variable \(hy3_{t-1}\), and \(\epsilon_t\) is the error term capturing the deviations from the predicted values.
The goal is to find the best-fitting line through the data points that minimizes the sum of the squared vertical distances (errors) of the observed data points from the line. The coefficients \(\beta_0\) and \(\beta_1\) provide insight into the relationship being studied. Specifically, if \(\beta_1\) is close to 1, it suggests that the returns are consistent with the expectations hypothesis.

Statistical Significance

Statistical Significance is a concept used to determine if the observed results in a study are unlikely due to random chance. In econometrics, it helps in understanding whether the coefficients obtained from a regression are meaningful in explaining the relationship among variables.
In the exercise, a t-test is employed to assess if the coefficient \(\beta_1\) is statistically different from 1. The t-value is calculated by:
\[t = \frac{\beta_1 - 1}{SE(\beta_1)}\]where \(SE(\beta_1)\) is the standard error of \(\beta_1\).
Depending on the significance level (like 1% or 5%), you compare the calculated t-value against a critical value from the t-distribution. If the t-value exceeds the critical value, you reject the null hypothesis. In this case, the t-value was 2.67, exceeding the critical value of 2.576, thereby indicating that \(\beta_1\) is statistically different from 1.

Hypothesis Testing

Hypothesis Testing is a statistical method that is used to make inferences or draw conclusions about a population based on sample data. It involves setting up two hypotheses: the null hypothesis \(H_0\), which represents a status quo, and the alternative hypothesis \(H_1\), which is what you want to test against the null.
For instance, when testing whether \(\beta_1 = 1\), \(H_0\) would be:\
\(H_0: \beta_1 = 1\)
and \(H_1: \beta_1 eq 1\).
Using statistical tests, such as the t-test, allows you to determine if the evidence against \(H_0\) is strong enough to reject it. The outcome guides whether the initial assumption can be kept or should be rejected in favor of the alternative hypothesis.
Calculating the test statistic, comparing it to the critical value, and judging based on significance levels define this process. Successful hypothesis testing informs decisions in finance, econometrics, and beyond, by relying on objective data analysis.