Dozens of economic research presents the negative correlation between economic cycle and term spreads.
Why Term spreads matter?
Term spreads is capable of predicting changes in the real economy. Empirical research shows term spread contains effective information on economic operation and highly efficient in predicting changes in the real economy. Kessel (1965) found that the trend of term spread is opposite to the fluctuation of the business cycle. Estrella and Hardouvelis (1991) indicated a positive yield curve slope predicts future expansion of real economic avtivity. Haubrich and Dombrosky (1996) presented the interst rate spread was an excellent predictor of economic growth over the next four quarters. Estrella and Mishkin (1996) showed a flattening of the yield curve is a strong signal of an impending recession. Rudebusch and Williams (2009) argue that negative term spreads predict low output and a high probability of recession. Wang and Yang (2012) believe that the inversion of term spread is a non-equlibrium phenomenon occuring after interest rates and output are impacted. It is a leading indicator of economic recession, not the cause of economic recession.
U.S. historical experience shows that inversions in term spreads indicate recessions. Since July 1961, there have been 9 inversions of the 10-year Treasury bond and 2-year Treasury bond maturity spreads in U.S. history, and 8 of them experienced recessions 6-24 months later. Only in 1966 After the inversion of the term spread, the economy did not fall into recession within 2 years. Although there was no recession, the U.S. real GDP growth rate (annualized on a month-on-month basis) fell sharply in 1967, from 10.1% in the first quarter of 1966 to 0.2% in the second quarter of 1967, a decrease of 9.9 percentage points. In view of the fact that the term structure of interest rates contains a lot of useful information, central banks such as the Federal Reserve and the Bank of England have included them in the leading economic sentiment index, and regularly announced changes in long- and short-term interest rate spreads. Market personnel will also track the spread between the 10-year and 2-year Treasury bond yields to predict future economic trends. In particular, be wary of the inversion of the term interest rate spread, in order to prepare for a possible economic recession in advance.
Economists mainly use the 10Y-2Y bond term spread for forecasting. There are two ways to choose the term: one is to understand the main information contained in the interest rate of each term based on the interest rate term structure theory, and select the corresponding term according to the research purpose; the other is result-oriented, comparing the forecast results and choosing the best forecast effect. that set of deadlines. In reality, there are many ways to choose the term spread, such as 10-year treasury bond rate - 2-year treasury bond rate (10Y-2Y), 10-year treasury bond rate - 1-year treasury bond rate (10Y-1Y), 10-year treasury bond rate Treasury bond rate-3-month Treasury bill rate (10Y-3M), 10-year Treasury bond rate-Federal funds rate (10Y-Fd), and 5-year Treasury bond rate-3-year Treasury bond rate (3Y-3Y), etc. Compared with the 10Y-3M Treasury bond term spread, the 10Y-2Y Treasury bond term spread inverts earlier and more frequently. Therefore, market researchers mainly use the 10-year Treasury bond rate - the 2-year Treasury bond rate.
The concept of Propensity score system was first proposed by Rosenbaum and Rubin. They define the propensity score as the conditional probability that an individual is affected by an independent variable after controlling the observable “confused” variable. The causal relationship between the phenomena obtained by controlling the propensity score can eliminate the influence of the “confusing” variable and pbtain the “net effect” between the two, thus ensuring the reliability of the conclusion. From a philosophical point of view, PSM is a clever use of “control” ideas in sociological research. From a statistical point of view, it is based on the counterfactual framework and technologically controls many confusing variables.
Now, we examine this method based on two perspectives of philosophy and satistics.
The philosophical perspective of propensity score matching
All sociological studies emphasize that only by controlling other variables can we really derive the causal relationship between the two varibales of interest. When there is only one confusing variable, such as only the “ability” variable may bias the causal relationship between education and income, the general practice of controlling ability is to subdivide the ability variable into different levels and guarantee that during each level, the ability of samples are the same or close. In this way, we can examine the relationship between education and income at each level. Which is a common practice for controlling confusing variables. However, as the number of confusing variables that need to be controlled increases, this method of directly controlling confusing variables becomes more and more difficult. When we have two confusing variables, we can divide the variables into \(2\times 2\) interactive groups, and only need to observe the relationship between education and income within the group, then figure out the overall effect.
However, when the number of variables to be controlled is increased to 5 or 6, then equal amount of interactive groups will be generated, which will make the grouping no longer easy, and there may also be some other problems, such that a group without individual may be created due to the limitation of sample size. At this point, PSM is able to subtly reduce the dimension of the confusion variable by means of tendency scoring. Instead of focusing on the specific value of the confusion variable, it focuses on the propensity score obtained by substituting these confounding variables into the logistic regression equation. Under this situation, you can control all of the obfuscated variables simply by ensuring that the propensity scores match. Therefore, no matter how many confusing variables thare are, we can still control them through PSM to get the ideal causal relationship. From the perspective of “control”, PSM solves the problem of controlling multiple confounding variables well, so that a “purer” causal relationship can be obtained.
The statistical basis of the propensity score matching method
From a statistical point of view, let \(Y_{if},Y\) represent the dependent variables of experimental group and the control group. W is a binary variable, \(w=1\) means the individual is in the experimental group, \(w=0\) means the individual is in the control group. When an individual belongs to the experimental group,
\(E\left ( Y_{t}|w=1 \right )\) is observable and counterfactual. We cannot observe what if a well educated person without being educated at the time. Similarly, for the control group, \(E\left ( Y_{0}|w=0 \right )\) is observable, and \(E\left ( Yj|w=0 \right )\) is counterfactual and unobservable. The causal relationship we hope to obtain is a weighted average of the differences between the “facts” and “reverse facts” of individuals in the experimental group.
Where n is the proportion of all subjects in the experimental group. Since counterfactuality cannot be observed, a specific group of people can only be in the experimental group or in the control group, the following non-confusion assumptions must still be met when making causal inferences:
\[E\left ( Y_{1}|w=0 \right )=E\left ( Y_{1}|w=1 \right )\] \[E\left ( Y_{0}|w=0 \right )=E\left ( Y_{0}|w=1 \right )\]That is, another group in the control group can represent the “counterfactual” status of the individual in the experimental group. Thus, equation(1) can be simplified as:
\[T=E\left ( Y_{j}|w=1 \right )-E\left ( Y_{0}|w=0 \right )\]Under randomized experimental conditions, since the experimental individuals were assigned to the experimental and control groups in a random manner,
\[E\left ( Y_{1}|w=0 \right )=E\left ( Y_{i}|w=1 \right )\]\(E\left ( Y_{0}|w=0 \right )=E\left ( Y_{0}|w=1 \right )\) assumptions hold. Based on the fact that the observed test cannot guarantee randomization, it is necessary to control the confusion variable as much as possible so that \(w\) is independent from \(Y_{0}\) and \(Y_{1}\).
\[E\left ( Y_{1}|w=0,x \right )=E\left ( Y_{1}|w=1,x \right )\] \[E\left ( Y_{0}|w=0,x \right )=E\left ( Y_{0}|w=1,x \right )\]Among them, it is a confusion variable. As long as the aliasing variables can be found and controlled, it is approximated that \(w\) is independent of \(Y_{0},Y_{1}\)(Rosenbaum and Rubin, 1983), i.e. \(\left ( Y_{0}, Y_{1} \right )\perp w\mid x\)
At this point, the confounding variable obtains a specific propensity value p by logistic regression, which results in:
\[E\left ( Y_{1}|w=0,p \right )=E\left ( Y_{1}|w=1,p \right )\] \[E\left ( Y_{0}|w=0,p \right )=E\left ( Y_{0}|w=1,p \right )\]In summary, the non-confused assumptions can be satisfied “approximate” to obtain the desired causality inference.
