In the last two decades, international financial markets have integrated to an extent remarkable in history. This process has profound implications for the transmission of shocks, both across financial asset prices and to the real economy. Therefore, the role of asset prices including interest rates, stock returns, dividend yields and exchange rates are considered as predictors of inflation as well as growth.

1. The Yield Curve as a Predictor of U.S. Recessions
Based on (Estrella & Mishkin 1996) Work
Lin Jibin and Verny Tania
Universit´e Paris 1 Panth´eon Sorbonne
Dissertation submitted to MOSEF, Faculty of Economics, Universit´e Paris 1
Panth´eon Sorbonne, as a partial fulfilment of the requirement for the Master 1 of
Economie Quantitative.
May 2015

2. Contents
List of Figures ii
1 Introduction 1
2 Probit Model 4
3 Economic Indicators 6
3.1 Treasury Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 S&P500 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Consumer Price Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Crude Oil Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Unemployment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Comparisons of the forecasted probability of recession of different indicators 29
4.1 Comparison of Spread & Consumer Price Index . . . . . . . . . . . . . . . . 30
4.2 Comparison of Spread & Crude Oil Price . . . . . . . . . . . . . . . . . . . . 31
4.3 Comparison of Spread & S&P500 . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Comparison of Spread & Unemployment Rate . . . . . . . . . . . . . . . . . 33
5 Conclusion 34
Bibliography 35
i

3. List of Figures
3.1 Means of the Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Statistical Description of the Indicators . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Statistical Distribution of spread . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Treasury Spread: Ten-Year Bond Rate minus Three-Month Bill Rate Monthly
Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.5 Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Trea-
sury Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.6 Statistical Distribution of S&P500 index . . . . . . . . . . . . . . . . . . . . . 14
3.7 S&P500 monthly index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.8 Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the S&P500
index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.9 Statistical Distribution of CPI . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.10 CPI monthly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.11 Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Con-
sumer price index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.12 Statistical Distribution of crude oil price . . . . . . . . . . . . . . . . . . . . . 22
3.13 Crude oil monthly price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.14 Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Crude
oil price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.15 Statistical Distribution of unemployment rate . . . . . . . . . . . . . . . . . . 26
3.16 Unemployment rate monthly . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ii

4. LIST OF FIGURES
3.17 Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the unem-
ployment rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Graph of Forecasted probability of recessions of Spread and CPI . . . . . . . 30
4.2 Graph of Forecasted probability of recessions of Spread and Crude oil price 31
4.3 Graph of Forecasted probability of recessions of Spread and S&P500 . . . . 32
4.4 Graph of Forecasted probability of recessions of Spread and Unemployment
rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
iii

5. Chapter 1
Introduction
In the last two decades, international financial markets have integrated to an extent re-
markable in history. This process has profound implications for the transmission of shocks,
both across financial asset prices and to the real economy. Therefore, the role of asset prices
including interest rates, stock returns, dividend yields and exchange rates are considered
as predictors of inflation as well as growth.
Recently, numerous empirical studies have been carried out in order to evaluate the
usefulness of spreads between long and short-term interest rates as leading indicators of
real economic activity. In most these studies linear regression-based techniques are applied
to forecast output growth rate and also even though, some well-known authors have done
probit estimations in order to calculate the likelihood of future economic recessions. In
such probit models the dependent variable is a recession dummy that equals one if the
economy is in recession and zero otherwise, whereas the explanatory variable is a lagged
potential recession predictor. This particular model translates the steepness of the yield
curve at the present time into a likelihood of a recession sometime in the future. Thus, we
need to identify three components: a measure of steepness, a definition of recession, and
a model that connects the two. The approach which is in fact the most appropriate is the
probit equation as mentioned previously make use of the normal distribution to convert
the value of a measure of yield curve steepness into a probability of recession one year
ahead. The input to this calculation is the value of the term spread, that is, the difference
between long and short-term interest rates in month t. The output is the probability of a
recession occurring in month t + 12 from the viewpoint of information available in month
1

6. t. Both of these variables, however, need to be defined more precisely that is, we need to
specify what we mean by a recession and which long and short term interest rates we will
use to produce the spread that constitutes our measure of steepness.
Moreover, recent survey (Stock & Watson 2003) have determined that the slope of the
yield curve is referred as one of the main asset prices studied and the one which has proved
most useful for forecasting. The latter has come into particular focus in the recent pe-
riod, as its inversion in the US triggered a lively debate as to whether it would signal a
recession. In this framework, the usefulness of the slope of the yield curve as a predic-
tor of future growth has been challenged vigorously ((Greenspan 2005); (Estrella 2005);
(Bernanke 2006)). The yield curve is confirmed to be a quite reliable recession predictor
across the evaluated countries, because on average it signals recessions a considerable time
before they actually begin, and produces only a few signals that falsely indicate business
cycle turning points. (Estrella & Hardouvelis 1991) and (Estrella & Mishkin 1997) pro-
vide various evidence for the United States that the yield spread significantly outperforms
other popular financial and macroeconomic indicators in forecasting recessions, particu-
larly with horizons beyond one quarter. In his recent study, (Dueker 1997) confirms the
US results presented by (Estrella & Mishkin 1998) using a modified probit model which
includes a lagged dependent variable and additionally allows for Markov-switching coef-
ficient variation.
Building upon the expectations hypothesis, two straightforward arguments explaining
why the yield curve contains information about future recessions. The first argument re-
lates to the role of monetary policy. When a central bank raises short-term interest rates,
agents may view this contraction as temporary and, consequently, raise their expectations
of future short-term rates by less than the observed current change in the short rate. From
the expectations theory it follows that long-term rates rise by less than the short-term rate,
resulting in a flat or inverted yield curve. Since the real sector of the economy is affected
by monetary policy measures with a considerable time lag, agents expect future real eco-
nomic growth to decline. Hence, the monetary tightening flattens the yield curve and
simultaneously increases the likelihood of a recession onset. The second one focusses on
inflationary expectations that are contained in long-term interest rates. Since recessions
are generally associated with low inflation rates, an anticipation of a recession probably
results in a falling long-term rate. Consequently, when the short rate does not change, the
2

7. yield curve flattens or inverts.
Using information across the whole yield curve, rather than just the long maturity
segment, may lead to more efficient and more accurate forecasts of GDP. But in an OLS
framework, since yields of different maturities are highly cross-correlated, it is difficult to
use multiple yields as regressors because of the occurrence of collinearity problems. This
collinearity suggests that we may be able to condense the information contained in many
yields down to a parsimonious number of variables. We would also like a consistent way
to characterize the forecasts of GDP across different horizons to different parts of the yield
curve. With OLS, this can only be done with many sets of different regressions. These
regressions are clearly related to each other, but there is no obvious way in an OLS frame-
work to impose cross-equation restrictions to gain power and efficiency.
However, the reliability of the yield curve’s predictive ability has been challenged
lately. (Greenspan 2005) argues that many factors can affect its slope, including the gap
between near-term and long-term inflation expectations or near-term and long-term risk
premia. Yet, all these factors do not have similar implications for future growth. For in-
stance, as he recalls, the yield curve flattened sharply from 1992 to 1994, shortly before
the US economy entered its longest expansion of the post-war period. Consequently, a
flattening of the yield curve might also well signal a deceleration in inflation accompa-
nied by a favorable growth outlook, e.g. once the impact of an adverse oil price shock has
dampened.
In our study, we have decided to base our work on those of (Estrella & Mishkin 1996)
as well as adding additional variables such as five indicators over 10 years as monthly that
is the spread, the price of crude oil, the rate of unemployment, the S&P500 index and the
consumer price index. An outline of the report is given as follows: In chapter 2, we give
the explanation of the probit model. Chapter 3 describes briefly the five indicators, their
graphical representations of forecasted probability during different quarters. In chapter
4, we identify the comparisons between the indicators through graphical representation.
Finally, we conclude our study in chapter 5.
3

8. Chapter 2
Probit Model
In order to quantify the predictive power of the variables examined with respect to future
recessions, the probit model is being used. The name come from PRObability probIT. This
model is a particular type of regression where the dependent variable can only take two
possible values that is whether the economy is or is not in a recession. It is defined in
reference to a theoretical linear relationship of the form
In statistics, a probit model is a type of regression where the dependent variable can
only take two values, for example married or not married. The name is from probability +
unit. The purpose of the model is to estimate the probability that an observation with par-
ticular characteristics will fall into a specific one of the categories; moreover, if estimated
probabilities greater than 1/2 are treated as classifying an observation into a predicted
category, the probit model is a type of binary classification model.
y∗
t+h = αi + βixt + t
where y∗
t refers to an unobservable variable that determines the occurrence of a recession
at time t, h refers to the length of forecast horizon, β is the vector of coefficients, t is
the normally distributed error term and xt is a vector of independent variables. So , the
observable recession indicator Recessiont form part of this model by:
Recessiont = 1 if y∗
t > 0 and,
Recessiont = 0 otherwise.
4

9. The form of the estimated equation is given as follows:
P(Recessiont+h = 1) = F(βixt),
where F refers to the cumulative normal distribution function corresponding to − .
The probit model is being estimated by the maximum likelihood which is defined by
the following:
L = (ΠRecessiont+h=1)F(βixt)(ΠRecessiont+h=0)(1 − F(βixt))
In practice, the recession indicator is given from the standard National Bureau of Economic
Research recession dates, that is,
Recessiont = 1 if the economy is in recession in quarter t,
= 0 otherwise.
In the following chapters, we has decided to apply the probit model on different eco-
nomic indicators and hence give a significant interpretation on each results obtained through
statistical descriptions tables as well as graphical representations.
5

10. Chapter 3
Economic Indicators
In our study for the forecasting of crisis, we have decided to choose monthly data from
01/04/1953 to 01/03/2015 as well as additional variables with those of (Estrella & Mishkin
1996) and they are the Treasury Spread: Ten-Year Bond Rate minus Three-Month Bill Rate
Monthly Average, the S&P500 index, the Consumer price index, the crude oil price and
additionally the unemployment rate. To be able to assess how well each indicator variable
predicts recessions, we use the so-called probit model described previously, which, in our
application, directly relates the probability of being in a recession to a particular explana-
tory variable such as the yield curve. In order to understand more clearly, let see a brief
explanation on each on them as well as their graphical representations of forecasted prob-
ability at h = 1, 2, 4 that is at different horizons so as to see more clearly the difference of
forecasting at different quarters.
Firstly, we start the application on the whole data that is from 01/04/1953 to 01/03/2015
by using the probit model and we obtained the following:
Figure 3.1: Means of the Indicators
6

11. Figure 3.2: Statistical Description of the Indicators
Conclusion: We notice that only the consumer price index and the unemployment
rate are rejected at a critical value of 0, 05 whereas the other indicators are under the null
hypothesis that is presence of correlation is there with the recession. Now, we will study
furthermore, with forecasting under different horizons h as mentioned before.
7

12. 3.1 Treasury Spread
3.1 Treasury Spread
The interest rates generally used to compute the spread between long-term and short-term
rates on the yield curves predictive power. For example, market analysts usually choose
to focus on the difference between the ten-year and two-year Treasury rates, while some
academic researchers have favored the spread between the ten-year Treasury rate and the
federal funds rate. In our research, we have decided to choose Ten-Year Bond Rate and
Three-Month Bill Rate Monthly. We then applied the probit model at different horizons
and obtained the following results:
8

13. 3.1 Treasury Spread
(a) Treasury Spread: Ten-Year Bond Rate minus Three-
Month Bill Rate Monthly Average; one quarter ahead
(b) Treasury Spread: Ten-Year Bond Rate minus Three-
Month Bill Rate Monthly Average; two quarters ahead
(c) Treasury Spread: Ten-Year Bond Rate minus Three-
Month Bill Rate Monthly Average; four quarters
ahead
Figure 3.3: Statistical Distribution of spread
Conclusion: We can see that whatever the horizon used that is the h, the impact of the
spread on the recession remains highly correlated. The p-value is extremely high that is
p-value= 0, 1269 with h = 1, p-value = 0, 6436 with h = 2 and p-value = 0, 3164 with
h = 4, then we do not reject the null hypothesis, thus it means that there is the presence of
correlation during periods of recession.
9

14. 3.1 Treasury Spread
Conclusion: It is similar there between the short-term and long-term. We can deduce
through the graphs 3.4 and 3.5 the high presence of correlation during recession times. We
can see the steepness when we do the forecast at h = 4 and clearly see that a high signal
occur when approaching near the recession period.
10

15. 3.1 Treasury Spread
(a) Treasury Spread: Ten-Year Bond Rate minus Three-
Month Bill Rate Monthly Average; one quarter ahead
(b) Treasury Spread: Ten-Year Bond Rate minus Three-
Month Bill Rate Monthly Average; two quarters ahead
(c) Treasury Spread: Ten-Year Bond Rate minus Three-
Month Bill Rate Monthly Average; four quarters
ahead
Figure 3.4: Treasury Spread: Ten-Year Bond Rate minus Three-Month Bill Rate Monthly
Average
11

16. 3.1 Treasury Spread
(a) One quarter ahead (b) Two quarters ahead
(c) Four quarters ahead
Figure 3.5: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Treasury
Spread
12

17. 3.2 S&P500 Index
3.2 S&P500 Index
So we applied again the probit model on the S&P500 index at different horizons and ob-
tained the following results:
13

18. 3.2 S&P500 Index
(a) S&P500 index; one quarter ahead (b) S&P500 index; two quarters ahead
(c) S&P500 index; four quarters ahead
Figure 3.6: Statistical Distribution of S&P500 index
Conclusion: We can notice that it is similar to the consumer price index, the forecasting
is not efficient when the horizon is small that is when h = 1, the p-value is 0, 2573, when
h = 2, the p-value is 0, 6315. When h = 4, the forecast is very efficient and its p-value is
0, 0009.
14

19. 3.2 S&P500 Index
(a) S&P500 index; one quarter ahead (b) S&P500 index; two quarters ahead
(c) S&P500 index; four quarters ahead
Figure 3.7: S&P500 monthly index
Conclusion: According to the graphs 3.7 and 3.8, we can clearly notice that the signal is
high when the economic condition change. In fact, the reversal of the curve of the S&P500
before crisis period allow a better forecasting of recession, therefore it indicates that it is a
good indicator for forecasting recession.
15

20. 3.2 S&P500 Index
(a) One quarter ahead (b) Two quarters ahead
(c) Four quarters ahead
Figure 3.8: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the S&P500
index
16

21. 3.3 Consumer Price Index
3.3 Consumer Price Index
We applied again the probit model on the consumer price index at different horizons and
obtained the following results:
17

22. 3.3 Consumer Price Index
(a) Consumer price index; one quarter ahead (b) Consumer price index; two quarters ahead
(c) Consumer price index; four quarters ahead
Figure 3.9: Statistical Distribution of CPI
Conclusion: For the consumer index price, we can notice that the forecasting done
with the horizon h = 1 and h = 2 with p-value 0, 5559 and 0, 7259 respectively allow to
prevent at short-term the risk of recession. However, it is not efficient when the horizon is
large for example at h = 4.
18

23. 3.3 Consumer Price Index
(a) Consumer price index; one quarter ahead (b) Consumer price index; two quarters ahead
(c) Consumer price index; four quarters ahead
Figure 3.10: CPI monthly
Conclusion: The graphical representations 3.14 and 3.13 shows clearly that the the
data of the CPI with h = 4 are correlated with period of recession, we can notice that the
reversal of the curves of the CPI forecasting the arrival of crisis.
19

24. 3.3 Consumer Price Index
(a) One quarter ahead (b) Two quarters ahead
(c) Four quarters ahead
Figure 3.11: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Con-
sumer price index
20

25. 3.4 Crude Oil Price
3.4 Crude Oil Price
We applied again the probit model on the crude oil price at different horizons and obtained
the following results:
21

26. 3.4 Crude Oil Price
(a) Crude oil price; one quarter ahead (b) Crude oil price; two quarters ahead
(c) Crude oil price; four quarters ahead
Figure 3.12: Statistical Distribution of crude oil price
Conclusion: We reject the null hypothesis at horizon 1 which has a p-value of 0, 0879
with a critical value at 0, 05 and at h = 2, 4, we do not reject the null hypothesis with
p-value of 0, 2233 and 0, 6176 respectively.
22

27. 3.4 Crude Oil Price
(a) Crude oil price; one quarter ahead (b) Crude oil price; two quarters ahead
(c) Crude oil price; four quarters ahead
Figure 3.13: Crude oil monthly price
Conclusion: The graphs 3.14 and 3.13 show clearly that there is a signal effect before
the recession with the oil price and this increase as h increases. Consequently, we deduce
that the price of oil is a good indicator because it refers to one which is easily affected by
the change of financial markets.
23

28. 3.4 Crude Oil Price
(a) One quarter ahead (b) Two quarters ahead
(c) Four quarters ahead
Figure 3.14: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Crude
oil price
24

29. 3.5 Unemployment Rate
3.5 Unemployment Rate
So we applied again the probit model on the unemployment rate at different horizons and
obtained the following results:
25

30. 3.5 Unemployment Rate
(a) Unemployment rate; one quarter ahead (b) Unemployment rate; two quarters ahead
(c) Unemployment rate; four quarters ahead
Figure 3.15: Statistical Distribution of unemployment rate
Conclusion: We can see that whatever the horizon h, there will be correlation that is
the p-value will be large, thus it refers to a suitable indicator.
26

31. 3.5 Unemployment Rate
(a) Unemployment rate; one quarter ahead (b) Unemployment rate; two quarters ahead
(c) Unemployment rate; four quarters ahead
Figure 3.16: Unemployment rate monthly
Conclusion: However, through the graphical representations 3.17 and 3.16, the corre-
lation is extremely high at h = 1. But the fact that the rate of unemployment is a good
indicator of long-term and that it can be the results of crisis, this can call into question its
predictive power of crisis.
27

32. 3.5 Unemployment Rate
(a) One quarter ahead (b) Two quarters ahead
(c) Two quarters ahead
Figure 3.17: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the unem-
ployment rate
28

33. Chapter 4
Comparisons of the forecasted
probability of recession of different
indicators
Now we will combine some indicators together in order to see the difference between those
indicators chosen during recession periods.
29

34. 4.1 Comparison of Spread & Consumer Price Index
4.1 Comparison of Spread & Consumer Price Index
Figure 4.1: Graph of Forecasted probability of recessions of Spread and CPI
Conclusion: The figure 4.1 shows that the forecasting of the two indicators do vary in a
similar case but the only thing that is different is their degree of amplitude. Indeed, the
spread’s variation is less important in relation of the consumer price index. Moreover, the
spread has become more reactive than the CPI, that is, it is always in advance according to
the crisis than the CPI.
30

35. 4.2 Comparison of Spread & Crude Oil Price
4.2 Comparison of Spread & Crude Oil Price
Figure 4.2: Graph of Forecasted probability of recessions of Spread and Crude oil price
Conclusion: This graphical representation 4.2 demonstrates that the two curves are rela-
tively different. This means that it does not vary when the economic conditions is good but
when a recession approaches, there is convergence of the variations which we can defined
them as good indicators.
31

36. 4.3 Comparison of Spread & S&P500
4.3 Comparison of Spread & S&P500
Figure 4.3: Graph of Forecasted probability of recessions of Spread and S&P500
Conclusion: This graph 4.3 is clearly similar to that of the spread and CPI where the sen-
sibility of the S&P500 index seems to be larger than the spread. Indeed, there are higher
peaks which can be observed through the S&P500 curve than the spread. This means that
it is more volatile and thus conclude that both indicators give better forecasting at the same
times.
32

37. 4.4 Comparison of Spread & Unemployment Rate
4.4 Comparison of Spread & Unemployment Rate
Figure 4.4: Graph of Forecasted probability of recessions of Spread and Unemployment
rate
Conclusion: Here the forecasting seems to be non significant compared to the spread
which is more significant. But we can suppose that it refers to a long-term indicators and
hat it does not forecast crisis efficiently.
33

38. Chapter 5
Conclusion
After our research study on the yield curve as a leading indicator based on the work of
(Estrella & Mishkin 1996) with additional variables, we can notice that better indicators do
exist for the forecasting of the arrival of periods of recessions either in a partial way or an
efficient one.
In fact, the spread is considered to be the variable that represents the very high capacity
to forecast growth at a horizon of 12 months. On the other hand, other indicators as the
S&P500, the price of crude oil have a fundamental impact on short-term and indicators
such as consumer price index and the unemployment rate are more long-term. According
to the important economic condition and their different features, we have obtained results
which satisfied the forecasting at different horizons chosen that is at h = 1, 2, 4. However,
it seems that the forecasting cannot be done immediately, it would need four quarters
for the overall indicators to be able to measure appropriately the crisis. Nevertheless, it
also seems that their predictive power is not the same for all of them. In relation to their
different sensibility, we can conclude that the choice of indicator is fundamental in order to
anticipate future crisis and even though to avoid repeating the same history, thus to reduce
the effect when anticipating the arrival of it.
So, the question to ask is whether the financial variables are useful to anticipate the
economic growth? (L. 2010).
34

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35