Prévisions des crises

The Yield Curve as a Predictor of U.S. RecessionsBased on (Estrella & Mishkin 1996) Work

Lin Jibin and Verny Tania

Universite Paris 1 Pantheon Sorbonne

Dissertation submitted to MOSEF, Faculty of Economics, Universite Paris 1

Pantheon Sorbonne, as a partial fulfilment of the requirement for the Master 1 of

Economie Quantitative.

May 2015

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

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

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

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

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

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

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 + β′

ixt + εt

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, εt 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

The form of the estimated equation is given as follows:

P (Recessiont+h = 1) = F (β′

ixt),

where F refers to the cumulative normal distribution function corresponding to −ε.

The probit model is being estimated by the maximum likelihood which is defined by

the following:

L = (ΠRecessiont+h=1)F (β′

ixt)(ΠRecessiont+h=0)(1 − F (β′

ixt))

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

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

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

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

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 quartersahead

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

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

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 quartersahead

Figure 3.4: Treasury Spread: Ten-Year Bond Rate minus Three-Month Bill Rate MonthlyAverage

11

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 TreasurySpread

12

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

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

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

3.2 S&P500 Index



Figure 3.8: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the S&P500index

16

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


(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


(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




Figure 3.11: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Con-sumer price index

20

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

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

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

3.4 Crude Oil Price



Figure 3.14: Probability of U.S. Recession at h = 1, 2, 4 ahead, as Predicted by the Crudeoil price

24

3.5 Unemployment Rate


So we applied again the probit model on the unemployment rate at different horizons and

obtained the following results:

25


(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


(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



(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

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

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

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

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

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

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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of New York, mimeo .

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Journal of Finance pp. 555–576.

Estrella & Mishkin (1996), ‘The yield curve as a predictor of u.s. recessions’, Federal Reserve

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35

Economy & Finance

Prévisions des crises