**4. Empirical results and analyses**

Using the *CDR* as the switched factor to identify the business cycle periods, Henry et al. [9]

1, , 1, , 1 , , 1

*ry rs CDR*

( ) 0

coefficients; *εkit* is the error term. *k* = 1, 2. Substituting the CDR of Eq. (3) with the NCDR specified in Eqs. (2a) and (2b) and defining Δ, < 0 the depression subperiod and Δ, ≥ 0

1, , 1, , 1 , , 2 , 2


*ry rs if CDR CDR*

 e

1, , , 2 <sup>4</sup> <sup>1</sup>

2, , , 2

*j it j i t*

( )2 2 0

*rs DV if CDR*

, ( )1 1 0

*j it j i t*

e



*i j it j j it j it it*

<sup>ï</sup> ++ + + > <sup>ï</sup>

 p

*i j it i j it j it i t*

+++ = ï

 p

2, , 2, , 2 , , 1


 e

 e


*ry rs CDR*

0 ,

1 20

(3)

(4)

, *ϕ, βkj*, and *δkj*

A better way to

, *ϕ, βkj*, and *δkj* are

construct the single‐variate nonlinear panel data model:

ab

af

*i t*

ì

342 Nonlinear Systems - Design, Analysis, Estimation and Control

ï = í

î

*ry*

,

ì

ï ï

*i t*

6

avoided.

( )

af

ab

4 4

å å

1 1 , 4 4

= =

*j j*

 b

1 1

= =

where , denotes the economic growth rate; , is the stock return; *αi*

4

å

*j i j it j i t*

=

*ry rs DV if CDR*

1

=

where , denotes the economic growth rate; , denotes the stock return; *αi*

otherwise. *K* = 1~2. Eq. (4) is the primary empirical model of this chapter.

subperiods from the recession period, rather than specifying three regimes.

and the dynamic panel data characteristic at the same time in the estimation.6

å

*j*

2, , 2 , <sup>4</sup> <sup>1</sup>

<sup>ï</sup> ´ <sup>&</sup>gt; <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> ++ + <sup>í</sup> <sup>+</sup> <sup>ï</sup> <sup>ï</sup> <sup>ï</sup> ´ <sup>&</sup>gt; <sup>ï</sup> <sup>î</sup> <sup>î</sup>

are the coefficients; *εkit* is the error term; *CDR*1 and *CDR*2 are the switched factors; DV*k*( ) is the dummy variable, where DV*k* = 1 if the condition inside the parenthesis holds, DV*k* = 0,

The readers might be curious why not to construct a three‐regime DPDM. The primary reason is that a three‐regime DPDM would lead to many differences from the model of [9] to compare the empirical findings. In addition, the three‐regime model costs lots of degrees of freedom. Because of these two reasons, what is done here is to derive the depression and recovery

To estimate the nonlinear DPDM, it will be too complicated if one considers the nonlinearity

do is to use the two‐step method to estimate the nonlinear DPDM. First, the exogenously given switched factors (the CDR or NCDR) are employed to divide the regimes; in the meantime, the dummy variable can reveal the nonlinear relationship between the variables. Second, the

 In this paper, the model is a linear panel data model with the characteristic of nonlinearity given by the dummy variables. Therefore, the GMM estimation still can be used to estimate the DPDM and the heteroskedastic residual problem can be

d

d

*i j it i j it j it i t i t*

+ + + == ï

the recovery subperiod, one can revise the model of Henry et al. [9] as

 p

4 4

å å

1 1

= =

*ry*


*j j*

*j*

<sup>ï</sup> <sup>=</sup> <sup>í</sup> <sup>ì</sup>

=

å

 b å å

*j j*

To avoid the spurious regression problem, one should examine whether the panel data are stationary by conducting the unit root test.7 The result of the panel data unit root test is shown in **Table 1**, shows that the variables are stationary.


Notes: The "*ry*" is the economic growth rate, and "*rs*" is the stock return. The above five types of panel unit root tests: Levin et al. [22], Breitung [23], Im et al. [24], Fisher‐type tests using ADF and PP tests (Maddala and Wu [25] and Choi [26])."\*", "\*\*", and "\*\*\*" denote the 1% significant level.

**Table 1.** Panel unit‐root test.

At this moment, the CDR (or NCDR) is employed as the switched factor in the nonlinear model to divide between the expansionary and recession periods. The switched factor has been chosen and the switched point has been determined. The lagged period is 4,8 same as the setting of [9]. The only thing that is adjustable here is the delay period. We find the 1 and 2 periods delay CDR are well, which indicates that it is appropriate to use the delay CDR as the switched factor. In this chapter, the delay period is set to 2 to meet the 5% significant level condition. Because of this, if an economy has a sequence of negative economic growth rates in two seasons, then the economy could be viewed as entering the recession period.9

<sup>7</sup> For the spurious regression problem, please refer to Ref. [21].

<sup>8</sup> There is reason to choose four lagged periods. Because the data are quarterly data, so four lagged periods could cover the whole year and capture the seasonal characteristics.

<sup>9</sup> The way to identify whether an economy is entering a recession period is to see whether GDP is decreasing in sequentially two seasons. If it is, then this economy is said to experience recession.


Notes: The "*ry*" is the economic growth rate, and "*rs*" is the stock return; *i* indicates the country and *t* the time. "CDR" is abbreviated from current depth of recession that proposed by [14]. The "Obs" is observation number. "\*", "\*\*", and "\*\*\*" denote the 10%, 5%, and 1% significant levels.

**Table 2.** Dynamic panel data OLS estimation with CDR switched factor.

After all the parameters have been decided, one can proceed to estimate Eq. (3). For compar‐ ison, both the OLS and AB‐GMM estimations are performed and the results are listed in **Tables 2** and **3**, respectively. For the coefficient joint test result, in the expansionary period, the stock return cannot explain the economic growth in **Table 3**, which is conflict with the result in **Table 2**. In the recession period, the stock return can significantly explain the economic growth in both **Tables 2** and **3**. Please note that the estimation result in **Table 3** is more consistent with what are found in the literatures (including [9]). In addition, the result in **Table 3** is obtained with the AB‐GMM estimation, which can avoid the biased estimation caused by the OLS estimation.10

<sup>10</sup> When there are lagged dependent variables in the regressor, the model becomes the DPDM. If one still estimates the model with the OLS estimation, then there would be biased estimation results.


Notes: When one estimates Eq. (3) with AB‐GMM estimation, since the variables will be first differenced, the constant term will disappear. The "*ry*" is the economic growth rate, and "*rs*" is the stock return; *i* indicates the country and *t* the time. "CDR" is abbreviated from current depth of recession that proposed by [14]. The "Obs" is observation number. Under the null hypothesis that the over‐identifying restrictions are valid, the reported *J*‐statistic is simply the Sargan

statistic, 2 , where *k* is the number of estimate coefficients and *p* is the instrument rank. "\*", "\*\*", and "\*\*\*" denote the 5%, and 1% significant levels.

**Table 3.** Dynamic panel data AB‐GMM estimation with CDR switched factor.

**Whole Sample (Expansionary period) (Recession period) CDRi,t-2=0 CDRi,t-2>0 Variable Coefficient p-value Variable Coefficient p-value**

**<sup>A</sup> 0.04 (0.87)** + **2.37\*\*\* (0.00)**

, <sup>1</sup> ‐0.92\*\*\* (0.00) , <sup>1</sup> ‐0.88\*\*\* (0.00)

, <sup>2</sup> 0.02 (0.67) , <sup>2</sup> ‐0.17\*\* (0.02)

, <sup>3</sup> 0.63\*\*\* (0.00) , <sup>3</sup> ‐0.23\*\* (0.02)

, <sup>4</sup> ‐0.25\*\* (0.03) , <sup>4</sup> 0.03 (0.44)

, <sup>1</sup> 0.04\*\* (0.04) , <sup>1</sup> 0.06\*\* (0.03)

, <sup>2</sup> 0.01 (0.61) , <sup>2</sup> 0.11\*\* (0.05)

, <sup>3</sup> ‐0.01 (0.61) , <sup>3</sup> 0.05\*\* (0.05)

, <sup>4</sup> 0.01 (0.41) , <sup>4</sup> ‐0.09 (0.23)

Wald Test 0: 11 = 12 = 13 = 14 = 0 0: 21 = 22 = 23 = 24 = 0

Notes: The "*ry*" is the economic growth rate, and "*rs*" is the stock return; *i* indicates the country and *t* the time. "CDR" is abbreviated from current depth of recession that proposed by [14]. The "Obs" is observation number. "\*", "\*\*", and

After all the parameters have been decided, one can proceed to estimate Eq. (3). For compar‐ ison, both the OLS and AB‐GMM estimations are performed and the results are listed in **Tables 2** and **3**, respectively. For the coefficient joint test result, in the expansionary period, the stock return cannot explain the economic growth in **Table 3**, which is conflict with the result in **Table 2**. In the recession period, the stock return can significantly explain the economic growth in both **Tables 2** and **3**. Please note that the estimation result in **Table 3** is more consistent with what are found in the literatures (including [9]). In addition, the result in **Table 3** is obtained with the AB‐GMM estimation, which can avoid the biased estimation caused by the OLS

10 When there are lagged dependent variables in the regressor, the model becomes the DPDM. If one still estimates the

= 1

2 <sup>=</sup>0.13

4

1 <sup>=</sup>0.05 ∑

Chi‐square 8.39\* Chi‐square 13.38\*\*\* p‐value (0.08) p‐value (0.01)

**Table 2.** Dynamic panel data OLS estimation with CDR switched factor.

model with the OLS estimation, then there would be biased estimation results.

**Obs. 1245 Obs. 1449**

344 Nonlinear Systems - Design, Analysis, Estimation and Control

∑ = 1

"\*\*\*" denote the 10%, 5%, and 1% significant levels.

estimation.10

4

The AB‐GMM method is used to estimate Eq. (4). **Table 4** reports the full sample estimation result. In the recession period, in the depression or the recovery subperiods, the coefficient joint test result is significant, which indicates there is no difference between the two subperiods and that the stock return can significantly explain the economic growth in two subperiods.


Note: The NCDR proposed by Bradley and Jansen [12] is employed as the switched factor in the model. "\*" and "\*\*\*" denote the 10% and 1% significant levels.

**Table 4.** Dynamic panel data AB‐GMM estimation with NCDR switched factor.

In the following, the estimation method of **Table 4** is repeated on the estimation of groups A to C. The estimation result of group A to C is combined listed in **Table 5** (divide into three part), group A in the upper, group B in the middle, and group C in the lower.

**whole sample(Expansionary period) (Recession period)**

Variable Coefficient p‐value Variable Coefficient p‐value , <sup>1</sup> ‐0.87\*\*\* (0.00) , <sup>1</sup> ‐1.37\*\*\* (0.00) , <sup>2</sup> 0.30\*\*\* (0.00) , <sup>2</sup> 0.78\*\* (0.02) , <sup>3</sup> 1.18\*\*\* (0.00) , <sup>3</sup> 0.37 (0.29) , <sup>4</sup> 0.14 (0.32) , <sup>4</sup> 0.28\* (0.06)

, <sup>1</sup> 0.02 (0.14) , <sup>1</sup> \* 0.38\* (0.07) , <sup>2</sup> 0.02 (0.12) , <sup>2</sup> \* 0.07 (0.52) , <sup>3</sup> 0.00 (0.97) , <sup>3</sup> \* 0.16 (0.20) , <sup>4</sup> ‐0.01 (0.50) , <sup>4</sup> \* 0.28\*\* (0.02)

1 = 0.03 ∑

Chi‐square 4.89 Chi‐square 10.27\*\* p‐value (0.30) p‐value (0.04)

104 Instrument

J‐statistic 110.41 J‐statistic 117.85 p‐value (0.15) p‐value (0.11)

**Table 4.** Dynamic panel data AB‐GMM estimation with NCDR switched factor.

"\*" and "\*\*\*" denote the 10% and 1% significant levels.

Wald Test 0: 11 = 12 = 13 = 14 = 0 0: 11 = 12 = 13 = 14 = 0

rank

Note: The NCDR proposed by Bradley and Jansen [12] is employed as the switched factor in the model.

**Obs. 1231 Obs. 1389**

346 Nonlinear Systems - Design, Analysis, Estimation and Control

∑ = 1

Instrument rank

4

**CDR1i,t-2=0 CDR2i,t-2=0 CDR1i,t-2>0 CDR2i,t-2>0**

1 <sup>2</sup> > 0)

2 <sup>2</sup> > 0)

1 = 0.89 ∑

0: 21 = 22 = 23 = 24 = 0

= 1

2 <sup>=</sup> 0.03

4

, <sup>1</sup> \*20.00 (0.99) , <sup>2</sup> \* 0.26\* (0.09) <sup>3</sup> \* 0.05 (0.81) , <sup>4</sup> \* ‐0.34\*\* (0.03)

= 1

Chi‐square 11.19\*\* p‐value (0.02)

112

4


Note: Table 5 just show the estimation results of three groups partly. "\*" and "\*\*\*" denote the 10%, and 1% significant levels.

**Table 5.** Dynamic panel data AB‐GMM estimation with NCDR switched factor.

From **Table 5** (upper part), one can see that the coefficients are significantly positive only in the recovery subperiod. This tells that for group A (five Asian emerging countries), the stock return can explain the economic growth only in the recovery period. **Table 5** (middle part) shows that the coefficients are significantly positive only in the depression subperiod, which indicates that for group B (the G7 countries), the stock markets will go down before the economies start to grow.

The economic rationale behind this is as follows. Since the stock return and economic growth are positively correlated in the G7 countries, when they enter the depression subperiod, the stock market would go down to reveal the upcoming depressions. When the G7 countries enter the recovery subperiod, their production and consumption will increase. At this time, the G7 countries will place many orders on the Asian emerging markets, and this will help the Asian emerging markets grow and their firms perform well. These outcomes will be reflected by the stock markets in these emerging markets; with more foreign investments from the developed countries, these stock markets will stay in the bull status for a long time. This is why the stock market can significantly explain the economic growth in the recovery subperiod. Although both the Asian emerging markets and the G7 countries are all in the recovery subperiod, the economies will not grow as fast. Moreover, because developed countries tend to invest in high return foreign stock markets, there is no significant relationship between the stock return and economic growth in the G7 countries in the recovery subperiod.

**Table 5** (lower part) reports the estimation result for group C (12 OECD members), the coefficients are significantly positive only in the depression subperiod, same as the result in **Table 5** (middle part). In addition, in the expansionary period, the stock return can significantly explain the economic growth, which is different from the results of other subperiod estima‐ tions. The results in **Table 5** (middle and lower parts) can be used to derive the results of **Table 4** that the stock return can explain the economic growth in both the depression and recovery subperiods.

The results of **Tables 5** are not quite the same as the result of **Table 4** (the whole sample estimation), which indicates that one cannot apply the conclusion of **Table 4** to every case. Some of the effects may be "cancelled out" by pooling all the countries into one sample.

The empirical findings of this chapter could benefit the corporations, financial companies, as well as regular investors. The contribution of this chapter can be summarized as follows. First, the empirical findings could avoid corporations from misunderstanding the recession period. In the recession period, the government tends to reduce the interest rate and enhance the government spending to stimulate the economy. When making future operation and finance decisions, if the decision maker can seize the chance to adjust the factory size or to raise corporate debts in the recession period or to finance in the expansionary period, it would be beneficial to the corporation. Second, the findings help financial companies or financial supervisors better understand the business cycle. Macroeconomic analyses and business cycles are crucial factors to investment decisions to maximize capital gains and to minimize risks from market fluctuations. Third, the findings help regular investors better understand the business cycle. The business cycle information can help regular investors with medium or long term investment decisions and avoid capital loss from short term fluctuations in the market. Fourth, the empirical findings prove that there does exist useful information in the recession period.
