**3. Research methodology**

problems associated with the use of the cross country data as well, for example, ignoring the impact from the time. To avoid these problems, many researches employ the panel data. This data structure has both the time and the section dimensions; therefore, the empirical result can capture the difference among sample countries and the dynamic changes from time to time. In addition, the two dimensional characteristic also increases the observation numbers, which

In the present chapter, the empirical model is modified from the dynamic panel data model (DPDM) of [9]. For the DPDM, if one uses the traditional fixed effect method to estimate the model, it may lead to biased estimation results, because of the correlation between the lagged explained variables and the residual, a problem not addressed in [9]. One way to conquer this problem is to estimate the DPDM with the generalized method of moment (GMM) estimation

The empirical results of this study show that in the recession periods, the stock return significantly impacts the economic growth. In addition, in some of the Asian emerging markets, the stock return is the leading indicator of the economic growth only in the recovery subperiod. As to the developed countries, the stock return is the leading indicator of the economic growth only in the depression subperiod. The empirical results have the following implications and contributions. First, for the international companies, the results can be used to avoid the misunderstandings of the recession process and the relationship between the stock return and economic growth. Second, for the investment organizations and the financial professionals, the results can help them better understand the business cycle and teach the investors the true meaning of the CDR. Third, this study contributes to the filed by further

This chapter is organized as follows. Section 1 is the introduction. Section 2 discusses and analyzes the data. The research methodology is listed in Section 3. Section 4 shows the

The data set of this chapter contains quarterly data of stock index, Gross Domestic Product, and consumer price index (CPI) (or Gross Domestic Product deflator) of 26 countries, including the G7 countries (The USA, UK, Canada, France, German, Italy, and Japan), five Asian countries (the Philippines, Singapore, Hong Kong, Korea, and Taiwan), 12 OECD countries (Australia, Austria Belgium, Denmark, Finland, Mexico, the Netherlands, New Zealand, Norway, Spain, Sweden, and Switzerland), Israel, and South Africa. In addition, we divided the sample countries into three groups, one can see the impact of the development level on the relationship between the stock return and economic growth. Group A has five Asian countries; group B has the G7 countries; group C is contains the 12 OECD members. All the data come from the International Financial Statistics (IFS) database of International Monetary Fund (IMF) and the AREMOS database. The longest sample period is from the first quarter of 1982 to the fourth

quarter of 2009. Please refer to Appendix A for detailed descriptions of the data.

investigate the recession periods with well‐established research methods.

enhances the degree of freedom.

340 Nonlinear Systems - Design, Analysis, Estimation and Control

proposed by Arellano and Bond [20].

empirical results. Section 5 is the conclusion.

**2. Data analysis**

Beaudry and Koop [14] propose the CDR indicator to capture the stages of business cycle:

$$CRR\_{i,t} = \max\left\{ Y\_{i,t-s} \right\}\_{s \ge 0}^{\prime} - Y\_{i,t} \ge 0 \tag{1}$$

where *Yi,t* is the output of period *t*. Eq. (1) tells that the CDR is the output difference between the largest output of period *t* − *s* and the output of period *t*. 5 Although the CDR is very sensitive at capturing the recession period, the indicator can only differentiate between the expansionary and recession periods.

Bradley and Jansen [12] argue that the recession period captured by CDR > 0 is a mixed, rather than a pure, recession period; therefore, the authors proposed a new CDR, the NCDR, to capture the pure recession period. Utilizing the output growth rate (Δ*Yt* ), the NCDR divides the mixed recession period into two subperiods, the depression period where the economy is approaching the trough and the recovery period where the economy is approaching the next peak. Bradley and Jansen [12] name the NCDR established the CDR1 > 0 for the depression subperiod and the CDR2 > 0 for the recovery subperiod. The CDR1 and CDR2 indicators are specified as follows:

$$\text{CDRI}\_{i,t} = \max \left\{ Y\_{i,t-s} \right\}\_{i \ge 0}^{\prime} - Y\_{i,t} \ge 0 \quad \text{if } \Delta Y\_{i,t} < 0,\tag{2a}$$

$$CDR\mathcal{Z}\_{i,t} = \max\left\{ Y\_{i,t-s} \right\}\_{s\geq 0}^{\prime} - Y\_{i,t} \geq 0 \quad \text{if}\ \Delta Y\_{i,t} \geq 0. \tag{2b}$$

In the case that *CDR*1 = 0 and *CDR*2 = 0, it indicates that the economy is in the expansionary period, same as the case that CDR = 0 in [14]. Because the NCDR has the benefits of capturing the pure recession period, in the empirical model of this chapter, the NCDR is employed to divide business cycle stages.

<sup>5</sup> If the CDR > 0, that mean the economy is entering the recession period. With the same rationale, if a country is in the expansionary period, then the CDR = 0.

Using the *CDR* as the switched factor to identify the business cycle periods, Henry et al. [9] construct the single‐variate nonlinear panel data model:

$$\mathbf{r}\mathbf{y}\_{i,t} = \begin{cases} \mathbf{z}\_{i} + \sum\_{j=1}^{4} \mathcal{B}\_{1,j} \mathbf{r} \mathbf{y}\_{i,t-i} + \sum\_{j=1}^{4} \pi\_{1,j} \mathbf{r} \mathbf{s}\_{i,t-j} + \mathcal{E}\_{\mathbf{i}t,t} & \text{CDF}\_{i,t-1} = \mathbf{0} \\\\ (\mathbf{z}\_{i} + \boldsymbol{\phi}) + \sum\_{j=1}^{4} \mathcal{B}\_{2,j} \mathbf{r} \mathbf{y}\_{i,t-j} + \sum\_{j=1}^{4} \pi\_{2,j} \mathbf{r} \mathbf{s}\_{i,t-j} + \mathcal{E}\_{2i,t} & \text{CDF}\_{i,t-1} > \mathbf{0} \end{cases}, \tag{3}$$

where , denotes the economic growth rate; , is the stock return; *αi* , *ϕ, βkj*, and *δkj* are 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 the recovery subperiod, one can revise the model of Henry et al. [9] as

$$\mathbf{y}\_{i,i} = \begin{cases} \alpha\_i + \sum\_{j=1}^4 \beta\_{1,j} \mathbf{y}\_{i,i-1} + \sum\_{j=1}^4 \pi\_{1,j} \mathbf{r}\_{i,i-j} + \mathfrak{e}\_{i1,i} & \text{if } CDR1\_{i,i-2} = CDR2\_{i,i-2} = 0 \\\\ (\alpha\_i + \phi\_1) + \sum\_{j=1}^4 \beta\_{2,j} \mathbf{r}\_{j,i-j} + \begin{cases} (\sum\_{j=1}^4 \delta\_{1,j} \mathbf{r}\_{i,i-j}) \times DV1 & \text{if } CDR1\_{i,i-2} > 0 \\\\ (\sum\_{j=1}^4 \delta\_{2,j} \mathbf{r}\_{i,i-j}) \times DV2 & \text{if } CDR2\_{i,i-2} > 0 \end{cases} \end{cases} \tag{4}$$

where , denotes the economic growth rate; , denotes the stock return; *αi* , *ϕ, βkj*, and *δkj* 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, otherwise. *K* = 1~2. Eq. (4) is the primary empirical model of this chapter.

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

To estimate the nonlinear DPDM, it will be too complicated if one considers the nonlinearity and the dynamic panel data characteristic at the same time in the estimation.6 A better way to 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

<sup>6</sup> 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 avoided.

GMM estimation, named AB‐GMM afterward, is utilized to deal with the "dynamic" panel data characteristic that is caused by the inclusion of the lagged dependent variable in the regressors. When one estimates Eq. (4) with AB‐GMM estimation, since the variables will be first differenced, the constant term will disappear.
