Product outsourcing policy for a sustainable flexible manufacturing system with reworking and green investment

Raj Kumar Bachar; Shaktipada Bhuniya; Santanu Kumar Ghosh; Ali AlArjani; Elawady Attia; Md. Sharif Uddin; Biswajit Sarkar; Raj Kumar Bachar; Shaktipada Bhuniya; Santanu Kumar Ghosh; Ali AlArjani; Elawady Attia; Md. Sharif Uddin; Biswajit Sarkar

doi:10.3934/mbe.2023062

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 1: 1376-1401. doi: 10.3934/mbe.2023062

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Research article Special Issues

Product outsourcing policy for a sustainable flexible manufacturing system with reworking and green investment

1.
Department of Mathematics, Kazi Nazrul University, Asansol 713340, West Bengal, India
2.
Department of Mathematics & Statistics, Banasthali Vidyapith, Rajasthan 304022, India
3.
Industrial Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
4.
Mechanical Engineering Department, Faculty of Engineering (Shoubra), Benha University, Cairo, Egypt
5.
Department of Mechanical and Industrial Engineering, College of Engineering, Prince Sattam Bin Abdulaziz University, Al-kharj 11942, Saudi Arabia
6.
Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh
7.
Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul 03722, South Korea
8.
Center for Transdisciplinary Research (CFTR), Saveetha Institute of Medical and Technical Sciences, Saveetha University, Velappanchavadi, Tamil Nadu 600077, India

Academic Editor: Jeffrey Yi-Lin Forrest

Received: 16 July 2022 Revised: 19 September 2022 Accepted: 23 September 2022 Published: 28 October 2022

Production of defective products is a very general phenomenon. But backorder and shortages occur due to this defective product, and it hampers the manufacturer's reputation along with customer satisfaction. That is why, these outsourced products supply, a portion of required products for in-line production. This study develops a flexible production model that reworks repairable defective products and outsources products to prevent backlogging. A percentage of total in-line production is defective products, which is random, and those defective products are repairable. A green investment helps the reworking process, which has a direct impact on the market demand for products. A classical optimization solves the profit maximization model, and a numerical method proves the global optimal solutions. Sensitivity analysis, managerial insights, and discussions provide the highlights and decision-making strategies for the applicability of this model.

Keywords:

Citation: Raj Kumar Bachar, Shaktipada Bhuniya, Santanu Kumar Ghosh, Ali AlArjani, Elawady Attia, Md. Sharif Uddin, Biswajit Sarkar. Product outsourcing policy for a sustainable flexible manufacturing system with reworking and green investment[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 1376-1401. doi: 10.3934/mbe.2023062

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Abstract

1. Introduction

Since introduction of the measure of policy uncertainty by Baker et al. (2016), the impact of policy uncertainty on macro variables have gained momentum. While Baker et al. (2016) demonstrated its adverse effects on economic activity, Wang et al. (2014) assessed the response of corporate investment to the new uncertainty measure. On the other hand, Pastor and Veronesi (2013), Ko and Lee (2015), and Brogaard and Detzel (2015) investigated response of risk premium and market returns to uncertainty. Other macro variables which are said to be affected by policy uncertainty include oil prices which was subject of an analysis by Kang and Ratti (2013) as well as Bahmani-Oskooee et al. (2018a). Additionally, while Bahmani-Oskooee et al. (2016) assessed the impact of policy uncertainty on the demand for money in the U.S., Bahmani-Oskooee and Ghodsi (2017) investigated the response of house prices in each state of the U.S.

The new policy uncertainty measure is now constructed on a monthly basis by the Policy Uncertainty Group for countries other than the U.S.¹ It is an index based on the volume of news associated with any uncertainty. Recently, Bahmani-Oskooee and Saha (2019) assessed the short- run and long-run effects of the policy uncertainty measure on stock returns in Canada, Japan, Korea, U.S., and U.K. These were the five countries for which a multivariate model could be estimated since monthly data were available for all the variables for these countries. Their main finding was that in all countries, increased uncertainty has adverse short-run effects but not long-run effects on stock prices. The main assumption included in Bahmani-Oskooee and Saha's (2019) study as well as other studies is that they have assumed that the effects of policy uncertainty on macro variables are symmetric. The symmetry assumption implies that if an x% increase in uncertainty hurts a variable by y%, an x% decrease in uncertainty will boost that variable by y%. However, this may not be the case since investors' reaction could be different to an increase in policy uncertainty as compared to a decline in uncertainty. If due to an x% increase in uncertainty investors shift their assets from stock to safer assets by y%, an x% decline in uncertainty may induce them to shift their portfolio into stock market by less than y%, if they expect decreased uncertainty to be short-lived, hence asymmetric effects. Similar arguments were recently made by Bahmani-Oskooee and Maki-Nayeri (2019) who assessed the asymmetric effects of policy uncertainty on domestic investment.

¹ For more details on constructing this index visit: www.policyuncertainty.com.

Therefore, the main purpose of this paper is to extend Bahmani-Oskooee and Saha's (2019) study from symmetric analysis to an asymmetric analysis of policy uncertainty on stock prices for the same five countries. To that end, we introduce the model and the methodology in Section Ⅱ that is followed by the results in Section Ⅲ. A summary is provided in Section Ⅳ and data definition and sources are cited in an Appendix.

2. The models and methodology

The model adopted here closely follows the model by Bahmani-Oskooee and Saha (2019), who followed the literature and used the following specification:²

² The same model is used by Boonyanam (2014) and Moore and Wang (2014) with no policy uncertainty measure. Other variants of the model without policy uncertainty measure is also used by Granger et al. (2000), Ansari and Kolari (2001), Nieh and Lee (2001), Phylaktis and Ravazzolo (2005), Yau and Nieh (2006), Pan et al. (2007), Richards et al. (2009), Liu and Tu (2011), Tsai (2012), Lin (2012), Tsagkanos et al. (2013), Groenewold et al. (2013), and Bahmani-Oskooee and Saha (2016).

L n S P_{t} = a + b L n E X_{t} + c L n I P I_{t} + d L n C P I_{t} + e L n M_{t} + f L n P U_{t} + ε_{t}

$L n S P_{t} = a+b L n E X_{t}+c L n I P I_{t}+d L n C P I_{t}+e L n M_{t}+f L n P U_{t}+\varepsilon_{t}$

(1)

where SP denotes the stock prices, EX is the nominal effective exchange rate, IPI is a measure of output proxied by Index of Industrial Production (IPI), CPI is the Consumer Price Index as a measure of the price level, M is a measure of nominal money supply, and finally PU is the measure of policy uncertainty.

Specification (1) is a long run model and coefficient estimates are long-run estimates. Once the long run estimates are obtained, an estimate of b is expected to be negative or positive depending upon if firms associated with the specific stock are export or import oriented. Since an increase in economic activity measured by IPI is expected to increase the demand for stocks, an estimate of c is expected to be positive. An estimate of d is also expected to be negative or positive depending on whether inflation hurts firm costs and profit margin or if stocks are considered a hedge against inflation in the long run.³ An estimate of e is also expected to be positive or negative, depending whether increase in money supply leads to lower interest rates and economic growth or inflation. Finally, if increase in uncertainty is to hurt the markets, we expect an estimate of f to be negative.

³ See Fama (1981) for the negative effects and Anari and Kolari (2001) for the positive effects.

As mentioned above, specification (1) is a long-run model and in order to assess the short-run effects of right-hand side variables, we must incorporate the short-run dynamics. Bahmani-Oskooee and Saha (2019) followed Pesaran et al.'s (2001) ARDL (Auto Regressive Distributed Lag) bounds testing approach and considered the following error-correction model:

\begin{array}{l} Δ L n S P_{t} = α + \sum_{i = 1}^{n 1} β_{i} Δ L n S P_{t - i} + \sum_{i = 0}^{n 2} δ_{i} Δ L n E X_{t - i} + \sum_{k = 0}^{n 3} φ_{i} Δ L n I P I_{t - i} + \sum_{i = 0}^{n 4} θ_{i} Δ L n C P I_{t - i} \\ + \sum_{i = 0}^{n 5} π_{i} Δ L n M_{t - i} + \sum_{i = 0}^{n 6} ω_{i} Δ L n P U_{t - i} + λ_{1} \ln S P_{t - 1} + λ_{2} L n E X_{t - 1} + \\ + λ_{3} L n I P I_{t - 1} + λ_{4} L n C P I_{t - 1} + λ_{5} L n M_{t - 1} + λ_{6} L n P U_{t - 1} + μ_{t} \end{array}

$\begin{array}{l} \Delta LnS{P_t} = \alpha + \sum\limits_{i = 1}^{n1} {{\beta _i}} \Delta LnS{P_{t - i}} + \sum\limits_{i = 0}^{n2} {{\delta _i}} \Delta LnE{X_{t - i}} + \sum\limits_{k = 0}^{n3} {{\varphi _i}} \Delta LnIP{I_{t - i}} + \sum\limits_{i = 0}^{n4} {{\theta _i}} \Delta LnCP{I_{t - i}}\\ + \sum\limits_{i = 0}^{n5} {{\pi _i}} \Delta Ln{M_{t - i}} + \sum\limits_{i = 0}^{n6} {{\omega _i}} \Delta LnP{U_{t - i}} + {\lambda _1}\ln S{P_{t - 1}} + {\lambda _2}LnE{X_{t - 1}} + \\ + {\lambda _3}LnIP{I_{t - 1}} + {\lambda _4}LnCP{I_{t - 1}} + {\lambda _5}Ln{M_{t - 1}} + {\lambda _6}LnP{U_{t - 1}} + {\mu _t} \end{array}$

(2)

In (2), the short-run effects of each variable is judged by the coefficient estimates attached to the first-differenced variables and the long-run effects by the estimates of λ₂–λ₆ normalized on λ_1.⁴ However, for the long-run estimates to be meaningful, cointegration must be established. Pesaran et al. (2001) propose two tests: the F test to establish joint significance of lagged level variables in (2) and the t-test to establish significance of λ₁ which must be negative.⁵ They demonstrate that the distribution of both tests are non-standard, hence they tabulate new critical values which also account for degree of integration of all the variables in a given model. Their upper bound critical values should be used to establish cointegration. These values are valid, even if some variables are I(0) and some I(1) and this is one of the main advantage of this approach since almost all macro variables are either I(0) or I(1).⁶

⁴ Note that once normalization is done we will have $\frac{{{{\hat \lambda }_2}}}{{ - {{\hat \lambda }_1}}} = \hat b, \frac{{{{\hat \lambda }_3}}}{{ - {{\hat \lambda }_1}}} = \hat c, \frac{{{{\hat \lambda }_4}}}{{ - {{\hat \lambda }_1}}} = \hat d, \frac{{{{\hat \lambda }_5}}}{{ - {{\hat \lambda }_1}}} = \hat e, {\rm{ }}and\; \; {\rm{ }}\frac{{{{\hat \lambda }_6}}}{{ - {{\hat \lambda }_1}}} = \hat f.$

⁵ Note that the estimate of λ₁ in (2) is the same as estimate of coefficient attached to lagged error-correction term in Engle and Granger (1987) error-correction specification. For empirical demonstration of this, see Bahmani-Oskooee (2019) and Bahmani-Oskooee and Ghodsi (2019).

⁶ Another advantage of this approach is that since short-run adjustment process is included in estimating the long-run effects, any feedback effects among variables are allowed to exert themselves which help reduce multicolinearity or endogeniety (Pesaran et al. 2001).

Shin et al. (2014) modified Pesaran et al.'s (2001) approach so that it could be used to assess asymmetric effects of any variable of concern. Since our concern is to assess asymmetric effects of policy uncertainty measure, PU, we take the following steps. First, we form ΔLnPU which includes positive as well as negative changes. Second, we separate increases in uncertainty from declines and form two new time-series variable as follows:

P O S_{t} = \sum_{j = 1}^{t} max (Δ L n P U_{j}, 0), N E G_{t} = \sum_{j = 1}^{t} min (Δ L n P U_{j}, 0)

$PO{S_t} = \sum\limits_{j = 1}^t {\max } \left({\Delta LnP{U_j}, 0} \right), \quad NE{G_t} = \sum\limits_{j = 1}^t {\min } \left({\Delta LnP{U_j}, 0} \right)$

(3)

Where POS_t is the partial sum of the positive changes and reflects only increased uncertainty and NEG_t is the partial sum of negative changes in uncertainty and reflects only the decline in uncertainty. Next, we move back to the error-correction model (2) and replace the LnPU variable by the two partial sum variables to arrive at:

\begin{array}{l} Δ L n S P_{t} = α + \sum_{i = 1}^{n 1} β_{i} Δ L n S P_{t - i} + \sum_{i = 0}^{n 2} δ_{i} Δ L n E X_{t - i} + \sum_{i = 0}^{n 3} φ_{i} Δ L n I P I_{t - i} + \sum_{i = 0}^{n 4} θ_{i} Δ L n C P I_{t - i} \\ + \sum_{i = 0}^{n 5} π_{i} Δ L n M_{t - i} + \sum_{i = 0}^{n 6} ω_{i}^{+} Δ P O S_{t - i} + \sum_{i = 0}^{n 7} ω_{i}^{-} Δ N E G_{t - i} + λ_{1} L n S P_{t - 1} + λ_{2} \ln E X_{t - 1} + \\ + λ_{3} L n I P I_{t - 1} + λ_{4} L n C P I_{t - 1} + λ_{5} L n M_{t - 1} + λ_{6}^{+} P O S_{t - 1} + λ_{6}^{-} N E G_{t - 1} + μ_{t} \end{array}

$\begin{array}{l} \Delta LnS{P_t} = \alpha + \sum\limits_{i = 1}^{n1} {{\beta _i}} \Delta LnS{P_{t - i}} + \sum\limits_{i = 0}^{n2} {{\delta _i}} \Delta LnE{X_{t - i}} + \sum\limits_{i = 0}^{n3} {{\varphi _i}} \Delta LnIP{I_{t - i}} + \sum\limits_{i = 0}^{n4} {{\theta _i}} \Delta LnCP{I_{t - i}}\\ + \sum\limits_{i = 0}^{n5} {{\pi _i}} \Delta Ln{M_{t - i}} + \sum\limits_{i = 0}^{n6} {\omega _i^ + } \Delta PO{S_{t - i}} + \sum\limits_{i = 0}^{n7} {\omega _i^ - } \Delta NE{G_{t - i}} + {\lambda _1}LnS{P_{t - 1}} + {\lambda _2}\ln E{X_{t - 1}} + \\ + {\lambda _3}LnIP{I_{t - 1}} + {\lambda _4}LnCP{I_{t - 1}} + {\lambda _5}Ln{M_{t - 1}} + \lambda _6^ + PO{S_{t - 1}} + \lambda _6^ - NE{G_{t - 1}} + {\mu _t} \end{array}$

(4)

Since the construction of the partial sum variables introduces nonlinear adjustment of policy uncertainty variable into our specification, Shin et al. (2014) refer models such as (4) as nonlinear ARDL model whereas, (2) is commonly referred to as the linear ARDL model. However, both models are subject to the same estimation, interpretation, and diagnostic tests.⁷ Once (4) is estimated, a few asymmetric assumptions could be tested. First, short-run effects of policy uncertainty will be asymmetric if at any given lag order, i, $\hat \omega _i^ + \ne \hat \omega _i^ -$ . However, if the Wald-test rejects the null hypothesis of $\sum {\hat \omega _i^ + } = \sum {\hat \omega _i^ - }$ , short-run cumulative or impact asymmetry will be established. Second, short-run adjustment process will be asymmetric if ΔPOS and ΔNEG take a different lag order, i.e., if n6 ≠ n7. Finally, long-run effects of policy uncertainty on stock prices will be asymmetric if we reject the null hypothesis of $\frac{{\hat \lambda _6^ + }}{{ - \hat \lambda _1^{}}} = \frac{{\hat \lambda _6^ - }}{{ - {{\hat \lambda }_1}}}$ . Again, the Wald test will be used for this purpose.⁸

⁷ Shin et al. (2014) even argue that the critical value of the F test should stay at the same high level when we move from (2) to (4) even if (4) has one more variable.

⁸ For some other application of these and other nonlinear models see Apergis and Miller (2006), Wimanda (2014), McFarlane et al. (2014), Gogas and Pragidis (2015), Durmaz (2015), Baghestani and Kherfi (2015), Al-Shayeb and Hatemi-J (2016), Lima et al. (2016), Nusair (2017), Aftab et al. (2017), Arize et al. (2017), Gregoriou (2017), Bahmani-Oskooee and Kanitpong (2018), Olaniyi (2019), and Istiak and Alam (2019).

3. The results

Since the linear model (2) has already been estimated by Bahmani-Oskooee and Saha (2019) for Canada, Japan, Korea, U.K., and the U.S. using monthly data, we restrict ourselves to estimating only the nonlinear model (4) and compare our findings to theirs. We have updated the data and used monthly data over the period January 1985–October 2018. Following their approach we impose a maximum of eight lags on each first-differenced variable and use Akaike Information Criterion to select the optimum number of lags. In any case, if there was evidence of serial correlation, following Bahmani-Oskooee and Fariditavana (2019) we added additional lags of dependent variable to reduce autocorrelation. Using required critical values reported in the notes to each table, we identify significant statistics by * at the 10% level and ** at the 5% level. The results are reported in five tables, one for each country.

From the short-run estimates in Panel A of each table, we gather that either ΔPOS or ΔNEG variable carry at least one significant coefficient in every optimum model, implying that changes in policy uncertainty has short-run effects on stock prices in all the five countries. This was also the case in the linear model. However, the new results reveal that except in the case of Japan, short-run effects are asymmetric since at the same lag order, the estimate attached to the ΔPOS variable is different than the one attached to the ΔNEG variable. Thus, in the four remaining countries, short-run effects of changes in policy uncertainty are asymmetric. However, sum of the coefficients attached to the ΔPOS variable is different than the sum attached to the ΔNEG variable only in the results for the U.S., since the Wald test reported as Wald-S in Panel C of each table is significant only in Table 5 for the U.S. In sum, while the short-run effects of policy uncertainty are asymmetric in four countries, evidence of cumulative or impact asymmetric effects is discovered only in one country (the U.S.).

Table 1. Full information estimate of the nonlinear ARDL model for Canada.

Panel A: short-run coefficient estimates
Lag order	0	1	2	3	4	5	6	7	8
ΔLnSP
ΔLnEX	0.50
ΔLnEX	(3.65)**
ΔLnIPI	0.21	0.47
ΔLnIPI	(0.93)	(2.17)**
ΔLnCPI	0.76	0.03	-0.67	-0.31	-1.56
ΔLnCPI	(1.30)	(0.06)	(1.15)	(0.53)	(2.72)**
ΔLnM	0.001
	(0.93)
ΔPOS	-0.06
ΔPOS	(5.30)**
ΔNEG	-0.007	-0.02	0.03
ΔNEG	(0.56)	(1.39)	(2.18)**
Panel B: long-run coefficient estimates
Constant	LnEX	LnIPI	LnCPI	LnM	POS	NEG
12.94	0.41	0.09	-1.84	0.01	-0.18	-0.25
(0.83)	(1.34)	(0.12)	(1.86)*	(0.03)	(2.27)**	(3.08)**
Panel C: Diagnostics
F	t-test	LM	RESET	${{\bar R}^2}$	CUSUM (CUSUMQ)	Wald-L	Wald-S
2.57	-0.08	0.02	12.15**	0.19	S (S)	3.77*	1.33
	(4.25)**
Notes: a. Numbers inside the parentheses are absolute value of t-ratios. , * indicate significance at the 10% and 5% levels respectively. b. The upper bound critical value of the F-test for cointegration when there are five exogenous variables is 3.35 (3.79) at the 10% (5%) level of significance. These come from Pesaran et al. (2001, Table CI, Case Ⅲ, p. 300). c. The critical value for significance of ECM_t-1 is -4.04 (-4.38) at the 10% (5%) level when k = 6. These come from Pesaran et al. (2001, Table CII, Case Ⅲ, p. 303). d. LM is the Lagrange Multiplier statistic to test for autocorrelation. It is distributed as χ² with 1 degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. e. RESET is Ramsey's test for misspecification. It is distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. f. Both Wald tests are also distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level.

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Table 2. Full information estimate of the nonlinear ARDL model for Japan.

Panel A: short-run coefficient estimates
Lag order	0	1	2	3	4	5	6	7	8
ΔLnSP
ΔLnEX	-0.01
ΔLnEX	(0.39)
ΔLnIPI	0.04	0.36	-0.32
ΔLnIPI	(0.22)	(2.17)**	(1.88)*
ΔLnCPI	-0.65
ΔLnCPI	(2.62)**
ΔLnM	-2.71
	(1.73)*
ΔPOS	-0.002
ΔPOS	(2.01)**
ΔNEG	-0.002
ΔNEG	(2.47)**
Panel B: long-run coefficient estimates
Constant	LnEX	LnIPI	LnCPI	LnM	POS	NEG
66.48	-0.25	-1.00	-12.34	0.17	-0.03	-0.03
(0.92)	(0.41)	(0.63)	(1.98)**	(0.06)	(1.94)*	(2.79)**
Panel C: Diagnostics
F	t-test	LM	RESET	${{\bar R}^2}$	CUSUM (CUSUMQ)	Wald-L	Wald-S
2.98	-0.05	1.08	0.00	0.07	S (S)	0.34	1.93
	(4.48)**
Notes: a. Numbers inside the parentheses are absolute value of t-ratios. , * indicate significance at the 10% and 5% levels respectively. b. The upper bound critical value of the F-test for cointegration when there are five exogenous variables is 3.35 (3.79) at the 10% (5%) level of significance. These come from Pesaran et al. (2001, Table CI, Case Ⅲ, p. 300). c. The critical value for significance of ECM_t-1 is -4.04 (-4.38) at the 10% (5%) level when k = 6. These come from Pesaran et al. (2001, Table CII, Case Ⅲ, p. 303). d. LM is the Lagrange Multiplier statistic to test for autocorrelation. It is distributed as χ² with 1 degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. e. RESET is Ramsey's test for misspecification. It is distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. f. Both Wald tests are also distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level.

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Table 3. Full information estimate of the nonlinear ARDL model for Korea.

Panel A: short-run coefficient estimates
Lag order	0	1	2	3	4	5	6	7	8
ΔLnSP	-	0.05	-0.10	0.15
ΔLnSP	(0.71)	(1.47)	(2.30)**
ΔLnEX	-0.42	-0.20	-0.06	-0.08	0.40
ΔLnEX	(1.94)*	(0.85)	(0.27)	(0.36)	(2.48)**
ΔLnIPI	0.42	0.41
ΔLnIPI	(1.95)**	(1.96)**
ΔLnCPI	-4.12	0.31	-1.07	-2.85	-0.88	-4.58	0.74	-3.39
ΔLnCPI	(3.24)**	(0.24)	(0.82)	(2.24)**	(0.70)	(3.87)**	(0.62)	(2.98)**
ΔLnM	-0.07
	(0.38)
ΔPOS	-0.04
ΔPOS	(2.05)**
ΔNEG	-0.0003
ΔNEG	(0.13)
Panel B: long-run coefficient estimates
Constant	LnEX	LnIPI	LnCPI	LnM	POS	NEG
25.32	0.36	-1.27	1.23	-0.61	0.06	-0.002
(0.54)	(0.73)	(1.36)	(0.34)	(0.38)	(2.47)**	(0.13)
Panel C: Diagnostics
F	t-test	LM	RESET	${{\bar R}^2}$	CUSUM (CUSUMQ)	Wald-L	Wald-S
6.84**	-0.13	2.36	0.24	0.31	S (U)	1.92	1.02
	(7.00)**
Notes: a. Numbers inside the parentheses are absolute value of t-ratios. , * indicate significance at the 10% and 5% levels respectively. b. The upper bound critical value of the F-test for cointegration when there are five exogenous variables is 3.35 (3.79) at the 10% (5%) level of significance. These come from Pesaran et al. (2001, Table CI, Case Ⅲ, p. 300). c. The critical value for significance of ECM_t-1 is -4.04 (-4.38) at the 10% (5%) level when k = 6. These come from Pesaran et al. (2001, Table CII, Case Ⅲ, p. 303). d. LM is the Lagrange Multiplier statistic to test for autocorrelation. It is distributed as χ² with 1 degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. e. RESET is Ramsey's test for misspecification. It is distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. f. Both Wald tests are also distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level.

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Table 4. Full information estimate of the nonlinear ARDL model for the U.K.

Panel A: short-run coefficient estimates
Lag order	0	1	2	3	4	5	6	7	8
ΔLnSP
ΔLnEX	-1.19
ΔLnEX	(2.58)**
ΔLnIPI	1.99	1.67
ΔLnIPI	(2.91)**	(2.57)**
ΔLnCPI	0.23
ΔLnCPI	(0.35)
ΔLnM	-0.69	-0.67	-1.29	-1.50
	(1.10)	(1.00)	(1.94)*	(2.32)**
ΔPOS	-0.04
ΔPOS	(3.70)**
ΔNEG	-0.01
ΔNEG	(4.36)**
Panel B: long-run coefficient estimates
Constant	LnEX	LnIPI	LnCPI	LnM	POS	NEG
-83.52	-2.55	4.87	1.52	2.60	-0.25	-0.07
(4.10)**	(1.57)	(2.61)**	(0.36)	(3.82)**	(3.17)**	(4.45)**
Panel C: Diagnostics
F	t-test	LM	RESET	${{\bar R}^2}$	CUSUM (CUSUMQ)	Wald-L	Wald-S
3.69*	-0.14	0.31	1.73	0.18	S (S)	3.28*	0.13
	(5.05)**
Notes: a. Numbers inside the parentheses are absolute value of t-ratios. , * indicate significance at the 10% and 5% levels respectively. b. The upper bound critical value of the F-test for cointegration when there are five exogenous variables is 3.35 (3.79) at the 10% (5%) level of significance. These come from Pesaran et al. (2001, Table CI, Case Ⅲ, p. 300). c. The critical value for significance of ECM_t-1 is -4.04 (-4.38) at the 10% (5%) level when k = 6. These come from Pesaran et al. (2001, Table CII, Case Ⅲ, p. 303). d. LM is the Lagrange Multiplier statistic to test for autocorrelation. It is distributed as χ² with 1 degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. e. RESET is Ramsey's test for misspecification. It is distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. f. Both Wald tests are also distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level.

| Show Table

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Table 5. Full Information Estimate of the Nonlinear ARDL Model for U.S.A.

Panel A: short-run coefficient estimates
Lag order	0	1	2	3	4	5	6	7	8
ΔLnSP	-	0.05	0.03	0.12
ΔLnSP	(1.00)	(0.59)	(2.32)**
ΔLnEX	-0.33
ΔLnEX	(2.28)**
ΔLnIPI	-0.93	0.75	0.61
ΔLnIPI	(2.49)**	(2.06)**	(1.61)
ΔLnCPI	0.25
ΔLnCPI	(1.80)*
ΔLnM	-0.79	-0.98
	(1.14)	(1.40)
ΔPOS	-0.04	-0.03	0.005	0.03
ΔPOS	(3.77)**	(2.21)**	(0.39)	(2.45)**
ΔNEG	-0.01	0.004	0.03
ΔNEG	(1.35)	(0.39)	(2.27)**
Panel B: long-run coefficient estimates
Constant	LnEX	LnIPI	LnCPI	LnM	POS	NEG
-23.17	-0.34	2.00	3.29	0.36	-0.05	-0.01
(1.18)	(0.65)	(2.92)**	(1.86)*	(0.63)**	(2.28)**	(0.94)
Panel C: Diagnostics
F	t-test	LM	RESET	${{\bar R}^2}$	CUSUM (CUSUMQ)	Wald-L	Wald-S
2.98	-0.06	0.29	12.03**	0.16	S (S)	0.67	4.83**
	(4.51)**
Notes: a. Numbers inside the parentheses are absolute value of t-ratios. , * indicate significance at the 10% and 5% levels respectively. b. The upper bound critical value of the F-test for cointegration when there are five exogenous variables is 3.35 (3.79) at the 10% (5%) level of significance. These come from Pesaran et al. (2001, Table CI, Case Ⅲ, p. 300). c. The critical value for significance of ECM_t-1 is -4.04 (-4.38) at the 10% (5%) level when k = 6. These come from Pesaran et al. (2001, Table CII, Case Ⅲ, p. 303). d. LM is the Lagrange Multiplier statistic to test for autocorrelation. It is distributed as χ² with 1 degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. e. RESET is Ramsey's test for misspecification. It is distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level. f. Both Wald tests are also distributed as χ² with one degree of freedom. The critical value is 2.70 (3.84) at the 10% (5%) significance level.

| Show Table

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Short-run effects of policy uncertainty last into the long-run significant effects in all countries. In all the countries, either the POS variable or the NEG variable carry a significant coefficient that is reported in Panel B of each table. These long-run effects are meaningful since cointegration is supported in all five countries either by the F or by the t-test, reported in Panel C. This was not the case in Bahmani-Oskooee and Saha (2019), who failed to find significant long-run effects of policy uncertainty on stock returns in any of the five countries, except Canada. These new significant long-run effects must be attributed to the nonlinear adjustment of policy uncertainty. Furthermore, the long-run effects are asymmetric in the results for Canada and the U.K., since the Wald test reported as Wald-L in Panel C is significant in these two cases, rejecting the equality of normalized coefficients attached to the POS and NEG variables.

In each table we have also reported additional diagnostic statistics. To test for serial correlation, the Lagrange Multiplier (LM) statistic is reported in Panel C and since it is insignificant in all models, there is no evidence of serial correlation. Ramsey's RESET test is also reported to check for misspecification. It is insignificant in three models and significant in two models. We have also applied the CUSUM and CUSUMSQ tests to the residuals of each optimum model to establish stability of estimated short-run and long-run coefficients. The stable coefficients are indicated by "S" and the unstable ones by "U". As can be seen, all estimates are stable.

4. Summary and conclusion

In a recent study by the same authors they asked, "are these adverse effects of different types of uncertainty on stock prices transitory or do they have long run implication?". By using the policy uncertainty measure and a linear ARDL model, they concluded that indeed the effects are short run only.

In this paper, when we separated the increases in uncertainty from the declines and estimated a nonlinear multivariate model for the same five countries, we found that, short-run effects of policy uncertainty lasts into the long run which we attribute to the nonlinear adjustment of policy uncertainty. More precisely, we found that policy uncertainty has short-run asymmetric effects in Canada, Korea, U.K., and the U.S. but not in Japan. Short-run effects translated into significant and meaningful long-run effects in all five countries. However, the results are country-specific. While in the cases of Canada, Japan, and the U.K., we found that increased policy uncertainty hurts stock prices, decreased uncertainty boosts them, though at different rates. In the U.S. however, we found that while increased uncertainty hurts stock returns, decreased uncertainty does not have any long-run effects, a clear sign of long-run asymmetric effects. All in all, it appears that reducing uncertainty could benefit stock returns in the short run as well as in the long run.

Acknowledgements

Valuable comments of two anonymous referees are greatly appreciated. Remaining errors, however, are ours.

Conflict of interest

We, hereby, declare that the submitted paper is not associated with any kind of conflict of interest.

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沈阳化工大学材料科学与工程学院沈阳 110142

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Mathematical Biosciences and Engineering

Product outsourcing policy for a sustainable flexible manufacturing system with reworking and green investment

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Abstract

1. Introduction

2. The models and methodology

3. The results

4. Summary and conclusion

Acknowledgements

Conflict of interest

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Abstract

1. Introduction

2. The models and methodology

3. The results

4. Summary and conclusion

Acknowledgements

Conflict of interest

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Mathematical Biosciences and Engineering

Product outsourcing policy for a sustainable flexible manufacturing system with reworking and green investment

Related Papers:

Abstract

1. Introduction

2. The models and methodology

3. The results

4. Summary and conclusion

Acknowledgements

Conflict of interest

References

This article has been cited by:

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通讯作者: 陈斌, bchen63@163.com

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Abstract

1. Introduction

2. The models and methodology

3. The results

4. Summary and conclusion

Acknowledgements

Conflict of interest

References