# Capital Markets Integration and Cointegration: Testing for the Correct Specification of Stock Market Indices

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Testing for Common Trends in an Integrated Framework

#### 2.1. Treating the Variables as I(1)

_{t}, a stationary component and the initial values.

#### 2.1.1. Scenario I: One Cointegrating Vector, Four Common Stochastic Trends

#### 2.1.2. Scenario II: Two Cointegrating Vectors, Three Common Stochastic Trends

#### 2.2. Treating Prices as I(2)

#### 2.2.1. Scenario I: One Cointegrating Vector, Three I(1) and One I(2) Common Stochastic Trends

#### 2.2.2. Scenario II: Two Cointegrating Vectors, Two I(1) and One I(2) Common Stochastic Trends

## 3. Econometric Methodology

## 4. Empirical Evidence

#### Determination of the Cointegration Rank and the Order of Integration

_{1}= 0, s

_{2}= 5} we tested successively less and less restricted hypotheses according to the Pantula (1989) principle. The last column of Table 4 reports the standard Johansen trace test, ${Q}_{r}$. Therefore, the first hypothesis that we were unable to reject was {r = 1, s

_{1}= 3, s

_{2}= 1} for the U.S./U.K., U.S./Germany and U.S./Japan cases, which implies that there are I(2) components. However, when we also apply a small sample adjustment the first hypothesis that we were unable to reject was {r = 1, s

_{1}= 4, s

_{2}= 0} which means that there is one linear cointegrating relation and four I(1) common trends in the multivariate framework.9

^{,}10

_{1}= 4, s

_{2}= 0} while for the US/Germany case there is a disagreement. The formal test indicates two cointegrating vectors and three unit roots, while the companion matrix is also consistent with one cointegrating vector and four unit roots. The overall evidence leads us to concentrate on the I(1) case under which the preferred model is {r = 1, s

_{1}= 4, s

_{2}= 0}.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | This argument follows from the evidence linking cointegration and error correction that leads to the predictability of at least one of the asset prices. Those results apply strictly however on total returns (i.e., cum dividend) or otherwise on non-interest/dividend paying assets (Richards 1995). The efficiency hypothesis can still be preserved, in the presence of cointegration, if the error correction is a proxy for a risk premium. |

2 | The general tendency is to draw a distinction between tests for capital integration and those for the existence of common trends. The important implication of integrated capital markets is the equalization among countries of marginal rates of substitution in consumption both inter-temporally and across states of nature (Lucas 1982; Kasa 1995). But even then the existence of a common model of required returns is not sufficient to generate cointegrating relationships. The cumulative stochastic errors, if they are found to be I(1), of each one of the assets return series must be cointegrated as well (Richards 1995). |

3 | In this paper we focus on results derived from the multivariate cointegration analysis of Johansen (1988). However, the first strand of papers examining cointegration between stock prices employed the Engle and Granger (1987) methodology (see e.g., Arshanapalli and Doukas 1993; Chan et al. 1992). |

4 | Serletis and King (1997) have transformed their data into “real deutschemark” units. In other studies, the stock indices are expressed in local, nominal, currency terms (e.g., Calvi 2010). |

5 | In this case it might be more realistic to restrict the impact on the U.S. price index and have the exchange rate absorb all of the shock (i.e., ${c}_{51}=0,{c}_{21}={c}_{31}$). As a matter of fact, the common trends in (2) are over-identified and up to three restrictions can be imposed without changing the likelihood function. |

6 | Since the two prices indices are excluded from the cointegrating vector. Therefore, deviations from PPP have no impact, in the long-run, on the behavior of the two equity market indices. |

7 | In general if ${z}_{t}~I(2)$ then the unrestricted linear regressor, ${\mu}_{1}t$, allows for cubic trends while the constant regressor, ${\mu}_{0}$, allows for quadratic ones. Rahbek et al. (1999) show that to guarantee linear trends in all linear combinations of ${z}_{t}$ we must impose restrictions on both ${\mu}_{1}$ and ${\mu}_{0}$. Finally, Rahbek et al. (1999) provide a likelihood ratio (LR) test, which is asymptotically ${\chi}^{2}(r)$ distributed, to examine whether the linear trend significantly enters the cointegrating vector. |

8 | Gonzalo (1994) shows that the performance of the maximum likelihood estimator of the cointegrating vectors is little affected by non-normal errors. Lee and Tse (1996) have shown similar results when conditional heteroskedasticity is present. |

9 | The calculations of all tests as well as the estimation of the eigenvectors have been performed using the program CATS 2.0 in RATS 9.0 developed by Katarina Juselius and Henrik Hansen, Estima Inc. Illinois, 1995. |

10 | |

11 | It should be noted however that the falling number, over time, of the underlying stochastic trends governing the system can be attributed to the convergence of the trace statistics to their long-run values as the sample expands (i.e., the power of the test increases). Garcia Pascual (2003) presented evidence on estimates of the trace statistic obtained from constant sample sizes that are being rolled over each time to the next period. It is shown there that the trace statistics do not present any upward trend and this is interpreted as running against the proposition that an increased number of cointegrating vectors among the stock price indices is an indicator of an increasing integration among the capital markets (Rangvid 2001). |

**Figure 1.**Trace Test 1 is the 5% significance level. (

**a**) U.K.–U.S. case. (

**b**). Germany–U.S. case. (

**c**) Japan–U.S. case.

**Figure 2.**The Test of Constancy of Beta 1 is the 5% significance level. (

**a**) U.K.–U.S. case. (

**b**) Germany–U.S. (

**c**) Japan–U.S.

**Figure 3.**The Eigenvalue Test U.K.–U.S. case. (

**a**) Test for lambda 1 and for lambda 2 Germany–U.S. case. (

**b**) Test for lambda 1 and for lambda 2 Japan–U.S. case. (

**c**) Test for lambda 1 and for lambda 2.

Direction | Dimension | Stationary Process |
---|---|---|

${\beta}_{0}^{{}^{\prime}}{z}_{t}$ ~ I(0) | $r-{s}_{2}$, if $(r>{s}_{2})$ | |

${\beta}_{1}^{{}^{\prime}}{z}_{t}$ ~ I(1) | ${s}_{2}$ | ${\beta}_{1}^{{}^{\prime}}{z}_{t}+k{}^{\prime}\mathsf{\Delta}{z}_{t}$ ~ I(0) |

${\beta}_{\perp 1}^{{}^{\prime}}{z}_{t}$ ~ I(1) | ${s}_{1}$ | ${\beta}_{\perp 1}^{{}^{\prime}}\mathsf{\Delta}{z}_{t}$ ~ I(0) |

${\beta}_{\perp 2}^{{}^{\prime}}{z}_{t}{}_{t}$ ~ I(2) | ${s}_{2}$ | ${\beta}_{\perp 2}^{{}^{\prime}}{\mathsf{\Delta}}^{2}{z}_{t}{}_{t}$ ~ I(0) |

USA | UK | GER | JAP | |
---|---|---|---|---|

USA | 1.0 | |||

UK | 0.750 | 1.0 | ||

GER | 0.705 | 0.709 | 1.0 | |

JAP | 0.422 | 0.445 | 0.419 | 1.0 |

U.S.–U.K. | |||||||
---|---|---|---|---|---|---|---|

Eq. | ${\mathit{\sigma}}_{\mathit{\epsilon}}$ | LB(36) | ARCH(4) | ${\mathit{\eta}}_{3}$ | ${\mathit{\eta}}_{4}$ | NORM(4) | ${\mathit{R}}^{2}$ |

$\mathsf{\Delta}p$ | 0.006 | 28.14 | 9.71 | 0.39 | 1.19 | 100.71 * | 0.703 |

$\mathsf{\Delta}i$ | 0.054 | 23.12 | 6.43 | −0.68 | 4.19 | 86.44 * | 0.363 |

$\mathsf{\Delta}e$ | 0.039 | 34.09 | 1.59 | −0.05 | 0.88 | 9.59 * | 0.285 |

$\mathsf{\Delta}{p}^{f}$ | 0.007 | 24.81 | 2.94 | −0.28 | 1.76 | 102.25 * | 0.603 |

$\mathsf{\Delta}{i}^{f}$ | 0.045 | 23.65 | 5.22 | −0.71 | 2.15 | 73.61 * | 0.362 |

U.S.–Germany | |||||||

Eq. | ${\mathit{\sigma}}_{\mathit{\epsilon}}$ | LB(36) | ARCH(4) | ${\mathit{\eta}}_{\mathbf{3}}$ | ${\mathit{\eta}}_{\mathbf{4}}$ | NORM(4) | ${R}^{2}$ |

$\mathsf{\Delta}p$ | 0.004 | 31.28 | 3.51 | −0.78 | 6.78 | 59.39 * | 0.446 |

$\mathsf{\Delta}i$ | 0.024 | 14.31 | 3.37 | −0.76 | 5.87 | 55.79 * | 0.365 |

$\mathsf{\Delta}e$ | 0.010 | 29.44 | 5.15 | −0.08 | 3.56 | 7.04 | 0.402 |

$\mathsf{\Delta}{p}^{f}$ | 0.003 | 31.16 | 8.15 | −1.36 | 1.55 | 102.21 * | 0.667 |

$\mathsf{\Delta}{i}^{f}$ | 0.018 | 30.14 | 10.38 | −0.97 | 5.01 | 73.33 * | 0.209 |

U.S.–Japan | |||||||

Eq. | ${\mathit{\sigma}}_{\mathit{\epsilon}}$ | LB(36) | ARCH(4) | ${\mathit{\eta}}_{\mathbf{3}}$ | ${\mathit{\eta}}_{\mathbf{4}}$ | NORM(4) | ${R}^{2}$ |

$\mathsf{\Delta}p$ | 0.003 | 22.39 | 8.62 | −0.35 | 0.01 | 19.45 * | 0.553 |

$\mathsf{\Delta}i$ | 0.023 | 31.35 | 11.39 | −0.37 | 0.08 | 18.97 | 0.288 |

$\mathsf{\Delta}e$ | 0.011 | 28.16 | 7.32 | −0.31 | 0.33 | 8.62 * | 0.346 |

$\mathsf{\Delta}{p}^{f}$ | 0.003 | 32.11 | 6.73 | −1.37 | 0.01 | 105.22 * | 0.591 |

$\mathsf{\Delta}{i}^{f}$ | 0.018 | 24.12 | 10.07 | −1.02 | 0.05 | 73.02 * | 0.371 |

Testing the Joint Hypothesis $\mathit{H}({\mathit{s}}_{1}\cap \mathit{r})$ U.S.–U.K | |||||||
---|---|---|---|---|---|---|---|

p-r | r | $Q({s}_{1}\cap r/{H}_{0})$ | ${Q}_{r}$ | ||||

5 | 0 | 901.2 206.1 | 691.0 174.3 | 486.9 146.4 | 346.8 123.1 | 216.5 103.8 | 126.7 88.6 |

4 | 1 | 483.6 141.5 | 302.8 115.8 | 166.5 94.2 | 73.4 76.8 | 59.8 63.7 | |

3 | 2 | 164.7 89.0 | 68.6 69.4 | 50.9 53.9 | 37.2 42.7 | ||

2 | 3 | 49.7 48.5 | 29.7 34.98 | 20.8 25.7 | |||

1 | 4 | 18.5 20.0 | 8.28 12.44 | ||||

s_{2} | 5 | 4 | 3 | 2 | 1 | 0 | |

U.S.–Germany | |||||||

p-r | r | $Q({s}_{1}\cap r/{H}_{0})$ | ${Q}_{r}$ | ||||

5 | 0 | 867.1 206.1 | 603.1 174.3 | 493.1 146.4 | 327.2 123.1 | 229.6 103.8 | 112.4 88.6 |

4 | 1 | 444.1 141.5 | 299.6 115.8 | 172.3 94.2 | 64.3 76.8 | 64.1 63.7 | |

3 | 2 | 171.5 89.0 | 66.4 69.4 | 51.7 53.9 | 38.7 42.7 | ||

2 | 3 | 41.2 48.5 | 28.3 34.98 | 17.8 25.7 | |||

1 | 4 | 15.3 20.0 | 6.59 12.44 | ||||

s_{2} | 5 | 4 | 3 | 2 | 1 | 0 | |

U.S.–Japan | |||||||

p-r | r | $Q({s}_{1}\cap r/{H}_{0})$ | ${Q}_{r}$ | ||||

5 | 0 | 802.4 206.1 | 598.9 174.3 | 452.2 146.4 | 319.7 123.1 | 208.8 103.8 | 107.7 88.6 |

4 | 1 | 426.7 141.5 | 287.1 115.8 | 170.6 94.2 | 68.1 76.8 | 63.3 63.7 | |

3 | 2 | 167.1 89.0 | 68.6 69.4 | 35.0 53.9 | 27.9 42.7 | ||

2 | 3 | 34.7 48.5 | 25.2 34.98 | 13.2 25.7 | |||

1 | 4 | 12.4 20.0 | 4.16 12.44 | ||||

s_{2} | 5 | 4 | 3 | 2 | 1 | 0 |

**Table 5.**Statistical Properties and Misspecification Tests of the Model. (a) Tests for long-run exclusion, stationarity, and weak exogeneity.

(a) Tests for Long-Run Exclusion, Stationarity, and Weak Exogeneity | |||||||||
---|---|---|---|---|---|---|---|---|---|

Long-Run Exclusion | Stationarity | Weak Exogeneity | |||||||

US/UK US/GE | US/JP | US/UK | US/GE US/JP | US/UK | US/GE | US/JP | |||

$p$ | 3.39 * | 15.32 * | * | 20.60 * | 22.47 * | 11.71 * | 2.77 | 4.69 * | 0.62 |

$i$ | 0.97 * | 11.61 * | 6.63 * | 14.77 * | 23.01 * | 24.11 * | 9.08 * | 4.43 * | 14.78 * |

$e$ | 5.84 * | 0.05 * | 14.02 * | 40.83 * | 25.88 * | 15.52 * | 3.30 | 7.27 * | 1.52 |

${p}^{f}$ | 1.08 * | 16.73 * | 7.22 * | 11.76 * | 18.50 * | 12.72 * | 13.33 * | 9.43 * | * |

${i}^{f}$ | 2.47 * | 0.16 * | 9.29 * | 25.16 * | 19.525 * | 14.73 * | 1.67 | 3.49 | 8.72 * |

**Table 6.**Statistical Properties and Misspecification Tests of the Model. (b) Multivariate Residuals Diagnostics.

(b) Multivariate Residuals Diagnostics | ||||
---|---|---|---|---|

Case | L-B(117) | LM(1) | LM(4) | ${\mathit{\chi}}^{2}\text{}\left(10\right)$ |

U.S–U.K | 1960.46(0.00) | 34.39(0.69) | 40.92(0.26) | 1023.49(0.00) |

U.S.–Germany | 1784.33(0.00) | 36.95(0.41) | 38.12(0.31) | 490.69(0.00) |

U.S.–Japan | 1000.49(0.00) | 35.19(0.62) | 29.84(0.46) | 351.53(0.00) |

**Table 7.**Tests of overidentified restrictions on ${z}_{t}=\left[{p}_{t},{i}_{t},{e}_{t},{p}_{t}^{f},{i}_{t}^{f},cons\mathrm{tan}t\right]$. Scenario I: one cointegrating vector, four I(1) common stochastic trends. Case 1(a).

U.S.–U.K. | |
---|---|

${\beta}^{{}^{\prime}}=\left[1,1,\text{}-1,\text{}-1,\text{}-1,\text{}0.002\right],$ | Q(5) = 53.91(0.0000) |

${\beta}^{{}^{\prime}}=\left[0,\text{}1,\text{}-1,\text{}0,\text{}-1,\text{}-0.009\text{}\right]$ | Q(5) = 54.31(0.0000) |

${\beta}^{{}^{\prime}}=\left[\begin{array}{cccccc}-1,\text{}1,\text{}& 0& & 1& -1& 0.05\end{array}\right]$ | Q(5) = 9.51 (0.0642) |

${\beta}^{{}^{\prime}}=\left[0,\text{}1,\text{}0,\text{}0,\text{}-1,\text{}0.08\text{}\right]$ | Q(5) = 8.46 (0.1752) |

U.S.–Germany | |

${\beta}^{{}^{\prime}}=\left[1,1,\text{}-1,\text{}-1,\text{}-1,\text{}0.010\right],$ | Q(5) = 29.83 (0.0000) |

${\beta}^{{}^{\prime}}=\left[0,\text{}1,\text{}-1,\text{}0,\text{}-1,\text{}-0.015\right]$ | Q(5) = 33.35 (0.0000) |

${\beta}^{{}^{\prime}}=\left[\begin{array}{cccccc}-1,\text{}1,\text{}& 0& & 1& -1& 0.056\end{array}\right]$ | Q(5) = 7.31 (0.0941) |

${\beta}^{{}^{\prime}}=\left[0,\text{}1,\text{}0,\text{}0,\text{}-1,\text{}0.03\right]$ | Q(5) = 9.02 (0.0833) |

U.S.–Japan | |

${\beta}^{{}^{\prime}}=\left[1,1,\text{}-1,\text{}-1,\text{}-1,\text{}0.012\right],$ | Q(5) = 44.18(0.0000) |

${\beta}^{{}^{\prime}}=\left[0,\text{}1,\text{}-1,\text{}0,\text{}-1,\text{}-0.022\right]$ | Q(5) = 41.29(0.0000) |

${\beta}^{{}^{\prime}}=\left[\begin{array}{cccccc}-1,\text{}1,\text{}& 0& & 1& -1& 0.058\end{array}\right]$ | Q(5) = 6.38 (0.1348) |

${\beta}^{{}^{\prime}}=\left[0,\text{}1,\text{}0,\text{}0,\text{}-1,\text{}0.011\text{}\right]$ | Q(5) = 10.28 (0.057) |

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**MDPI and ACS Style**

Agoraki, M.-E.K.; Georgoutsos, D.A.; Kouretas, G.P. Capital Markets Integration and Cointegration: Testing for the Correct Specification of Stock Market Indices. *J. Risk Financial Manag.* **2019**, *12*, 186.
https://doi.org/10.3390/jrfm12040186

**AMA Style**

Agoraki M-EK, Georgoutsos DA, Kouretas GP. Capital Markets Integration and Cointegration: Testing for the Correct Specification of Stock Market Indices. *Journal of Risk and Financial Management*. 2019; 12(4):186.
https://doi.org/10.3390/jrfm12040186

**Chicago/Turabian Style**

Agoraki, Maria-Eleni K., Dimitris A. Georgoutsos, and Georgios P. Kouretas. 2019. "Capital Markets Integration and Cointegration: Testing for the Correct Specification of Stock Market Indices" *Journal of Risk and Financial Management* 12, no. 4: 186.
https://doi.org/10.3390/jrfm12040186