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This paper investigates the cross-country correlation between stock markets and its implications. It does so by introducing a new measure called the Scaled Covariance Difference ( SCD ), which captures the difference between the covariance of short term returns and longer term returns. This measure has practical implications for portfolio optimization, as well as in testing for the joint efficiency of markets. Our focus in this paper is on including the off-diagonal terms of the variance-covariance matrix in the analysis so as to develop a test for joint market efficiency, unlike the univariate tests for market efficiency which only make use of information along the main diagonal of the variance-covariance matrix. We also demonstrate how to implement the test for joint market efficiency using data on weekly stock returns from the Nifty and S&P 500 indices.

Stock market efficiency is one of the most fundamental topic of research in finance. In a broad sense, market efficiency depends on the degree to which prices of stocks and other securities reflect all available information in the market. Bachelier [

Lo and MacKinley [

The variance ratio test using the L-M statistic put forward by Lo and MacKinley [

The above-mentioned studies analyze the variance-covariance matrix to examine the cross-country correlation over time. The first contribution of our paper is that we examine the correlation and its determinants of short term returns as well as the long term returns (ranging from 1-week return to 8-week returns) of the US and Indian stock markets over the time period under study. Secondly, rather than using the variance-covariance (VCV) matrix, we observe k-week correlation, scaled covariance and V-ratio. More importantly we introduce the measure, Scaled Covariance Difference (SCD) to check if there exists any significant difference between 1-week covariance and k-week covariance. This measure serves beneficial in portfolio optimization as well as to test the joint efficiency of two markets. In this paper we limit our effort on making use of SCD to develop a joint test of market efficiency from an investor’s perspective.

We consider weekly returns from Nifty and S&P500 index for empirical analysis. What we find is that the factors under study (k-week correlation and k-week scaled covariance) appear more or less flat before and after the global financial crisis. However, the level of these factors has changed subsequent to the crisis. The V-ratio remains insignificant for Nifty and S&P500 across all samples. The SCD remains insignificant for all k-week across all samples. This suggests that the information transmission between markets happens within the same week.

The rest of the paper is organized as follows. Section 2 summarizes the Methodology by explaining the theory (2.1), equations and hypotheses (2.2). Section 3 explains empirical analysis. Section 4 presents empirical results. The 5th section concludes.

First we obtain weekly log returns from stock index X and then generate k-week log returns using the formula given below.

S x t ( k ) = ∑ i = 0 k − 1 x t − i , [ 1 ≤ k ≤ 8 ] (1)

For an investor from country A with stock index X, returns from country B with stock index Y_{B} is:

Y = ln ( 1 + R s ) + ln ( 1 + R f x )

where R s = Y B t − Y B ( t − 1 ) Y B ( t − 1 ) and R f x = ( Exchange rate t − Exchange rate t − 1 Exchange rate t − 1 )

We compute k-week log returns using the following equation.

S y t ( k ) = ∑ i = 0 k − 1 y t − i , [ 1 ≤ k ≤ 8 ] (2)

where, x t is the weekly log return (from stock index X), y t is the weekly log return (from stock index Y after taking the fluctuations into account), k is the number of weeks to be summed up (as a moving window). That is to say to obtain longer term returns, we add up x_{t} or y_{t} (1-week returns) over k-week moving window, such as when k =1, the original series is taken as it is and consists of weekly return data. When k = 2, observations 1 & 2, 2 & 3 etc. are added to form a new series of 2-week returns etc.

We compute the k-week correlation, k-week scaled covariance and k-week V-Ratio using the following formulae.

k -week correlation : ρ x , y = Covariance ( x ( k , t ) , y ( k , t ) ) σ x ( k , t ) σ y ( k , t )

k -week Scaled covariance = k -week Covariance ( x t , y t ) k

k -week V-Ratio = V a r ( k -week return ( x t ) ) k ∗ V a r ( 1 -week return ( x t ) )

If the V-Ratio is not significantly different from 1, then the weekly stock prices follow a standard normal distribution asymptotically. That is to say, the stock prices tend to follow a random walk. This acts as an indicator of market efficiency.

We are more interested in exploring the valuable information available in the off-diagonal elements of the variance-covariance matrix. Let Ω(k) denote the variance-covariance matrix of k-week returns of say two series of returns—x and y.

Ω ( k ) = [ σ x ( k , t ) 2 Cov ( x ( k , t ) , y ( k , t ) ) Cov ( x ( k , t ) , y ( k , t ) ) σ y ( k , t ) 2 ] { VCV matrix of k -week returns }

S C D = 1 k Ω ( k ) − Ω (1)

This is a natural generalization of the Variance Ratio (LM statistic) proposed by Lo and MacKinlay [

Test for Joint Market Efficiency: An Illustrative Example.

Two markets are said to be jointly efficient if the stock prices in the two markets follow a two dimensional random walk. The returns from the two markets can be correlated during the same week but not over time.

Assume, X_{t} is iid so that the CUMSUM of X_{t} follows a random walk. Suppose, Y t = X t − 1 ; It follows Y_{t} has the same distribution as X_{t}

Consider the covariance between two-weekly returns of X_{t} and Y_{t}

Cov ( X t + X t − 1 , Y t + Y t − 1 ) = σ 2

Consider covariance between three-weekly returns of X_{t} and Y_{t}

Cov ( X t + X t − 1 + X t − 2 , Y t + Y t − 1 + Y t − 2 ) = 2 σ 2

In general,

Cov ( k -week return of X t , k -week return of Y t ) = ( k − 1 ) σ 2 ^{ }

Scaled Covariance = 1 k ( k − 1 ) σ 2 forall k ≥ 1

S C D = 1 k ( k − 1 ) σ 2 ^{ }

As k becomes large, SCD converges to σ^{2}.

Covariance between one week returns of X_{t} and Y_{t}

Cov ( X t , Y t ) = C o v ( X t , X t − 1 ) = 0

Because X_{t} series is iid.

Therefore, if we look at univariate tests for random walk behavior in asset Y, we probably will find no rejection because the return series is iid. The univariate tests of the random walk hypothesis are not powerful enough to detect this sort of a deviation from joint market efficiency. Our point is that we need to make use of the information in the off-diagonal elements of the variance-covariance matrix to test for joint market efficiency.

If k-week SCD is significant, it means that scaled k-week covariance is significantly different from scaled 1-week covariance. This suggests that there is a lag in information/shock transmission between the markets under consideration. SCD being insignificant indicates that the k-week covariance is not statistically different from 1-week covariance. This means that there is no lag in information or shock transmission between the markets. Therefore SCD serves as an indicator to test the joint efficiency of markets.

There are other practical implications for the SCD other than test for joint efficiency. In portfolio optimization, estimation of the variance-covariance matrix is of paramount importance. Scaled cov-diff helps to identify circumstances when k-week covariance depends on k. Knowing this is important for figuring out what values to use as parameter inputs in mean variance optimization framework. If the scaled k-week covariance is significantly different from 1-week covariance, then the input for mean-variance optimization depends on the value of k. That is to say, the SCD helps to identify the circumstances when k-week covariance depends on k.

We consider Equation (1) and Equation (2) from Section 2.2. We test the following hypotheses

H 1 : ρ x , y = 0 ( H 1 a : ρ x , y ≠ 0 )

H 2 : k -week Scaled Covariance ( x , y ) = 0 ( H 2 a : k -week Scaled Covariance ( x , y ) ≠ 0 )

H 3 : V - R a t i o = 1 ( H 3 a : V-Ratio ≠ 1 )

H 4 : S C D ( x , y ) = 0 ( H 4 a : S C D ( x , y ) ≠ 0 )

We make use of data of weekly closing price of S&P500 and NIFTY index and data on USD-INR from Bloomberg to compute weekly log returns using the formula:

x t = ln ( Closing index level on week t ) − ln ( Closing index level on week t − 1 )

Data from January 1996 to December 2016 is used for the analysis. There is no time-overlap between the indices considered for the study. The data is time aligned i.e. the weekly data is sorted and matched according to the date. The whole sample was further sub-divided into 2 sub-samples to examine the pre-crisis and post-crisis behavior. Sub-sample#1 consists of data of weekly log returns from January 1996 to December 2007. Sub-sample#2 consists of data of weekly log returns from January 2009 to December 2016.

It is important to discuss why weekly data is considered for the analysis. The most crucial reason for conducting this study using weekly data is to avoid biases arising from non-trading days, asynchronous prices etc. Lo and MacKinlay [

In order to test Hypothesis I and II (H_{1}:ρ_{x}_{,y} = 0 & H_{2 }:Scaled Cov_{x,y} =0) we compute weekly log return of S&P500 and Nifty index using the data and generate series of k-week returns using the equation specified in Section 2.2 for sub-sample#1 and sub-sample#2. The null hypothesis for each sub-sample is based on the assumption that the two series of weekly returns are independent of each other. To test this hypothesis we make use of the bootstrapping technique to generate 1000 bootstrap samples of k-week returns of S&P500 (adjusted for ) independently of the 1000 bootstrap samples of k-week returns of the Nifty index. For each of the 1000 bootstrap samples, we then compute the k-week correlation, scaled k-week covariance. We also compute the bootstrap standard error for each k-week, t-statistic with respect to 0 and upper and lower limits of the confidence interval.

To test Hypothesis III (H_{3 }:V-Ratio = 0), we compute the V-Ratio of Nifty and S&P500 during sub-sample#1 and sub-sample#2. Bootstrapping technique is used in order to generate reliable standard errors. The result consists of V-Ratio, Standard error, t-stat with respect to 1 and the upper and lower limits of the confidence interval.

A different bootstrapping technique is employed to test Hypothesis IV (H_{4}: Scaled Cov-diff = 0). In this case, we do not generate bootstrap samples for the series by considering them independent of each other. The weekly returns of Nifty and S&P500 (adjusted for ) are taken as inputs to compute SCD. Thousand bootstrap samples are generated such that each pair of returns belong to the same calendar week. The output consists of k-week SCD, standard error, t-stat with respect to 0 and upper and lower limits of the confidence interval. Our empirical findings are summarized in the next section.

The output of the analysis is summarized in the following 10 tables. Each table contains six columns that represent the k-week, correlation/scaled covariance/V-ratio/scaled cov-diff, standard error, t-statistic with respect to 0 (t-statistic with respect to 1 for V-ratio), upper and lower limits of the confidence interval. The bootstrap procedure has been made use of in order to get reliable standard errors based on the finite sample distribution of the test statistics. Figures corresponding to each table are also given.

To test the hypothesis-H1 stated in Section 2.3, we undertake t-tests, with the size of the test being α = 5%. For the sub-sample#1 (

In order to check the pattern of covariance between the k-week return series of Nifty and S&P500, we computed the scaled covariance between the k-week returns of Nifty and S&P500.

k week | Correlation | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 2.08E−01 | 3.94E−02 | 5.27E+00 | −7.72E−02 | 7.72E−02 |

2 | 2.26E−01 | 5.13E−02 | 4.40E+00 | −1.00E−01 | 1.00E−01 |

3 | 2.41E−01 | 5.95E−02 | 4.04E+00 | −1.17E−01 | 1.17E−01 |

4 | 2.34E−01 | 6.67E−02 | 3.50E+00 | −1.31E−01 | 1.31E−01 |

5 | 2.31E−01 | 7.37E−02 | 3.13E+00 | −1.44E−01 | 1.44E−01 |

6 | 2.25E−01 | 8.03E−02 | 2.80E+00 | −1.57E−01 | 1.57E−01 |

7 | 2.27E−01 | 8.79E−02 | 2.58E+00 | −1.72E−01 | 1.72E−01 |

8 | 2.24E−01 | 9.00E−02 | 2.49E+00 | −1.76E−01 | 1.76E−01 |

k week | Correlation | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 2.80E−01 | 4.86E−02 | 5.78E+00 | −9.52E−02 | 9.52E−02 |

2 | 2.92E−01 | 5.88E−02 | 4.97E+00 | −1.15E−01 | 1.15E−01 |

3 | 3.01E−01 | 7.09E−02 | 4.25E+00 | −1.39E−01 | 1.39E−01 |

4 | 2.91E−01 | 8.17E−02 | 3.57E+00 | −1.60E−01 | 1.60E−01 |

5 | 2.73E−01 | 9.09E−02 | 3.01E+00 | −1.78E−01 | 1.78E−01 |

6 | 2.55E−01 | 9.78E−02 | 2.60E+00 | −1.92E−01 | 1.92E−01 |

7 | 2.42E−01 | 1.09E−01 | 2.21E+00 | −2.14E−01 | 2.14E−01 |

8 | 2.26E−01 | 1.13E−01 | 2.01E+00 | −2.21E−01 | 2.21E−01 |

k week | Scaled Covariance | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 1.64E−04 | 3.11E−05 | 5.37E+00 | −6.09E−05 | 6.09E−05 |

2 | 1.79E−04 | 4.02E−05 | 4.46E+00 | −7.89E−05 | 7.89E−05 |

3 | 1.94E−04 | 4.71E−05 | 4.15E+00 | −9.24E−05 | 9.24E−05 |

4 | 1.90E−04 | 5.26E−05 | 3.61E+00 | −1.03E−04 | 1.03E−04 |

5 | 1.83E−04 | 5.79E−05 | 3.14E+00 | −1.13E−04 | 1.13E−04 |

6 | 1.75E−04 | 6.31E−05 | 2.80E+00 | −1.24E−04 | 1.24E−04 |

7 | 1.76E−04 | 6.85E−05 | 2.57E+00 | −1.34E−04 | 1.34E−04 |

8 | 1.69E−04 | 7.03E−05 | 2.42E+00 | −1.38E−04 | 1.38E−04 |

k week | Scaled Covariance | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 1.53E−04 | 2.52E−05 | 6.08E+00 | −4.94E−05 | 4.94E−05 |

2 | 1.55E−04 | 3.34E−05 | 4.64E+00 | −6.55E−05 | 6.55E−05 |

3 | 1.61E−04 | 3.73E−05 | 4.32E+00 | −7.30E−05 | 7.30E−05 |

4 | 1.48E−04 | 4.48E−05 | 3.30E+00 | −8.77E−05 | 8.77E−05 |

5 | 1.35E−04 | 4.92E−05 | 2.74E+00 | −9.64E−05 | 9.64E−05 |

6 | 1.22E−04 | 5.03E−05 | 2.43E+00 | −9.87E−05 | 9.87E−05 |

7 | 1.13E−04 | 5.98E−05 | 1.89E+00 | −1.17E−04 | 1.17E−04 |

8 | 1.02E−04 | 6.43E−05 | 1.58E+00 | −1.26E−04 | 1.26E−04 |

k week | V-Ratio (Nifty) | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 1.00E+00 | 0.00E+00 | 1.00E+00 | 1.00E+00 | |

2 | 1.09E+00 | 4.00E−02 | 2.3545 | 9.22E−01 | 1.08E+00 |

3 | 1.13E+00 | 5.95E−02 | 2.2156 | 8.83E−01 | 1.12E+00 |

4 | 1.16E+00 | 7.69E−02 | 2.0725 | 8.49E−01 | 1.15E+00 |

5 | 1.16E+00 | 8.85E−02 | 1.7604 | 8.27E−01 | 1.17E+00 |

6 | 1.14E+00 | 9.73E−02 | 1.4786 | 8.09E−01 | 1.19E+00 |

7 | 1.13E+00 | 1.11E−01 | 1.1718 | 7.82E−01 | 1.22E+00 |

8 | 1.11E+00 | 1.18E−01 | 0.9057 | 7.70E−01 | 1.23E+00 |

k week | V-Ratio (Nifty) | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 1.00E+00 | 0.00E+00 | 1.00E+00 | 1.00E+00 | |

2 | 1.01E+00 | 4.73E−02 | 0.2866 | 9.07E−01 | 1.09E+00 |

3 | 1.07E+00 | 7.11E−02 | 0.9339 | 8.61E−01 | 1.14E+00 |

4 | 1.04E+00 | 9.23E−02 | 0.4323 | 8.19E−01 | 1.18E+00 |

5 | 1.04E+00 | 1.09E−01 | 0.3384 | 7.86E−01 | 1.21E+00 |

6 | 9.97E−01 | 1.28E−01 | −0.0242 | 7.49E−01 | 1.25E+00 |

7 | 9.81E−01 | 1.35E−01 | −0.1433 | 7.35E−01 | 1.26E+00 |

8 | 9.64E−01 | 1.41E−01 | −0.2565 | 7.24E−01 | 1.28E+00 |

afterwards. V-ratio for each k-week is insignificant during sub-sample#2. This can also be observed from

The scaled k-week V-ratio of S&P500 index during sub-sample#1 and sub-sample#2 is summarized in

k week | V-Ratio (SP) | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 1.00E+00 | 0.00E+00 | 1.00E+00 | 1.00E+00 | |

2 | 8.99E−01 | 4.03E−02 | −2.5019 | 9.21E−01 | 1.08E+00 |

3 | 8.99E−01 | 5.95E−02 | −1.7006 | 8.83E−01 | 1.12E+00 |

4 | 8.96E−01 | 7.38E−02 | −1.4036 | 8.55E−01 | 1.14E+00 |

5 | 8.64E−01 | 9.18E−02 | −1.4764 | 8.20E−01 | 1.18E+00 |

6 | 8.39E−01 | 1.01E−01 | −1.5836 | 8.01E−01 | 1.20E+00 |

7 | 8.43E−01 | 1.14E−01 | −1.3790 | 7.77E−01 | 1.22E+00 |

8 | 8.18E−01 | 1.20E−01 | −1.5174 | 7.65E−01 | 1.24E+00 |

k week | V-Ratio (SP) | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 1.00E+00 | 0.00E+00 | 1.00E+00 | 1.00E+00 | |

2 | 9.60E−01 | 5.01E−02 | −0.7904 | 9.02E−01 | 1.10E+00 |

3 | 9.41E−01 | 7.61E−02 | −0.7775 | 8.51E−01 | 1.15E+00 |

4 | 8.89E−01 | 9.20E−02 | −1.2046 | 8.20E−01 | 1.18E+00 |

5 | 8.35E−01 | 1.06E−01 | −1.5551 | 7.92E−01 | 1.21E+00 |

6 | 8.08E−01 | 1.18E−01 | −1.6279 | 7.68E−01 | 1.23E+00 |

7 | 7.70E−01 | 1.33E−01 | −1.7249 | 7.39E−01 | 1.26E+00 |

8 | 7.28E−01 | 1.38E−01 | −1.9757 | 7.30E−01 | 1.27E+00 |

k week | SCD | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |

2 | 1.41E−05 | 2.28E−05 | 6.20E−01 | −4.46E−05 | 4.46E−05 |

3 | 2.82E−05 | 3.40E−05 | 8.28E−01 | −6.67E−05 | 6.67E−05 |

4 | 2.30E−05 | 4.14E−05 | 5.56E−01 | −8.11E−05 | 8.11E−05 |

5 | 1.65E−05 | 5.17E−05 | 3.19E−01 | −1.01E−04 | 1.01E−04 |

6 | 8.49E−06 | 5.68E−05 | 1.49E−01 | −1.11E−04 | 1.11E−04 |

7 | 9.57E−06 | 6.21E−05 | 1.54E−01 | −1.22E−04 | 1.22E−04 |

8 | 3.28E−06 | 6.78E−05 | 4.84E−02 | −1.33E−04 | 1.33E−04 |

k week | Scaled Cov−diff | Standard Error | t w.r.t 0 | C.I Min | C.I Max |
---|---|---|---|---|---|

1 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |

2 | 7.76E−07 | 1.99E−05 | 3.90E−02 | −3.90E−05 | 3.90E−05 |

3 | 6.06E−06 | 2.88E−05 | 2.11E−01 | −5.64E−05 | 5.64E−05 |

4 | −7.42E−06 | 3.67E−05 | −2.02E−01 | −7.20E−05 | 7.20E−05 |

5 | −2.02E−05 | 4.29E−05 | −4.72E−01 | −8.41E−05 | 8.41E−05 |

6 | −3.27E−05 | 5.01E−05 | −6.53E−01 | −9.82E−05 | 9.82E−05 |

7 | −4.24E−05 | 5.46E−05 | −7.77E−01 | −1.07E−04 | 1.07E−04 |

8 | −5.51E−05 | 5.90E−05 | −9.35E−01 | −1.16E−04 | 1.16E−04 |

We observe that the scaled cov-diff is not statistically different from 0 for k-week as k goes from 2 to 8 during sub-sample#1 and sub-sample#2 (

In this study, we have introduced a new measure called the Scaled Covariance Difference (SCD), which captures the difference between the covariance of short term returns and longer-term returns. We have made use of information in the off-diagonal terms of the variance-covariance matrix in the analysis so as to develop a test for joint market efficiency. We have demonstrated how to implement the test for joint market efficiency using data on weekly stock returns from the Nifty and S&P 500 indices. What we find is that, the k-week SCD between the US and the Indian market remains insignificant for all values of k. This suggests that the information transmission between these markets occurs within the same week. This provides strong evidence for the joint efficiency of Indian and American stock markets.

The authors declare no conflicts of interest regarding the publication of this paper.

Viswanathan, L., Maheswaran, S. and Balasubramanian, G. (2019) A Test for Joint Market Efficiency from an Investor’s Perspective. Theoretical Economics Letters, 9, 1518-1533. https://doi.org/10.4236/tel.2019.95098