Nonlinearity everywhere: implications for empirical finance, technical analysis and value at risk

We show that expected returns on US stocks and all major stock world market indices are non-linearly dependent on previous returns. The expected sign of returns tends to reverse after large price movements and trends tend to continue after small movements. This property can be captured by a simple polynomial model. Incorrectly fitting a simple linear model to the data leads to substantial bias in coefficient estimates and the polynomial model can be used to eliminate trends in the data. In addition, well known technical trading rules may be substantially driven by the non-linear behavior observed.

BackgroundIn binary logistic regression data are ‘separable’ if there exists a linear combination of explanatory variables which perfectly predicts the observed outcome, leading to non-existence of some of the maximum likelihood coefficient estimates. A popular solution to obtain finite estimates even with separable data is Firth’s logistic regression (FL), which was originally proposed to reduce the bias in coefficient estimates. The question of convergence becomes more involved when analyzing clustered data as frequently encountered in clinical research, e.g. data collected in several study centers or when individuals contribute multiple observations, using marginal logistic regression models fitted by generalized estimating equations (GEE). From our experience we suspect that separable data are a sufficient, but not a necessary condition for non-convergence of GEE. Thus, we expect that generalizations of approaches that can handle separable uncorrelated data may reduce but not fully remove the non-convergence issues of GEE.MethodsWe investigate one recently proposed and two new extensions of FL to GEE. With ‘penalized GEE’ the GEE are treated as score equations, i.e. as derivatives of a log-likelihood set to zero, which are then modified as in FL. We introduce two approaches motivated by the equivalence of FL and maximum likelihood estimation with iteratively augmented data. Specifically, we consider fully iterated and single-step versions of this ‘augmented GEE’ approach. We compare the three approaches with respect to convergence behavior, practical applicability and performance using simulated data and a real data example.ResultsOur simulations indicate that all three extensions of FL to GEE substantially improve convergence compared to ordinary GEE, while showing a similar or even better performance in terms of accuracy of coefficient estimates and predictions. Penalized GEE often slightly outperforms the augmented GEE approaches, but this comes at the cost of a higher burden of implementation.ConclusionsWhen fitting marginal logistic regression models using GEE on sparse data we recommend to apply penalized GEE if one has access to a suitable software implementation and single-step augmented GEE otherwise.

Lagged dependent variables (LDVs) have been used in regression analysis to provide robust estimates of the effects of independent variables, but some research argues that using LDVs in regressions produces negatively biased coefficient estimates, even if the LDV is part of the data-generating process. I demonstrate that these concerns are easily resolved by specifying a regression model that accounts for autocorrelation in the error term. This actually implies that more LDV and lagged independent variables should be included in the specification, not fewer. Including the additional lags yields more accurate parameter estimates, which I demonstrate using the same data-generating process scholars had previously used to argue against including LDVs. I use Monte Carlo simulations to show that this specification returns much more accurate coefficient estimates for independent variables (across a wide range of parameter values) than alternatives considered in earlier research. The simulation results also indicate that improper exclusion of LDVs can lead to severe bias in coefficient estimates. While no panacea, scholars should continue to confidently include LDVs as part of a robust estimation strategy.

Quantitative comparative social scientists have long worried about the performance of multilevel models when the number of upper-level units is small. Adding to these concerns, an influential Monte Carlo study by Stegmueller (2013) suggests that standard maximum-likelihood (ML) methods yield biased point estimates and severely anti-conservative inference with few upper-level units. In this article, the authors seek to rectify this negative assessment. First, they show that ML estimators of coefficients are unbiased in linear multilevel models. The apparent bias in coefficient estimates found by Stegmueller can be attributed to Monte Carlo Error and a flaw in the design of his simulation study. Secondly, they demonstrate how inferential problems can be overcome by usingrestrictedML estimators for variance parameters and at-distribution with appropriate degrees of freedom for statistical inference. Thus, accurate multilevel analysis is possible within the framework that most practitioners are familiar with, even if there are only a few upper-level units.

As an emerging research area, application of satellite-based nighttime lights data in the social sciences has increased rapidly in recent years. This study, building on the recent surge in the use of satellite-based lights data, explores whether information provided by such data can be used to address attenuation bias in the estimated coefficient when the regressor variable, Gross Domestic Product (GDP), is measured with large error. Using an example of a study on infant mortality rates (IMRs) in the People’s Republic of China (PRC), this paper compares four models with different indicators of GDP as the regressor of IMR: (1) observed GDP alone, (2) lights variable as a substitute, (3) a synthetic measure based on weighted observed GDP and lights, and (4) GDP with lights as an instrumental variable. The results show that the inclusion of nighttime lights can reduce the bias in coefficient estimates compared with the model using observed GDP. Among the three approaches discussed, the instrumental-variable approach proves to be the best approach in correcting the bias caused by GDP measurement error and estimates the effect of GDP much higher than do the models using observed GDP. The study concludes that beyond the topic of this study, nighttime lights data have great potential to be used in other sociological research areas facing estimation bias problems due to measurement errors in economic indicators. The potential is especially great for those focusing on developing regions or small areas lacking high-quality measures of economic and demographic variables.

K-means clustering algorithm has been used to classify major world stock market indices as well as most important assets in the Warsaw stock exchange (GPW). In addition, to obtain information about mutual connections between indices and stocks, the Granger-causality test has been applied and the Pearson R correlation coefficients have been calculated. It has been found that the three procedures applied provide qualitatively different kind of information about the groups of financial data. Not surprisingly, the major world stock market indices appear to be very strictly interconnects from the point of view of both Granger-causality and correlation. Such connections are less transparent in the case of individual stocks. However, the “cluster leaders” can be identified which leads to the possibility of more efficient trading.

On the upsurge of foreign exchange reserves in India

Rejoinder to “Endogeneity bias in marketing research: Problem, causes and remedies”

Although much market-based accounting research is based on regressions of abnormal returns on contemporaneous unexpected earnings, many have despaired about the intrinsic ability of accounting earnings to explain stock returns. These regressions exhibit low R2, lower than expected coefficients on unexpected earnings (ERC's), and various unusual features including non-linearity, lower R2 and response coefficients for loss firms, and lower R2 and response coefficients for high-growth and high-tech firms. Some improvement in explanatory power has been achieved by including various proxies for information that is currently available about future period earnings. This paper contributes to that line of research by deriving a specification, from the abnormal earnings model, that extends the traditional ERC regression by including current period forecast revisions of future period earnings. Relative to the traditional regression, the full specification increases R2 substantially, reduces the bias in coefficient estimates (caused by omitted correlated variables), and mutes the three unusual features mentioned above.

There exists a small sample bias in predictive regressions, when a rate of return is regressed on a lagged stochastic regressor, and the regression disturbance is correlated with the regressors’ innovations. Although this bias can be a serious concern in time-series predictive regressions, it is not significant in panel data setting. By using simulations and stock level data, we document that as the number of cross sections used in the panel data increases the bias in coefficient estimates becomes negligible.

Reporting error in weight and its implications for bias in economic models

The presence of chaos in the financial markets has been the subject of a great number of studies, but the results have been contradictory and inconclusive. This research tests for the existence of nonlinear patterns and chaotic nature in four major stock market indices: namely Dow Jones Industrial Average, Ibex 35, Nasdaq-100 and Nikkei 225. To this end, a comprehensive framework has been adopted encompassing a wide range of techniques and the most suitable methods for the analysis of noisy time series. By using daily closing values from January 1992 to July 2013, this study employs twelve techniques and tools of which five are specific to detecting chaos. The findings show no clear evidence of chaos, suggesting that the behavior of financial markets is nonlinear and stochastic.

In a rapidly globalizing world, understanding the relationships between major stock markets is of paramount importance for investors and financial analysts. This study explores the interdependence and cointegration of stock markets in Japan, India, and the USA, and explores the dynamics of global financial markets as well as the survival of a long-term and short-term link between these three indices. These leading stock markets were selected because of the researchers’ desire to learn more about the connections between them. From April 2012 through March 2022, we used monthly data from three major stock market indices: the NIKKEI (Japan), theBSE SENSEX (India), and the NASDAQ (USA). Stock market performance in both the United States and India tend to move together. Additionally, the GC test is utilized in an effort to ascertain if the markets have any form of forecasting ability. Based on the results of the tests conducted, it was determined that the NASDAQ index can accurately predict the SENSEX index, but the NIKKEI index. The United States and the Indian stock markets are highly correlated. To further investigate the markets’ potential for foresight, the Granger causality test is applied. Tests showed that while the NASDAQ index predicted the SENSEX index with high precision, the NIKKEI index did not. After a causal relationship has been established, we then look for evidence of a short- and long-term connection.

The Indonesia Stock Exchange (IDX) is one of the fastest growing capital markets. The relatively large proportion of foreign investment in the IDX is expected to cause a high frequency of inflows and outflows of funds from the IDX. There is a possibility that the inflows and outflows of the IDX will move to the stock exchanges of other countries which provide better profit opportunities. This study is to find out how big the correlation and cointegration of the world's major stock market indices, namely the European stock market represented by the London stock market (FTSE-100), the American stock market represented by the New York stock market (DJI), the Asian stock market represented by stock market in Singapore (STI) and Hong Kong (HKEX) against the composite stock price index on the Indonesia Stock Exchange (IDX). The conclusions obtained are (i) there is a positive (weak to moderate) and significant correlation between FTSE-100 with IDX, DJI with IDX, STI with IDX and HKEX with IDX, (ii) there is cointegration between FTSE-100 with IDX, DJI with IDX , STI with IDX and HKEX with IDX. Cointegration between the IDX composite stock price index and the stock market index in four other countries minimizes the possibility for investors to gain arbitrage profits by investing in foreign exchanges. (iii) FTSE-100, DJI, STI, HKEX and IDX do not have a unit root test, this means that the data in period t-1 does not affect the data in period t. This also means that the stock market in this study is a random walk
 Keywords: correlation, cointegration, arbitrage profit, Indonesian stock exchange, world's major stock market, random walk

Nobel Price for Father of the Efficient Market Hypothesis